881 lines
74 KiB
HTML
881 lines
74 KiB
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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<p lang="en-US" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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">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, <font color="#0000ff"><u><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCaL.htm" target="Lo"><font face="Arial, sans-serif">Change</font></a></u></font>,
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which is the subject matter of science. </font></font></font>
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</p>
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<p lang="en-US" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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,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" 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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
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holds in every region. Or as we assumed (provisionally) in the
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foundation, each part of space has the same kinds of relations to all
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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
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truth of geometry by its correspondence to space does not depend on
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geometry being stated as an axiom system.</font></font></font></p>
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<p lang="en-US" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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
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are all deduced from them (and definitions of terms introduced to
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simplify the statement of geometrical propositions). The former
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propositions are called "axioms," and the latter are called
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"theorems." This way of organizing geometrical propositions
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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
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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
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the way of premises. Geometry lends itself to axiomatization because
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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
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certain basic aspects of that structure, it is possible to combine
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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
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constructions from simpler truths are the derivation of theorems in
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geometry. </font></font></font>
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</p>
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<p lang="en-US" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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
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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>
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</p>
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<p lang="en-US" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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">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>
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</p>
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<p lang="en-US" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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
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was that there are different ways of axiomatizing geometry. That is,
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different geometrical propositions can be used as axioms, and still
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all the rest follow logically. Thus, there is no necessary order by
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which some should be taken as implying others. </font></font>
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</p>
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<p lang="en-US" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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">Second, and
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more importantly, it became clear that the deductive relationship
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cannot, by itself, establish any truth about the world. The truth of
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the theorems depends on the truth of the axioms. But the truth of the
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axioms cannot be shown within the deductive system. The axioms
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contain terms which are not defined within the system, or so-called
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"primitive terms," and thus, the truth of the axioms
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depends on what those terms refer to. And there are other objects
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that will make the axioms of geometry true (the set of whole numbers,
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if nothing else, according to the Löwenheim-Skolem theorem). The
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deducibility of the theorems from the axioms means that the theorems
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will be true of whatever objects make the axioms true, but unless the
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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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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
|
||
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 <i>ultimate members </i>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 <font color="#0000ff"><u><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCaL.htm" target="Lo"><font face="Arial, sans-serif">Change</font></a></u></font>.)
|
||
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 <font color="#0000ff"><u><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCaL.htm" target="Lo"><font face="Arial, sans-serif">Change</font></a></u></font>).
|
||
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. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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><i> </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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; margin-top: 0.49cm; margin-bottom: 0.49cm; line-height: 100%; widows: 0; orphans: 0">
|
||
<font color="#000000"><font face="Times New Roman, serif">That is,
|
||
Gödel’s incompleteness theory does not give us any reason to
|
||
believe that there are true arithmetic propositions <i>about the
|
||
world</i> 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
|
||
<font color="#0000ff"><u><font face="Arial, sans-serif"><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></font></u></font>), 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.) </font></font>
|
||
</p>
|
||
<p lang="en-US" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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" align="left" style="margin-left: 1.27cm; margin-right: 2.54cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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" align="left" style="margin-left: 2.54cm; margin-right: 1.27cm; text-indent: 0cm; 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 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 <font color="#0000ff"><u><a href="/F:/Philosophy/Existentialism/The%20Wholeness%20Of%20the%20World/www.twow.net/Lo/LoOtkCaL.htm"><font face="Arial, sans-serif">Change</font></a></u></font><font face="Arial, sans-serif">
|
||
</font>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. </font></font></font>
|
||
</p>
|
||
<p lang="en-US" align="left" style="margin-left: 3.81cm; margin-right: 2.03cm; text-indent: 0cm; 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>
|
||
</body>
|
||
</html> |