Overweening Generalist

Sunday, May 29, 2011

Reflections on Mathematics

In Eric Temple Bell's classic Men Of Mathematics, first published in 1937, a passage has long puzzled me into wonderment. Near the end of the book, on p.572, Bell writes, "At this point we may glance back over the whole history of mathematics [...] and note two modes of expression which recur constantly in nearly all mathematical exposition." Bell then gives two broad examples of phrases which different mathematicians have used. 

Here's one:
"We can choose a number less than n and greater than n - 2."

The second type of phrase is of this sort:
"There exists a number less than n and greater than n - 2."

Bell insists these two examples are not "merely stereotyped pedantry," and that mathematicians literally mean what they say when they use either formulation, and, even though they may seem different to an inconsequential degree, each phrase is pregnantly revelatory about the kind of philosophical disposition of the thinker. The differences are not "trivial."

Now I quote the section that fills me with a sort of giddy wonder in its implications:

"These two ways of speaking divide mathematicians into two types: the 'we can' men believe (perhaps subconsciously) that mathematics is a purely human invention; the 'there exists' men believe that mathematics has an 'extra-human' existence of its own, and that 'we' merely come upon the 'eternal truths' of mathematics in our journey through life, in much the same way that a man taking a walk in the city comes across a number of streets with whose planning he had nothing whatever to do." - p.573

Far be it for such a horribly rank amateur math enthusiast such as myself to take a side here (and yet just watch as I do take as side: typical? perhaps overweening?); however, the uncanny usefulness of math (I think I'm paraphrasing the great physicist of the quantum Paul Dirac there?) does seem to make one wonder...if, as most of the great mathematicians seemed to think, there exists some sort of Platonic Pure World of math, numbers, form...somewhere? this seems to a person of my own disposition as poetic, romantic, and frankly sort of crazy, math's incredible usefulness notwithstanding. 

What I'm saying is I'm with Bell's "we can"men. I have no proofs that math is a human invention, but I think cognitive scientists like George Lakoff and Rafael Nunez (co-authors of Where Mathematics Comes From) seem probably on the right track. We are incredibly agile symbol-making critters embodied with nervous systems who evolved systems of counting that became more and more diffuse and baroque and abstruse...and wonderful. And aye: useful. At this point in my life, I can't get behind the "there exists" people with their "extra-human existence"s. But I do find it odd in the extreme that so many profoundly great mathematicians have indeed believed in what mathematician-novelist John Casti called "the one true Platonic heaven."

I'd like to be so adept at math that my constructions...sorry!: my findings so enchanted myself that I believed in a Purer World somewhere else. Lord (and Pythagoras?) knows, I'd have great company...

It seems to me that, when I cash out the "we can" vision of how math works, I'm effectively saying what Robert Anton Wilson said of math: it is "pure fiction." 

In this sense, it seems a cousin to the Modernist poet Marianne Moore's definition of poetry: poetry was "imaginary gardens with real toads in them." Because if math is pure fiction, it has functioned as the language that undergirds/underwrites/instantiates all of science and technology, including the magickal gadget you're using to read this right now. 

I find it difficult to wrap my neurons around this weird problem (which is why I return to it in my private thoughts so often: I seem of the temperament of loving NOT being "sure" about anything). Other temperaments surely differ, and I may be too close to this wonderful problem to sort it out adequately, which is truly fine by me. I'm okay with the indeterminacy of it all. I have my guesses, but they may be wrong, or crucially inaccurate in some way. Possibly I lack "imagination" with regard to this vexing and passing strange conundrum (which seems not a conundrum at all for the hard core "we can" and "there exists" folk).

The present blogger-critter has found that any further probes into either basic orientation towards the ontological status of mathematics only yields more uncanny wonderment. Perhaps more on this some other day, but I should like to leave you with this quote from one of the giants of 20th century physics, Paul Dirac (ponder the implications!):

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. This is only natural and to be expected. What however was not expected by the scientific workers of the last century was the particular form that the line of advancement of mathematics would take, namely it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation."
Paper on Magnetic Monopoles (1931)

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