The Continuum: Mainstay or Menace? (erweitert)

The Continuum:  the original sin, from whose fecund loins
came all that is non-constructive in mathematics.

-- Anon.

Kronecker dismissed mathematical entities beyond the natural numbers as “Menschenwerk”.  An average practicing mathematician (who uses such entities all the time) may  agree with him to this extent:

(1)  Our intuitions about the natural numbers are clear and solid.   So long, indeed, as one deals only with some set of actual numbers (thus, a finite set), nothing especially surprising  or even all that interesting  turns up.  If we extend our horizon to the actual infinite of the set of all natural numbers, we meet some concepts that take getting used to (Hilbert's hotel):  but once we’ve done so, they seem natural enough.

(2) The rationals and negative integers  definitely, the algebraic numbers  probably, pretty much come along for the ride (that is, you can hardly exclude them once you’ve accepted N), and they still bring in no paradox – being, after all, of the same cardinality as the natural numbers themselves.  Though, a case could be made that these are not “entities” of the same standing as the integers, which in a sense we can hold in our hands (embodied in oranges, say), but rather abbreviations for operations on integers.  Thus, we cannot hold minus-two oranges in our hands; minus-two is not a thing, but a bookkeeping device. 

(3)  The real numbers, by contrast, are … a piece of work.  Maybe even Menschen-work, except that one could hardly imagine Menschen coming up with anything so intricate and even bizarre.  Their very cardinality baffles intuition  -- and the independence of the continuum hypothesis  shows that we are right to be baffled.  [Note:  The simple infinity of the integers already baffles *untutored* intuition;  but eventually you get the idea.  Click on the Label "Hilbert's Hotel" for further exemplification.  Whereas, the cardinality of the continuum is more like... Hilbert's Nightmare...] All sorts of queasy consequences arrive for simple quantification (cf. Quine re.  objectual vs. substitutional quantification).  The reals were invented (discovered?) for purposes of analysis, which in turn was developed largely for the sake of physics: but it now appears that physics (whether in its quantum cast, where Uncertainty provides a certain indissoluble granularity; or in the Wolframesque finite-automata approach) might not actually require, or afford, a continuum.

And yet standard mathematics speaks indeed ontologically of the reals, not merely pragmatically.  Thus for instance, Rudin’s standard text (Principles of Mathematical Analysis, 3^rd edn. 1976, p. 8):

We now state the existence theorem [emphasis in original] which is the core of this chapter.

Theorem. There exists an ordered field R which has the least-upper-bound property.

The author then mentions that the proof actually constructs the Reals out of the Rationals.  This is, of course, the most solid sort of proof of all – not one of those Cantorian diagonalization thingies that has you winding up assenting to the Infinite Woodchuck, without ever quite knowing how you got into such a fix.  It gives you an actual recipe for the construction of these extended numbers, as concrete and explicit as for baking a cake.  And yet… all kinds of things can be thus “constructed”, at will, including items which presumably are not part of the furniture of the universe, in the sense that angels actually sit on them.

~

A roaring vote of confidence in the continuum  is voiced by the noted mathematician René Thom:

“God created the integers and the rest is the work of man.”  This maxim spoken by the algebraist Kronecker  reveals more about his past as a banker who grew rich through monetary speculation  than about his philosophical insight.  There is hardly any doubt that, from a psychological and, for the writer, ontological point of view, the geometric continuum is the primordial entity.

-- “’Modern’ Mathematics: An Educational and Philosophic Error?”, in American Scientist (1971), repr. in Thomas Tymoczko, ed., New Directions in the Philosophy of Mathematics (1986, rev. 1998), p. 74.

That is in-your-face Platonism, with which, quâ Realism, we have no quarrel.  But the psychological claim seems dubious:  Our intuition of the continuum is probably no more than a vague notion of a smear (and not very infinite at that, neither going out nor going down).   And as for the ontology … When we first meet the Real numbers mathematically (that was the very first thing we did in first-year calculus, with the opening chapter of Spivak’s text), we conceive them as the completion of the rationals.  And such they are indeed:  only, with respect to the metric provided by the absolute value.   With a p-adic valuation, you get a different completion of the rationals, the p-adic numbers.   Lastly, the surreal numbers augment the continuum in yet a different unexpected direction.  (I have less than no intuition about any of this.)

The physicist Schrödinger is less sure:

The idea of a continuous range, so familiar to mathematicians in our days, is something quite exorbitant, an enormous extrapolation of what is really accessible to us.

-- Erwin Schrõdinger, “Causality and Wave Mechanics”, repr. in translation in: James R. Newman, ed. World of Mathematics (1956), p. 1059

And from an Intuitionist (close kin to a physicist):

This could be done  by seeing the continuum as something that is infinitely becoming, instead of already being.

-- Dennis Hesseling, Gnomes in the Fog:  The Reception of Brower’s Intuitionism in the 1920s (2003), p. 333

(Compare our old friend the actio/actum distinction.)
Might be fine for physics, doesn’t work for math.  ‘See’ it however you like; that uncompleted-account doesn’t jibe well with Cantor-style constructions.

