Every discipline has its crackpots: Stories of mathematics

I'm sure every discipline has its share of crackpots. I suspect the physicists get it the worst, starting with the old standbys of perpetual motion machines and faster-than-light travel, then tossing in quantum mechanics and nuclear physics, and stirring with large quantities of long rambling text that makes no sense. But my experience is with mathematics.

When I was in college, the mathematics department would post "interesting letters" onto the bulletin board. The ones I remember:

• A claimed proof of Fermat's Last Theorem by means of music theory. It started with 3² + 4² = 5², then somehow converted this into musical notation, transposed it into another key, and then concluded that the proof was complete. There were many "proofs" of Fermat's Last Theorem on the bulletin board.
• A letter from an inmate at a correctional facility who had developed a system for winning the lottery and merely needed a printout of all possible ways of choosing 6 numbers from a pool of 46.

A number theorist working on factorization explained to me that he would frequently receive "manuscripts" from people who claim to have found a high-speed algorithm for factoring large numbers. At first, he would take the time to study these manuscripts, and each time he would determine that the algorithm boiled down to trial division, often cleverly-disguised trial division, but trial division nevertheless. (Though when this was pointed out, the authors often rejected his analysis.) Eventually, he realized that he could separate the wheat from the chaff very easily by simply replying with the following message:

Thank you for your fascinating manuscript on the factorization of large numbers. There are some numbers that have been giving me difficulty of late. I would be most appreciative if you could use your technique to factor this one for me.

Upon which he would include a number whose factorization would take years on generally-available computational hardware at the current state of understanding. He never heard back from these people.

Another of my professors told a story of one "correspondent" who was convinced that the speed of light could be overcome. (Yes, that's actually a physics question, not a mathematics question, but it was sent to the math department anyway.) The professor started by trying to explain the principles of special relativity to his new pen-pal but quickly realized that wasn't going to lead anywhere. The correspondence was quite pleasant; the other person was a retired gentleman who gardened and enjoyed going for walks when he wasn't working on pushing the envelope of modern physics. At one point, the correspondent wrote back a multi-page letter consisting of crayon drawings that proved that the speed of light could be exceeded. It went something like this:

The first page consisted of a drawing of the earth with a little rocket ship in orbit around it.

Consider a rocket ship that circles the earth once a day. This rocket is travelling at 463.831019 meters per second.

Say what you will, but these people never suffer from the problem of too few significant digits.

Now imagine that each day, the time it takes to circle the earth SHRINKS IN HALF.

So that on the SECOND DAY it takes only twelve hours to circle the earth at a speed of 927.662037 METERS PER SECOND.

The rocket ship on the second day has a few extra zoom-lines on it.

By the THIRD DAY it takes only SIX HOURS to circle the earth at a speed of 1855.32407 METERS PER SECOND.

The rocket ship on the third day is going a little faster.

Each page consisted of a daily status report on our little rocket ship, illustrated in glorious crayon, each drawing more elaborate than the last. As the rocket ship goes faster and faster, the report on its speed gets bigger and bigger. I'll skip ahead a bit.

 On DAY TWENTY the rocket ship circles the earth in 164.794922 MILLISECONDS at a speed of 243,181,037 METERS PER SECOND.

Finally, the hammer falls:

 On DAY TWENTY-ONE the rocket ship circles the earth in only 82.3974609 MILLISECONDS at a speed of 486,362,075 METERS PER SECOND. IT HAS BROKEN THE LIGHT BARRIER.

The professor realized the jig was up. He wrote back, "Yes, it looks like you've done it."

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1. muro says:

I think the explanation is quite simple – the more you know, the more you exact and thorough you want to be, so you’re not embarrassed later.

If you know nothing about the problem, the more fantastic can be your crackpot solutions.

2. BryanK says:

All possible ways of choosing 6 numbers from 46 numbers (without replacement)…  hmm…

Well, if order matters, then there are "only" 6744109680 (about 6.7 billion — go Lisp bignums; I can figure out 46-factorial if needed!) ways of doing that.  If order does not matter, then there are "only" 9366819 (~9.4 million).  I’m not sure exactly how lotteries work, but I’d guess that order does matter.

So how much paper can 6 billion choices of 6 numbers from 46 possibly take up?  ;-)

3. David Candy says:

6 from 46 sounds like a lotto thing not a lottery. If so then order is not important.

I suggest you stick to horses. You WILL win sometimes in your lifetime unlike lotto/lotteries which you can expect not to win at all.

My first real program I wrote was a horse rating program for 4yo and up and a betting program.

