On Tarski’s “Convention T” (expanded)

Accolades:

Convention T embodies our best intuition as to how the concept of truth is used.

-- Donald Davidson

My lone dissent:

More boredom thou shalt never see

than Tarski’s Truth Convention T.

“Snow’s white” is true -- let’s get this right --

if, but only if,  snow is white.

A.T.,  logician and skirt-chaser

[Footnote]

When Tarski’s logical machinery is used to provide languages with an interpretation, it should not be seen as giving us a definition of truth. [Emphasis in original.]  When used in this way, it gives us a theory that, for each of the infinitely many sentences of the language, assigns a condition that obtains if and only if that sentence is true -- where truth is something we takes ourselves to understand antecedently, without definition.

-- Scott Soames, Philosophical Analysis in the Twentieth Century (2003), vol. II, p. 293

For a major empirical research program, investigating the validity of Convention T, click here.
 

 ~

A fellow dissenter:

See for example the panegyrics in Wallace:  “It may strike the reader that Convention T is an astonishingly powerful intellectual device…” (etc.)  Wallace’s expressed admiration for the achievements of modern logic  is of course sincere, and I share it (not always for his reasons).  But these achievements should not be used to terrorize the reader …

-- Saul Kripke, “Substitutional Quantification”, in Evans & McDowell, eds., Truth and Meaning (1976), p. 340

Ontological Epigrams

Okhamians like Quine (for so I shall refer to him, out of deference to my former professor, so as to avoid the loathsome connotations of Nominalist) are concerned not to clutter up the ontology with extra entities like facts or meanings which parallel and move in unison with statements  without really adding anything useful.  Strawson (a reliable nutshell-stuffer) puts it in a nutshell  thus:

Facts are what statements (when true) state;  they are not what statements are about.

-- P.F.Strawson, “Truth” (1950; often anthologized)

~     ~     ~

Quantum superposition is ontological, not epistemological.  It is not that we don’t know what state the cat is in.

(Unfortunately I have forgotten who said this.  Mark Twain?)

~     ~     ~

The ontological touchstone for British analytic philosophy, whether of the Idealist or the Empiricist inclination, has ever been the Tree in the Quad.    For several thousand years, this noble oak has been under uninterrupted observation by a rota of pre-Chalcolithic druids, lest it pop out of being, unobserved.  What a shame that would be!  It was such a nice tree!

~     ~     ~

The first question of ontology is:  What is there?

The answer to this, as Quine put it: Everything.
(Yet adds the stern proviso:  But no more than that.)

The second is, of each thing:  What is it?

To answer this, we may revise good Bishop Butler thus.

=>  Everything is, what it is  --  along with several other things as well.

Thus: a diamond is:

* A costly gem, proverbial for hardness

* A face-centered cubic crystaline allotrope of carbon

* A girl’s best friend

* Forever.

Given this, the question, What is Being? (überhaupt or tout court), must be categorized as meta-ontological.

~     ~     ~

Meinong’s bizarre deviation(**) … was … jenseits von Sein und Nichtsein.  Oddly enough, I find this idea a good one, provided that we bolster it with Bentham’s theory of fictions.  Contextual definition, or what Bentham called paraphrasis, can enable us to talk very considerably and conveniently about putative objects  without footing an ontological bill.  It is a strictly legitimate way of making theories in which there is less than meets the eye.

--- Quine, “Existence and Quantification”

The concluding epigram of that paragraph  is particularly relished by Minimalists.

(** Note:  The reference is to that philosopher’s ontological exuberence,  and not to any possible undue familiarities with capybaras during off-hours.)

~     ~     ~

Thus far the apophthegmatic epitome.  If these have whetted your appetite, click here.

[Orthoepic update]  Since folks keep clicking on this post, I'll add this:
The fifty-dollar first word of this essay  refers to the Nominalist, William of Okham.  That last word is pronounced OCK-um.  But the adjective, Okhamian (which I possibly just made up, but on a proper model) would be pronounced ock-HAY-mee-an.
Try to work this into your next singles-bar conversation.  It's guaranteed.