~

One might say:  The continuum feels unproblematic enough, so long you take it for granted, as just some kind of smooth dense slippery thing, like mud.  Yet so soon as you pause to enquire more nearly, you are back in Saint Augustine’s predicament with regard to Time: “Quid est tempus? Si nemo a me quaerat, scio …”

~

Even in a universe which (like Wolfram’s) abjures the continuum, the continuum might turn out to be mathematically indispensible for its treatment.   Cf. the indispensible role of “imaginary” numbers in electromagneticsm or quantum mechanics, even though all observables must be real-valued.

Cream for your Coffee

[The following paragraph has just been added to our essay, "A New Proof of the Existence of Coffee-Cups".]

Having at length satisfied ourselves as to the reality, or at least reliability, of coffee-cups, would should not  on that account sink back into an attitude of Moorean complacency (“I’m all right, Jack;  I’ve got hands”).  For our commitment to these  suggests yet further commitments, which we had not realized were there to assess.  Such as :  Realism with regard to quantum state vectors.

The question of ‘reality’ must be addressed in quantum mechanics -- especially if you takes the view that the quantum formalism applies universally to the whole of physics -- for then, if there is no quantum reality, there can be no reality at any level.

-- Roger Penrose,  The Road to Reality (2004), p. 508

And:

The question of the objective existence of the objects of mathematics … is an exact replica of the question of the objective existence of the outer world.

-- Kurt Gödel, “What is Cantor’s continuum problem?”, in American Mathematical Monthly, 1947.

A complex Schrödinger equation,

after the Collapse of the Wave Packet

In for a penny, in for a pound.

A New Proof of the Existence of Coffee-Cups

But the math had not clarified one basic question:  what the hell did it all mean?  How could the world of atoms be nothing but a puff of probabilities, and yet conglomerations of those atoms could create something as strong and unbending as the  chair on which he sat?

-- David Kaiser, How the Hippies Saved Physics (2011), p.  54

[Update to an earlier essay.  For the original, with readers’ comments, click here.]

Re the authors of “The Stability of Many-Particle Systems” (1966):

They couldn’t explain why it was  that ordinary everyday objects wouldn’t just go up in smoke.  It was quite anomalous.  Inert matter just lies there in a kind of steady state, ticking over silently in a metaphysical sort of way, but this ordinary and plain fact of experience  was incomprehensible to physics.

-- Ed Regis, Who Got Einstein’s Office? (1987), p. 185

There comes a time in the life of every young person, when they doubt the existence of coffee-cups.  So here comes jolly old Doctor J, to prove it for you;  and more importantly, to reassure you that, even should you someday forget the details of the proof, you may continue  to place entire faith in the existence of these useful objects, asking no questions for conscience’ sake.

(We earlier sketched the idea of the proof, but there the argumentation was informal.)

So!  It’s a quiet Sunday, the desk and the mind are clear, let us pour out a generous helping of this fresh-ground, new-brewed, steaming darkling French roast, into the convenient receptacle which (to all appearances, at any rate) sits within easy arm’s reach upon the solid oaken desktop, to stimulate the grey cells, and set ourselves to this great task  of confounding the nominalists and solipsists.

[Excursus:  Historique du problème

Dr Johnson famously proved the existence of stones  by kicking one (a martyr to science -- he reckoned without his gout).  The Cambridge philosopher of banality, G. E. Moore, satisfied himself (I choose the verb with care) of the existence of physical reality, by noticing his own hands.  (The choice was not a happy one, being potentially seen as solipsistically self-centered, not to say onanistic;  further, the phenomenon of phantom limbs complicates the picture in irrelevant ways.)
Accordingly, we shall proceed by proving the existence of coffee-cups:  both because they form the prototypical physical object for philosophers (apart from mats, which are ignored unless they have cats on them), and because, for mathematicians, who in general care not a fig for the physical, do care crucially about this one concrete object, since coffee is their essential fuel.]

But before embarking upon the abstract part of this argument (since some folks are uncomfortable with abstractions -- though really, they need not be;  what follows makes a lot more sense than most of what gets uttered around the water-cooler), I shall offer what critics have called the objective correlative:  a description of the actual coffee-cup whose space-time coordinates are My-Here and My-Now [** But now see below!].  (Rather than the ambiguous “here and now”;  for such coordinates must indeed be relativized to some observer, though not necessarily a mortal observer.)  After all, we wouldn’t want you to wonder whether we might not be just making the whole thing up.

[**Horrified update, August 2019:  Testing the link, I clicked, winding up  indeed at the sales-site for the "Catholic Attitude" mug;  but -- It has been censored!  Shorn of its knight, along with his attitude! -- Fortunately I -- Fahrenheit-451-fashion -- saved the original image, and included it in this essay; vid. inf.]

From the standpoint of the chemist, the object in question would appear to consist in some sort of ceramic, whatever that may be, though I really couldn’t say -- anyhow, something sturdily non-porous.  In general outline, it forms a cylinder, sealed off at the bottom and evacuated at the top.  Somewhat spoiling this sleek Platonic profile, a handle protrudes from the side, a concession to the physical infirmities of hominoidal incarnation.

