It would be nice if they described Euclid's argument in its original geometric form rather than converting it back to post 1700 algebra notation.
Euclid's Proof that √2 is Irrational
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Re: Euclid's Proof that √2 is Irrational
#92Earlier quoted context omitted.
This depends on your philosophy of mathematics. If you believe god gave us the nonnegative integers then in a very natural way one is led to the notion of algebraic integers. Whether this would be god’s creation or man’s creation depends on your view of who gets the credit in such a situation.
I think next time I’d better make the apologies to Kronecker explicit.
Sorry. I missed it. It was too clever for me!
Re: Euclid's Proof that √2 is Irrational
#93Earlier quoted context omitted.
If you took abstract algebra (which presumably you did as a math major), you certainly encountered these at least in the exercises as groups of the form ax + b where x is some irrational number (or imaginary) and a and b are integers are a staple of chapter 1–2 proofs. Gaussian integers ( ai + b ) are a special case that are loads of fun to play with it. They are not unique factorization domains like the integers (e.…
Nit: while it is not generally the case that rings of algebraic integers must be unique factorization domains, it is the case for Gaussian integers! In your example, 5 is uniquely factorizable up to units as (1-2i)(1+2i).
Indeed, the integers have the same limitation -- factorization is unique only up to units. 1 = -1 * -1
In elementary mathematics, people wave away "-1" by saying silly things like "positive integers", before Gaussian integers arrive and force us to figure out precisely what we are trying to say without silly ideas from analysis like "ordering". :-)
Re: Euclid's Proof that √2 is Irrational
#94Earlier quoted context omitted.
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).
This seems a bit harsh. Mathematicians tend to be mostly unambiguous when they write. This isn't even on the same planet as the empirical disciplines.
Mathematicians (with books in hand) tend to have the ability to disambiguate when confusion arises. But they rarely do, instead relying on context and the intelligence of the reader to derive meaning.
Re: Euclid's Proof that √2 is Irrational
#95Earlier quoted context omitted.
That assumes that every fraction has a unique simplest form. The first section of the article makes no such claims about the existence of a simplest form of fraction. The proof uses just algebraic manipulation, the fact that a sequence of strictly decreasing positive integers is finite in length, and the definition of a rational number (there exist integers p, q (q != 0) such that the number can be expressed as p/q).
The article explicitly claims > Rational numbers or fractions must have a simplest form. They make no claim about uniqueness, but that is not needed in the argument.
t0mek sais "the simplest form", but the comment is fixable by changing that to "a simplest form". (For example, if hypothetically a/b and c/d where somehow the same number, and yet somehow there is no x such that a/b = xc/xd, the argument about how 2 divides into a and b also applies to c and d.)
Re: Euclid's Proof that √2 is Irrational
#96Earlier quoted context omitted.
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.)
You take a set S and partition it into two subsets, A and B, so that each element uniquely belongs to either of them. For each element of S it is true that "not-not-in-A -> in-A". Can this be shown to be true in MLTT? I don't know much about setoids to show it, but if it can't, that's the problem of MLTT, not mine. In the context of this proof, it is true.
Re: Euclid's Proof that √2 is Irrational
#97Earlier quoted context omitted.
I was going to say this. The proof, unfortunately, sidelines into some very deep mathematics that it didn’t need to. I imagine but don’t know for sure that Euclid stopped where you do. I’m not sure when infinite descent would be considered to have been formally proven to a modern mathematician but I bet it wasn’t in Euclid’s time!
I think the idea is that any strictly decreasing sequence of positive integers starting at N is a subsequence of (N, N-1, N-2, ..., 1) which has N elements, so any such sequence must have a finite number of elements. So if you manage to produce an infinite sequence of strictly decreasing positive integers starting at a particular positive integer, then you've reached a contradiction.
Yes, I understand it intuitively, as would I think most non-suspicious people. It sounds very reasonable! As does the axiom of choice. We could even “prove” it using high school induction methods.
In practice though, concepts with infinity are very tricky, and the Greeks definitely did not have a strong accurate theory of infinity, Aleph-zero even, much less (as is the point talking about the sqrt(2)), the reals.
In this case, I think embedded in a high school proof would be the assumption that for any integer you can list there are not an infinite number smaller than them. This is manifestly true about the integers, but it is not true for the rationals, despite the two sets having the same cardinality, and a solid proof would need to be able to distinguish the reasons why. It’s this part that I think would be beyond the Greeks.
Re: Euclid's Proof that √2 is Irrational
#98And then in 1737 Euler (another name starting with Eu, definitely a good name) showed that the constant e is irrational. His proof exploited the fact that the continued fraction representation of any rational number terminates. The CF representation for e does not.
> Eu, definitely a good name I see what you did there.
The worst non-Euclidean geometries should be called Dysclidean. I think Lovecraft missed an opportunity here.
Re: Euclid's Proof that √2 is Irrational
#99Earlier quoted context omitted.
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.
>>>> As I said, according to the three sources above, which are the first sources I clicked on which didn't seem like blogspam, the phrase "proof by contradiction" is a term of art which means " uses the law of excluded middle to conclude the truth of a statement given a proof that its negation is false" (quote from you; my emphasis)
In that case your "It's much dumber than that, since he's invoking the law of the excluded middle to use contradiction at all." is simply false: you can use contradiction to refute a proposition without proving that proposition, without using LEM.
Re: Euclid's Proof that √2 is Irrational
#100Earlier quoted context omitted.
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.
excluded middle: the logical axiom that, for any proposition P, (P v ¬P) is a tautology/valid/always holds. contradiction: a proof of ⊥. The definition of ⊥ does not matter (it just means false), thanks to the ex falso quodlibet principle. A proof by contradiction: proving P by showing that (¬P -> ⊥). Notice that I haven't defined the ¬ operator. This is due to the fact that its definition differs between classical l…
Sure, I agree with everything you've said; I was sloppy in saying "Euclid's proof" (whose baroque language I haven't actually bothered to wade through) when I meant "the proof in the OP", though I was precise in saying "it can't use LEM" where you've interpreted it as "a mathematician can't use LEM".
(I asked for an explanation because I resented being called "dumb" by someone I'm pretty sure doesn't actually understand the distinction I'm talking about.)