My understanding of cracking lotto is that it relies on small differences in the balls making it slightly more likely that some balls will fall more often than others. Some people run stats on results and believe they find patterns. They may be right to an insignificant degree but you still won’t win in a lifetime.

Horses are far more sensible. A tote coralates very well. EG horses assessed by the tote at 2/1 (33.3%) do in fact win a third of the time. To win with horses one inflates the margin. To take the 2/1 chance (that your rating has assessed) one only backs it at 3/1 or higher (I used to inflate the pool for the ratings by 20% giving me a 20% profit margin). The downside is there are plenty of times when one must just do nothing which is boring.

How one calculates ratings is relativly unimportant. While one needs more than one significant factor to make a rating, research shows including too many factors (and too many isn’t that many) reduces the accuracy.

Sceptics need to note ratings cannot predict anything. That merely show the chance of a horse winning. It also doesn’t matter if they are accurate or not for an individual horse as many races cancels errors.

The last point is one needs to inflate the horses price to account for taxation.

A sample formula may be

PrizeMoney/NumberOfStarts

While the above is only a measure of class not current form or handicapping (another form of rating) it was the heart of my formula (I factored in current form and weight as well).

Betting truely randomly will mean you will lose approx 20% to taxation/profit margins.

4. Csaboka says:

BryanK: I don’t know about other countries, but in the Hungarian lottery, the order doesn’t matter (our version needs choosing 5 numbers from 90).

About the original post: This gave me a fantastic idea that would solve all problems with slow computers. Imagine a CPU running at 1MHz initially. Now, imagine it doubles its clock speed every second. Eventually, it will be faster than any other computer. Do I get a Nobel prize or what? :P

5. JamesW says:

Yes, Physics departments get crank letters. The head of department at the University of Glasgow told us about them during one lecture. As head of department all correspondence ended up with him. It fell into certain categories: bats–t insane; genuine queries and proof that although flawed were of some interest.

He gave an example of the latter category. It came from a carpet factory: they had a formula for calculating the length of a roll carpet. I don’t remember the details but the formula was of the same form as one of the equations of motion – quite possible ‘s = ut + 1/2at^2’. The carpet workers had unrolled a roll of carpet, measured it and found the results disagreed slightly with theory. They extrapolated that the equation of motion was incorrect too and wrote to the University. The answer was that the formula was approximate in the case of the carpet as it didn’t take the thickness of carpet into account. Still the professor applauded them on their curiosity and testing the formula against experiment.

He said with the insane letters he couldn’t win. If he replied pointing out flaws it would just generate more nonsense by return post. Ignoring it completely resulted in accusatory letters claiming that the professor had stolen their theories and would be claiming the credit for their scientific breakthroughs.

BTW you are too stupid to understand timecube!

http://www.timecube.com

6. David R says:

Please allow me to suggest the crackpot index, at

http://math.ucr.edu/home/baez/crackpot.html for a more quantitative method of evaluating these sorts of proofs.

7. A Finn from Finland (nem beszélek magyarul) says:

Csaboka’s idea of a computer that can double its speed won’t win him the Nobel Prize for two reasons. First of all, there’s no Nobel Prize in Computer Science. Secondly, the idea is known to have very serious ramifications – something that is analyzed in this hilarious 1995 post to  sci.math:

8. Martin says:

At Cambridge (uk) we got more than our fair share of these. The solution was to reply "I’m not qualified to judge your interesting work, you should contact <insert name of colleague/rival>" people would play "crackpot tag" for years!

9. Roger says:

Crackpots in the field of Mathematics is particular problematic, because you’d have to read it through line by line to refute the proof. And then of course they’ll come out with another refutation of their own.

There are still people who are trying to proof Pi=3, or you can square the circle…etc.

In computer science, you can just ask them to write the program or make a computation and that’s the end of that.

10. Anon says:

What I find really sad is when you get gibberish from people who seem to have been reasonably intelligent once.

E.g. some of the people on usenet. They dropped out of university, and you can see that they’ve lost the plot now. But amongst all the paranoia and incoherence there are signs that they must have known quite a lot of physics at some point.

It’s painful to read their stuff really.

11. Eric TF Bat says:

One of the first programs I ever wrote from scratch (in Commodore PET Basic 4.0 on a 32Kb CBM-8032 in about 1984) was a lotto simulator.  I fed in my Mother’s standard six lotto numbers and then had it generate a 40-cell grid with a random draw, highlighting the number of matches with Mum’s numbers.  It also told you what week it was, so you could keep track.  Then I held down the spacebar and let it keep regenerating over and over.  Eventually, after watching for a while, it got as high as three (or maybe four) matches out of the six, but no more in the time I was experimenting, and the number of elapsed weeks was disturbing.

