Ten Signs a Claimed Mathematical Breakthrough is Wrong | Hacker News Reader

Ten Signs a Claimed Mathematical Breakthrough is Wrong

scottaaronson.com

125 pointsgnosis55 comments

One of the 'fun' aspects of running a math blog is that you get to see Word documents containing "proofs" of unresolved problems or the existence of God or Allah. A true glimpse into madness.

I made the mistake of writing a short series of blog posts about unsolved math problems a few years ago. It seemed innocent enough at the time, but I still get "proofs" submitted today. I'm a programmer, not a mathematician, so in a lot of cases I don't even know enough to be able to tell people why they're wrong (other than the fact that they submitted their proof of Goldbach's conjecture to a programming blog).

Here's part of the saga of the blogger who got into a similar situation when he explained that 0.999... is equal to 1: http://polymathematics.typepad.com/polymath/2006/06/no_im_so...

> note: this post has been closed to comments; comments about it on other pages will be deleted!

That's not a good sign. :)

That was a great read. It reminds me of the Monty Hall problem fiasco that arose from the "Ask Marilyn" column about it. I appreciate the author of Polymath actually refuting all of the arguments against his post (to a reasonable degree).

Coding Horror describes it here: http://www.codinghorror.com/blog/2009/06/monty-hall-monty-fa...

Reasonable people get in arguments over Monty Hall because the problem is usually stated incompletely. The odds depend on what the show's host is doing.

Reasonable people do usually get in arguments (and are wrong) over the Monty Hall problem even when the problem is correctly stated (as the Vos Savant or Erdős' stories show).

Wow, that expert mathematicians refused to accept the solution is quite amazing to me...

When I was originally asked the Monty Hall problem, I asked "Does the host know where the prize is?" to which they replied "it doesn't matter". I also used simulations to show it does.

The host has to know where the prize is, otherwise how could he open one of the two other doors after the player's first choice?

I've seen it worded poorly where it is stated that contestant picks door 2 and then the host opens door 1 revealing an empty stage.

The problem becomes to 50/50 if you change the rules that the host always opens the left most door not picked by the contestant.

If the host always opens the left-most door not picked by the contestant, the host would open the door on the prize one out of every three times.

Indeed, and these cases are the ones we don't examine.

Because it may be that the host opened a random door, which happened not to have the prize in it.

If that's the case then you're only examining a subset of the probability distribution, which means you're getting skewed results.

Yes, and being asked about the nature of these skewed results.

Ah, my favorite example of a pons asinorum:

http://mathworld.wolfram.com/PonsAsinorum.html

> An elementary theorem in geometry whose name means "asses' bridge," perhaps in reference to the fact that fools would be unable to pass this point in their geometric studies. The theorem states that the angles at the base of an isosceles triangle (defined as a triangle with two legs of equal length) are equal and appears as the fifth proposition in Book I of Euclid's Elements.

Generalizing the concept a bit, a pons asinorum is a critical concept in any field, such that if you don't grasp it you'll never comprehend the field as a whole. 0.999... = 1 is the critical point for the idea of the infinite series and, thus, an important part of how mathematicians use infinity.

And the concept of infinity seems to be the stumbling block for a lot of people, in the Internet arguments I've seen about this. A fair number of people seem to hold a religious fear of infinity, at least to the extent they apparently believe that it cannot be handled in a logical fashion; it is sacred and not to be mixed with such profane ideas as 3 or 12. Mix that with a half-formed notion of infinitesimals and some cod-Platonism and you have people seriously arguing that math is wrong and that they have inspired access to a true mathematical revelation.

Or they deny that the argument is mathematical at all and retreat into increasingly self-contradictory pseudo-philosophical arguments.

There was a nice article called "An editor recalls some hopeless papers" about all the "proofs" the editor had received alleging to have disproved Cantor's Theorem, so I'd say you're exactly right that many people have problems with infinity.

[1] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.6...

"cod-Platonism"?

Cod-Latin is fake Latin so cod-Platonism is fake Platonism which comes from a simplistic understanding of Platonism.

Ah, right, thanks.

Using logic to "prove" God exists isn't totally crazy, just mostly crazy. Gödel himself tried his hand at it [1].

[1]: http://en.wikipedia.org/wiki/G%C3%B6del's_ontological_proof

Yeah, but Gödel was legitimately crazy towards the end of his life. He ended up dying because he was too paranoid to eat!

This is one of the bizarre comforts I feel when I acknowledge that I will never be a math super-genius... the fact that there is a high correlation between math super-genius and "overflowing" to insanity. Godel, Nash...

I've noticed that you can roughly date logical proofs for God's existence by looking at how complicated and confusing they are: the newer the proof, the more epicycles get added.

When I finally prove the Riemann Hypothesis it will be presented in HTML5 and Javascript with interactive graphs.

Make sure to include a tl;dr, 'cause ain't nobody got time for that.

i may have come up with a truly clever proof but it won't fit in an HN comment.

my homepage amirhirsch.com is an interactive graph of the zeta function (and partial sums) along a horizontal slice with spacings of .05 between 0<Re(s)<1

> Make sure to include a tl;dr

You mean the Abstract?

Yes but label it "TL;DR" instead of "Abstract"

Browsers will still support HTML5 and JavaScript when you finally get around to proving RH? Optimistic! ;)

Site is overloaded, so here is the Coral cache:

http://www.scottaaronson.com.nyud.net/blog/?p=304

See also the famous Crackpot Index:

http://math.ucr.edu/home/baez/crackpot.html

This is from 2008. Could you add (2008) to the title?