It is thus strictly speaking a ‘mug’ rather than a cup;  but still I say “coffee-cup” because, for philosophers (who don’t get out much), that is the prototypical object in the cosmos.  (At least this is true for hypercaffeinated Americans; more leisurely Oxonians look to "the tree in the Quad").  In similar fashion, the propotypical Contingent Truth for the Oxbridge set is:  “The cat is on the mat.”  (That last one is not true at the moment, b.t.w.;  she seems to have wandered off.)

That is pretty much all you need to be  to call yourself a coffee-cup.  But these days, most cups are not quite so minimalist:  most of them sport some wacky slogan of office lore, or the logo of a sports team.  This particular specimen displays the image of a medieval knight, standing in contrapposto, his tabard emblazened with a cross pommée.  On his face is an expression difficult to read, some blend of troubled arrière-pensées and manly determination.  He is set largely within a blue circle, which his mailed headgear slightly overtops; at the bottom -- a detail I just noticed only now -- the end of his belt (bulging slightly at the tip) laps down and over, in a way that, hm,  might be misinterpreted.  And written boldly above him, the words:

Catholic  Attitude

Well.  So much for the objective correlative.  Now for the proof, deo volente -- while mindful of the strictures of Kant:

It remains a scandal for philosophy  and for human reason in general, that the existence of things outside us  must be taken only on faith,
and that, if it occurs to someone to doubt it,
we can produce no counter-argument  sufficient to prove it.

-- Kant, Critique of Pure Reason

~ ~ ~

We must confess at the outset (well, it’s a bit late for that) that we cannot actually prove the existence of coffee-cups, by the lofty standards of proof first intuited by Euclid, and refined  in our own day  to a high sheen.   Outside indeed of mathematics, such a capability appears not to exist:  Whenever I try to read a physics article reporting recent research, it always appears at some point to be hand-waving, “You’ll just have to trust us on this.”   And within mathematics itself (trade secret; don’t whisper this to Muggles) we seldom explicitly set out all the steps, even in the final published results.   As for the actual process of mathematical discovery, it normally bears no relation at all to Hilbert’s formalist program;  at its deepest and most mysterious, it is more like…. (and here we’ll have to whisper very softly) … Revelation…

Anyhow, we obviously can’t actually prove the existence of coffee-cups, because logically, strictly speaking, they might not exist:  We might all be just brains in a vat, hallucinating the whole thing.   So to clear the air, we frankly state our Auxiliary Assumption:

(AUX)  We’re not just brains in a vat.

If you personally reject that assumption -- well, happy marinating.  For the rest of us -- to proceed.

[Excursus:  Among the merits of d’Abro’s history The Rise of the New Physics (1939), is that he repeatedly points out the (sometimes unacknowledged) auxiliary assumptions that were required in the development of even the most successful programs in the physical sciences, by the most celebrated researchers, during physics’ Golden Age.

Almost nothing we do in everyday life  is free of unspoken assumptions, sheer force of habit, and random whims.  So, simply by listing such propositions as (AUX), we are making some modest progress.]

The principal premises of the proof, and the only ones that would occur to most people, are evidential.   This is where the hard work of science is done.  Fortunately, in the case of perceptions of ordinary middle-size objects, Nature does it for us, that we need not continually trouble our little heads:  all sorts of cross-connections and epistemological assumptions are hard-wired into our brains.  (These brains, b.t.w., reside in a cranium, vice a vat.)  We shall nevertheless lay out some of the perceptions and observations that serve to buttress the conclusion to objectual existence -- not so much to convince you yet further that your mug is real, but rather, virtually the contrary:  by such explicit exposure of our evidential grounds, to make plain their essential poverty, absent certain grounding metaphysical principles, along the lines of (AUX) though more substantial -- or rather again, more abstract and as it might be insubstantial, since (AUX), though it has the look of a metaphysical assumption, might in some circumstances be actually demonstrable, and thus empirical, as happens in the movie “The Matrix”.

(PE) Perceptual Evidence

(1) I seem to see before me a colour-patch (Note to the lay reader: These colour-patches compose the whole of the world for the positivist empiricists, who never quite manage to convince themselves that coffee-cups are real, and therefore cannot drink the coffee, and die of thirst), roughly cylindrical in outline -- though the exact projection upon the retina depends on the viewing-angle in complex ways (blah blah blah; insert usual empiricist verbiage here).

(2) When I reach out with my hand (for reassurance as to the existence of your own hands, consult the works of G. E. Moore), groping in the general direction of the above-named coloured patch, I abruptly encounter what seems to be a solid object -- a bit too abruptly, it turns out, as some sort of dark hot liquid is now pouring into my lap.

(3) (Insert more such evidence here -- crucially, cross-modally, involving the sound of the mug as you strike it with your pen; the smell and the taste of the coffee within).

(4)  When I pour a small amount of coffee into this apparent container, it does not quantum-tunnel out:  thus recalling, by uniformity and analogy, such similar objectual posits as the Water-Glass.

(5)  Jones here -- a stout fellow of sound mind -- affirms that, egads, he too perceives a coloured patch in what, calculating the parallax by triangulations, dum-de-dum, doing the math, appears to correspond to the same space-time locus that I myself have identified.

