Anyway i was wondering, does the following happen in research?:that the backward exercise of recognising when two proofs are or not equivalent(and in which meaning they can be) seem possibly useful in forward direction of selecting a strategy of a proof when one has to consider many possible ways, and has already found that some of them are failing, and here can be useful to efficiently detect wich apparently new strategy is in some sense equivalent to the failing ones(…meaning that if this would work, then it would give a proof, basically equivalent to the one you’d obtained from the failing strategy wich is incorrect), and so has to be rejected without actually doing it.

It seems that the trade on could be of becoming much faster in discarding a lot of useless approach, but the trade off can lie in the ambiguity of the concept of equivalence:maybe the seemingly equal approach had some very little detail that turned in a different picture, and one was unable to see this from the high level, and so loose an actual proof(…or some unexpected results).

Are there example of this from your research?(i’m an almost graduate student…so i’ve just example from my problem solving session, that sounds pretty silly).

]]>0=(a-bw)(c+dw)=ac+adw-bcw-bdw^2=ac-2bd+w,

thus w is an integer. ]]>

What Is a Logic, and What Is a Proof?

Lutz Straßburger

AbstractI will discuss the two problems of how to define identity between logics and how to define identity between proofs. For the identity of logics, I propose to simply use the notion of preorder equivalence. This might be considered to be folklore, but is exactly what is needed from the viewpoint of the problem of the identity of proofs: If the proofs are considered to be part of the logic, then preorder equivalence becomes equivalence of categories, whose arrows are the proofs. For identifying these, the concept of proof nets is discussed.

23 October 2006

Logica Universalis—Towards a General Theory of Logic, pp. 135–152, Birkhäuser, 2007

PDF of the paper is here: http://www.lix.polytechnique.fr/~lutz/papers/WhatLogicProof.pdf

Site that I got this link from, that has a ton of related stuff: http://alessio.guglielmi.name/res/cos/index.html

]]>– Cantor’s original proof, as quoted by Jonah Sinick above, is taking the real interval as they are i.e. a compact and complete metric space, and then detects at least one missing (i.e. not contained in the counting) real number by sequential compactness i.e. Bolzano-Weierstrass. Viewing the real numbers in binary expansion with base 2, the two binary representations of all binary rationals coincide, e.g.

– Whereas the diagonal argument models the reals set-theoretically as a space of sequences of digits: In the example of base 2, the sequences (0,0,1,1,1,..) and (0,1,0,0,0,…) are not the same. This is no coincidence as the topology is now very different: taking a natural base , the space of sequences of natural numbers has the desired cardinality and is a compact and totally disconnected space in the product (Tychonoff) topology. For base this space of 0-1 sequences is homeomorphic to the classical Cantor set.

Both spaces are compact, metrizable and complete in their metric topology. But they are far from being homeomorphic: the real interval is connected, and the Cantor set is totally disconnected.

Thus the a priori purely set-theoretical statement of uncountability of the reals is proven by two methods familiar from the categories of topological and metric spaces, but based on two models of the set of reals which are topologically quite different.

It would say that two proofs are the same if they induce the same function between the proposition types.

Tom

]]>The situation is somewhat better with “these two theorems are equivalent.” As Terry said, in reverse mathematics one posits a very weak “base theory” (RCA_0 is a common choice) and then sometimes you can show that two theorems A and B are equivalent in the sense that (1) both A and B are independent of the base theory and (2) their equivalence can be proved in the base theory. There are many successful examples of this. On the other hand, it doesn’t always capture our intuitions perfectly either. Intuitively one might argue that Sperner’s lemma and Brouwer’s fixed-point theorem are equivalent, but Sperner’s lemma is provable in RCA_0 while Brouwer’s theorem isn’t.

By the way, here’s another proof of the irrationality of . Let be the smallest positive integer such that is an integer. Let . Then is an integer yet is a strictly smaller positive integer than ; contradiction. Is this the same proof as one of those already given? Note that this proof immediately generalizes to show that the square root of an integer is either an integer or an irrational number (i.e., it can’t be a non-integral rational number), if you let . The traditional proof of the irrationality of does not generalize so quickly without a detour into unique factorization or something.

