Paul P. Mealing

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Saturday, 19 February 2011

Metaphysics in mathematics revisited

I recently wrote a post on E. Brian Davies’ book, Why Beliefs Matter (Metaphysics in mathematics, science and religion). Davies is Professor of Mathematics at Kings College London, so his knowledge and erudition of the subject far outweighs mine. I feel that that imbalance was not represented in that post, so this is an attempt to redress it.

Davies’ book is structured in 5 parts: The Scientific Revolution; The Human Condition; The Nature of Mathematics; Sense and Nonsense; and Science and Religion.

Davies addresses mathematical Platonism in 2 parts: The Human Condition and The Nature of Mathematics. Due to the nature of my essay, I believe I gave him short thrift and, for the sake of fairness as well as completeness, I seek to make amends.

For a start, Davies discusses Platonism in its wider context, not just in relation to mathematics, but in its influence on Western thought, regarding religion as well as science. Many people have argued that Aquinas and Augustine were both influenced by Platonism, to the extent that Earth is an imperfect replica of Heaven where the perfect ‘forms’ of all earthly entities exist. There is a parallel view expressed in some interpretations of Taoism as well. Note that one doesn’t need a belief in ‘God’ to embrace this viewpoint, but one can see how it readily marries into such a belief.

Davies discusses at length Popper’s 3 worlds: World 1 (physical); World 2 (mental); and World 3 (cultural). Under a subsection: 2.7 Plato, Popper, Penrose; he compares Popper’s 3 worlds with Penrose’s that I expounded on in my previous post: Physical, Mental and Mathematical (Platonic). In fact, Davies concludes that they are the same. I’m sure Penrose would disagree and so do I.

There is a relationship between mathematics and the physical world that doesn’t exist with other cultural ideas. Even non-Platonists, like Paul Davies and Albert Einstein, acknowledge that the correlation between mathematical relationships and physical phenomena (like relativity and quantum mechanics for example) is a unique manifestation of human intelligence. In his book, The Mind of God (a reference to Hawking’s famous phrase) Paul Davies devotes an entire chapter to this topic, entitled The Mathematical Secret.

On the other hand, Brian Davies produces compelling arguments that mathematics is cultural rather than Platonic. He compares it to other cultural entities like language, music, art and stories, all of which are products of the human brain. In one of his terse statements in bold type he says: Mathematics is an aspect of human culture, just as are language, law, music and architecture.

But, as I’ve argued in one of my previous posts (Is mathematics evidence of a transcendental realm? Jan. 08) there is a fundamental difference. No one else could have written Hamlet other than Shakespeare and no one else could have composed Beethoven’s Ninth except Beethoven, but someone else could have discovered Schrodinger’s equations and someone else could have discovered Riemann’s geometry. These mathematical entities have an objectivity that great works of art don’t.

Likewise I think that comparisons with language are misleading. No one has mathematics as their first language, unless you want to include computers. Deaf people can have sign language as a first language, but mathematics is not a communicative language in the same way that first languages are. In fact, one might argue that mathematics is an explanatory language or an analytic language; it has no nouns or verbs, subjects and predicates. Instead it has equalities and inequalities, propositions, proofs, conjectures and deductions. Even music is more communicative than mathematics which leads to another analogy.

Is music the score on the page, the sounds that you hear or the emotion it creates in your head? Music only becomes manifest when it is played on a musical instrument, even if that musical instrument is the human voice. Likewise mathematics only becomes manifest when it is expressed by a human intelligence (and possibly a machine intelligence). But the difference is that mathematical concepts have been expressed by various cultures independently of each other. Mathematical concepts like quadratic equations, Pascal’s triangle and logarithms have been discovered (or invented) more than once.

Davies makes the point that invention is a necessary part of mathematics, and I wouldn’t disagree. But he goes further, and argues that the distinction between invention and discovery cannot be readily drawn, by comparing mathematics to material inventions. He argues that a stone axe may have been the result of an accidental discovery, and Galileo’s pendulum clock was as much a discovery as an invention. I would argue that Galileo discovered a principle of nature that he could exploit and people might say the same about mathematical discoveries, so the analogy can actually work against Davies’ own argument if one rewords it slightly.

In my previous post, I did Davies an injustice when I referred to his conclusion about mathematical Platonism being irrelevant. In section 3.2 The Irrelevance of Platonism, Davies explains how some constructivist theories (like Jordan algebras) don’t fit into Platonism by definition. I don’t know anything about Jordan algebras so I can’t comment. But the constructivist position, as best I understand it, says that the only mathematics we know is what we’ve created. A Platonist will argue that the one zillionth integer of pi exists even if no one has calculated it yet, whereas the constructivist says we’ll only know what it is when we have calculated it. Both positions are correct, but when it comes to proofs, there is merit in taking the constructivist approach, because a proof is only true when someone has taken the effort to prove it. This is why, if I haven’t misconstrued him, Davies calls himself a mathematical ‘pluralist’ because he can adjust his position from a classicalist to a formalist to a constructivist depending on the mathematics he’s examining. A classicalist would be a Platonist if I understand him correctly.

I still haven’t done Davies justice, which is why I recommend you read his book. Even though I disagree with him on certain philosophical points, his knowledge is far greater than mine, and the book, in its entirety, is a worthy contribution to philosophical discourse on mathematics, science and religion, and there aren’t a lot of books that merit that combined accolade.

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