The Crackpot Offer

"So I found this counterexample, and saw that my attempted disproof was false, along with my dreams of fame and glory."

I know how that feels. When I was 14 or so, I took a course on cryptography, and the textbook proclaimed that modular inverses were the basis of public-key algorithms like RSA. I felt that modular inverses were crackable, and I plodded along on the problem for a few weeks, until I finally discovered a polynomial-time algorithms for doing modular inverses. It turned out that I had reinvented Euclid's algorithm, and the textbook authors were idiots.

Not to draw attention away from your main argument, but how does 1101 map onto {0, 2, 3}? It's probably obvious, but I don't see it.

Phil: Build the set from the used exponents of the powers of two. For instance, 1101[2] = 20 + 22 + 2**3

So I found this counterexample, and saw that my attempted disproof was false, along with my dreams of fame and glory.

Feels familiar - when I was younger, I proved the Poincaré conjecture, and Fermat's last theorem (twice). I generally managed to slay my proofs by myself, though I felt not regret at being wrong, just frustration and anger at myself.

Even now, as a mathematical researcher, it's very hard to give up a nice result that can't be proved. But I manage. And I do feel that there is a silver lining: greater, more confident accuracy.

After the fact you could see you made a mistake. But the key question is: what were the clearest signals at the time, the sort of signals you had a chance to notice and recognize? What is the warning to others? Presumably it is not to give up after your first failure.

But the key question is: what were the clearest signals at the time, the sort of signals you had a chance to notice and recognize?

In my case, it was the fact that brilliant mathematicians had tried to prove these results for generations. No matter how brilliant I think myself, it would be unlikely for me to have found a simple proof where everyone else had failed.

Minor quibble: since binary 0.1111... is 1, you need a number like 0.1010101... to get an actual counterexample.

It seems like one of the key factors in your story, Eliezer, is that you had read that book on math cranks. You were able to make the leap from your project of disproving Cantor and see its implications for the rest of your life thanks in part to having the example of the math crank in your mind.

Seeking evidence outside the immediate domain of inquiry can be tricky because it might lead one to include evidence that has no bearing on the actual problem, but because human endeavors don't happen in a vacuum, it's a great way of checking yourself for more general errors (like tilting at windmills).

I was not displaying ... any ... virtue

Most math teachers would be delighted if a student was able to understand Cantor's proof, think critically enough to search for a counter-proof, think creatively enough to describe a counter-proof (and based on different mathematical constructs at that), even though the proof was wrong at some critical steps.

This would be quite an achievement even for those who do not go on to the crucial last step of thinking self-critically enough to find the mistake in that "proof."

Minor quibble: since binary 0.1111... is 1, you need a number like 0.1010101... to get an actual counterexample.

Afaict, the original post doesn't contain any mention of binary fractions. An infinite binary sequence consisting entirely of ones doesn't represent any finite integer.

It seems to be a common childhood experience on this list to have tried to disprove famous mathematical theorems.

Me, I tried to disprove the four-color map conjecture when I was 10 or 11. At that point it was a conjecture, not a theorem. I came up with a nice moderate size map that, after a apparently free initial labelling and a sequence of apparently forced moves, required a fifth color.

Fortunately the first thing that occured to me was to double-check my result, and of course I found a 4-color coloring.

I expect that many people who grew up to be scientists and mathematicians attempted to create famous proofs when they were young, but I also expect that for many engineers such as myself our youthful folly went more along the direction of perpetual motion machines. I'd actually like to see some research on what the correlations really are.

LOL. Color me for both, Andrew. Perpetual motion using magnetic levitation in a vacuum at 10. Attempting to come up with a simple proof of Fermat's Theorem at 20 (if there was an easy way to determine n-roots of non-primes, I'd have been SET! :-) )

Something seems out of kilter about this, Eliezer.

When I was 13, I thought I had a proof in principle that there must be a minimum possible distance-- because to move is to move a finite distance, but no sum of infitesimal distances can compose a finite distance. I shared my idea with a professional physicist, who dismissed my idea using an appeal to authority. I don't care how fabulous the authority was, nor how ignorant I may have been, it was a terrible thing to for him to do that. It killed my enthusiasm for questioning physics, or math, at the time.

Reasoning, even mathematical reasoning, is not about just about right and wrong. It's also about how we model the world and apply our models to it. See Imre Lakatos's wonderful Proofs and Refutations for a look at how proofs are not just proofs, they are assertions about what's worth talking about and what we mean by our words.

