1835 lines
171 KiB
HTML
1835 lines
171 KiB
HTML
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<html>
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<title>Relations</title>
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<meta name="author" content="Amr Gharbeia">
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<body lang="en-GB" text="#99ccff" link="#0000ff" dir="ltr" style="background: transparent">
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<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font color="#ff0000"><font face="Verdana, sans-serif"><b>R<img src="data:image/png;base64,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" name="OdjRRelation_up" align="right" hspace="5" width="100" height="40" border="0">elations.</b></font></font>
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Having considered the properties that substances have in a
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spatiomaterial world, the next step in demonstrating necessary truths
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about the world from these ontological assumptions is to determine
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the kinds of relations that substances have in the world. Relations
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are different from properties only in that relations hold of (or are
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true of) more than one substance at once. Thus, relations will be
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explained ontologically as aspects that hold of more than one
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substance, just as properties were explained as aspects of substances
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taken separately. In short, relations are aspects of the world. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">It
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is because of how substances exist together as a world that there are
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relational aspects of substances. But relations have been introduced
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in two different ways. The relations among points (that is, the parts
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of space) are part of the essential nature of space as postulated by
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this ontology. And there are relations among bits of matter, because
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each coincides with some part of space or other. (Spatial relations
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among bits of matter is one of the basic aspect of the natural world
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that was used as evidence for spatiomaterialism.) Both kinds of
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relations are part of our ontology, and both will be used to explain
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other kinds of relations. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Ontological
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philosophy proves that propositions about relations are true by
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deriving them from spatiomaterialism. That is, it shows how the
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relations are constituted by its basic substances, space and matter,
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given their essential natures and their basic relationship as parts
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of the same world. Propositions that follow from the best ontological
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explanation of the natural world are ontologically necessary and,
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thus, prior to what we know about what actually happens in the world
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from experience . That is what it means to say that they are
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"necessary truths," according to ontological philosophy. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">These
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ontologically necessary propositions about the basic relations in a
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spatiomaterial world include what is usually called mathematics. That
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is, the basic relations that hold among points (or that can hold
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among bits of matter at any moment) are, as we shall see, the subject
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matter of mathematics. There are other ontologically necessary
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relations in the world, such as those that derive from substances
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being in time and from further aspects of the essential natures of
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matter and space. They are merely complications of these basic
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relations, which will be taken up in the next chapter, </span></font></font></font><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCaL.htm" target="Lo"><font color="#0000ff"><font face="Arial, sans-serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><u>Change</u></span></font></font></font></a><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">,
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which is the subject matter of science. </span></font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">The
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ontological explanation of the truth of mathematics and science
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involves a different set of necessary truths from those already
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discussed. Unlike the truths about the intrinsic and extrinsic
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essential natures of substances, these further truths depend on
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substantivalism about space. The relations among points are part of
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the essential nature of space. Nor would there be any relations among
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bits of matter without space to help constitute them. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Explaining
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relations as aspects of a world constituted by space and matter is
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straightforward enough, but it is not the traditional way of
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explaining the truth of mathematics. Epistemological philosophy takes
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relations to be objects of knowledge, and obstacles to explaining how
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the basic relations are known to give rise to philosophical problems
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about the nature of mathematical truth. But the critique of
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epistemological philosophy is a consequence of ontological
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philosophy, and so let us begin by considering what can be said about
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relations as aspects of a spatiomaterial world. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font color="#800000"><font face="Verdana, sans-serif"><b>R<img src="data:image/png;base64,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" name="OdjRAsAspect_up" align="right" hspace="5" width="108" height="43" border="0">elations
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as aspects of substances.</b></font></font> In a spatiomaterial
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world, the relations that hold among particular substances are of two
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kinds, the relations that hold among points (or parts of space) as
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part of the essential nature of space, and the relations that hold
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among bits of matter because each coincides with some part of space.
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Since the ontological foundation of geometry is space, let us
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consider what holds simply because of its nature before we see what
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that implies about the relations among bits of matter contained by
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space. That will put us in a position to take up the ontological
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explanation of arithmetic.</font></font></font></p>
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<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Verdana, sans-serif"><b>G<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFEAAAAhCAMAAACWc7fFAAAAwFBMVEX////38PDv4ODg4ODn0NDMzMzjx5vfw5jfwMDWu5LTuZDXsLDDq4XMmZnHkJDGk3KumHe/gIC3cHCSgGSvYGCoXEh8bFSmUFCfVUKUSzpsXkqeQEBhVUKHPzFbUD6ZMzN8NipgOi10LiSOICBuKSCGEBB+AAArJR0QEBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABDu07rAAABdUlEQVR4nO2TwVLEIAyGAeuaFVsNoqhZrYqgff8XNFC2teuMdkYOHjaHTIHky59AxWadbYdheN+uiRQbsc5Oh4+TNXFnTIw1TZxXJ15UJ15VJ15XJ95WJz5UJ+6qE58mYm8QXahH7EF12DXCrk5F+JEYpM7yyKUF0Xjakx8/AvXF5SAKvxONmPs1EhrJWnulQALvi7YB2SKv2sRTDYguYvqHMQqjBaBKeQq/EmEu6BQLczJE1bEuYAgnRj+6PnrJDSQ3ahTapw5ZgJXhGxFyVQ3IJqjPsp3koDSEvTMqHbOeQswDSiJHiROxk2UKyPCUgp7yjSW/IGImIi2ILLJInIieJ1OIXRlAyMHYHBBdyVwQuU6ROL8eK8ESdUz0UjsykHRbQuEOiLFpLNmWohWO30QhBrkvNL9wbzT3m86DAWjTu7Qa2qzTLxxva8P5HObGnemiF8S/2X6K9YiTxGrE2Y7EI/HfEesaEy9v7u4fn1/f6tjL7hOHcEB9GAOEDwAAAABJRU5ErkJggg==" name="OdjRGeo_up" align="right" hspace="5" width="81" height="33" border="0">eometry.</b></font>
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Geometry describes the structure of space. Space, as we have assumed,
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is made up of many particular substances whose essential natures
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include their being related to one another in the way described by
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three-dimensional (Euclidean) geometry. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">The
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parts of space are particular substances, according to
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spatiomaterialism, but in geometry, they are called points. Points
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are identified by their locations in space, since that is how they
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differ from one another, and they are recognized to be simple, that
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is, without length, width, breadth, or any possibility of being
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divided into parts. </font></font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif">The
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propositions of geometry include the following: Any two points
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determines a straight line, where a straight line is the path of the
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shortest distance between them. Any straight line can be extended
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continuously in a straight line. A straight line and any point not on
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it determines a plane. Intersecting lines have only one point in
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common, and when the angles determined by them are equal, the angles
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are "right angles" and the lines are perpendicular. Through
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any point, there are exactly three mutually perpendicular straight
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lines. There is a metric to the distances between points, so that
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things equal to the same thing are also equal to one another. And so
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on . . . There is no need to state all the propositions of geometry
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here, since they are well known. </font></font>
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</p>
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Since
|
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geometry has been used to help define the essential nature of one of
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the two basic substances postulated by spatiomaterialism, ontological
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philosophy can explain why geometry is true of the parts of space by
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the correspondence between geometrical propositions and space as a
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substance. Those propositions describe an order among the parts of
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||
space, and since space is homogeneous, the order is universal and
|
||
holds in every region. Or as we assumed (provisionally) in the
|
||
foundation, each part of space has the same kinds of relations to all
|
||
the other parts of space as every other part of space has to parts
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||
others than itself. But it is relevant to notice that explaining the
|
||
truth of geometry by its correspondence to space does not depend on
|
||
geometry being stated as an axiom system.</font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Geometry
|
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as an axiom system.</i> The propositions of geometry can be stated as
|
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a system in which some are treated as assumptions, and all the rest
|
||
are all deduced from them (and definitions of terms introduced to
|
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simplify the statement of geometrical propositions). The former
|
||
propositions are called "axioms," and the latter are called
|
||
"theorems." This way of organizing geometrical propositions
|
||
was discovered by the ancient Greeks. It was worked out in some
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||
detail by Euclid. It aims at an optimal arrangement among the
|
||
proposition in which some of the simplest and most intuitive
|
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propositions are singled out and used to generate all the rest, that
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||
is, producing the most in the way of consequences using the least in
|
||
the way of premises. Geometry lends itself to axiomatization because
|
||
it describes a simple structure that contains implicitly many complex
|
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relations. The relations among the parts of space is a kind of order
|
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that makes the whole uniquely simple, and when the axioms describe
|
||
certain basic aspects of that structure, it is possible to combine
|
||
those relations in ways that describe all the other relations that
|
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must also hold among points, lines, angles, and the like. Such
|
||
constructions from simpler truths are the derivation of theorems in
|
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geometry. </font></font></font>
|
||
</p>
|
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<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
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<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">The
|
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significance of this deductive arrangement among the propositions of
|
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geometry has long been understood epistemologically, that is, as a
|
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way of knowing that geometrical propositions are true. Deductive
|
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inferences preserve the truth of the premises, and since the axioms
|
||
of geometry seem to be self-evidently true, it seemed that deriving
|
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them from the axioms would prove that they are also true. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">This
|
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epistemological approach became less attractive, however, as two
|
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facts about such axioms systems became known. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
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<font color="#000000"><font face="Times New Roman, serif">The first
|
||
was that there are different ways of axiomatizing geometry. That is,
|
||
different geometrical propositions can be used as axioms, and still
|
||
all the rest follow logically. Thus, there is no necessary order by
|
||
which some should be taken as implying others. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Second, and
|
||
more importantly, it became clear that the deductive relationship
|
||
cannot, by itself, establish any truth about the world. The truth of
|
||
the theorems depends on the truth of the axioms. But the truth of the
|
||
axioms cannot be shown within the deductive system. The axioms
|
||
contain terms which are not defined within the system, or so-called
|
||
"primitive terms," and thus, the truth of the axioms
|
||
depends on what those terms refer to. And there are other objects
|
||
that will make the axioms of geometry true (the set of whole numbers,
|
||
if nothing else, according to the Löwenheim-Skolem theorem). The
|
||
deducibility of the theorems from the axioms means that the theorems
|
||
will be true of whatever objects make the axioms true, but unless the
|
||
primitive terms in the axioms refer to points and their relations,
|
||
the theorems of geometry will have nothing to do with the structure
|
||
of space. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Thus,
|
||
even though it is possible to <i>come to know </i>that some
|
||
geometrical propositions are true by deriving them from others that
|
||
are true, that does not explain <i>why they are true</i>. It merely
|
||
shows that they are true, if the premises are true. Hence, the truth
|
||
of both depends on how the premises are true. Ontological philosophy
|
||
is not bothered by the aforementioned discoveries, because it
|
||
explains why both kinds of geometrical propositions are true in the
|
||
same way, that is, by virtue of their correspondence to the world. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">If geometry
|
||
is formulated as an axiom system, then the primitive terms, which are
|
||
not defined within the system, are taken as referring to the
|
||
substances it postulates or to aspects of them. The axioms are,
|
||
therefore, descriptions of the essential nature of one of the two
|
||
basic substances postulated by spatiomaterialism. But so are the
|
||
theorems derived from them. They are also descriptions of the
|
||
essential nature of space. Apart from being entailed by the axioms,
|
||
what makes the theorems different is that they can be stated without
|
||
introducing any new basic terms (that is, any terms that are not
|
||
defined by those used in the axioms). </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Euclidean
|
||
Geometry.</i> In the nineteenth century, however, the deductive view
|
||
of the truth of geometry suffered another blow, because it was
|
||
discovered that several axiom systems can be constructed for geometry
|
||
that are alike in making the most out of the least even though they
|
||
differ from one another in one of the axioms, the so-called parallel
|
||
axiom. Euclid’s fifth postulate holds, in effect, that through any
|
||
point not on a line, one, and only one, parallel line can be drawn in
|
||
the same plane as the first line. But Lobatchevsky and Bolyai showed
|
||
that this axiom could be replaced by one holding that more than one
|
||
line through such a point could be extended infinitely in the plane
|
||
without intersecting the first line and the resulting geometry would
|
||
be just as rich in implications. Later Riemann showed that the axiom
|
||
could be replaced by one holding that there are no parallel lines at
|
||
all, because any line drawn in the plane through a point not on the
|
||
same line will intersect with the first line in two points. Both of
|
||
these new geometries were just as rich in theorems as Euclid’s. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">The
|
||
existence of such non-Euclidean geometries shows that it is <i>possible
|
||
</i>that space is curved (that is, that geometry is consistent even
|
||
with carious artificial, new distance functions). But that is not of
|
||
much consequence to ontological philosophy, for it explains how
|
||
geometry is true, not by the deducibility of theorems from the axioms
|
||
of geometry, but rather by the correspondence of the axioms (and,
|
||
thus, the theorems) to the structure of space. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">The
|
||
correspondence theory of truth does, of course, force us to decide
|
||
which geometry describes the space we are postulating. And that
|
||
depends on the nature of the space that we find in the world, for we
|
||
are following the empirical method in deciding which ontology to
|
||
believe. That is, we choose the simplest ontological explanation that
|
||
will explain the basic features of the world. Since the simplest is
|
||
obviously Euclidean geometry, the space we postulate has a
|
||
three-dimensional Euclidean structure. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">To be sure,
|
||
since it is an empirical claim, it could turn out that space is not
|
||
Euclidean. In that case, ontological philosophy would have to start
|
||
over again with non-Euclidean space of some kind — or else give up
|
||
spatiomaterialism and go back to epistemological philosophy. But as
|
||
it turns out, there is no good reason to doubt that space is
|
||
Euclidean. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">What has
|
||
led naturalists to give up Euclidean space is Einsteinian relativity.
