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Euclid's Proof that √2 is Irrational

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Re: Euclid's Proof that √2 is Irrational

#51

Post by Smaug123 »

thaumasiotes wrote: post #43
wazdra wrote: post #29

Earlier quoted context omitted.

> but it has standardised on that phrase! To me, the only formal distinction you can make between the two lies in the use of the excluded middle. However, this distinction has not standardised in mathematics, as many mathematicians simply do not care for intuitionistic logic. Such a mathematician could see the above proof as: I want to show ¬P by contradiction. Therefore I assume ¬(¬P) which is just P to me (the unin…

> To me, the only formal distinction you can make between the two lies in the use of the excluded middle. It's much dumber than that, since he's invoking the law of the excluded middle to use contradiction at all.

Could you please explain this? Clearly we both understand something completely different by either the term "excluded middle" or "contradiction". Note that Euclid's proof is intuitionistically valid, so it can't use excluded middle.

Re: Euclid's Proof that √2 is Irrational

#52

Post by Smaug123 »

LudwigNagasena wrote: post #36
Smaug123 wrote: post #7

1) it's not a proof by contradiction, it's a proof of a negation :grump: 2) I am not a fan of this phrasing "we can't simplify forever". Why can't we? It's obvious if you phrase it in the usual way as "the denominator is strictly smaller than it was before", but the "simplify" operation is kind of complex! They don't even mention "decreasing" until the very final Note box where they say offhand that actually it's an…

It's both a proof by contradiction and a refutation by contradiction because it's the same thing in this context. Positing "P = is rational and ¬P = is irrational" is as valid as positing "P = is irrational and ¬P = is rational".

No, "is not irrational" isn't the same as "is rational" without excluded middle; that's the whole point. (Equality of real numbers is not computable, so there is nonconstructive content to the implication "if not irrational, then rational".)

(I will retract a whole bunch of my worldview if you can inhabit the type "not-not-rational -> rational" in something like MLTT.)

Re: Euclid's Proof that √2 is Irrational

#53

Post by Smaug123 »

thaumasiotes wrote: post #47
Smaug123 wrote: post #7

1) it's not a proof by contradiction, it's a proof of a negation :grump: 2) I am not a fan of this phrasing "we can't simplify forever". Why can't we? It's obvious if you phrase it in the usual way as "the denominator is strictly smaller than it was before", but the "simplify" operation is kind of complex! They don't even mention "decreasing" until the very final Note box where they say offhand that actually it's an…

> They don't even mention "decreasing" until the very final Note box where they say offhand that actually it's an infinite descent (which is a critical part of the proof they've otherwise handwaved). That isn't actually a critical part of the proof; you can just assume that your initial two integers are relatively prime and then derive a contradiction directly.

Then why didn't they! Of course the proof can be fixed, we all know sqrt(2) is irrational, but why get so close to proving it and then just not finish the job?

Re: Euclid's Proof that √2 is Irrational

#54

Post by p-e-w »

kevinventullo wrote: post #34
gerdesj wrote: post #33

Earlier quoted context omitted.

Maths is always a bit boggling. You say that root two can be considered an integer despite being irrational. What does "usual integer" mean?

By “usual integer”, I mean what people usually refer to as an integer: …, -2, -1, 0, 1, 2, … As opposed to “algebraic integer”, which is a more general notion.

If I wasn't familiar with that concept already, then I would probably assume that math is no more rigorous than psychology after encountering this thread.

The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and "algebraic integers" are far from the only such case).

Re: Euclid's Proof that √2 is Irrational

#55

Post by Smaug123 »

jfengel wrote: post #38
wging wrote: post #20

It's an interesting exercise to find the right generalization of this proof to sqrt(n) for arbitrary numbers n that are not perfect squares, and for kth roots for m >= 2. I.e. prove that if kth_rt(n) is rational, then n is a perfect kth power (or equivalently, that if n is not a perfect kth power, then kth_rt(n) is irrational). (I'm talking about adapting the ideas of this divisibility-based proof. abstractbill's pos…

I remember being extremely struck by how general this proof was. In fact it made me suspicious that it would even apply to perfect squares. Running the proof on the square root of 4 took a little while to sink in.

(Important thing to do while understanding any theorem! Test the boundaries, discover what happens when you relax every hypothesis.)

Re: Euclid's Proof that √2 is Irrational

#56

Post by bsaul »

DiggyJohnson wrote: post #5
cornstalks wrote: post #3

> Likewise if a number is even and is a square of an integer, then its square root must be even. The proof would be more compelling if this was proven instead of being taken as an obvious fact.

Even x Even results Even Even x Odd irrelevant if squaring Odd x Odd results Odd

except sqrt(2) x sqrt(2) is even ( i know we're talking about numbers in Z in this case, and sqrt(2) definitely isn't in Z, but still)

Which made me wonder if the original sentence isn't already assuming something about sqrt(2) and even/odd properties.

(i stopped at the same step as OP wondering if this is as trivial as it seemed)

Re: Euclid's Proof that √2 is Irrational

#57

Post by lIl-IIIl »

DiggyJohnson wrote: post #5
cornstalks wrote: post #3

> Likewise if a number is even and is a square of an integer, then its square root must be even. The proof would be more compelling if this was proven instead of being taken as an obvious fact.

Even x Even results Even Even x Odd irrelevant if squaring Odd x Odd results Odd

>Odd x Odd results Odd

This isn't obvious and can't be taken for granted. The explanation posted above (2k+1)^2 by Smaug123 explains this part.

Re: Euclid's Proof that √2 is Irrational

#58

Post by mananaysiempre »

p-e-w wrote: post #54
kevinventullo wrote: post #34

Earlier quoted context omitted.

By “usual integer”, I mean what people usually refer to as an integer: …, -2, -1, 0, 1, 2, … As opposed to “algebraic integer”, which is a more general notion.

If I wasn't familiar with that concept already, then I would probably assume that math is no more rigorous than psychology after encountering this thread. The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and "algebraic integers" are far from the only such case).

The red herring principle[1] is unfortunately popular enough in mathematical terminology to have a name and a page about it.

Roughly, a fooish bar will frequently be something like a bar except fooish, so not actually a bar. (Algebraic integer, multivalued function, manifold with boundary, etc.) On the other hand, a nonfooish baz when baz is normally fooish often means a not necessarily fooish baz, so a particular one might be fooish but we can’t assume that. (Noncommutative ring, nonassociative algebra, the very field of noncommutative geometry, etc.)

[1] https://ncatlab.org/nlab/show/red+herring+principle

Re: Euclid's Proof that √2 is Irrational

#59

Post by RGamma »

p-e-w wrote: post #54
kevinventullo wrote: post #34

Earlier quoted context omitted.

By “usual integer”, I mean what people usually refer to as an integer: …, -2, -1, 0, 1, 2, … As opposed to “algebraic integer”, which is a more general notion.

If I wasn't familiar with that concept already, then I would probably assume that math is no more rigorous than psychology after encountering this thread. The exact disciplines are doing themselves a terrible disservice by muddying up established terminology like this (and "algebraic integers" are far from the only such case).

Ah yes, good ol' clopen sets.

Re: Euclid's Proof that √2 is Irrational

#60

Post by chx »

CaptainNegative wrote: post #28

This proof has almost nothing to do with Euclid. The Pythagoreans knew about it more than a century before his birth (Hippasus was apocryphally killed for divulging this proof), and the proof is widely believed to have only been inserted into Elements by others after Euclid's death.

Yeah, this can't be done in the geometrical way Euclid worked

Mostly because you need the Archimedean property for it which can not be derived from Euclid's axioms.

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