22. Mathematicians on Wikipedia distort and exaggerate Wiles' proof of Fermat's Last Theorem by (i) concealing how it relied on the controversial Axiom of Choice and by (ii) omitting the widespread initial criticism of it.

I am not making this up.

Now, I don't know Wiles' proof anywhere near well enough to comment on its use of the axiom of choice. But the

*placement*seems to imply that the people behind Conservapedia think the proof of FLT is somehow "liberal" mathematics. Why they single out FLT is unclear. Apparently conservative mathematics mentions God a lot. (The previous link goes to descriptions of math courses at a Christian high school, which I previously mentioned back in August.)

By the way, the "God" of my blog's title is

*not*the God of those course descriptions. My blog's title is an answer to Einstein's famous aphorism "I shall never believe that God plays dice with the universe" -- and the Einsteinian God has basically nothing in common with the Judeo-Christian God except name.

## 29 comments:

That's... totally odd. But that's Conservapedia.

According to a recent article in Toronto Star,

The New York Times reported recently that mathematicians believe in God at a rate 2 1/2 times that of biologists, quoting a survey of the National Academy of Sciences. Admittedly, that's not saying much: Only 14.6 per cent of mathematicians embraced the God hypothesis, versus 5.5 per cent of biologists...I find this statistic rather telling, reflecting a strong ideological flavor of mathematics and mental rigidity of many of its practitioners, making them an easier target for religious indoctrination. Well, maybe familiarity breeds contempt...

Is it possible that that item was added by an anti-Conservapedia vandal / parodist?

As far as I know, the Axiom of Choice has been proven independent of ZF. That means, if you can prove that x^n+y^n=z^n don't have integer solutions for n>2, using Axiom of Choice, there cannot exist any solution to that equation (if there was, AC wouldn't be independent of ZF). Thus, the truth of Fermats Last Theorem cannot "rely" on AC (but AC can be used in the proof).

Mark,

that's a good point. But it seems that that particular change appeared in the article for the first time (23:11, 20 December 2007) thanks to the editing of Aschlafly, who is apparently the founder of the site. (That's right, I'm using Wikipedia as a source for info on Conservapedia.)

I'd delete the FLT reference at Conservapedia, except:

1. they don't allow anonymous edits, and that's really not a site I want to have an account at;

2. they don't allow editing between 1 AM and 6 AM (US Eastern time), and I'm not going to care about this in the morning;

3. the article is arguably better

withthe FLT reference in it, because it makes it more obviously stupid.With regard to misha's comment: I wonder if part of this statistic is attributable to set theorists. I'm thinking of Kronnecker's comment about the integers; the search for large cardinals to infallibly resolve those pesky independent axioms in the "real" (Platonic) universe; and the fact that the question of continuum hypothesis always seemed to me, in a fairly precise mathematical sense, to be equivalent the medieval question of "How many angels can dance on the head of pin?" 8^)

Barry, I'm curious, how do you take Kronecker's comment?

I always, until about two months ago, had taken it to be nothing more than wit. But recently I've gotten into a couple of discussions with people who thought it was an indication of bonafide theism.

Andy Schlafly's "understanding" of math is, basically, one long story of delusion. This was pretty apparent from the get-go; see Mark Chu-Carroll's analysis in February 2007.

Oh, and by the way, the project name is properly spelled

Conservapædia,with an æ ligature (HTML æ). Well, for certain values of "proper".Misha: as a formerly liberal christian, I ultimately abandoned my faith because it couldn't stand up to the logical standards that I was confronted with in my real analysis courses. In effect, I converted to mathematics, like some pagan overawed with a more impressive doctrine.

Ironically though, this skeptical turn of my thoughts circled back on itself, and I went from being a born-again mathematical fundamentalist to something more of a mathematical existentialist...

I always took Kronnecker's comment to fairly straightforward, given his time. He believed in God. He believed that the integers actually existed in an ideal Platonic realm. As for all the rest being the work of man, I consider the concept of God to be the work of man also.

In fact, it does not really matter whether the integers "exist" in this sense, we can behave as if they do. The number 3 is a convenient fiction.

I have not investigated the question of Kronnecker's beliefs in primary sources. I presume that the percentage of 19th century German mathematicians that believed in God was much higher than that in the NY Time poll.

Johann Goodflag and Barry, doesn't the proof of Bolzano-Weierstrass lemma by bisection strike you as cheating, because in practice we can't tell which of the subintervals contains an infinite part of our sequence, which is a huge hole in the proof that everybody keeps quiet about? This makes the traditional notion of compactness look like wishful thinking.

"...reflecting a strong ideological flavor of mathematics and mental rigidity of many of its practitioners, making them an easier target for religious indoctrination. "Misha- That's one way to take it. You could also say that mathematicians are trained to think very carefully about things they cannot touch, taste, or see; and are prone to believe these things are real even though they do not correspond directly to physical analogues.

So instead of saying that mathematicians have some sort of intellectual weakness that makes them "susceptible to indoctrination", you might also say that mathematicians are more capable than others of apprehending a truth that exists in the spiritual world.

By the way, I grew up just down the street from Castle Hills First Baptist. Passed it every day on my way to high school.

Place gives me the heebiejeebies. No lie. I knew students there socially.

Misha,

Why's that cheating? I mean, it's just like saying if I have two numbers, one of them is bigger. We know which subinterval we're dealing with: it's the one with infinitely many terms!