|
||
Einstein’s general theory of relativity holds that spacetime is
|
||
curved, and that means that it is not Euclidean. But the <i>curvature
|
||
of spacetime </i>is quite a different thing from the <i>curvature of
|
||
space </i>as a substance enduring through time, and as we have
|
||
promised, spatiomaterialism offers a perfectly intelligible
|
||
interpretation of what Einstein’s general theory calls "curved
|
||
spacetime" on the assumption that substantival space is
|
||
Euclidean. That removes any empirical reason for doubting that space
|
||
is Euclidean, and thus, we are free to believe the simplest geometry
|
||
that explains the categorical features of what we find in the world.</font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>What
|
||
geometry corresponds to. </i>Geometry holds of space in a
|
||
spatiomaterial world, because the space it postulates is a substance
|
||
whose essential nature is defined as making geometry true of it. The
|
||
relations among points, that is, the simplest parts of space, are
|
||
geometrical. But given how we explain the spatial relations among
|
||
bits of matter, geometry also most hold of them (except for
|
||
limitations that may be imposed by bits of matter having a finite
|
||
sizes in space), because they coincide with parts of space. Thus, the
|
||
propositions of geometry are true not only of the relations among
|
||
parts of space, but also of the relations among bits of matter.</font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">In
|
||
both cases, geometry is ontologically necessary, because it is part
|
||
of the ontology that we are taking to describe the basic nature of
|
||
existence. That means that it is prior to what is known about what
|
||
happen in the world by experience, and that is the sense in which
|
||
ontology if prior to science and other ordinary ways of knowing about
|
||
the world. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">However,
|
||
this proof the the ontological necessity of geometry involves a
|
||
genuine ontological explanation only when its propositions are taken
|
||
as applying to bits of matter. In that case, they describe facts
|
||
about the world that depend on both ontological causes, space and
|
||
matter. There is no genuine ontological explanation of why geometry
|
||
holds of space itself, because its geometrical nature is what is
|
||
assumed about just one of the basic substances being used as an
|
||
ontological cause. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Verdana, sans-serif"><b>A<img src="data:image/png;base64,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" name="OdjRArith_up" align="right" hspace="5" width="81" height="34" border="0">rithmetic.</b></font>
|
||
Besides the relations among points and bits of matter that describe
|
||
the structure of space, bits of matter and points have a more
|
||
abstract relationship to one another. They are all parts of a single
|
||
world in way that allows them to be picked out individually and,
|
||
thus, to be grouped together. Space is also an ontological cause of
|
||
this more abstract relationship, for it comes from particular
|
||
substances having spatial relations that all fit together
|
||
geometrically. Thus, arithmetic is no less ontologically necessary
|
||
than the relations that make geometry true. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Arithmetic
|
||
is, basically, the theory of numbers. The basic numbers are whole
|
||
numbers, or integers, and arithmetic includes the laws governing
|
||
their addition, multiplication, subtraction and division. Arithmetic
|
||
can be taken broadly as including all the propositions about the
|
||
numbers (except those that have to do with what numbers refer to and
|
||
how propositions about them are true).</font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Given the
|
||
arithmetic of whole numbers, it is possible to construct rational
|
||
numbers, negative numbers, irrational numbers, and complex numbers
|
||
and to show that these numbers also obey the laws of addition,
|
||
multiplication, subtraction, and division. With the use of set
|
||
theory, transfinite number can also be introduced, though special
|
||
laws govern operations on them. Taken broadly, therefore, arithmetic
|
||
includes algebra, the calculus, and analysis.</font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Even
|
||
geometry can be included, for its propositions can be generated by
|
||
way of analytic geometry, or the "algebra of geometry," as
|
||
Descartes showed. The contemporary attitude is to take arithmetic as
|
||
more basic than geometry, though that is to reverse the ancient Greek
|
||
assumption.</font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i><b>S<img src="data:image/png;base64,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" name="OdjRSet_up" align="right" hspace="5" width="72" height="36" border="0">et
|
||
theory.</b></i> It is possible to give an ontological explanation of
|
||
the truth of all these propositions at once, because they can all be
|
||
derived from set theory. Set theory provides the foundation that
|
||
mathematicians currently use to prove the truth of arithmetical
|
||
propositions, taken broadly. But there are various ways of
|
||
axiomatizing set theory, just as there are for geometry. The most
|
||
widely used by mathematicians is the Zermelo-Fraenkel system, and its
|
||
axioms will be used here to show how the truth of arithmetic (and
|
||
mathematics generally) can be explained ontologically. (A similar
|
||
argument could be constructed for other axiomatizations of set
|
||
theory.) </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Set
|
||
theory is a formal system in which the axioms are simply assumed to
|
||
be true. Though its axioms describe the nature of sets, "set"
|
||
is a primitive term, and so the axioms are an implicit definition of
|
||
that term. Thus, <i>if we can show that the substances that
|
||
constitute the spatiomaterial world satisfy the axioms of set theory,
|
||
that will show that all the propositions of arithmetic are true of
|
||
them</i>. Furthermore, since nothing exists in a spatiomaterial world
|
||
but those substances, it will also show that this interpretation of
|
||
set theory includes all possible interpretations of its axioms, and
|
||
thus, that it includes all the ways that set theory can be true by
|
||
virtue of corresponding to the world. Thus, this is, in effect, to
|
||
derive the truth of mathematics from the spatiomaterialist ontology,
|
||
which shows that mathematics is a necessary truth of ontological
|
||
philosophy.</font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Let
|
||
us consider, therefore, whether the substances in a spatiomaterial
|
||
world satisfy the axioms of Zermelo-Fraenkel set theory. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Axiom
|
||
1.</i> The first axiom defines "sets," in effect, by
|
||
holding that <i>two sets are identical when they have the same
|
||
members</i>. To explain its truth ontologically, we must say what the
|
||
members of sets are and what the sets themselves are.</font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">Sets
|
||
can be members of sets, but unless there is something else the most
|
||
basic sets are sets of, only the empty set can exist. Set theory says
|
||
nothing about the nature of the </span></font></font></font><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><i>ultimate
|
||
members </i></span></font></font></font><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">of
|
||
sets except to assume that they are all distinct and can be
|
||
distinguished from one another. But in a spatiomaterial world,
|
||
nothing exists at any moment except all the parts of space and all
|
||
the bits of matter, which it contains. Hence, those substances and
|
||
what they constitute are the only possible ultimate members of sets
|
||
that exist wholly at any moment. (We will see how arithmetic can be
|
||
extended to cover different moments in time in </span></font></font></font><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCaL.htm" target="Lo"><font color="#0000ff"><font face="Arial, sans-serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><u>Change</u></span></font></font></font></a><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">.)
|
||
Particular points in space can be picked out by their locations, and
|
||
so can particular lines, figures, and other geometrical constructs,
|
||
since they are constituted by such points. Likewise, let us assume
|
||
that bits of matter can also be picked out by their locations in
|
||
space, though we will not explain the sense in which it is true until
|
||
we take up the concrete nature of matter (in </span></font></font></font><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCaL.htm" target="Lo"><font color="#0000ff"><font face="Arial, sans-serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><u>Change</u></span></font></font></font></a><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">).
|
||
And if ordinary material objects are constituted by elementary bits
|
||
of matter and parts of space, as spatiomaterialism holds, they can be
|
||
picked out in a similar way. Indeed, any collection of points in
|
||
space and/or bits of matter can be picked out as an individual in
|
||
such a way. These are all the substances, elementary and compound,
|
||
that can exist at any moment in a spatiomaterial world, and thus,
|
||
they include all possible ultimate members of sets in such a world. </span></font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">The
|
||
sets of such members are, however, distinct from the substances,
|
||
which are their ultimate members, and in order to explain
|
||
ontologically how the axioms of set theory are true, there must also
|
||
be something to which the term "set" refers. What explains
|
||
the existence of sets in a spatiomaterial world is the fact that all
|
||
its substances have spatial relations to one another. That is the
|
||
aspect of the world that makes it <i>possible </i>to pick our
|
||
particular substances and group them together. Since their
|
||
possibility is entailed by the essential nature of a
|
||
spatiomaterialist world, every possible set actually exists as an a
|
||
distinct aspect of the world. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">To be sure,
|
||
sets would not be recognized to exist without rational beings like us
|
||
to pick out their members and actually group them together. And we
|
||
shall see how rational beings (with the spatiotemporal and rational
|
||
imagination required to construct such sets) come to exist in a
|
||
spatiomaterial world. But rational subjects are not essential to the
|
||
existence of sets, since sets are aspects of the world (though I may
|
||
refer to sets by saying that rational beings pick individuals out and
|
||
group them together). </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Substances
|
||
may be grouped together in many different ways, by using various
|
||
properties to define them, but <i>every </i>such class can, in
|
||
principle, be constructed by the spatial relations of the substances
|
||
making it up. (They must have spatial relations, since every
|
||
substance is constituted by a set of basic substances, according to
|
||
ontological philosophy.) Spatial relations make it possible not only
|
||
to pick out each substance as distinct from all the rest, but also to
|
||
group any substances together. Space is a whole of which they are all
|
||
already parts, and being parts of it, substances can be parts of
|
||
lesser wholes. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">To be sure,
|
||
merely being parts of the same world also makes them part of a single
|
||
whole. But that does not make it possible to group them together,
|
||
because if "the world" is defined as merely all the
|
||
substances that exist, it would not even be possible to distinguish
|
||
among particular substances (of the same kind), much less to relate
|
||
some of them to one another in a way that others are not related. But
|
||
having spatial relations means that each substance has a unique
|
||
relationship to all the others and, at the same time, that each is
|
||
part of a single whole, three dimensional space with them. (Though a
|
||
bit of matter and the part of space containing it have the same
|
||
spatial relations to every other substance in the world, they can be
|
||
distinguished from one another by the kind of substances they are.) </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Thus, space
|
||
is an ontological cause of every set, for it is the wholeness of
|
||
space that explains the existence of sets. Thus, groups constructed
|
||
by grouping substances (elementary or composite) together can be
|
||
taken as the basic sets of Zermelo-Frankel set theory. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">The
|
||
first axiom of Zermelo-Fraenkel set theory holds that <i>two sets are
|
||
identical if they have the same members</i>. It is true of sets in a
|
||
spatiomaterial world, given this ontological interpretation of sets
|
||
and their ultimate members. It is true of the basic sets, because the
|
||
substances that wind up together in a set do not depend on how they
|
||
are grouped together, but on which substances they are, for that is
|
||
the aspect of the world that constitutes the existence of the set.
