[This is a continuation of a thread begun
here.]
The fundamental theorem of enumeration, independently discovered by
several anonymous cave dwellers, states that the number of elements in a
set is the sum over all elements
of that set of the constant
function 1.
-- Doron Zeilberger, “Enumerative
and Algebraic Combinatorics”, in Timothy Gowers, ed., The Princeton
Companion to Mathematics (2008), p. 67
The very simplest thing one can do with the natural numbers, beyond simply admiring them, is to add two of them together. Do we understand how to do this?
[Update: My mistake.
The very simplest thing you can do is, given one of them, take its successor. Addition is a binary relation; whereas
Counting-one-more is a unary operation in the set of numbers.
-- Andrew Gleason, Fundamentals of Abstract Analysis (1966), p. 88]
I don’t mean practically: that we occasionally get our sums wrong, no more shows that we can’t add – let alone that there is any paradox at the heart of addition – than an occasional stutter or solecism shows that we can’t talk – always provided that, in each case, we recognize our error when it is pointed out. (“You’re right; I meant to say ‘357’/’palimpsest’.") I mean, conceptually: Do we truly understand what addition is, beyond our comforting successes up to this point, in stacking one smallish number on top of another?
Time for a parable. Farmer John pulls into his long driveway, rolls to a stop, turns off the ignition, and announces that he knows how to drive; been driving for up’erds o’ forty years, in fact. To all appearances, we must agree. But then we learn that he has never driven over 5 mph, never used reverse gear nor indeed any gear beyond the first, never driven on a highway nor indeed any route but that half-mile stretch between his driveway and the barn, and thus never had to use turn signals (or headlights, or windshield wipers, etc.), or the brakes. We might then say that he doesn’t quite know how to drive. He then retorts: But this is driving, this is what I mean by the word, just this and nothing more; and I do it perfectly.
At this point one could make the usual observations about language-games, you say tomahto I say tomayto, all that, but I wish to point to something quite different: to something not linguistical at all, nor a “matter of semantics”, nor of convention: something out there, and quite real: the automobile itself. A top-of-the-line Mercedes, as it happens (seems rather wasted on our farmer friend). John does what he does with it, and may call that what he likes; but we may note as an objective fact -- true, as it were, across all possible worlds – that he has not exhausted the capabilities of the actual machine. Can you say that you have eaten, if you have chewed, but not swallowed?
In his Wittgenstein, Kripke tries to outline a skeptical problem about addition, by defining a new operation, quus, which corresponds to plus sometimes and not others, and getting all into a lather about that. The move is much like Nelson Goodman’s celebrated/execrated blue vs. grue, a nice enough puzzle that was funny the first time. Essentially, quus is plus when the summands are small enough, and collapses thereafter. It’s the sort of artificial move that can give skepticism a bad name. (Nor am I here to clear that name. Doubt the existence of your own head, if you must; but go do it behind the barn.)
Now in fact, real arithmetical puzzles do exist, even in the matter of addition, without the invention of any artificial operations. (I shall now sit down and sup with the devil of skepticism; but observe this long spoon.) If you go on long enough, addition does totter – and almost falls; yet at that point where the nominalist bellows, “Ist gerichtet!”, a sweeter and yet mightier voice calls: “Ist gerettet!”; as we shall hear.
Meanwhile back in parable country, Farmer Jim has been introduced to a horseless carriage for the first time. He admires its sleek lines, its metallic glint, its rumble when the engine is turned on. He gets in, rolls forward one foot, and gets out. “Nice,” he says, “very nice. But I can go farther on my horse.”
Likewise with addition. Although the core and essence of addition is indeed simply that of tacking one number onto another, it is of the essence of math, as of language, that the operation is recursive: having done it you can do it again, with the output of the first addition an input to the second one; and so forth, for a while; then stop.
Now we could in fact stop here, with no further concepts or developments, and have a perfectly coherent, and quite useful, operation of addition. Had we not been created but a little lower than the angels, we probably would. Every sum, let us say of a, b,c, d, and e, is to be performed thus:
(((( a+b) + c) + d) + e)
This model for addition we may dub that of the Downs Syndrome Grocery Clerk (a familiar figure). The items to be tallied come along a conveyor belt, seriatim, and are rung up one by one until the items run out. The details of the arithmetic have been exported to the cash register, just as the details of definite integrals are often exported to computers or math tables. Let's not have any Searlian "Chinese Room" nonsense now: So long as the clerk punches in the integer written on each item as it comes, he is indeed adding; he has the entire system under his belt, as far as it goes.
Consider now a more advanced grocery clerk. After some glitch on the conveyor belt, two items (let’s keep it simple) arrive together. Which shall he ring up first? At this point, the Downs Syndrome model breaks down; we need something a little stronger. Well, infinitely stronger, in fact, but let’s not emphasize that point just yet. We add the proviso that addition of natural numbers is commutative: take the addends in whatever order you like. Moreover, this clerk – who, let us say, has no cash register, but must do the sums in his head or on paper --sometimes saves himself some trouble, thus: Presented with
apple (25 cents) plus banana (30 cents) plus ten oranges @15 cents each
he does not add each orange successively to the previous sum of the apple and banana, but adds their own total sum to what proceeded: .25 + .30 + 1.50 To justify this, we require that addition be associative: for instance, (a + b) + c = a + (b + c).
