Sunday, January 19, 2014

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

For the latest in fine reading, check this out:

For more about abstraction, here:

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