A man hath sapiences thre
Memorie, engin and intellect also
- Chaucer
When Professor McConkey asked me to lecture in this
series, I knew I had to do it. Chances to share insights obtained
from a career in science with a general audience in a formal setting
do not occur often. Doubts came later, especially when an article
headed "Area Celebs to Lecture on Memory," and leading
off with "So what inspires some of the best and brightest
. . ."1 appeared in the Ithaca Journal.
Ultimately, though, and like many professors, I am better at stilling
doubts about being honored than at overlooking being overlooked.
What am I going to tell you? That creativity in the mathematical sciences is both very much easier and very much harder than in other human endeavors, similar in some ways and very different in others, and that memory is as essential to creativity here as elsewhere. Much will be based on personal experience, and anecdotal rather than researched. Along the way, I hope to convey something of the nature of the scientific enterprise, particularly as it is carried out by theoretical physicists.
"What is physics? Physics is a body of knowledge and a point of view. Both have grown out of the astonishing discovery in seventeenth-century Europe that the observed workings of inanimate nature, from how the planets move to how a prism makes a rainbow, can be accurately summarized in mathematical terms. Why the world is thus regulated and why we have evolved to a point a little beyond cats from which we can perceive these regularities are questions for which science has no answers.2
"That mathematical relationships ("Laws")
do exist is demonstrably true. The physical laws that have been
discovered - by a mixture of observation, intuition, and a desire
for a concise and therefore in some sense beautiful description
- do not merely organize experience, they organize it in a manner
that encourages disprovable predictions. The wave theory of light
predicted a bright spot in the center of the shadow cast by a
circular screen, a result so apparently absurd that it made believers
of disbelievers when it was found. . . . When there are disagreements,
it is often unclear whether the observations, the mathematics,
or both are at fault. The resolution of conflicts is emotional
and often heated. More than in most fields of scholarship, though,
when the dust settles, it settles for good. This suggests that
the physical study of nature is an act of uncovering a hidden
structure. That we are inextricably part of the structure makes
its study all the more fascinating."3 As you can see, I agree
with Einstein that "the eternal mystery of the universe is
its comprehensibility."
My epigraph, from the second nun's Canterbury tale, rather nicely attributes three qualities to the mind: memory, ingenuity which for the present purposes we can take to be synonymous with creativity, and the ability to reason. The ordering is also good: memory at the root of everything. Indeed, without memory there can be no scientific creativity. Isaac Newton wrote to Robert Hooke, "If I have seen further, it is by standing on the shoulders of Giants." The memory brought to bear on a scientific question need not be encyclopedic. But, even if truly creative scientists are likely to be quite selective about keeping up with the froth of intellectual fashion, every such person I have known has had a large number of facts and their consequences stored in the memory. The first moment of creativity usually comes in the form of an analogy with something previously understood. This has always been my experience. . .
The final working out of a creative thought requires
good training and technique: "intellect," I suppose.
This is not to suggest that a single "aha" ends the
truly creative part of scientific discovery. Einstein describes
as "the happiest thought of my life" his realization
that "if a person falls freely, he will not feel his own
weight."4 It took seven more years of false starts and other
creative insights before he reached his geometrical theory of
gravitation. Over this period, though, the threefold division
is not far-fetched, and the memory of such things as Newtonian
gravity and Riemannian geometry - the latter brought to Einstein's
attention by his friend Grossmann - was crucial.
So what, then, is different about creativity in the mathematical sciences, if a fourteenth-century quote - and, at that, a rendering into early English of a certainly much older thought - describes its structure? It is the fact that simple natural phenomena, being self-evidently mathematical, can be described in mathematical language, so that reasonable people can agree when something has been understood, and when a mistake has been made. Rather than talking in abstract, I shall tell you a story.
In the summer of 1962, six months after I came to
Cornell, I was invited to spend a month at the Bell Labs. During
that month I learned from P. W. Anderson about a discussion earlier
that summer between a Cambridge graduate student, Brian Josephson,
and John Bardeen. (These will be just names to some of you, but
they are famous names to condensed-matter physicists: all have
since won Nobel prizes, Bardeen his second; he already had one
at the time for the transistor, the device that started the "silicon
age.") The discussion was about a calculation done by Josephson
early that year, predicting that an electrical current could pass
without resistance between two superconductors separated by a
thin insulating layer. Bardeen had expressed the opinion, and
somewhat later said in print, that the calculation was incorrect.