My conclusion has remained with me ever since: given that there’s a vague chance that someone might die and leave you a lotto ticket in their will, or you might happen to find one lying on the ground: your chance of winning your money back on lotto is not significantly affected by whether you play or not.

Lotteries are a tax on people who don’t understand mathematics.

12. Manip says:

Your calling people that even flirt with the idea of faster-than-light travel crackpots? …

Wow… By that definition everyone who has ever questioned popular science is also a crackpot and no doubt you would have been one of the people that called Einstein a crackpot in his lifetime.

The great thing about science, even today, that our understanding isn’t completely solid — there are holes. Which likely means that a few of our widely held assumptions are incorrect, particularly the ones that can’t be proven.

If nobody questioned popular science then we could be walking into a new dark age in science… Ironically it is these ‘crackpots’ that make the most significant discoveries… Not the people that stick to the status quo.

13. Norman Diamond says:

The computer industry has just as many crackpots.

For example there are postings in public newsgroups (actual public ones, not specially hosted by any particular company or server).  Wise people who knew that BSODs couldn’t be generated by the drivers that the BSODs were telling me they were generated by.  There were usually two simple obvious reasons:  the drivers were signed and certified, and the drivers hadn’t crashed on the wise people’s machines because the drivers hadn’t been installed on the wise people’s machines.  So they knew I was the crackpot, looking at BSODs which couldn’t possibly exist.

Joe Newcomer has more interesting tales on his site.  One was a programmer who knew that the syntax of arguments to printf() had to obey rules of English not rules of C.  When her program didn’t compile, obviously the compiler was defective.  If I recall correctly, Dr. Newcomer was particularly affected by this one because the programmer had graduated from his university.

14. Anonymous Coward says:

A crackpot doesn’t just question the idea that the speed of light is an absolute limit. Instead they patent the FTL drive they built in their garage, in the hope that someone else will invent the fuel required before their patent expires. Strangely, the fuel never seems to get invented, at least partly because it’s usually some form of anti-gravity perpetual motion atom, and there seem to be issues with finding this stuff no matter how many noble crackpots question physics.

My favourite software crackpots are the ones who invent compression algorithms that can compress absolutely any file, including every compressed file that their algorithm creates. (i.e. every possible file can be successively compressed to a size of exactly one byte.) It’s always funny when you ask them to prove it and they get around to writing the decompression application only to discover that they had invented lossy data deletion instead of compression. Some of them never get it, though.

People who invent trivially crackable encryption aren’t crackpots though, they’re just sad.

15. Vince P says:

This reminds me of Monty Python:

"How to Rid the World of All Known Diseases":

Alan (John Cleese): Well, last week we showed you how to become a gynecologist. And this week on "How to Do It" we’re going to show you how to play the flute, how to split an atom, how to construct a box girder bridge, how to irrigate the Sahara Desert and make vast new areas of land cultivatable, but first, here’s Jackie to tell you all how to rid the world of all known diseases.

Jackie (Eric Idle): Hello, Alan.

Alan: Hello, Jackie.

Jackie: Well, first of all become a doctor and discover a marvelous cure for something, and then, when the medical profession really starts to take notice of you, you can jolly well tell them what to do and make sure they get everything right so there’ll never be any diseases ever again.

Alan: Thanks, Jackie. Great idea.

16. BryanK says:

OK, so there are still just under ten million different combinations of lottery numbers; how much paper would that take?  ;-P

Manip — they’re not crackpots because they question the constancy of c in all reference frames (since that is, IIRC, what makes it impossible for anything with mass to ever have a velocity equal to c).  Rather, they’re crackpots because of their methods of reaching or exceeding c:  "imagine that a rocket doubles its speed every day" is not a valid way to *actually* *go* the speed of light or faster.

17. Simon Witkiss says:

At my Maths department, ‘crackpots’ were dealt with by forwarding their "proof" on to the person before them who’d sent in a ‘crackie’ (in-house slang for a letter from a crackpot).  In that way this special community would form their own bonds and relationships!!

18. markp says:

I would hope that everyone is familiar with the the experience of being so embarrassingly wrong about something. I’m an intelligent person, but that doesn’t mean that my reasoning is always sound. Eg here are two embarrassing things I have wasted hours of my life on, thinking I might have stumbled on to some amazing secret:

1) Superfast way to generate prime numbers (wrong of course, but it kept me awake for hours trying to correct for all the ways it failed)

2) A way to win at a 50/50 game of chance (using a more subtle variant of the Martingale system, this one is at least fun to test)

I think what makes people crackpots is not their dumb ideas, rather it’s their unwillingness to accept new information and leave those dumb ideas behind. Maybe if they realized that even the most successful people make embarrassing mistakes it would be easier for them?

