David Foster Wallace on abstraction

In his book Everything and More: A Compact History of Infinity (he had a way with titles), David Foster Wallace has some wise things to say about abstraction.  To orient ourselves, let’s start with the definition of “abstract” he quotes from the O.E.D.: “Withdrawn or separated from matter, from material embodiment, from practice, or from particular examples.  Opposed to concrete.”  So, for example, a billiard rack or a dinner bell is a concrete, particular material object.  Triangularity, by contrast, is a general pattern we mentally abstract or separate out from such objects and consider apart from their individualizing material features (being made of wood or steel, being brown or silver, weighing a certain amount, and so on).

Mathematics is a paradigmatic example of a discipline that deals with abstractions, and it is the one that is the focus of Wallace’s book (though he has some things to say about philosophy too).  He notes how children are introduced to numbers:

First they are given, say, five oranges.  Something they can touch or hold.  Are asked to count them.  Then they are given a picture of five oranges.  Then a picture that combines the five oranges with the numeral ‘5’ so they associate the two.  Then a picture of just the numeral ‘5’ with the oranges removed.  The children are then engaged in verbal exercises in which they start talking about the integer 5 per se, as an object in itself, apart from five oranges.  In other words they are systematically fooled, or awakened, into treating numbers as things instead of as symbols for things.  Then they can be taught arithmetic, which comprises elementary relations between numbers. (pp. 8-9)

This little narrative nicely illustrates the sense in which an Aristotelian would regard numbers as abstract objects.  (To be sure, Wallace himself was not an Aristotelian, but it turns out that some of his examples and remarks are readily adaptable to Aristotelian purposes.)  Numbers are objects in the sense that we can refer to them, attribute features to them, etc.  They are abstract objects in the sense that they exist only as abstracted out from concrete reality by the intellect.  There can be five apples in mind-independent reality, or five dogs, or five donuts, but not five itself, considered as a kind of freestanding entity.  The number five is not an “abstract object” in the Platonic sense of the term common in contemporary philosophy (viz. an object that exists in some third kind of way over and above intellects on the one hand and concrete particulars on the other), and could not be.  For what is abstract is, for the Aristotelian, what results from abstraction, and abstraction is a mental activity of considering one aspect of a thing in isolation from other aspects.

The example also indicates, at least obliquely, both the power and the perils of abstraction.  That mathematical reasoning is powerful goes without saying, and it would not be powerful if it didn’t get at something deep in the nature of objective reality.  The intellect doesn’t make up mathematical truth out of whole cloth, but pulls it out from the concrete reality into which it is, as it were, mixed.  The peril is that we can easily be “fooled” (as Wallace puts it) into treating mathematical objects as if they existed outside the mind in precisely the abstract way in which they exist within the mind. 

Nor is it merely that abstract objects – numbers, geometrical figures, universals, etc. – do not exist in this third, Platonic way.  Another peril is that we are tempted to take what are really just artifacts of the abstractive exercise to be features of mind-independent reality.  Commenting on Zeno’s dichotomy paradox, Wallace writes:

[T]here’s obviously some semantic shiftiness going on here… [which] lies in the implied correspondence between an abstract mathematical entity – here an infinite geometric series – and actual physical space… [T]raversing an infinite number of dimensionless mathematical points is not obviously paradoxical in the way that traversing an infinite number of physical-space points is… [T]he translation of an essentially mathematical situation into natural language somehow lulls us into forgetting that regular words can have vastly different senses and referents.  (p. 70)

As the example of Zeno shows, the tendency to confuse abstractions with concrete reality is the source of many metaphysical errors.  To be sure, as Wallace immediately goes on to say of the specific muddle he calls our attention to:

Note… that this is exactly what the abstract symbolism and schemata of pure math are designed to avoid, and why technical math definitions are often so numbingly dense and complex.  You want no room for ambiguity or equivocation.  Mathematics… is an enterprise consecrated to the ideal of precision.

