Richard Jernigan -> RE: The Beauty and Mystery of Mathematics (Nov. 20 2013 1:18:52)
|
It's been a while since I last read Hardy's autobiography. I don't remember whether Hardy was a declared Platonist or not. Certainly Platonism was more prevalent among Hardy's generation than among more recent ones. During Hardy's youth belief in the supernatural was far more prevalent than nowadays. For instance, in the biography of R.L. Moore commissioned and published by the Mathematical Association of America, Moore's tactic while an undergraduate for becoming better acquainted with a young woman was to invite her to attend church with him. Moore was a young undergraduate, having matriculated at 16. Some time later, by his early twenties Moore experienced a crisis of faith, as evidenced in correspondence with his earliest mathematical mentor, Halsted. I suspect Moore may have undergone a similar experience with Platonism. During my time as a student with Moore he certainly behaved as though mathematical concepts partook of "reality", though he was careful to skirt the issue. Indeed he changed from referring to the Axiom of Choice as "that fact" to carefully pointing out in the second edition of his American Mathematical Society Colloquium Publication that it was an axiom, not a fact. The Axion of Choice occupies a position similar to Euclid's parallel postulate. There are systems of geometry which satisfy all of Euclid's axioms except the parallel postulate. Similarly there are systems of logic which obey all the Zermelo-Frankel axioms of set theory, some incorporating the Axiom of Choice, others negating it. The Classical Greeks were pretty much Platonists in geometry, mistaking the intuitive appeal of Euclid's axioms for reality. At present the Non-Euclidean Minkowskian geometry of General Relativity is taken as a better representation of reality. The discovery of non-Euclidean geometries in the mid-nineteenth century led to a weakening of the Platonist position. The problem was that any geometry that obeyed Euclid's axioms necessarily harbored within itself a model for non-Euclidean geometry. Which of the geometries was "reality"? Two strong supports for Platonism are the "universality" of mathematical concepts, and the intense feeling of discovery occasioned by the extension of these concepts to new realms, or the discovery of the proof of a difficult proposition, or the solution of a hard problem. The "universality" argument is undermined by the unresolved controversies on the foundations of mathematics. Russell and Whitehead tried to develop all mathematics from set theory, an effort effectively dynamited by Gödel. The Intuitionists counseled a radical retreat, and so on. For those interested, Morris Kline's "Mathematics, the Loss of Certainty" is an informative read. The emotional feeling of discovery brings forth an analogy on my part. During Newton's lifetime, the great majority of educated Europeans believed the Bible story of language. It was a gift from God at the Creation. Human hubris at the attempted construction of the Tower of Babel annoyed their Creator, who confounded the universal language of mankind up until then, resulting in the polyglot world. The modern story of language is that it is an innate, genetically determined aptitude of humans, produced by a long running course of evolution. Toddlers learn with striking efficiency to speak and understand because their brains are built to do so. The diversity of languages was produced, and continues to be produced by the evolution that brought it into being in the first place. Details of the aptitude and its development are the subject of a minor industry of debate, but few linguists publicly adhere to the supernatural origin of language. Personally, I find believable a similarly modern story for the human aptitude for mathematics, and the ready adoption of its concepts by students. To me the relationship of Newtonian mechanics to the ability of a dedicated adolescent to become a pool shark is quite clear. Newtonian mechanics is the mathematization of universal intuitive concepts of motion. The rapid development of Newtonian mechanics and classical electromagnetic theory ironically led to the ability to detect their inadequacies. The mathematical developments that enabled the elaboration of classical physics formed the basis for newer, more accurate theories. Counting and measurement are absent from only a very few studied societies. The application of measurement to motion was a ball buster for centuries. Much of the difficulty was overcome by Newton and Leibnitz, finally tidied up by Cauchy, Dedekind et al in the 19th century. Mathematical analysis is a centuries long conscious work of extending the ideas of counting and measurement and their application. I find it both hard to conceive of, and unproductive to believe in the supernatural existence of the Platonic entities. This doesn't diminish the esthetic and emotional pleasures, the sense of discovery and wonderment in mathematics, and indeed in all of science. I was raised a Christian. I fell away from major components of belief at the age of 18. Still the drill, ceremonies and rhetoric of the Anglican liturgy have a strong emotional resonance for me. It speaks to important human issues. I don't need to divorce myself from the pleasures of Platonism in order to doubt its supernatural claims. RNJ
|
|
|
|