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.(adsbygoogle = window.adsbygoogle || []).push({});

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 perfecttabula 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 [tex]y=x^2[/tex], 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 simplya priorisyntheses 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.

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# Kant, Geometry, and Physical Theory

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