|
||
Sets with the same members will be constituted by the same
|
||
substances. And it holds of sets of sets, because if sets are
|
||
constructed by grouping substances in this way, sets of sets are just
|
||
groups of groups formed in this way, and two groups of the same
|
||
groups will be constituted by the same groups of substances. There is
|
||
no ontological difference between the two sets. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Axiom
|
||
2.</i> The second axiom holds that <i>the empty set exists</i>. The
|
||
empty set does exist in a spatiomaterial world in the same sense as
|
||
any set. The same aspect of the world that makes it possible to group
|
||
substances together also makes it possible to form a group without
|
||
any members. Whether or not it has any members, the grouping itself
|
||
depends on how space makes the world whole, that is, on how space
|
||
itself is whole and how everything contained by space is related in
|
||
its three dimensions. That aspect of the world is not constituted by
|
||
substances taken separately, but by how they exist together as a
|
||
world, and that aspect is what explains the existence of the empty
|
||
set. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Axiom
|
||
3.</i> The third axiom holds that <i>if </i>x <i>and </i>y <i>are
|
||
sets, then the unordered pair {</i>x,y<i>} is a set</i>. That is to
|
||
say that sets can be members of sets as well as basic substances, and
|
||
the truth of this axiom has already been explained. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Sets
|
||
exist in the sense that spatial relations allow substances to be
|
||
grouped in all possible ways. But sets that exist in that sense can
|
||
themselves be grouped in a similar way into groups. For the same
|
||
reason, it is possible to group sets of sets into sets, and sets of
|
||
sets of sets into sets, and so on. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Axiom
|
||
4.</i> The fourth axiom holds that <i>the union of a set of sets is a
|
||
set</i>, that is, that a set can be formed from all the distinct
|
||
substances that are members of at least one set included in the set
|
||
of sets. That axiom is true in a spatiomaterial world, because sets
|
||
are just groups of substances. Any substance can be picked out by its
|
||
spatial relations. And if a substance is a member of more than one of
|
||
the member sets, it will not become two substances in the union of
|
||
the sets, because its identity with a substance in the other sets can
|
||
be determined by its spatial relations. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Axiom
|
||
5.</i> The fifth axiom holds that <i>the infinite set exists</i>,
|
||
including transfinite cardinals. The obstacle to taking the axiom of
|
||
infinity to be a truth about the natural world has been doubts about
|
||
the bits of matter in the world being infinite in number. Even if
|
||
spatiomaterialism did not (yet) take a stance on that issue, it would
|
||
entail the existence of infinite sets, including transfinite
|
||
cardinals, because it takes space as well as matter to be a
|
||
substance. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Space
|
||
may not be infinite in extent, but since any finite line is
|
||
infinitely divisible, there are infinite sets of points (for example,
|
||
the points determined by cutting a line in half, cutting the
|
||
half-line in half, cutting the quarter-line in half, etc.). Such sets
|
||
are denumerably infinite, because they can be put in a one-to-one
|
||
relation with whole numbers. And if the world <i>is </i>infinite, the
|
||
bits of matter in the world can also be put in one-to-one relations
|
||
with the whole numbers. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">But
|
||
substantivalism about space also entails the existence of transfinite
|
||
sets of substances, for the number of points on a finite line is
|
||
indenumerably infinite. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Axiom
|
||
6.</i> The sixth axiom of Zermelo-Frankel set theory is that <i>any
|
||
property that can be formalized in the language of the theory can be
|
||
used to define a set</i>. The truth of this axiom is entailed by this
|
||
ontological explanation of the world, because properties are aspects
|
||
of substances and all properties are explained by showing how they
|
||
are constituted by substances. Since properties can all be explained
|
||
by the substances whose aspects they are, it holds for all the
|
||
properties that can be formalized in the language of the theory. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Axiom
|
||
7.</i> The seventh axiom holds that, <i>for any set, the power set
|
||
can be formed</i>; that is, that the collection of all subsets of any
|
||
given set is a set. This follows from our ontological explanation of
|
||
the existence of sets, for it implies that all sets that can be
|
||
formed of the particular substances in the world exist, and that
|
||
includes all the subsets of any set formed, that is, its power set.
|
||
(What makes this axiom so important is that the power set is itself a
|
||
set, and another set can be formed of its subsets, over and over
|
||
again indefinitely.) </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Axiom
|
||
8.</i> The eighth axiom is the so-called "axiom of choice,"
|
||
which holds that <i>from any collection of non-empty, non-overlapping
|
||
sets, a new set can be formed by selecting one member from each set</i>.
|
||
This axiom is clearly true, if sets are all ultimately made up of
|
||
substances as members (that is, are complex substances), because
|
||
substances exist. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Despite
|
||
being used in many mathematical proofs, this axiom has not been
|
||
considered self-evident, because there seems to be no way to assure
|
||
that it is possible to pick out a particular member of every set.
|
||
However, it is always possible, given the ontological explanation of
|
||
the truth of this axiom. Since the ultimate members of every set are
|
||
points in space, bits of matter, or determinate combinations of basic
|
||
substances, it is possible to pick out a specific member of each set
|
||
by its spatial relations. For example, select the particular
|
||
substance from each set which is closest to a given point, or in
|
||
cases of ties, the first in an ordered set of directions in three
|
||
dimensions from a given point. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>Axiom
|
||
9.</i> The ninth axiom holds that <i>no set is a member of itself</i>.
|
||
This axiom avoids certain paradoxes that can arise from taking sets
|
||
to be members of themselves, for example, Russell’s paradox about
|
||
whether the set of sets that are not members of themselves is a
|
||
member of itself. (If it is not a member of itself, it must be a
|
||
member of the set; but if it is a member to the set as defined, it is
|
||
a member of itself.) But this is not just a device to avoid
|
||
paradoxes. It is a fact about sets, if sets are formed by grouping
|
||
substances or groups ultimately made up of substances together,
|
||
because it is not possible to include the group one is currently
|
||
constructing as a member of the group. It does not yet exist, and so
|
||
rational beings having nothing to group together with the members.
|
||
Thus, no set is a member of itself. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">These
|
||
are the axioms of Zermelo-Fraenkel set theory, and as we have seen,
|
||
they are true of a spatiomaterial world, if the ultimate members of
|
||
sets are substances and sets exist in the sense that substances (and
|
||
groups of them) can be grouped together. Since deduction preserves
|
||
the truth of its premises, all of mathematics that can be derived
|
||
from them (including arithmetic, algebra, the calculus, and analysis)
|
||
is also true of the natural world, if spatiomaterialism is true.
|
||
Hence, the truths of arithmetic are not only true, but also
|
||
ontologically necessary, that is, prior to empirical science. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i><b>S<img src="data:image/png;base64,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" name="OdjRSol_up" align="right" hspace="5" width="71" height="36" border="0">olutions
|
||
of puzzles about set theory.</b></i> There are further advantages of
|
||
the ontological explanation of the truth of arithmetic, because it
|
||
solves several puzzles that have cast doubt on mathematics in the
|
||
twentieth century. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>T<img src="data:image/png;base64,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" name="OdjRTot_up" align="right" hspace="5" width="60" height="32" border="0">otality.</i>
|
||
It is remarkable that all the truths of arithmetic can be generated
|
||
by Zermelo-Fraenkel set theory without countenancing the
|
||
all-inclusive set, that is, the set of all sets. That was required in
|
||
order to avoid paradoxes, because the all-inclusive set would be a
|
||
member of itself. But in terms of set theory itself, it is puzzling
|
||
how sets could exist without all the sets being a set, for they are
|
||
all parts of the same world. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">On
|
||
this ontological explanation of the truth of set theory, however,
|
||
there is no puzzle. All the sets do exist together, because they are
|
||
aspects of a single world, in the sense that they can all be
|
||
constructed by grouping substances or groups of substances together.
|
||
That explains how all of the sets can exist without there being a set
|
||
of all sets. The totality is the world itself. And the set of all
|
||
sets cannot be formed. As we have seen, it is not possible for a
|
||
rational subject to group the set he is constructing as a member of
|
||
the set he is constructing, for it does not yet exist. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>C<img src="data:image/png;base64,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" name="OdjRConsis_up" align="right" hspace="5" width="68" height="29" border="0">onsistency.</i>
|
||
This ontological explanation of the truth of set theory and the
|
||
arithmetic theorems that follow from it proves that they are
|
||
consistent. That is important, because mathematicians want assurance
|
||
that their deductions will not generate paradoxes, that is,
|
||
contradictions. In 1931, Kurt Gödel (1906-1978) showed that any
|
||
formal system that is complex enough to generate the propositions of
|
||
arithmetic cannot be shown to be consistent on the basis of set
|
||
theory or logic alone. The inability to prove the consistency of
|
||
arithmetic has been a source of embarrassment and consternation,
|
||
because mathematicians now look to formalizations, such as set
|
||
theory, as the foundation for their mathematical proof. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">It
|
||
is, however, possible to show the consistency of a formal system by
|
||
giving an interpretation (or model) of it that is assumed to be
|
||
consistent. That is how the consistency of non-Euclidean geometries
|
||
was demonstrated. The axioms of Lobachevskian and Riemannian geometry
|
||
were shown to hold of geometrical objects that were constructed
|
||
within Euclidean geometry, and that proved that those non-Euclidean
|
||
geometries were both consistent, because Euclidean geometry was
|
||
assumed to be consistent. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Although
|
||
the consistency of arithmetic cannot be shown by logical means, it
|
||
can be shown ontologically. The reason no one doubted the consistency
|
||
of Euclidean geometry is that it holds of the structure of the world
|
||
and the world actually exists. There cannot be any contradiction in
|
||
propositions that merely describe the nature of something that
|
||
actually exists. That was an ontological proof of the consistency of
|
||
Euclidean geometry, and that is the kind of proof that
|
||
spatiomaterialism gives of the consistency of arithmetic. If set
|
||
theory is understood as a description of the groups that can be
|
||
formed of substances in a spatiomaterial world ((by rational beings
|
||
in that world), then the existence of that world shows that set
|
||
theory and all the theorems that follow from it are consistent. There
|
||
can be no paradoxes. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>C<img src="data:image/png;base64,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" name="OdjRComp_up" align="right" hspace="5" width="77" height="30" border="0">ompleteness.</i>
|
||
Another embarrassment to basing arithmetic on set theory was also
|
||
contained in Gödel’s 1931 paper, namely, his incompleteness
|
||
theorem. He showed that there are propositions in arithmetic that
|
||
cannot be proved. (And what is more, he showed by further, less
|
||
formal, means that those propositions are true.) That is, Gödel
|
||
proved by the use of arithmetic that, if any formal system that is
|
||
complex enough to include arithmetic is consistent, then it is
|
||
incomplete. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">His proof
|
||
depended on using numbers (Gödel numbers) to represent not only
|
||
propositions in arithmetic, but also <i>propositions about logical
|
||
relations </i>among arithmetic propositions. By representing both
|
||
arithmetic and a formal system for describing logical relations in
|
||
arithmetic by numbers, Gödel was able to construct a sentence within
|
||
arithmetic that says, when interpreted, "This sentence is not
|
||
provable." </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Now, is
|
||
this sentence provable in arithmetic? If it is not provable, it is
|
||
true. But it must be true, if arithmetic is consistent, because if it
|
||
were provable, it would be false, and arithmetic would not be
|
||
consistent. Hence, there is a true statement in arithmetic that
|
||
cannot be proved. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">What
|
||
Gödel showed was the <i>logical incompleteness </i>of arithmetic and
|
||
set theory. But that does not necessarily mean that the propositions
|
||
of arithmetic are not a complete set of truths about the numbers and
|
||
their properties. That is true only if mathematical truth is taken to
|
||
be mere provability within set theory (or any other formal system).
|
||
But that is what ontological philosophy denies. It explains the truth
|
||
of arithmetic ontologically, that is, as correspondence to the world.