]]>More generally, I think Joe brings up a very good point: the definition of “proof homotopy” can be contingent on the current state of mathematics as opposed to anything that could be considered an independent truth. If, for example, two proofs of a result like the Fundamental Theorem of Algebra come from two distinct branches of mathematics and do not appear to relate easily to each other, then perhaps the reason they appear so different is that some deep correspondence between those two branches has not yet been uncovered, and this hypothetical correspondence would reveal those two proofs as essentially the same by placing them in context we don’t yet have. So I guess what I’m suggesting is that if a result appears to have genuinely distinct proofs, then perhaps we do not really understand it!

]]>“Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side.”

Wagon offers 14 different proofs in “Fourteen proofs of a result about tiling of a rectangle” and he even classifies them according to how they generalize…

]]>Here is an elementary explanation:

Multiply both sides by ,

By setting , we get

Now here is a complex explanation:

Consider

Where is a closed contour to be determined.

We know

Set be a circle of radius around respectively, and use the Cauchy’s integral formula. Again, by the same process, we get:

]]>So what does this mean? One might say that this means that if we assume the first we can prove the second and vice verse. But along these lines any two true statements are equivalent. I would protest this by saying that “The Pythagorean theorem is equivalent to the Prime Number Theorem.” Now one might try to correct this by saying that we mean that A and B are equivalent if the proof of B uses A and the proof of A uses B. But, in the Prime Number Theorem example, certainly there are proofs of the Prime Number Theorem that do not make use of the fact that Re(s)=1. Now if I had tried to point to an example (say the Selberg-Erdos “Elementary” proof, or Wiener’s using his Tauberian Theorem) someone might dive into a discussion about how the zeta function is really lurking in the background of these. But even if someone presented a case that every known proof of the prime number theorem relied on that fact that the zeta function doesn’t vanish on the line Re(s)=1, this certainly doesn’t imply that there doesn’t exist a proof the proceeded without use of the fact. Moreover while I’m not sure what we do mean when we say “A is equivalent to B” I’m pretty sure it should be a mathematical assertion and not a statement about the literature in existence on the result.

]]>Can’t leave it unsaid that the second proof mentioned for the irrationality of $\sqrt{2}$ can be seen in a beautiful manner:

Take an actual triangular piece of paper of sizes p,p,q (assuming $p^2=2q^2$). You can now fold one of the sides on the diagonal, and get yourself a smaller triangle of sizes $p-q,p-q,2q-p$, revealing that $2(p-q)^2=(2q-p)^2$, and contradicting the minimality of $p,q$ if you assumed that in advance.

]]>Actually the first two proofs are quite similar if you think about valuations: a valuation is a multiplicative function from a field to that respects the triagular inequality, ie and , and only if . It can be proved that all such valuations over the rational number are (up to equivalence) the usual absolute value () and given by prime numbers (, where and are prime with ), so they can be thought of as a suitable generaliation of prime numers and ideals, providing additionally a “prime at infinity”. All valuations share important properties, for instance they induce a topology with respect to which you can take the completion, etc.

In the first proof of the irrationality of , you are taking a fraction where and minimize the valuation at the prime 2 (by dividing the numerator and denominator by the biggest common power of 2), and you get an absurd because cannot possibly be equal to . In the second proof, you are choosing and with smallest absolute value (and so smallest valuation at ), and then apply a tranformation (an element of ) that leaves the value of the fraction, supposed to be equal to , unchanged, but decreases the valuation at of the denominator, absurd by minimality. So, by what i’m understanding, most of the difference in the proofs comes by how the “non-archimedean” valuation at 2 and the “archimedean” one at work, possibly an experienced number theorist may give us a better explaination of this fact.

]]>I think the first proof shows in passing, as part of its argument, that the group is cyclic, whereas with the second proof, you have to go back at the end and look at the group you’ve proved you have, and check that it’s cyclic. I’m not sure whether that’s a real distinction between the proofs. My rather poor point about the quaternions was just that needn’t be cyclic there, so there can’t be any generalization of the first proof there that uses some other definition of minimal element. Now that I’ve looked at some of your web pages, I see that I was thinking vaguely along the lines of “quaternions are another model for the complex numbers”, in a way slightly reminiscent of your page about why it isn’t obvious that the integers are bounded in the reals.

]]>I also don’t quite understand what it means to say that the second proof doesn’t show that the group is cyclic: if it shows that it’s the group of th roots of unity then surely it does.

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