Copyright © 1998, Allan Randall

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 synthetica prioriif 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 syntheticasdisciplines 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 mathematicsorlogic--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.

"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."Encyclopedic Dictionary of Mathematics, Vol. I., "Geometry", p. 685, The MIT Press, 1987.

Kant´s unique contribution
was his claim that one could thus have *a priori* truths that, being
synthetic, were not merely logical or definitional. One can have *a priori*
truths that are contingent, and so could be imagined to be otherwise. Mathematical
truths, according to Kant, fall into this category.

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 *a
priori* 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 *a priori *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 *from* an empirical given *to* 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.

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
*a
priori* and synthetic.

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 *human *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 *a priori* synthetic, but *a posteriori*
synthetic. On the other hand, if space were merely an arbitrary mental
construct, geometry would be *a priori* analytic.

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 *a priori *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 *as *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".

But even aside from those analytic
propositions used in the service of geometry, there is another sense in
which *all *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 [CPR 259]).
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.

**Fig. 1. Judgement: synthesis
performed in the mind via intuition.**

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).

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* a priori* 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 *particular *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." [CPR 259]
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 [CPR 65],
which is why Kant says that "all our knowledge begins with experience."
[CPR 41]

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 home
page). 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_{1},
o_{2}, ...}. 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 *process*
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."
[CPR 65]

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" [CPR66].

**Fig. 2: Pure and impure intuition
as information processing filters.**

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 *only* to the formal
concept F() and not to the data D.

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 *all *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, *not *the particular one we
have constructed in imagination. So our reasoning about the triangle is
truly *a priori* 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 *a priori* if it does
not depend on experience.

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 *a priority*. 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 *a priori*. So, although
we noted earlier that "all knowledge begins with experience" (seeing that
nothing can be cognized except *as* an experience), we see now that
"it does not follow that it all arises out of experience," [CPR41]
since our intuition filters allow us to filter out the empirical component.

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 *a priori* and *a posteriori* 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:

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 supplySo far we have seen two kinds of truth that can be realized in our judgements:a priori. [CPR 67]

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" [CPR 67]. Time is the pure form of inner intuition, since it is presupposed even in purely internal thought that is not directed outward [CPR74-75]. It is space that we will concentrate on here, since that is what is most relevant for geometry.

"But," we might venture to further
protest, "what if space is, as you say, intuitive and not empirical, but
nonetheless still not a *necessary *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" [CPR 68].
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 *does
*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.

"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"
[CPR 69].
But concepts are general forms to be instantiated by synthesis with particular
empirical data, *not *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.

Space, of course, is *also *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 *intuitive *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.

Kant is *not* 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
[Kant 1747]. While he did not pursue this idea in
the *Critique*, 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).

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 [CPR69-70]
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 *merely* a concept"
[CPR 70].

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" [CPR74].
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 *know
*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 "*empirical
reality* of space, as regards all possible outer experience; and yet
at the same time ... its *transcendental ideality*" [CPR72].

"It is, therefore," says Kant, "solely
from the human standpoint that we can speak of space, of extended things,
etc." [CPR 71]
Synthetic *a priori *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 *for humans*,
or perhaps somewhat more generally, for any cognizing creatures with the
same basic innate faculties as humans.

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 *a priori*, 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 *is *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.

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. [CPR53]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

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 *Prolegomena*, Kant gives an everyday example of a geometric
"necessary" truth for humans: that a left and right hand are incongruent
[Prolegomena, sect.
13]. 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.

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 *are* 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.

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 *a priori *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 *can *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 *does* 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 *logically* arbitrary), but
one based on the brute facts concerning who we are and how we are built
to cognize."

**Fig. 3: Incongruent counterparts
demonstrate the syntheticity of geometry.**

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 *from the point of view of humans*.

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,
*unless
*the introduction of space is done in a completely *general
*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.

**Figure 4: Using non-Euclidean
space to demonstrate the syntheticity of geometry.**

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) [Prolegomena, sect. 13]. 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.

It is commonly--and mistakenly--claimed
that Kant rejected the whole idea of non-Euclidean geometries (see [Palmquist
1990] 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 *that
*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.

So again we see that a geometric
truth of Euclidean 3-space disappeared when we generalized to non-Euclidean
geometry. But which is the *a priori* "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.

In both cases (*a posteriori*
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." [Nogelo 1996]

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
*per
se* 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.

