Albert Einstein is famously quoted as saying
"How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?" - Albert Einstein
He's wrong.
Mathematics is what happens when you observe that the world is regular and works in a regular fashion. Two eggs, and two eggs more always make four eggs, except when you drop one. Addition, subtraction, multiplication, division, fractions, Algebra, Calculus...all these are things we SEE in the world, and were developed (theoretically) in order to correspond to the world we live in. Why is it that all these crazy scientists/mathematicians think that just because we took an idea (algebra -- what's still true if I don't tell you what numbers you're using) and extended it a little (abstract algebra--what happens when I don't tell you which operators/number systems you're using) that it wouldn't still be talking about the elements of the world? Are they surprised to find out that because the things we work (eggs, apples, $) with and the operators we normally care about (+,-,*) exhibit regularity (Commutativity, Associativity), that there are other parts of the world (electromagnetism, Relativity, QM, QCD) that exhibit the same conformance to mathematics that mathematics was originally designed to reproduce? It's not like we can't learn almost everything we needed to know about Groups/Rings/Cohomologies from basically looking at numbers of eggs or apples. Reimann geometry is just regular Geometric theorizing, applied to a sphere (or hyperbola) rather than to rectilinear space. So what's shocking?
What's shocking to me is the number of people who think that deduction is more certain/reliable than induction, or that induction isn't the base of our collective comprehension of the world. I mean, I suppose that there are some folks who gain some status from assuming that their abstract theorizing is somehow higher/better/purer than the observational induction made by a car mechanic. But their status aside, it doesn't make it true.
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4 comments:
In that case explain String Theory.
That some maths apply to the real world is distinct from saying that all maths apply to the real world. My claim is only about the former and only that this result is (contra Einstein) no more electrifying than cold oatmeal.
he he.
String theory is interesting in this discussion because it's really beautiful from a mathematical standpoint. Like Athena sprung fully formed from Zeus's forehead, it's fully self consistent. It's almost like mathematical fine art. However, it makes no predictions and can't be tested, so it has as much to do with Physics as Greek Mythology.
Then again, Einstein was not self consistent either. He didn't really "get" quantum mechanics. his famous God does not play dice... quote, but the photo-electric effect clearly demonstrates statistical quantum mechanics. You don't get 1000 electrons for 1000 photons, you get a percentage (called the quantum efficiency) and the number varies as the square root of the count.
God dies play dice, but the dice are rigged. (for suitably anthropomophised definitions of God of course)
Yeah, evolution gives you induction for free and then we induce the rules of deduction, and induce the probability that deductive conclusions are true.
Though by now humans have probably picked up a few evolved rules of deduction as well.
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