Experimental mathematics
Friday 2007 May 4
I moved the discussion from the page “about” to this page.
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Tuesday 2007 May 29 at 1:58 am
Well, I’m a sculptor with a blog. I also have not decided what it is for – so welcome to the club
Congratulations on the Fields btw….
Tuesday 2007 May 29 at 5:36 pm
May I ask what you think of so-called “experimental mathematics?” In particular, do you agree with Doron Zeilberger (c.f. the current issue of the MAA journal, FOCUS) that we should expect and welcome the advent of software that will relieve us of the “burden” of proving things?
Oh, and once we’re relieved of this “burden,” will we really have very much left to do?
After all, Zeilberger maintains that existing software already plays a large (if often unacknowledged) role in the discovery and isolation of interesting new mathematical phenomena, and this software, like all software, will presumably only get better. So it’s not as though we can insist, with Picasso, that computers are useless because they can give us only answers. Odds are they’ll get pretty good at giving us questions, too — indeed, if you believe Zeilberger, they already are.
So should we then expect and welcome a steadily diminishing human role in the whole mathematical enterprise? Will we eventually have software that generates results like those in Ramanujan’s notebooks — together with their proofs! — at every punch of the “Return” key? And if so, how should we feel about this? Equipped with such software, would we really sit there by the hour, punching that “Return” key, and marveling at the results? Even if they inspired in us the same wonderment as Ramanujan’s results, would we really attach the same value to them, and indeed, to mathematics itself, if we got them so cheaply?
Henri Poincare has observed that “…to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the ‘logic piano’ imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.”
This, of course, is a view of mathematics automated, and it seems to me to echo down the decades in the language of Zeilberger’s arguments. Donald Knuth, in his preface to Zeilberger’s book (with Wilf and Petkovsek) A = B, says that “Science is what we understand well enough to explain to a computer. Art is everything else we do.” Zeilberger and his fellow experimental mathematicians seem to look forward eagerly to a day when all art in mathematics is entirely obviated.
Now, I’m hardly an enthusiastic mystic. In general, I welcome the transformation of human understanding into a form transparent enough to be explained to a computer, and I agree that the transformation of human mathematical understanding into such a form would be a tremendous accomplishment.
But I confess I still recoil a bit at the thought of mathematics reduced to something very much like the legendary Chicago machine. I suppose what I’m saying is that I value the art itself almost as much as the understanding that results from its practice. I would mourn its passing pretty keenly.
How, though, do you feel about this?
Wednesday 2007 May 30 at 2:53 am
Doron Zeilberger has provided work which is delightful and deep. But my background [takes off math hat briefly] as a Science Fiction author makes me see the human-machine future in a more nuanced way.
I prefer the definition and examples of Experimental Mathematics by Jonathan Borwein et al. What is see is lovely, creative, and not sausage-making. It is, analogously to the space program, or a symphony orchestra, good teamwork between humans and machines.
See the definitional material, examples, and editorial board of:
Experimental Mathematics
http://www.expmath.org/
I do not see computers through a Terminator lens. I prefer Utopia to Dystopia. Software, I hope, will not leave us “With Folded Hands.”
Wednesday 2007 May 30 at 1:32 pm
Computers might be able to do real math eventually, but they still have a very long way to go. They are really good at certain restricted problems, such as running algorithms to evaluate special classes of sums and integrals (as in Zeilberger’s work) or checking lots of cases (as in the 4 color theorem or the Kepler conjecture) or even searching for proofs in very restricted first order theories, but none of these problems come anywhere near finding serious mathematical proofs in interesting theories such as Peano arithmetic.
Rather than find proofs by themselves, computers might be quite good at finding formal proofs with human assistance, with a human guiding the direction of the proof (which computers are bad at), and the computer filling in tiresome routine details (which humans are bad at). This would be useful for something like the classification of finite simple groups, where the proof is so long that humans cannot reliably check it.
Wednesday 2007 May 30 at 2:43 pm
Yes. I have discussed this recetly with Michael Aschbacher, at Caltech where I got my Math degree, he being author of “The Status of the Classification of the Finite Simple Groups”, Notices of the American Mathematical Society, August 2004. They’ve apparently (I have to take their word for it) filled the gaps of the proof initiated so long agao, and when John H. Conway was on the team (I’d spoken to him then).
Coincidently, I was at a Math tea at Caltech yesterday, joking about the 26 sporadic groups being a “kabbalistic coincidence” — or perhaps examples of some incompletely glimpsed set of more complicated mathematical objects which are forced to be Groups for reasons not yet clear. Some people deny that there are “coincidences” in Mathematics. Gregory Chaitin insists that Mathematics is filled with coincidences, so that most truths are true for no reason. That sounds like the beginning of a human-guided deep theorem-proving project to me. Humans supply gestalt intuition that we don’t know how to axiomatize or algorithmize. Humans did not stop playing Chess when Deep Blue became able to beat a world champion. The computer is crunching fast; the human looks deep. The human has the edge in Go, which takes deeper search.
