Kant, Geometry, and Physical Theory

[glengarry]

I want to dig a little bit into the idea of geometry and how it relates to physical theorization. It is my feeling that there is profound ambiguity as regards the "ontological status" of geometry, and that this ambiguity only serves to muddy the waters as far as the way in which mathematics applies to theoretical physics.

On the first hand, contemporary civilization is blessed to have received the gift of Euclidean geometry. For, it is only through the kind of axiomatic, deductive method that was used throughout Euclid's Elements that the idea of "mathematical truth" has been able to survive the onslaught of religious dogmatism that characterized the Middle Ages.

But due to the success of the Euclidean method, there has been an ever-present questioning of its "ultimate" foundation: i.e. Is geometric space already inside of us, or is it something that can only be derived from experience? And it was by way of this essential question that the Kantian system of thought began to take root.

It was Kant's position that space and time are "forms" of the faculty of intuition, with space as the "external form" and time as the "internal form." And it may well be argued that this notion, perhaps more than any other, strikes at the very heart of what it means for anything to be known as a "physical theory."

Many scientists are afraid that the identification of physical dimensionality with the intuitive faculty is a recipe for the subjectification of any possible physical theory. However, I submit that the only alternative--that is, the thought that space and time can, in themselves, somehow be objectively "known"--only serves to undermine the very possibility of objective truth.

So let us try to decipher what Kant was trying to say. By way of placing the formal aspects of geometry and dynamics squarely in the seat of the subject, the problem of the mechanism whereby forms could be known was thus dissolved. That is, to think of the subject as initially being a perfect tabula rasa ("blank canvas"), the ideas of the spatial forms themselves become purely contingent on the nature of one's [accidental] sensual experiences.

But if these experiences must necessarily conform to one's inherent intuitive faculty, then we are in a much better position to say, for instance, that the forms represent the very possibility of all objective thought. That is, if forms merely "come to us" from "out there," then there does not exist any kind of necessary "conformity" of object to subject, and all that exists are just subjective thoughts. And in this way, there is no right for any subject to say that the forms that are given bear any kind of necessary relation to the forms that are given to anyone else. Taken to its logical conclusion, there is therefore no possibility for the kind of objective truth that is necessary for the existence of any mathematical proof.

So, Kant developed the idea that the internal intuition (time) is nothing other than the subjective sense of self-continuity, and furthermore, that this continuous internal state conditions the possibility that the manifold external sensations can combine into representations. And given that these sensations must themselves conform to the [external form of the] intuition, the representations can thus be understood to be objective.

But if we assume that the physical dimensions are given apart from the subjective condition, then all knowledge is reduced to purely immediate contingency. That is, only "particulars" can only ever be known, and this knowledge can only manifest at the very instant of the experience of the "particle" in question. Thus, there is no longer any possibility of developing synthetic systems of thought that must necessarily combine remembered concepts into a logical unity.

But there have been many influential thinkers since Kant who have disavowed themselves of the idea that the dimensional forms are the necessary subjective conditions for the possibility of objective thought, to the point that there is said to exist an infinite number of geometries, and Euclidean geometry is understood to merely be the one that humans happen to experience, with the conclusion being that Euclidean geometry turns out to be the one that is most "convenient" to use.

It is said that the problem with Euclidean geometry is that it does not allow for the existence of "curved" geometries. But what does it mean for geometry, itself, to be "curved"? After all, isn't any mathematical definition of curvature wholly dependent upon a space within which all successive points are necessarily equidistant? That is, how can it be possible to dependably represent, for instance, the curve $$y=x^2$$, if the distances between any two adjacent numbers that are used to define this curve are not necessarily identical?

We can start to develop the idea, then, that Euclidean geometry is essentially a concrete specification of the abstract integral space that must exist inside all of us in order to apprehend the basic arithmetic operations, upon which all mathematical concepts depend. So, we can thus say that any possible n-dimensional geometric form is nothing other than a manifold that must necessarily depend on an n+1-dimensional mathematical space within which all successive points are separated by an identical unit. And it is within this spirit of mathematical space (i.e. the form of the external intuition) that we may come to understand the significance of the geometry that is called "Euclidean."

And so, we can understand hyperbolic and Riemannian n-dimensional "geometries" simply as being manifolds within an n+1-dimensional Euclidean space. That is, if Euclidean space is a necessary condition for the possibility of objective thought, then these manifolds are simply a priori syntheses that are understood merely to be conditioned objects of thought, rather than necessarily existing "external realities" that are only accidentally related to the subject, via the empirical sensations.

