Flickr

  • www.flickr.com
    Abiola_Lapite's photos More of Abiola_Lapite's photos

« Reactionary Idiocy of the Day | Main | Is Occam's Razor Logically Justifiable? »

March 09, 2006

Comments

Chuckles

Hmmm - I had a look at the test and I thought it was satire - I thought the author had the same points you just made in mind. But maybe not. But of course you are right - and I think the relevance of this to I.Q. tests has been discussed before.

Lary

I'm that idiot. =)

I put the "Outside the box" questions there just for the purpose of preventing people from taking it *too* seriously, and OEIS is great - it even had the "subway" sequence, and as a NY-er, I probably could've changed to another line to make it obscure, but figured it's just there..

I didn't create it expecting it to be somehow the "meme of the moment", but most of the mathy series are relatively well-known series that combinatorics people actually recognize.. or while not as well know as Fib, they do come in handy if you know your sequences, which, as they say, is good enough for me!

(One example is the Catalan Numbers.. it shows up in the most "random" places, so it's very nice to know!)

For the ones with a name (not just "find the formula"), I actually came across them enough that if a sequence starts that way, I'm going "guess" the sequence - saved me bunch of time before!

Of course, anyone taking this seriously.. well, needs to be just a little bit introspective and worry about the right things.. =P

Abiola

"I'm that idiot. =)"

Good to see you have a sense of humor about the whole thing!

"I would say that for any particular problem, there is a somewhat objective way to rank the solutions"

Oh really? And what might that be? Again, I suggest you look around in the OEIS database yourself and see just how many "obvious" sequences have equally logically compelling alternative answers.

"the problem can be considered a good problem if there aren't multiple "best" solutions."

Integer sequence "problems", and similar puzzles of the sort used in IQ tests, fail this criterion by definition. Read your Hume and Popper: induction is not a logically rigorous or all that trustworthy mode of thinking.

Heck, here's a little challenge for you: what numbers come next in the following series, and why?

1,2,3,4,5,?,?,?

When you're done giving your answer and a justification for it, proceed to the following URL and tell me why your answer deserves to be favored over all the others in the first 10 options.

http://tinyurl.com/r642y

Abiola

"because that extrapolation relies on the "simplest" pattern"

What exactly do you mean by "simplicity"? Can you give a rigorous definition of it and establish that it necessarily gives a unique answer?

"all the other patterns are implicitly dependent on the ordering of the integers."

As is the one you think "simplest".

"There is no need to bring Hume and Popper into this."

Judging by what you have to say next, there clearly is.

"I think that the aesthetic judgment one makes in saying, "This is the best solution to the sequence" is somewhat objective (not entirely so)"

On what basis do you make this claim, other than gut feeling? "Aesthetic judgment" sounds suspiciously to me like, "this is the sequence I personally find most appealing, for idiosyncratic reasons I can't possibly hope to logically justify."

Gdr

>>> What exactly do you mean by "simplicity"? Can you give a rigorous definition of it and establish that it necessarily gives a unique answer?

Kolmogorov–Chaitin algorithmic complexity is the measure that people usually use for "simplicity". It has a rigorous definition and gives a unique answer.

Chuckles

[...Kolmogorov–Chaitin algorithmic complexity...]

But this is part of the problem, isnt it? Stochastic complexity is environment bound, in so far as it admits description. Thus there could be 1 billion simplicities involved.

[...that people usually use...]

Ah Ha! What people, pray tell?

Abiola

"Kolmogorov–Chaitin algorithmic complexity is the measure that people usually use for "simplicity"."

No, it is just *one* of many possible definitions of such.

"It has a rigorous definition and gives a unique answer."

The last claim is pure bollocks. You clearly don't really have a clue about the subject beyond what can be gleaned in the popular press.

"I find it hard to believe that someone can have even a mild interest in mathematics and really think that mathematical aesthetic judgments are completely arbitrary, unless they can be formally defined and quantified."

I find it hard to believe anyone would be stupid enough to stuff absurd claims into my mouth upon being challenged to make their assertions more rigorous.

"I can't imagine why someone with that attitude would even bother to study mathematics. Maybe because it's a subject that lends itself well to being used to lord over others?"

Ah yes, nothing like a bit of ad hominem to bowl others over with your intellectual rigor. Does it occur to you that the *reason* I ask you such questions is that I have a far keener sense of what mathematics is *really* about than you do, and am therefore not in the least inclined to go with "obvious" or "intuitive" rationalizations?

"If you took a poll to see which sequence people thought was simpler -- of eight-graders, of college students, of professional mathematicians, of Fields Medal recipients -- what do you think the results would be?"

No one ever told you argumentum ad populum was a logical fallacy? Why don't we use a poll to settle the Poincare Conjecture while we're at it?

http://philosophy.lander.edu/logic/popular.html

"But this is a little silly."

No, the only thing silly here is YOU, for descending to such depths of foolishness when faced with even the slightest probing of your ridiculous assumptions.

Andrew

Given that Barbar and Abiola are arguing about essentially aesthetic judgments of 'elegance' and 'simplicity' and not logical judgments of a proof's validity, I find it strange that Abiola is taking a hard line against the idea that one mathematical idea could be seen non-arbitrarily as 'simpler' than another, when he has no trouble making non-arbitrary aesthetic judgments about what is more conventionally viewed as art.