19. Chris L says:

"They laughed at Columbus, they laughed at Galileo, but they also laughed at Bozo the Clown."

20. On the "contact name of rival" approach, I’m sure I read something about that w.r.t. Fermat. Someone who received lots of Fermat "proofs" would write back with "I am not competent to judge your proof. Please contact another pre-eminent expert, N", where N was the *last* chap to send in a flawed proof of Fermat…

21. Alun Jones says:

My all-time favourite crackpot is Eugene Terrell.  He frequently submits Internet Drafts (the documents that RFCs start from), and it’s sad that the IETF doesn’t keep all old versions of these around, because he’s got some great ideas on how we can improve the Internet.

My favourite was "we can increase the address space by understanding that binary mathematics as expressed in the IPv4 addressing standards is incorrect".  The first draft essentially stated that you could get more address space by writing the octets in decimal, rather than binary.

22. Gerard says:

The book A Budget of Trisections (http://www.amazon.com/gp/product/3540965688/sr=8-2/qid=1149017743/ref=sr_1_2/002-3652404-6380862?%5Fencoding=UTF8)

is full of examples of people who thought they had  found a way to trisect an angle using only ruler and compass – the professor who wrote it collected examples of these proofs for many years, and even visited and talked to many of the so-called "cranks".  This is one of the famous Greek classical mathematics problems – it was shown to be impossible in 1837.  The book is fascinating and talks a lot about the mentality of these sort of people – the major conclusion, if I recall, being that most of them did not really understand the nature of mathematical proof.  The other interesting thing I remember is that many of the people thought that if they could find a way to trisect the angle that it would be of great use to science and technology – of course it wouldn’t be of any use, even the Greeks knew how to trisect an angle as long as you were allowed to mark the ruler!  So in practice it is simple to do, the classical problem being only a theoretical issue of finding a "non-mechanical" method.  More info here:

http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trisecting_an_angle.html

Unfortunately it seems this book is out of print, and the used copies on Amazon are quite expensive – I had found it in my university library years ago.

23. Eugene Terrell’s IP4 RFC draft:
http://tinyurl.com/qgqax

[Count how many times he says, “Needless to say”. -Raymond]
24. BryanK says:

Not just "Needless to say" — count how many times he Capitalizes Things He Apparently Thinks Should Be Proper Names, when in fact they are nothing of the sort.  (That *really* makes it hard to read…)

(Can anyone explain what, exactly, that guy is trying to propose there?  Is he just counting the same address space multiple times with different subnet masks?  I read a few paragraphs, skipped to the end, and then gave up, but I bet someone here has more stupidity tolerance than I do.)

25. David Walker says:

Wow.  (Eugene Terrell, that is.)  Along with many grammatical no-nos, he likes to say "underlining" when he means "underlying".  As in "the underlining fact".

"Notwithstanding the

hurried imposition of market demands, the crying voice of the consumer,

and the shortage of the insightful."

Yes, a shortage of the insightful, indeed.

David

26. James says:

This reminds me of a discussion I had at Cambridge (UK) a few years ago about primes. One of us said something about "an algorithm for factoring large primes in linear time", to which another replied "actually, I can do it in constant time". (Which is completely true, and equally pointless.)

On a related note, MarkP: I have an algorithm which does indeed generate primes extremely quickly! (The downside? They’re mixed in with non-primes, but all you need to do is filter them out…)

27. Poofias says:

The funniest post I’ve ever read :-)

28. David Walker says:

Lotteries:  I have read about a scheme that points out that the sequence 1,2,3,4,5,6 is vary rare (which is true) so it throws out sequences where there are several consecutive numbers.

The sets of numbers you’re left with after throing out certain unlikely sets, are supposed to be the winners to bet on.

Dumb.

29. If your goal is just to win the lottery, then no scheme will help (given assumptions about the randomness of the winning numbers.)

But if your goal is not merely to win the lottery, but in addition you want to maximize the chance that you are the ONLY winner (so you don’t have to split the prize with anyone) then there are all kinds of sensible tactics.

For example, many people use birthdays… so avoiding numbers 31 or lower can help.

And some people tend to favor prime numbers… so avoid those;

And some people will avoid numbers ending in 0… so favor those;

etc., etc.