The Ladder of Abstraction (with added rungs)

The following  logically belongs in the “Abstraction” section of our essay Consilience in Mathematics.  But as that effort is growing overlong,  we begin to cultivate here a particular idea  building upon that of abstraction simpliciter :  namely, the tendency, in modern mathematics -- and indeed this may serve virtually as the defining characteristic of modern (even: modernist) mathematics -- to abstract from any given abstraction, layer upon layer, rise upon rise, to a virtual (topless/cloud-topped) Babel, reaching to the Beyond.

(Oh, and here again we have a term from the arts, Modernism, which, as it includes “abstract art”, metaphorically applies to mathematics.  Compare our earlier essay on Minimalism in Mathematics.)

In normal practice, mathematicians mostly talk to one another -- and indeed, mostly just to those within their own hyperspecialized neck of the woods.  But occasionally, one writes an undergraduate textbook, and thus must descend to earth, if only for the nonce, and address the laity.  Thus:

This “intrinsic” formulation of Calculus, due to its greater “abstraction”, and in particular  to the fact that, again and again, one has to leave the initial spaces, and to climb  higher and higher  to new “function spaces” (especially when dealing with the theory of higher derivatives), certainly requires some mental effort, contrasting with the comfortable routine of the classical formulas.  But we believe that the result is well worth the labor, as it will prepare the student to the still more general idea of Calculus on a differentiable manifold.

-- Jean Dieudonné, Foundations of Modern Analysis (1960), p. 141

We dub this the “ladder of abstraction”, taking the phrase from our teacher of yore. Referring likewise to ascent into functions-of-functions, and function spaces, and functions from one function space to another, and to the duals of all that:

Detached from any context, this construction is a pointless formality.  But as we move up the ladder of abstraction, we find that constructions such as this  become commonplace …

-- Andrew Gleason,  Fundamentals of Abstract Analysis (1966), p. 43

The metaphor of ascent is well attested.  Gödel speaks of

the infinite series of ever stronger axioms of infinity, each of which expresses a new idea or insight.

-- quoted in Hao Wang, From Mathematics to Philosophy  (1974), p. 325

A mathematician writes of

the inferential staircase  leading from the laws of physics  to the world that lies about us …

-- David Berlinski, “The End of Materialist Science”, collected in:  The Deniable Darwin (2009), p. 160

~

Saunders MacLane,  in his book Mathematics:  Form and Function (1986), p. 36ff, has a section called “Mathematical Activities”, structured somewhat like our own in the Consilience essay.  Some of the topics are the same (analogy, abstraction, generalization), while others, not relating to consilience especially, differ (conundrums, axiomatization, proof).  One, intrinsic structure, seems to relate to consilience, but is only briefly developed; and the last, completion, we have treated under the more Quinean label of rounding out.

Now, abstraction and generalization are related notions, but neither entails the other.  MacLane acutely adduces the example of group theory.  Originally, this grew out of the concrete examples known as groups of transformations.  Later, algebraists abstracted into abstract groups.  Whether a generalization has thereby been achieved, is (as Chomsky likes to put it) “an empirical question”;  and in this case, it turns out, it has not. “No new groups turn up in this process, in view of the famous theorem of Cayley, which asserts that every (abstract) group is isomorphic to a group of transformations.”  Thus, in this case, the ladder of abstraction has only one rung.  (By contrast, abstract rings do turn out to generalize upon their original model, rings of integers.)

~

Seeking analogues of the Ladder of Abstraction  outside of mathematics proper, I happened upon this:

Quine suggests that levels of abstractness, modeled on Russell’s Theory of Types, might be established.  “In the beginning  there are only concrete objects.”  These constitute type zero and are the values of bound individual variables.  “To be is to be a value of a variable.”  Next comes first-order classes and relations:  they constitute entities of type 1 and are the values of bound predicate variables.  Classes of classes, and relations, constitute entities of type 2;  and so on.