**Figure 6: Synthetic and Analytic
Geometry.**

So what would Kant say to the modern
claim made by some that synthetic geometry, while purer than the *a priori*
analytic variety, becomes *a posteriori* in the process? Geometry
is literally an empirical science, a branch of physics, not just a way
of visualizing analytic geometry, since it is *a posteriori*, and
thus constrained by the results of physics, which tells us that space is
Einsteinian. Does this not contradict Kant´s *a priori* claim
for the nature of space?

"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."

**Fig. 7: Einsteinian space: an
analytic tool for understanding Euclidean space.**

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* *be understood
as new physical laws about the behaviour of empirical objects within Euclidean
space. [Bell 1976] However, even he does not
actually privilege the Euclidean perspective, but suggests that we use
*both
*perspectives to achieve a closer match to the truth.

"I am still not convinced," our imaginary
sceptic replies, "of the transcendental ideality of space (and hence the
*a
priori* 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 *rationally*
deciding that this innate idea of something Euclidean outside of us is
*wrong.
*How, for instance, would Kant react if empirical evidence showed that
hands really *do *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!"

In order to answer this and continue
to maintain that the incongruency of the mirror-counterparts is indeed
an *a priori* 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 *is *Euclidean, the incongruency
is necessarily *still *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 *not *as archaic ether stresses, but truly in terms
of a non-Euclidean (and hence analytic-geometric) apparatus, the truth
of which is *empirical*. 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, *for us*. 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 *object *acting *within*
space."

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.

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. [CPR 52-53]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

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 *one *particular
conceptualization of it that is definitively "recursive". For instance,
Turing machines, the l-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
[Randall 1997] 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.

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 *as geometry and independent
of algebra *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 *process* of adding the two numbers
together. There is no definitive *result *to the addition, unless
we consider "5+7" as a function that then passes its result to *another*
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).

So arithmetic is in the same boat
as geometry. But what about logic, surely *that *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 *not* 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 l-calculus.
However, this makes it equivalent to mathematics, since the l-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.

Such a change in terminology would
not affect the essentials of Kant´s system, since "intuition-space"
would still be a transcendentally ideal *a priori* 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 *a priori*, since they would be based on intuition-space,
not empirical space. What is sometimes called *a posteriori* 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 *in *space, and that seems to yield a rather less intuitive
use of the word space.

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.

So is synthetic geometry in fact
*a
priori*? 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).

It is important that we learn to
distinguish between the analytic and synthetic components of mathematics
(geometry, arithmetic *and *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
*and*
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 *both *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.

Immanuel Kant (1781).Critique of Pure Reason(2nd. Ed.), Norman Kemp Smith (Trans.). http://csmaclab-www.uchicago.edu/philosophyProject/sellars/kant/kant.html, 1781, 1787, 1985, 1996. [Cited as "CPR pp".]Immanuel Kant (1783).

Prolegomena to Any Future Metaphysics, Paul Carus (Trans.). http://www.utm.edu/research/iep/text/kant/prolegom/prolegom.htm, 1783, 1902, 1997. [Cited as "Prolegomena".]Immanuel Kant (1747).

Thoughts on the True Estimation of Living Forces(selections), Andrew N. Carpenter (Trans.). http://socrates.berkeley.edu/~phlos-ad/forward.html, 1747, 1998.Stephen Palmquist (1990). "Kant on Euclid: Geometry in Perspective",

Philosophia Mathematica II5:1/2, pp.88-113, http://www.hkbu.edu.hk/~ppp/srp/arts/KEGP.html, 1990.Alexis Nogelo, et al. (1996).

Geometry Updated.http://www.stg.brown.edu/projects/classes/ma8/papers/anogelo/hist4dim.html, 1996.J.S. Bell (1976). "How to teach special relativity," In:

Speakable and Unspeakable in Quantum Mechanics, pp. 67-80. Cambridge University Press, Cambridge, 1976, 1987.Kurt Gödel (1931). "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I," In:

Gödel's Theorem in focus, S.G. Shanker (Ed.), pp. 17-47. Routledge, London, 1931, 1967, 1988.Allan F. Randall (1997).

Quantum Phenomenology. http://www.allanrandall.ca/Phenomenology/, Dept. of Philosophy, York University, Toronto, 1997.

Acknowledgement: without the help and wisdom of Robert Hanna, this paper could not have been written.Go to In Defence of Transcendental Idealism. Go Back to Allan Randall's Home Page.