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Why is non-Euclidean geometry, as encountered in relativity theory, always used as the prime counterexample to rationalism – and regarded as pretty much decisive?

The parallel postulate was controversial from the start and Euclid himself avoided invoking it whenever possible; so it seems far from being a victim of a psychological illusion, the insight of the early geometers into the structure of space was so reliable, that they got – except this one issue – everything right and even correctly singled out the “later refuted”1 axiom as “problematic”.

Is the strong association of rationalism with Kant (who thought – or at least so it is claimed – that space for us could only be Euclidean) the reason behind this?


1 sloppily phrased, of course. Also, afaik there is no way to directly show the non-Euclidean nature of space. It follows from an appeal to the simplicity of the theory, that non-Euclidean geometry is to be preferred and therefore correctly describes space. Which ironically seems a pretty “rationalist” line of argument, too.

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    How exactly is it used as such an example, and where? Greek geometers got geometry from Egyptian rope stretchers, who were engaged in a plainly empirical practice, so how would geometry be an argument for rationalism in the first place? Even for Kant geometry requires sensibility, "synthesis in pure intuition", pure reason is not enough. – Conifold Apr 10 '18 at 19:51
  • Argument against rationalism? No of course not. Non-Euclidean geometry is an argument against the claim that math equals physics. Nothing more. – user4894 Apr 10 '18 at 20:45
  • @Conifold that's a mischaracterization of rationalism. Leibniz: “Now all the instances which confirm a general truth …, are not sufficient to establish the universal necessity of this same truth, for it does not follow that what happened before will happen in the same way again. … From which it appears that necessary truths, such as we find in pure mathematics, & particularly in arithmetic & geometry, must have principles whose proof does not depend on instances, nor consequently on the testimony of the senses, although without the senses it would never have occurred to us to think of them” – wolf-revo-cats Apr 10 '18 at 23:45
  • I am actually unsure what the characterization is, even with Leibniz's quote. Is it the existence of "necessary truths" independent of the senses? Then Kant wouldn't be a rationalist. And why should reliable insight into something be rational rather than synthetic a priori, historically cumulative or mystical, say? Why should it even be "insight" rather than a framing invention? I think the question will become clearer if you lay out in the post how you understand "rationalism", how the argument from geometry against it is supposed to go, and how you propose to defend it. – Conifold Apr 11 '18 at 1:59
  • @Conifold rationalism = “by appeal to reason alone, we may justify belief in the truth of certain propositions” … it doesn't mean that sense experience isn't necessary for forming the concepts – as long as sense experience isn't appealed to as justification. – wolf-revo-cats Apr 11 '18 at 2:44

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