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<font face="Times New Roman,Times"><font size=-1>http://www.allanrandall.ca/Geometry.html</font></font>
<br><font face="Times New Roman,Times"><font size=-1><a href="copyright1998.html">Copyright</a>
© 1998, Allan Randall</font></font>
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<center>
<h2>
<font face="Times New Roman,Times"><font size=-1> </font> <b>A Critique
of the Kantian View of Geometry</b></font></h2></center>
<center><b><font face="Times New Roman,Times">Allan F. Randall</font></b>
<br><b><font face="Times New Roman,Times">Dept. of Philosophy, York University</font></b>
<br><b><font face="Times New Roman,Times">Toronto, Ontario, Canada</font></b>
<br><b><font face="Times New Roman,Times"><a href="mailto:arandall@ican.net">research@allanrandall.ca</a>,
<a href="http://www.allanrandall.ca/">http://www.allanrandall.ca/</a></font></b></center>
<br>
<center>
<h3>
<b><font face="Times New Roman,Times">Abstract</font></b></h3></center>
<blockquote><b><font face="Times New Roman,Times"><font size=-1>A survey
of Kant's views on space, time, geometry and the synthetic nature of mathematics.
I concentrate mostly on geometry, but comment briefly on the syntheticity
of logic and arithmetic as well. I believe the view of many that Kant's
system denied the possibility of non-Euclidean geometries is clearly mistaken,
as Kant himself used a non-Euclidean geometry (spherical geometry, used
in his day for navigational purposes) in order to explain his idea, which
amounts to an anticipation of the later discovery of the general concept
of non-Euclidean geometries. Kant's view of geometry and arithmetic as
synthetic was, I believe, essentially correct, in that geometry and arithmetic
are both synthetic <i>a priori </i>if considered as branches of mathematics
independent of the rest of mathematics. However, the view that somehow
logic is analytic, while mathematics is synthetic for Kantian reasons,
is mistaken. All three disciplines--logic, arithmetic and geometry--are synthetic
<i>as
</i>disciplines independent from one another. However, they have a common
basis, recursion theory, which I prefer to identify with mathematics as
a whole. As a result, I do not say, as is often considered to be the Kantian
view, that mathematics is synthetic while logic is analytic. Rather, I
prefer to say that mathematics is analytic, while logic is synthetic. This
is perfectly consistent with Kant's system, since it was arithmetic and
geometry individually that he argued were synthetic. What Kant called the
analytic is recursion theory, which could be considered as a basic formulation
of mathematics <i>or</i> logic--or better, both mathematics and logic could
be recognized as essentially the same discipline. However, if "logic" is
taken to mean "predicate logic", as is often the case in modern times,
then it is mathematics that is closer to Kant's analytic, not logic. Such
ambiguities, of course, can be avoided by simply associating Kant's analytic
with recursion theory, and avoiding the controversies as to what counts
as mathematics or logic.</font></font></b></blockquote>
<blockquote>
<blockquote>
<blockquote>
<blockquote><font face="Times New Roman,Times"><font size=-1>"Geometry
has now permeated all branches of mathematics, and it is sometimes difficult
to distinguish it from algebra or analysis. The importance of geometric
intuition, however, has not diminished from antiquity until today."</font></font></blockquote>
</blockquote>
</blockquote>
<center><font face="Times New Roman,Times"><font size=-1><i><a href="http://www.amazon.com/exec/obidos/ISBN=0262090260/allanfrandallA/">Encyclopedic
Dictionary of Mathematics</a></i>, Vol. I., "Geometry", p. 685, The MIT
Press, 1987.</font></font></center>
<br> </blockquote>
<center>
<h3>
<b><font face="Times New Roman,Times"><font size=+1>I. Introduction</font></font></b></h3></center>
<font face="Times New Roman,Times">In the <i>Critique of Pure Reason</i>
[<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a>] and the <i>Prolegomena to Any Future Metaphysics
</i>[<a href="http://www.allanrandall.ca/Geometry.html#prolegomena">Prolegomena</a>],
Kant argues that the truths of geometry are synthetic <i>a priori </i>truths,
and not analytic, as most today would probably assume. Truths of logic
and truths that are merely true by definition are "analytic" because they
depend on an "analysis", or breakdown, of a concept into its components,
without any need to bring in external information. Analytic truths are
thus necessarily <i>a priori</i>. In other words, they are nonempirical
and independent of sensory input. Synthetic truths, on the other hand,
require that a concept to be "synthesized" or combined with some other
information, perhaps another concept or some sensory data, to produce something
truly new. When a concept is thus combined with empirical data, the synthetic
"judgement" that results is <i>a posteriori,
</i>as it depends on sensory
input.<i> </i>When the synthesis is performed purely in imagination, however,
and does not depend at all on any particular sensory input, the resulting
judgement is both synthetic and <i>a priori</i>.</font>
<p><font face="Times New Roman,Times">Kant´s unique contribution
was his claim that one could thus have <i>a priori</i> truths that, being
synthetic, were not merely logical or definitional. One can have <i>a priori</i>
truths that are contingent, and so could be imagined to be otherwise. Mathematical
truths, according to Kant, fall into this category.</font>
<p><font face="Times New Roman,Times">This remarkable claim flies in the
face of the most popular current conceptions of mathematics, in which the
truths of mathematics are generally assumed to be so airtight that one
could not possibly imagine them as otherwise. However, Kant´s argument
can be quite convincing, and we will examine it in detail, in order to
see if this remarkable claim of the synthetic nature of mathematics holds
up. In particular, we will look at Kant´s discussions on the <i>a
priori</i> syntheticity of geometry. Kant argues that synthetic a priori
geometric truths are possible if (and only if) space is what he calls a
transcendentally ideal <i>a priori </i>form of sensory intuition. In brief,
this means that space, which much be assumed for any geometry to take place,
is also a necessary presupposition for human cognition in general. Our
notion of Euclidean, three-dimensional space is built into us. We cannot
help but cognize things in terms of it. It is thus a necessary precondition
for our empirical experience (this argument is called "transcendental",
because it argues <i>from</i> an empirical given <i>to</i> its necessary
preconditions). Kant calls such a precondition for experience an "intuition".
So geometry presupposes something which is a general condition for all
experience, namely, three-dimensional Euclidean space. For this reason,
it cannot be analytic.</font>
<p><font face="Times New Roman,Times">This can be seen by trying to imagine
higher dimensional or non-Euclidean spaces, which would make many of Kant´s
geometrical "truths" to be false. Such geometric truths (i.e., those that
depend on three-dimensional Euclidean space) are not "logically true" or
"true by definition", but "necessarily true for us, as humans". And since,
as Kant argues, three-dimensional Euclidean space is a necessary presupposed
form for all human experience, it follows that geometrical truths are both
<i>a
priori</i> and synthetic.</font>
<p><font face="Times New Roman,Times">It further follows from all this,
that space is not a "thing in itself" (or at least, can never be known
as such). Since it is imposed on sensory data as a necessary condition
for <i>human </i>cognition, it is thus a feature of the human mind, rather
than of an external reality. Thus, this whole transcendental argument implies
the "ideality", or mind-dependence, of space (and hence geometry as well,
which might be called the science of space in general). If space were not
an innate feature of the human mind, but an external thing out there in
the world, geometry would not be <i>a priori</i> synthetic, but <i>a posteriori</i>
synthetic. On the other hand, if space were merely an arbitrary mental
construct, geometry would be <i>a priori</i> analytic.</font>
<p><font face="Times New Roman,Times">I examine this entire argument in
more detail, in terms of a somewhat modified, more modern computational
version of Kant´s constructive philosophy of mathematics. I will
also try to explain why Kant´s conclusions about the nature of geometry
and its relation to space are troubling to some people, and I present some
possible problems with his argument. Finally, I will comment briefly on
the issue of the syntheticity of mathematics as a whole. Kant's view of
geometry and arithmetic as synthetic was, I believe, essentially correct,
in that geometry and arithmetic are both synthetic <i>a priori </i>if considered
as individual branches of mathematics, independent of the rest of mathematics.
