Tanasije Gjorgoski wonders what it would mean for math to be empirical.
I think we can stipulate that mathematical theorems are “empirical”
all and only to the extent that they are construed as statements that
apply to (among other things) empirically describable states of
affairs. So the proposition ‘2+2=4′ is an empirical fact just to the
extent that it is taken to mean that a collection of any two things
added to another collection of any two things will in every case yield
a collection of four things. (This may be what Einstein had in mind.)
Now, '2+2=4' even construed as an empirical claim admittedly seems like a case of necessary truth. But perhaps it's only a case of what I’ll call necessary truthiness. (Apologies to Stephen Colbert.) How would we know the difference?
A possibly instructive example come from (where else?) quantum mechanics — viz., the case of quantum indeterminacy. Try to conceive of a particle that does not have a jointly determinate position and momentum. Can you do it? Probably not. In fact, I think doing so may be psychologically impossible. And yet, if the predominant interpretation of quantum uncertainty is to be credited, particles do not have a determinate position-momentum.
This confounding empirical discovery, combined with the necessary-truthiness of determinate position-momentum, is probably what led Feynman to declare that “no one understands quantum mechanics.” I think we’d be in a similar situation with respect to mathematics, were such a compelling series of countervailing discoveries to undercut it: No one would understand it.
It’s no surprise, then, that we can’t imagine how it might be so.
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* I’m setting aside several possible objections here, among them: that the quasi-necessary status I attribute ex ante to the thesis of determinate position-momentum is contestable; that notwithstanding the consensus view, some scientists and philosophers insist that uncertainty is epistemic and not ontological (though this may be further evidence that the intuition underlying the determinacy thesis is irresistible); and that unlike mathematical statements, the determinacy thesis is not a formal theorem.
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