-- Harold Lee, “Discourse and Event”, in: Hahn & Schilpp, eds., The Philosophy of W. V. Quine (1986), p. 297

The resemblance to Russell’s Theory of Types had not escaped me, but I rejected mention of it, since, rather than leading -- as the Ladder does -- to ever greater depth (the metaphor is here in distress -- maybe think of it as a ladder down a mineshaft), it seems to lead mostly to More of the Same.  In other words, forming those strata, as described above, is less like the dizzying and ethereal Abstract Ascent of mathematics, than simply forming new sets via the Power Set operation (a new and larger set consisting of all the subsets of the original set).  Now this, if we start with a finite set, leads absolutely nowhere.  It’s just like counting.  If you start with the whole of the Natural Numbers, now the Power Set operation does become more powerful, leading to new and incomparable levels of infinity.    Whether this leads to true new depth, or is rather a mere formal exercise, I do not know, since I lack all intuition of any infinities beyond the countable, let alone the Power of the Continuum or Measurable Cardinals.  Perhaps it does;  espresso-sodden Berkeley conversations about Quality emerging out of Quantity, return to mind.

Still, I am inclined to doubt it.  The very fact that the fellow can say “and so on”  virtually proves as much.  For there is no “and so on” to true mathematical abstraction.   There is nothing mechanical about such ascent -- it is more like a miracle.  You can proceed only one step -- nay rather, one leap at a time;  and the interval between leaps may take decades or even centuries.   Above the calculus lies Function Theory;  above that, Topology.  Above that, Algebraic Geometry. Far, far above us, hovers Category Theory, unreachably aloft.  And far, far above and beyond that, soars Topos Theory.  What comes next  is known only to angels.

To vary Nestroy’s celebrated epigram -- “Bis die Topologie gehts noch, aber von da bis sheaf theory  zieht sich der Weg.”

Additionally, Quine introduced the term semantic ascent.  There is some similarity to Gleason’s ladder of abstraction, but the ascent doesn’t go very high, and Quine himself -- perhaps surprisingly for a logician -- is wary of the upper reaches, preferring basic-level entities  behaviourally grounded.

Here the Russian author A. D. Aleksandrov, instead of envisaging a ladder,  uses the metaphor of layers  or (appropriately enough) of nesting, like Russian dolls, in the procession to affine or projective geometry and on to topology:

The properties of space are stratified … with respect to their depth and stability.  The ordinary Euclidean geometry was created by disregarding all properties of real bodies other than the geometrical;  here we perform yet another abstraction within geometry.

-- Aleksandrov et al, eds, Mathematics: Its Content, Methods, and Meaning (publication in the original Russian: 1956;  Eng. tr. publ. 1963), vol. III, p. 133

~

The more I think about it, the more this Ladder of Abstraction idea seems possibly fruitful.  Not so much as in the Theory of Types, but as in the scala naturae, which encompasses angelology.  (Compare also graded algebras.)

By contrast, mere ungraded “abstractness” in itself is of little interest. Thus, to take MacLane’s Group Theory example:  the so-called “abstract” groups (MacLane himself uses the sneer-quotes here) mean to lift aloft from Groups of Transformations, in that they retain the laws (associativity, inverses, and all that) while becoming agnostic as to the nature of the elements of the group.  But, first of all, groups of transformations are, compared with, say, pickles, already quite Abstract;  so the word adds, really, nothing.  Indeed, as soon as you say that two apples plus two apples are four apples, and that in the same sense  two penguins plus two penguins make four penguins (well, and a few more, after a while, if the sex mix is right), you are already indulging in such abstraction.

~

I tried looking up “abstraction” in the index of the various math textbooks and philosophy treatises on my shelves, and basically came up with  bupkes.  Thus, in Dummett’s omnibus volume, Truth and Other Enigmas (1978), we find no reference to abstraction per se, let alone to the Ladder of Abstraction, but only to “abstract objects” -- i.e., pickles versus the Meaning of ‘Pickle”,  the Idea of a Pickle, the set-containing-a-pickle, the… sandwich containing a pickle, the -- but enough.  Mathematics is so far beyond this, no comment is required.

~

The ethic -- even, the aesthetic -- of abstraction for its own sake, sociologically chronicaled here (“On Vulgar Numbers”), eventually evoked a backlash.