However, the view that somehow logic is analytic, while mathematics is
synthetic for Kantian reasons, is mistaken. All three disciplines--logic,
arithmetic and geometry--are synthetic <i>as </i>disciplines independent
from one another. However, they have a common basis, recursion theory,
which I prefer to identify with mathematics as a whole, not logic, and
certainly not "predicate logic".</font>
<center>
<h3>
<b><font face="Times New Roman,Times"><font size=+1>II. The Nature of Synthesis</font></font></b></h3></center>
<font face="Times New Roman,Times">We will start by looking at just what
it means for geometry to be synthetic. It is important that Kant <i>does
</i>believe
geometry to have both a synthetic and analytic component, and that there
are analytic truths involved in geometry. To claim otherwise would be trivially
absurd, since geometry obviously employs concepts that can be broken down
and analyzed. Kant is not saying that geometry <i>as practised</i> is not
at all analytic, he is simply saying that <i>as geometry,
</i>it is synthetic.
No "<i>fundamental</i> proposition of <i>pure</i> geometry", he says, is
analytic [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/3.html#052">53</a>,
emphasis mine]. Furthermore, "some few fundamental propositions, presupposed
by the geometrician, are, indeed, really analytic" [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a><a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/3.html#052">54</a>].
He gives "a + b > a" and "a = a" as examples. But such analytic components
of geometry, while certainly important to the science, are no longer "geometric"
if divorced from the synthetic judgements they must at some point be employed
in the service of (or else, we would not say they were part of geometry
at all).</font>
<p><font face="Times New Roman,Times">But even aside from those analytic
propositions used in the service of geometry, there is another sense in
which <i>all </i>geometric propositions have an analytic component in Kant´s
system, the same sense in which all synthetic judgements have an analytic
component and all analytic judgements have a synthetic component. Fig.
1 illustrates the general principle (as described by Kant in many places,
but see in particular [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/11.html#257">259</a>]).
All judgements, it turns out, whether primarily analytic or primarily synthetic,
involve synthesis: the combining of a formal concept with some empirically
given object, which we might call the incoming sensory data.</font>
<center>
<p><img SRC="Kant7.gif" height=130 width=460>
<br><b><font face="Times New Roman,Times">Fig. 1. Judgement: synthesis
performed in the mind via intuition.</font></b></center>
<p><font face="Times New Roman,Times">The "formal" concept is itself not
an objective concept, but an abstract form that could be applied to many
different empirical particulars. Since it is purely conceptual, any judgements
that flow purely out of this concept, that do not rely on an empirical
object, are "analytic". But note that without synthesis, we cannot use
the concept at all, or make any judgement about it in the first place.
This is because, for humans, general concepts are always thought via some
kind of particular instantiation involving synthesis with empirical data
(although this may be imagined data, and the synthesis thus performed in
imagination).</font>
<p><font face="Times New Roman,Times">So even purely analytic judgements
are always thought via some kind of synthesis, by imagining or even actually
generating some kind of empirical experience. When I prove to myself that
the shortest distance between any two points is a straight line, it seems
obviously<i> a priori</i> and universal. Yet how did I prove it? I certainly
did not draw every possible pair of points and then draw every possible
path between them and measure their distances! No, instead I either imagined
or actually drew a <i>particular </i>set of points and a particular line
and then proceeded to reason about the resulting empirical objects. Without
the empirical object, my general forms or concepts are useless: "In the
absence of such [empirical] object, it [the formal concept] has no meaning
and is completely lacking in content, though it may still contain the logical
function which is required for making a concept out of any data that may
be presented." [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/11.html#257">259</a>]
The formal concept is not an objective concept on its own, until it is
synthesized with an empirical object. Likewise, the empirical object is
only an objective object when considered as synthesized with some concept.
It is the fusion, or synthesis, of the two that generates an actual experience
in us [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#065">65</a>],
which is why Kant says that "all our knowledge begins with experience."
[<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/3.html#041">41</a>]</font>
<p><font face="Times New Roman,Times">This process of synthesis, as depicted
in figure 1, is achieved in and through the cognitive faculty that Kant
calls "intuition". Formal concepts, which cannot even be thought on their
own, are like logical or mathematical functions, which require some kind
of data to be realized as anything concrete (for more information on recursion
theory and my own reasons for believing a function requires input data
for objective validity, see my <a href="http://www.allanrandall.ca/">home
page</a>). The function F() is merely an abstraction. To actually think
about F(), we need to talk in terms of its application to some kind of
data, as in F(x). Even to think about the concept (i.e., function) completely
in general, we can still only do this with respect to some domain of empirical
objects that can be synthesized with the concept, as in F(x), x:{o<sub><font size=-2>1</font></sub>,
o<sub><font size=-2>2</font></sub>, ...}. Kant´s conception of this
melds very naturally with modern notions of recursive functions (i.e.,
computation), and so I will talk in those terms. A judgement, in fact,
is like a computation in that it can only be understood as a <i>process</i>
of combining things, not as something static on its own. In fact, it can
be very useful to imagine synthesis in terms of modern computation theory.
"Intuition" then becomes a kind of information processing "filter" through
which we perceive the world. No empirical object is ever perceived as it
is in itself, but always through this filtering process via which we apply
our concepts to incoming sensory information. Any cognition requires some
kind of cognizing equipment, which will necessarily impose structure on
the incoming sensory data. "All thought must," says Kant, "directly or
indirectly ... relate ultimately to intuitions, and therefore, with us,
to sensibility, because in no other way can an object be given to us."