(6)  (etc. etc. etc.)

(C ) Conclusion

Coffee-cups really do exist.

(They do, but this one's just an image of such a cup)

Now:   The thoughtful reader will already have noticed how grotesquely speckled with gaps  such reasoning is -- why, it shouldn’t convince a child.  For, though it pretends to consist of but simple reports of perceptions -- “observation sentences” in the lingo -- rather than any abstract reasoning that might be open to critique, it actually smuggles in a great deal of unbuttressed assumptions.  Thus -- what ties (1) and (2) into any sort of connection with each other?  Why,  the unexamined assumption that the visual coordinates of (1) map smoothly and without controversy to the kinesthetic coordinates of (2):  a fact by no means obvious  -- intellectually, that is;  of course, the identity is more or less hard-wired, though it may take baby a certain amount of groping and spoon-dropping to get the respective ordinates and abscissae to finally match up.  Nor is the correlation in any sense necessary -- indeed, it can be easily overturned in simple experiments involving funny spectacles.

And as for that “Jones” there -- you are assuming the existence of Other Minds, about which great vats of ink have been spilled!  (There is, in fact, one sense in which philosophers at any rate  are brains-in-a-vat.)  And indeed you need vastly more than that -- for the Other Minds projected by mere analogy, might be only Other Monads, and incommunicable among themselves.  All right, wire them one to another -- still not enough:  you are assuming that you each mean ‘the same thing’ when you utter the same (or: “similar”) syllables:  an assumption demonstrably false in every political summit or marital argument.  All right, plow ahead and assume that:  you’re still not there.  With all the intelligence and good-will in the world, your private knowledge might not be transmissable intact: witness the celebrated case of determining whether or not your extragalactic pen-pal resides in a world made of matter or of anti-matter, or whether he is right- or left-handed.  -- And here my grey-cells throw in the towel; but were you (younger, and keener) to pursue this line of speculation further, things would probably only get worse.

No, for the various clauses of (PE) to have any cohesion and probative force at all, we require a vast apparatus of non-evidential, metaphysical assumptions, almost never made plain.  The above syllogism,  (PE) => (C ), is thus in reality an enthymeme .  Suppressed is what we might dub the enthymematic assumption, which we now state here:

(EA -- full version)  Background metaphysical premises

(Insert the entire body of philosophy here, along with the whole of science.  Do not omit to mention the Assumption of Cosmic Uniformity (spatial; temporal; spatio-temporal), the Problem of Induction, the Reliability of Deduction, along with some still-unnamed rules of thumb concerning our right to ignore pesky quantum-mechanical paradoxes when speaking of mid-level objects, etc. etc.)

Now, that is rather a tall order, and even were it somehow to be accomplished for this particular case (which might or might not "go over" towards founding the existence of pickles, say), no individual mind could survey the results and verify their correctness and inter-consistency.   And yet, this is the heavy machinery that we need to conclude to so modest a proposition as the existence of coffee-cups.  So let us superadd, as a concession to our feeble powers of ratiocination and memory, the following Concise Version:

(EA -- concise version)  God is Good.

This one works for me;  but if you’re an epistemological stickler, you can probably get by with this weaker assumption, as stated by Einstein, and applied, not only in his life, but in the practice of his physics:

(EA -- weakened version)  Boshaft ist er Nicht.

Which is to say, anglicê:  at least He is not plain mean.   By which he meant:  The cosmos is not just some vast joke, jerry-rigged to baffle our senses.  Overall, it ultimately makes sense -- even if what we perceive here below, is but the thread-gnarly underside of the grand patterned carpet.

(Note:  The basic point is more logical than theological;  still, the realms do tend to intertwine.)

Historical footnote:

Descartes derived the principle of inertia from two premises:

* the homogeneity of the straight line, and

* the immutability of God, of which the constant quantity of motion in the world is an expression.

-- Charles Gillispie, The Edge of Objectivity (1960), p. 90

Note incidentally that the first of these Cartesian premises  is at least as fraught as the second (the problems and paradoxes of The Continuum).

~

~ For a theo-philosophical fantasia, try this:

Murphy and the Magic Pawnshop

[Kindle] [Nook]

~

~ ~ ~

To add some punch to our insistence on the Importance of the Enthymeme, with all its tacit metaphysical assumptions, let us instance another, less familiar syllogism.

(P) Evidential Premises:

1. Something funny’s going on here.

2. What’s that smell?

3.  I can’t find my car keys -- probably somebody swiped them.

4.  What’s that noise?!

5.  So where does Jones get the money for a fancy car like that?

6. No way I’m letting that bastard merge into my lane.

7. Why are all those people staring at me?

8. Ouch!

9. It’s a conspiracy.

(C ) Conclusion:  The world is ruled by giant lizards.

That conclusion is an actual doctrine, firmly held by some people:  non-institutionalized people, actual citizens who have the right to vote (and who seemingly exercise that right with disproportionate frequency).  I’d provide a link to the relevant Web sites, but fear lest these poison your computer.

What, though, must be the missing enthymeme, which licenses this deduction?  Apparently something like this:

(E)  The cosmos is ruled by Satan.

