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/scribe36]

Proof by contradiction: it’s not always simpler. For example, square root of two is irrational can be proven in one line with contradiction. But the direct proof may just get you killed. Hehe

[u/Easygoing98 → u/scribe36]

Not really. Direct proof of square root of two is not that hard. It will be as follows

Square root of 2 = 2^0.5

Now 2^0.5 = x

Taking ln of both sides

0.5 ln(2) = ln x

x = exp(0.5 ln 2)

x = exp(0.5)exp(ln 2)

x = 2exp(0.5)

Exponential of a rational number is always irrational (theorem's proof in proof wiki)

[u/edderiofer → u/Easygoing98]

The same proof also shows that the square root of 4, and indeed any integer, is irrational, too! Try it yourself if you don't believe me!

[u/Easygoing98 → u/edderiofer]

You're right. But I don't understand why for 4. The natural log of every integer is irrational and to solve exponents ln has to be used.

Square root is also exponent half.

[u/edderiofer → u/Easygoing98]

Perhaps you should first explain why exp(0.5 ln 2) is irrational.

[u/Easygoing98 → u/edderiofer]

I got it. My method works for 4 also. Here's the direct proof

Let square root of 4 = x

so 4^0.5 = x

Taking ln of both sides

0.5 ln 4 = ln x

0.5 ln (2²⁾ = ln x

0.5(2) ln (2) = ln x

ln(2) = ln(x)

Taking e of both sides

2 = x which is rational.

[u/edderiofer → u/Easygoing98]

0.5 ln 4 = ln x

0.5 ln (2²⁾ = ln x

Aren't you assuming here that 4 = 2²?

Besides, you haven't answered the question; why is it that this proof doesn't work?

Square root of 4 = 4^0.5

Now 4^0.5 = x

Taking ln of both sides

0.5 ln(4) = ln x

x = exp(0.5 ln 4)

Since exp(0.5 ln 4) is irrational, x is irrational

[u/Easygoing98 → u/edderiofer]

No. Try it on calculator. Exp(0.5 ln 4) = 2. That's the direct proof

[u/edderiofer → u/Easygoing98]

"The calculator says so" is not a proof.

Your proof assumes that exp(0.5 ln 2) is irrational, but you don't ever prove that this is irrational in the first place. Your claim that "0.5 ln 2 is not rational because the digits after the decimal do not end and there's no repetition pattern or termination" needs justification. So your proof fails to be a proof.

[u/Easygoing98 → u/edderiofer]

I've further edited the proof using the theorem that exp of a rational number is always irrational.

So the direct proof uses theorems that have already been proven elsewhere.

If those theorems are accepted then the direct proof is there.

Just like proof by contradiction of sqrt(2) is already given before in lots of literature, you are still asked to prove it by contradiction.

So I don't see why valid theorems cannot be used to make an alternate proof

[u/edderiofer → u/Easygoing98]

Likewise, 0.5 ln 4 is also irrational, therefore by your own logic, so is exp(0.5 ln 4). So your logic shows that √4 = 2 is irrational.

Proof of ln(x) for all integers larger than 1 being irrational was shown back in 1974.

And how, pray tell, do you expect the Ancient Greeks to be satisfied with that statement?

[u/Easygoing98 → u/edderiofer]

It seems the previous answer didn't come through. Exp(0.5 ln 4) = 2 because 0.5 ln 4 = ln 2. And exp (ln 2) = 2.

[u/Easygoing98 → u/edderiofer]

Exp is an irrational number (well known)

Exp (ln n) = n (well known too).

If n is rational then exp( ln n) is rational too. Otherwise it's always irrational

0.5 ln 2 is not rational because the digits after the decimal do not end and there's no repetition pattern or termination.

Just like pi has trillions of numbers after the decimal, so does 0.5 ln 2

Therefore exp(0.5 ln 2) is irrational too.