The curious thing about this disagreement is that Josephson was
using the BCS theory of superconductivity (developed by Bardeen,
Cooper, and Schrieffer in 1957), which had finally and very beautifully
explained the then-forty-six-year-old puzzle of how certain materials
("superconductors") conduct electricity without friction
at low temperatures. Before your eyes glaze over, let me move
from the details to the bigger picture. A young man is predicting
a new and striking effect; his reasoning is being disputed by
the most respected figure in the field.
Bardeen was my hero. Not only because I had studied
the BCS theory but also because as a graduate student I had been
asked to give a few seminars on the background to the subject,
and for that purpose had read a long review article written by
him just before the new theory appeared. That review made it clear
that he had followed with great care and tenacity all the clues
that experimental research had been revealing. His own reasoning
was very hard to follow, and some of it, in hindsight, clearly
wrong. But it was wrong in a deep way: his wrong arguments had
the seeds of how matters did in fact work out when the pieces
were put together correctly. So, when I got back to Cornell, I
said very grandly to my first graduate student, Alexis Baratoff,
"Bardeen is having a disagreement with some Englishman. Bardeen
is always right. Let's find out why."
It took us most of the fall term to acquire the necessary
background. Meanwhile, Josephson's paper appeared. Yet, somehow,
it was difficult for us to decide one way or the other about Bardeen's
objection, because the key question of the distance over which
two superconductors can affect each other was buried in the formulas.
In the midst of this confusion, it occurred to me that it might
be useful to translate Josephson's steps into another (equivalent)
mathematical language - developed by L. P. Gor'kov and other Russian
physicists - which I remembered
as allowing for an easier visualization of spatial dependences.
No sooner was this done than it became clear to us that Josephson
was right!
It was not clear what general interest there might
be in our epiphany, so we delayed submitting it for publication.
Meanwhile early in 1963, I privately circulated the calculation
which earned me letters from Bardeen (saying that our calculation
was elegant but wrong) and Anderson (saying that he had always
believed Josephson). Some weeks later, we decided that since Josephson
had not calculated the temperature dependence of his current -
which would be a useful diagnostic prediction - we should do that,
thereby making our work more clearly new and interesting.
At this point my story takes an interesting turn.
Our work appeared in the June 1 issue of the journal Physical
Review Letters. It contained a calculational
error, so that the published curves were not a correct representation
of what the theory actually predicts! Shortly thereafter I received
a letter from a French physicist, P. G. de Gennes, in which he
said that he liked our work very much, but that we had gone astray
at the very end. He was right, so we corrected the calculation
and submitted an erratum - which appeared in the July 15 issue
of the same journal - in which we acknowledged our debt to de
Gennes. (Incidentally, de Gennes has since also won the Nobel
Prize.)
That August (1963) there was a conference at Colgate
University here in upstate New York. All of the players in this
game were there. By this time, the Josephson current had been
detected, by Rowell and Anderson at the Bell Labs, and Bardeen
had come around to agreeing that the calculation was correct,
but there had been no prior report of measurements of the temperature
dependence. At the conference, I met an experimental physicist
from the GE laboratories in Schenectady, Milan Fiske. Quite unknown
to me, he had been measuring precisely what we were claiming to
predict. I do not know whether he had started his work before
or after our letter appeared, but he told me that he was taking
data in the month of June, during which time it became clear to
him that there was a discrepancy between what he was seeing and
our published curves. Since it was apparently going to be necessary
for him to report a failure of the theory, he made new and careful
measurements which were just completed when the erratum came out.
His results, taken in ignorance of our mistake, agreed with the
corrected calculation.
It is worth reflecting on this episode, because it
brings out the fact that in the mathematical sciences one is engaged
in a kind of archaeology, an uncovering of hidden artifacts. This
wonderfully constrains the creative imagination, as it is possible
to be clearly wrong. In my story, the error was of a rather simple
kind: "memorie" and "engin" worked well; but,
as with W. S. Gilbert's little tomtit on a tree by a river, there
was a - luckily less fatal -"weakness of intellect."