The Bourbaki group sought to present the entire abstract structure of all mathematical concepts in one set of volumes, the Eléments de Mathématique. In that treatise, the real numbers, which most of us regard as a starting point, only appeared midway into the series, as a special “locally compact topological group”.

An opposing idea, promoted especially in the Russian school, is that a few well-chosen examples can illuminate an entire field.

-- David Mumford, Forward to Mircea Pitici, ed., The Best Writing on Mathematics 2012, p. xv

~

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Uniform Spaces

[The following does not rise even to the level of an essay-in-progress;  more like a thought-in-progress, or even (saving your presence) a difficult bowel-movement.   But the hordes of typist-elves in the cavernous warehouses of WDJ  have yet to present anything brought to perfection this morning, and I wished not to disappoint the milling crowds that swarm this site each weekend, bringing the whole family, Sister Sue and Fido too, gawking at the glittering thoughtfronts -- the polemics, the poems, the darling little monostichs (these we can all afford) -- while shaking their heads sadly at the Trinitarian Minimalism and Cantorian Realism (out of our price-range) -- all  save one diminutive child towards the back of the bunch, eyes riveted on the prize, instinct with penetrating understanding…]

We saw here the dialectic of mathematical invention (not trying to be too Hegelian here -- think of it as an ensouled pendulum) whereby, beginning with the everyday world we live in -- I almost wrote ‘space’, but that would be to get ahead of our tale -- we abstract from the clutter of minute-to-minute experience, and conceive of it all happening within a space.   We then formalize that space with the Euclidean axioms.   We then familiarize ourselves with this new mind-environment, solving tricky problems and whatnot for a couple of thousand years, then -- since we have long effectively been working in the World of the Unseen -- very lightly generalize to Euclidean spaces of any finite dimension  -- a bit of a stretch biologically, but where, mathematically, everything works pretty much as before.

Meanwhile independently, mathematical analysis had proceeded apace, not necessarily concerned with the geometrical substrate as such, but piling up its own increasingly intricate problematics.   Then by an ideational leap which is of the essence of mathematics, and into which simply listening to lectures and slogging through the problem-sets at the end of the chapters, gives you no insight at all (executive summary:  Mathematicians are like gods), a clutch of bold spirits, bearing in mind certain delicate problems such as infinite sequences of functions and their convergence, generalized the stage on which such pageants play out, from the Euclidean to the general topological.   (The history has here been brutally telescoped.)  Something of the sort was in any case needed to save the Euclidean picture itself, since infinite-dimensional spaces were now required (even by physics),  and the finite-dimensional structures would not generalize in any straightforward way.

General topological spaces being a wildly assorted bag, various restrictions are put on them, for one purpose or another, to allow deduction and calculation.  One of these is metrizability, which we examined in the essay on Urysohn.   That has the advantage of preserving much of our hard-won familiarity with the Euclidean metric, while allowing a vast array of new metrics for particular purposes. (For example:  the by-now-familiar Lorentz metric of Einsteinian spacetime.  Once mind-boggling, yet now -- in this vaster context -- almost cuddly.)  These in turn can be slightly re-generalized, by considering pseudometrics; or further regimented, with the concept of a norm, which in turn may be relaxed into a seminorm;  and so it goes.

~

A quite different and likewise fruitful generalization of metric spaces  is the notion of a Uniform Space, introduced by algebraic geometer André Weil, in “Sur les espaces à structure uniforme et sur la topologie générale” (reprinted in volume I of his Collected Papers as [1937]).   He broaches it with a bang:

La notion de distance  est utilisée dans de nombreux travaux de topologie, [mais] l’on s’explique mal qu’elle soit venue à jouer un pareil rôle  dans une branche des mathématiques  où elle n’est, à proprement parler, qu’une intruse…

On voit apparaître ici  cette hypothèse du dénombrable (dite aussi, on ne sait pourquoi, de séparabilité),  malfaisant parasite qui infeste tant de livres … dont il affaiblit la portée  tout en nuisant à une claire compréhension des phénomènes.  … La conscience d’un mathématicien, s’il en possède [!], doit répugner à faire intervenir une hypothèse superflue …

Strong words !   The notion of metric, he claims, is not simply too restrictive, but is the wrong sort of notion for topology -- a cuckoo’s-egg in the nest.   And indeed, minus the polemics, James Dugundji makes the same point (Topology, p. 200):

A metric … can be regarded a providing a measure of nearness that is applicable throughout the space  … This notion of uniform smallness is not a topological concept :  equivalent metrics specify different sets as being equally small.