[<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#065">65</a>]</font>
<p><font face="Times New Roman,Times">This information filtering process
produces in the mind a "representation." No matter what the nature of our
filter isÑwhether we developed our intuition filters gradually over
time or were born with themÑit is impossible to perceive anything
except through some such filter. In humans, Kant observes two basic kinds
of intuition filters, which produce in us two different kinds of representations:
those that are "impure" and include empirical information and those that
are "pure" in which "there is nothing that belongs to sensation" [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a><a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#065">66</a>].</font>
<center>
<p><img SRC="Kant8.gif" height=406 width=560>
<br><b><font face="Times New Roman,Times">Fig. 2: Pure and impure intuition
as information processing filters.</font></b></center>
<p><font face="Times New Roman,Times">Figure 2 shows the way these kinds
of intuition work as information filters. Incoming sensory data D can be
synthesized with a formal concept F() via pure intuition to produce an
objective experience F(D) that is free of empirical content, such as a
mental image of a triangle. The intuition filter is considered pure in
this case only if the resulting triangle we hold in our minds is sufficiently
free of dependence on the empirical object. This will not necessarily hold
of all mental images of triangles. Just because the synthesis occurs in
imagination does not guarantee that it is pure. One can imagine a triangle
in one´s mind without actually drawing one on paper or looking at
a triangular physical object, but that mental triangle might still be dependent
on the empirical data. It could be coloured blue, for instance. This empirical
dependence may be due to the fact that we constructed our F(D) experience
out of some sensory data D, which we failed to filter out. Or it could
be due to the fact that the concept we used depended on past impure empirical
experiences, having been built up from them over time. To call the synthesis
pure, the intuition filter must remove all traces of dependence on D, so
that the resulting F(D) has content pertaining <i>only</i> to the formal
concept F() and not to the data D.</font>
<p><font face="Times New Roman,Times">This may seem impossible, for how
can we get rid of D, so that we can think solely about F() on its own,
if F() cannot be thought on its own without some synthesis with some kind
of D! It might seem we are caught in a circle. We can cognize F() only
if given D, yet with D, we are cognizing F(D) and not F(). But some reflection
will reveal that our pure intuition filters, however they may work, do
seem to have this ability to filter out dependence on D. When we perform
geometric proofs by the construction of figures, for example, we may imagine
a particular triangle in our minds, while at the same time performing our
reasoning so that the particular mental triangle we have constructed is
a stand-in for <i>all </i>triangles in general. So long as we have processed
our triangle experience in a "pure" fashion, we are in effect reasoning
about the general concept of triangle, <i>not </i>the particular one we
have constructed in imagination. So our reasoning about the triangle is
truly <i>a priori</i> in that it concerns only what flows solely from the
general concept itself, and of course, whatever may be added by the intuition
filter. We will ignore for now just what if anything gets added by the
filter, and whether this can itself be further filtered. For now, we will
just note that in either case, the result is <i>a priori</i> if it does
not depend on experience.</font>
<p><font face="Times New Roman,Times">Just as it is possible to perform
synthesis in imagination alone and still obtain a priori results, it is
also possible to perform it with paper and pen, or other physical objects,
and still retain the <i>a priority</i>. One could just as well perform
a geometric proof with pen and paper as with a mental image, and as long
as our pure intuition filter has removed dependence on the empirical element,
we are nonetheless content that the result is <i>a priori</i>. So, although
we noted earlier that "all knowledge begins with experience" (seeing that
nothing can be cognized except <i>as</i> an experience), we see now that
"it does not follow that it all arises out of experience," [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a><a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/3.html#041">41</a>]
since our intuition filters allow us to filter out the empirical component.</font>
<p><font face="Times New Roman,Times">As can be seen from figure 2, pure
and impure intuition can trade information, and experience can be built
up that combine the two. In fact, most everyday reasoning and experiencing
involves both <i>a priori</i> and <i>a posteriori</i> components. But what
exactly is the pure intuition filter? Much of how we filter and process
information is obviously empirical, which must be ignored. Likewise, much
is conceptual, and this must also be removed if we are to be left with
just pure intuition. Kant describes the basic method:</font>
<blockquote><font face="Times New Roman,Times">We shall, therefore, first
isolate sensibility, by taking away from it everything which the understanding
thinks through its concepts, so that nothing may be left save empirical
intuition. Secondly, we shall also separate off from it everything which
belongs to sensation, so that nothing may remain save pure intuition and
the mere form of appearances, which is all that sensibility can supply
<i>a
priori</i>. [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#067">67</a>]</font></blockquote>
<font face="Times New Roman,Times">So far we have seen two kinds of truth
that can be realized in our judgements:<i> a priori</i> and<i> a posteriori</i>.
When we perform the above reduction, we should be left with whatever it
is that distinguishes those <i>a priori</i> truths that are necessary from
those that are contingent. Truths that flow solely out of the preexisting
content of the concepts themselves have somehow managed to detach themselves
from dependence on both the empirical data <i>and </i>the structure of
the intuition filter, (whatever that may be). Such judgements are analytic.
Truths that manage to filter out dependence on the empirical component,
but nonetheless retain a dependence on the nature of the information filters
themselves, are not necessary, but contingent, since it is conceptually
imaginable for it to be otherwise. Further filtering would be necessary
to remove the dependence on human intuition if generalization to a necessary
truth were desired (such generalization may or may not be possible).</font>
<p><font face="Times New Roman,Times">Kant does not provide a rigorous
proof as to what is left when pure intuition is thus isolated. Instead,
he provides numerous arguments that essentially invite the reader to look
for anything else, other than the two general forms he presents, that cannot
be determined to have a conceptual or empirical component. These general
forms of human sensory experience, according to Kant, are space and time.
Space he calls the pure form of outer intuition, since "by means of outer
sense, a property of our mind, we represent to ourselves objects as outside
us, and all without exception in space" [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#067">67</a>].
Time is the pure form of inner intuition, since it is presupposed even
in purely internal thought that is not directed outward [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a><a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#074">74-75</a>].
It is space that we will concentrate on here, since that is what is most
relevant for geometry.</font>
<center>
<h3>
<b><font face="Times New Roman,Times"><font size=+1>III. The Transcendental
Ideality of Space</font></font></b></h3></center>
<font face="Times New Roman,Times">"But," we might put it to Kant, "couldn´t
it be that space, as ubiquitous as it is in our experience, is nonetheless
still just a concept built up by example, via induction, rather than some
kind of innate <i>a priori</i> necessary form for all human cognition?"
Kant responds that "space is not an empirical concept which has been derived
from outer experiences. For in order that certain sensations be referred
to something outside me ... the representation of space must be presupposed"
[<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#067">68</a>].
Space cannot be built up from external empirical data, since the very cognizing
of any such data presupposes space in the first place. We need space, then,
even to begin the induction process. So the general notion of space must
be innate.</font>
<p><font face="Times New Roman,Times">"But," we might venture to further
protest, "what if space is, as you say, intuitive and not empirical, but
nonetheless still not a <i>necessary </i>condition for human cognition.
Perhaps it is possible to have cognitions in which space does not play
a role." Kant dismisses with this by pointing out that "we can never represent
to ourselves the absence of space, though we can quite well think it as
empty of objects" [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#067">68</a>].
If we cannot represent an objectified concept without space, then space
is indeed universally part of any cognition. Note that while it is possible
to conceptualize objects, such as logical forms, that are not in space,
this is only by generalizing from a cognition that necessarily <i>does
</i>involve
space. The concept of "justice" may be nonspatial, but we cannot actually
cognize any instance of justice without imagining things as being in space.</font>
<p><font face="Times New Roman,Times">"But," we continue to pester Kant,
" might space not itself be a universal concept, rather than an intuition?"
Kant objects to this because "we can represent to ourselves only one space"
[<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#067">69</a>].
But concepts are general forms to be instantiated by synthesis with particular
empirical data, <i>not </i>particular things themselves. Although a concept
can act as the data for another concept, something that is absolutely singular,
like our notion of space, can only ever act as a particular to be applied
to a concept, and not the other way around. The general notion of "space"
cannot be instantiated. We only ever have a notion of one.</font>
<p><font face="Times New Roman,Times">Space, of course, is <i>also </i>a
concept, so Kant is not saying that we have no general concept of it at
all. It may even be possible conceptually to imagine "other spaces". This
constitutes an analytic extension, or generalization, of our innate intuition.
However, our <i>intuitive </i>notion of space remains singular, and any
analytic extensions will always be synthesized somehow in terms of our
natural tendency to visual a singular infinite 3-D space. For how could
you imagine "other spaces"? The only way the human mind can really deal
with such an abstract concept is to picture numerous different "spaces",
or universes, laid out in some kind of super-space. This super-space would,
of course, have to be at least four-dimensional. But do we actually picture
such a thing? No, because our intuition is not up to the task. We will
tend to employ whatever tricks we need, such as collapsing the multi-dimensional
space down into a three-dimensional projection, in order to render the
concept in terms that our innate intuition can handle.</font>
<p><font face="Times New Roman,Times">Kant is <i>not</i> saying, then,
that multi-dimensional spaces or "other spaces" are impossible. It is sometimes
suggested that Kant was closed off to the very idea of anything but three-dimensional
space. But this is a misunderstanding. In fact, the early pre-critical
Kant speculated about the possibility of multi-dimensional spaces as a
way that the universe might contain more than one three-dimensional world
[<a href="http://www.allanrandall.ca/Geometry.html#forces">Kant 1747</a>]. While he did not pursue this idea in
the <i>Critique</i>, it certainly shows that the idea of multi-dimensional
spaces was not alien or absurd to him. He simply believed that any such
extension would take us beyond what is necessary for human cognition, into
the larger sphere of concepts in general. For Kant, this takes us beyond
geometry per se (i.e., it is no longer pure geometry).</font>
<p><font face="Times New Roman,Times">As a singular, infinite manifold,
space contains an infinity of possible "parts" that it can at least potentially
be broken down into (i.e., places or locations). Kant points out [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a><a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#067">69-70</a>]
that this also distinguishes it from a mere concept, which, while being
a part of the infinite manifold of conceptual possibility, cannot itself
be viewed as contained an infinity of further representations or concepts.