And reasoning is also about honing our skills. We must develop the guts to recognize when we are wrong, but also the guts not worry so much about being wrong that we give up before we learn very much.

I once discovered a way to trisect an angle with a compass and straight edge. This has been proven to be impossible, apparently, but I did it. Later I discovered that I used an operation that wasn't "allowed" (an approximation maneuver), even though I performed the maneuver with only a compass and straight edge. To me, the proof that it can't be done is obviously incorrect, by any practical standard. Show me an angle and I can trisect it to an arbitrarily high degree of accuracy with my mechanical procedure. I challenge the "rules" set out by whomever thinks he's the know-all on what can be done with a compass and straight edge.

I hope other 13 year-olds don't read your essay and decide that the rational attitude is never to try to reinvent or challenge the Ancient Ones.

"I challenge the "rules" set out by whomever thinks he's the know-all on what can be done with a compass and straight edge."

I would be interested to see what you can get out of a compass and straightedge if you change the allowable operations. You could wind up with something much more complex than the things the ancient Greeks studied (think of how much more complex a Riemannian manifold is than a Euclidean n-space, once you remove a few of Euclid's axioms).

The ancient Greeks themselves played around with the rules. Archimedes used a "marked straightedge" to trisect an angle.

The first hit on google for trisect an angle is about ways to do it, not discussions of impossibility.

It seems to me that unless Eliezer was unusual in some other important way not described, he was not at close risk of becoming a math crank.

I'm also getting that impression, Robin. I'd say, "But there may be a selection effect in the people who comment at Overcoming Bias", but perhaps that would be, well, clinging.

This of course begs the question of where math cranks do come from.

While many people have mentioned similar disappointments, no one has echoed "I'll get that theorem eventually...even though my first try failed!" That was what seemed like a really bad sign when I read the essay before the comments. But I think we're really bad at communicating feelings, so I don't know how the feelings relate, how strong they were, and especially, how the commenters see the parallels with their reactions.

This argument that one should admit when they're wrong doesn't generalize beyond the exact reasoning of mathematical proofs and the like. In probablistic reasoning one can be, indeed usually is, wrong but close. The whole Bayesian worldview is predicated on the assumption that being a little bit wrong, or less wrong than the next guy, means you are probably on a more correct track towards the truth. But it doesn't and can't prove that, given just a few more important bits of information, the guy who's currently "more wrong" is right after all. So just how far from 100% probability must one be before one should admit that one is wrong? At what point does searching for more data relevant to a low-probability hypothesis become crackpottery? Should there not be more than just a single probability figure by which one makes this decision?

Would any regular commenters/readers object if I deleted comments like those from "a woo just like you"? I've always been nervous around censorship, especially where it carries the appearance of conflict of interest, but lack of censorship also carries its penalties. If I don't get any requests not to do so, I'll delete the comment tomorrow.

As I'm not much of a contributor, you can take my suggestion with a grain of salt but: Why not file away all deleted non-spam comments to a place where they can be read, but are out of the way? That way, moderators don't have to worry so much about censoring people and can instead focus on keeping discussions civil/troll-free.

I would much prefer that, but I don't think this blog has the technology.

Do you remember the title of the book? It sounds interesting, speaking as a lapsed mathematician.

"Mathematical Cranks"

Not sure if this is cranky or not, but when I was youthful I noticed that the Lorentz transformation of space-time due relativistic effects, square root of one minuc v squared over c squared, implies an imaginary solution for an v greater than c, that is for traveling faster than the speed of light. Now, most sci fi stories suggest that one would go backwards in time if one exceeded the speed of light, but I deduced that one would go into a second time dimension.

Of course the problem is that as long as Einstein is right, it is simply impossible to exceed the speed of light, thereby making the entire speculation irrelevant.

I don't like the formulation: "A thought you should never have thought."

I'd prefer, "An idea you should have quickly rejected."

I suspect that many genuine innovations might first appear to be mistakes or unwarranted challenges to the prevailing wisdom. They should be thought. And they should be considered and criticized. But, we should be ready to reject them if they don't survive the criticism.

Don't know what your blogging software allows, but richarddawkins.net now has a separate thread for off-topic posts; you click on a label at the end of the article to get to the off-topic thread.