|
||
And there is no reason to doubt that arithmetic, founded on set
|
||
theory, is ontologically complete. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><span lang="en-US">That
|
||
is, Gödel’s incompleteness theory does not give us any reason to
|
||
believe that there are true arithmetic propositions </span></font></font><font color="#000000"><font face="Times New Roman, serif"><span lang="en-US"><i>about
|
||
the world</i></span></font></font><font color="#000000"><span lang="en-US">
|
||
</span></font><font color="#000000"><font face="Times New Roman, serif"><span lang="en-US">that
|
||
are not provable in arithmetic. The statement Gödel constructed,
|
||
which said, in effect, "This statement is not provable,"
|
||
depended on interpreting the numbers in terms of the symbols used in
|
||
arithmetic and in a formal system for describing logical relations
|
||
among propositions in arithmetic. That is not a reference to
|
||
substances in the world, but a reflective reference to formal systems
|
||
as they are understood by the rational beings using them, and as we
|
||
shall see when we explain the nature of reason (in </span></font></font><font color="#0000ff"><font face="Arial, sans-serif"><span lang="en-US"><u><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCbGeRRS09.htm" target="Lo">Change</a><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCbGeRRS09.htm">:
|
||
Stage 9</a></u></span></font></font><font color="#000000"><font face="Times New Roman, serif"><span lang="en-US">),
|
||
a far more complex ontological explanation is required to spell out
|
||
the nature of formal systems in terms of the substances constituting
|
||
the natural world.) </span></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Far
|
||
from being a puzzle about mathematical truth, therefore, Gödel’s
|
||
incompleteness theorem is a reason for believing that the truth of
|
||
mathematics should be explained ontologically. There is no reason to
|
||
doubt the ontological necessity of mathematical truth, that is, its
|
||
priority to what is known by empirical science about the world on the
|
||
basis of experience of what happens there. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>D<img src="data:image/png;base64,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" name="OdjRDet_up" align="right" hspace="5" width="71" height="30" border="0">eterminacy
|
||
of reference.</i> Determinacy of reference. A further puzzle was
|
||
posed by the Löwenheim-Skolem theorem. It holds that a formal system
|
||
constructed to generate propositions about one kind of mathematical
|
||
object can always be given another interpretation in which they are
|
||
true of an entirely different set of objects. For example, any
|
||
consistent set of axioms constructed to generate all the theorems
|
||
about real numbers, which are non-denumerable, can be given another
|
||
interpretation in which they are true of sets which are denumerable,
|
||
such as the integers. Likewise, axioms designed to derive all the
|
||
theorems about the whole numbers can be given an interpretation in
|
||
which they are true of non-denumerable sets. Indeed, every consistent
|
||
set of axioms has a countable model. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">No
|
||
puzzles are posed by the Löwenheim-Skolem theorem, however, if the
|
||
truth of mathematics is explained ontologically. Indeed, such a
|
||
theorem is just what just what should be expected, if mathematics is
|
||
true because of its correspondence to the world. A formal system,
|
||
such as set theory, has primitive terms, which are not defined in the
|
||
system, and what makes it possible to give other interpretations in
|
||
which those axioms are true is assigning different referents to those
|
||
primitive terms. But when the truth of arithmetic propositions is
|
||
explained as correspondence to the world, the primitive terms of the
|
||
axioms of set theory are introduced as references to substances and
|
||
the groups that can be formed of substances in a spatiomaterial
|
||
world, and there is no possibility of another interpretation. All of
|
||
mathematics that follows from set theory refers to certain aspects of
|
||
the world. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">And we must
|
||
distinguish between geometry generated as analytic geometry and
|
||
geometry as explained above, because the correspondence to the world
|
||
in the latter restricts the interpretation of such terms as "line,"
|
||
"angle," and the like to only certain possible sets in the
|
||
world.</font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>The
|
||
usefulness of mathematics in science.</i> This ontological
|
||
explanation of the truth of arithmetic and geometry may also make it
|
||
possible to solve other problems (for example, by showing that there
|
||
is no good reason to believe that the continuum hypothesis is true),
|
||
but enough has been said to illustrate its significance. There is,
|
||
however, one final consequence that is worth noting, though it is as
|
||
much a problem about science as about mathematics. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">The
|
||
assumption that the truth of mathematics comes down to provability
|
||
within a formal system has made it seem puzzling that mathematics
|
||
should be so useful in science. Indeed, that is the most unsettling
|
||
puzzle about mathematics in the view of contemporary philosophers,
|
||
who take these puzzles as casting doubt on mathematics as the model
|
||
of true knowledge. But it is not at all puzzling, given this
|
||
ontological interpretation of the truth of mathematics. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">It
|
||
is not puzzling that mathematics is so useful in science, when its
|
||
propositions are understood to be about the most basic aspects of the
|
||
world, namely, how the world is made up of many distinct, particular
|
||
substances and how, being related to one another spatially, they can
|
||
be grouped together in all possible ways. Such sets include all the
|
||
quantitative aspects of substances, from distances and times to
|
||
masses and forces. Thus, it is hardly surprising that sets in that
|
||
sense and the ontologically necessary propositions that hold of them
|
||
because they are substances in a spatiomaterial world are relevant in
|
||
explaining what happens in the world. Their relevance will become
|
||
even more clear in </span></font></font></font><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCaL.htm"><font color="#0000ff"><font face="Arial, sans-serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><u>Change</u></span></font></font></font></a><font color="#000000"><span lang="en-US">
|
||
</span></font><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">when
|
||
we take time into consideration and describe the concrete nature of
|
||
matter and space. The basic laws of physics describe quantitatively
|
||
precise regularities about how bits of matter move and interact, and
|
||
since mathematics holds of the sets picked out for those purposes,
|
||
there is no wonder that mathematics describes relations that are
|
||
relevant in those descriptions. </span></font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">It is not
|
||
easy for contemporary physicists to see this, however, because the
|
||
twentieth century revolutions in physics have forced them to abandon
|
||
the expectation of an intuitive understanding of what their highly
|
||
mathematical theories are about. Though the intelligibility of
|
||
scientific theories in terms of spatial imagination was taken for
|
||
granted in classical physics, it is now generally assumed that it is
|
||
beyond our grasp. But the ontological explanation of the truth of
|
||
contemporary physics will show that that is not necessarily the case.
|
||
</font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0; page-break-before: always">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Verdana, sans-serif"><b>R<img src="data:image/png;base64,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" name="OdjRAsObj_up" align="right" hspace="5" width="108" height="41" border="0">elations
|
||
as objects of knowledge.</b></font> Ontological philosophy explains
|
||
relations as aspects of the world that exist because of the essential
|
||
nature of space and how space contains bits of matter at any moment,
|
||
and correspondence to them explains, as we have seen, how
|
||
mathematical propositions are true. That means that mathematics is
|
||
prior to empirical science in the sense of being <i>ontologically
|
||
necessary</i>. However, necessity in the sense of being <i>certain</i>
|
||
is what has traditionally been thought to make mathematics different
|
||
from empirical science. Certainty is what is relevant about
|
||
mathematics when the project is justifying belief in certain
|
||
propositions by how they are related to what is known in other ways.
|
||
Thus, epistemological philosophy approaches mathematical objects as
|
||
objects of knowledge, rather than as aspects of the world, and it is
|
||
not obvious that what mathematics is about are the most basic
|
||
relations that hold in the world. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Verdana, sans-serif"><b>T<img src="data:image/png;base64,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" name="OdjRProblem_up" align="right" hspace="5" width="103" height="44" border="0">he
|
||
problem of mathematical knowledge.</b></font> When the certainty of
|
||
mathematics is taken for granted, the problem of mathematical
|
||
knowledge is to explain how such certainty is possible, that is, why
|
||
it is more certain than what is known by ordinary experience of what
|
||
happens in the world.<sup><a class="sdendnoteanc" name="sdendnote1anc" href="#sdendnote1sym"><sup>i</sup></a></sup></font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">It
|
||
is somewhat misleading to think of the certainty of mathematics only
|
||
as a problem, for in the beginning, that is what inspired belief in
|
||
epistemological philosophy. In ancient Greece, mathematics was taken
|
||
as an example to show the possibility of philosophy as a superior
|
||
kind of knowledge of the world, one that revealed necessary truths.
|
||
In the </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><i>Meno,</i>
|
||
for example, Plato describes Socrates as asking a slave boy a series
|
||
of questions about some lines he draws in the sand which lead the boy
|
||
to recognize the truth of a special case of Pythagoras’ theorem
|
||
(that the square built on the diagonal of a square is twice the area
|
||
of the first square). That put the slave boy in a position to defend
|
||
what he knew rationally, and Plato used that story to illustrate how
|
||
<i>knowledge </i>is different from <i>true belief</i>. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Beliefs
|
||
about whose truth one can be certain are what philosophy pursues out
|
||
of its love of wisdom, according to Plato. Above the entrance to
|
||
Plato’s Academy, the first university, it was written that no one
|
||
should enter who does not know mathematics. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">It
|
||
is hard to overstate how important mathematics has been to the
|
||
credibility of philosophy’s claim to provide a kind of knowledge of
|
||
that is superior to our ordinary ways of knowing what happens in the
|
||
world through experience. But given its role in epistemological
|
||
philosophy, the issue about how the certainty of mathematical
|
||
knowledge is possible becomes the issue of how realism is possible. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Verdana, sans-serif"><b>T<img src="data:image/png;base64,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" name="OdjRTheo_up" align="right" hspace="5" width="105" height="45" border="0">heories
|
||
of mathematical knowledge.</b></font> To set the stage for
|
||
considering the received explanations of the certainty of
|
||
mathematics, let us consider briefly what ontological philosophy
|
||
implies about the <i>knowledge of </i>mathematics. We will then take
|
||
up the epistemological theories. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i><b>O<img src="data:image/png;base64,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" name="OdjROnto_up" align="right" hspace="5" width="98" height="31" border="0">ntological
|
||
theories of mathematical knowledge.</b></i> We have explained why
|
||
mathematics is true by showing how its propositions correspond to
|
||
relations as basic aspects of a spatiomaterial world. Geometry
|
||
corresponds to the structure that space has as (part of) its
|
||
essential nature as a substance, and that explains why the
|
||
propositions of geometry hold of bits of matter in space as well as
|
||
points. Arithmetic holds of the particular substances postulated by
|
||
spatiomaterialism, because they all have spatial relations to one
|
||
another, making it possible to pick out particular substances and to
|
||
group them together in sets. But that does not explain how it is
|
||
possible for rational beings like us to know that these propositions
|
||
are true — and to know that they are true in a way that makes them
|
||
certain in comparison to empirical science and other ordinary ways of
|
||
knowing about the world. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">The
|
||
short answer is that mathematics is not certain, but merely prior to
|
||
empirical science. Mathematical propositions are among the necessary
|
||
truths proved by ontological philosophy. They are <i>ontologically
|
||
</i>necessary, because they are entailed by the best ontological
|
||
explanation of the natural world, namely, spatiomaterialism. That is
|
||
the foundation that ontological philosophy uses to prove that
|
||
propositions are necessarily true about the world, and mathematical
|
||
propositions are among them, because they correspond to basic aspects
|
||
of any spatiomaterial world. But to prove that propositions are
|
||
ontologically necessary is not necessarily to prove that they are
|
||
certain, that is, <i>epistemologically </i>necessary. Since
|
||
spatiomaterialism itself is an empirical truth, the justification of
|
||
what follows from it is ultimately empirical and, thus, falsifiable
|
||
by experience. It is nevertheless prior to empirical science, because
|
||
ontological explanations are prior to efficient-cause explanations.