… Notice that, even in metric spaces, a continuous map may be uniformly continuous if one pair of metrics is used, but not uniformly continuous when another pair of equivalent metrics is used;  uniform continuity is therefore  not a topological concept.

(“Equivalent” metrics in the sense that they generate the same roster of open sets, which define the topology.)

Contrast a different -- and very fruitful -- restriction on general topological spaces, that of being compact Hausdorff.  This notion is strictly topological in spirit.

Footnote:   For another instance of Gallic arithmophobia, cf. the remarks of Weil’s countryman  Jean Dieudonné, in Foundations of Modern Analysis (1960), p. 141:

The fundamental idea of Calculus [is] the “local” approximation of functions by linear functions.  In the classical teaching of Calculus, this idea is immediately obscured  by the accidental fact that, on a one-dimensional vector space, there is a one-to-one correspondence between linear forms and numbers, and therefore the derivative at a point is defined [horresco referens !] as  number instead of a linear form.

In defense of Sir Isaac Newton, it must be observed, that our worthy ancestor was  quite understandably  interested in how fast something was going, at each moment:  to answer which question, he needed to invent the differential calculus.  Dieudonné, from the vantage point of centuries of progress, is looking ahead to function-spaces and dense subsets of special functions and like that.

~

The passages immediately above  evoke, unbidden, an untoward echo  characteristic of their times (the Thirties; the Sixties):  “unAmerican” and (failure to adhere to) “Chairman Mao’s Correct Line”.   But “topological” is not an all-or-nothing concept;  and we return to sanity  with jolly John Kelly (General Topology), in the chapter titled “Uniform Spaces”:

We deduce from a topological premise (that the space is compact) a non-topological conclusion (that a function is uniformly continuous).  This chapter is devoted to a study of quasi-topological results of this sort.

Even more telling is the remark by George Simmons, author of the superbly pedagogical Introduction to Topology and Modern Analysis (1963):

Some writers deal with the theory of metric  spaces as if it were merely a fragment of the general theory of topological spaces.  This practice is no doubt logically correct, but it seems to me to violate the natural relations between these topics, in which metric spaces motivate the more general theory.

Thus, it is scarcely fair, or psychologically realistic, to denounce the notion of metric as an “intruder” in topology, as Weil does.  Similarly:  you shouldn’t start off with categories and functors  before learning about  ordinary numbers and sets, even if categories prove ultimately more foundational.

That said, there does come a point where actual everyday examples impel one to consider such things as convergence and compactness  in a setting more general than a metric space.  As: pointwise convergence, which is a perfectly familiar non-exotic sort of convergence, but which cannot be seen as convergence with respect to a metric.

~     ~     ~

We have thus seen uniform space as a gentle generalization of metric spaces.  Since the point of the latter is often concerned largely with matters of limits and convergence, all we really need to know is what it means to get “closer and closer”;  we don’t need to put a number on how close, each step of the way.   This aspect was highlighted by André Weil, when he debuted the idea of uniform spaces, as a kind of intellectual hygiene.   But in practice,  quite as important to the introducer of uniform spaces is their natural application to topological groups, which come ready-made with a structure amenable to notions of nearness.

But there is more.   John Kelley, in his General Topology (1955), who devotes an entire chapter to uniform spaces, writes:

It should be emphasized that this is by no means the only framework in which uniformity can be studied.  It is possible to study a set X  together with a distinguished family of pseudo-metrics for X, or to distinguish a collection of covers of X where are to be uniform covers (roughly in the sense of the Lebesgue covering lemma).  One may also consider “metrics” with values in a structure less restricted than that of the real numbers.  All of these notions are essentially equivalent.