Put these two operations together, then, given that there is no upper limit on the (finite) number of things that can be added, you have now added either one mighty fire-breathing rule involving advanced quantification over infinite sets and strings, or else an infinite number of finite rule-schemata, each of which is an instance of the taboo’d fire-breather. Either way, you’ve made a huge step, and you’re still just a grocery clerk.
This strengthened system of addition is adequate to all the needs of the grocery. But now we step out to the playing fields, where Achilles is racing a tortoise (who was given an advance lead). A philosopher who (here with some reason) doubts his own head, points out that Achilles can never catch up with the tortoise. We point out that he can – indeed look, he just has – but to do so we had to add up an infinite series,
1 + 1/2 + 1/4 +1/8 + ….
Now we are facing yet another sort of infinity: not any actual infinite number (we shall still shun that, for now), nor infinitely many rule-schemata describing finitary processes, but a procedure with infinitely many terms. So, are we cool with that? We’d better be; because look: Achilles won.
Now the finitist pounces. “Does your grocery store allow rebates?”, he asks, innocently enough.
“Why, yes.”
“A-ha! Then you must allow negative numbers in your sums.”
“Well, yes, that can be done. Our cash register is actually programmed for that.”
“Good. For now I’ve got you.” And he shows us the infinite sum
1 – 1 + 1 – 1 + 1 – 1 + 1 ….
Now, as it stands, that expression is ambiguous – though no more so than “1 + 2 + 3…”. We allow ourselves to make do with expressions like the latter, because we agreed that you may group the terms however you like; it makes no difference. Only now it does:
(1 – 1) + (1 – 1) + …
yields partial sums 0, 0, 0, … and so converges to 0;
1 (-1 + 1) (-1 + 1)
yields partial sums 1,1,1,… and so converges to 1; whereas
((…((1) – 1) +1) -1) ……………..
yields partial sums 1, 0, 1, 0, and so doesn’t converge at all.
And worse is to come. In steps the concierge of the Hilbert Hotel, and reassigns the guests to new rooms: each guest in a room of even number k, is moved to room 2k. Now the sum looks like this:
1 + 1 -1 + 1 + 1 – 1 + 1 + 1 – 1 ….
which, suitably grouped by the threes of the minimal ecurring pattern, yields
1 + 1 + 1 + ….
which diverges to infinity.
So much (our finitist cries in triumph) for your easy accomodation of infinities – it has led you right over a cliff! Be ye content therefore with finite sums, with finite everything. Let Achilles forever lag behind that tortoise, in this finite life; abjure for aye the everlasting; and worship ye the finite godling, Mbumbo, lord of all the dumbos, creator of all things visible and that’s it.
At this point, we really are properly chastened; we do not know what to reply. But let us look back, to earlier testaments, and see if they provide guidance.
Often in history, mathematicians have shrunk back, with something like horror, upon encountering something ontologically unprecedented. So it was with the irrationals, the imaginaries, the non-Euclidean geometries, the infinitessimals (here the shrinking was much delayed, and the unshrinking rather recent), and much else. And had they experienced a permanent failure of will – or of trust in the creations (or as it may be, lineaments) of our infinite Father – and never returned to the subject, the initial shock might have attained, in time, the force of a precept, an Awful Warning. Yea, it is related, that in the distant past, a sailor proclaimed the irrationals, and was cast into the sea. Yea and another, in a farther age, did espouse the imaginaries, and was crucified head-downwards.
It turns out, though, that the problem of indefinitely iterated addition can be tamed, at least partially. Doing so involves a conceptual detour, to the notion of “absolute convergence”. Once that is understood, infinite sums separate essentially into sheep and goats: the sheep-series converge as nicely as you like, with shepherding (re-arrangements) allowed; the goat series stink, and are to be shunned (save as further techniques may allow us to herd a few of these).
This fable is reasonably reflective of the actual developments, though it has been compressed, and truncated before subsequent ingeniosities like Cesaro-summation. An even clearer instance of a case, in which we thought we understood a concept, and had even become rather adept at slinging it around, only to discover that we didn’t know how to proceed when we came to the edge of a certain cliff, is provided by integration. Here the infinite process was present from the start (even for a bounded function on a compact domain): the ever-shrinking rectangles of the Riemann integral. Thus, the case is not like that of mere addition, where indeed we might have planted our flag without ever crossing over the line (and how soon it came! right in the grocery store!) into infinite collections of rule-schemata or infinite processes. (There is, so to speak, no analogous Downs Syndrome Theory of Integration, and those suffering from that affliction are advised to steer clear of integrals.) Yet some otherwise-lovable functions are not Riemann-integrable; others still are, yet their collection can converge to a function that is not. The perplexities led to a new kind of integral, called the Lebesgue integral: and thus to a realization that what we had thought was integration tout court, was really only one species of what turns out to be a more general idea.
The latter saga has an after-fable: for it was this general perplexity that impelled Cantor down a path that led eventually to something quite unlooked-for, and which still lifts the bristles at the back of the neck: the topless tower of uncountable infinities. These were thus sired not of a fever, nor an opium-dream, but from (at origin) the quite practical matter of adding stuff up. If you open Cantor’s closet, that’s what falls out.
And so, to reply to the imaginary speaker of the title: You might like to add just one thing, but, unless you shut your eyes to the bloom and truth of the Creation, you’ll wind up adding much more. In for a penny, in for a pound; in for an integer, in for
the infinite.