Again, we see that "space is an intuition not <i>merely</i> a concept"
[<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#067">70</a>].</font>
<p><font face="Times New Roman,Times">The move from experience to the necessary
precondition of space is transcendental. Kant argues further from this
that space is transcendentally "ideal", meaning that the transcendental
technique reveals only a property of the mind, not anything about the "things
in themselves", which remain "quite unknown to us" [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a><a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#074">74</a>].
Space cannot possibly be known as a "thing in itself" for Kant, since it
is synthetically imposed on incoming sensory data by us. Kant´s whole
way of looking at cognition precludes any real "knowledge" of a thing in
itself apart from our minds, since all we ever experience are filtered
appearances, never the "really real" thing behind it. This does not mean
that the mind somehow constructs the empirical objects it perceives in
space, or that they are illusory. It just means that we cannot <i>know
</i>one
way or the other what their "real" natures are, so we must simply accept
the object of appearance as the "real" thing. Kant maintains the "<i>empirical
reality</i> of space, as regards all possible outer experience; and yet
at the same time ... its <i>transcendental ideality</i>" [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a><a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#067">72</a>].</font>
<p><font face="Times New Roman,Times">"It is, therefore," says Kant, "solely
from the human standpoint that we can speak of space, of extended things,
etc." [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/4.html#067">71</a>]
Synthetic <i>a priori </i>truths are therefore contingent only from a logical
perspective. There is, after all, a sense in which the notion of necessity
is still there, as such truths are necessarily true <i>for humans</i>,
or perhaps somewhat more generally, for any cognizing creatures with the
same basic innate faculties as humans.</font>
<br>
<center>
<h3>
<b><font face="Times New Roman,Times"><font size=+1>IV. Geometry:
Analytic, Synthetic <i>A Priori</i>,<i> </i>or Synthetic <i>A Posteriori</i>?</font></font></b></h3></center>
<font face="Times New Roman,Times">So we have discovered several kinds
of "truth" that synthesized judgement can provide: (1) <i>a priori</i>
analytic, (2) <i>a posteriori</i> synthetic and (3) <i>a priori</i> synthetic.
Logical truths, like (P v ~P), and definitional truths, like "all bachelors
are unmarried", are analytic. Although they cannot be cognized without
synthesis, their truth is not dependent on either the empirical or intuitive
components of that synthesis, and can thus be said to be solely about concepts.
It is simply not possible to even imagine it being otherwise, since the
analytic truth of the proposition is already contained implicitly in the
very concepts it speaks of. Truths of physics are obviously <i>a posteriori</i>
synthetic, since they are not at all independent of the sensory component
of the judgement. These truths are purely inductive, built up from experience
after watching many different cases. Kant´s observation that space
and time are the sole general forms of human cognition is itself an <i>a
posteriori</i> synthetic truth, since one could imagine it being otherwise
(this is true even if you cannot quite imagine <i>how </i>it could be otherwise).</font>
<p><font face="Times New Roman,Times">But what about the truths of geometry?
The example we gave earlier of a geometric proof performed on a triangle
is a case in point. Does the proof that the sum of the angles of any triangle
is 180 ely on empirical data? It would seem not, since our constructed
imaginary (or pen and paper) triangle is operated on in such a way as to
ensure complete independence from any particular empirical content. So
geometric truths are probably <i>a priori</i>, but are they analytic or
synthetic? Well, Kant might suggest at this point, are they necessary truths
or are they contingent (it being possible to imagine otherwise)? At first
it may seem completely obvious that they are necessary. After all, how
can one possibly imagine a triangle whose angles, say, sum to more than
180 But Kant argues that in fact, it <i>is </i>possible to imagine that
a geometric truth could be otherwise. Geometric truth in general relies
on human intuition, and requires a synthetic addition of information from
our pure intuition of space, which is a three-dimensional Euclidean space.
Kant does not buy the idea that such intuition can be reduced out to make
the truth analytic. This kind of reduction of geometry to the analytic
was pioneered by Ren? Descartes, usually considered the father of analytic
geometry. Descartes attempted to reduce geometry to algebraic computations
on co-ordinate systems, constructs which in theory are completely detachable
from our intuitive ideas of space. But Kant believed this analytic reduction
to be incomplete. There remained, he believed, a synthetic component to
geometry completely tied into the way humans are innately built to perceive
the world, i.e., in terms of space.</font>
<blockquote><font face="Times New Roman,Times">That the straight line between
two points is the shortest, is a synthetic proposition. For my concept
of straight contains nothing of quantity, but only of quality. The concept
of the shortest is wholly an addition, and cannot be derived, through any
process of analysis, from the concept of the straight line. [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a><a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/3.html#052">53</a>]</font></blockquote>
<font face="Times New Roman,Times">But if this is true, it should be possible
to imagine, conceptually, that geometric truths might be otherwise, since
they are synthetic and hence contingent. But can one imagine that a straight
line between two points might <i>not </i>be the shortest? While this does
indeed fly in the face of our most instinctive intuitions, it <i>is</i>
indeed possible. If we drop the (synthetic) assumption that space is Euclidean,
for instance, truths like this are up for grabs. Likewise for "the sum
of the angles of any triangle is 180 In non-Euclidean geometries, the sum
is often greater than 180 In spite of what is sometimes claimed, Kant´s
philosophy was not hostile to the development of what we now call non-Euclidean
geometries. In fact, they are virtually a consequence of his philosophy
of mathematics, since as analytic generalizations to synthetic geometry,
they show Euclidean geometry to be synthetic, providing that "other way
it could be" that we require of synthetic judgement. </font>It is interesting
to note here that, since analytic judgements require synthesis, just like
synthetic ones, it is purely an inductively drawn conclusion that analytic
judgements actually do manage to detach themselves from innate intuition,
thus becoming completely generalized, formal and conceptual. One can only
assume that Kant would have to concede that it is always possible, however
unlikely, that someone could yet demonstrate some "other way things could
be" for analytic truths as well, in which case they would turn out to be
actually synthetic, after all. It will be granted, however, that this seems
exceedingly unlikely.
<p><font face="Times New Roman,Times">Another way to analytically generalize
geometry is to extend past three dimensions to four or beyond. As already
mentioned, this was not a foreign idea to Kant, and some of his examples
of the synthetic nature of geometry can be interpreted in terms of four-dimensions.
In the <i>Prolegomena</i>, Kant gives an everyday example of a geometric
"necessary" truth for humans: that a left and right hand are incongruent
[<a href="http://www.allanrandall.ca/Geometry.html#prolegomena">Prolegomena</a>, <a href="http://www.utm.edu/research/iep/text/kant/prolegom/pro-1st.htm">sect.
13</a>]. The notion of "hand" here need not be understood as the empirical
object "hand"; we can assume that our pure intuition filter has adequately
abstracted our hand-experience into something detached from its empirical
component, so we are merely dealing with a three-dimensional geometric
figure shaped like a hand. By "incongruent", the geometer simply means
that no matter how we move one figure around in relation to the other,
we cannot get the two figures to coincide, to match up perfectly.</font>
<p><font face="Times New Roman,Times">Try it with your own hands and see.