Abiola

Barbar,

I'm fed up with your rubbish. You are nothing more than a mannerless, braindead anonymous coward who makes all sorts of grandiose claims about himself no one can verify. Go bother someone else from now on.

"I find it strange that Abiola is taking a hard line against the idea that one mathematical idea could be seen non-arbitrarily as 'simpler' than another, when he has no trouble making non-arbitrary aesthetic judgments about what is more conventionally viewed as art."

Blame that on your own inability to distinguish claims about human psychology from claims about the objective nature of reality and reason.

Abiola

Oh, and by the way, Andrew, the following statement of yours

["Given that Barbar and Abiola are arguing about essentially aesthetic judgments of 'elegance' and 'simplicity' and not logical judgments of a proof's validity"]

must either be the product of overhasty reading on your part or evidence your brain has gone on holiday. What the hell else could I be arguing about other than logic in saying there's no good, rigorous justification for preferring one sequence continuation over another? Making such sloppy and manifestly self-contradictory statements just makes me apply a steep intellectual discount to everything else you have to say. "Barbar" is a moron, and it would be unfortunate if I had to lump you in with the likes of him.

Andrew

Perhaps I phrased that infelicitously. It seems to me that Barbar was making an argument about taste and aesthetic judgment, and you were countering that with an argument about logic, even though I would have thought from other arguments you've made about aesthetics in, say, literature that you might agree with Barbar's arguments about taste and aesthetic judgment. Maybe it's just a case of two people talking past each other. [I was focused on your statement, '"Aesthetic judgment" sounds suspiciously to me like, "this is the sequence I personally find most appealing, for idiosyncratic reasons I can't possibly hope to logically justify."' and interpreted it to mean that you were saying aesthetic judgments are necessarily arbitrary (i.e. idiosyncratic) but now see the emphasis on "logically justify." So I guess there's no contradiction with your arguments about aesthetics in literature. Mea culpa.]

Abiola

If Barbar had simply made an argument about which sequences seemed most psychologically plausible given human nature and left it at that, rather than descending into ad hominems upon my noting the difficulty of making rigorous the claims he was trying to make, we might have had a worthwhile discussion. One could make a reasonable elaboration on Kant's "rose tinted spectacles" argument to the effect that "6,7,8" seems most "right" as a succession to "1,2,3,4,5" because human minds have been shaped by evolution to see the natural number sequence most readily - though given research I've seen within the last year about the relationship of number sense to language, not even this much is necessarily true - but that isn't "Barbar" was doing, just slinging insults about my mathematical judgment and schooling from the safety of his anonymous little rathole, as if I ought to automatically accord respect and credibility to any nameless troll who comes on here and claims to be Bill Gates and Albert Einstein merged into a single body.

Returning to the core issue behind this whole post, the thing with sequence guessing tests is that they are woefully underspecified, and contrary to what many naive reasoners might believe, the notion that there is a single algorithm or criterion by which one can solve all such problems is an impossible dream, a reality machine-learning researchers are familiar with ("no free lunch" theorems). Saying that there is a single, clearcut criterion by which we can sort all "better" answers from "worse" ones is the same as saying that we can have a free lunch, and even psychologically appealing criteria like "simplicity" (however defined) can easily be shown to fail under many perfectly sensible circumstances: Newtonian mechanics is simpler than general relativity even under the Kolmogorov–Chaitin criterion, but no one will argue that this makes it a "better" theory; similarly, saying "God did it" is a *much* simpler "explanation" for everything one sees than having to mess about with quantum field theories, Calabi-Yau manifolds and all that stuff. We are forced to toss aside the "God did it" type simple arguments by the test of empirical reality, but no such constraint binds us when we ask what comes next in any arbitrary numerical sequence.

Barbar

OK Abiola, I'm not welcome here, so I'll stop posting (after this). Seems that most of your problem with me generated from my poke at you that your interest in math was generated by your desire to lord over others rather than an appreciation of mathematical elegance (I would quote myself exactly but my posts have been deleted).

I sincerely apologize, I didn't really mean it, it was a barb at someone who seems to rather enjoy calling other people idiots and morons, obviously it seemed to just eliminate the possiblity of rational discourse between the two of us.

Anyway, I didn't think my posts were so bad that they deserved to be deleted while I wind up being labelled a moron (I think Andrew didn't have any problem discerning what I was saying), but obviously this is your blog and if you want to delete this post and continue to call me an idiot you can go ahead, I'm not too worried about my anonymous reputation. I would recommend Hofstader's "Metamagical Themas" for a better explication of some of the things I was getting at in this thread (also, I think you would genuinely enjoy it).

dsquared

I think Andrew is referring to your post on Paul Dirac's theories of aesthetics, although IIRC Dirac was talking about an objective (or at least intersubjective with respect to mathematicians) aesthetic criterion for equations of fundamental physics rather than arbitrary number sequences.

http://foreigndispatches.typepad.com/dispatches/2005/06/wise_words_from.html

The comments to this entry are closed.

Notes for Readers