|
||
What follows from spatiomaterialism could be false, because
|
||
spatiomaterialism could be false. But if what follow from it is
|
||
false, we must give up our otherwise empirically well-founded belief
|
||
about the basic nature of existence and deny that the world is
|
||
constituted by its two, opposite kinds of basic substances. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">Nevertheless,
|
||
mathematics </span></font></font></font><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><i>seems
|
||
</i></span></font></font></font><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">to
|
||
be certain. It was not without reason that traditional philosophy
|
||
took an epistemological approach to necessary truths. And the long
|
||
answer to the question about why beings like us believe that
|
||
mathematics is true and believe that it is more certain than science
|
||
has to do with the nature of reason. Reason is a cognitive capacity
|
||
that evolves in certain animals, and as we shall see (in </span></font></font></font><font color="#000000"><font face="Arial, sans-serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><u><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCbGeRRS09.htm">Change</a><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCbGeRRS09.htm">:
|
||
Evolutionary stage 9</a></u></span></font></font></font><font color="#000000"><span lang="en-US">
|
||
</span></font><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">and
|
||
following), reason has an ontologically necessary nature which
|
||
involves two forms of imagination. But it will be easier to explain
|
||
the received, epistemological philosophies of mathematics if we
|
||
anticipate that explanation with a brief account of them here. </span></font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Animals
|
||
become rational as they evolve the use of language, and in a world of
|
||
space and matter in time, it is plausible to suppose that those
|
||
animals already have a spatial imagination by which they can
|
||
understand the structure of space. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">By "spatial
|
||
imagination", I mean a brain mechanism (a system of
|
||
representation in what will be called the "animal behavior
|
||
guidance system") that uses spatial images of objects and
|
||
temporal sequences of them to represent objects, their spatial
|
||
relations to one another in three dimensions, and how their spatial
|
||
relations change as a result of motion or being manipulated. At its
|
||
core, it is a memory mechanism that records the locations of objects
|
||
by lining up images of them in the order they would appear as a
|
||
result of locomotion in each direction in space, and since "covert
|
||
locomotion," that is, motor commands for moving the body that
|
||
are not actually executed, can call up those images in sequence, it
|
||
serves as a form of imagination that gives animals a nonlinguistic
|
||
way of thinking about the basic geometrical structure of space and
|
||
the effects of motion on their relations. (Spatial imagination is
|
||
this brain mechanism that makes it possible for computers to generate
|
||
what is called "virtual reality.") </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Furthermore,
|
||
in animals that can manipulate objects, such as primates, spatial
|
||
imagination also includes an ability to think about the geometrical
|
||
structures of objects and how they interact when being manipulated.
|
||
Acts of imagination call up spatial images of objects in sequences
|
||
that represent the effects of manipulating them in various ways.<sup><a class="sdendnoteanc" name="sdendnote2anc" href="#sdendnote2sym"><sup>ii</sup></a></sup></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Spatial
|
||
imagination gives even nonlinguistic animals an intuitive way of
|
||
understanding the structure of space, that is, that spatial relations
|
||
among objects. And as suggested in the last chapter, since such brain
|
||
activity involves a form of matter whose intrinsic nature registers
|
||
what is happening throughout the forebrain, spatial imagination is
|
||
what makes it appear that sensory qualia are located in phenomenal
|
||
space. That is, its structure is what gives rise to complex
|
||
phenomenal properties and what we are calling the unity of
|
||
consciousness. In this context, however, it phenomenal appearance
|
||
explains the faculty of intuition on which epistemological philosophy
|
||
typically bases its theory about the nature of reason. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">It
|
||
is not surprising that such a cognitive faculty evolves in a
|
||
spatiomaterial world, given that animals acquire food by ingesting
|
||
other objects in space, for it gives them more power over objects in
|
||
space. Indeed, we shall see that its evolution is inevitable in
|
||
worlds where evolution can occur at all. But this nonlinguistic
|
||
understanding of the spatial and temporal aspect of the world is
|
||
inherited by animals in which language evolves, and in such animals,
|
||
spatial imagination comes under the control of verbal behavior. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">In order to
|
||
understand a sentence about objects in space, users of language must
|
||
construct its meaning in imagination. Spatial imagination makes it
|
||
possible to connect words to particular material objects in space,
|
||
and thus, learning the meanings of words involves the development of
|
||
"abstract images," which correspond to properties and
|
||
relations, or the aspects of objects in space that are called
|
||
"abstract objects." (As we shall see, they develop in the
|
||
brain as states that represent many different particular objects of
|
||
certain kinds indifferently.) </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Furthermore,
|
||
learning to combine such words grammatically involves the development
|
||
of complex representations, in which properties are related to the
|
||
objects that have them and states of affairs are represented. Thus,
|
||
language is a second system of representation. The capacity of
|
||
language to represent basic aspects of a spatiomaterial world derives
|
||
therefore, from the spatial imagination of the (mammalian) animal
|
||
system of representation. (This is the role of what I will later call
|
||
"natural sentences.")</font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">However,
|
||
rational imagination, as I will call it, depends on another kind of
|
||
linguistic representation, in addition to the linguistic
|
||
representations based on spatial imagination (or "natural
|
||
sentences') and the representations of spatial imagination itself.
|
||
The use of language, as we shall see, eventually makes the animals in
|
||
which it evolves reflective. (This further stage in the evolution of
|
||
language introduces what I will call "psychological sentences.")</font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">The ability
|
||
to use more complex sentences enables language-using animals to
|
||
represent to themselves the (brain) states (such as perceptions,
|
||
memories, beliefs, desires, and intentions) that occur in the process
|
||
of perceiving and thinking about the natural world and to think about
|
||
the roles that such states play in causing behavior and beliefs.
|
||
Thus, these animals can reflect on the causes of their beliefs and
|
||
behavior. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">But in
|
||
reflective animals, such reflective (brain) states can themselves be
|
||
causes of the conclusions they draw about how to behave or what to
|
||
do, and thus, they have earned a special name. They are called
|
||
"reasons." In other words, reasons are basically just
|
||
causes of conclusions that are represented as causes as an essential
|
||
part of the process of causing such conclusions. Considering how
|
||
language depends on spatial imagination to connect words to objects
|
||
in the world, the control that language has over spatial imagination
|
||
transforms the animal faculty of imagination into rational
|
||
imagination, a capacity to think about the possible reasons for
|
||
certain conclusions.</font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">These
|
||
three elements — the animal's spatial imagination, how it connects
|
||
linguistic representations to the world, and how language eventually
|
||
enables the animals in whom it evolves to reflect on the reasons for
|
||
their beliefs and intentions — are essential to reason, and they
|
||
explain why it seems that mathematical truths are certain. Spatial
|
||
imagination is an intuitive way of understanding the structure of
|
||
space, and thus, if spatial relations among substances are the basic
|
||
subject matter of mathematics, it is an intuitive understanding of
|
||
mathematical propositions. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">There
|
||
is, of course, a longer story to be told about how reflection on its
|
||
operations evolves into explicit knowledge of geometry and
|
||
arithmetic. But for now, let us simply notice that, as rational
|
||
beings with such knowledge reflect on the causes of their beliefs, a
|
||
difference between mathematics and empirical science will inevitably
|
||
appear. Though it is possible to know the propositions of geometry
|
||
and arithmetic by perception, in the same way as other facts about
|
||
nature, it is eventually noticed that they have reasons for believing
|
||
mathematical propositions that do not depend on perceiving what
|
||
actually happens in the world. They seem forced to believe, for
|
||
example, that a straight line is the shortest distance between two
|
||
points and that two plus three is five by their very understanding of
|
||
those propositions. Those beliefs seem especially compelling, because
|
||
those facts about the world are built into the structure of their
|
||
spatial imagination. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Thus, when
|
||
epistemological philosophers reflect on how they know that
|
||
mathematical propositions are true, the first hypothesis is that
|
||
geometrical objects and numbers are objects of a special kind which
|
||
are revealed only to rational intuition (or what ontological
|
||
philosophy explains as the subjective, phenomenal appearance of
|
||
rational imagination). That is basically Platonism about mathematics.</font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Given how
|
||
theorems about geometrical figures and numbers can be derived from
|
||
axioms, however, another possible hypothesis is that mathematical
|
||
propositions are a result of logic or reasoning. That leads to forms
|
||
of anti-realism about mathematics.</font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">In either
|
||
case, however, there there seem to be reasons for believing
|
||
mathematical propositions that are sufficient, but which do not
|
||
depend on perceiving what happens in the natural world. That explains
|
||
the apparent certainty of mathematics. It can be known in a way that
|
||
does not seem to be vulnerable to what is learned about the world
|
||
through perception. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">These
|
||
epistemological theories do not lead to errors in mathematics,
|
||
because what seems certain in this way actually holds universally in
|
||
a spatiomaterial world. That is, what spatial imagination corresponds
|
||
to is the basic aspect of the world in which rational beings find
|
||
themselves. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">It is
|
||
interesting to notice that, since that basic aspect of the world is
|
||
its spatial structure, or the aspect of the world that, more than
|
||
anything else, makes the world whole, mathematics is a way of knowing
|
||
about the wholeness of the world. And since it is known by subjects
|
||
who are part of that world, mathematics is the part's knowledge of
|
||
the basic nature of the whole of which it is part. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">This
|
||
ontological explanation of the apparent certainty of mathematical
|
||
knowledge is the foundation for its critique of epistemological
|
||
philosophy of mathematics. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><b>E<img src="data:image/png;base64,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" name="OdjREpist_up" align="right" hspace="5" width="98" height="32" border="0"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMoAAAAUCAMAAAD7j1LFAAAAYFBMVEVksf9epu9Xm99RkM9Lhb9Fea8+bp84Y48yWH8sTW8lQl8fN08ZLD8SIS8MFh8AAIAGCg8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACpI6u9AAAC7klEQVR4nO1WAa/jIAhWRIvU9ur//7P3gWu399q37ZJLLlmOrK4gAh+CNvz6GAr/OoC/R4ASPoL+Q7miGN9UpJMkPZ++8HWhdIdC2kVk3h6UaHo3uhCm1i6UC3+X8LymrxLSjQ9/YPJLX7DxFErgbmMqD4Fs6bziJ2pyIVz1JJJ2iqw/+DuYZ3S2ES6glAsl5Gt+af4EJcplqTyBcmJ+otdQmpEpFm0zI5ZmAcnMvG0t2atOPl3bnFibWkmVKlp2KDvDVeY1x+xMmapyoCbVFuxhUF14bkrmmMzg8OdQqEqtmElYQm5NUeusWloTt+HeaRITKJ+h2FgRfYeZspWYIZlLiMn9J2sGrWNaOuQz9mqxYHUZUHYm2yZClUyoqFKWkNEDqncokXuJUbymEJ5swf1ZFHmBp4SmQeOmGKpVadkoyppRH4spu3cswH9pF7uyb1U3mNPmEtkMtPnXTEWqjGlXhrQsHhZMI+qDaWYgdsdX6r4L2Kn2UBzDH9LhL9zvz+qdL2toK7JG3c+Tprel2zRePCZ7+BUUsC5JdZt9WWslHdM7FPHoLWT8DsYAB1pdLgMK2REiZyhgT1B63uez9okHFHPWbr4OKLRx2u2dT7DlZkrUJWRp9mXTaibpKxRy2BGHIzwcTKzCBdVvQvZjkDyg+hWK2WvlDEVdZVJzhlpd/UBBpXuqIqrugIKta3SCEqfOXNSi6UvmOkdIEqG+SkG1CYe6of0QW59MOeFCWAh1LZznjBSgp3eGVGXyvEBYNhXVqKvospAvukGZoSzDlg37E6K2zKIxw6skNKkywzEaFBlqeTgehuL9uH+AwoO8ghKTS5DQxJa8lIaKhYDRp4hNCwrJx6HnTJ7yUtE/N+E+FYnptuhWwTbhtmzYn+HJq4pHH9D4lzYicBs3Q8tx4l9+uPTTHf3HhK3JL+6it26QL3S+TnK932ZXUKTVd76DnlJSkedfPbhLrr4PnhDr/O0Cp0cTn/Zl/Cn0QVB+A104TsShuPf8AAAAAElFTkSuQmCC" name="EpistCmt" align="right" hspace="5" vspace="10" width="202" height="20" border="0">pistemological
|
||
theories of mathematical knowledge.</b> The approach of
|
||
epistemological philosophy is just opposite to ontological
|
||
philosophy. Instead of starting with ontology and showing that
|
||
mathematical truths are ontologically necessary, epistemological
|
||
philosophy starts by reflecting on how we know about the truth of
|
||
mathematical propositions and tries to show that they are necessary
|
||
in the sense of being certain, or epistemologically necessary. The
|
||
basic form of success in epistemological philosophy of mathematics is
|
||
realism about entities beyond what is known by ordinary experience of
|
||
the natural world, and as we have seen, the fate of epistemological
|
||
philosophy is sealed, because its realism involves metaphysical
|
||
dualism. The problems of metaphysical dualism eventually leads to
|
||
anti-realism. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">As
|
||
the example of Socrates and the slave boy in the <i>Meno </i>suggests,
|
||
mathematical knowledge was the original inspiration for philosophy’s
|
||
claim to have a superior way of knowing about the world. It was the
|
||
first way philosophy ever claimed to prove there are necessary
|
||
truths. Since epistemological philosophy began with Plato's use of
|
||
the certainty of mathematics to illustrate the success of realism,
|
||
realism in the philosophy of mathematics is now called "Platonism."