~     ~     ~

« Ontology can be saved  with a sufficient ideology. »

-- Saul Kripke, “Is There a Problem about Substitutional Quantification?”,

in: Evans & McDowell, eds., Truth and Meaning (1976), p. 343

Not sure what that means, exactly;  but it has a lilt to it, like “Die Welt ist alles, was der Fall ist”;  of which, perhaps, it is a paraphrase.

Naming & Necessity

In a celebrated lecture by that title, later reprinted as a booklet, the philosopher Saul Kripke argues for an account of proper names as being in origin ostensive/baptismal, handed down then  to the generations  by a sort of isnâd (to use the term of Islamic traditionaries).   This, by contrast with the more traditional understanding of names achieving their reference via matching some understood description -- you manage to pick out Aristotle from amidst the hordes of shades, by his embodying certain characteristics --  teacher of Alexander, author of the Organon, known to his homeys as “the Stagirite”.

The essay sparked an extensive debagte, to which I would not presume to attempt to contribute;  save that one signal example, supportive of Kripke’s a-posteriori necessitarian position, has never, to my knowledge, been cited in the literature.   Namely:  How (on the descriptive account) do you pick out an individual who has no characteristics? 

I refer, of course, to the Mann ohne Eigenschaften, to whom the novelist Robert Musil  has dedicated an extensive tome (widely praised, though seldom read).

To be sure:  Lacking all characteristics  is itself a kind of (meta-)characteristic, distinguishing its (non)bearer much like the “Invisible Man” of Monty Python.

A critic might object, How do you know you have managed to pick out precisely that individual by that (meta-)description, “the Man without Qualities” -- rather than a whole grey horde of characterless/uncharacterizable wannabes and also-rans? 

This conundrum brings to mind some mysterious passages in Quine’s essay “Ontological Relativity”, which basically posit a kind of non-identity of indiscernibles:  things you might quantify over, on the substitutional sense of quantification, but at the same time slur over, since they lack individual names, and may be indistinguishable from any of the named entities.

Well -- nunc dimittis, I have said enough;  time for a nightcap, and so to bed.   Perhaps these puzzles may be unraveled by our grandchildren.

The (In)scrutability of Reference

In our post below (Doctor Justice’s Petting Zoo), we offered an ontological fable, inspired by Quine’s celebrated demonstration, in Word and Object and later writings, that an investigator faced with the problem of “radical translation” cannot determine, by behavioristic methods alone, whether the expression in the exotic language refers to rabbits as unified individual enduring entities, or as (potentially contingent) conglomerates of undetached rabbit parts, or as ontological instantons (what you see before your eyes right now), or individual doses of the mass-term Rabbit Fusion (the way puddles instantiate water), or as incarnations of the Urkaninchen, or what have you:  the reference of the expression is insofar inscrutable.   From this potential modus tollens  he decides to hang on to the behaviorism  and to ditch various purported mental entities instead -- a process which, if carried to what some (Scott Soames, for one) deem its logical conclusion, then  “Quine’s position has several consequences that are so unpalatable as to make it …  self-undermining.”  (Philosophical Analysis in the Twentieth Century (2003), II 282).  But the spice of the tale is in the telling, and Quine tells it exceedingly well.

Likewise delightful is his fable of referential ontogenesis, a tale broached in Word and Object  and taken up again in Roots of Reference.  Here we are given poignant glimpses of Quine’s tortured infancy (“More mama!  More water!  More red!), and offered careful, psychologically plausible reconstructions of how we acquire mass terms, collectives, quantifiers, relative pronouns and so forth, along with their associated ontologies.

In “Reference and its Roots” (in: Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 523), Strawson has a bone to pick with Quine’s (psycho)logical account of the acquisition of singular versus general reference:

It must be part of [the learner’s] mastery of ‘Fido’, that Fido is unique.   A plurality of Fidos simutaneously soliciting his attention  must be ruled out by his understanding of the term -- semantically ruled out. … But if it is ruled out, then what becomes of the claim that ‘Fido’ is semantically simpler than ‘dog’? The answer  at least  is simple:  we just reverse the terms. … ‘Fido’ is a more, not a less, sophisticated acquisition than ‘dog’.  ‘Dog’ has individuation;  ‘Fido has individuation plus.

