I’d vote against “infinity” being considered a number, because it tears great big holes in the fields, groups, and ordered sets that most other numbers belong to. The argument, “Zero is a special case” doesn’t carry much weight. Zero is a special case, and the definition of a group takes it explicitly into account. Numbers are closed under addition, because of zero. If infinity is a number, then the numbers are no longer closed under addition. You get problems like i + 1 = i. Working with standard algebra, that can be used to show that 0 = 1. It’s messy.
Infinity is better left aside as a kind of extreme boundary condition. It’s an idea, like an asymptote. An asymptote isn’t a number, but an abstract idea. So is infinity.
I know a bloke who is a mathematical constructivist. He insists that a number must be algorithmically constructible to be meaningful. 1 is a number. 3/4 is a number. pi is not a number (he says) because you can only construct an approximation.
I don’t agree with him, either: once more, such an idea destroys the closure property of numbers. He holds that, for instance, gigantically large numbers – like 10^ 20 – cannot be constructed, and are “not numbers.” This means that there is a number, n, such that n+1 is “not a number.” Boom. So much for groups and fields.
Yes, and zero tears holes in multiplicative groups. For that matter, complex numbers tear holes in ordered fields. Even lowly negative one tears holes in one of the most basic principles of the first systems of arithmetic, “x <= x + y”. So it is. Some structures importantly satisfy such principles [e.g., the natural numbers] and some don’t [e.g., integers]. We can study them all.
Refraining from saying “infinity is not a number” doesn’t mean you cannot choose to focus your attention on other kinds of numbers when appropriate.
I don’t follow what this has to do with anything. Yes, the definition of a group includes an identity element, which, if the group is considered additive, would correspond to zero. What of it? The definition of a monoid with absorbing element also takes explicitly into account an element with special properties which, if viewed additively, corresponds to infinity. And some structures are monoids with absorbing elements and some are not, just as some structures are groups and some are not. We can study them all.
The positive integers are closed under addition, in the sense that whenever x and y are positive integers, so is x + y. Yet they don’t have zero.
It’s true that they are not closed under addition in the stronger sense of having zero, but then, this is like observing the natural numbers aren’t closed under addition in the even stronger sense of not having subtraction.
So what? These are all structures of interest, some closed under some things, and some closed under other things. We can study them all.
There’s no one grand unified system of numbers. But certainly there are arithmetic systems with infinity which are “closed under addition”. The extended numbers [0, infinity] with infinity + x = infinity are closed under addition in the sense that, well, addition is a total operation.
Even the extended numbers [-infinity, infinity] or the projective numbers with a single unsigned infinity are still “closed under addition” in the sense that whenever x + y = z, and x and y are among such number systems, so is z. [This sense of closure is tautological; it’s only really relevant relative to an ambient super-structure with its own account of addition]
Yes, you cannot add infinity to -infinity in such systems. This is a gap in the totality of some arithmetic operation. If you are worried by the fact that such systems are not “closed under addition/subtraction”, then why don’t you worry just as much over the fact that fields are therefore not “closed under division”?
Yes, and working with an algebraic rule that lets you pass from x * y = x to 1 = y would also be messy if you had noninjective multiplication by zero around. Accordingly, what people did when they began to formalize fields was… not demand such a rule in that context, while continuing to understand that rule in other contexts. Which is the reasonable thing to do. We can study everything, rather than just the one sort of thing.
Infinity can be made a number in just as much a sense as anything else. “Abstract idea” and “number” are not conflicting descriptions. All numbers are abstract ideas.
Note to him that this depends on what one takes an algorithmic construction to be. One can certainly write a computer program which will tell you, of any rational, whether pi is above or below it.
Well, he might reasonably feel that the closure properties you are demanding are not ones he is interested in, and that his system has all the closure properties he cares about.
He may not feel there is a number n such that n + 1 is not a number. And he may feel your attempt to prove that there is, simply from 0 being a number and some putative 10^20 (which, remember, he doesn’t believe exists in the first place) not being one fails; that this argument is precisely like saying to him “How can you not believe F is a number? If you don’t believe F is a number, then you must think that there is some number n below F at which things transition from numberhood to non-numberhood”, where F is the number of steps some program P takes to halt which has never actually been proven to halt, and which, for all anyone knows, might never.