|
||
Given the fate of epistemological philosophy, Platonism eventually
|
||
leads to anti-realism. But in the case of mathematics, even most
|
||
anti-realists affirm the certainty of its propositions. There is,
|
||
however, a form of anti-realism that denies the certainty of
|
||
mathematics by assimilating it to empirical science, that is, by
|
||
denying that there is any basic difference between mathematics and
|
||
science. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">A
|
||
brief account of the history of epistemological philosophy of
|
||
mathematics follows, and having seen how ontological philosophy can
|
||
explain why mathematics appears to be certain to those who reflect on
|
||
how they know it, I will use the ontological theory of reason to show
|
||
not only what is true and false in the traditional theories of
|
||
mathematics, but also how the philosophical problems caused by the
|
||
epistemological approach are solved by ontological philosophy. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i>R<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEcAAAAkCAMAAADsDtfZAAAAwFBMVEX////w8Png4PPg4ODQ0O3MzMzjx5vfw5jAwOjWu5LTuZCwsOLDq4WgoNywm3iQkNank5qumHeei5SAgNCSgIuXhWeSgGRwcMqCcoCDc1l8bFRgYMRtX3FzZU5QUL5pXEhkV0RcUGZhVUJbUD5AQLhPRV1HPlhJQE9EO1YwMLIgIKwrJR0QEKYcGRMAAJkQEBAAACcAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABlznWSAAAB6ElEQVR4nOXVe5PTIBAAcIp4t5VD1FWjotQHPsB1vXz/L+eS2jRNLmnnzB83486UyQD58dxUXZ2L7a+2/bk910tdqXPxuP396Fyfa3H430M9Wcm5Wcl5vpLzeiXn3amTU0rTXj5wycvO7tQBQK/juBcgB7/sfB45KKMDc/FgY30El7vaFJgbgIYZkwMkBEdD58vUsZ5pkzjLLxKj7moRuNFE4ihPpKGwdUuOBi0DNRoRtcwCwaiDEzZN3TuV9sNJzeJ8nJFJGNnvVOImUuodUc2GLnYqlJXsCmU0zKF3MjEJcrHDYCgaAyYSGOt6J0qVv3Bd943/wMmWqMuuPJsCk9Trs2XgQOak0RtKh4NAHL0GPA6bxk6Sa14Fj1ImW9NHkrZIIiFxig7kDWRytQEjeOpqCowdn/aOa6TMhRvHziVycnWc3BXKkmeKbcMps4psbVfDQCOnwrIu162LGmugW1dtVIcnxc7G0j0e2jDd5RgZrM7KNFW7w+Hg5DOw5LiyX1dXKuIgjpyGbMJRVJzrBg6d+tqJE3DgBA0OuBjIGUAOsnfQgOzI0SEzOXcz/L5dGhgmTrwHFP5O5yHn18Nx1ghxnr14+eb9p6/ffszE7e1cyzG+y//p9fbm6au3H3YfZ6Jt51oGsfsDgb5hbIqdf/EAAAAASUVORK5CYII=" name="OdjRReal_up" align="right" hspace="5" width="71" height="36" border="0">ealism:
|
||
Platonism about mathematics. </i>For philosophers who argue from how
|
||
we know to what can be known, success comes from showing that we have
|
||
knowledge of the real existence of entities of some kind beyond a
|
||
kind of knowledge that is taken for granted, that is, knowledge of
|
||
reality beyond appearance. In the philosophy of mathematics, realism
|
||
is called "Platonism," after its founder. But Platonism
|
||
takes different forms in the ancient and modern worlds. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Arial, sans-serif">A<img src="data:image/png;base64,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" name="OdjRAncient_up" align="right" hspace="5" width="56" height="28" border="0">ncient
|
||
Platonism about mathematics. </font>Plato’s explanation of what the
|
||
slave boy learned from Socrates is that beings like us have a faculty
|
||
of reason that makes us aware of objects that are fundamentally
|
||
different from the objects of perception. That is how all genuine
|
||
knowledge (as opposed to mere belief) was explained by Plato, and it
|
||
is the model for Platonism in mathematics. Numbers and geometrical
|
||
objects are part of a reality that Platonists believe lies beyond
|
||
appearances in natural world. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">According
|
||
to Plato, the Forms in the realm of Being are different from natural
|
||
objects, which are known by perception (that is, empirical
|
||
knowledge), because the Forms do not change and never appear
|
||
differently from what they really are. What enables us to know about
|
||
them is rational intuition, which Plato repeatedly contrasted to
|
||
perception, as knowledge to mere belief. But it is the difference in
|
||
the natures of the objects being cognized that was supposed explain
|
||
the certainty and necessity of mathematical truths. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Rational
|
||
intuition of mathematical objects does involve appearances, according
|
||
to ontological philosophy, for there is a faculty of rational
|
||
imagination in the brain and its activity has an appearance to the
|
||
subject by way of phenomenal properties (by generating bits of matter
|
||
whose intrinsic natures register brain activity). But that is not the
|
||
appearance of objects that are outside space and time, and the belief
|
||
that the objects being grasped are Platonic Forms involves an
|
||
insuperable problem, namely, Platonic ontological dualism. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Mathematical
|
||
propositions hold of objects found in the natural world, and in order
|
||
for Platonists to explain how our knowledge of such propositions is
|
||
certain, they need a way to explain why truths about abstract
|
||
entities in a realm beyond nature reveal something about objects that
|
||
exists in nature. The main problem with platonic realism, as Plato
|
||
himself recognized, is that there is no way to explain how objects
|
||
outside space and time can have any effect on objects in the natural
|
||
world.<sup><a class="sdendnoteanc" name="sdendnote3anc" href="#sdendnote3sym"><sup>iii</sup></a></sup></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Ontological
|
||
philosophy avoids the problems of Platonic realism by taking
|
||
mathematical objects to be aspects of the natural world, rather than
|
||
abstract entities that exist in a transcendent realm. But it can also
|
||
explain why they appear to be abstract entities. In both cases,
|
||
abstract entities are reifications of concepts based on spatial
|
||
imagination.<sup><a class="sdendnoteanc" name="sdendnote4anc" href="#sdendnote4sym"><sup>iv</sup></a></sup></font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Though
|
||
geometrical structures are always concrete parts of space, they seem
|
||
to be universal, because space exists everywhere with the same
|
||
three-dimensional structure. Since reflective subjects with spatial
|
||
imagination recognize such geometrical structures in many different
|
||
particulars, it is not surprising that they think of them as
|
||
universals or abstract entities. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Likewise,
|
||
though the material objects they count are concrete particular
|
||
substances existing independently of one another, their spatial
|
||
relations are what makes it possible for them to be grouped together,
|
||
and since that makes the results of arithmetic operations the same
|
||
everywhere, numbers seem to abstract entities. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Arial, sans-serif">M<img src="data:image/png;base64,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" name="OdjRModern_up" align="right" hspace="5" width="57" height="30" border="0">odern
|
||
realism (Platonism) about mathematics. </font>With the rise of modern
|
||
philosophy, the problems with Platonism about mathematics were
|
||
transformed, but not solved. Plato was a naive realist about both
|
||
perception and reason. He believed that the objects of both forms of
|
||
intuition (that is, perception and rational imagination) exist
|
||
independently of the subject, but are nevertheless immediately
|
||
present to the subject. The modern period began with the recognition
|
||
that perception is mediated by appearances (or "ideas’) that
|
||
are part of the mind, and that meant that rational intuition is
|
||
likewise just another kind of appearance in the mind (what Descartes
|
||
called clear and distinct ideas). That eliminated the problems caused
|
||
by Plato's attempt to explain the relationship between the objects of
|
||
perception and the objects of reason as the relationship between
|
||
Forms in a realm of Being and visible objects in the realm of
|
||
Becoming. But modern philosophers were still Platonists, in effect,
|
||
because they believed that what makes knowledge of mathematics
|
||
certain, in contrast to empirical knowledge, is that it is about
|
||
abstract objects that exist independently of both the subject and the
|
||
natural world. But instead of existing in a realm of Being,
|
||
mathematical objects were taken to exist as ideas in the mind of God.
|
||
In short, as an offspring of the marriage of Platonism and
|
||
Christianity in the medieval period, the modern era had inherited a
|
||
rationalistic theology in which God played the role of the realm of
|
||
Being. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Modern
|
||
philosophy still had to explain, however, why mathematics is true of
|
||
the natural world. Indeed, the question was even more pressing,
|
||
because the new science discovered laws of nature that are highly
|
||
mathematical. Those laws described precise quantitative relationships
|
||
among properties, such as distance, mass, time, and velocity, and
|
||
since relations among different quantities of the same property are
|
||
arithmetical, those physical descriptions required the truth of
|
||
mathematical propositions. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Modern
|
||
philosophy had, however, a ready solution, at least, until doubts
|
||
about theistic supernaturalism late in the eighteenth century,
|
||
because the objects of mathematics were assumed to be ideas in God’s
|
||
mind. God created the natural world according to a rational plan, and
|
||
since God had used mathematics to create a world governed by natural
|
||
laws, the discovery of those laws was basically seeing into God’s
|
||
mind. In <i>The Assayer</i> (1610), for example, Galileo described
|
||
nature as a book that God had written in the language of mathematics.
|
||
And Descartes used God to prove that our clear and distinct ideas
|
||
about geometry corresponded to extension, the essential nature of the
|
||
bodily substance. In other words, it was possible for rational beings
|
||
to recognize the truth of mathematical propositions, because
|
||
rationality comes from their being created in God’s image. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><i><b>A<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAlCAMAAAA5mzTPAAAAwFBMVEX////w8Png4PPg4ODQ0O3MzMzjx5vfw5jAwOjWu5LTuZCwsOLDq4WgoNyynHqQkNaumHeAgNCYhWiSgGRwcMqFdFt8bFRgYMR2Z1BQUL5sXkpmWkZhVUJbUD5AQLgwMLIgIKwrJR0QEKYcGRMAAJkQEBAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAmLBivAAABYElEQVR4nO3V4W6CMBQF4EvX6dXaYSezg2nbdXe8/yPuohOXmIKQZr88CRhs/ULrIcBiJKuvtv1cjc3iwAJG8tx+P43N4SxZohyBdTZpk03aZpN22aR9NqlJSy7cfBMHpENSqkH+uTLIJ/QD0jEpKQT+YTBOYU0BhTFkbu7yHimA15oXBJaPELV0jsDNkUpkIfLRzXLn1fFniVhOlAqBCNWNNP2ebMGnSlwlOVeS3Roi1BcpoJQz92lq/lnyOFzv+6Vur4bqnZCc5VqTVVhymRRidZa43l6j9jxuUflT8UckU6BxRkXSSN5TEPYk8d9WVORrMsJRxcX3EMakroKFNkYDxUpjYS6SUHU4j1+qcYcEleOQ5NVgL7HL7ERJqW7HCGqKspfc6WGcKEUtUCqyAvVVUhKlnyDNzUN6SL9SnrC0XG+2u31zOCbTtumxPvw2X642L69v781HMm2bHuvT/ADbETaF737pSAAAAABJRU5ErkJggg==" name="OdjRAnti_up" align="right" hspace="5" width="73" height="37" border="0">nti-realism
|
||
about mathematics.</b></i> The problems of supernaturalism eventually
|
||
made Platonism in either its ancient or modern form untenable. There
|
||
is simply no way to prove the existence of entities existing beyond
|
||
the natural world. But anti-realism about mathematics takes two
|
||
fundamentally different forms, because mathematics still seems to be
|
||
certain, even if realism is doubtful. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">One form
|
||
continues to accept the certainty of mathematics and tries to explain
|
||
how there can be such self-evident truths without having to prove the
|
||
existence of entities beyond what is given to ordinary experience of
|
||
the world. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">The other
|
||
form takes the denial of the existence of platonic entities beyond
|
||
the natural world to mean that mathematics must be about the natural
|
||
world, and by assimilating mathematics to empirical science, denies
|
||
that mathematics has the kind of certainty that is taken for granted
|
||
by realists and other anti-realists. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Arial, sans-serif">A<img src="data:image/png;base64,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" name="OdjRAffirm_up" align="right" hspace="5" width="63" height="31" border="0">nti-realism
|
||
that affirms the certainty of mathematics. </font>The nineteenth
|
||
century was a transitional period in the history of mathematics. Not
|
||
only did the rise of naturalism made Platonism less attractive, but
|
||
developments in mathematics itself also made it less plausible that
|
||
mathematics describes the essential nature of a reality beyond the
|
||
subject, natural or supernatural. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Though
|
||
Euclidean geometry had once made it seem obvious that one kind of
|
||
rational certainty somehow reveals the inherent nature of the natural
|
||
world, the discovery of non-Euclidean geometry cast doubt on that
|
||
assumption. The certainty of geometry seemed to depend more on the
|
||
deducibility of theorems from axioms. And recognition that the
|
||
arguments (about infinity) on which the calculus had been based were
|
||
logically faulty focused mathematicians on the project of making
|
||
mathematical proofs more rigorous. Though physics undoubtedly
|
||
required mathematics for its spectacularly successful descriptions of
|
||
regularities in the natural world, it was, by the beginning of the
|
||
twentieth century, plausible to hold that the certainty of
|
||
mathematics does not come from knowing a special kind of object that
|
||
exists independently of the subject. Instead, it seemed possible to
|
||
explain its special certainty as deriving from the nature of the
|
||
rational subjects themselves. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Various
|
||
theories of the certainty of mathematical truth have been proposed in
|
||
the twentieth century, and disputes among them tend to be technical.