If you place them palms together so that thumb touches thumb, etc., they
largely match up, except that they are facing in opposite directions. But
turn one around so it faces the other way, and the thumbs no longer line
up. The incongruence is due to the fact that the two hands are mirror-images
of each other. However, since they <i>are</i> mirror images, they are each,
considered by themselves, identical objects! There is nothing about the
left hand, considered solely in terms of its internal parts and their relations
to each other, that distinguishes it from the right hand. Only when viewed
as related to each other in space can the two hands be distinguished as
incongruent.</font>
<p><font face="Times New Roman,Times">Figure 3 illustrates the case in
two dimensions, and I will speak in those terms to make visualization easier.
As shown in the diagram, a synthetic cognition includes the human standpoint,
locating the hand in "space", in order to see any incongruency, which is
not there from the purely logical, analytic point of view. So far, however,
our case is not yet that convincing. A sceptic can just come along and
point out that Cartesian analytic geometry can still preserve the concept
of congruency. "Whatever is provided <i>a priori </i>by your precious intuition
filter," the sceptic sneers at us, "can just be brought into the analytic
system as an axiom, and we can still derive the incongruency." This is,
of course, true, as testified to by the great success of analytic geometry.
But Kant might well reply: "Ahh, yes, you <i>can </i>incorporate the intuitive
preconditions as analytic axioms if you like, perhaps, but as axioms they
become completely arbitrary and can just as well be replaced by new and
different axioms, yielding a completely different 'geometry' (perhaps non-Euclidean
or multi-dimensional, both perfectly plausible conceptually). But geometry
in its pure sense truly <i>does</i> give priority to the human perspective
of three-dimensional Euclidean space. This is not just an arbitrary choice
of perspective or axioms (although it is <i>logically</i> arbitrary), but
one based on the brute facts concerning who we are and how we are built
to cognize."</font>
<center>
<p><img SRC="Kant9.gif" height=310 width=597>
<br><b><font face="Times New Roman,Times">Fig. 3: Incongruent counterparts
demonstrate the syntheticity of geometry.</font></b></center>
<p><font face="Times New Roman,Times">Imagine one of the 2-D mirror image
hands in figure 3 being lifted up off the page and flipped over, then placed
back down next to its counterpart. Now both hands are congruent! What no
number of rotations and translations in 2-D space could accomplish was
accomplished quite simply by extending the space into the third dimension
and performing a simple rotation. Of course, the reason we could thus make
the hands congruent without leaving 3-space was because we simplified things
to two dimensions to make life simpler. But if we go back to the original
example of three-dimensional mirror-image hands, the same principle applies.
The hands can be made congruent by taking one, extending out into the fourth
dimension, flipping it over, and putting it back. But now, we are being
asked to go beyond the limits of human spatial intuition, which is restricted
to three dimensions, and we find the situation impossible to visualize.
Nonetheless, we understand the concept in terms of a generalization of
the 2-D/3-D case to the 3-D/4-D case, and we can rationally understand
that the 3-D/4-D case is conceptually possible. Yet we somehow cannot think
about it without imagining it in the simplified form of the 2-D/3-D case.
This simply proves the point that the incongruence, while not logically
necessary, is necessarily true <i>from the point of view of humans</i>.</font>
<p><font face="Times New Roman,Times">Also shown in figure 3 is the situation
after congruence has been achieved, and the two hands are now both left
hands. Even now, however, it is possible to see that the imagined situation
is synthetic, since from a purely analytic perspective such as that Leibniz
would have adopted, the two hands are indiscernible and hence, there is
logically only one object. The notion of having two identical hand figures
presupposes space, without which, there is only one hand, which cannot
even be said to be located anywhere. As a logical concept, it is not in
space at all. Place it in space, and it automatically becomes synthetic,
<i>unless
</i>the introduction of space is done in a completely <i>general
</i>analytic
fashion so that three-dimensional Euclidean space is given no preferred
status or privilege. Even then, Kant would probably point out that if space
is generalized into completely analytic terms, it becomes equally appropriate
to talk about it in those terms (purely algebraic or computational). The
notion of "space" has been generalized to the point that it is no longer
especially about space anymore, and might just as well be characterized
in other terms. We are no longer doing "pure" geometry.</font>
<center>
<p><img SRC="Kant10.gif" height=384 width=353>
<br><b><font face="Times New Roman,Times">Figure 4: Using non-Euclidean
space to demonstrate the syntheticity of geometry.</font></b></center>
<p><font face="Times New Roman,Times">Figure 4 shows the situation that
Kant actually presents as his main example (the hand example being given
secondarily as a more everyday case easier to visualize) [<a href="http://www.allanrandall.ca/Geometry.html#prolegomena">Prolegomena</a>,
<a href="http://www.utm.edu/research/iep/text/kant/prolegom/pro-1st.htm">sect.
13</a>]. We are asked to visualize two mirror-image spherical triangles
(triangles drawn from the arcs of great circles on the surface of a sphere)
on opposite hemispheres of a sphere. If drawn on flat two-dimensional paper
like the hands in figure 3, we could use the three dimensional pull-through
trick to achieve congruence. But when drawn on the surface of a sphere,
the trick no longer works, since the curvature of the sphere gives the
triangles curvature into the third dimension, so if we attempt to pull
the back triangle through the third dimension (by pulling it through the
center of the sphere as in the action labelled 'A'), we end up with incongruent
triangles. We can´t flip one over, because then the spherical curvature
will fail to match up.</font>
<p><font face="Times New Roman,Times">It is commonly--and mistakenly--claimed
that Kant rejected the whole idea of non-Euclidean geometries (see <a href="http://www.allanrandall.ca/Geometry.html#palmquist">[Palmquist
1990]</a> for a critique of these views), yet here we see that Kant is
using the closest thing he had in his time to a non-Euclidean geometry
to prove his point. In fact, from the point of view of an imaginary creature
living on the surface of the sphere, these triangles are drawn in a non-Euclidean
2-D space. The only way the 2-D creatures can try to match the figures
up is by doing things like action B, pulling the triangle along the surface
of the sphere around to the other side, only to find that they do not match
up. While pulling one triangle through the third dimension (action A) also
fails to achieve congruence, that was only because we assumed a Euclidean
3-D space. But the triangles are already, in a sense, drawn in a non-Euclidean
2-D space, so why not pull the back triangle through an extension of <i>that
</i>space?
Note that in the 2-D spherical geometry, these triangles are actually flat!
So action A, if interpreted as being through a 3-D extension of the sphere-space,
will match the two figures perfectly. This is because, as a 3-D extension
of the 2-D space where the triangles are flat, we can ignore the curvature
due to the sphere.</font>
<p><font face="Times New Roman,Times">So again we see that a geometric
truth of Euclidean 3-space disappeared when we generalized to non-Euclidean
geometry. But which is the <i>a priori</i> "truth", that the triangles
were congruent, or that they were incongruent? Obviously, neither if we
are talking purely logically, since it depends on our choice of axioms,
which determines what flavour of (analytic and impure) geometry we are
dealing with. But, Kant points out, there is still something that is "true"
about the 3-D Euclidean case that has some kind of priority over the other
cases. Synthetically, then, it is necessarily true that the figures are
incongruent, since the choice of viewpoint in figure 4 is in point of fact
no choice at all! The other logical choices, such as 2-D and 3-D Spherical,
are non-human perspectives. Perhaps there are creatures with built-in spherical
intuition filters, but we are simply not such creatures. We have no choice
but to cognize via 3-D Euclidean filters.</font>
<center>
<h3>
<font face="Times New Roman,Times">V. Some Objections and Replies</font></h3></center>
<font face="Times New Roman,Times">While it is easy to be convinced that
3-D Euclidean space has a privileged status as the uniquely human perspective,
it is not at all clear that this should be what geometry is about. Today,
in the twentieth century, mathematicians have accepted that the move to
make geometry purely analytic was unwise, and that synthetic methods have
a place. Hence, geometry is divided into analytic and synthetic or "pure",
which is sometimes called the "science of space". However, modern synthetic
geometers see their science in a somewhat difference light than Kant, even
though they recognize it as synthetic. The difference is that modern synthetic
geometry is often, although not always, considered to be either simply
a visual tool for doing analytic geometry, or literally <i>a posteriori</i>.
