\nCommunications on Pure and Applied Mathematics 13 (1): 1\u201314, 1960.<\/p>\n
including two follow-up articles:<\/p>\n
Richard Hamming, “The Unreasonable Effectiveness of Mathematics<\/a>“, and:<\/p>\n Sundar Sarukkai, “Revisiting the \u2018unreasonable effectiveness\u2019 of mathematics<\/a>“, Now, after reading it, I must say I found the article by Wigner rather disappointing; it asks an interesting question that coincides with its amazing title, but it offers not much substance in the way of further positioning this question or answering it. He does give a number of examples from the history of physics of cases in which mathematical formulations of laws of physics were based on very few observations indeed, laws which afterwards proved very accurate quantitatively nonetheless. That is interesting. But the main reason I now read these articles was because of reading this book:<\/p>\n Gregory Chaitin<\/a>, ‘Meta Math!, The Quest for Omega<\/a>‘, Pantheon, New York, 2005.<\/p>\n It is a very entertaining book ! full of exclamation marks ! because Gregory Chaitin is passionate about his subject ! And it is one of the rare popular-scientific books where things are made more easy to understand without watering down the content too much ! He refers quite often to fellow-‘digital philosophers’ like Fredkin<\/a>, Toffoli<\/a> or Wolfram<\/a>, very interesting writers that I tend to mistrust because of their crypto-platonist tendencies ! But he is different !<\/p>\n I can not agree more with him when he writes that the best way to learn mathematics is to read the history of mathematics; Not that I know a lot of math so that I can speak out of experience, but I know that what I like about it are the philosophy and the ideas when they are still wet and fresh, not the dry formulas and routines that are left over after hundreds of years. The most interesting part of this book for me was chapter five, ‘The Labyrinth of the Continuum’, where he gives an overview of the history of ‘real’ numbers, including the very Borgesian ‘know-it-all’ number of Emile Borel<\/a>:<\/p>\n ‘The idea of being able to list or enumerate all possible texts in a language is an extremely powerful one, and it was exploited by Borel in 1927 in order to define a real number that can answer every possible yes\/no question! and<\/p>\n ‘Borel\u2019s often-expressed credo is that a real number is really real only if it can be expressed, only if it can be uniquely defined, using a finite number of words. It\u2019s only real if it can be named or specified as an individual mathematical object. And in order to do this we must necessarily employ some particular language, e.g., French. Whatever the choice of language, there will only be a countable infinity of possible texts, since these can be listed in size order, and among texts of the same size, in alphabetical order. (these quotes are actually from this paper<\/a> but cover topics also adressed in the book)<\/p>\n What I like about this is the way it completely contradicts any idea of an ‘unreasonable effectiveness’ of mathematics. We can not even name or specify pretty much all the numbers we somehow imagine in a small line segment between 0 and 1. It is as if we think all is mathematics because that is the only thing we can think about ! Here is an unexpected thinker on the ‘unreasonable effectiveness’ of mathematics; the current Pope<\/a>, paraphrasing the same Galileo one of his precursors accused of heresy:<\/p>\n ‘Yet the human mind invented mathematics in order to understand creation; but if nature is really structured with a mathematical language and mathematics invented by man can manage to understand it, this demonstrates something extraordinary. The objective structure of the universe and the intellectual structure of the human being coincide; the subjective reason and the objectified reason in nature are identical. In the end it is “one” reason that links both and invites us to look to a unique creative Intelligence.’<\/p>\n What I like so much about Chaitin’s book is that it shows how much there is we do not know, and how much there actually is that is even unknowable. Nevertheless we seem more than able to construct<\/a> a coherent world based on very little. In my view, this ‘unique creative intelligence’ that links world and mathematics is our own, so no need for a capital ‘i’ !<\/p>\n","protected":false},"excerpt":{"rendered":" I finally got around to reading a famous article with one of the most intriguing titles ever: Eugene Wigner, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences“, Communications on Pure and Applied Mathematics 13 (1): 1\u201314, 1960. including two follow-up articles: Richard Hamming, “The Unreasonable Effectiveness of Mathematics“, The American Mathematical Monthly 87 (2), … Continue reading the unreasonable effectiveness of mathematics !<\/span>
\nThe American Mathematical Monthly 87 (2), 1980.<\/p>\n
\nCurrent Science 88 (3), 2005.<\/p>\n
\nRichard Hamming<\/a> (yes, the one from the ‘Hamming window’, if you are a FFT-geek) wrote a polite and modest response: ‘I shall spend relatively more time trying to explain the implied question of the title. But when all my explanations are over, the residue is still so large as to leave the question essentially unanswered’. But he offers much more in the way of possible answers, of which I find two are interesting:
\n– we see what we look for
\n– science in fact answers comparatively few problems, almost all of our experiences in this world do not fall under the domain of science or mathematics.<\/p>\n![\"chaitin_metamath.jpg\"](\"https:\/\/www.joostrekveld.net\/wp\/wp-content\/uploads\/2010\/02\/chaitin_metamath.jpg\")
<\/p>\n
\nYou simply write this real in binary, and use the nth bit of its binary expansion to answer the nth question in French.
\nBorel speaks about this real number ironically. He insinuates that it\u2019s illegitimate, unnatural, artificial, and that it\u2019s an \u201cunreal\u201d real number, one that there is no reason to believe in.’<\/p>\n
\nThis has the devastating consequence that there are only a denumerable infinitely of such \u201caccessible\u201d reals, and therefore, as we saw in Sec. 2.2, the set of accessible reals has measure zero.
\nSo, in Borel\u2019s view, most reals, with probability one, are mathematical fantasies, because there is no way to specify them uniquely. Most reals are inaccessible to us, and will never, ever, be picked out as individuals using any conceivable mathematical tool, because whatever these tools may be they could always be explained in French, and therefore can only \u201cindividualize\u201d a countable infinity of reals, a set of reals of measure zero, an infinitesimal subset of the set of all possible reals.
\nPick a real at random, and the probability is zero that it\u2019s accessible\u2014 the probability is zero that it will ever be accessible to us as an individual mathematical object. ‘<\/p>\n