the silent gray weeks roll on
Set theorist Saharon Shelah's 1991 talk on "The Future of Set Theory" is a nice exploration of where the field is these days, following the proof in the sixties that its central problem, the Continuum Hypothesis, is inherently unsolvable.
From another angle, I was amazed to find that many of my colleagues, including some of the best minds in the field of set theory, feel apologetic about their subject. Many are apologetic toward mathematicians, (implying somehow that there are mathematicians and there are logicians, as if they are disjoint species) working in fields which are surely deeper, harder, more profound and meaningful, etc., and so we have to justify our existence by finding applications of "logic" to "mathematics." This leads to putting great store on category A.2 [applications to mathematics], as in the Abraham Robinson school. Now, I love to prove theorems in as many areas of mathematics as I can, but I do not like this servile attitude (this says nothing concerning the attitude of, say, a number theorist to this).
Many others put great store on the role of foundations, and philosophy. Again, I do not have any objection to those issues per se, but I am suspicious. My feeling may be akin to that of many authors who, while acknowledging the role of literary critics in cultural life, think that heeding their dicta will lead to boring works - but that theirs, of course, will shine forever because of their intrinsic beauty.
Still others mourn the loss of the "good old days" when the proofs were with ideas and were not so technical. In general I am not a great fan of the "good old days" when they treated your teeth with no local anesthesia, and technical is a red flag for me, as it is many times used not for the routine business of implementing ideas but for the parts, ideas and all, which are just hard and many times contain the main novelties.
My feeling, in an overstated form, is that beauty is for eternity, while philosophical value follows fashion.
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A disgusted reader may shout: "Beauty? You found in your mess some trace of beauty?" I can only say that I hear the music of those spheres or that every one likes his own dirt (the difference is small).
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The future - the reader may well remind me - what will be the future of set theory? Being optimistic by nature, and proving theorems which look to me reasonably satisfying, I am not at all gloomy. More seriously, looking at the last hundred years, repeatedly old mysteries have been clarified by deep answers, dark interludes were followed by the opening of new horizons; some directions require a substantial amount of preliminary study while others can be approached with little; and I find the old lady as fascinating as ever.
