The wisdom of Gian-Carlo Rota (1932-1999)
From www.rota.org:
Graph theory, like lattice theory, is the whipping boy of mathematicians in need of concealing their feelings of insecurity.
Mathematicians also make terrible salesmen. Physicists can discover the same thing as a mathematician and say ‘We’ve discovered a great new law of nature. Give us a billion dollars.’ And if it doesn’t change the world, then they say, ‘There’s an even deeper thing. Give us another billion dollars.’
When an undergraduate asks me whether he or she should major in mathematics rather than in another field that I will simply call X, my answer is the following: “If you major in mathematics, you can switch to X anytime you want to, but not the other way around.”
Flakiness is nowadays creeping into the sciences like a virus through a computer system, and it may be the greatest present threat to our civilization. Mathematics can save the world from the invasion of the flakes by unmasking them, and by contributing some hard thinking. You and I know that mathematics, by definition, is not and never will be flaky.
Note: Quotation here does not necessarily imply endorsement by Shtetl-Optimized LLC or any of its subsidary enterprises.
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Comment #1 April 9th, 2007 at 7:25 pm
Rota was wise indeed. He said combinatorics was ‘honest’. Unfortunately, there are some good physicists who don’t get a cent of the many billions of dollars being wasted on stupid experiments.
Comment #2 April 10th, 2007 at 2:57 am
Rota says: “Mathematics can save the world from the invasion of the flakes by unmasking them, and by contributing some hard thinking.”
Serge Lang? Alexander Grothendieck? Isaac Newton?
These mathematicians are so great, that even their undeniably “flaky” non-mathematical writings are quite interesting to read. 🙂
Comment #3 April 10th, 2007 at 7:43 am
Mathematics can save the world from the invasion of the flakes by unmasking them, and by contributing some hard thinking
If mathematicians CAN do it, then why aren’t they doing it?
The most outrageous example is darwinism. The whole theory is based on assumption that evolution is driven by random mutations. Take away this “randomness” – and the ideological message of darwinism will collapse. Why don’t you, complexity theorists, explain to public that the statement of randomness is meaningless? Why don’t you require a mathematical model that demonstates the claims of darwinists? One of the biggest surprises for me after moving to the West was the discovery that mathematical elite here absolutely supports dawinism, apparently for ideological reasons. (Most Soviet mathematicans were extremely skeptical about it; darwinism was rightfully treated as integral part of marxism there – as laughable as the rest of it).
It’s exactly the assumption of “randomness” that allowed undeniable facts regarding evolution to be cast as cornerstone of materialistic dogma. Why don’t you argue with that assumption? There’s certainly some inconsistency here, because in other cases, that don’t have ideological overtones yet, you are arguing openly (e.g. DWave affair).
[ BTW, the readiness with which the idea of global warming,
or global cooling, is embraced – absolutely uncritically – by
scientific establishement – is equally worrisome – but I don’t want to return to this topic ]
Comment #4 April 10th, 2007 at 8:50 am
“Why don’t you require a mathematical model that demonstates the claims of darwinists?”
Vasily, I took a course in evolutionary biology last semester, and it was almost entirely math-based. Most of what working evolutionary biologists do these days requires a lot of math.
“Why don’t you, complexity theorists, explain to public that the statement of randomness is meaningless?”
Or perhaps you could explain it to us? Since have considered this carefully, please explain in what sense mutations are non-random.
Comment #5 April 10th, 2007 at 9:04 am
Scott, didn’t you read the text in Rota’s click-through license agreement? By quoting Rota, you and all of your subsidiaries are endorsing him.
Comment #6 April 10th, 2007 at 9:13 am
Quote:
Here we can add a remark by I.M. Gel’fand: there exists yet another phenomenon which is comparable in its inconceivability with the inconceivable effectiveness of mathematics in physics noted by Wigner – this is the equally inconceivable ineffectiveness of mathematics in biology
End quote
This is an excerpt from article by V.Arnold, “On teaching math”, which is very interesting by itself, and can be found
here
In this article, you can find a lot of stuff related to the subject of this thread.
Comment #7 April 10th, 2007 at 9:33 am
Vasily, what do you want from us?
Comment #8 April 10th, 2007 at 9:43 am
You are conflating evolutionary biology with the narrow model for historical reasons called the darwinian model. But even this narrow model is more than “random mutations”. The smallest model is variation (independent of selection), selection and hereditary of characteristics. Variation of itself doesn’t work fast, it is the direction which selection gives that enhances evolution.
By adding up some of the earliest known evolutionary mechanisms (now there are many more), an evolutionary biologist described it in a mathematical simile:
“Science educators need to help their students gain a better understanding of how powerful selection operating in nature can be. One of the pedagogical strategies that educators can use involves simple mathematics. The four genetic consequences of selection operating in nature can be represented in mathematical terms.