Compared with (E), (C) begins to look relatively inviting.

~  ~  ~

But enough of such stuffy studies!   The sun, renewed, has broken through the clouds, the bird-sounds rise in chorus, and a breeze bestirs itself among the leaves.   Let us then fare forth, into the wide world, whose splendor bears the imprint of its Maker, as plainly as had He signed it, John-Hancock-style, with His celestial pen.

~  ~  ~

[Afterthought]  But what then, of the java, that flows from the mug whose reality we have just proved???  Why, it flows to fuel the brains of mathematicians! 

[Appendix, Jan 2015]  Having at length satisfied ourselves as to the reality, or at least reliability, of coffee-cups, would should not  on that account sink back into an attitude of Moorean complacency (“I’m all right, Jack;  I’ve got hands”).  For our commitment to these  suggests yet further commitments, which we had not realized were there to assess.  Such as :  Realism with regard to quantum state vectors.

The question of ‘reality’ must be addressed in quantum mechanics -- especially if you takes the view that the quantum formalism applies universally to the whole of physics -- for then, if there is no quantum reality, there can be no reality at any level.

-- Roger Penrose,  The Road to Reality (2004), p. 508

And:

The question of the objective existence of the objects of mathematics … is an exact replica of the question of the objective existence of the outer world.

-- Kurt Gödel, “What is Cantor’s continuum problem?”, in American Mathematical Monthly, 1947.

In for a penny, in for a pound.

~

This notion of ‘enthymeme’ deserves further comment, and here is not really the place;  yet I can find no other home for it, among these essays.  Thus:

This is characteristic of ancient informal logic -- that is, of the logic of proof or of thought-experiment …. ; we regard it as enthymematic  only through hindsight:  it was only later that an increase in content  became a sign, not of the power, but of the weakness, of an inference.

-- Imre Lakatos, Proofs and Refutations (1976), p. 81

For more from this pen, try this:
http://www.linguasacrapublishing.com/justice.html

On “Rounding Out”

The shortest and best way between two truths of the real domain  often passes through the imaginary one.

-- Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), p. 123

We have noticed Quine’s grudging acceptance of the irrationals given his unquestioning acceptance of the rationals, a process we alluded to as “rounding out”.

But it is really not so much rounding out as filling out -- or rather, filling in: filling in the gaps between the rationals.  And the result is, unfortunately, not so rational as the rationals themselves.  The rationals -- that is, fractions -- are forced upon you by Nature already in nursery school:  How shall we divide these two cupcakes among the three children? (Answer:  Each gets two-thirds.)  But the Reals are (we admit this, despite our Realism) a bit unreal.  Full of all manner of set-theoretic paradox.  Inscrutable.  You can still work with them in practical terms, because the rationals, which are well understood, are, though no more numerous than the integers, dense in R, providing a sort of well-defined ladder or footbridge along which we may proceed.

Here in any event  is the testimony of a first-rate mathematician,  to the effect that the transition to the full reals  is essentially a forced move:

We shall show how to construct a complete ordered field  from a simple chain [Think:  the natural numbers].  This … proves that any contradiction inherent in the postulates for a complete ordered field -- that is, the real number system -- is latent in the postulates for a simple chain, which is a far less complicated structure  whose consistency is almost guaranteed by our intuition.

Note that we do not discuss the existence of the simple chain.  In spite of its intuitive simplicity, a simple chain carries within itself  the germs of all the difficulties in logic and mathematics;  we are obliged to take its existence as axiomatic.

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p.  112

Such is the very axiom -- we need but two -- which we seized upon to begin this whole series of essays.   We quoted it in the form made famous by the Nominalist (and our otherwise-foe) Kronecker, which invokes the deity -- possibly casually or ironically, but perhaps more pertinantly than he knew:  for whence this “intuition” of which we are so sure?

~

The next step beyond the reals were named the “imaginary” numbers, and the name stuck.  Adjoin these, and you generate the Complex Plane.   And, unforced and arbitrary as this move might initially feel, it is on the Complex Plane  that you at last truly get a notion of a Natural Setting.  Everything just suddenly… works, and works better than you ever thought it could.  You have, all of a sudden, a Circle of Convergence -- sounds like something out of Lord of the Rings, and it is just as good.  Differentiable functions turn out to be perfectly smooth, and with a natural notion of their own domain.  (Try to define them on too small a region, and they will propagate themselves by analytic extension till they are nice and fat.)

On the complex plane, things are rounder.  (Note:  Round is good.) In R, a ‘ball’ is a line-segment, and a ‘sphere’ (the surface of a ball) is two points.  In C, they’re a disk and a circle respectively.  And you can round out or rather round off the complex plane yet further, by adjoining a single ‘point at infinity’, which is the limit of any ray pointing in any direction.  The plump, rotund result:  the Riemann Sphere.  This is homeomorphic to the surface of a penguin,  the world’s most perfect shape.

As Penrose puts it:

It is as though Nature had herself entrusted to these numbers  the operation of her universe.

(Again, note the theistic language which, all unbidden, surges forth at such a time, from even the driest of nibs.   It is a very early and natural theology, such as is depicted in that fine chapter of The Wind in the Willows, "The Piper at the Gates of Dawn".)

