I'm not the only one who gets mail from crackpots!

A kind reader forwarded me *yet another* bit of Cantor crackpottery. It never ceases to amaze me how many people virulently object to Cantor, and how many of them just spew out the same, exact, rubbish, somehow thinking that they're different than all the others who made the same argument.

This one is yet another in the representation scheme. That is, it's an argument that I can write out all of the real numbers whose decimal forms have one digit after the decimal point; then all of the reals with two digits; then all of them with 3 digits; etc. This will produce an enumeration, therefore, there's a one-to-one mapping from the naturals to the reals. Presto, Cantor goes out the window.

Or not.

As usual, the crank starts off with a bit of pomposity:

Dear Colleague,

My mathematic researshes lead me to resolve the continuum theory of Cantor, subject of controversy since a long time.

This mail is made to inform the mathematical community from this work, and share the conclusions.

You will find in attachment extracts from my book "Théorie critique fondamentale des ensembles de Cantor",

Inviting you to contact me,

Francis Collot,

Member of the American mathematical society

Membre de la société mathématique de France

Member of the Bulletin of symbolic logic

Director of éditions européennes

As a quick aside, I love how he signs he email "Member of the AMS", as if that were something meaningful. The AMS is a great organization - but *anyone* can be a member. All you need to do is fill out a form, and write them a check. It's *not* something that anyone sane or reasonable brags about, because *it doesn't mean anything*.

Anyway, let's move on. Here's the *entirety* of his proof. I've reproduced the formatting as well as I could; the original document sent to me was a PDF, so the tables don't cut-and-paste.

The well-order on the set of real numbers result from this remark that it is possible to build, after the comma, a set where each subset has the same number of ordered elements (as is ordered the subset 2 : 10 …13 … 99).

Each successive integer is able to be followed after the comma (in french the real numbers have one comma after the integer) by an increasing number of figures.

0,0 0,10 0,100 0,1 0,11 0,101 0,2 0,12 0,102 … … … 0,9 0,99 0,999 It is the same thing for each successive interger before the comma.

1 2 3

So it is the 2 infinite of real number.

For this we use the binary notation.

But Cantor and his disciples never obtained this simple result.

After that, the theory displays that the infinity is the asymptote of the two branches of the hyperbole thanks to an introduction of trigonometry notions.

The successive numbers which are on cotg (as 1/2, 1/3, 1/4, 1/5) never attain 0 because it would be necessary to write instead (1/2, 1/3, 1/4, 1/4 ).

The 0 of the cotg is also the origin of the asymptote, that is to say infinite.

The beginning is, pretty much, a typical example of the representational crankery. It's roughly a restatement of, for example, John Gabriel and his decimal trees. The problem with it is simple: this kind of enumeration will enumerate all of the real numbers with *finite length* representations. Which means that the *total* set of values enumerated by this won't even include all of the rational numbers, much less all of the real numbers.

(As an interesting aside: you can see a beautiful example of what Mr. Collot missed by looking at Conway's introduction to the surreal numbers, On Numbers and Games, which I wrote about here. He specifically deals with this problem in terms of "birthdays" and the requirement to include numbers who have an infinite birthday, and thus an infinite representation in the surreal numbers.)

After the enumeration stuff, he really goes off the rails. I have *no idea* what that asymptote nonsense is supposed to mean. I think part of the problem is that mr. Collot isn't very good at english, but the larger part of it is that he's an incoherent crackpot.

What was the last "original" bit of Cantor crackpottery you've seen? The last fundamentally new "proof", the last original representation or enumeration?

I also got this mail, and I am wondering where Mr. Collot got my address. I was also amused by the fact that he listed himself as a "member of the Bulletin of symbolic logic"... He might receive the Bulletin of Symbolic Logic through the Association for Symbolic Logic (a membership of which also tells you nothing), but it's kind of hard for a human to be a member of paper bound together.

I'm more interested in trying to figure out what that last paragraph is supposed to convey. I'll have a go at the beginning:

"It is the same thing for each successive interger before the comma.

1 2 3

So it is the 2 infinite of real number."