[…]
(1) The SELECTIVE ELIMINATION (subtraction) of harmful traits, that is, the removal of maladaptive genetic information;
[…]
(2) The SELECTIVE ACCUMULATION (addition) of new adaptive genetic information in the form of new mutations each generation.
[…]
(3) The SELECTIVE MULTIPLICATION of adaptive genetic information each generation brings about an EXPONENTIAL increase in the frequency of adaptive genes in a population. (multiplication)
[…]
(4) The SELECTIVE RECOMBINATION of adaptive genetic information each generation. (division and selective addition)”
( http://scienceblogs.com/evolvingthoughts/2007/04/what_makes_natural_selection_a.php )
Turning from the simple simile, there are plenty of real mathematical population models that describes the outcomes of the above processes when we include later genetics to describe hereditary. Statistician and evolutionary biologist R.A. Fisher laid much of the groundwork. He has been described as “the greatest of Darwin’s successors”. (Wikipedia.)
Btw, I think it is a remarkable fact that evolutionary biology gives the greatest precision of all natural sciences. The basic structure that common descent with variation predicts is the phylogenetic tree. There are many ways to arrange these trees, yet analysis can pick out a small subset of likely candidates for later confirmation.
“Nevertheless, a precision of just under 1% is still pretty good; it is not enough, at this point, to cause us to cast much doubt upon the validity and usefulness of modern theories of gravity. However, if tests of the theory of common descent performed that poorly, different phylogenetic trees, as shown in Figure 1, would have to differ by 18 of the 30 branches! […] However, as illustrated in Figure 1, the standard phylogenetic tree is known to 38 decimal places, which is a much greater precision than that of even the most well-determined physical constants.” ( http://www.talkorigins.org/faqs/comdesc/section1.html#independent_consilience )
Comment #9 April 10th, 2007 at 10:19 am
[…] Scott Aaronson provides quotes from someone else whose lectures I attended around the same time, Gian-Carlo Rota, who taught at MIT, including one that ends “You and I know that mathematics, by definition, is not and never will be flaky”. I kind of agree with the sentiment in the full quote, but my experience with Rota back then was a rather weird one. For some misguided reason I had decided that since category theory was the most abstract kind of mathematics I had heard of, it would be a good idea to take a course on it. The only course on the subject was a graduate course down at MIT offered by Rota, so I started going down there to sit in on it. A few lectures into the course Rota all of a sudden announced that he had decided that there only those students actually enrolled should be taking the course, and that the several of us who were just auditing should leave. So we did, somewhat mystified (it’s not like the room was over-packed or anything). To this day, I still don’t know what that was about, perhaps Rota knew that he was doing me a favor by stopping me from thinking about category theory at that point in my education, when in retrospect it seems likely that it really would have been somewhat of a waste. […]
Comment #10 April 10th, 2007 at 10:32 am
Vasily, you are confusing many things. Evolution does not require random mutations, or even uniformly distributed mutations. Merely getting _some_ mutations in a reproducing organism will be sufficient to have evolution select for a different population. It’s incredibly difficult to construct a situation where it wouldn’t happen.
Additionally, scientists, especially experts, are not so quick to judge that the earth is warming or cooling or whatever it will be, but in the face of massive and growing evidence, and consistent predictions from 19th, 20th and 21st century physics, it’s a bit hard to stay agnostic. At minimum the atmosphere is changing in makeup at an alarming rate, and likely these changes will cause problems for very simple to understand reasons. This is equivalent to a no-go theorem for the skeptics.
Comment #11 April 10th, 2007 at 11:01 am
“Most Soviet mathematicans were extremely skeptical about it; darwinism was rightfully treated as integral part of marxism there – as laughable as the rest of it”
Vasily, you are wrong on this point as well. If you think that darwinism was an “integral part” of soviet ideology, you may be confusing it with Lysenkoism, which you can read about on wikipedia if you’d like to educate yourself.
Comment #12 April 10th, 2007 at 11:58 am
Vasily Shirin. Great troll or greatest troll? You decide.
Comment #13 April 11th, 2007 at 7:56 am
Science Fiction about Mathematical universes – flakey Math or not, you decide. And, in another part of the multiverse, you don’t decide!
This is simultaneously peer reviewed, a rant, and as crazy as any Science Fiction novel by Greg Egan, Rudy Rucker or Vernor Vinge – but in a good way! I’m not including the URLs at arXiv.com nor MIT, but do go look! This might deserve a thread of its own in Shtetl-Optimized.
The Mathematical Universe
Authors: Max Tegmark
(Submitted on 5 Apr 2007)
Abstract: I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters and initial conditions to broader issues like consciousness, parallel universes and Godel incompleteness. I hypothesize that only computable and decidable (in Godel’s sense) structures exist, which alleviates the cosmological measure problem and help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems. Comments:
28 pages, 5 figs; more details at this http URL
Subjects:
General Relativity and Quantum Cosmology (gr-qc)
Cite as:
arXiv:0704.0646v1 [gr-qc]