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Two Times Two Is Not Four

Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight

Johann Wolfgang von Goethe
--Quoted in Newman's The World of Mathematics



Comments

user-pic

I like this quote much. It is especially interesting since Goethe was kind of a contemporary with Kant, who made the distinction between a priori synthetic judgements and a priori analytic judgements. In the later category would fall mathematical truths, which are simply conclusions from certain assumptions (i.e. given what we agree 2 and 4 are, we can say with certainty that 2+2+4). In the former would fall the conclusion that there is a moral law which categorically obliges me. Kant of course believed in both.

I mention this because as I understand it, today most Anglo-American philosophers claim that only analytic a priori judgments are possible. So it is a matter of course for many that contemporary philosophy as we know it seems to have the limitations that Goethe describes here. Perhaps Chris might have something to say about this.

Actually, most modern Anglo-American philosophers think Quine's seminal paper "Two Dogmas of Empiricism" dismantled the idea that there could be analytic truths. http://www.ditext.com/quine/quine.html It would be quite scary if philosophers were under the same limitations Goethe said mathematicians have to deal with. After two thousand years all we could say with much certainty are things like "no bachelors are married." That would be tragic.

user-pic

I will take a look at this, since I don't understand fully what you mean at this point. Thanks for the link, though.

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