cohenp_1.jpgYou’ve probably never heard of him unless you are a logician or logic groupie (Muffti has yet to verify the existence of the latter!), but Paul Cohen was a leading logician and mathematician. He passed away at 72. He spent most of his career teaching at Stanford. In his time, he won two of the most prestigious math awards in the world: the Bocher prize (1964) and the Fields Medal (1966). He showed that (augmenting work by Kurt Godel) that the Continuum Hypothesis was independent of Zermelo Frankel set theory, along with other important work (on the axiom of choice, for example). He was inventor of a mathematical technique known as ‘forcing’.

The Stanford obituary can be seen here.

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  • Because I hate to see any post here go uncommented upon. Via (quoted in total, with attribution, from ‘Barkley’):

    “March 31, 2007

    Paul J. Cohen freed infinity in two papers: “The Independence of the Continuum Hypothesis,” Proceedings of the National Academcy of Sciences of the USA, Dec. 15, 50(6) pp. 1143-1148, and in a followup of the same title with “II” in the same journal in Jan. 1964, 51(1), pp. 105-110. He laid the whole thing out and his view of it and its implications in his 1966 book, Set Theory and the Continuum Hypothesis, W.A. Benjamin.

    One of the curious things about pure mathemetics is that while many of its practitioners are atheists or agnostics, many of them also believe in Platonic idealism in connection with mathematics. Mathematical truth is “out there” and theorems are “discovered,” not “invented.” This Platonic idealism becomes most evident and semi-theological (the Roman Catholic Church officially thinks so, anyway) in connection with the mathematics of infinity. The material universe is finite, but infinity exists in pure mathematics in its classical formalist form, although the competing minority school of constructivism seeks to do math without any reference to infinity. This proves to be difficult.

    The mathematician who really got the math of infinity going seriously was Georg Cantor in the late nineteenth century, who did much of his most important work while in a mental institution. Cantor’s most brilliant achievement was to prove that there is more than one level of infinity; infinity is not just this big formless blob of endlessness. It has levels.

    He did this by first showing how one could show that two infinite sets are at the same level. One does this by showing that the sets can be arranged in a one-to-one relationship. Thus, even though it looks like there are more natural numbers (1, 2, 3, …) then positive even numbers (2, 4, 6, …), they are at the same level of infinity, aleph zero, or countable infinity. This is because one can line them up as 1 with 2, 2 with 4, 3 with 6, and so forth, with all of them used up in both sets. This is the basis of the old story of the infinite hotel never being full. It can be full, but if a new customer arrives, you just move everybody over to the next room, freeing up the first room.

    The kicker came when Cantor realized that one cannot make such a lineup between the countably infinite natural numbers and the real numbers. No matter what one does, there will always be some real numbers left over that one failed to count. In this fundamental sense, the real numbers are at a higher level of infinity. Cantor then went on to prove that there is no highest level of infinity. If one thinks one has found a set at the highest level, one can always generate a set of a higher level by considering the set of all subsets of the set one was just thinking about, the so-called power set. It is this infinity of the levels of infinity that gets some people thinking theologically about all this.

    It was Cantor who first proposed the continuum hypothesis (CH). It states that there is no level of infinity between that of the natural numbers and that of the real numbers. The Hebrew letter aleph is used to denote these levels of infinity, and if the CH is true, then aleph null is the level of the natural numbers and aleph one is the level of the real numbers. The generalized CH says that the power set of an infinite set is always at the next higher level of infinity.

    The question arose whether or not the continuum hypothesis is “true,” or to put it differently, can it be proven (or disproven) from the standard axioms of set theory, known as the Zermelo-Fraenkel axioms (ZF), that appear to underlie standard arithmetic and mathematics. In 1900, David Hilbert posed this question as the most important unresolved question in mathematics.

    It was thinking about this problem that led Kurt Godel to prove his famous Incompleteness Theorem in 1931. I note that Godel, who would starve to death eventually out of paranoia that all food being given to him was poisoned, ended up at the Institute for Advanced Study in Princeton, where he became a good friend of Albert Einstein’s. John von Neumann was also there at that time.

    In 1940, Godel would prove that the CH could not be disproven using the ZF axiom system. Both Godel and Cohen were Platonic idealists in their views and wanted to think of the possibility of there being more levels of infinity, ones that could exist between that of the natural numbers and the real numbers. If the CH can be false, then infinity is free, or at least freer.

    What Cohen proved in 1963 was that the CH could not be proven using the ZF system. This is the completion of proving its independence from the standard set of mathematical axioms. One can accept or not accept the continuum hypothesis as one pleases. Infinity is free, for what that is worth!

    Cohen’s proof is also famous because it involved the first use of a new method of proof, which he invented (or “discovered”) for this. This method is known as “forcing,” and is now widely used in mathematics, but I shall not get into trying to explain it, as I have already probably lost most usual readers of this blog, and it is pretty hairy. If you are curious, Wikipedia gives a not-too bad explanation of forcing and its variations.

    Now, the constructivist position on all this is that the whole thing is a bunch of hooey because infinity does not exist. This debate goes back to the time of Cantor, when his great critic was Kronecker, the founder of the constructivist school, which argues that all math proofs must be constructed out of finite arguments. A crucial part of the argument involves proof by contradiction, which most people first encounter in the theorem showing that the square root of two is not a rational number. Cantor’s proof that the natural numbers and the real numbers are not at the same level of infinity involved such a proof by contradiction, with most of this latter series of arguments coming out of that. Proofs by contradiction assume the idea that things are either true or false, the so-called “excluded middle” proposition. One proves something is true (false) by assuming it is false (true) and then finding a contradiction. The constructivists allow for things to be both true and false, thereby disallowing the use of proof by contradiction. This may be true (that things can be both true and false), but it does make proving things in math a lot more difficult, which may be why classical formalism remains the dominant school.

    This classical formalism reached its peak with the French Bourbakist school in the mid-20th century. It influenced economics at the time as well, with the most important figure carrying over the Bourbakist influence being the late Gerard Debreu, with the existence proof of general equilibrium with Kenneth Arrow being the supreme example.

    I note that I have personal connections to all this. My late father, whose centennial is this year, was a mathematical logician who proved a variation of Godel’s Incompleteness Theorem and knew all those people. He attended the first public presentation by Cohen of his proof in the summer of 1963, which took place on the Berkeley campus. We were spending the summer in LA (I was fifteen then), and were in the process of moving from Ithaca, NY, where I was born and my father was in the math department at Cornell, to Madison, WI, where he would be on the faculty at the University of Wisconsin-Madison. My mother still lives there at age 93.

    Anyway, we drove up to the Bay area from LA specifically for this event, although we did a bunch of sight seeing as well. However, I was brought into the auditorium with my sister after some sightseeing partway through Cohen’s presentation. I did not follow his presentation, although I had had a rough explanation of what it was about. My father was up in the front row with the other panjandrums of mathematical logic who were there (Godel was not) to see if Cohen could pull it off and how he did it, given that he was using this new proof method.

    Anyway, it was universally agreed that he did pull it off. So, Paul J. Cohen became the man who freed infinity. May he rest in peace.
    Posted by Barkley at March 31, 2007 12:15″ PM


    Cheers, ‘VJ’

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