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Monday, September 2, 2013

Laboring Over Formalism


Happy Labor Day to US readers... and here's a little reading you can labor over:

A piece (not sure how old it is???) from non-Platonist Timothy Gowers on philosophy and mathematics, focusing on Platonism, logicism and formalism (good stuff, BUT ONLY if you're already inclined toward philosophical underpinnings):

https://www.dpmms.cam.ac.uk/~wtg10/philosophy.html

It's brimming with interesting ideas, including (in the "#6 Truth and Provability" section) the notion that somewhere in the decimal expansion of pi there ought surely be a string of a million 7's, on the basis of it being a "normal number."

Here's a little bit of his wrap-up to the longread:
"...the point remains that if A is a mathematician who believes that mathematical objects exist in a Platonic sense, his outward behaviour will be no different from that of his colleague B who believes that they are fictitious entities, and hers in turn will be just like that of C who believes that the very question of whether they exist is meaningless...
"So why should a mathematician bother to think about philosophy? Here I would like to advance a rather cheeky thesis: that modern mathematicians are formalists, even if they profess otherwise, and that it is good that they are...

"When mathematicians discuss unsolved problems, what they are doing is not so much trying to uncover the truth as trying to find proofs….

"I also believe that the formalist way of looking at mathematics has beneficial pedagogical consequences. If you are too much of a Platonist or logicist, you may well be tempted by the idea that an ordered pair is really a funny kind of set -- the idea I criticized earlier. And if you teach that to undergraduates, you will confuse them unnecessarily. The same goes for many artificial definitions."


1 comment:

.mau. said...

while I agree that for a Platonist «Mathematical concepts have an objective existence independent of us», and so "2" and "4" exist even if nobody have defined them, I am not really sure that «"2+2=4" is true because two plus two really does equal four». Even an hardcore Platonist accepts that there exist deductive rules: he just don't believe that *the rules* are the basic level, and that they work even if no meaning is attached to them. What do you think?