I enjoyed the author's Beijing/Chinatown metaphor in point 10.

I probably disagree on the Tex one. I know several people with PhDs in engineering and science disciplines who have never used Tex. Several of them know enough math to at least make a reasonable stab at a few of the problems, even if they are unlikely to be successful.

So what you're saying is that these non-TeX users are unlikely to successfully solve P=NP? I'd say that sort of upholds his premise.

It would be reasonable to say users, in general, are unlikely to prove P = NP. So I'd say it doesn't affect the premise at all. :)

The SIGHCI/CHI community is noted for using Word more often than not. It is ironic for a field that is supposedly well connected to design to have such horrible looking papers on average.

Note also signs 11 and 12 in the comments[1], and the later "Eight Signs A Claimed P≠NP Proof Is Wrong"[2]. (OK, so there's a bunch of overlap.)

[1] http://www.scottaaronson.com/blog/?p=304#comment-8957

[2] http://www.scottaaronson.com/blog/?p=458

Most of the points require reading the whole paper first, isn't it?

Skimming, sure. But really reading the paper, ie confirming its soundedness, requires going through most if not all the ideas and steps presented. Depending on the paper it can take weeks or even months. The ten tests in the OP take at most a few hours, I think.

Lightly reading a paper and working to understand it are two different things, especially for a potentially groundbreaking work.

No, not really. One thing that makes an expert an expert is their ability to filter signal from noise. Sometimes they can prematurely bucket ideas into relevant/not-relevant, but usually not.

Have you ever seen a novice programmer trying to find information on a StackOverflow page? The answer might be staring them in the face, but they still scroll past it once, twice, three times, etc. until I point it out.

A novice has no sense of what's important and what's not so they tend to think everything is whereas an expert has a set of heuristics (like the ones in this blog) which make them much faster. They'll look for familiar ideas, themes, likely mistakes, etc. first before going into full-on, read-this-thing-line-by-line mode.

>One thing that makes an expert an expert is their ability to filter signal from noise.

Certainly, but will you be able to spot the breakthrough insight if all your doing is filtering it against your notion of signal vs noise. Breakthroughs tend to come at a problem from a totally different angle & generally it takes a bit of time to understand why its brilliant rather than "noise".

Like I said, "Sometimes they can prematurely bucket ideas into relevant/not-relevant, but usually not." I wrote that hoping to preempt your exact response. Alas! ;)

Even if I grant your premise (which I don't), just because a breakthrough is likely to be written in crazy-talk doesn't mean something written in crazy-talk is likely to be a breakthrough. At that point it becomes a question of opportunity cost, which is exactly where heuristics come in.

In fact, what you said could be true and it could still be true that

  P(breakthrough | does not seem crazy) > P(breakthrough | seems crazy)

So, if I were on the hunt for breakthroughs I'd still be better off ignoring the crazy-seeming things.

Regarding your premise, can you name, say, ten mathematical breakthroughs since the Enlightenment that came about the way you described?

Pretty sure we're talking past each other, but I'll give it a shot anyway.

>Even if I grant your premise (which I don't), just because a breakthrough is likely to be written in crazy-talk

Who said anything about "crazy talk"?. My point is simply that your signal/noise filter is by necessity driven by information derived from the status quo. The status quo part being a really big problem if you're hunting for something new.

I wonder if there is a context in which preparing a document in TeX would be a strong indicator that you don't know what you're talking about?

Maybe history. Or another of the humanities. :) But TeX takes time to learn to use properly and only people that expect to use it regularly go through the trouble I'd expect.

A conference on Microsoft Office?

I probably wouldn't make a scrapbook with it.

Are mathematicians going to be using TeX in a hundred years? Two hundred? Five hundred? Two thousand?

Math is timeless. TeX may be, but it's too soon to say. It's a recent invention. It's only been around for 35 years.

It wouldn't surprise me if they still use it in a hundred years. It just fits thinking in math. By then TeX will be on version number 3.1415926535897932384626433832795028841971693993751 :-).

IIRC, it's supposed to be set exactly equal to pi upon Knuth's death.

And Emacs (with AUCTeX ) will have reached version 2.7182818284590452353603...

Perhaps you're correct, but I would say this article is the author explaining his criteria, and asserting that it might be useful for some other people, on both sides of the line.

Given that, I suspect that in one, two, five hundred years neither he nor his blog will be around. Nor anybody who read his blog. Even if his text had survived (perhaps someone goes digging though the NSA archives in a few hundred years), I'm sure that the person reading it will just intelligently substitute in the succeeding technology.

So, what I'm trying to say is that the point about TeX is just a point of practicality. It happens to be approximately true right now, and the author has no information to guide him to an alternate conclusion.

TeX is evolving with LuaTeX appearing to get some momentum. Given that almost no one uses TeX directly, I'd expect the biggest change would involve a gradual "improvement" in the macro library. It may even result in an embedded TeX à la sweave.

Wolfram is hammering against the interactive element with CDF and I could see TeX being extended to support a more dynamic electronic document format if HTML rendering engines continue to be typographic messes. Perhaps an LLVM-type backend to generate executable ebooks would be an interesting project, but I'm not sure I'd start with TeX except for math markup.

I would hope my prediction of the death of TikZ and PGF would come true soon enough, though.

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