Misha,

the fact that the proof of Bolzano-Weierstrass is not constructive does not strike me as cheating, it strikes me as classical. If you prefer constructive mathematics, go for it. It is the best way to understand many problem domains. Any discussion of which flavor of mathematical reasoning is

right, are too philosophical to amuse me any more.When I studied modal logics in the early 1970s, much of the literature was written by philosophers. They tended to write papers arguing why their particular set of axioms yielded the one

notion of strict implication or possibility or necessity or whatever. The mathematicians just categorized all the possible variants, how to model their semantics, and where the same patterns occurred in other parts of mathematics. From my perspective, this was a much more interesting way to look at things.trueBecause to decide which interval to take you have to examine an infinte number of the members of your sequence, and it's not a feasible task. Remember, the lemma says that ANY bounded sequence has a convergent subsequence. It doesn't hold water in such a general formulation. See chapter 13 of Foundations of Mathematical Analysis by J.K. Truss. It is featured on google books. Classical analysis is full of holes like this.

It's not a philosophical problem at all, because you have to deal with it when you want to apply your mathematical theorizing to some practical calculations.

I donno, I mean, it's not constructive, but at each step there's only two choices. So you've got a countable number of binary choices. Not so bad, in the scheme of non-constructive proofs. Objecting to that is a stronger form of constructivism than, say, objecting to the use of the excluded middle to conclude existence.

But to make a choise you have to examine an infinte number of the members of your sequence! The process of making even just one of these binary choices is potentially infinite!

I want to make clear that I'm not entirely against the parts of classical analysis that I would call mythological, such as Bolzano-Weierstrass. They help us understand some practical matters, the same way that religious tales and fables give us some ethical guidance and help people cope with the brutishness of their daily existence. I'm just urging everybody to be aware and earnest about the nature of such theorems and not to take these mathematical myths too literally.

Now I hope the parallel I am trying to draw between (at least some parts of) mathematics and religion is becoming clear.

Misha, you actually replied to yourself with most of the reply I would have posted to an earlier comment. Mathematics can be viewed as a set of convenient fictions, mythologies, or religion. We form abstractions to model the natural world. The abstractions become quite complex, so that much of mathematics consists of simplifying, generalizing, and providing more useful generalizations. The danger lies in reifying our abstractions, thinking that these abstractions that we use to model the real world are the

realreal world.Plato succumbs to this fallacy. His story about the shadows in the cave is the classic description of this fallacy. To my mind, has the

realandidealworlds reversed in his metaphor. Spherical balls are our abstractions of certain properties of apples, billiard balls, and planets. They are shadows of real world objects we project on the wall with the magic lantern of our intellect. The real apples, billiard balls, and planets do not want to become "ideal" spherical balls.There a number of problems with classical analysis and set theory. I would cite a very fundamental one as being the entire concept of discrete static objects. It is a convenient fiction rooted deeply in our animal ancestry. A primitive predator swimming in an ancient sea derives considerable advantage from regarding his prey as a discrete object he can track, chase, and calculate related rates problems to catch his prey. However, in reality, once caught, the distinction between the predator and the prey swiftly vanishes. Some of the prey becomes the predator, some becomes parts of scavengers, other parts go back to other parts of the sea.

The fiction is so convenient and so deeply rooted in our evolution, it is no wonder Kronnecker thought God created the integers.

I don't know whether or not Kronnecker believed in God in any traditional sense. My strong suspicion is that he used God mostly metaphorically, like Einstein, who always found the opinion that he was deeply religious in any traditional sense (based on his references to God) to be totally ridiculous.

Actually I wasn't talking about "reality" of mathematical objects at all. I was talking about applicability of Bolano-Weierstrass and other shaky statements to practical calculations. On the surface of it, complactness guarantees the attainment of the absolute minimum of a continuous function, but in practice it doesn't help us much in finding where this minimum is actually attained. So the existence guaranteed by compactness is a shadowy kind of existence. Kronnecker, on the other hand, insisted that everything that we are dealing with in mathematics should be explicitly constructed by finite algorithms, except the integers, that he took for granted, and that is the true meaning of his aphorism.

Maybe it's just xenophobia that makes him dislike Wiles's proof. A Britt* proving a theorem conjectured by a Frenchman? That has to be wrong.

*Remember that he once thought allowing brittish spelling was an example of liberal bias.

Johan

Not 'once'. He still thinks that.

And building upon the Japanese ideas!

an æ ligature (HTML æ)Of course, if you're on a Mac you can just type

option-': æ

:D

Did you notice that the Conservapedia article about FLT has got all the dates of Wiles work wrong?

Even if Wiles's proof uses the Axiom of Choice, there must also then be a proof of FLT in ZF without AC-- due to the fact that FLT is an arithmetical sentence (i.e. quantifiers range over positive integers). The relevant notion from set theory is called "absoluteness" --roughly, for any finite set B of axioms of ZFC, ZF proves that B holds in the constructible universe L, and so by taking B to be the set of ZFC axioms used in the proof of FLT, we get that ZF proves that FLT holds in L. But ZF also proves that [FLT holds in L IFF FLT holds in the universe] (since FLT is arithmetical). So ZF proves FLT.

"I find this statistic rather telling, reflecting a strong ideological flavor of mathematics and mental rigidity of many of its practitioners, making them an easier target for religious indoctrination."

i love math, politically moderate, and not so religious... i dont know if math-types are easier "targets", but, at least for the pure mathematicians (i'm more finance-applied), there is a sort of sentiment and romanticism in the work that mathematicians do

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