Friday, August 20, 2010

Mathematics promotes a scientific temper?

Our honourable President, Smt. Pratibha Patil while inaugurating the International Congress of Mathematicians (the largest and most prestigious congregation of mathematicians of the world, held every four years and for the first time in India) claimed that the study of mathematics leads to the development of a scientific temper.

To my mind this statement needs to be proved (though of course not in the sense my mathematician friends might want). Since neither she nor I are mathematicians, we are perfectly placed to comment on mathematics and mathematicians! The importance of mathematics is not in doubt. As a physicist, the "unreasonable effectiveness of mathematics" is obvious to all practitioners of the subject. However using mathematics as a tool is very different from doing mathematics itself which is what mathematicians do. And therein lies the rub.

Mathematics, to put it somewhat simplistically, is purely Platonic -- that is, complete understanding is to be achieved, and in fact, is achieved through pure thought. Pure mathematicians needs no knowledge of the physical world to prove their theorems. In fact, many of them (like G. H. Hardy) would be horrified at the thought that their work had any connection, dependence or relevance to the real world. Unlike physics, which is based on observation and experiment (leading to a theory) and eventual falsification (leading to a new theory, in Popper's famous description) a correct mathematical theorem is for eternity. No new observation of the physical world, no new insights into nature's working can have the slightest effect on its correctness. The only thing that can topple a theorem is a realisation that there was a flaw in its proof. 'Laws' of physics, on the other hand, are continuously falling by the wayside, replaced by new ones, based on newer and better experimental observations of the real world. (Even String Theory, the most esoteric and mathematical of all physical theories needs to pay obeisance to some fundamental symmetries of nature).

In the absence therefore, of a scientific principle or method, pure mathematics cannot be classed as a science. It is a pure art form, existing by and of itself with no supporting scaffolding from nature. It therefore requires no scientific bent of mind from its practitioners (in the sense that it does not require that its practitioners demand rational explanations of all natural phenomenon). A stone thrown into the air at an angle could well follow a parabolic path because God so decreed it, rather than the laws of physics. It would make not an iota of difference to any theorem past, present or future. (In recent years, this supremacy of mathematics has been partly dented by its somewhat intricate dependence on other branches of science -- Jones Polynomials and Chern-Simons theory, or the proof of the four colour map theorem which required the use of computers to eliminate a few remaining counter examples).

There is therefore no evidence, in my humble opinion, that the study of mathematics either promotes or even requires a scientific temper. This is also the reason why mathematicians as a community are far more religious (Ramanujan being the classic but by no means the only example) than their counterparts in the physical sciences -- not because mathematics promotes a belief in the supernatural, but because it does not require you to relinquish your belief in it. This has also been my personal experience, though I obviously do not have statistics to prove this claim.

Disclaimer: I do not claim that physicists are not religious -- many are, surprisingly. However it is a matter of statistics. As a fraction, fewer physicists in my opinion are overtly religious or believe in non-rational explanations of natural phenomena, compared to their mathematics colleagues.

Note added: Morris Kline discusses some of these issues and many more and twentieth century mathematics in his book -- Mathematics -- The Loss of Certainty. In particular he discusses what he calls 'The Authority of Nature" in the last chapter and in and around page 333. I thank Rahul Siddharthan for acquainting me with this book.

21 comments:

AmOK said...

I am about to lose my "Scientfc" temper at your spell-checker.

Be that as it may, if you had read the Vedas you would have known that indeed a scientific temper and mathematics are intricately connected. To bring you up to date, OLO, may I humbly present this.

Madhavan said...

I dispute your narrow definition of science. I would define what a mathematician does as seeking the explanation behind patterns, just like any scientist. Unlike the natural sciences, in which these patterns arise in the world we observe around us, mathematics typically deals with patterns "observed" in abstract entities such as numbers and more complex mathematical artifacts. A mathematician notices a pattern, seeks a rule that governs this pattern and verifies it as a theorem. Mathematicians would like this verification to be purely analytic (a proof based only on logical reasoning), but sometimes it is partly experimental (e.g. the 4 colour theorem).

The process of finding an explanation for the pattern and verifying the rule is not so different as you make out from a physicist building models to explain natural phenomena. I would say that this process is as much through "experimentation" as in any other science, even if the "experiments" are performed on abstract entities. The fact that the rules are more "absolute", a posteriori, does not matter. And, indeed, absoluteness is a matter of opinion: consider non-Euclidean geometry, non-standard analysis, quantum computation. Mathematics, like any other "science", has to start from some "self-evident truths" and some of these truths turn out, later, to not be so self-evident after all.