Which all sounds very nice, except it turns out that there is also immense ambiguity – formal, logical, metaphysical – in many of the basic terms and concepts of math itself.  In fact the more fundamental the math concept, the more difficult it usually is to define.  This is itself a characteristic of formal systems.  Most of math’s definitions are built up out of other definitions; it’s the really root stuff that has to be defined from scratch.  Hopefully… that scratch will have something to do with the world we all really live in. (pp. 70-1)

Wallace seems to think that the way people study math at more advanced levels tends to exacerbate the problem:

The trouble with college math classes – which classes consist almost entirely in the rhythmic ingestion and regurgitation of abstract information, and are paced in such a way as to maximize this reciprocal data-flow – is that their sheer surface-level difficulty can fool us into thinking we really know something when all we really ‘know’ is abstract formulas and rules for their deployment.  Rarely do math classes ever tell us whether a certain formula is truly significant, or why, or where it came from, or what was at stake.  There’s clearly a difference between being able to use a formula correctly and really knowing how to solve a problem, knowing why a problem is an actual mathematical problem and not just an exercise. (p. 52)

Now, this is an extremely important point, which applies well beyond mathematics itself.  The sheer difficulty of reasoning about abstractions can lead us to overestimate the significance of the payoff, especially when the payoff is indeed significant in some respects.  Nowhere is this truer than in modern physics.  As Wallace writes:

The modern transition from geometric to algebraic reasoning was itself a symptom of a larger shift.  By 1600, entities like zero, negative integers, and irrationals are used routinely.  Now start adding in the subsequent decades’ introductions of complex numbers, Napierian logarithms, higher-degree polynomials and literal coefficients in algebra – plus of course eventually the 1st and 2nd derivative and the integral – and it’s clear that as of some pre-Enlightenment date math has gotten so remote from any sort of real-world observation that we and Saussure can say verily it is now, as a system of symbols, “independent of the objects designated,” i.e. that math is now concerned much more with the logical relations between abstract concepts than with any particular correspondence between those concepts and physical reality.  The point: It's in the seventeenth century that math becomes primarily a system of abstractions from other abstractions instead of from the world.

Which makes the second big change seem paradoxical: math’s new hyperabstractness turns out to work incredibly well in real-world applications.  In science, engineering, physics, etc. (pp. 106-7)

But this tremendous effectiveness can be misleading (and I should note that in what follows I go beyond anything Wallace himself says).  Modern physics is very difficult indeed; unlike mathematics, it is concerned with the physical world; and the payoff in predictive and technological success is enormous.  The temptation is strong to conclude that everything in the mathematical model of the world presented by physics corresponds to something in physical reality, and that there is nothing in physical reality that isn’t captured by the model presented by physics.

But it simply isn’t so, and the mathematical abstractness of physics is precisely what guarantees that it isn’t so.  Abstraction by its very nature leaves out much that is in concrete reality, and the more abstract the model arrived at (as when, to use Wallace’s nice phrase, we are dealing with “a system of abstractions from other abstractions”), the more that is left out.  Physics can no more tell you everything there is to know about the material world than learning how to count oranges can tell a child everything there is to know about oranges.  As Bertrand Russell put it in a passage I’ve often quoted:

It is not always realised how exceedingly abstract is the information that theoretical physics has to give.  It lays down certain fundamental equations which enable it to deal with the logical structure of events, while leaving it completely unknown what is the intrinsic character of the events that have the structure… All that physics gives us is certain equations giving abstract properties of their changes.  But as to what it is that changes, and what it changes from and to – as to this, physics is silent. (My Philosophical Development, p. 13)

Or, as Russell put it more pithily and wittily elsewhere:

Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract.  Only mathematics and mathematical logic can say as little as the physicist means to say.  (The Scientific Outlook, p. 82)

Like Zeno, contemporary popularizers of science, and sometimes scientists themselves, confuse mathematical abstractions with concrete physical reality and draw absurd metaphysical conclusions.  This is precisely what happens when it is claimed that relativity has shown that time and change are illusory, as Lee Smolin and Raymond Tallis have recently pointed out.  Indeed, as Tallis emphasizes, the tendency in modern physics is to abstract from the notion of time whatever isn’t space-like, but also to abstract from the notion of space everything but pure quantity – and thus to treat time as an abstraction from an abstraction, in other words.

But again, now I’m going beyond Wallace, so I’ll stop.  Much more on these subjects forthcoming in the philosophy of nature/science book I am currently working on.

Further reading:

Concretizing the abstract

Think, McFly, think!

Progressive dematerialization

Rucker’s Mindscape

Five Proofs of the Existence of God, Chapter 3