This decision is only in force for a year;then it will probably be reversed. The reasons for dropping the AMS membership are simple:The main benefit for me of an AMS membership are the magazines: the Notices and the Bulletin.If I intend to go to a conference I will renew my membership then.
The AMS Notices does occasionally carry interesting articles, although most of them are what one might call "bureaucratic". Every now and then a notable mathematician segues to the next lifetime, and one can read a predictable spate of eulogies about his research, his teaching ability and so on. Some of these obituaries do contain insights into the real living being behind the research;most of them are predictable boiler-plate, the kinds of things one says on public occasions in which it is understood that there will be no negative vibes in any utterances vis-a-vis the corpse. If I really want to read an article in the AMS Notices I can pay a visit to the Wesleyan Science Library.
The situation with regard to the Bulletin of the AMS is very different. Every issue has excellent articles about modern mathematics: survey articles, original research, book reviews. The menu has improved greatly in recent years, and the AMS is no longer bashful about publishing articles that may have real-world applications, those which might stain its escutcheon of "mathematical purity".
The problem with the articles in the BAMS is this: either I haven't got the background to get past the first half dozen pages; or, if they are accessible and valuable for me, I can never find time to digest them. The issues pile up in my room for an entire year, during which time I've been able (apart from the review) to fully absorb perhaps half a dozen articles. After that the accumulation of issues may be carried up to the math department of Wesleyan University . This really doesn't need to be done, as the lounge of that department is already submerged in AMS Bulletins. Otherwise (I'm ashamed to say) they are recycled in the local landfill. I know this isn't right, but who can I find to pass them along? The recovering drug addicts and alcoholics in my subsidized rental apartment building? The clientele of the soup kitchen around the corner? The local merchants? The members of the local grassroots activist organizations? The local art galleries?
When I was living in New Paltz, New York in the 1980's , a librarian once approached me in the town library asking for help. When the library had opened that morning, he'd discovered a box filled with a year's supply of the University of Michigan's Mathematical Reviews! Apparently someone in the mathematics department at the State University of New York (SUNY;up the hill) , burdened with guilt for the great loss to human knowledge that he might cause by tossing them out, had imagined that the local library would find some use for them. On the contrary, the librarian asked me: "What the hell am I supposed to do with this stuff?"
Once again, if there is an interesting article in the Bulletin, I can and do go to the Wesleyan Science Library, although nowadays most of my extra-curricular math/physics reading comes from articles on the Internet.
The above observations are a preamble to a contrast: just this afternoon I received a copy of the Mathematics Monthly, magazine of the MAA. The table of contents on the cover gives ample evidence that the level of the Math Monthly never fluctuates from the pedestrian. 'Pedestrian' does not necessarily mean inferior or without value. Although the mathematics in them rarely goes beyond the 19th or early 20th century, some of the articles hold interesting ideas. Here is a sampling:
(1)Drilling for Polygons : Designing a drill that will produce polygonal holes
(2) Prime Divisors on Thin Sequences:Simple additive number theory
(3) Explicit Solutions to a certain third order recurrence relation: Very elementary
(4)How Nature's Speed Limit on communication relates to quantum mechanics:The article is in the language of quantum computation. I've the impression that its' very long-winded pages (complete with exercises!) could be cut 50%.
(5)A Generalization of the Cayley- Hamilton Theorem: Extremely elementary
(6)Tossing coins to guess a secret number Elementary game theory
By asserting that the contents of the Math Monthly are pedestrian, I'm implying that it appears to me to be without aspirations. It is for people who are already satisfied with what they already know. It can be read and understood by scientists, engineers and technicians, (the SET subset of STEM ) with perhaps a pages or two of explanations of unfamiliar terms. Not without interest, it is without ambition.
Perhaps my situation is special: I left mathematics for 25 years to pursue a literary career, and only picked it up again in my 40's. Obviously I'm going to be out of the loop in several fields of modern mathematics. Yet I suspect that many people feel the way I do, and that it is not a matter of personal backgrounds but a comment on the state of the sciences today.
One knows, for example, that in the field of academic literary criticism, the unpalatable linguistics of Deconstructionism was used as a way of slamming shut the doors of this field for any candidate for a PhD (or tenure) who found its' 'hermeneutics' unacceptably rebarbative. This manifestation of linguistic imperialism collapsed when it was discovered that Paul de Man, the inventor of said hermeneutics, was a deeply stained anti-Semitic journalist and collaborator in Belgium in WWII. However for a decade or so between the 70's and 80's, one of the qualifications for an advanced degree in literature at many of the major universities was that one hated the actual activity of reading fiction and poetry!
I don't think that one can extend such a pejorative description to mathematicians: that is to say, I don't believe that most of them are inventing far-out or far-fetched jargon just to make themselves incomprehensible to all but their own graduate students! Yet the results are often similar. With each shift in the linguistics of mathematics, caused by new ways of looking at things and changing fashions, there is an entire sub-class of mathematically active persons (Paul Erdös would have said, "those whose brains are not asleep") who are forced to drop out because they can't understand the new textbooks, let alone the articles in the journals. These shifts are occurring more and more frequently, leading to the extreme Balkanization of mathematics with which we are all familiar.