Such a situation illustrates a recurring intellectual theme of this series of essays (with both philosophical and mathematical applications), tagged as “Rome by different roads”.   There is a section on this notion in our essay Consilience in mathematics (indeed, in one sense, the entire notion of consilience in general  is related to this idea).

Theologia Mathematica: I

THEOLOGIA MATHEMATICA

 [Synopsis:

            Beginning with a parsimonious outset of only two Postulates,

            (1) Die ganzen Zahlen hat der liebe Gott gemacht … [Kronecker]

            (2) …visibilium omnium et invisibilium [the Credo]

we conclude to the Realist position in mathematics, associated with Cantor and Gödel. We note the nice fit with theism.]

I

The great Laplace, having presented his Mécanique Céleste, and having been asked by the Emperor, "Mais où est Dieu dans tout cela ?", notoriously replied:

"Sire, je n'ai pas eu besoin de cette hypothèse."

The cheese-eating atheist, feeling pleased with himself

(I am reminded of the austere style of his countryman Lagrange, who boasted that his Mécanique analytique contained no pictures to help make things plain; and later their compatriot Dieudonné, who in the preface to his celebrated Foundations of Modern Analysis, warns his audience that the tome will contain no such sweetmeats as pictures or diagrams, as that would only encourage the reader.)

            Perhaps people read too much into that oft-quoted remark (or in a sense, too little).  It is often taken, I suspect, as a dismissive, not to say smart-alecky reply:  a snook cocked at theists.  But really the context is both richer and more narrow.  The famous remark is more austerely analytical, I believe, and comparable to Newton’s celebrated “hypotheses non fingo”.

            Newton discovered the basic clockwork of the planetary scheme, but it was not at the time apparent, whether that system were ultimately stable under the various perturbations that mass is heir to.   And if it turned out in fact not to be so, then how to explain its evident stability over all geological time (itself of a vastness only recently appreciated – long enough to let *us* evolve, for instance)?  Before inertia was discovered (or, again rather, in a way more like posited, but still: based upon a more systematic survey of the phenomena), a  traditional perspective had angels impelling the planets in their paths by constantly puffing on them from behind (a pleasant thought); now it seemed as though these same angels might have to be called out of retirement, not to impel the planets, but to herd them from time to time, lest they wander off like lost sheep.  It was Laplace’s great achievement to prove stability by sheer mathematical means – as noble a use of the imaginative faculty  as ever graced the Sistine Chapel, and by no means a snub to the Watchmaker.  Laplace, in demonstrating that planets (‘wanderers’ in Greek) were (like the a-toms) misnamed,  proved we may dispense with the hypothesis of shepherding angels –  for this.

            But there is more beneath the firmament than the placid planets.  Beyond the deductive system of classical mechanics, there is the inductive panoply of actual life.  This too Laplace addressed, in his Essai philosophique sur les probabilités.  And again, he made no use of the supernatural.  But perhaps he took too much comfort from the pleasing and positive example of the stability of the solar system.  There was an overconfidence, a sunny bumptiousness, taken to task at length by Keynes in his Treatise on Probability (a work too little known, and which I commend to your attention).  Keynes descries a witless wizard behind the mathematical manipulations that pretend to deduce so much, and who ultimately throws us back -- surreptitiously -- on human intuition to make sense of events.  (Pay no attention to the man behind the screen.)

            Since Keynes’ time, the problems of induction and prediction  have only grown worse – or rather, they remain the same as ever, but our awareness of their depth and paradox has grown.  We have now become familiar with examples of dynamical chaos, even within the heart of the classical theory; and learned that such systems are, theoretically, rather the rule than the exception.  Even that simplest paradigm of all, the fabled billiard table, affords examples.  There are tables so shaped that, given a desired degree of knowledge of the trajectory by some future time, one may attain it by sufficiently precise initial conditions, whose precision is in some sense reasonably related to that of the required precision of prediction;  and there are tables so shaped that one may not.  (That is, the relation of the intransigent epsilon to the hapless delta, is in one case that of a banker, with perhaps rather stiff interest rates, and in the other, that of a highway robber.) This directly refutes the Laplacian determinist vision, which once extended to the universe as a whole, and which now fails right in the pool hall.