One word of caution. My story may have given the
impression that in physics, theory leads experiment. This is very
far from being the case. Nature is at root mathematical, but her
ways are with very rare exceptions more inventive than our most
imaginative dreams.
If I have convinced you that there is something different
about creativity in the mathematical sciences, does this mean
that there is some great cultural abyss between science and other
human endeavors? I once heard Richard Feynman, perhaps the most
creative scientist of the latter half of our century, say something
like: "I met a painter once, and I made a deal with him.
You teach me painting; I'll teach you physics. I learned how to
paint. He didn't learn physics. I don't know why." Now, I've
never seen any of Feynman's paintings, but there are examples
enough of scientists as pretty good sculptors or poets or musicians
to suggest that C. P. Snow in The Two
Cultures was talking more about high-table
conversation in post - World War II Cambridge than about a general
phenomenon. Crossings in the other direction are rarer; the age
of the gentleman scientist is over. It is close to impossible
to dabble in science; it is too big and intense an enterprise.
Does this mean that nonscientists have to be totally in the dark about the excitement of my calling? There are books without equations which sell in great numbers, so they must be filling a need. Most such books talk about the most recent and speculative ideas. You have to tell me what they communicate. Typically, I find that they cannot be understood. Feynman, somewhere, complains about this, and in his own semipopular writings goes to considerable pains to give complete and correct explanations. The difficulty is that for even moderately deep scientific understanding a different language is needed, and this has to be studied.5 Now it takes at least two years and a lot of reading to acquire a foreign language, and few people apart from scientists put that sort of effort into studying science.
An achievable educational goal may be to try to communicate the excitement of some corner of physics to the interested nonspecialist. When physics is taught seriously, it is usually taught to nonscientists in the same way as it is to budding professionals: the first two years are a preparation for a (in this case nonexistent) second two. For what I am talking about, the answer may be to go further into special topics, and with mathematics so that it is not a sequence of fairy tales. Various attempts in this direction are made in the physics department at Cornell. My own attempt is to be found in my book.3
Finally, some avuncular remarks. Deni-grating memory has gone too far.6 Some may tell you that you can find out everything you need by logging on to the Internet, and surfing the World Wide Web. Don't count on it. And what about your own home page: is that to be a compilation of everyone else's home pages? I will assert that who you are is determined by what you have in your very own and not your computer's memory. If there is a poem that has caught your fancy, memorize it. If not, find one and memorize it. It will serve you, as will logical thought, in really tough times.
Footnotes:
1 The book title The Best and the Brightest, from which the current wide use of this formula seems to have sprung, has always struck me as metrically deficient. When I mentioned this to my great friend Scott Elledge, Goldwin Smith Professor of English Literature Emeritus, he immediately retrieved a metrically sound quote from Kipling,
Very great our loss and grievous-
So our best and brightest leave us,
which in its context (the poem "files") makes clear that this year's best and brightest are next year's unknowns. So perhaps with a literate memory as guide the phrase fits uncannily well.
2 If you are interested in reading about attempts at answers, the key words Anthropic Principle will get you started, at your own risk, in a good library.
3 Reasoning about Luck: Probability and Its Uses in Physics, V. Ambegaokar, Cambridge University Press, 1996.
4 See Subtle is the Lord, A. Pais, Oxford University Press, 1982.
5 Galileo Galilei (1564Ð1642), arguably the first modern physicist, said about the "book" of Nature: ". . . it cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language."
6 See the inspiring essay by Clara Claiborne Park, The Anatomy of Memory, J. McConkey, editor, Oxford University Press, 1996
Reasoning about Luck: Probability and Its Uses in Physics (Cambridge University Press, 1996), by Vinay Ambegaokar (physics), grew out of a course he has taught five times during the last fifteen years to students fulfilling a general-education requirement in science and quantitative reasoning. Ambegaokar assumes students can add, subtract, multiply, and divide with confidence. He then builds usable mathematics along the way. His goal is to teach quantitative topics without "dispensing watery baby-food" and to help other physics teachers do the same.
This article is Copyright © 1996 Vinay Ambegaokar. All Rights Reserved.