http://www.allanrandall.ca/Geometry.html
Copyright © 1998, Allan Randall
 
 

  A Critique of the Kantian View of Geometry

Allan F. Randall
Dept. of Philosophy, York University
Toronto, Ontario, Canada
research@allanrandall.ca, http://www.allanrandall.ca/

 

Abstract

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 a priori 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 as 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 or 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.
"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.

 

I. Introduction

In the Critique of Pure Reason [CPR] and the Prolegomena to Any Future Metaphysics [Prolegomena], Kant argues that the truths of geometry are synthetic a priori 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 a priori. 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 a posteriori, as it depends on sensory input. 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 a priori.

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".

II. The Nature of Synthesis

We will start by looking at just what it means for geometry to be synthetic. It is important that Kant does 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 as practised is not at all analytic, he is simply saying that as geometry, it is synthetic. No "fundamental proposition of pure geometry", he says, is analytic [CPR 53, emphasis mine]. Furthermore, "some few fundamental propositions, presupposed by the geometrician, are, indeed, really analytic" [CPR54]. 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).

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:{o1, o2, ...}. 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 supply a priori. [CPR 67]
So far we have seen two kinds of truth that can be realized in our judgements: a priori and a posteriori. When we perform the above reduction, we should be left with whatever it is that distinguishes those a priori 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 and 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).

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.

III. The Transcendental Ideality of Space

"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 a priori 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" [CPR 68]. 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.

"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.
 

 IV. Geometry: Analytic, Synthetic A Priori, or Synthetic A Posteriori?

So we have discovered several kinds of "truth" that synthesized judgement can provide: (1) a priori analytic, (2) a posteriori synthetic and (3) a priori 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 a posteriori 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 a posteriori synthetic truth, since one could imagine it being otherwise (this is true even if you cannot quite imagine how it could be otherwise).

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 not be the shortest? While this does indeed fly in the face of our most instinctive intuitions, it is 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. 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.

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.

V. Some Objections and Replies

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 a posteriori. 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).

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."

VI. Arithmetic and Logic

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 [Gödel 1931], 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.

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 not 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 is such a truth, but that truth is only part of 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 per se.

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.

VII. Conclusion

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 a priori "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 a priori form of appearances in 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.

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.
 

References

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 II 5:1/2, 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.