To me "scientific temper" means two things: recognizing the need to "explain" why things are the way they are (rather than blindly accepting them as unalterable "facts of life") and accepting that such explanations need to be validated and may, indeed, turn out to be wrong (rather than blindly believeng what an "authority" has to declare on the matter). Each scientific community has its own culture of validating its explanations and it is probably human nature that each community believes its own way of doing things is immeasurably superior to everyone else's.

Venkataraman said...

OLO,

I now understand why my grandfather's degree says B.A. mathematics instead of B.Sc. mathematics.

YHR

Anant said...

It is very curious that the expression "scientific temper" seems to be known only in Indian english, see Current Science, July 25, 2005 issue, letter by Anuradha Ravi. Maybe off topic, but this brings us to the question as to why our leaders harp on this issue? As those who preside over functions where coconuts are routinely broken, astrologers readily consulted, God men feted...maybe you can cast some light OLO.

Sourendu said...

I'm pretty astonished at the unreasonable effectiveness of the English language, and Bengali in some cases, at expressing scientific thoughts. Since the use of mathematics in physics is little more than compact linguistic notation, (people who live near Kalakshetra, or a building called Lilavati, probably known that maths can also be written in verse or, less precisely, expressed in dance; thus making the relationship to language clearer) I'm not more surprised about the effectiveness of maths in describing physical reality.

If Mrs. Patil's speech writers knew all this, surely she would have talked about poetry improving your temper.

gaddeswarup said...

I am not sure what scientific temper is. At one time I was more interested in Physics than Mathematics but had difficulty in understanding what seemed to be some leaps. Harish Chandra expressed similar feelings in his comments on Dirac:
"This remark confirmed my growing suspicion that I did not have the mysterious sixth sense which one needs in order to succeed in physics and I soon decided to move to mathematics"
(From "Bhabha and his Magnificient Obsessions" by G. Venkataraman, page 78)

Raj said...

I would think that your case rests on a somewhat narrow definition of scientific temper. To me the more relevant distinction between doing physics and mathematics (or, equivalently, being a good physicist as opposed to a good mathematician) is the requirement of the ability to make creative guesses about the way nature functions that cannot be logically described in an adequate way. This is also what Gaddeswarup above points out.
As far as experiment is concerned, i do not immediately see that using the computer in mathematics is very different from the physicist using his lab. So again, I would think that this distinction rests on how broad a definition of experiment one uses. Admittedly, tho, mathematicians use computers much less frequently than physicists use labs. But maybe that will change in the near or far future, as people run out of conjectures that can be proved by logic alone.

Rahul Basu said...

Madhavan, Raj: Both of you have tried to widen my definition of the scientific method and that is your prerogative. However, I think the idea of using observation to decide your conclusions and hence the theory, is uniquely scientific rather than 'mathematical'. I accept that these days the line between pure and applied mathematics is getting blurred and so is the rigidity of this stance, but the hard core pure mathematicians still balk at the idea of using computers or other such external artifices to construct proofs. To take a somewhat elementary example, the proof that the square root of two is irrational does not depend on any external observation -- it is just 'it' . On the other hand, even apparently sacrosanct notions like Lorentz or Gauge symmetry in physics have the possibility of being violated perhaps in some part of the universe, which we may notice someday.

Of course things are changing. Words from physics like statistical mechanics, Ising model etc. pepper the Fields Medal Laudationes and the speech of Laci Lovász actually mentioned the Large Hadron Collider! (A colloquium given in our institute by the well known mathematician V. S. Varadarajan also talked about the LHC and in fact asked young mathematicians to look at the new notions of space time that the LHC is expected (maybe) to discover !!) . Thus the stratospheric purity of 'pure' mathematics is getting diluted by its interaction with physics :-)

Rahul Siddharthan said...

For some reason I missed this post when it was written. I think some mathematicians themselves dispute the narrow interpretation of mathematics that you have given. For example, the recently deceased V I Arnold. And I have recently been reading Morris Kline's "Mathematics: The loss of certainty" which is, to a large degree, a strong indictment of twentieth-century mathematics and in particular of the axiomatisation programme and the idea that mathematical structure and concepts can exist independently of the physical world. Jaynes observes that atrocious theorems like the Banach-Tarski paradox should be viewed as "reductio ad absurdum" results on the "foundations" of set theory, rather than as the paradoxical but true results that they claim to be (he puts it more strongly, as I remember). Hermann Weyl, Murray Gell-Mann and others said similar things.