This situation can only get worse. One can predict an eventual polarization between (a) Teachers who will remain 50 years behind the times and (b) Small Balkan states known as "geometric analysis", "Khavanov co-homology", "inaccessible cardinal theory", "post-Grothendieckian algebra", "post-Perelmannian low dimension topology", "post-Wilesian number theory", and so on.
(1) physicists rarely need or use more mathematics than what was available at the end of the 19th century;<2>the fact that most of the work in the other sciences don't even need that much! One might cite Fractals and Chaos as a modern branch of mathematics that has transformed all the sciences. Yet the mathematics for this subject was fully developed by the end of the 19th century. The impetus to apply it to the other sciences was given by the computer revolution. What Lorentz, Mandelbrot and others discovered was how ideas that were half a century old in mathematics could be applied to the sciences in general.
Likewise for Linear Algebra, the rudiments of which were known and used by the middle of the 19th century, yet which astonished Heisenberg when it was explained to him in the 1920's that he'd really been working with old-fashioned mathematical objects known as "matrices" !
Most physicists I know will readily admit to little familiarity with even the most basic ideas of Topology (though this is changing). What need then do they have for Transfinite arithmetic? Gödel's Theorems? Transcendental numbers? Algebraic Geometry? Category Theory? The Poincaré Hypothesis? Non-Standard Arithmetic?
Here is a quotation from a recent, recommendable an theology: Deep Beauty (editor Hans Halvorson; Cambridge University Press, 2011) . It's a collection of articles describing ways of applying Category Theory, n-Categorical Theory, Topos Theory, generalized Operator Algebras, and other advanced mathematical notions to the profoundly intractable quandries of contemporary physical theory. This quote is on page 28, in an article by John Baez and Aaron Lauda:
"..Yang and Mills did not know about bundles and connections when formulating their theory. Yang later wrote..:
"What Mills and I were doing in 1954 was generalizing Maxwell's Theory. We knew of no geometrical meaning of Maxwell's theory, and we were not looking in that direction. To a physicist, gauge potential is a concept rooted in our description of the electromagnetic field. Connection is a geometric concept which I only learned in 1970."
Books like Deep Beauty may lead to some deeper understanding of the structure of physical theory, but are unlikely to lead to actual experiments, like the ones at CERN designed to unearth new elementary particles.Time will tell.
As for those branches of theoretical physics which have hopped onto the higher mathematics band-wagon, such as String Theory, they've yet to produce anything one might call physics.
Another example: A few years ago I began and completed the translation of two books of interviews with René Thom. His ideas about biology are considered very radical by biologists, though to a mathematician they consist simply of descriptions of scenarios of morphogenesis and development using the language and concepts of differential topology and singularity theory.
Most biologists haven't studied these subjects. One would have thought that this would be a opportunity for bringing modern mathematics into biology, a subject that normally resists the kinds of abstraction and quantification one finds in mathematics. This did not happen. What happened was that the biologists said, "This stuff doesn't make predictions. We've never studied it and there's no reason why we should." Most biologists are more than happy to remain in the comfortable nature park of 19th century mathematics.
The field of modern mathematics that really has led to major advances in physics is Group Theory. Yet what use do 99% of all physicists have for Rings, Modules, Fields, Categories, Clones, Clans, Groupoids, Monoids, Lattices (perhaps in Ôquantum logic'), Schemes, Motifs, Sheaves..? General Relativity, admittedly does use differential geometry, yet only up to the point where it was developed by Levi-Civita at the end of the 19th century.
As for the other sciences, almost all research in them is done with basic statistics, differential equations, partial differential equations, numerical approximations, combinatorics, Fourier Transforms and perhaps a few other things. Indeed, Claude Levi-Strauss is to be commended for bringing lattice theory into Anthropology!
Summarizing, here is the contemporary relationship of Mathematics to science in general, as I see it: in Mathematics itself one has a sharp division between research mathematicians who are "up to date", working with ideas and above all with language that is usually no more than a decade old, perhaps only a few years old, perhaps only a few weeks old ! Because of this phenomenon, more and more of us are destined to be left behind, particularly when old subjects (like 19th century Galois Theory) are clothed in modern dress (Artin reformulation), or classical mechanics is treated as a sub-branch of differential topology. (These two examples are low-key: with a bit of effort it is possible to make the transition.)
The other sub-class of mathematicians are the teachers of mathematics, the targeted readership of the MAA's Mathematics Monthly. They've accepted their role as 2nd class citizens in a world that is destined to remain incomprehensible to all but a few.
As for the other sciences: Physics uses the most advanced mathematics, yet even the physicists can continue to do most of their work using pre- World War I mathematics. As for the others, biology, chemistry, economics, psychology, geology, astronomy, and the human sciences, they really don't need any 20th century mathematics at all. These divisions, (1) between research and teaching in mathematics, then (2) between math and physics, (3) physics and the other sciences, and (4) all of these and the general public, find their reflection in the lives and work of myself and every scientist I know, I suppose it is the price one pays for living in an age in which science and technology advance more rapidly than anyone is able to keep up with them.
It may be this which, before nuclear holocaust, global warming, or surprise asteroids, will lead to the demise of the human race. Space ships of the future visiting our lifeless planet may discover nothing more than enormous piles of junked machines, books of data and incomprehensible textbooks !