Topics Mathematics

Question I am interested in how mathematical propositions relate to objects in the world; that is, how math and its concepts somehow correspond to the physical world. I have thought a bit about the issue, and realize that what happens, say, with numbers when we do some kind of mathematical operation with them may be the same as when we deduce one proposition in logic from another (If there is a number 2 and an operation "+", and an operation "=", then the result of using 2 + 2 = 4); but my question is this: does the proposition 2 + 2 = 4 mean the same thing as taking two objects and placing two more objects alongside of them, and then counting that there are four objects?

Accepted:
October 17, 2013

Comments

Philosophers continue to debate the relationship of mathematics to the physical world, including why mathematics is so effective at describing the physical world. The SEP entry on "Explanation in Mathematics," available at this link, contains much useful discussion as well as many references to further reading. At least one of the articles cited in the bibliography is available online: The Miracle of Applied Mathematics, by Mark Colyvan. I hope these prove helpful.

Strictly speaking, the proposition that 2 + 2 = 4 can't mean the same thing as the process of taking two objects, placing two more objects alongside them, and then counting that there are four objects in total. Propositions and processes belong to different categories. Moreover, one might doubt that the proposition that 2 + 2 = 4 even entails that whenever you take two physical objects and place two more physical objects alongside them, there will be four physical objects to count up. Why? Because the proposition that 2 + 2 = 4 is necessarily true: true in all possible worlds whatsoever. Yet there are strange-but-possible worlds where any two objects of a certain kind vanish whenever you try to put two more objects of that kind alongside them; even in those strange worlds, 2 + 2 = 4.

Indeed, even in the actual world 2 + 2 = 4 doesn't infallibly guide us about what happens when we put things together. Notoriously, if you mix 2 liters of water and 2 liters of ethanol, you get less than 4 liters of solution. Good to know if you're planning a party.