I find it a fascinating subject. Let me just toss off a hand-grenade here: I believe that, if Alan Turing had been born 100 years earlier (and had, magically, intuited his ideas on computation in the early 19th century), the history of mathematics would have been different. In particular, there would have been no need for "real numbers" as usually defined: all the numbers that one actually needs to use are "computable numbers" by Turing's definition. In fact, it is impossible even to specify a non-computable number using a finite amount of text. (This is related to the fact that a "real number", as usually defined, contains an infinite amount of information, since it has an infinite amount of digits with no pattern to predict them; but any real number that we need, such as pi or e, can be defined precisely in a rather short sentence, so contains a finite amount of information).

Now, the computable numbers are countable. So if Turing had defined them in the early 19th century, nobody would have needed uncountable sets, and all the usual paradoxes of set theory would never have arisen...

Rahul Basu said...

Rahul S: Yes, I am aware of Arnold's somewhat strong views on this -- for example http://pauli.uni-muenster.de/~munsteg/arnold.html where he starts by calling mathematics a part of physics -- not an idea that would endear him to many mathematicians. But then, he would probably be called a mathematical physicist!

I am not an expert on this subject so what is say is somewhat off the cuff (like my post). However, yesterday I happened to watch Simon Singh's documentary on Andrew Wiles and Fermat's Last Theorem. It's beautifully made (its available on Google videos) and there is something very touching and deeply moving about the fact that he was proving something that can never be overturned -- it's a truth for all eternity. No physicist can ever boast of something like this. In that sense I would say it is close to art than to science, though one can dispute that. My view is intrinsically tied up with the idea of falsifiability of science. Of course this absence of falsifiability is completely different from something like string theory. That is a theory of the world which in the way it is constructed is not falsifiable, at least at present day energies (but is in principle if we can reach those energies -- or someone will someday come up with some experiment that can be done today). Fermat's Last Theorem is not in that category -- it is not falsifiable because it is an eternal truth (unless someone finds a flaw with the proof itself but that is a different matter). So pure mathematics is indeed different from doing science. Proving a beautiful theorem is like creating a work of art that will last for ever, if you will pardon the somewhat overblown imagery. It therefore can be practiced by anyone without necessarily developing a scientific 'temper'.

Rahul Siddharthan said...

I don't think there is such a difference from Wiles and physicists as all that. Wiles showed that, if one accepts the axioms of number theory (I am not an expert but I'd guess he assumes the Peano axioms, at least, and perhaps a lot more), then Fermat's last theorem is true. A physicist can show that, for example, if Newton's law of gravitation is true and there are no other forces, a planet follows an elliptical orbit around the sun (in reduced coordinates, anyway). The latter statement is as "true" as Fermat's theorem. Falsifying, to a physicist, would mean rejecting the underlying axiom (Newton's law). And of course that's what eventually happened.

Whereas, when a mathematician "proves" something like the Banach-Tarski theorem, he does not decide that this nonsensical result means the axiom of choice (on which the proof depends) must be rejected. Indeed, as Kline details in his book, this axiom was controversial long before Banach-Tarski. And in 1963, Cohen proved that it is "undecidable" within the ZF axiomatic system of set theory. So to accept it is more a matter of faith than anything else. As far as I can tell, it is retained by most mathematicians only because, if you throw it out, you throw out an absolutely enormous chunk of twentieth-century mathematics.

Perhaps you can argue that "mathematics" means starting from axioms and proving theorems, but not caring about the "truth" of the axioms. Physicists care about the assumptions they make (the theories may be false) but, assuming the theories, results in physics are just as "concrete" as in mathematics. The question is whether this view of "mathematics" is appropriate, or have we been brainwashed by the set-theorists and the Bourbakists...

Rahul Basu said...

Rahul S: You have answered the question yourself -- ..." "mathematics" means starting from axioms and proving theorems, but not caring about the "truth" of the axioms. Physicists care about the assumptions they make"...Indeed I would think that caring about the truth of the axioms is what sets physicists apart from mathematicians. Anyway, this is turning into a discussion between two physicists -- pity that we cannot get more mathematicians into this discussion. It would be interesting and of course more diverse (and diverting!)

kapil said...

Like RahulS, I missed this non-controversial post since I had no
fish to fry being too far from the sea. :-#

Two mathematicians (Madhavan and Anand) have already made some remarks
so let me make a few orthogonal ones.

