>> “although the implications of the mathematics for physics would surely still be debated.”

That, and also the search for a new physical mechanism of which Bell’s theorem is a limiting case.

–Ajit

[E&OE]

–Ajit

[E&OE]

I once thought i solved a problem about computational geometry. I though that i had managed to prove the problem np-hard by finding a polynomial reduction from PARTITION to this problem. Well the thing is that PARTITION has polynomial algorithms when the numbers are written in unary. And guess what, the pieces in my problem all had fixed length, so they were unary numbers. So i was actually doing an exponential reduction between PARTITION and my problem.

I thought during some days i had solved the problem until someone pointed the error to me.

After some months, i found a way to reduce 3-PARTITION to this problem.

And this is the story

]]>In the main post, you say:

“you don’t earn any epistemic virtue points unless the errors you reveal actually embarrass you.”

Ummm… That’s fine by me, but strangely, you omit to *quantify* the extent of the embarrassment that should be felt before a comment could be posted. … Quite unlike you, Scott.

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OK. Since I had spent a little time (a couple of hours or so) on the coffee-related thingie in the past (c. 2011), might as well now note that I now feel (only so slightly) embarrassed that I had not thought of what Brent Werness did.

Namely, that there would be this contribution of convection apart from that of diffusion. Brent’s “shearing” of the “whole regions” would precisely mean convection, and your “long range correlations” probably would mean more or less the same.

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Why is the embarrassment which I do feel so mild?

Because, I think, I could, probably, have got what Brent did, if only I were to work on it. [… Guys, don’t hit me so hard on the head; instead, please read on….]

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Could I really have?

Yes, I think, chances are bright—provided I were to actually spend some serious time on it, i.e., if I were to write and run simulations on computer (instead of hurriedly working out a small example) and attempt to write at least a draft version of a paper (instead of a mere comment on someone else’s blog).

Two reasons: (i) NS equations are the precisely kind of stuff that I (or “mechanical” engineers like me) spend (at least) some time on, and the fluxes in the NS equations involve, on the “ma” side, *only* these two mechanisms—diffusion and convection. If I were to actually begin simulating even an abstract version of a multi-phase fluid phenomenon, I very, very probably couldn’t have escaped wondering also about convection. It would be hard to think of only one form of the flux and not the other. (ii) Within CFD, particles-based approaches like SPH (smoothed particle hydrodynamics) and LBM (lattice Boltzmann method) are right up my ally.

Of course, I was not at all active in CFD back in 2011 (or even in 2014), so the counter-chances also are appreciably strong.

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As to *counter*-chances, there does remain the issue of diffidence. *Even* in *my* case, there *exists* a small but definite probability that while talking with the TCS folks, I could/would have relegated my fluids-knowledge to the background. May be out of a fear of sounding not enough intelligent, abstract, mathematical, or even plain relevant. The probability in my case is admittedly small, very small, but it is finite. So, even *I* could have done that—not bring up convection to your notice.

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But still, on the whole, I *feel* that, if I were to *actually* do some research on this topic (or even if I were only to participate in it in a personal settings) I *do* feel that at some time or the other, in some form or the other, at least as an informal/stupid/brain-storming kind of a query to you maths/physics/TCS folks, I would have ended up bringing up convection (or groups of coffee/milk pixels moving together as blocks) to your attention. Guess I would have done that. (In fact, *any* CFD guy would have done that.)

As to converting these physical insights into a suitable maths?… LOL! (One feels embarrassment only over something he is sure he *could* have got.)

All the same, good that you brought up this topic again. (I mean the coffee, not embarrassments.)

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Hmmm… So the take-away point is: there is some kind of a correspondence between the differential non-linearity of convection and the information complexity of strings. … Sounds right. Non-linearity is the basic reason behind chaos.

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[One last point: Here, in writing this comment, I have happily chosen to skip over the “colleagues” and “mathematical” parts of your requirements, Scott.]

[One more “last” point: Something is to be said in favor of inter- and multi-disciplinary researches.]

OK. Enough is enough. I now await your completed (revised) paper.

Best,

–Ajit

[E&OE]

#1: The computer-assisted proof of the NP-hardness of minimum weight triangulation, or related hardness proofs with extremely complicated gadgets. Note that for gadget reduction proofs the polynomial running time is usually obvious, so the formalization would only be needed to ensure the correctness of the gadgets, but paradoxically and fortuitously the correctness would likely be the easier part to formalize. Of the 12 problems with uncertain NP-hardness status listed by Garey and Johnson in their famous book, minimum weight triangulation was the 10th to be resolved after being open for several decades. Since JACM published it I’m not questioning the proof’s correctness or implying anything negative about the authors’ clearly impressive work, but it would surprise me if more than a handful people on Earth have taken the trouble to verify every detail. The proof involved a bunch of complicated funny looking wire gadgets, long decimals floating everywhere, and many computer-assisted optimizations.

#2: Another place where formalization might be called for is in requiring a proof assistant script to earn the million dollars for solving P versus NP, given the number of past mistakes.

#3: Finally, the proof of Bell’s theorem is simple enough that a formalization probably wouldn’t take very long, assuming it hasn’t already been done. A formalized proof would permanently end hidden variable counterexamples and objections to this apparently controversial theorem from people like Joy Christian, although the implications of the mathematics for physics would surely still be debated.

]]>**Scott** observes [without necessarily endorsing] “My Singularity friends assure me that, almost immediately after the first AGI is created, it will bootstrap itself up to become incomprehensibly smarter (or at least faster) than the smartest humans.”

It has never been clear (clear to me at least), that there exists any very strong reason to foresee that “bootstrapping to incomprehensible smartness” is feasible even in principle.

To the extent that the anatomy of the brain is dual to the structure of cognition, then isn’t the demonstrably parcellated anatomy of the human brain (which is the subject of a very great deal of ongoing fundamental and clinical research) isomorphic to an inherently parcellated structure of human cognition in general, and mathematical cognition in particular?

To borrow Terry Tao’s parcellation of mathematical appreciation (per comment #20 and Tao’s arXiv:math/0702396), gains in the cognitive capacity for “insight”, “discovery”, “vision”, “taste”, “beauty”, “elegance”, and “intuition” (etc.) are not obviously mutually correlated by any presently-known anatomic or cognitive parcellation of brain or mind.

Enough is known, however, to be reasonably confident that human mathematical cognition is not accomplished solely by the processes of deductive ratiocination that the Singularitarians contemplate.

Fortunately, even in the face of these unknowns, we can all be entirely confident that by the time these tough questions are settled (if they ever are), plenty of good mathematics, science, engineering, medicine, and even art will have been created. Because here the entire STEAM enterprise comes into play. Which is good! 🙂

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Please let me say, too, that I am very grateful to James Smith (of #22,26,32, and 33) for his links and remarks on Thomas Hales works, which are wonderful. And in regard to the formal status of the Axiom of Choice, constructive versus nonconstructive proofs, Grothendieck Universes (etc.), Freek Wiedijk’s list of formal proofs (see #20 for a link) italicizes those proofs that are formally constructive. Obviously plenty of questions remain open; Colin McLarty’s “What does it take to prove Fermat’s last theorem? Grothendieck and the logic of number theory” (2010) is a good introduction to these exceedingly difficult, intrinsically “multiparcellate”, mathematical questions.