|
||
But a rough sketch of two opposite approaches and their problems will
|
||
put us in a position to see why naturalists now seem to have little
|
||
choice but to treat mathematics as a species of empirical, scientific
|
||
truth. Both of the following views give up the belief in
|
||
independently existing, abstract entities, and both explain its
|
||
certainty by holding that it is a kind of truth that is discovered
|
||
within the mind. And both are just what would be expected of
|
||
epistemological philosophers, given how ontological philosophy
|
||
explains the ability of rational beings to know the truth of
|
||
mathematical propositions. One takes account of the role of spatial
|
||
imagination and attempts to reduce all of mathematics to objects of
|
||
rational intuition, and the other takes account of the role of
|
||
language in expressing those intuitions and attempts to reduce
|
||
mathematics to logic or the structure of language. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Arial, sans-serif"><i>I<img src="data:image/png;base64,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" name="OdjRIntuit_up" align="right" hspace="5" width="75" height="29" border="0">ntuitionism.
|
||
</i></font>Intuitionism derives historically from Kant, and it
|
||
reflects the assumptions of modern philosophy. Kant argued that
|
||
mathematics is <i>a priori</i> knowledge about the natural world
|
||
because it describes the structure of the forms of intuition (space
|
||
and time) in which nature itself is presented in experience. Proofs
|
||
of mathematical propositions involve the construction of mathematical
|
||
objects in imagination, and thus, they must conform to the mind’s
|
||
pure forms of intuition, space and time. But that means that
|
||
mathematical truth hold necessarily and universally in experience of
|
||
the natural world, because the two forms of intuition are also
|
||
conditions of possible experience. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Kant
|
||
was describing the process by which rational beings do actually come
|
||
to accept the certainty of mathematical propositions, according to
|
||
our ontological explanation of reason. The role that Kant ascribed to
|
||
space and time as forms of intuition in understanding mathematics is
|
||
explained by spatial imagination, and that accounts for knowledge of
|
||
geometrical and arithmetical propositions. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">The use of
|
||
a language to control and to reflect on the structure of spatial
|
||
imagination gives one a nonlinguistic understanding of what is meant
|
||
by such concepts as "point", "line", "plane",
|
||
and "sphere," and thus, one can "see" that there
|
||
is a shortest distance between two points and that a line and a point
|
||
not on it determines a plane. One can also recognize the truth of
|
||
simple propositions, such as that exactly three lines intersecting at
|
||
a point can be mutually perpendicular, that three planes can be
|
||
mutually perpendicular, and that any closed plane figure with just
|
||
three internal angles has three sides. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Ontological
|
||
philosophy confirms Kant's theory of arithmetical knowledge in a
|
||
similar way. Spatial imagination enable reflective subjects to think
|
||
about the operations of singling objects out, combining them as
|
||
groups, adding and subtracting members, and the like, and thus, they
|
||
can recognize the truth of arithmetic axioms and construct theorems
|
||
of arithmetic in imagination. That makes it seem that such truths
|
||
about the world can be known prior to discovering their truth by
|
||
perception, because what makes arithmetic true is the way in which
|
||
space makes different bits of matter parts of the same world and that
|
||
aspect of the world is represented in spatial imagination.</font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">It also
|
||
seems, in a similar way, that they can know the truth of theorems in
|
||
other mathematical systems constructed from arithmetic and geometry,
|
||
such as calculus, prior to discovering their truth by perception.</font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Though
|
||
Kant did not develop his constructivist approach to mathematical
|
||
propositions in much detail, intuitionism was taken up by many
|
||
mathematicians in the twentieth century (including Henri Poincaré,
|
||
1854-1912) and given a detailed defense by L. E. J. Brouwer
|
||
(1881-1966). </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">In the end,
|
||
however, intuitionism was not acceptable to most mathematicians,
|
||
because the requirement that all mathematical objects be constructed
|
||
in imagination required giving up too much of mathematics. (Brouwer
|
||
rejected the axiom of choice, actually infinite sets, Cantor’s
|
||
transfinite numbers, and any arguments for the existence of
|
||
mathematical objects based on the law of excluded middle.) </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Even if it
|
||
were possible for intuitionists to construct all of mathematics,
|
||
however, this way of explaining the certainty of mathematics implies
|
||
that its truth comes from the structure of thought. Kant believed
|
||
that nature is just the phenomenal world, which is "in the
|
||
mind," so to speak, and though he never doubted there is a
|
||
noumenal world (or things in themselves) beyond the phenomenal world,
|
||
he denied that mathematical truths hold of it (or them). That may be
|
||
plausible to Kantians, but it is not plausible to naturalists.
|
||
Naturalists believe that what exists independently of the subject is
|
||
a world of material objects with spatial relations that change over
|
||
time, and intuitionism does not imply that mathematics is true of
|
||
that world. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Arial, sans-serif"><i>L<img src="data:image/png;base64,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" name="OdjRLogic_up" align="right" hspace="5" width="82" height="30" border="0">ogicism
|
||
and formalism. </i></font>The other way of explaining the certainty
|
||
of mathematics in epistemological philosophy without accepting
|
||
Platonism tries, in effect, to reduce mathematics to language.
|
||
Historically, it has taken two main forms, logicism and formalism. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Logicism
|
||
holds that all of mathematics is derivable from logic. Its
|
||
philosophical roots are in Leibniz, but it was developed more
|
||
rigorously by Gottlob Frege (1848-1925) and Bertrand Russell
|
||
(1872-1970) around the turn of the twentieth century. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">It turned
|
||
out, however, that the laws of logic needed to generate number theory
|
||
involved several axioms that hardly seemed to be laws of logic at
|
||
all. They included, for example, the axiom of reducibility (which
|
||
holds that propositions about higher types, or sets of sets, could be
|
||
reduced to propositions about first order members of sets), the axiom
|
||
of infinity (which affirms the existence of infinite sets), and the
|
||
axiom of choice (which says that from any set of non-empty,
|
||
non-overlapping sets, it is possible to form a set of one member from
|
||
each). If one were to insist that these are laws of logic and that
|
||
they are known by rational intuition of some kind, logicism would be
|
||
a kind of Platonism in which the laws of logic, rather than the
|
||
mathematical objects themselves, have an independent existence as
|
||
abstract objects outside of space and time. But to many, these axioms
|
||
seemed more doubtful than the propositions about numbers that were
|
||
derived from them. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">"Formalism"
|
||
is the name of the project pursed by David Hilbert (1862-1943) in
|
||
order avoid the problems of logicism. He did not believe that
|
||
mathematics could be reduced to the laws of logic. He held that each
|
||
branch of mathematics requires its own axioms and rules of inference.
|
||
But he believed that logicism was on the right track in taking
|
||
logical entailments to be what explains mathematical truth. Thus,
|
||
Hilbert set out to prove the certainty of mathematics by
|
||
reconstructing each branch of mathematics as a formal system with its
|
||
own axioms, rules of inference, and theorems. But these statements
|
||
were to be stripped of any meaning outside the formal system and
|
||
treated as meaningless symbols, mere marks on paper, which were
|
||
written down in sequence according to strict rules. Each formal
|
||
system would include all the propositions in some branch of
|
||
mathematics, and the rigor of these symbolic manipulations was
|
||
supposed to prove the certainty of its theorems. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><span lang="en-US">The
|
||
formalists’ explanation of mathematical certainty, however,
|
||
required systems constructed in this way to be free of
|
||
contradictions, and so Hilbert saw the main challenge as
|
||
demonstrating their consistency. For this purpose, he developed a
|
||
special formal system for describing formal systems, a
|
||
"metamathematics," which was supposed to be beyond
|
||
reproach. In the end, however, it was not possible to demonstrate the
|
||
consistency of arithmetic, or even of set theory, as Gödel showed.
|
||
(This is the origin of the puzzles encountered by set theory that
|
||
were solved in the ontological explanation of the truth of arithmetic
|
||
in </span></font></font><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/L/LoOtjR06.htm" target="Lo"><font color="#000000"><font face="Arial, sans-serif"><span lang="en-US"><u>Relations:
|
||
Solutions to puzzles</u></span></font></font></a><font color="#000000"><font face="Times New Roman, serif"><span lang="en-US">.)</span></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">What
|
||
logicists and formalists are getting at can be understood from our
|
||
sketch of the nature of rational subjects. They are also
|
||
epistemological philosophers reflecting on how we know the truth of
|
||
mathematical propositions. But by contrast to intuitionists, they
|
||
abstract from spatial imagination and focus on the use of language.