So, as I say, “yes.” I agree with you that we should each (human and machine) be doing what we’re best at, together. After all, that’s what the right-brain / left-brain hemisphere architecture does. When John Mauchley (and J. Presper Eckert) built the BINAC under top security for the USAF, delivered 1949, it was the first dual processor. Mauchley told me that the brain hemisphere structure had evolved, and was probably good for more than we knew.
He and I were introduced by Ted Nelson, father of Hypertext, in 1973, while I was developing the first hypertext for PC’s (before Appple, IBM
and Tandy made PCs). We demoed our system at the world’s first personal computer conference in Philadelphia, 1976. So the human-computer teamwork is something I’ve been working in for 40 years. Do you suspect that a human/computer partnership (Including you, of course) will get to the bottom of quantum field theory?
Wednesday 2007 May 30 at 2:59 pm
They print, Jon. You just need an escape sequence. < for <, > for >, and & for & are the most common. It’s because those characters are used in HTML code
Wednesday 2007 May 30 at 3:37 pm
John Armstrong is right, of course. I’d made a typo that gave a character that showed as a lower-case epsilon, which acted like an HTML operator of some sort. The “H” in HTML is an explicit nod to Ted Nelson’s invention of Hypertext. To me, the “killer app” of the Web is its use as collaborationware. Blogs by Fields medalists, Nobel laureates, and the Online Encyclopedia of Integer Sequences are valuable, in establishing dialogue between people who have something to say. That’s embedded, of course, in a vast sea of people gossiping while indicating what pop song they’re listening to and their mood (by emoticon). Vladimir Nabokov taught that there were 3 levels of novel reading. This, I believe, applies to blogs and emails as well. I paraphrase, not having his textbook in front of me. The lowest, trivial level, is to read a novel as story about characters, equivalent to a soap opera. The second level is to, by exegesis and induction, glimpse the techniques of language that the writer must have used. The third level is the absorb onself in deep study of the specific objects and words used by the writer, perhaps with misdirection, that lead 2nd order readers. I suspect that he concealed a 4th level in his own work, for further literary disinformation theory. See, for instance:
* Zina’s Paradox: The Figured Reader in Nabokov’s Gift by Stephen H. Blackwell
Author(s) of Review: Priscilla Meyer
Slavic Review, Vol. 60, No. 3 (Autumn, 2001), pp. 684-685.
Blackwell’s approach leads him to identify seven levels of narration (three more than critics have so far discussed) that produce a “multiplicity of ways …” [combinatorics, eh?].
50 years after Lolita
by Robert Fulford
(The National Post, 20 September 2005)
“… The story works on at least three levels. As a picture of America it’s brilliant satire; as a depiction of Humbert’s character it’s harsh psychological comedy; and as a narrative it’s a tragedy of misplaced love. Tonally, it’s always complicated. Nabokov shifts effortlessly from one genre to another — detective story, revenge drama, legal brief, road story, confession and fairy tale…”
Genres exist in Mathematics and Physics as well, and there must be narrative within them, specific to the traditions of the genre, and pressing against the genre boundaries in exception work, and informed by the collaborative effort of craftsmen within the genre. See, particularly:
How Mathematicians May Fail to be Fully Rational, by Dr. David Corfield –
http://www.dcorfield.pwp.blueyonder.co.uk/
HowMathematicians.pdf
Friday 2007 June 1 at 6:43 pm
>computers might be quite good at finding formal proofs with human assistance, with a human guiding the direction of the proof (which computers are bad at), and the computer filling in tiresome routine details (which humans are bad at).
The software that is able to do such things (or something very close) exists today , so why is it so rare that it is used in research? I think there are two reasons. The first one is that creating real (formal) proofs is tedious. It will still be tedious even if software supporting that was much better than today. Machines can not guess human intentions. Those intentions have to be communicated to them in a precise way. For complicated ideas this is tedious and there is no way around that.
The second reason is a the lack of libraries of formaly verified facts that provide the background for original research. To push the boundaries of known mathematics you need to get there first in a formalized way. Creating such standard libraries would be a substantial effort and would not be considered original research. I don’t see who in the mathematical community would be sufficiently motivated to do this on a larger scale.
I think if using computers for creating and verifying mathematics ever becomes common, it will first happen for the purpose of software and hardware verification in industry, not in academia. This is where there is motivation due to huge cost of trivial mistakes.
Thursday 2007 December 6 at 5:00 pm
Nature (including “mathematical nature”) seems to be more imaginative than we are (microbes shaped like screws, the Mandelbrot set….) What a waste to refuse to look–like Galileo’s churchmen, for example. I think this is the main reason to expect experimental mathematics to be fruitful.
Monday 2008 May 26 at 4:48 pm
Well, we could use a computer that would solve nonlinear and time-variant equations involving complex functions. For instance, the exponent of a vector can be a also be complex exponential function. This would most likely require some new mathematics, such as a convolution of a convolution and/or the transform of a transform, rather than a series or product expansion.
