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).
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.
> 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.
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
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 :-).
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.
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.
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.
40 comments
[ 1345 ms ] story [ 277 ms ] threadThat's not a good sign. :)
Coding Horror describes it here: http://www.codinghorror.com/blog/2009/06/monty-hall-monty-fa...
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 problem becomes to 50/50 if you change the rules that the host always opens the left most door not picked by the contestant.
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.
[1] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.6...
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
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.
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.
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.
[1]: http://en.wikipedia.org/wiki/G%C3%B6del's_ontological_proof
You mean the Abstract?
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.
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".
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
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?
>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.