Another indication of the greater naturalness of the complex plane as a nursery for functions:  A real function may be C-infinity (infinitely differentiable) at a point, yet somehow “off” at this point, a fact revealed by the fact that its complex analogue is not there analytic.   Thus, as one writer put it, (complex) analytic functions are “smoother” than real functions.

[Example:  exp(-1/x), for x > 0; 0 at x = 0.  That last point is artificially “tacked on”, and in the complex picture, it shows.]

 

This Complex Plane  is a real find; it is not just a waystation to something better yet.  (David Berlinski calls complex numbers "instruments that providence had provided for the recovery of lost symmetries," a neatly postlapsarian formulation.)  There is very little beyond this, by way of fields suitable for the calculus -- certainly nothing that approaches the leap that the complex numbers represented beyond the reals.   There are the quaternions, which have their points, but are a very poor cousin indeed: the theory is poorer, not richer, for the extra generating elements, since the field of quaternions offers no analogue of holomorphic functions. ( “Quaternions have more or less dropped by the wayside.” -- Thomas Hankins, Sir William Rowan Hamilton (1980), p. 325)

Then there are the octonions, for which no-one has ever found much of a use.  And there’s an end to it.

~

Other mathematical instances of “rounding out”:

*  The adjunction of zero to the natural numbers, and of the empty-set to the world of sets.  Both function exactly like their less spectral congeners.

And a nice aesthetic note -- both are represented by a round symbol: respectively, a goose egg, and a goose egg barre sinistre.

* There are various elaborate ways of constructing things out of other things, like a Stone-Cech compactification.  But in “taking the power set”, we just stand back and let it happen.  Again and again.  Yielding the “beth numbers”, and more infinities than most folks know what to do with.

* The mathematics of string theory adds extra tiny “compactified” spatial dimensions to the three of everyday experience; in these, you just go round and round.  But this isn’t rounding-out, really, since the large spatial dimensions may themselves be compact, in which any sufficiently long journey circles back on itself.  (“Compact” doesn’t mean “tiny”;  it’s a topological, not a metrical notion.)  Space could even be flat, yet finite -- thus having the topology of a three-torus.

~

Footnote:

It is well-known that it took mankind a long time to recognize zero as itself a number.  Less well known is that “not until modern times was unity considered a number” (D.E. Smith, History of Mathematics, vol. II, p. 26.)  Or that the negative numbers were long qualified as "false".

Compare the uncertainty over whether white qualifies as a “color”.  (And if it does, what about black, or grey?)

~

So where is Minimalism in all this?  Are we just tacking on turrets and wing-additions to some increasingly sprawling McMansion?

Not a bit of it.  The operative word here really is round.  For, round things are minimal surfaces -- indeed, the very simplest class of these -- in the sense of using-up a minimal area to enclose a prescribed volume.   Our purpose is, indeed, to group like with like, and to enclose them in some stable structure.  This is no multiplication of entities for their own sake -- the itchy-clutching witchfingers of insensately proliferating fractals, which is the very architecture of the dungeons of Hell.   In rounding out, the mathematician is seeking a coherent minimal structure to regiment and account for what he has hitherto seen:  one which, upon acquaintance, may become more intuitive than the partial structures initially encountered.  (The “upon acquaintance” part may of course require a bunch of Ph.D.’s and several hundred years.)

            And the things which we have seen, and which need explanation -- or at least for agencement into some larger and more natural whole -- do keep arising.  Connections are detected among them which cry out for elucidation.  So we ascend to a yet loftier bird’s-eye -- eagle-eye -- phoenix-eye view.  To arrive, it may be, eventually at Topos Theory, or the Lord of Hosts.

(For the latter, though note:  that ladder reaches only so high.  We quote the saint:

Remaneret igitur humanum genus, si sola rationis via ad Deum cognoscendum pateret, in maximis ignorantiae tenebris.
-- Thomas Aquinas,  Contra Gentiles, lib. 1 cap. 4 n. 4 )

~

The examples we gave were mathematical, merely for clarity.  But the principle of Rounding Out  applies to any field with structure.

These vague words ‘capable’ and ‘normal’  allow the grammarian scope for shaping his task to suit his convenience.  Seeking simplicity, he will round out and round off.

-- Quine, “Reply to Harmon”, in The Philosophy of W.V.O. Quine (1986)

~

Footnote re the irrationals:

Dedekind sttressed the distinction of category  between cut and number  in 1888; against the view of his friend Heinrich Weber  that “the irrational number is nothing other than the cut itself”, he explained that “as I prefer it, to creat something New distinct from the cut, to which the cut corresponds.  We have the right to grant ourselves such power of creation”,  and cuts corresponding to both rational and irrational numbers were examples.

-- Grattan-Guinness, The Search for Mathematical Roots 1870 - 1940 (2000), p. 87

A seemingly slight, even pedantic distinction;  but like many another such, it might have its point.   Cf. my astonished delight in junior high-school, upon meeting the distinction between  x (the thing itself) and ‘x’ (the name of x) -- already adequately foreshadowed in Alice in Wonderland, but encountered now in a new context.  Likewise the difference between  x and {x} (the singleton-set of x).

