It never occurred to Euclid or indeed anyone back then that this was something that requires proof. In fact, it required the construction of the real number continuum, which came centuries later. So in this case, filling in the missing details was a lot more than a few little niceties.

This does not render Euclid’s arguments logically incorrect. They were not. But just because they were logically correct does mean they are rigorous *proofs*, since they omit steps that the intended reader (say, a professional mathematician) would find non trivial.

If we are going to allow proofs that miss out significant steps, then I can give you a logically correct proof of Fermat’s Last Theorem that will fit in a tweet, let alone a margin. Here it is:

*Let n>2. Suppose there are positive integers x, y, z such that x^n + y^n = z^n. Then 0 = 1. This is a contradiction, proving FLT.*

That sequence of statements is logically correct. It just happens to miss out some steps. Some fairly hefty steps, as it turned out.

Most of us would say my tweet is not a *proof*, and rightly so. But now we are talking about providing enough steps for the reader to be able to follow it, perhaps with some effort, filling in any missing steps as she goes. Andrew Wiles likely requires a lot less filling in of missing steps than the rest of us. I would need fewer steps than the average high school student. And so on.

In Euclid’s time, it was not recognized there were any missing steps in his arguments, so what he wrote was regarded as a proof. Back then, it was. But it surely was not a proof in our present day sense. We would probably say it is a strategy for constructing a proof. It can be built out to yield a proof. The standard mathematical way is to construct the real (Euclidean) plane.

I don’t believe I am disagreeing with what you say. But I think the different interpretation I put on it is significant if we want to understand what mathematicians today (really) mean by *proof*.

So we are still in the realm of stories told by mathematicians to mathematicians, but in some cases only a handful are able to fully understand those stories.

There’s also a leap of trust involved with a computer proof. Let’s not forget the Pentium FDIV math bug.

Certainty is still restricted to death and taxes. Mathematics does not make the cut.

]]>BTW, I have never read either paper. I take it on trust that the theorem is proved.

]]>The paper, which is 500 pages long, ……”

Both that and Wiles’ contradict the notion that a proof be understandable by “someone versed in the subject” – as there is no-one else with those mathematician’s grasp of the subject.

Suppose we change the idea to read “… and I understand and accept each step, so therefore I accept the conclusion”.

People being fallible (mathematicians included), he just might be wrong about step 798 – just as the guy who calculated pi to lots of places, but made an error somewhere around the 700th digit.

http://mathworld.wolfram.com/ has a good selection of artices on proofs.

Here’s a telling quote from a math site (http://m-phi.blogspot.com/):

“However, in the sense that in fact matters for mathematicians, Mochizuki’s ‘proof’ is not (yet) a prof because it has not been able to convince the mathematical community of its correctness;…”

Prof. Devlin: you’ve given the world a new bumper sticker:

“Mathematics is about the why, not the what.”

That opens a lot of doors here. Even when I’m doing elementary calculus, I’m just following rules, manipulating symbols – something computers do rather well.

]]>I think you misread my article, which is about proofs as understood and used in modern mainstream mathematics. Namely, they are devices we use to understand properties of, and relationships between, mathematical objects.

True, the theorem-proof fashion in which we typically present mathematical discoveries gives the impression that the focus is on what it true, but it’s really about why something is true.

In short, mathematics is about the why, not the what.

Proofs (as used by mathematicians) are stories humans tell to humans to explain the why.

Computer constructed formal proofs are something entirely different. They can provide us with a high degree of certainty that something is true, but by their nature they don’t satisfy our thirst to understand.

For instance, the general reaction in the mathematical community to the computer proof of Appel and Haken’s proof of the Four Color Theorem was, “That’s nice, but it just shows that the question was not a good math problem.”

The computer proof of the 4CT provided a mathematical fact, but no new insights into coloring networks. (That had already come from the initial reduction.) Likewise, the subsequent formal proof did not increase our understanding of the problem. It was not intended to, the goal was to increase our confidence in the result. It certainly did to me. But then, I follow those kinds of development (as an interested outsider).

These reactions by the mathematical community don’t constitute a dismissal. They are just an acknowledgement that formal proofs simply do not interest (most) mathematicians.

A computer proof generator or proof checker can increase our confidence in the truth of an assertion. But mathematicians view proofs as a means of understanding, which means the moment they suspect a problem might have to be attacked by a computer, they are likely to look for another problem.

Hales’ proof of Kepler’s Conjecture is an obvious exception, but he backed himself into that. There will always be problems that yield partially to an understanding-driven investigation, but result in having to check a set of possibilities too large to be done by a human. I for one am glad Hales kept going at that point (with the Flyspeck Project). But the way his result was published, and the difficulties he encountered, illustrates very well how the mathematical community views computer-aided proofs. (See his articles referenced here.)

Hales is not the only example, of course. But mathematically, they are all outliers. I predict that the points of productive contact between mainstream mathematics and the formal (computer) proofs community are going to remain few and far between for the foreseeable future. Mathematicians are not going to fire up a proof generator or proof checker when they start work each day. That’s not how the mathematical community works.

The best we can do is meet for beer on a regular basis. After all, we both share the predicament that no one else really understand what we do.

]]>What happens in practice is that once several experts in the domain have examined the argument and declared themselves to be satisfied it is correct, then the rest of the mathematical community accepts that fact and moves on. The more people look at it, the more confident we get.

However, there have been times when this process was eventually found to have gone wrong. Arguments that have been accepted for some time are subsequently found to have a fatal flaw. That’s why mathematicians are reluctant to ever say something is totally correct.

There is nothing wrong with this situation. Mathematicians don’t lose sleep worrying if “proved” really means “true”. We proceed as if it does. We actually do stake our livelihood on the fact (actually, it’s an assumption!) that all the theorems in the standard undergraduate canon are correct.

There’s a spectrum here. No mathematician would question Euclid’s theorem that there are infinitely many primes. But some very good mathematicians do still question whether the Poincare Conjecture is true. That may change over time. (The second one, not the first.)

For sure, our knowledge of mathematical “truth” is more accurate and reliable than the “truth” in any other parts of our lives. But if you live by the sword of precision, you should be prepared to die by it. As mathematicians, we have to admit we can never be totally sure a proof is correct.

]]>At this point, it can be of some comfort to learn that Euclid screwed up big-time when he penned his famous geometry proofs in Elements..

I think this overstates the case. Indeed, the Elements certainly passes the standard of rigor modern logicians require.

See Avigad, Dean and Mumma’s paper “A Formal System for Euclid’s Elements” (in the Dec 2009 issue of the Review of Symbolic Logic). In this paper, they give a formal calculus (named E) for Euclidean reasoning, and show that (a) it is sound and complete for ruler-and-compass constructions, and (b) nearly everything in the Elements can be directly transcribed into their calculus.

The only exceptions they found are that Euclid is sometimes cavalier about considering all the cases in a case analysis (a fact already noted in the fifth century by Proclus), and that proposition I.9 needs a few more words of justification for its construction than is given in the Elements.

]]>