But here Quine is surely correct, as regards the sequential ontologenesis.  Initially, Mama (for let us revert to her, for dignity, leaving Fido in his kennel)  simply is what she is, as presented (well, and as supplemented by instinct, but in this connection  the point is a distraction, though we revert to it later):  Die Mutti ohne Eigenschaften, as it were (nurse, nurse, nurse).  There is initially no reason to posit that she, or Fido, are representatives of any  larger Natural Kind, any more than is Yuri the Unicorn (the family pet, and unique of his species, as it happens).  Later we learn and add layers to this initial baldly-presented UrMutti:  she turns out to be an example of the class of Women, and moreover a painter, and a Clinton Democrat, and foreign-born (which accounts, in retrospect, for certain oddities of her accent, though it sounded fine to us as an infant);  so eventually the term Mama does become enriched beyond that of woman;  it just happens that we met her prior to such enrichment, unlike the case of such other specializations as, say, topologist as against person, where we don’t meet the former term until our including ontologies are well in place.  -- The situation is quite general, not especially connected with singular terms.  As, there is a certain commonality in the notion ‘atom’ that has persisted since the days of Democritus, yet the concept has repeatedly deepened, from the periodic table and the notion of valence, to the Rutherford model, then the Bohr model, and eventually SU(3).

*

Let us now take leave of Quine’s idealized account of the logic of language acquistion -- take leave of philosophy altogether, and glance at actual acquisition,  in all its oddity.

There was a time, I am told, by one in a position to know (I myself have no memory of it), when I uttered “Daddy” in the presence of any bald man.  What did I mean by this?   Did I actually mistake the gentlemen in question for my own father?  Was the word used as a metonym or simile -- “Lo, a man who reminds me of my father!”   Was it a way of saying (the word bald not yet being in my young vocabulary) “Behold, a bald man!”  Or is the reality behind the reference  truly inscrutable, my infantile mind not being yet up to the task of distinguishing among these variant interpretations.  (Such, indeed, was some of the thrust of my “Petting Zoo” parable:  Quine’s sophisticated picture of temporal rabbit-slices competing with Rabbithood and the Rabbit Fusion  for primacy, is comically at variance with his scenario of savages in the bush.)

The first hypothesis (debility in recognizing my own father) seems unlikely (certainly I blush to imagine it), and yet a somewhat similar situation occurred involving a friend of mine, who at the time was eight years old -- not eight months.  The lad was somewhat familiar with a friend of mine named Nick;  and one day, spotting a fellow who sported a sandy moustache like Nick’s, referred to him as “Nick”.   I said No, that’s not Nick, they just look somewhat alike.   To my surprise, the boy insisted that he was not mistaken, and was quite upset that I continued to contradict him.

Now, such anecdotes have a curious attraction, as offering rare insights into the no-longer-accessible logic of our own young minds.  (For another such, click here: Temps et durée chez une enfant de quatre ans.) But more significant -- and more disturbing -- are referential inscrutabilities that persist (often undetected) into our adult life.

The clearest examples are provided by the shifting, mixed figures of our own dreams.   Very often an individual in your dream is a sort of quantum superposition of a number of people you have known (and some, perhaps, that you have not precisely known personally, but are archetypes or something of the sort).  You seldom get “Daddy” in a pure state.  -- In film, such ambiguities were classically depicted in “Persona”.

By now we are on the boggy ground, not of Frege and Quine, but of Freud.   And the latter’s analysis of the Psychopathologie des Alltaglebens  is relevant here as well.  We can see this in standard-issue misspeakings (Versprechen).  One my wife spotted just an hour ago, when, referring to a phone conversation she had had with our son Steven concerning the Shakespeare course he’s now taking, I said, “Did you mention that you and Steven used to attend the Ashland Shakespeare festival?” -- but I should have said, “You and I”.  And this was indeed no mere unmotivated “slip of the tongue”; its psychological grounding was immediately apparent to analysis (for I live an examined life).   What was surprising was simply that the mixed quantum state here involved a melding of myself with my son, whereas the usual melding in my mind (originally pre-conscious, now familiar) is between my son and my younger brother.