In the latter case, our built-in 3-D Euclidean faculties are seen as providing
a "synthetic method" of construction much like Kant´s except that
it is subject to empirical evidence. Often cited in favour of this view
is the fact that space has turned out, according to Einstein´s theory
of relativity, to be Einsteinian, not Euclidean, and having at least 4
dimensions (and in quantum mechanics, there are an infinity of dimensions).</font>
<p><font face="Times New Roman,Times">In both cases (<i>a posteriori</i>
and analytic), synthetic methods are used to build an intuition for nonintuitive
geometries in terms of the more intuitive 3-D Euclidean space. For multi-dimensional
space, a projection is usually made onto either 2 or 3 dimensions, and
then scaled up to higher dimensions, while attempting to retain the intuitive
understanding and reasoning tools that were developed in the lower dimensions.
"Synthetic geometry requires that one build up one's knowledge dimension
by dimension, one at a time, while analysis is developed for an arbitrary
dimension, so that once something is known for any dimension, one is ready
to consider n dimensional objects."</font> <a href="http://www.allanrandall.ca/Geometry.html#nogelo">[Nogelo 1996]</a>
<p><font face="Times New Roman,Times">Figure 5 shows the analytic/synthetic
distinction in computational versus visual terms. On the left is a computer
program (drawn as a dataflow diagram) that accepts a description of the
geometry it is to use (in this case Euclidean), and a series of commands
that tell it how to manipulate the objects in the "space". At each step,
the old co-ordinates are fed back into the system and it computes new co-ordinates.
When understood computationally (i.e., algebraically or analytically),
there is nothing special about 3-D Euclidean geometry. Extending to higher
dimensions or changing axioms is no big deal, as opposed to the synthetic
case, where it is a big conceptual leap. The synthetic style of geometry
assumes and is constrained by the human standpoint, even though analytic
methods will also invariably be used. The analytic method, however, tries
its best to rid itself of the human standpoint, and view geometry as algebra
or computation. As such, Kant in a way is right, that it is no longer geometry
<i>per
se</i> at all. Almost any mathematical system could be considered a kind
of geometry in this sense. To retain any distinction between geometry and
everything else, we need to retain for geometry a strong synthetic component.</font>
<center>
<p><img SRC="Kant11.gif" height=344 width=574>
<br><b><font face="Times New Roman,Times">Figure 6: Synthetic and Analytic
Geometry.</font></b></center>
<p><font face="Times New Roman,Times">So what would Kant say to the modern
claim made by some that synthetic geometry, while purer than the <i>a priori</i>
analytic variety, becomes <i>a posteriori</i> in the process? Geometry
is literally an empirical science, a branch of physics, not just a way
of visualizing analytic geometry, since it is <i>a posteriori</i>, and
thus constrained by the results of physics, which tells us that space is
Einsteinian. Does this not contradict Kant´s <i>a priori</i> claim
for the nature of space?</font>
<p><font face="Times New Roman,Times">"But", Kant might response were he
around, "look at figure 7. Here, we have modified our analytic geometry
in terms of Einsteinian space, a relatively minor adjustment consisting
of changing a few equations. There is now no trace left in our analysis
of the old-fashioned 'Euclidean' geometry. We did not have to program our
computer so as to appear to perform Euclidean functions for us, while really
computing Einsteinian ones on the sly. Yet look at the modifications to
the synthetic formulation. We have had to introduce all kinds of "tricks"
to maintain an understanding consistent with our intuition filters. We
now need to visualize objects shrinking and expanding, and clocks slowing
down, as if caught in some kind of molasses. In other words, the only way
to visualize non-Euclidean geometries is either to project them straightforwardly
onto a Euclidean space, or to change the empirical laws of physics governing
the behaviour of the objects within the space. The analytic non-Euclidean
"geometry" is really a nongeometric analytic tool to aid in the empirical
study of objects in Euclidean space. This interpretation is perfectly consistent
with relativity theory."</font>
<center>
<p><img SRC="Kant12.gif" height=334 width=657>
<br><b><font face="Times New Roman,Times">Fig. 7: Einsteinian space: an
analytic tool for understanding Euclidean space.</font></b></center>
<p><font face="Times New Roman,Times">There is something to be said for
this view. We can never look around and "just see" things from an Einsteinian
perspective, since that is a non-human perspective.The "Einsteinian" effects
can most intuitively be understood as properties of empirical objects in
3-D Euclidean space, undergoing such things as ether-stress that deforms
them at high speeds, and causes "drag" on the objects, making them move
sluggishly. The great physicist John Bell has taken this Kantian perspective
quite seriously, and seems to believe it almost a truism that any empirical
discovery of non-Euclidean features of space can always<i> </i>be understood
as new physical laws about the behaviour of empirical objects within Euclidean
space. <a href="http://www.allanrandall.ca/Geometry.html#bell">[Bell 1976]</a> However, even he does not
actually privilege the Euclidean perspective, but suggests that we use
<i>both
</i>perspectives to achieve a closer match to the truth.</font>
<p><font face="Times New Roman,Times">"I am still not convinced," our imaginary
sceptic replies, "of the transcendental ideality of space (and hence the
<i>a
priori</i> syntheticity of geometry), since I certainly feel that it is
unintuitive, to say the least, to have to resort to fortuitously arranged
ether stresses to account for something that seems much more easily accounted
for by concluding that our intuitive faculties are simply mismatched with
reality. Our intuition for space is of something outside of us, regardless
of whether that something is actually anything in itself. Either way, our
innate faculties are directed outward by this notion of Euclidean space.
While Kant is absolutely right that we cannot help but cognize things in
terms of this innate form, that does not stop us from <i>rationally</i>
deciding that this innate idea of something Euclidean outside of us is
<i>wrong.