|
||
They identify brain states by the linguistic representations they
|
||
involve, and they use logical relations to keep track of the causal
|
||
roles that brain states play in drawing conclusions about what to
|
||
believe. By focusing exclusively on the formal relations among brain
|
||
states identified in that way, whole systems of mathematical proofs
|
||
can be reconstructed as formal deductive systems. The logical
|
||
structure of language represents the elements in such reasoning
|
||
completely enough that there are formal tests of the validity of
|
||
those inferences, making it seem that their truth can explained by
|
||
their deducibility from certain axioms and definitions. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Though
|
||
the validity of deductive relationships in formal systems does afford
|
||
a certain concept of certainty, it does not explain how mathematics
|
||
is true. Even logicists complained that formalism cannot account for
|
||
the truth of the axioms or the usefulness of the definitions that are
|
||
assumed. But neither do the axioms used in geometry and arithmetic
|
||
follow from the laws of logic. Indeed, any consistent set of
|
||
sentences could be used as axioms and definitions, because as far as
|
||
formal logic is concerned, deductive systems are just rule-governed
|
||
ways of transforming assumptions as inscriptions that preserve their
|
||
truth. Formalism has no explanation of why the axioms used in
|
||
mathematics should be singled out as true. Nor does it explain why
|
||
they, or the theorems derived from them, should be useful in
|
||
describing the natural world.</font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><font face="Arial, sans-serif">A<img src="data:image/png;base64,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" name="OdjRDeny_up" align="right" hspace="5" width="61" height="32" border="0">nti-realism
|
||
that denies the certainty of mathematics.</font> Applicability to the
|
||
natural world is, however, as crucial to the nature of mathematics as
|
||
its apparent certainty, and since neither the intuitionists nor
|
||
logicists/formalists were able to explain its certainty in a way that
|
||
would also explain its applicability to nature, naturalists could not
|
||
help being attracted to the view that mathematical objects are
|
||
somehow part of the natural world. That would be, like Platonism, a
|
||
kind of realism about mathematical objects. But since our way of
|
||
knowing about the natural world is perception, it would be more like
|
||
scientific realism, for there would be no basic difference between
|
||
mathematics and empirical science. And it would have to deny the
|
||
certainty of mathematics.</font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">The
|
||
view that mathematics is a form of empirical knowledge was first
|
||
defended by John Stuart Mill in the nineteenth century, but it was
|
||
renewed in 1983 by Philip </span></font></font></font><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Kitcher"><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><u>Kitcher</u></span></font></font></font></a><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">.</span></font></font></font><font color="#000000"><sup><span lang="en-US"><a class="sdendnoteanc" name="sdendnote5anc" href="#sdendnote5sym"><sup>v</sup></a></span></sup></font><font color="#000000"><span lang="en-US">
|
||
</span></font><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">Kitcher
|
||
rejected what he called "apriorism", the belief that the
|
||
certainty of mathematical knowledge comes from its being
|
||
epistemologically prior to experience of nature, and proceeded to
|
||
explain mathematics as a species of scientific knowledge. Kitcher
|
||
bases knowledge of mathematics on perception, by thinking of
|
||
arithmetic operations as "idealizations" of publicly
|
||
observable manipulations of natural objects.</span></font></font></font><font color="#000000"><sup><span lang="en-US"><a class="sdendnoteanc" name="sdendnote6anc" href="#sdendnote6sym"><sup>vi</sup></a></span></sup></font><font color="#000000"><span lang="en-US">
|
||
</span></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">The price
|
||
of explaining how mathematics is true about nature seems to be giving
|
||
up the belief that it has a certainty that is basically different
|
||
from natural science. Kitcher explains the appearance of certainty by
|
||
the extremely general character of the regularities described by
|
||
mathematical hypotheses. But since there is no essential difference
|
||
between mathematics and scientific hypotheses, he agrees that they
|
||
are confirmed in basically the same way. </font></font>
|
||
</p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt">Ontological
|
||
philosophy agrees with Kitcher in rejecting apriorism. It also takes
|
||
mathematics to be a form of empirical knowledge in the end. But the
|
||
end does not come so quickly as it does for Kitcher, because
|
||
ontological philosophy recognizes two levels of explanations
|
||
(ontological-cause explanations and efficient-cause explanations)
|
||
and, accordingly, two levels of empirical truths (empirical ontology
|
||
and empirical science). In other words, instead of taking mathematics
|
||
to be knowledge of very general regularities about what happens in
|
||
the world, it sees mathematics as knowledge about the most basic (or
|
||
categorical) features of what exists in the world, namely, how space
|
||
makes the world whole. That means that mathematics is still prior to
|
||
empirical science in a philosophically relevant way. But the priority
|
||
is ontological rather than epistemological.</font></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">Furthermore,
|
||
when this explanation of mathematical truth as ontologically
|
||
necessary is combined with what ontological philosophy holds about
|
||
the nature of reason, there is even a sense in which mathematics is
|
||
<i>epistemologically more certain </i>than empirical science. As we
|
||
have seen, it holds that mathematical knowledge is not merely a
|
||
correspondence of linguistic representations to the world, but also
|
||
involves a correspondence of representations in the brain’s spatial
|
||
imagination to the world. Thus, unlike Kitcher’s theory, it can
|
||
explain the role that constructions in imagination play in proving
|
||
mathematical truths according to intuitionists as well as the role of
|
||
formal deductive relationships among sentences that logicists and
|
||
formalists take to be basic.<sup><a class="sdendnoteanc" name="sdendnote7anc" href="#sdendnote7sym"><sup>vii</sup></a></sup></font></font></p>
|
||
<p lang="en-US" class="western" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; margin-top: 0.49cm; margin-bottom: 0.49cm; background: #66ccff; border-top: 6.75pt double #000080; border-bottom: 6.75pt double #808080; border-left: 6.75pt double #808080; border-right: 6.75pt double #808080; padding: 0.28cm 0.46cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">But as we
|
||
have seen, what explains the truth of both geometry and arithmetic
|
||
ontologically are the spatial relations that particular substances
|
||
have in a spatiomaterial world. Thus, since the rational subject is
|
||
part of the world, mathematical knowledge involves a relationship
|
||
between subject and object that is a correspondence between the
|
||
structure of spatial imagination in a part of the world and the basic
|
||
structure of the whole world of which he is part. It is within that
|
||
basic correspondence that rational being discover what happens in the
|
||
world by perception, and thus, if this is a spatiomaterial world,
|
||
mathematics is not only ontologically necessary, but
|
||
epistemologically certain. </font></font>
|
||
</p>
|
||
<div id="sdendnote1">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdendnotesym" name="sdendnote1sym" href="#sdendnote1anc">i</a><span lang="en-US">For
|
||
an accessible discussion of the problem of certainty in the
|
||
philosophy of mathematics, see </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Kline"><font color="#0000ff"><span lang="en-US"><u>Kline</u></span></font></a><span lang="en-US">
|
||
(1980). A somewhat more technical, but still readable discussion of
|
||
issues about infinity is </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Lavine"><font color="#0000ff"><span lang="en-US"><u>Lavine
|
||
</u></span></font></a><span lang="en-US">(1994).</span></p>
|
||
</div>
|
||
<div id="sdendnote2">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdendnotesym" name="sdendnote2sym" href="#sdendnote2anc">ii</a>Covert
|
||
manipulation also makes it possible to combine images of effects of
|
||
motion from locomotion imagination into a single geometrical
|
||
structure in manipulative imagination to think about all the
|
||
relations of the objects in some territory at once, like a map.</p>
|
||
</div>
|
||
<div id="sdendnote3">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdendnotesym" name="sdendnote3sym" href="#sdendnote3anc">iii</a><span lang="en-US">In
|
||
a much discussed paper, </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Benacerraf73"><font color="#0000ff"><span lang="en-US"><u>Benacerraf</u></span></font></a><span lang="en-US">
|
||
(1973) argues against Platonism on the ground that it is not
|
||
compatible with a causal theory of mathematical knowledge. </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Bigelow88"><font color="#0000ff"><span lang="en-US"><u>Bigelow</u></span></font></a><span lang="en-US">
|
||
(1988) nevertheless takes universals to be the objects of
|
||
mathematics, and he avoids this problem with abstract entities by
|
||
following </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Armstrong83"><font color="#0000ff"><span lang="en-US"><u>Armstrong</u></span></font></a><span lang="en-US">
|
||
(1983) and assuming, in effect, that universals are just spatial
|
||
relations of bits of matter, which are always instantiated (that is,
|
||
as so-called "tropes").</span></p>
|
||
</div>
|
||
<div id="sdendnote4">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdendnotesym" name="sdendnote4sym" href="#sdendnote4anc">iv</a><span lang="en-US">This
|
||
is similar to the "skeptical fictionalist" view of
|
||
mathematical truth defended by naturalists like </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Field80"><font color="#0000ff"><span lang="en-US"><u>Field
|
||
</u></span></font></a><span lang="en-US">(1980) and </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Papineau93"><font color="#0000ff"><span lang="en-US"><u>Papineau</u></span></font></a><span lang="en-US">
|
||
(1993, pp. 193-197). They think that they must deny the existence of
|
||
mathematical objects because such objects are abstract. But that is
|
||
because they do not recognize the ontological role that space plays
|
||
in making arithmetic true. They are implicitly materialists (taking
|
||
space to be just spatial relations) who are nominalists about the
|
||
concepts used in science, and though they recognize that mathematics
|
||
can facilitate complex inferences about the natural world, they
|
||
believe that all those inferences could, in principle, be made
|
||
without referring to numbers or geometrical figures as abstract
|
||
entities. Thus, they take such numbers and geometrical figures to be
|
||
useful fictions and are skeptical about their existence. But that
|
||
makes it just as puzzling why mathematics holds of the natural world
|
||
as Platonism does. However, if space as a substance containing all
|
||
the bits of matter is the ontological cause of geometrical figures
|
||
and the groupings of material objects called numbers, there is an
|
||
alternative, non-fictionalist defense of geometry and arithmetic
|
||
truth. Though there is every reason to be skeptical about the
|
||
existence of mathematical objects that are abstract entities, that
|
||
is no reason to believe that numbers are just useful fictions. They
|
||
could describe something concrete -- a very general, ontological
|
||
effect of the structure of space on the bits of matter it contains.</span></p>
|
||
</div>
|
||
<div id="sdendnote5">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdendnotesym" name="sdendnote5sym" href="#sdendnote5anc">v</a><span lang="en-US">Kitcher's
|
||
approach is endorsed by other philosophers of mathematics, such as
|
||
J. </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Bigelow88B"><font color="#0000ff"><span lang="en-US"><u>Bigelow</u></span></font></a><span lang="en-US">
|
||
(1988, p. 3). Bigelow holds that mathematics is about universals,
|
||
but he follows </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Armstrong83B"><font color="#0000ff"><span lang="en-US"><u>Armstrong's</u></span></font></a><span lang="en-US">
|
||
(1983) "a posteriori realism" in taking universals to be
|
||
physical, and thus, in the terms used here, he is a materialist.
|
||
Indeed, the subtitle of his book is "A Physicalist's Philosophy
|
||
of Mathematics".</span></p>
|
||
</div>
|
||
<div id="sdendnote6">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdendnotesym" name="sdendnote6sym" href="#sdendnote6anc">vi</a><span lang="en-US">The
|
||
traditional theory about mathematical truth that comes closest to
|
||
Kitcher's is abstractionism, the view originally defended by
|
||
Aristotle that mathematical objects are abstractions from perceived
|
||
objects. See </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Körner60"><font color="#0000ff"><span lang="en-US"><u>Körner</u></span></font></a><span lang="en-US">
|
||
(1960, pp. 18ff). Kitcher has a more sophisticated theory about how
|
||
mathematics is derived from perception than Aristotle. According to
|
||
his "evolutionary theory of mathematical knowledge" (p.
|
||
92), the abstractions come from idealizing the operations of
|
||
arithmetic, though Kitcher insists that this is compatible with
|
||
saying that "arithmetic describes the structure of reality"
|
||
(p. 109).</span></p>
|
||
</div>
|
||
<div id="sdendnote7">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-right: 6.64cm; margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdendnotesym" name="sdendnote7sym" href="#sdendnote7anc">vii</a><span lang="en-US">The
|
||
assumption that the use of language makes spatio-temporal
|
||
imagination a cognitive capacity of reflective subjects enables
|
||
spatiomaterialists to answer objections that </span><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/ObjText/#Kitcher83C"><font color="#0000ff"><span lang="en-US"><u>Kitcher</u></span></font></a><span lang="en-US">
|
||
(1983, pp. 50ff) raises to intuitionism (or "constructivism').
|
||
Contrary to Kitcher, it is possible to distinguish essential
|
||
properties from those that are accidental, because imagination is
|
||
not just "pictures in the mind", but images that
|
||
reflective subjects construct and manipulate within its structure.
|
||
When a geometrical figure, such as a triangle, is defined, it is
|
||
constructed in imagination, and thus, assuming that language-using
|
||
subjects can reflect on what they are doing, they can see the
|
||
effects of varying triangles in all possible ways on their
|
||
inferences about them. Second, although Kitcher is right to insist
|
||
that infinite sequences of operations, such the division of a line,
|
||
cannot be carried out in practice, subjects who can reflect on what
|
||
they do in imagination and its effects can come to see that what
|
||
will happen each time is limited in a certain way and, thus, infer
|
||
what would, and would not, happen if the operations were taken to
|
||
infinity. Finally, the problems about exactness that may arise with
|
||
Kitcher's "mental pictures" do not arise with
|
||
spatio-temporal imagination. For example, imagined straight lines
|
||
cannot be crooked, for they are constructed according to the
|
||
understanding of the structure of space that is built into the
|
||
structure of spatio-temporal imagination, that is, as the path of
|
||
the shortest distance between two points. In short, since
|
||
ontological philosophers postulate space as a substance containing
|
||
all the matter in the world, they need only recognize the basic role
|
||
that a spatio-temporal imagination would play in the reflective
|
||
subject's knowledge of the world to explain how reflective subjects
|
||
have a priori knowledge of mathematical truth, because its structure
|
||
corresponds to the structure of space. Indeed, without the capacity
|
||
to see what is given in perception against the background of what
|
||
imagination tells us is possible in three dimensional space, it is
|
||
hard to see how we could perceive a line as straight, a set of three
|
||
lines as a triangle, or anything as a mathematical object.</span></p>
|
||
</div>
|
||
</body>
|
||
</html> |