Harvard Law School Makes Online Zero-L Course Free for All U.S. Law Schools Due to Coronavirus


For Kennedy School Fellows, Epstein-Linked Donors Present a Moral Dilemma


Tenants Grapple with High Rents and Local Turnover at Asana-Owned Properties


In April, Theft Surged as Cambridge Residents Stayed at Home


The History of Harvard's Commencement, Explained



has become frightening to intrude because of the difficulty of distinguishing between sophisticated and naive interpretations of events which seem in the first instance absurd. Newman's presentation of representative work from these writers, though it has its flaws is an excellent survey. Two relevant points that appear neither in the quoted work nor in his own descriptions might have been added. One is that modern physics has been unable to design an experiment which "proves" that matter is of corpuscular form, and also says something about its wave nature. Put another way, the contradictions of wave-particle duality have been derived from the positive results of separate experiments; we have not been able to unite them in one experiment. If the experiment can't be constructed, it becomes reasonable to ask whether the contradiction derived from two experiments is of the same order as the explicit contradiction in a proof of one theory coupled with the simultaneous proof of the other--or, whether a truth is as true without a simultaneous disproof of its converse.

To admit this hypothesis would mean revising truth-function theory. Yet this is no less conceivable than the other point: that if we come up against an irreconcilable logical contradiction, we might consider replacing the dyadic notion of truth, by which a proposition takes only the values "true" or "false," with a triadic or n-adic notion. W. V. Quine mentions this possibility in the introduction to his Methods of Logic: Newman, as a writer with his finger on the mathematical pulse of science ought to transmit to non-scientists what he can about notions like these.

ONE can take issue, then, less with what Newman has done than with what he has left undone. It should not be unfair to go to Mathematics and the Imagination for a better example of what I mean. In a passage on the notion of a simple curve, the authors write:

The theorems are found in a queer branch of top mathematics called "topology." A French mathematician, Jordan, gave the fundamental theorem of this study: every simple curve has an inside and an outside. That is, ever simple curve divides the plane into two regions, one inside the curve, and one outside.

Why this patently obvious fact is fundamental to one of the most difficult branches of mathematics, why it is fairly hard to prove in a general form, and how it relates the geometric and the algebraic in topology, are nowhere to be found. These questions are not impossible to answer in a way that would satisfy Newman's readers.

Science and sensibility does have at least one particularly self-contained piece; unfortunately Newman claims he had little to do with it. This is G. H. Hardy's description of his experiences with Ramanujan, certainly the most fascinating article I found in the two volumes. "My job has been merely to copy, paraphrase, and select" writes Newman. Hardy is describing his five-year acquaintance with the unknown Hindu clerk who had, before he was 25 and with the help of just one obscure work of higher mathematics, independently divined answers to problems that occupied Europe's best mathematicians. Hardy writes of his first letter from Ramanujan, and reproduces 15 typical theorems of the 120 which Ramanujan sent him from India:

I had proved things rather like (1.7) myself, and seemed vaguely familiar with (1.8). Actually (1.8) is classical; it is a formula of Laplace first proved properly by Jacobi; and (1.9) occurs is a paper published by Rogers in 1907. I thought that, as an expert in definite integrals, I could probably prove (1.5) and (1.6), and did so, though with a good deal more trouble than I had expected....

The formulae (1.10)--(1.13) are on a different level and obviously both difficult and deep ... (1.10)--(1.12) defeated me completely; I had never seen anything in the least like them before. A single look at them is enough to show that they could only be written down by a mathematician of the highest class. They must be true because, if they were not true, no one would have had the imagination to invent them.

The last two formulae, stand apart because they are not right and slow Ramanujan limitations, but that does not prevent them from being additional evidence of his extraordinary powers....

Hardy has conveyed in an extraordinary way the essence of scientists and science. Everything is there: the intellectual response, the implications, the historical situation. He may seem in this way to describe a hero, though scientists are usually the first to claim that there are few heroes in science. James Newman claims in a preface not to believe in them, but Science and Sensibility is full of his heroes--many of them as heroic as anyone can be, nevertheless a little embarrassed by Newman's subjectivity.

In the end. Newman succeeds as a reviewer where he fails as essayist: you want to read what he is interested in, to read Hardy or Russell instead of Newman. But two volumes are a lot of appetizer and appetizer to an awful lot. They shouldn't be started unless you're also prepared to spend next year in science biographies.

Want to keep up with breaking news? Subscribe to our email newsletter.