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971 lines
97 KiB
HTML
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<title>Relations as objects of knowledge</title>
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<meta name="author" content="Amr Gharbeia">
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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>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
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as objects of knowledge.</b></font> Ontological philosophy explains
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relations as aspects of the world that exist because of the essential
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nature of space and how space contains bits of matter at any moment,
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and correspondence to them explains, as we have seen, how
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mathematical propositions are true. That means that mathematics is
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prior to empirical science in the sense of being <i>ontologically
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necessary</i>. However, necessity in the sense of being <i>certain</i>
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is what has traditionally been thought to make mathematics different
|
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from empirical science. Certainty is what is relevant about
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mathematics when the project is justifying belief in certain
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propositions by how they are related to what is known in other ways.
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Thus, epistemological philosophy approaches mathematical objects as
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objects of knowledge, rather than as aspects of the world, and it is
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not obvious that what mathematics is about are the most basic
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relations that hold in 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: 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>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
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problem of mathematical knowledge.</b></font> When the certainty of
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mathematics is taken for granted, the problem of mathematical
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knowledge is to explain how such certainty is possible, that is, why
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it is more certain than what is known by ordinary experience of what
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happens in the world.<sup><a class="sdfootnoteanc" name="sdfootnote1anc" href="#sdfootnote1sym"><sup>1</sup></a></sup></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">It
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is somewhat misleading to think of the certainty of mathematics only
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as a problem, for in the beginning, that is what inspired belief in
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epistemological philosophy. In ancient Greece, mathematics was taken
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as an example to show the possibility of philosophy as a superior
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kind of knowledge of the world, one that revealed necessary truths.
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In the </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"><i>Meno,</i>
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for example, Plato describes Socrates as asking a slave boy a series
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of questions about some lines he draws in the sand which lead the boy
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to recognize the truth of a special case of Pythagoras’ theorem
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(that the square built on the diagonal of a square is twice the area
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of the first square). That put the slave boy in a position to defend
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what he knew rationally, and Plato used that story to illustrate how
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<i>knowledge </i>is different from <i>true belief</i>. </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">Beliefs
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about whose truth one can be certain are what philosophy pursues out
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of its love of wisdom, according to Plato. Above the entrance to
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Plato’s Academy, the first university, it was written that no one
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should enter who does not know mathematics. </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 hard to overstate how important mathematics has been to the
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credibility of philosophy’s claim to provide a kind of knowledge of
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that is superior to our ordinary ways of knowing what happens in the
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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>
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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">
|
||