In the case of an algebraic number like √2, a simple number staring you in the face out of a hypotenuse  versus the infinite train of rational pilgrims (never quite arriving at their destination) of a Dedekind cut,  one is reminded of the variety of definitions of something so familiar as a tangent:  the slope of a curve (at a point); the closest linear approximation to the curve (at that point); versus the distressing definition in Loomis & Sternberg as an infinite equivalence-class of curves (through that point).

From Finitude to Infinity

[Good heavens, it’s Sunday, and there isn’t a stitch to post !  Well -- le mieux est l’ennemi du bien, so here’s a stub -- what Malkiel would call a “torso” -- to be worked on further  if I am spared.  Think of this as a construction site, which the curious stroller may peer at through a knothole in the fence, checking back in a week or so, to see how the thing is coming.]

~

Thomas Nagel, The Last Word (1997), p. 71:

We draw this access to infinity  out of our distinctly finite ability to count, in virtue of its evident incompleteness.

Leave off, for the nonce, your incessant wrestling with the Riemann Hypothesis, and return to the simplicity of thought outlined in our parables of addition, one of the woodchuck, and one of Farmer John.

You awake in the night, in a cold sweat:  Might addition eventually break down?  For, aways down the number line -- somewhere we have never traveled, gladly beyond any experience -- there are these great big lumbering numbers -- the cube of googolplex, and whatnot.  Might they not eventually creak and crack beneath their own immensity?  Might not gigamegagoogolplex  simply refuse to have one digit more added to him, lest (like the glutton in Monty Python’s “The Meaning of Life”) he simply explode, splattering integers throughout the cosmos, in an arithmetical Big Bang ?

(Wittgenstein used to worry about that sort of thing, bless his heart.)

We do not, of course, believe anything of the sort;  though we’d be hard-pressed to explain just why we rule this out.  After all, some cosmologists, to account for certain perplexities in the red-shift, once put forward  in all seriousness  the hypothesis of “Tired Light”.  (Hey, if you had been stuck to the same old geodesic for thirteen billion years, wouldn’t you be tired too?  Or more likely, bored.)

Our metaphysical certainty  that finity is, so to speak, the same throughout, with no surprises down the line, is psychologically similar to various metaphysical assumptions in cosmology (uniformity in the large), but much more absolute.  The universe has, after all, so far proved lumpy at every scale, from the quark to galactic clusters and cosmic strings;  we can easily entertain the hypothesis that it might be gnarly all the way down.  Not so with the integers.  Our characterization of these as never ‘breaking down’  is a panoptical statement about their finished totality, thus about actual infinity.  Our faith in the well-behavedness of the finite  rests ultimately in our trust in the infinite.

Oh, and those photons ?  They do not get tired.  They whiz forever,
 tiny and trusting,

   communing with their Maker,

       in ways proportional to their understanding.

~

Other cases of proceeding from the finite to the infinite (if only heuristically):

The main purpose of the study of operator theory is to discover, formulate, and prove  the proper generalizations, valid for all Hilbert spaces, of the powerful results known in the finite-dimensional case.

-- Paul Halmos, Introduction to Hilbert Space (1951, 2^nd edn. 1957), p. 74

It turns out that, for instance, every compact operator on a Hilbert space is the norm-limit of some sequence of finite-rank operators.  But properties do change in the process:  thus, a compact operator need not have any eigenvalues.

~

A compact topological space is one in which every open cover has a finite subcover.  This turns out to be a very nice property indeed:  the classification theorem for compact surfaces (i.e., 2-manifolds) is one of the neatest and simplest and most satisfying and most startling in all of mathematics;  it completely settles that problem  once and for all.  All such shapes you can imagine  boil down to just:  a sphere; a sphere with handles; a sphere with cross-caps.  That’s it.

But as we leave compactness, we leave behind that finiteness -- and things get weird.

The classification theorem for compact surfaces  does not extend in any way to noncompact surfaces.  There is an incredible variety of noncompact surfaces.

-- Michael Henle, A Combinatorial Introduction to Topology (1979), p.  129

If, however, we retain compactness (and thus a kind of finiteness), but allow now surfaces with boundary, then everything settles back into place.  The choices are the same as before, only now you’re allowed to cut some holes.   As Henle puts it:

Every compact, connected surface with boundary  is equivalent to either a sphere or a connected sum of tori or a connected sum of projective planes, in any case with some (finite) number of disks removed.

~

A platonist conception of the infinite as a completed actual totality  misrepresents, according to [the intuitionists], the very nature of the infinite, by illicitly assimilating its character to that of finite collections.

Colin McGinn, “Truth and use”; in: Mark Platts, ed.  Reference, Truth and Reality (1980), p.  34

Note that that critique is not McGinn’s own, but is that of the sloth-like finitist/constructivist tribe known to the préfecture de police as the “Intuitionists”;  nor are the preternaturally sophisticated working mathematicians of today  guilty of any such puerile reductionist assimilation.  If anything (for all I know), the smart money now views individual integers as (more or less tragic) restrictions of an original infinitude : much as we creatures here below, though fashioned by His transfinite hands, yet mostly muck about the malls and brothels, not that different, all told, from our lesser cousins  feathered or furry.