Does scientific temper include the ability to distinguish between
trust and faith? In that case most mathematicians that I know
have that down in spades. In fact, more than any other discipline
mathematics teaches people to "verify and then trust" rather than
"believe as a matter of faith". Unlike the results of LHC, the
theorems of Wiles and Wiles-Taylor can be verified by investing
"polynomial time and space"; the LHC needs many hands at work
synchronously, a.k.a. "non-deterministic Turing machines".

The most renowned physicist of the twentieth century (Einstein)
expressed a belief in god, whereas other than your one example
(Ramanujan), I would be hard-pressed to find similar belief among
mathematicians of renown: Hilbert, Weyl, Cartan, Weil, Serre,
Grothendieck (or even Hardy whom you mention without mentioning his
atheism). They may have (or have had) some unusual philosophical ideas
but religion was not among them.

To get back to the original question "Does Mathematics promote
scientific temper?" My answer is, "Yes, if you do it like
mathematicians do." (i.e. if you work it out with a pencil). Real
science is about doing the calculation (see my Aug 11 post on
"Mast Kalander"), the rest is stamp-collecting.

Rahul Basu said...

Kapil: I find your claims deeply unconvincing :-) . First of all, it's always dicey to quote Einstein about God. His remarks about that entity were always shrouded in ambiguity and it was never clear he was really talking of some kind of a personal god that most religions urge us to believe in. Secondly, quoting a large number of mathematicians (and their beliefs or otherwise) is always risky -- first of all we don't quite know what they really thought unless they explicitly discoursed in this and most of them didn't except Weyl who also was more concerned about the success of mathematics in the real world. I could give counter examples from my (math) colleagues here, most of whom are actively religious. Neither of these would prove (or disprove) the question posed in this post. It is possible to believe in what you call"verify then trust" (in other words, 'anyone' with a certain level of (mathematical) competence could check with pencil and paper and lot of time, the proofs of Wiles or Wiles and Taylor or for that matter any other mathematical theorem) and yet also believe that some parts of nature are governed not by physical (observable) laws but by some other entity. This dichotomy is possible because what mathematicians do when they 'verify then trust" is different (or perhaps removed) from what is needed to "verify then trust" the laws of nature. Of course in some cases, nature's laws and mathematical theorems or results may be intimately related, but often there really are no direct connections. It is in this context that I proposed my idea of a scientific temper. If you expand the idea in the way Madhavan has, then it is different.

BTW on a slightly different but related note, I recall you once telling many of us long ago that mathematics was an art, not a science. Isn't that then a rather damaging admission?

As I said in my addendum to the post, Morris Kline address some of these issues in his book.

Anant said...

OLO: if there is a child genius who is good at solving chess puzzles, one trusts him or her only at chess. If there is a violin virtuoso, one appreciates that person for his or her violin playing. If someone is very good at cracking problems in mathematics, well, we trust him or her in mathematics. Why should one believe that a chess or violin whiz has any insights into epistemological questions more than any one else? Analogously why should a mathematician be trusted any more in these issues than any one else? Stated differently, these are all different corners of the brain, and an active area in one region of the brain does not necessarily say anything about other regions. Pray tell...Yhs.

Rahul Siddharthan said...

About Ramanujan: here's what Hardy says in his "Twelve lectures".

"I am sure that Ramanujan was no mystic and that religion, except in a strictly material sense, played no important part in his life. He was an orthodox high-caste Hindu, and always adhered... to all the observances of his caste. He had promised his parents to do so, and he kept his promises to the letter... I say [previously] '...his religion was a matter of observance and not of intellectual conviction, and I remember well his telling me (much to my surprise) that all religions seemed to him more or less equally true...'"

He concludes from that that Ramanujan was observant only for reasons for social nicety. However, it seems possible to me that Hardy misinterpreted the "all religions seem equally true" remark, which is much more likely to be said by a believing Hindu than by a believing Christian or Muslim. The Biblical religions are much more demanding of their believers.

In any case, I find examples like Ramanujan and Einstein unconvincing, as also the argument that mathematicians are less likely to be believers because they only care about proof from axioms, and not all truths may follow from axioms. I can't think of a physicist who was as articulate, in his criticism of religion, as Bertrand Russell, one of the major proponents of the logical/axiomatic approach to mathematics...

kapil said...

[q]uoting a large number of mathematicians (and their beliefs or otherwise) is always risky -- first of all we don't quite know what they really thought unless they explicitly discoursed in this and most of them didn't except Weyl who also was more concerned about the success of mathematics in the real world.