I believe he's saying that since all real numbers between 0 and 1 are countable (Z), and the same applies for all real numbers between n and n+1 for each n, the cardinality of the real numbers is Z x Z. Or in less sane terms "2 infinite of real number."

I got this email as well! I wonder what is the meaning of "Member of the Bulletin of symbolic logic"..

"the well-order on the set of real numbers"...

What well order? He says something wrong in the very first line!

Something told me Collot explained his idea much better than this, and indeed: Google informs us that

Collot wrote several books on this issue,

all self-published,

one of which is in the French National Library;

Franck Varenne's doctoral thesis in history mentions that Collot told him he is a surgeon (b. 1924) with a life-long interest in the mathematical description of human bone development and similar processes;

Google also finds one of Collot's articles on - what is if you ask me - the same subject.

Its abstract says the article proposes a way to well-order the powerset of natural numbers by mapping it to the real numbers; it proceeds to explain how such a mapping is constructed. I haven't read any further, so I don't know if any nonsense is claimed in the article, but at the least a little more care has gone into Collot's efforts than the above would suggest.

When criticizing someone for sloppy thinking, it's extremely sloppy to base your criticism on what is obviously an unrecognizably corrupted summary of that thinking.

Besides (regardless of the quality of Collot's work), I don't think these displays of contempt and ridicule do you or mathematics a service. It's fascinating to read about faulty mathematical reasoning; why don't you leave it at that?

What's going on in the columns of that table? Is that really an accurate transcription? If so, the ellipses are hiding a lot. IN column 1, there would seem to be ten rows, in column 2 90 rows, and in column 3 900 rows.

Oh, and where's .09?

I don't know what this is even about. I have no idea who Cantor is. But science blogs that call people "crackpots" really bother me. How about remembering we're all human beings with our own unique feelings and values. And just because someone disagrees with you or believes in something you don't, no matter why, doesn't mean they're a "crackpot". It just means the human race is a complicated and confusing tapestry, and life here sure would be easier if we weren't so negative to each other. You can easily state your disagreement without insulting other people's position or devolving into name calling. That's what my Christian values teach me anyway.

Hi Mark -

In thinking about this, I've come across an interesting construction of real numbers that at first seems to contradict Cantor. But then I found a way out of it... but the way out is pretty mind-blowing IMO. If I write it up in a blog post would you review it?

I think it's more interesting than those silly decimal sequences.

Max

@123, While I would normally agree, mathematics doesn't work like that. In math your argument is either right or wrong: starting from a few basic assumptions, you see what you can logically deduce. Given that he's talking about the real numbers (i.e. the number line), we know roughly what set of assumptions he's made (the 'standard ones' I guess) since they're all very far in the background of that theory. Given this, we can state with literally complete certainty that he is wrong. I'm sure Mr. Collot has his own unique feelings and values, and that's great, but it's completely irrelevant to the correctness of his mathematics. If he believes in his result, he is wrong. His proof makes a basic mistake, common among those who don't really understand Cantor's ideas, that was discussed above. There is literally no argument to the contrary.

And there are crackpots in science. It's not a random insult, it's a diagnosis of behaviour. Scientists hold themselves to the scientific method, almost by definition. Crackpots are those who don't, but claim to be scientists anyway. Take Paul Cameron for example: his work has been shown to have horribly and basically flawed reasoning behind it numerous times, but he doesn't respond to criticism with anything other than dismissal. This is *very* different from, say, the particle physicists at OPERA, who mad the recent mistake of finding "faster than light neutrinos". Those who make mistakes, but are very careful and willing to concede when they're wrong, are not crackpots.

I think there's a worthy difference between devolving into name calling, and pointing out that he made no sense whatsoever. Look at the wikipedia page for Crank (person). Mr. Collot fits it pretty well, considering how he's clearly been pushing his work for a while. Keep in mind that no one here would say the same thing about a *student* who made this mistake, because the student acknowledges they are learning and that they could (and probably would) be wrong, while the crackpot just ignores dissenters and keeps assuming their understanding is flawless.

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There are quite a few topics on Cantor at this link:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-556.html#post27247

Also two interesting comments on countable sets and Cantor's Diagonal Argument:

http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-505.html#post26575

Anyone seen these?