Mathematics

Is geometry purely mathematical or does it rely on spatiality which is beyond mathematics?

Comments

I take your question to be whether geometry can be axiomatized into a deductive system based on certain definitions, as some philosophers believe mathematics can be, or whether, because geometry is in some way related to space--unlike mathematics--it cannot so be axiomatized.

I begin by noting that there is disagreement among philosophers of mathematics whether mathematics can indeed be axiomatized in this way. Charles Parsons, for example, following Kant, believes that mathematics requires intuition. Since I don't know the details of Parsons's account, presented in his book Mathematical Thought and Its Objects, I draw instead on Kant's view, which inspired Parsons (who is also a great Kant scholar): consideration of Kant's view of mathematics will also lead us back to geometry.

According to Kant, both mathematics and geometry yield a body of necessary truths, truths which are, in Kant's terminology, 'a priori'; moreover, according to Kant, the truths of both mathematics and geometry extend our knowledge, and are, in his terminology, 'synthetic', in contrast to propositions that are 'analytic' and do not extend our knowledge, propositions derived directly from definitions. Now Kant thinks that since mathematics and geometry are synthetic, they must involve something empirical; since their truths are a priori, they cannot be directly derived from experience, and involve what he calls 'empirical intuition'. In the Transcendental Aesthetic of the Critique of Pure Reason, Kant argues that mathematics and geometry involve pure intuitions: mathematics involves a pure intuition of time; geometry involves a pure intuition of space. According to Kant, then, mathematics and geometry are not so different after all, although they do of course differ with respect to their objects.

Now at least one reason that Kant thought that geometry was based on an a priori intuition of space is because he conceived of geometry--Euclidean geometry, which was coextensive for Kant with geometry--as the science of space. The discovery of non-Euclidean geometries, however, requires that one limit Kant's claims about geometry to perhaps the geometry of visual or lived space. (For discussion of developments in geometry, click here.)

Later developments in geometry have led to a general consensus that geometry can be axiomatized--that geometry in general is not the science of space, and indeed that it can be defined in terms of arithmetic. However, it remains an open question as to whether mathematics is just a matter of deducing consequences from definition, whether it can be axiomatized, or to put it less misleadingly and frame the issue more clearly, whether mathematics can be logicized, or reduced to logic. (For discussion of various approaches to this issue, click here. §§. 2 & 3 are especially relevant in this context.)

So the question shifts: what now seems to be at issue is not whether geometry relies on some conception of space, but whether mathematics, and hence axiomatized geometry, relies on some other intuition. Does mathematics depend on intuition? That's a good question, about which I myself have no firm intuitions: perhaps another panelist will take it up.

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