I would agree that in the case of Serre and Cartan my remark is a step into the unknown. However, there is an extensive biography of Hilbert by Courant. Weil has written his own autobiography and Grothendieck has written extensively about his philosophy on "life, the universe and everything". (Grothendieck did one "get religion", but later grew out of it.)

BTW on a slightly different but related note, I recall you once telling many of us long ago that mathematics was an art, not a science. Isn't that then a rather damaging admission?

Only if you assume that I am consistent! :P

Anyway not being a science does not mean that "it does not promote scientific temper"; there are works of art that are meant (by their creators) to promote scientic temper.

Your argument hinges on the relationship of one's thinking to the physical world. In your view scientific temper requires one to verify one's thoughts through experimental interaction with the physical world. The thought processes themselves do not need to be rational --- perhaps rationality is a consequence that the physical world enforces on us.

A mathematician's view would be somewhat different. First of all, the thought process has to be a priori rational. Secondly, the physical world is a guideline or a source. Deep, deep, down most mathematicians know that "what we see is coloured by the glasses we wear whereas clear thought is a fountain of permanent truth."

In considering mathematics as an art, I will paraphrase Weyl to note that "Whenever I have had to choose between what is useful and and what is beautiful, I have chosen the beautiful." This is not an unscientific view, merely an un-technological one.

vbalki said...

I hope Rahul will not mind it if I reveal that he instigated me into making a comment on this issue, which has long fascinated me. I must thank him, as well as several of you (Rahul Siddharthan, Madhavan, Kapil, and the others) for your thoughtful remarks that have added to my education on the subject.

I think that, to a large extent, different people have addressed slightly different aspects of the topic, including the meaning of "scientific" and "scientific temper". Which is why there's truth in each of what appear to be different positions.
The question of whether mathematics is, ultimately, an art or a science falls, in my opinion, in this category.

But there does exist a deeper question. Let's agree, somewhat loosely, that any "science" looks for patterns in a broad sense in a certain methodical way that we can recognise when we see it, even if we can't fully codify it to cover all cases. The really deep question (that many people have written whole books about) then is whether the patterns of mathematics are "out there in reality", or whether they are constructs of the human mind. ("Is there 'pi' in the sky?") Avoiding the trite answer that the human mind itself is (presumably) a part of physical reality, I think the answer (at our present state of knowledge) is that mathematics is a bit of both.
I know you're going to say that this is essentially the same as saying that I don't really know the answer, which would be perfectly true.

I do think, however, that mathematicians have a specific psychological trait, presumably induced by the nature of the reasoning processes that their training imparts, that is somewhat different from that of other scientists. They tend to apply the rigorous yardsticks they use, for what constitutes a proof and what does not, even to casual statements that less rigorously logical minds would accept without argument, but with an automatic and more intuitive appraisal of the likelihood of their being valid or not. I hasten to add that I don't intend either any criticism or any commendation of this trait. I submit that it is worth recognising this difference as a fact of life, and allowing for it. Doing so would avoid many arguments!

Rahul Basu said...

Kapil: "there are works of art that are meant (by their creators) to promote scientic temper." -- now now, Kapil, isn't that pushing it a bit to explain your 'art' statement? And about Weyl, looking for beauty or elegance is not confined to the arts - it's true in the physical sciences. So that cannot settle the issue.

jv said...

Some statistics may help:

"We found the highest percentage of belief among NAS mathematicians (14.3% in God, 15.0% in immortality). Biological scientists had the lowest rate of belief (5.5% in God, 7.1% in immortality), with physicists and astronomers slightly higher (7.5% in God, 7.5% in immortality)."

From: Leading scientists still reject God; Nature, Vol. 394, No. 6691, p. 313

http://www.nature.com/nature/journal/v394/n6691/full/394313a0.html

I wonder, how basic and engineering sciences differ.

vbalki said...

In response to jv's query: yes, it would be very interesting to see if there's a significant difference in religious belief between practitioners of basic science and engineering. I'd suspect that the former might have a smaller number of believers, going by the general impression I have gathered over the years.

Judging by my immediate environment in an institute of technology, however, and going by the enthusiastic observance of things like ayudha puja by all sections of the staff at all levels, I'd say there's very little difference! Pretty much the same thing appears to hold good in the big govt. labs of DAE, ISRO, DRDO, etc.

The bottom line (since this blog started as one on scientific temper and mathematics) is that most technically trained people in this country have managed to neatly compartmentalise their minds into disjoint sets, each with its own premises and methodology. No mean feat, this!