Why not reason in this way, if you are interested in studying this sort of reasoning? After all, this is precisely the situation formalized in Parikh-style feasible arithmetic. The only problem with any of this is to deny anyone else their right to investigate their own interests in total exponentiation and the like.
For what it’s worth, if your friend accepts multiplication as total, then you can convince him 10^20 exists by writing out 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10. But if, say, he were to doubt the existence of 10 ^ (10 ^ 20), you would be out of luck on this score. You’d have to write out 10 ^ 20 many factors to convince him, and, let’s be honest, you never will; he would be write to retort to your saying “But I could! I just don’t have time right now.” by suggesting this is like the fellow who says “Of course there’s such a thing as the finite number of steps until P halts. Trust me, I could write down the full evaluation of P and thus demonstrate this to your satisfaction with enough patience; I just don’t have time right now”
I might also observe, since you like groups so much, that the operation a @ b = ab/(a + b), which relates to the harmonic mean in the same way as addition does to the arithmetic mean and multiplication does to the geometric mean, is a group operation on nonzero values, but requires projective infinity to serve as its identity element.
Now, this operation is just the conjugate of ordinary addition under the involution given by reciprocation. So you might want to say “Come now, this is just a perverted way of looking at addition”. But then, to make sense of this relationship between + and @, we need to understand reciprocation as an operator which is defined at zero, for which we still need… projective infinity! And why not? It’s a very convenient way to talk about this sort of thing.
In exactly the same way that 0 and negative numbers are useful to help simplify the study of addition [though any use of them can be translated away as only “an abstract idea”, not requiring us to reify them as numbers in themselves], and non-integral rationals are useful to help simplify the study of multiplication [though any use of them can be translated away as only “an abstract idea”, not requiring us to reify them as numbers in themselves], and irrational complex numbers are useful to help simplify the study of polynomials [though any use of them can be translated away as only “an abstract idea”, not requiring us to reify them as numbers in themselves], well, projective infinity is useful to help simplify the study of meromorphic functions [though any use of it can be translated away as only “an abstract idea”, not requiring us to reify it as a number in itself]. But avoiding dedicating mental energy to maintaining a hard distinction frees one up to think more fluently about such calculations.
The student who reasons “If f(n) = (1 - 1/n) * (1 - 2/n) * (1 - 3/n), then f(∞) = (1 - 1/∞) * (1 - 2/∞) * (1 - 3/∞) = (1 - 0) * (1 - 0) * (1 - 0) = 1” gets to just as much the right conclusion in just as valid a way as the one who makes sure only to ever say “The limit as n goes to ∞ of f(n) = (1 - 1/n) * (1 - 2/n) * (1 - 3/n) is (the limit as n goes to ∞ of 1 - 1/n) * (the limit as n goes to ∞ of 1 - 2/n) * (the limit as n goes to ∞ of 1 - 3/n) [by the product property of limits] = (1 - the limit as n goes to ∞ of 1/n) * (1 - the limit as n goes to ∞ of 2/n) * (1 - the limit as n goes to ∞ of 3/n) [by the subtraction property of limits] = (1 - 1 * the limit as n goes to infinity of 1/n) * (1 - 2 * the limit as n goes to infinity of 1/n) * (1 - 3 * the limit as n goes to infinity of 1/n) [by the fact that limits commute with multiplication by a constant factor] = (1 - 0) * (1 - 0) * (1 - 0) [by the fact that the limit as n goes to infinity of 1/n] = 1”, only with much more breath to spare at the end.
We torture calculus students by insisting that they only ever frame things as the latter student here has, in the misguided supposition that this is the mark of rigor, when we could just as well have taught them how to effectively work with the framing they are naturally drawn to instead.
By treating infinity as a number with appropriate rules for its manipulation, the former student has taken the route of abstracting a common pattern of reasoning into a more convenient mental representation. Great. They’d be an idiot not to. This is how mathematics progresses.