It is in this way that physical theorists are able to put to use their own intuitive faculties in the development of systems of thought that are fully objective (in the sense that the possibility of objectivity is conditioned by the forms of the intuition), and that do not have any [contingent] empirical elements mixed within.

 

[apeiron]

The mathematical objects or concepts which we use to model reality can be arrived at from two complementary directions - via bottom-up construction or top-down constraint.

So for instance, we can argue like Euclid and add up an infinite series of discrete zero-D points to create a 1-D line. Or we could come the other way, take a continuous 2D plane and successively constrain its dimensionality until we arrive at the infinitesimally slender region that makes our same 1-D line.

We can build up from below or shrink down from above, and asymptotically arrive at the same place.

Euclid went the bottom up route to modelling geometry. He took the continuity of the spatial world and found a way to construct it from the infinite sum of zero-D points (or by the successive motion of a point tracing out a line, a line sweeping our a plane, same thing).

Then non-Euclidean geometry was imagined/discovered via a different path - the successive relaxation of contraints. A process of generalisation. So flatness came to be seen as an extra constraint placed on curvature. Remove that constraint and let straight lines flex and you step up to a higher level of generality.

Remove enough constraints and eventually you go from geometry to topology. This is what category theory is all about. Tracing the hierarchy of mathematical generalisation by discovering/imagining the constraints still in place and then relaxing them.

In this way, mathematics has backed upwards to the most global or generalised view.

As to the ontological status of maths that results, I would argue that both the bottom-up and the top-down approaches are equally real. Or equally unreal, in the sense that we only can know the world via modelling.

So it is not the case that the events and objects are out there in the real world, the global organising concepts are synthetic creations of subjective experience.

Instead, psychologically, we started with a naive and direct modelling of the world around us - the anticipatory model a child develops by groping around at birth. Then first we broke this immediately available world down into a bottom-up construction - the atomistic view. Then we learned how to generalise, move upwards to higher levels of abstraction (as opposed to lower levels), by a systematic relaxation of constraints.

 

[Hurkyl]

Non-Euclidean geometry was developed as it was needed.


Saccheri (a priest, incidentally) developed a great deal of hyperbolic geometry in his unsuccessful attempt to prove the parallel postulate by contradiction.


Geometers conjectured and eventually constructed the projective plane to account for point-line duality -- the fact there is a wide class of theorems of Euclidean plane geometry that remained true if you swapped the words "point" and "line" and handled degenerate cases properly.


Differential geometry was developed for the purpose of studying the intrinsic geometry; e.g. to study a surface in Euclidean space without making any reference to the ambient space.


And so forth.

 

[glengarry]

The point that I was trying to make in the essay is that there is a profound difference between 1) the deductive method that was merely utilized by Euclid in Elements and 2) Kant's essential idea that the form of the external intuition can only be understood as "Euclidean." And it is through this distinction that I am trying to battle the brutish, leveling forces of philosophical empiricism, which, when allowed to run rampant on our intellectual culture, will only lead to the stagnation and the eventual decline of civilization as we know it.

What I am trying to say, then, is that Euclidean space, in the way that Kant was attempting to define it, is nothing other than the mathematical space that exists within all of us, in order for any kind of "intuitive science" (i.e. geometry) to be at all possible. And furthermore, that this mathematical space, in itself, is necessarily arbitrarily dimensional, such that all of the elements of any particular dimension are necessarily differentiated by precisely unity. For instance, in the third dimension, the elements of that dimension are two dimensional objects called planes, and the necessary condition for the possibility of any [3-dimensional] objective science is that the separations between any two successive planes are necessarily identical. The condition we call "flatness," then, is nothing other that the immediately intuitive quality that arises when empirical sensations conform to the necessary condition of objectivity (i.e. equidistance between successive elements), and it is not, therefore, merely an axiomatically given concept, and neither is it simply a particular "form" of curvature--for the reason that all objective determinations of curvature necessary depend upon the inter-elemental equidistance that conditions any given mathematical dimension.

That is, the fact that the several planes of E3 (3-dimensional Euclidean) space are identically separated is the same essential idea as the fact that the several points of E1 space are identically separated. In other words, the concept of "parallelism" deals with the question of the equidistance between the successive points of a numbered line, in the same essential way that it deals with the question of the equidistance between the successive lines of a numbered plane (or the planes of a numbered space, or the spaces of a numbered hyperspace, etc.).