</i>How, for instance, would Kant react if empirical evidence showed that
hands really <i>do </i>under certain circumstances get mirror-reversed
because of the non-Euclidean nature of space? In fact, there have actually
been Einsteinian models of the universe in which going on a very long trip
could potentially mirror-reverse your body!"</font>
<p><font face="Times New Roman,Times">In order to answer this and continue
to maintain that the incongruency of the mirror-counterparts is indeed
an <i>a priori</i> synthetic truth of pure geometry, which is crucial to
his argument, Kant needs to respond with something like: "well, since the
intuitive geometry of our innate faculties <i>is </i>Euclidean, the incongruency
is necessarily <i>still </i>true, from the human perspective. If Einsteinian
analytic computational models of empirical objects help us understand why,
during long trips through space, stresses tend to deform objects to the
extent that they become mirror-imaged, then fine. But here´s the
rub: we can even choose to flip our perspective and view these non-Euclidean
spatial features <i>not </i>as archaic ether stresses, but truly in terms
of a non-Euclidean (and hence analytic-geometric) apparatus, the truth
of which is <i>empirical</i>. We are free to take this perspective, and
still maintain the Euclidean synthetic geometry as our sole notion of the
"space" that all this is happening in, <i>for us</i>. The truths of the
analytic geometry we have employed will be empirical, and so not really
'spatial' in the pure sense. The non-Euclidean 'space' is thus more properly
viewed as a valid empirically determined <i>object </i>acting <i>within</i>
space."</font>
<center>
<h3>
VI. Arithmetic and Logic</h3></center>
<font face="Times New Roman,Times">It is often suggested that while geometry
is indeed synthetic, arithmetic (more often associated with mathematics
as a whole) is analytic. This was Frege's view for most of his life. My
main goal in this paper has been to critique Kant's view of geometry. I
believe the issue of the syntheticity of arithmetic is a more subtle issue,
requiring a full understanding of recursion theory, and in particular the
work of Kurt Gödel <a href="http://www.allanrandall.ca/Geometry.html#godel">[Gödel 1931]</a>, and thus
beyond the scope of the current paper. I plan to deal with these issues
fully in a forthcoming paper, but would like to briefly outline my views
here, in particular how they relate to what has already been said above
about geometry.</font>
<p><font face="Times New Roman,Times">Kant believed that arithmetic was
synthetic because "numbers" are constructed in the intuition, much as circles
and lines are. They do not analytically follow from the process of adding
or multiplying two other numbers. They have to do with our intuition about
objects in space and time.</font>
<blockquote><font face="Times New Roman,Times">We might, indeed, at first
suppose that the proposition 7 + 5 = 12 is a merely analytic proposition,
and follows by the principle of contradiction from the concept of a sum
of 7 and 5. But if we look more closely we find that the concept of the
sum of 7 and 5 contains nothing save the union of the two numbers into
one, and in this no thought is being taken as to what that single number
may be which combines both. The concept of 12 is by no means already thought
in merely thinking this union of 7 and 5; and I may analyze my concept
of such a possible sum as long as I please, still I shall never find the
12 in it. [<a href="http://www.allanrandall.ca/Geometry.html#CPR">CPR</a> <a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/3.html#052">52-53</a>]</font></blockquote>
<font face="Times New Roman,Times">The number "12", then, is an intuition
of an object in space and time. The analytic component to "5+7=12" should
therefore (if Kant is right) no longer contain "12" as an object distinct
from the process of adding 5 and 7. This means it should be possible to
come up with a "nonstandard" arithmetic, by refuting or changing some arbitrary
axioms in the system, just as refuting the Fifth Postulate of Euclid yields
a nonstandard geometry. Modern axiomitizations of arithmetic in terms of
set theory show that this is essentially correct, although the meaning
of the resulting arithmetics is harder to grasp than that of non-Euclidean
geometries (one does not notice the differences until one has to deal with
infinities). This of course does <i>not </i>mean that there is no analytic
truth at all behind "5+7=12" (or even that there is no analytic truth behind
the truths of transfinite arithmetic). There <i>is </i>such a truth, but
that truth is only <i>part of </i>what we mean when we say "5+7=12", or
whatever. Without the synthetic component, there might well be an analytic
component left, but if Kant is correct, this will no longer be an example
of arithmetic <i>per se.</i></font>
<p><font face="Times New Roman,Times">Modern recursion theory provides
something Kant lacked: a rigorous, precise foundation for both logic and
mathematics. As such, it is possible to develop his ideas within this context.
The important thing about recursion theory is that there is no <i>one </i>particular
conceptualization of it that is definitively "recursive". For instance,
Turing machines, the </font><font face="Symbol">l</font><font face="Times New Roman,Times">-calculus,
English and unrestricted predicate logic (or first-order predicate logic
plus set theory) are just a few examples of different formulations of recursion
theory. Recursion theory is not the study of any one such formulation,
but the study of what is common between them all. One can provide precise
translations between all these languages, which as a class are called "recursive"
or "Turing equivalent". Thus, anything that drops out when a structure
described in one language is translated to another can be considered a
synthetic component of that language. The entire class of languages can
thus be taken as a formalization of Kant's analytic. Any conceivable structure
can be described in recursion theory, even though we must intuit it in
terms of Euclidean space and time. We can filter out these synthetic components,
and be left with something closer to the underlying analytic truth (see
[<a href="http://www.allanrandall.ca/Geometry.html#randall">Randall 1997</a>] for more details). This filtering
is achieved by translating from one Turin-equivalent language to another.
Such translations are examples of the application of Kantian pure intution
filters, to render an initially empirical concept as a formal one.</font>
<p><font face="Times New Roman,Times">Arithmetic (at least as conceived
by the logicists who have attempted to reduce it to logic) is one example
of a Turing-equivalent language. As such, one could argue that it is analytic,
not synthetic. However, recall that geometry <i>as geometry and independent
of algebra </i>is not analytic, even though it can be viewed as one particular
formulation of algebra. Likewise, with arithmetic. It can be viewed as
a formulation of recursion theory, but as such loses what makes it arithmetic
as opposed to geometry or unrestricted predicate logic. One of the things
that disappears when arithmetic truths are translated to various other
recursive languages is precisely the idea of a number as an object independent
of the process of adding or multiplying that produces the number. "5+7=12"
is rendered in recursive theory as a <i>process</i> of adding the two numbers
together. There is no definitive <i>result </i>to the addition, unless
we consider "5+7" as a function that then passes its result to <i>another</i>
recursive function--the problem here is that splitting a recursive structure
like this into two arbitrarily defined substructures (i.e., "functions")
is itself completely synthetic (obviously).</font>
<p><font face="Times New Roman,Times">So arithmetic is in the same boat
as geometry. But what about logic, surely <i>that </i>is equivalent to
Kant's analytic? The problem here is determining what is meant by "logic".
Many philosophers take it to mean "first-order predicate logic". The problem
with this is that first-order predicate logic is <i>not</i> Turing equivalent.
It is thus, on its own, entirely synthetic! The "first-order" restriction
must be lifted (yielding unrestricted predicate logic) to get a truly recursive
language. This many philosophers prefer to call "set theory", rather than
logic. Under this view, logic is more strictly synthetic than arithmetic.
However, the more traditional definition of logic from which such philosophers
have drawn is simply that it is "correct reasoning" or somesuch. Thus,
in my opinion, those who associate logic with "first-order" or "predicate"
logic only are making a mistake. The former is not powerful enough to reflect
correct human reasoning, and the latter is specifically linguistic in nature.
Human reasoning need not be thought of as linguistic, so logic need not
necessarily be "predicative" in nature, and neither does it need to rely
on truth values ("TRUE" and "FALSE"). A pure calculus, Turing equivalent
but without built-in predicates or truth values, reflects all that is essential
to correct logical reasoning. Yet, for some, truth values and/or predicates
are needed for the calculus to really be a logic. (This includes many who
might at times claim otherwise and agree that truth values are synthetic
artifacts--old habits can be hard to break even when you know better.) The
notions of truth values and predication are so central to the way logic
is taught that it is hard to separate what is necessary to formal logic
from such dispensible synthetic components. In my opinion, logic, if it
means correct reasoning, should be taken to mean unrestricted predicate
logic, or better yet something without predicates like the </font><font face="Symbol">l</font><font face="Times New Roman,Times">-calculus.