[A note, since this essay is going out to a diverse audience:  The time is past, when an author need pull his punch, for fear some member of his audience may not have met this or that notion.  You are online, or you wouldn’t be reading this.  So if “dynamical chaos” is an unfamiliar term, simply Google it up in Wikipedia.  Wikipedia knows all that can be known to Man.  – Though incidentally this does *not* mean that we should actually *pray* to Wikipedia.  Just so you know.]

            So we are back to a paradox.  System after system is either itself chaotic, or is in contact with, and thus influenced by, chaotic systems:  The parson is a placid man, but he lives within the weather.  Why then does the center apparently hold?  Why does it not all eventually – spin out, or bubble down to a sort of porridge?  Does it, like the imagined planets, need a nudge from time to time, to get it back on track?  -- Well, whether it needs them or not, it gets them, but: from us.  For the nonce, let us leave God and the seraphim aside.  [Note: In what follows, we assume the three-level system of natural description which C.S. Lewis expounded in Miracles. Roughly: subnatural = quantum; natural = classical (including relativistic etc.);  supranatural = involving free will.]  For supra-natural intervention, we need look no farther than our own fingers – righting the wineglass that had started to topple, or pulling the baby back from the edge of the stage. (The paradigm case of supra-natural intervention is supernatural intervention, by ghosts or by God; but as a logical problem, this differs little from the intervention of human will, and thus may be dispensed with where parsimony suggests.)

            Most of what goes on the the universe is (let us call it) naturalistic – whether ruled by the (classical, relativistic) laws of the natural, or the (quantum, aleatory) laws of the subnatural, or some complexus of both. But most of what *we* experience, day to day – that is, experience in consciousness, as opposed to this or that enzyme oozing about – is generously admixed with the supra-natural.  I mean this in its familiar, almost ho-hum sense (except that, by dint of our ongoing ho-humming, we have become dulled to the fact that it is, strictly, miraculous.  Our simplest Saturday afternoon crackles with miracles like a fourth-of-July sky.)

            Now, this in itself need not point to God (let alone prove Him), any more than this or that billiard-ball impact proves classical mechanics, or some passing photon yields Maxwell’s equations.  Indeed, in so far merely as itself, it does not so much as indicate that this supra-natural capacity in ourselves is even rational or good:  picture (though only for a dreadful moment) a universe peopled exclusively by madmen and sociopaths, the free-willed equivalents of scorpions.  That is to say, an outside observer of our universe might detect its departures from plain (non-quantum, non-noetic) determinism, whether from sub-natural or supra-natural inputs; but lacking internal access to the lived experience, he could not tell whether these departures made sense.  Indeed, our hunch now is that the subnatural inputs do not “make sense” – that is, no moral sense, no sense beyond themselves.  It will all (within its own world) dutifully trot along in the path laid out by the Schroedinger equation, while its inputs to our world look to us like clowns piling out of an infinite Volkswagon: but it will not, pace philosophers from Protagoras to Penrose, supply or even heighten our humanity, our morality, our free will.  Indeed, from our present vantage, the subnatural is somehow even more alien to the noösphere, than is the shadow play of Newton, or the passion play of Darwin.  He who would seek the key to our humanity there, seeks the stars in a mudpuddle.

            Whereas we, in our priviliged observatory of our own shared experience (though of nothing else), can report:  Yep, it makes sense.  It’s often in practice too f***ed-up for words, but we’re not just ensouled scorpions, we’re … possibly fallen, anyhow substantially tattered angels.

Addendum:

 A distant kin to the imagery of the ushering angels, impelling the planets and keeping them on course, cropped up again in 1925, with de Broglie’s idea of “pilot waves”, guiding the electrons in their rounds while orbiting the proton; revived again in another context by David Bohm, in his hidden-variable theory of quantum mechanics.

[continued...]