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I'd like to talk about a couple of simple thought experiments invented by the mathematician and philosopher Gottlob Frege, which for me really nail the relationship between mathematics and the physical world, and at the same time raise some fundamental questions in the philosophy of mathematics.
Suppose you have a vessel containing 2cc of liquid. To this, you add a further 3cc of liquid. How much is in the vessel now? Well, 2+3 = 5, so obviously it's 5cc. But the liquid you added wasn't the same as the liquid that was in there: the two underwent a chemical reaction, emitting some gas, and so the volume of the liquid in the vessel is actually less than 5cc. Or suppose you have five things, and you then add two more half-things. You've got six things, right? Except the "things" were pairs of boots, and the half-things were individual boots, but both individual boots were left boots. You have twelve boots, but only five pairs.
In both cases, you have some physical system (liquid in a vessel, pairs of boots in a rack) that you're trying to model using some mathematical formalism (whole numbers and addition, fractions and addition). In each case, it turns out that the model isn't a very good one, as it incorrectly predicts the behaviour of the system. But this doesn't mean that 2+3 is not equal to 5, or that 5 + 1/2 + 1/2 is not equal to 6! Nor does it mean that there's no mathematical model that would fit - in both cases, it would be pretty easy to construct one. It just means that we've chosen the wrong models. Standard arithmetic still works fine, it just happens to be the wrong thing in this case.
( SCIENCE! )
( PHILOSOPHY! )
Frege's story's rather a sad one, as it happens. He did pioneering work in the philosophy of language and in logic (he's got a solid claim to be one of the three great logicians of all time, along with Boole and Aristotle, and without his work much of modern mathematics would be literally unthinkable). In 1903 the culmination of his life's work, the Basic Laws of Arithmetic (Grundgesetze der Arithmetik), was on the verge of publication: in it, he claimed to show that arithmetic (and thus all of mathematics) could be derived from the obvious truths of logic. As it was about to go to press, he received a letter from Bertrand Russell informing him of what is now known as Russell's Paradox. This blew the entire enterprise out of the water. Frege inserted a preface to the effect of "This doesn't work any more, but I hope you find it interesting", then went off and had a nervous breakdown.
1 If you can get hold of a copy of his (sadly out-of-print) Foundations of Arithmetic, I strongly recommend it, if only for the chapter contra Mill.
Suppose you have a vessel containing 2cc of liquid. To this, you add a further 3cc of liquid. How much is in the vessel now? Well, 2+3 = 5, so obviously it's 5cc. But the liquid you added wasn't the same as the liquid that was in there: the two underwent a chemical reaction, emitting some gas, and so the volume of the liquid in the vessel is actually less than 5cc. Or suppose you have five things, and you then add two more half-things. You've got six things, right? Except the "things" were pairs of boots, and the half-things were individual boots, but both individual boots were left boots. You have twelve boots, but only five pairs.
In both cases, you have some physical system (liquid in a vessel, pairs of boots in a rack) that you're trying to model using some mathematical formalism (whole numbers and addition, fractions and addition). In each case, it turns out that the model isn't a very good one, as it incorrectly predicts the behaviour of the system. But this doesn't mean that 2+3 is not equal to 5, or that 5 + 1/2 + 1/2 is not equal to 6! Nor does it mean that there's no mathematical model that would fit - in both cases, it would be pretty easy to construct one. It just means that we've chosen the wrong models. Standard arithmetic still works fine, it just happens to be the wrong thing in this case.
( SCIENCE! )
( PHILOSOPHY! )
Frege's story's rather a sad one, as it happens. He did pioneering work in the philosophy of language and in logic (he's got a solid claim to be one of the three great logicians of all time, along with Boole and Aristotle, and without his work much of modern mathematics would be literally unthinkable). In 1903 the culmination of his life's work, the Basic Laws of Arithmetic (Grundgesetze der Arithmetik), was on the verge of publication: in it, he claimed to show that arithmetic (and thus all of mathematics) could be derived from the obvious truths of logic. As it was about to go to press, he received a letter from Bertrand Russell informing him of what is now known as Russell's Paradox. This blew the entire enterprise out of the water. Frege inserted a preface to the effect of "This doesn't work any more, but I hope you find it interesting", then went off and had a nervous breakdown.
1 If you can get hold of a copy of his (sadly out-of-print) Foundations of Arithmetic, I strongly recommend it, if only for the chapter contra Mill.