The main point that I am attempting to make will be lost on all but the few remaining philosophically minded physical theorists--a situation that has arisen because of the fact that our modern scientific academies (university departments that deal in "objectivity") tend toward the advancement of profoundly non-critical modes of thought. It seems as though it was once a right of passage for a prospective physical theorist to attempt to come to grips with the Kantian system, but the academies are currently set up so that there is simply no longer any time for these sorts of students to engage in the meditative reflection necessary in order to develop the intellectual fortitude so that progress can indeed me made on the "front lines" of the battle that is known as theoretical physics.

 

[Hurkyl]

I don't mean to be mean, but when I read that, it looks like the short summary is simply the bald assertion that people cannot intuit anything but Euclidean geometry, followed by a complaint that people dare study other things.

 

[disregardthat]

I don't mean to be mean, but when I read that, it looks like the short summary is simply the bald assertion that people cannot intuit anything but Euclidean geometry, followed by a complaint that people dare study other things.

I don't interpret it as people cannot intuit anything but Euclidean geometry, but that euclidean geometry form the basis for our geometric intuition. Just as the natural numbers forms the conceptual basis for the notion of a number at all.

 

[apeiron]

The condition we call "flatness," then, is nothing other that the immediately intuitive quality that arises when empirical sensations conform to the necessary condition of objectivity (i.e. equidistance between successive elements), and it is not, therefore, merely an axiomatically given concept, and neither is it simply a particular "form" of curvature--for the reason that all objective determinations of curvature necessary depend upon the inter-elemental equidistance that conditions any given mathematical dimension.

There is also the second route to discovering flatness. The measured value of pi starts to vary as the constraint of flatness is eased. And this would seem to be more a direct result of relaxing the constraints on our geometric models of reality than a realisation that arises out of any Kantian experiential models of the world.

 

[Deveno]

I'm a little naive when it comes to such deep subjects, so i have little insight to offer. But what i find curious is this: apparently there was this "process", wherein a purely abstract geometry (whose name, in fact, comes from something like "earth-measurement") was developed from intuition informed by physical experience. At some point, this knowledge became "detachable" from experience, in the sense that one did not need to reference verifiable physical facts, in order to discover consequences of the fundamental postulates, and although the knowledge that sprang from the abstraction proved to be useful in our real, everyday world, it could be done in a purely formalist context.

OK, so far, so good. One is tempted to think that Euclidean geometry represents some Ideal, that has some kind of pure (a priori, perhaps?) existence, independent of our transitory, and incomplete physical one.

This is what mystifies me: further physical experiences led scientists to propose, eventually, that in fact, Euclidean geometry wasn't even a faithful abstraction of our real world, that the geometry of the universe we find ourselves in, is actually non-Euclidean. It seems reasonable to me, at any rate, that we may discover at some point, that even this modification to geometry will prove inadequate to describe what we consensually agree appears to be true. It appears our ability to abstract, may have led us astray, that out intuitions are somehow profoundly wrong. And yet...

When I actually DO mathematics, I approach it as, essentially, a game. I do not mean to trivalize it, thereby. What I mean is, as a practical matter, I start with a given set of rules, and I play by them, in hopes of being able to communicate something that represents a condensation of a lot of effort into a useful tool or result. Euclidean geometry, or non-Euclidean geometry, are different games. The ways I play them are similar, but not identical. The fact that they have different rules doesn't especially bother me, and the notion that it DOES seem to bother other people seems quaint to me.

Now, some may say, that non-Euclidean geometry does not, in and of itself, invalidate the point Mr. Kant was trying to make. I, for one, am undecided on this matter. I do feel a bit uncomfortable with the fact that the necessary machinery of developing a real number system, and metric topological vector spaces, depend on using a Euclidean norm to eventually investigate a non-Euclidean world. Perhaps we have somehow chosen the wrong starting point for investigation; perhaps the real numbers aren't so "real" after all, and we may someday find a better way of describing what we feel are more universal invariants.

I will note, in passing, that it does appear our world is "locally" Euclidean. In this respect, differential geometry represents some kind of reasonable compromise for accepting Euclidean geometry as a useful creation. If, however, exercises in logic or any kind of "pure" knowledge can only yield "local" truths, it gives one the rather bleak sensation that we will forever be profoundly ignorant of the nature of reality.