an + bn = cn, where integer n ≥ 3, has no non-trivial solutions.
This and, if all goes well, the next seven posts in this blog constitute an unprecedentedly ambitious project for me. I'll be attempting a very granular reading of the book I discussed in a more abstract fashion earlier, THE MURDER OF PROFESSOR SCHLICK: THE RISE AND FALL OF THE VIENNA CIRCLE.
Let us start with the "rise" of that subtitle. In 1922 Schlick was offered and accepted the chair for natural philosophy at the University of Vienna. Edmonds calls this simple hiring "a turning point in the history of twentieth-century philosophy."
The core of the study-and-discussion group that he seems to have formed almost immediately after that appointment consisted of Schlick himself, Neurath, and Hahn. Otto Neurath and Schlick had background and personality differences as far apart as the poles, as Edmonds tells it. Schlick was a gentile, Neurath a Jew. Schlick was soft-spoken and could happily spend an evening reading alone at home. Neurath was gregarious and sometimes loud. Schlick was apolitically interested in some very abstract ideas. Neurath saw himself as a workers' champion.
Hans Hahn seems to have been in between those poles in personality. Hahn would also, in 1929, become a (perhaps the) driving figure in the writing of the group's manifesto, titled THE SCIENTIFIC CONCEPTION OF THE WORLD. This document came together literally while Schlick was out of town -- indeed, Schlick was on another continent, enjoying a temporary position as a professor at Stanford University in distant California. When he got back to Europe, he received a bound copy of the new baby book.
The group had grown from 1922 to 1929 and would continue to grow for some time thereafter, aided by the public face the manifesto gave it.
I'd like to veer from straight-and-narrow narration for a moment here to say something about the philosophy of mathematics. As the Vienna Circlers knew well, Russell and Whitehead had produced an exhaustive three-volume treatise a few years before (in 1912) arguing that mathematics can be reduced to a development of formal logic.
But such logicism wasn't the only contender in the philosophy-of-mathematics space. Another was the "Intuitionism" of L.E.J. Brouwer. The label "intuitionism" is confusing, but has been generally accepted. Brouwer's view might better be called a moderate social constructivism. He contended that the law of the excluded middle (the notion that for any well-formed proposition X, we can know that X is either true or not true -- a notion integral to the Russell/Whitehead work) is itself false.
Brouwer (portrayed atop this entry) said, specifically, that a proposed mathematical theorem which has never been proven true or false, and for which there is as of yet no agreed upon procedure for proving it true or false, is in fact neither true or false (or, if you prefer, is both).
Note (because I formerly got this wrong somewhere), that this does NOT apply to questions like "is the one-millionth digit of pi after the decimal point 7 or not?" It may be the case that no one has ever computed it that far. But there is a straightforward of method of continuing the calculation for as long as one's computers still have their power source. We can say with assurance when we or the computers get there, it will be either 7 or not.
Consider, rather, Fermat's last theorem, that there is no set of four integers such that an + bn = cn, where integer n ≥ 3. [Technically one says there are no "non-trivial solutions," because one could 'cheat' by making a, b, and c all equal zero.] Now consider the case in Fermat's day, or in the Vienna Circle's day, or as recently as 1990. Not only was there no answer as to the truth of this theorem, there was no agreed-upon procedure for getting there. Brouwer's view was that at none of those times was this theorem either true or false. The proposition was perfectly well formed, yet the excluded middle did not hold.
Intuitionism, then, holds that Fermat's Last Theorem became true when Andrew Wiles completed his work constructing the proof in 1995. Note the gerund. He constructed a proof. He did not discover it.
Brouwer was NOT a member of the VC. I bring him up because Edmonds tells a fascinating story in which he plays a part. In March 1928, Brouwer gave a lecture on his Intuitionism in Vienna. Herbert Feigl, who WAS a member of the VC, was in attendance, as was Ludwig Wittgenstein.
After the lecture Feigl and Wittgenstein retired to a coffee shop. Wittgenstein, who over the preceding few years had been 'out of' philosophy, who had sought to live a stoic life and support himself as a gardener and schoolteacher, proceeded that night to hold forth for hours, stimulated by the case Brouwer had made. Feigl would later write simply, "Wittgenstein was a philosopher again" as of that night.
[Wittgenstein apparently agreed with Brouwer that the law of the excluded middle did not apply to such matters, although he disagreed with the reasoning.]
Enough about such abstract stuff. Tomorrow, we talk politics!
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