Something possibly a bit along those lines, in the supra-empyrean context of sheaf theory, is tantalizingly broached by Edward Frenkel  here:

The [finite] number 3 is a mere shadow of the [infinite] 3-dimensional space …

~

For a merely morphological look at this word finitude, try this:

     http://worldofdrjustice.blogspot.com/2012/07/on-finitude-and-truthitude.html

Integers are our Friends

[Further reflections on a topic treated here, here, and here.]

It may be legitimately objected, that in beginning with a grubstake of mere integers, and proceeding stepwise to full mathematical Platonism, I am here sneaking past the goalposts via the fallacy of the sorites.  The classical example: We know that bald men and the hairy-headed equally exist, although we cannot specify, in that excruciating Gedankenexperiment in which each hair of the hirsute is plucked out (stop that!) one by one, at whích point precisely our unfortunate subject becomes glabrous.  Thus, suppose we agree to side with Kronecker and to grant ourselves the integers; and even grant, say, Arithmetic (that is, number theory using only elementary methods): Still, somewhere short of Topos Theory and Noncommutative Geometry – you’re not sure where, exactly, maybe you can’t even say specifically on which side of the divide Analytic Number Theory should fall, fair enough – somewhere this side that stuff, some right-thinking citizen needs to draw the line.  Noncommutative geometry – who ordered that?  You feel as though you’ve been sold a bill of goods.  Frchrssks, look at him:  that dude is bald.

Thus  the methodological objection.  There is also – especially these days, with our penchant for deconstructing and debunking – a psychological.  You may suggest that I have swallowed such a prodigious amount of abstract soup, merely because of some pre-existent hunger for it.  Now, I don’t believe that it was always pre-existent, in this particular case.  When I played cowboys and Indians, my little mind was on other things.  But, l’appetit vient en mangeant; and the integers were the appetizer.

            More concretely:  I have assumed less than may seem.  I have not so much as assumed any particular ontological status for the number “2”: I have merely taken Kronecker at his word, then attempted to refute, or at least to nuance, the second half of his epigram  (“… the rest is the work of Man”), the refutation being based simply upon the logical consequences of the first.  If, on the other hand, you were to begin by stubbornly denying that  one, two, and three (and I don’t mean “one, two, three, … infinity”, I mean: 1; 2; 3) formed an any more necessary part of the furniture of the universe, than Humpty Dumpty or Porky Pig, then I would be unable to convince you of anything by argument, having then no materials to work with.  We only got as far as I think we did, because of the perfectly enormous initial concession by the skeptic Kronecker.  You grant us the necessary reality of the natural numbers – their necessity bestowed, indeed, by the Necessary One – you have conceded a heck of a lot.  You have (it turns out) given away the ontological store.

            In fact, let us retrace our steps, and traverse some of the same terrain less hastily, and with less hunger for depth.  We have agreed to accept, as necessary, the natural numbers, and the simplest thing we can do is to count them – not worrying about primality, or odd-versus-even, or whether one number’s twice the size of another – not even necessarily ‘keeping track’: but just, ticking them off as they go by, like a bored doorman, waving the arriving spectators in to the stadium.  Now you will notice, in such a procedure, a tendency to nod off.  The numbers become dimmer and dimmer.  Has it been a thousand, or maybe twice that amount?  And should you ‘skip ahead’, and try to visualize, say, 17^(8371^545), you really can’t begin to imagine it.  The integers gradually wane, for all practical purposes, invisible.  Yet it is clear that these numbers are every bit as real, as the ones you noticed before you nodded off.  Furthermore, there are a whole heck of a lot more ‘invisible’ integers, than those that are (even with practice), visualizable:  to be precise, countably-infinitely-many more.  And if you are starting to stammer some objection about the possible non-necessity of numbers beyond Praxo (defined as the largest number that our species will ever actually need for anything; though  come to think of it  there is an interesting application of Praxo-plus-one …), then you are trumped, for we hold in our hands Dr. Kronecker’s get-out-of-nominalism-free card:  Every one of those dim distant integers  is as real as a rock, straight from the Maker’s quarry. By the time we are asked to swallow some new kind of quantity – say, a fraction, like “one-half” – we shall have swallowed a literal infinitude of whole ones.

[Update 10 IX 11] The title of this post, as well as this one,  is an example of the faux-naïf  -- a somewhat idiosyncratic concept which, like that of Minimalism (to which it bears some affinity) is slowly to be developed in the course of these posts.  In the meantime, let it remain a bit of a mystery.

Additionally, it has come to my attention that someone just found this post by searching on the words

            integers in our world

This is touching.  Though indeed, that search does not work very well, since the phrase in question did not appear in this post until this very moment.   To aid such sincere and innocent searchers in future, herewith some phrases for Google to match:

            =>  Integers at home and school

            =>  My favorite integers

            => The Campfire Book of Integers

            => Jonathan Livingston Integer

            => O Integer, my Integer !

            => Integers I have known

 

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.