<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,iVBORw0KGgoAAAANSUhEUgAAAGkAAAAtCAMAAACalbVyAAAAwFBMVEX////38PDv4ODn0NDjx5vfw5jfwMDWu5LTuZDXsLDFrIbMmZnHkJCynHqvmniumHe/gIC6el+3cHCYhWiWg2avYGCFdFuoXEh8bFR2Z1CmUFCfVUJxY02USzppXEieQEBmWkZhVUKHPzFbUD6ZMzN8NipgOi10LiSOICBuKSCGEBB+AAAcGRM/AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAD7qT75AAAD3klEQVR4nM2Xa5fcJgyGuWS1YU1bwlnS1E2jTUsTgpv5/z8vEhjsuXrmNJkTfWBsJHiwMK814tUt9vjfbrd7vGlIN/FK3MUeHog03cHE69f3Ij093Yv02y9rUsbZYoSNkTFtzR0N2BXp+XlNigCgBDUWtx4U3EZAlg7HFend28PsuXJLpIyx9iDm8ptax2LNhWtX7TxYq/jwxznSoEAOfKU0CEpDBglK0YTCGwHlmZorSAAp5/FBlk7kNxtXpI8fzpEoKoo4JUkX3AxAK7WKxhjeIiJ1l+QszZlKYuTOcPRM/7yczR67cfLKkSlXp8vErislUndp7ecsTpPX3FpzO8mV6RzOAIELqbuyN1rM75mDub2ZNMp5tdqV7ryQuqtmrW5K7QR7O4lSEzAMhBQOg7TTQuouSh66viI9opHpChKWg5JK63jvgwHjaRocwITWOQVcXGGgE9hOcvb1Ju0fuBOkH2Q/F8mGGybMeNxXNPQyqb6xmwo3G9SX89hR+hZSUbkmdbFKVyfF+WBGnPc98UWa70j7D1aU6/iEJWKPJAZNKudmqQMFWgR638ncBHwraQmJLiRpEgpL0kanFQQnVmhQ5IYidGVWRzrI4/lnOCQ5Pnul4SnJ42V/JkPrM6ae3UwrL5oYSsOyytHs7tkbeVk5FgdL5z6Jn7832Vt6qL19ossoWHtg1pmlSW6gh12RTPsABoqm+c6TkjQe/REJC8mFA9IobKBDvSK1DdPaIV4kOT0P2Sfl9rnZJ5VPuFuTLMxhuc53noSS5EwJ3ouRXqtGmqz0OFp/QHKkc7T30zQoHBP3UE4QrS+f96PsFS3rzQjgI89PEjZWhUN+yyhJQP6iaEvjSA7ZnS1Aqt22SB+XLKnF/1xq9ANJcFQCnbKjqMsl4kmSOKGSJ4YeRl0uEbdITdpOlHolKiO2wFhJXRr7xQbJs26R3hVpa6VeER81sMakGiVJEfmttFzxCc4gayQsGrlFygOXkE6lqn+t1MtUy6EiCp9trmY4gEsUq3PNnqZllLPXNHKDFDT0jwaP76UeKdrgDRV4vpAMsDyJubSkyCjmcV0jN0hSmj1SL/WCyiKP1KRyC4XkaiY5sm7VWiM3SGPUnI5O6qVeFnbgndIVbNuala+RVRrXGrm1T1PmT18n9VJvMjyDE35qoj+iB/5OBfSslE5aB3KlkZdJRf+8r1pXRKuVerUYTF0gubYb2DHSlhWljI5+YeoaeZn0P4yPF6pTxdT3JtHHRmh/ynNvhb2LMenN73/+9fe/n79caV+/Xhu5tk/87/Ph8enX53fvXz5eZ7vdlYH79vL+G8fsUA2kMhp4AAAAAElFTkSuQmCC" name="OdjRTheo_up" align="right" hspace="5" width="105" height="45" border="0">heories
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of mathematical knowledge.</b></font> To set the stage for
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considering the received explanations of the certainty of
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mathematics, let us consider briefly what ontological philosophy
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implies about the <i>knowledge of </i>mathematics. We will then take
|
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up the epistemological theories. </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"><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"><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="sdfootnoteanc" name="sdfootnote2anc" href="#sdfootnote2sym"><sup>2</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,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" 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,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" 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="sdfootnoteanc" name="sdfootnote3anc" href="#sdfootnote3sym"><sup>3</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="sdfootnoteanc" name="sdfootnote4anc" href="#sdfootnote4sym"><sup>4</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,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" 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><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><a class="sdfootnoteanc" name="sdfootnote5anc" href="#sdfootnote5sym"><sup>5</sup></a></span></font></font></sup></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><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US"><a class="sdfootnoteanc" name="sdfootnote6anc" href="#sdfootnote6sym"><sup>6</sup></a></span></font></font></sup></font><font color="#000000"><font face="Times New Roman, serif"><font size="3" style="font-size: 12pt"><span lang="en-US">
|
||
</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; 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="sdfootnoteanc" name="sdfootnote7anc" href="#sdfootnote7sym"><sup>7</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="sdfootnote1">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdfootnotesym" name="sdfootnote1sym" href="#sdfootnote1anc">1</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="sdfootnote2">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdfootnotesym" name="sdfootnote2sym" href="#sdfootnote2anc">2</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="sdfootnote3">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdfootnotesym" name="sdfootnote3sym" href="#sdfootnote3anc">3</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="sdfootnote4">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdfootnotesym" name="sdfootnote4sym" href="#sdfootnote4anc">4</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="sdfootnote5">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdfootnotesym" name="sdfootnote5sym" href="#sdfootnote5anc">5</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="sdfootnote6">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdfootnotesym" name="sdfootnote6sym" href="#sdfootnote6anc">6</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="sdfootnote7">
|
||
<p lang="en-US" class="sdendnote-western" style="margin-right: 4.1cm; margin-top: 0cm; margin-bottom: 0.25cm">
|
||
<a class="sdfootnotesym" name="sdfootnote7sym" href="#sdfootnote7anc">7</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
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||
imagination tells us is possible in three dimensional space, it is
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hard to see how we could perceive a line as straight, a set of three
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||
lines as a triangle, or anything as a mathematical object.</span></p>
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</div>
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</body>
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