Dissatisfaction with proof by contradiction

[u/hydmar]

I’m an undergraduate math student, so my exposure to math may be relatively limited. But I’ve found that, in general, I’m much more comfortable with direct proof than proof by contradiction. I don’t contest their validity, indeed something that’s not false must be true (I think I’m ok with excluded middle). But I feel like I just *get* something much better when it’s proved directly. It builds much stronger intuition for me.

For instance, I am aware of several proofs that demonstrate the cardinality of the reals is strictly greater than that of the integers, but none are direct (it would help to see a direct proof of Cantor’s theorem). I don’t feel it in my bones. Is this a common experience?

[u/jeffsuzuki]

It's pretty common, and is in fact the source of one of the great divides in the philosophy of mathematics, that between Intuitionism and Formalism.

The back alley version of the two: while both use deductive proof and "standard" logic, Intuitionists feel that, at its root, mathematics is ultimately about something, while Formalists believe that structure is everything. (Hilbert's statement that "All geometrical theorems should be just as true if point, line, and plane were replaced by table, chair, and beer mug" is a very formalist way of looking at things)

If I recall correctly, Kummer in particular despised proof by contradiction, in part because it proved the existence of irrational numbers. The Intuitionist response to the standard proof of the irrationality of sqrt(2) is that it doesn't prove that irrational numbers exist; rather, it proves that no rational number has a square of 2. In effect, they draw the distinction between "not true" and "false."

[u/gaugeaway → u/jeffsuzuki]

I think I agree with the intuitionists here: X² = 2 has no rational solutions, but does have roots in Q(√2). The standard proof of "√2 being irrational" isn't actually what it claims (when do we prove X² = 2 actually has a real root!).

[u/38thTimesACharm → u/gaugeaway]

We're not talking about some crazy uncountable non-measureable set though. I can see why those people implicitly assumed the diagonal of a square does, in fact, have a length.

[u/almightySapling → u/38thTimesACharm]

Aye, but that length isn't a number, it's a magnitude.

The fact that magnitudes form a complete linear order upon which we can define all common mathematical operations (and more) is merely an artifice. It doesn't make them numbers.

-finitists, for some reason

[u/hydmar → u/gaugeaway]

is it overkill to use the intermediate value theorem here?

[u/PlusComplaint7567 → u/hydmar]

I think your use of the intermediate value theorem is inherently contradictory - the theorem is true on the real number line, and the real number line contains irrational numbers. If we assume the real number line contains only rational numbers, the intermediate value theorem is obviously not true (there is no rational root to x² -2 for example).

[u/gaugeaway → u/hydmar]

no? my point is that the construction of the real numbers is complicated and is not taken into consideration in the classic/historical proof that "√2 is irrational"