Sunday, 10 February 2013

On mathematical Platonism and faith

Mathematical Platonism is, roughly described, the belief that the entities considered in mathematics (numbers, functions, algebraic systems, etc) have a reality beyond their existence as neuronal configurations in our brains.  Non-mathematicians are often disposed to reject the belief ("Where do these entities exist?" seems to be a rather compelling objection) but many practising mathematicians are Platonists, and many more (maybe a majority, but I don't have any statistics to support this) behave as though they are Platonists.

Another way of expressing the belief is to say that mathematics is "discovered" rather than "invented".  In his short essay Barry Mazur calls this "The Question" that all mathematicians come to at some point in their metaphysical speculations.

Why does Platonism have such a strong foothold among mathematicians?  I am fascinated by this question because when I was a young pure mathematical researcher I would often maintain that the entities I thought about had an independent existence (and this was when I was only dimly aware of Platonism at all).  I had, for example, an absolute conviction that when humankind came across star-faring species they would have a mathematics in which simple groups (a particular interest of mine at the time) would play the same fundamental role that they play in terrestrial mathematics.  Why did I hold this view so strongly?  In this post I want to speculate about this psychological predisposition rather than whether it is defensible.

I suggest that Platonism has such a powerful grip on the mathematical mind because mathematical discourse is packed with the language of discovery rather than the language of invention.  This language biases us to subconsciously accept that we are finding out about things that already exist.  This tradition is very deeply rooted in our mathematical discourse and certainly appears in Euclid ("The sum of the three interior angles of a triangle equals two right angles": no doubt here that a triangle exists somewhere outside our minds).  In our modern mathematics how often do we say "There exists..."? - we even have a mathematical symbol for this phrase.  Mathematics is written in a certain style and this style abounds with phrases that suggest discovery of already existing objects.  For example here is paper published today that I selected at random: here is its abstract.
For every infinite sequence of simple groups of Lie type of growing rank we exhibit connected Cayley graphs of degree at most 10 such that the isoperimetric number of these graphs converges to 0. This proves that these graphs do not form a family of expanders.
I suggest that mathematicians cannot read that abstract without having their Platonic tendencies confirmed.  This and the entire manner in which mathematics is written explains why Platonism is the default subconscious belief of so many working mathematicians.

What this tells us is that a Platonist ought to consider his/her position in the light of these strong linguistic pressures that produce cognitive bias.  Maybe they can mount a metaphysical defence (and I confess to a wistful hope that they can).  But if they cannot it is only honest to admit that they are holding similar opinions to a committed theist.  And that brings me to the final point I wish to make.

A few studies have shown that, among scientists, mathematicians tend to hold a theistic stance more so than others.  A few years ago the US National Academy of Sciences conducted a survey about god beliefs among their members.  Apparently 14.6% of mathematicians believe in god whereas the figure for biologists is 5.5%.  I am sure we should be cautious about such figures but other surveys have also shown that mathematicians tend to belief in god in higher proportions than other scientists.

Is it not plausible that someone who has committed unthinkingly to a position of faith in the real existence of mathematical entities will be more prone to commit to a faith in the real existence of supernatural beings?

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