I love this site. Found it when looking at a piece of crackpot science on the internet and, wondering, typed "crackpot" into google. I am trying to argue with someone who's my nemesis in most every way, and I'm trying to do it honestly. I feel his vested interest in the preferred answer vastly biases his judgment & wonder what biases do I have, and how did they get there. You seem to address a key one I liken to tree roots, growing in deep and steadfast wherever you first happen to fall, whether it's good ground or not.

Not unlike that analogy, I landed here first, on your post, and found it very good ground indeed.

This is one of my favorite crackpot writings. It does seem plausible that held breath underwater swimming is really good exercise. http://www.winwenger.com/ebooks/guaran.htm

Since everyone is sharing their stories, here's mine. When I was around 10, a family friend introduced me to the four-color map problem. I spent months trying to draw a map that required five colors, and one time I thought I had it. I dreamed of fame and glory for a few hours, then I showed the map to a relative who colored it with four colors. Shortly after, I accepted that I wasn't going to get it and stopped.

what a terribly unfair test to visit upon a child of thirteen. That I had to be that rational, already, at that age, or fail.

I always find it odd that you seem to write as though there is no hope of redemption when one makes a mistake of this magnitude. Certainly, lifetimes can be lost to such mistakes. But then, sometimes, it only takes a week to realise our folly, neh?

I fear that I might be currently trapped in this error: I've always resented Gödel's Incompleteness Theorems. When I was about 17 I thought I'd disproved 1IT (turned out I'd just reconstructed the proof of 2IT and missed the detail that Con(T)≠ProvT(Con(T))). It took me about a year after that to realise that, no, I wasn't going to disprove the ITs no matter how much I wanted to, and I accepted that trying to disprove them anyway would be a crackpot thing to do. Since then I've been trying to construct a philosophical framework of mathematics in which the ITs become irrelevant. Have I, in fact, taken the Crackpot Offer?

I don't remember ever coming up with a false disproof in math, though I did manage to "solve" perpetual motion machines. I did successfully prove a trivial result in solving quadratic equations in modular arithmetic.

Eliezer, did you realize at the time that what you had done was construct the basic outline of the proof that 2^aleph0 = aleph1? There was an interesting gem hiding in your disproof, had you looked. Reversed stupidity is not intelligence, and all that :)

With all due respect to Eliezer, there exists an institution that can protect you from the danger described in the post. It's called "math school". Sometime in tenth grade, I came up with a proof of the continuum hypothesis that my teacher couldn't immediately overturn. We had a fun time finding the catch, then moved on to other things.

I think you gave up too soon. Cantor is the father of all mathematical cranks.

A disproof of the Diagonal Argument is found here:

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

I once thought I could prove that the set of all natural numbers is as large as its power set. However, I was smart enough to acknowledge my limitations (What‘s more likely: That I made a mistake in my thinking I haven‘t yet noticed, or that a theorem pretty much any professional mathematician accepts as true is actually false?), so I activly searched for errors in my thinking. Eventually, I noticed that my methods only works for finite sub sets (The set of all natural numbers is, indeed, as large as the set of all FINITE subsets), but not for infinite subsets.

Eliziers method also works for all finite subsets, but not for infinite subsets

What the hell, I'll tack mine on.

As a kid, I was introduced to an interesting contraption (that was supposed to be a party trick kind of thing) that I then dreamed for years of using to generate free energy (and therefore to be able to fight against entropy with). It was a ramp with the middle cut out so it looks like a "V" that you have a roller for made of 2 cones base-to-base. The cones look like they roll up the ramp. This does not work, but it took me half a decade to learn this.

My other one is that I heard about the angle trisection thing and "found a solution" on accident in highschool. For a little bit, I entertained the idea that I'd found it, then heard it had been proven impossible and believed that. For years it taunted me until I finally found someone willing to show me how it was wrong so I could feel like I had redeemed myself from crankdom (I promise that I didn't believe I was right, but it always bothered me that I didn't know).

perhaps I should apply Cantor’s Diagonal Argument to my clever construction, and of course it found a counterexample—the binary number (. . . 1111), which does not correspond to any finite whole number.

I’m not following despite having recently reviewed Cantor’s Diagonal Argument. I can imagine constructing a matrix such that the diagonal is all ones… but I don’t see how this connects up to the counterexample claim above.

Also, why worry that an infinite binary representation (of any kind) doesn’t correspond to a finite whole number? I suspect I’m missing something here. A little help please to help close this inferential distance?