Two things that computers lack: “emotion” and “intuition.
Wednesday 2009 May 6 at 4:15 pm
I have what one would call a conjecture. The only problem here is, does what I am about to present fall under the realm of ‘experimental mathematics’? I think it does since I am going to try to convince the reader that an interesting connection between mathematics and physics exists as a result of this specific case. In addition I am making an outlandish claim that this result if true represents a ‘needle in the haystack’ pathway through the Shang (Tsang) mountain pass of the mathematical Himalayas to a new vista from which interesting things will be marveled at and expounded upon
. Here goes. First of all I just want to say that I love playing around with very large numbers and that I am very good at it. In fact I am very good at dismissing most coincidental relations that pop up in large number space. There is a realm in large number space where such coincidences may not be easily dismissed and that is in range of numbers that exist around 10^40. This is an area that had fascinated physicists such as Dirac, Eddington and so on because of strange coincidences with large dimensionless physics calculations in this range presenting conundrums. The conjecture I have in mind is: hc/2piGm^2 = (640320^3 +744)^2 *70^2. The term on the left is a semiclassical physics form and using the latest codata values hc/2piGm^2 = 1.688497754…*10^38 (dimensionless) where m=neutron mass. If we invert this number we can see that it looks like the attractive Newtonian potential between two neutrons but with the features h and c which make the calculation semiclassical: 2piGm^2/hc ~10^-39.We could say that this would be a representative feature of weak gravity or how weak gravity is in comparison to the other gauge forces. What about the term on the right: (640320^3 +744)^2 *70^2 = 1.688684379…10^38 (which is an integer but I spare writing it out here). Actually 640320^3 +744 (an integer) is nearly the Ramanujan’s constant e^pi(sqr163) which is extremely near to being an integer. The number 70^2 is related to the construction of the Leech lattice through something called the norm 0 vector. I want to point out how close 1.688497754…*10^38 is to 1.688684379…*10^38. Could be a coincidence. But… look at this: hc/2piGm^2 = Mp^2/m^2 (where Mp=Planck mass and m=neutron mass). We get to a really nice compact symmetric looking form which involves squaring: (640320^3 +744)^2*70^2 = Mp^2/m^2. By the way I think the neutron gives the values we are looking for are due to something called the proton to neutron transition drip line in extremely dense matter states and because it is charge neutral. Briefly, let me show how this symmetry works to a large advantage in black hole microstates. The classical Bekenstein-Hawking entropy is: Sbh = 1/4 A 2pi*c^3/hG. We can define a large mass black hole using Mp^2/m^2: where Mbh = Mp^2/m^2* Mp2/e = 2.703857…10^30 kilograms (e is the base natural log 2.71828183…). It turns out that the Bekenstein-Hawking entropy for this black hole can be represented: Sbh = 4pi/e^2 (Mp^2/m^2)^2 = 4.848685…10^76 microstates. Because of this a very symmetrical compact form for the Bekenstein-Hawking entropy can be written as: Sbh = 4pi(number of matter fields)^2 * m^2/Mp^2, again where m^2/Mp^2 represents the inverse of 1.6866497754…10^38 and is probably representative of weak gravity action throughout the semiclassical gravity scale. The number of matter fields (binaries) can always be obtained by dividing the mass of the black hole in question by the neutron mass (never the proton mass) divided by 2 to obtain the number of ‘binary’ matter fields. This relation holds all the way through the semiclassical gravity before quantum gravity starts to dominate at entropies 4pi. If we want, we can replace all Mp^2/m^2 relations above with (640320^3 +744)^2 *70^2 and it does not affect the result in any significant way. As a matter codata values have uncertainties (maybe 9 or 12 decimal places max with some values) to the extent that it would be impossible to calculate hc/2piGm^2 or Mp^2/m^2 as an integer value out to 39 places. Will not happen. But if hc/2piGm^2 is related to elliptic modular forms? We could then merge: Sbh = 4pi(number of matter fields)^2 * 1/(640320^3 +744)^2 *70^2 and it gives excellent agreement with the same black hole values using Sbh = 1/4 A 2pi*c^3/hG. Again if this is true it represents a nexus where pure mathematics and physics become one and the same.
Friday 2009 May 8 at 9:34 pm
Just a quick note: I inadverdently left out a factor of 2 in the above thread so that, hc/2piGm^2 = ((640320^3 +744)^2 *70^2)/2. So that ((640320^3 +744)^2 *70^2)/2 = 1.68864379…*10^38 which would be an integer if written out to 39 places. Also,this makes the Bekenstein-Hawking form: Sbh = 8pi(number of matter fields)^2 /((640320^3 +744)^2 *70^2). The 8pi is a good sign that we are working within the symmetry of a Lorenztian metric. If this relation is true we can also determine the range of validity of the semiclassical gravity, that is when the entropy approaches 4pi since the additional factor 2 is not in the physics form. The real question however is, hc/2piGm^2 = ((640320^3 +744)^2 *70^2)/2 ? before it can be determined whether this is a safe approach.