However, this makes it equivalent to mathematics, since the </font><font face="Symbol">l</font><font face="Times New Roman,Times">-calculus
can equally well be taken as a formulation of mathematics or unrestricted
predicate logic. So predicate logic has both synthetic and analytic components,
in much the same way geometry does. Logical truths can be interpreted as
analytic only by giving up the distinction between logic and other things
like geometry and arithmetic. It is thus a matter of taste whether we say
that logic is a subfield of mathematics, or the other way around. Personally,
I prefer to call mathematics the more general field, partly because so
many have restricted the scope of "logic" to something linguistic and weaker
than recursion theory, and partly simply because the range and scope of
systems that in practise we tend to call mathematical are much broader
than those we tend to classify as logical.</font>
<center>
<h3>
<b><font face="Times New Roman,Times"><font size=+1>VII. Conclusion</font></font></b></h3></center>
<font face="Times New Roman,Times">This is the best case that I can make
for Kant at this time. I am not myself fully convinced, however, that the
term "space" is really appropriate for the intuition of outer sense. Too
much fancy footwork has to be issued in order to allow former <i>a priori</i>
"truths" to still hold for "space" even though they appear to be violated
in experience. Most of us would want, I think, to use the word "space"
in its common-sense fashion and view our intuition as only the <i>a priori</i>
form of appearances <i>in</i> space, while retaining for the term "space"
an empirical component. However, this is a terminology problem, not a fundamental
problem in Kant´s system, since we can simply make "space" an empirical
term and come up with another term for our intuition of outer sense, perhaps
"intuition-space", while retaining everything else Kantian.</font>
<p><font face="Times New Roman,Times">Such a change in terminology would
not affect the essentials of Kant´s system, since "intuition-space"
would still be a transcendentally ideal <i>a priori</i> form of sensory
intuition. Intuition-space would still be purely a matter of human perspective,
and space would be an empirical object that is subject to processing by
the intuition-space filter, just like any other empirical object. Neither
would be knowable as a thing in itself. Synthetic geometric truths would
still be <i>a priori</i>, since they would be based on intuition-space,
not empirical space. What is sometimes called <i>a posteriori</i> synthetic
geometry would simply be synthetic geometry used in an impure empirical
application. The only real difference is that we will have weakened the
strength of the connection between "intuition space", the form of outer
sense, and the empirical object known as "space", which no longer needs
to be Euclidean. In Kant´s terminology, on the other hand, what I
am here calling "space" would have to be considered an empirical object
acting <i>in </i>space, and that seems to yield a rather less intuitive
use of the word space.</font>
<p><font face="Times New Roman,Times">Keep in mind that Kant lived before
the discovery of non-Euclidean geometries, and well before the time of
special relativity. So although he did speculate about multi-dimensional
spaces, and used a primitive notion of non-Euclidean space, he probably
never dreamt that there would be any reason to separate the scientific
use of the word "space" from the intuitive use to the extent that the theory
of relativity seems to demand. As such, he can be forgiven for not slanting
his terminology towards such usage. However, I think the essentials of
his system survive the transition to twentieth century science quite well.</font>
<p><font face="Times New Roman,Times">So is synthetic geometry in fact
<i>a
priori</i>? I have to agree with Kant that it is. Those who wish to make
it an empirical science are giving up the very thing that makes it a branch
of mathematics in the first place. However, I have to allow that there
is also a such thing as "analytic geometry". Kant perhaps overestimated
the synthetic component of geometry. So much of mathematics can be understood
in terms that seem so closely tied to "pure" geometry that to refuse to
call it geometry would seem extreme. On the other hand, I will grant to
Kant that an analytic geometry cannot be clearly delineated from the rest
of mathematics, and so calling it "geometric" is an arbitrary matter of
interpretation. Thus, it seems reasonable to follow Kant in reserving the
term "pure" for synthetic geometry (which is in keeping with current usage).</font>
<p><font face="Times New Roman,Times">It is important that we learn to
distinguish between the analytic and synthetic components of mathematics
(geometry, arithmetic <i>and </i>logic). Most of mathematics is still being
carried out today from largely intuitive grounds, in spite of the fact
that those who carry it out believe they are being primarily analytic.
It is rare, except in the philosophy and foundations of mathematics, to
make much attempt to isolate what parts of a mathematical construction
depend on the human standpoint. In practise, geometry has both strong synthetic
<i>and</i>
analytic components, and today´s mathematicians are themselves in
a state of great confusion as to which enterprise is really their central
concern. The state of confusion in the study of other areas of mathematics
like arithmetic and logic is even greater. I favour following Kant´s
lead in his constructivist view of mathematics, while recognizing that
this leads to <i>both </i>an analytic and a synthetic "geometry", although
the purely analytic geometry that results is no longer about something
we would all necessarily agree to call by our everyday word, "space", while
the synthetic geometry is undeniably so. It is imperative that we learn
to separate these two geometries, as well as sort out the different kinds
of arithmetic and logic, as great confusion and paradox can result when
we think we are involved in one science while we go about our business
doing the other.</font>
<br>
<center>
<h3>
References</h3></center>
<blockquote><a NAME="CPR"></a>Immanuel Kant (1781). <i><a href="http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/kant.html">Critique
of Pure Reason</a></i> (2nd. Ed.), Norman Kemp Smith (Trans.). http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/kant.html,
1781, 1787, 1985, 1996. [Cited as "CPR pp".]
<p><a NAME="prolegomena"></a>Immanuel Kant (1783). <i><a href="http://www.utm.edu/research/iep/text/kant/prolegom/prolegom.htm">Prolegomena
to Any Future Metaphysics</a></i>, Paul Carus (Trans.). http://www.utm.edu/research/iep/text/kant/prolegom/prolegom.htm,
1783, 1902, 1997. [Cited as "Prolegomena".]
<p><a NAME="forces"></a>Immanuel Kant (1747). <i><a href="http://socrates.berkeley.edu/~phlos-ad/forward.html">Thoughts
on the True Estimation of Living Forces</a></i> (selections), Andrew N.
Carpenter (Trans.). http://socrates.berkeley.edu/~phlos-ad/forward.html,
1747, 1998.
<p><a NAME="palmquist"></a>Stephen Palmquist (1990). <a href="http://www.hkbu.edu.hk/~ppp/srp/arts/KEGP.html">"Kant
on Euclid: Geometry in Perspective"</a>, <i>Philosophia Mathematica II</i>
5:1/2, pp.88-113, http://www.hkbu.edu.hk/~ppp/srp/arts/KEGP.html,
1990.
<p><a NAME="nogelo"></a>Alexis Nogelo, et al. (1996). <i><a href="http://www.stg.brown.edu/projects/classes/ma8/%20papers/anogelo/hist4dim.html">Geometry
Updated.</a></i> http://www.stg.brown.edu/projects/classes/ma8/papers/anogelo/hist4dim.html,
1996.
<p><a NAME="bell"></a>J.S. Bell (1976). "How to teach special relativity,"
In: <i><a href="http://www.amazon.com/exec/obidos/ISBN=0521368693/allanfrandallA/">Speakable
and Unspeakable in Quantum Mechanics</a></i>, pp. 67-80. Cambridge University
Press, Cambridge, 1976, 1987.
<p><a NAME="godel"></a>Kurt Gödel (1931). "On Formally Undecidable
Propositions of Principia Mathematica and Related Systems I," In: <i><a href="http://www.amazon.com/exec/obidos/ISBN=0415045754/allanfrandallA/">Gödel's
Theorem in focus</a></i>, S.G. Shanker (Ed.), pp. 17-47. Routledge, London,
1931, 1967, 1988.
<p><a NAME="randall"></a><font face="Times New Roman,Times">Allan F. Randall
(1997). <i><a href="http://www.allanrandall.ca/Phenomenology/">Quantum
Phenomenology</a></i>. http://www.allanrandall.ca/Phenomenology/, Dept.
of Philosophy, York University, Toronto, 1997.</font>
<p>
<hr SIZE=1 WIDTH="80%">
<blockquote>
<center><i>Acknowledgement: without the help and wisdom of Robert Hanna,
this paper could not have been written.</i></center>
</blockquote>
<center><i>Go to <a href="http://www.allanrandall.ca/Kant/Idealism/">In
Defence of Transcendental Idealism</a>. Go Back to <a href="http://www.allanrandall.ca/">Allan
Randall's Home Page</a>.</i></center>
</blockquote>
</body>
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