Again, such blends are not restricted to persons.  Freud discovers many examples in Die Traumdeutung, under such designations as Mischbildung, Mischgebild, and Mittelding.  In extreme pathological cases, such blending can involve the rational and the inanimate, as in The Man who Mistook his Wife for a Hat.  (The most nightmarish title  in the history of literature.)

*

Back to Quine.

His radical-translation fable was framed as a tale of linguistic anthropology; of rabbit-hunting, but really he was after bigger game.  As he later stated (“Ontological Relativity”):  referential inscrutability begins at home.

I was reminded of this earlier today, while perusing Scott Soames’ account of Kripke’s Naming and Necessity and its aftermath.  All sorts of ingenuity has been applied to the case, to try to tame its oddities and paradox (problems with Peano, Thales, etc.):  actuality operators, neo-descriptivism …  And I couldn’t help thinking:  quelle besogne ingrate.   Here some highly trained minds are applying techniques appropriate to philosophy and mathematics, largely with a view to constructing abstract semantic castles in the air, yet in part to model actual human linguistic behavior, at least in the sense of being inspired and corrected and tethered to the same -- much as the theoretical physicist, spinning his gossamer webs of Strings, must occasionally glance at the actual facts of the cosmos, to justify his departmental affiliation.  But human linguistic behavior -- human any behavior -- is simply too messy and defective to admit of such austere systematic description.  (Kripke’s point was indeed partly just that, and he left his lecture informal.)   Certainly mathematicians, when asking foundational questions, do not worry about what Joe Sixpack thinks about Zorn’s lemma.

Epigram:  The enterprise of linguistic philosophy is like that of “Consider a spherical cow” -- good mathematics, bad biology.

The larger problem that confronts us (and it is a perspective I have been brought to only reluctantly) is the truth of the following two factual complexes, either surprising in itself, but jointly infuriating:

(a)  the Unreasonable Effectiveness of Human Communication

(b)  the Utter Impotence of Human Communication

[More to come, if reader interest warrants]

A New (Non-)Proof of the Existence of God

Actually an old one, of eighteenth-century vintage.  Let D. E. Smith tell it (History of Mathematics, 1923, vol. I, p. 523):

Diderot had somewhat displeased the Czarina  by his antireligious views, and so she persuated Eurler to assiste her in suppressing him.

Diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of God… Euler …said gravely, and in a tone of perfect conviction:

     Monsieur, (a + b^n)/n = x, donc Dieu existe; répondez !

Diderot, to whom algebra was Hebrew, was embarrassed and disconcerted, while peals of laughter arose on all sides …

For my own brilliant demonstration that modern syntactic theory implies the truth of the Nicene Creed, click here.

These are, of course, nonsense;  and are offered simply as counter-weights on the other side of the balance-pan, to equally ludicrous (though seriously meant) would-be scientific demonstrations of the opposite (chemicals go sploosh when you think of a tree, therefore we’re all just robots).

~

Denis Diderot, penning one of his zingers

Kripke offers a useful correction to the hoary anecdote:

In fairness to Diderot, it should be mentioned that the incident surely never took place as described.  In fact  Diderot was the author of several learned mathematical essays. 

If we do not take care, however, some of our philosophical discussions are in danger of coming to resemble the legendary confrontation.  I have seen cases where a very simple, almost mathematically trivial technical trick  has captured a philosopher’s imagination, and been used as if it were the key   that easily and mechanically unlocked  doors that were forever closed to ordinary philosophical investigation.

-- Saul Kripke, “Is There a Problem about Substitutional Quantification?”, in: Evans & McDowell, eds., Truth and Meaning (1976), p. 415.

Keynes makes the same point in some detail in his philosophically-attuned Treatise on Probability.

I’d Like to Add Just One Thing

[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.