Curious remark of D. Ravenel

Question

In his beautiful (but difficult) book "Complex cobordism and stable homotopy groups of spheres", concerned mostly with methods of computing homotopy groups of spheres, D. Ravenel describes a general method of producing elements on the $E_2$-page of Adams-Novikov spectral sequence. Later he discusses whether the so-called Greek letter elements descent to non-trivial elements in stable homotopy groups of spheres, ending with a rather curious remark which I quote in it's entirety.

In the intervening time there was a controversy over the nontriviality of $\gamma _{1}$ which was unresolved for over a year, ending in 1974 (see Thomas and Zahler [1]). This unusual state of affairs attracted the attention of the editors of Science [1] and the New York Times [1], who erroneously cited it as evidence of the decline of mathematics.

Can someone shed some more light on this matter? Why did the editors of New York Times become interested in the state of our knowledge of stable homotopy groups of spheres? Why would inability to determine whether a highly-complicated element (coming from an extremely complicated spectral sequence) is non-trivial be considered "decline of mathematics"?

Answer 1

The article in Science is entitled 'Mathematical Proofs: The Genesis of Reasonable Doubt'. It is essentially about proofs that are so long that "that they can never be written down, either by humans or by computers".

Regarding the $\gamma$ family, the relevant quote is just a short paragraph

Ronald Graham of Bell Laboratories in Murray Hill and others reply that they have more confidence in results that could be obtained by probabilistic methods such as Rabin's prime test than in many 400-page mathematical proofs. Such proofs can often be nearly impossible to check, as is evidenced by a debate over a particular result in homotopy theory, which is a subject in topology. One investigator came up with a proof of a statement and another came up with a proof of its negation. Both proofs were long and very complicated, hence the two investigators exchanged proofs to check each other's work. Neither could find a mistake in his colleague's proof. Now a third investigator has come up with still another complicated proof that supports one of the two original proofs. The verdict, then, is 2 to I in favor of one proof, but the problem is still not resolved.

This was followed by a letter by Zahler the next month.

To say that the proofs were so long and complicated as to be "nearly impossible to check" is also a red herring. Our proof, for example, takes 13 pages (not 400)and has been used and generalized by a number of other workers. Actually, the conflict persisted as long as it did only because just one outside person, J. F. Adams, took the trouble to verify the details of our proof independently.

Answer 2

Since the article linked in Tyler Lawson's comment seems to be restricted, here is a transcript. I have to say that everything in this article seems either too vague to evaluate, or a misunderstanding of an oversimplification (probabilistic proofs, for example), or a misunderstanding of proofs and the research process in general, or just obnoxious.

Crisis in Mathematics (New York Times, 1976-06-02)

Mathematics, school children are taught, is the most exact of the sciences. An answer to a mathematical problem is either right or wrong; maybe is excluded. The proof of a mathematical theorem is either correct or incorrect, and a good enough mathematician can always come to a firm conclusion. But now, according to Science magazine, such ideas may be obsolete. Mathematics, too, is in a state of crisis in which the old verities are at least suspect, if not actually destroyed.

Take the case of a certain statement in a branch of advanced mathematics called "homotopy theory," a subject we won't even try to pretend we know anything about. Anyway, one mathematician produced a long and complex proof that the statement was correct. About the same time another mathematician came up with a similarly complex and long proof that the statement was incorrect. The two investigators exchanged proofs and each sought to find an error in his rival's work. Neither succeeded.

Then there is the shattering discovery that some mathematical theorems require proofs that are so long that even computers can't work the proofs out in any acceptable period of time. An Israeli mathematician has suggested a possible way out. The trouble, he believes, is that mathematicians are too demanding; they won't accept the idea that a proof may be wrong once in a while. If mathematicians will just accept proofs which have even a slight probability — say, one in a billion — of being wrong, then, he thinks, a lot of impossible proofs can become possible.

At the root of this crisis, some mathematicians hold, is the fact that some of the long proofs now being published are pressing the limits of the amount of information a single human mind can handle. That may be, but we know a lot of people who thought that mathematicians had approached that limit a long time ago — about the time of Euclid, in fact.