Along the way he raises another question: how can one come to a justified belief in mathematics? Now he is speaking of pure mathematics, which is to say the mathematics that exists in the mind alone: whether or not it applies to circular objects in the world, or how imperfectly, the geometry of circles as an idea has a kind of logic to it. What he wants to defend is the idea that it is somehow already all there, and all you are doing is deducing what else you know from what you already know.
Some of you may find it pleasurable to work through this argument. If you don't, pass on; philosophy of math is not, in my experience, one of the things in life that grows on you.
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Somewhere I read the claim that you can distinguish reality and dream because reality can surprise you.
I haven't worked with any mathematicians in decades, but I had the sense that they generally assumed that mathematical objects were real -- in some sense. The objects certainly contain surprises.
Well, Plato certainly thought so, and he believed discovery was remembering. The Pythagoreans thought everything was number.
These ideas were common place among educated people until the Middle Ages, when William of Ockham and others invented nominalism.
Getting my second cup of coffee, it occurred to me that animals, even fish, have a kind of theory of ideas (forms). They are able to classify objects around them as either food/prey items or non-food/prey items. The classification scheme is crude, but the animals do have a sort of Idea as what constitutes food/prey.
Cats and dogs can count, and they can project where a prey item come out of hiding, and move to intercept.
Arguably the Pythagoreans are the proper ancestors of modern physicists, many of whom also seem to believe that reality is a sort of mathematics. The status of geometry becomes strange, though: Pythagoreans thought it really was what underlay the reality we observe, but now it looks like it's a kind of abstraction or perfection of the forms we encounter in conflicting ways. This is pure math in the same way that some earlier physical laws were pure math: the Ideal Gas Law doesn't govern any actual gases, but it's a good way of getting to the concept of how they really do behave. The fundamental particles don't organize themselves into circular orbits, and circles that perfectly obey geometry don't really exist, but they're good ways of approaching reality.
In a sense, that's what Tom is talking about here. What we get are platitudes, which aren't strictly true and yet contain enough truth to get on with. They can ultimately allow for a lot of progress even before proper definitions are fully worked out.
It's probably rabbit tracking, but...
There are two ways geometry appears in physics: used in models, which have some limits of applicability, and as a fundamental symmetry which implies some conservation law Noether's theorem
We have no model which applies everywhere (a Grand Unified Theory), and even if we did it wouldn't be practical everywhere, so we use other models like the Ideal Gas Law with the understanding that they don't apply outside certain limits, like "the density of molecules involved can't be too large or van der Waals effects become important."
XKCD 435
FWIW, I didn't find Plato's "remembering" demonstration very convincing.
"It's probably rabbit tracking, but..."
I don't think I know that expression, James.
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