......Perhaps most people would think (with Mill) that the meaning of '2 + 2 = 4' is that two objects plus another two makes four objects. Now, it might be supposed that negative numbers present a problem for that view (e.g. Mark Balaguer thinks it nonsense to think of minus two pebbles being added to minus two pebbles to make minus four pebbles), but think of a tiled floor, with one tile missing, as that is a pretty good picture of minus one tile. And positive numbers of tiles could be represented by tiles placed on top of the tiled floor (cf. holes and electrons in semiconductors).

......Note that positive numbers of tiles are not quite the same as numbers of tiles, in that representation, as there are tiles within the tiled floor. That is not a problem, however, because positive numbers

*are*different from unsigned numbers. The former exist within a mathematical space such as the integers, in which '2 - 4 = -2' makes sense (where +2 and +4 have, as usual, been written as '2' and '4' for convenience, because of the obvious isometry), whereas the latter do not, of course, allow you to take 4 away if you only have 2 to start with.

......Now, when we extend the negative numbers to include negative lengths (as we extend whole numbers to include fractions and other lengths), a better representation would have sea-level in place of the tiled floor, and consider extending positive heights of land above sea-level to include depths (negative heights) of the sea-floor. Or we might, of course, look at money, with negative money representing either a flow of money out of an account or, within the account, a debt. And we might, more generally, think of 'positive' and 'negative' as the names of two directions.

......That also fits nicely with the extension of the real number line to the complex number plane, in which there are not just two but infinitely many directions. Historically, the complex numbers (e.g. the square roots of minus one) were only taken seriously when mathematicians could picture them as extra directions. And now we find the complex numbers being as applicable in science (e.g. in quantum mechanics, the foundation of chemistry, electronics, lazers and so forth) as the negative numbers are in everyday situations, which indicates that we are here cutting nature at its joints, so to speak.

......Although Balaguer thinks we must use sets to get the negative (and similarly, the real) numbers, surely we use a concept of

*direction*to make sense of, e.g., 2 - 4 (and similarly, of the square roots of minus one), and surely that also accounts for how useful we find such numbers, as we make our ways in the world. (And incidentally, the real numbers show how our number concepts might be judged empirically, as Mill thought.) In other words, signed numbers and complex numbers may best be thought of as elementary vectors (another fundamental mathematical concept with many applications).

......And so the question of what we

*mean*by 'direction' arises. A very ordinary use of 'direction' is when one gives someone directions to get somewhere. When you get to the next crossroads, take the first left, for example. Note that the road to go down is picked out by a direction relative to where you will be when you have to go that way. And roads can be travelled in two directions (so I am not here thinking of Fregean directions, which are sets of parallel lines), and so I am wondering how deeply the concept of a direction is related to the concept of an option.

......Were there no choices, there would just be all those roads, where they are (much as sets are abstract objects that just sit there in the Platonic realm, so to speak), and little sense to logic. So we might expect choice to be a fundamental mathematical concept. Certainly the word 'choice' appears in the foundations of mathematics, there being an axiom of choice in standard set theory, and choice sequences replacing real numbers in constructive mathematics. But more deeply, our mathematical concepts are likely to be grounded in mathematical intuitions that we share with other intelligent animals (e.g. via the number sense), and intelligent animals make choices (of course).

(PS: This post is linked to in the Carnival of the Animals:)

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