Meta-abstraction in the physical and social sciences

One of the themes of Aristotle’s Revenge is the centrality of mathematical abstraction to modern scientific method, and the ways that it both affords modern physics tremendous predictive power but also, if we are not careful, is prone to generate philosophical fallacies and metaphysical illusions.  This is especially so where we are dealing with abstractions from abstractions – meta-abstractions, if you will. 

In his recent book on the philosophy of time, Raymond Tallis notes how this has happened in modern thinking about the nature of space and time.  First, physical space has come to be conflated with geometry.  Whereas the notions of a point, a line, a plane and the like were originally merely simplifying abstractions from concrete physical reality, the modern tendency has been to treat them as if they were the constituents of concrete physical reality.  But then a second stage of abstraction occurs when geometrical concepts are in turn conflated with values in a coordinate system.  Points are defined in terms of numbers, relations between points in terms of numerical intervals, length, width and depth in terms of axes originated from a point, and so on.  Time gets folded into the system by representing it with a further axis.  Creative mathematical manipulations of this doubly abstract system of representation are then taken to reveal surprising truths about the nature of the concrete space and time we actually live in.

You don’t have to be an Aristotelian to see the fallaciousness of all this.  From Berkeley to Whitehead to Lee Smolin, a diverse group of thinkers has cautioned against blithely reading off metaphysical conclusions from abstract models.  The predictive and technological successes of the abstractions have facilitated the fallacy, but it remains a fallacy all the same.  That’s why there is a longstanding debate in philosophy of science between realist, instrumentalist, and (the middle ground) structural realist interpretations of scientific theories.  The abstractions and predictive successes by themselves don’t settle anything.  Metaphysically revisionist arguments that begin “Relativity theory works, therefore…” are a bit like concluding, from the fact that you have found a certain road and highway map of Pennsylvania to be useful, that Pennsylvania must therefore really literally be nothing more than a perfectly flat white surface covered with black and red lines. 

In his book The Reformation in Economics, economist Philip Pilkington laments the ways in which the tendency to confuse what I am calling meta-abstractions with the concrete reality they are abstracted from can also infect social science and the policy recommendations based upon it.  He begins with some important points about abstraction in general.  In what follows I’ll note and expand on some of these points.

The nature of abstraction

First, following Kant, Pilkington notes that the concepts we arrive at via abstraction can have a greater or lesser degree of “homogeneity” with less abstract concepts, and with the concrete realities that fall under the concepts.  For example, the concept PLATE (as in a dinner plate) is closely homogeneous with the concept CIRCLE, as is a particular plate.  But it is less closely homogeneous with a more abstract concept like GEOMETRICAL FIGURE.  When we form concepts by abstraction, we mentally strip away concrete features of things and focus our attention on general patterns.  The more abstract the concept is, then, the more numerous are the features we are stripping away – and thus, the less there is in the way of actual concrete reality that we are capturing.  Now, Pilkington notes, the way economists represent the economy mathematically is very abstract indeed, and thus not closely homogeneous with the actual concrete economic facts. 

A second general point he makes, this time citing Berkeley, is that abstraction requires language, and, more generally, symbols.  For concepts cannot strictly be imagined.  Anything you can imagine – that is, form a mental image of – is going to be concrete rather than abstract.  To appeal to one of my stock examples, if you imagine a triangle, you are always going to form an image of some particular triangle, such as a blue right triangle.  But the concept TRIANGLE is completely universal, applying to red and green triangles as well as blue ones, acute and obtuse triangles as well as right ones, and so on.  More abstract concepts (like GEOMETRICAL FIGURE) are even further removed from anything we can form an image of.  At the same time, the way the human mind operates, we need to form some kind of image even when we entertain the most abstract of concepts, and need some kind of external sign by which we may record our thoughts about them and call the attention of others to them.  Hence we use words and other symbols.  For example, we represent the number four with the word “four,” or via the Arabic numeral 4, or the Roman numeral IV, or in stroke notation as ||||, or in some other way.

(Side note: There is a potential chicken-egg problem here insofar as nothing really counts as language in the first place – or at least, as language that goes beyond the expressive and signaling functions of which non-human animals are capable – apart from concepts.  Clearly, then, concepts and language must come about together.  But how does that happen?  Good question for another time, though for one possible answer I commend to you John Haldane’s “Prime Thinker” argument.)

Now, words and symbols are typically parts of systems of words or symbols – for example, languages, and the numeral systems just referred to – and such systems vary in their expressive power.  Pilkington cites the famous example of the advantages in mathematical reasoning that a numeral system containing 0 makes possible, compared to systems which lack any corresponding symbol.  As Berkeley writes in Alciphron, in a passage quoted by Pilkington:

But here lies the difference: the one, who understands the notation of numbers, by means thereof is able to express briefly and distinctly all the variety and degrees of number, and to perform with ease and despatch several arithmetical operations, by the help of general rules.  Of all which operations as the use in human life is very evident, so it is no less evident, that the performing them depends on the aptness of the notation… Hence the old notation by letters was more useful than words written at length: and the modern notation by figures, expressing the progression or analogy of the names by their simple places, is much preferable to that for ease and expedition, as the invention of algebraical symbols is to this, for extensive and general use.

End quote.  Now, here’s the thing.  The results we get when using a system of symbols to some extent reflect the system of symbols we’ve chosen, the things we’ve chosen to group together under a symbol, the rules that govern the system, and the ways we’ve manipulated the symbols according to those rules – rather than the objective reality being represented by the system. 

Here’s an analogy.  In a pen and ink line drawing, an artist can use thick lines to represent some contours, thin lines to represent others, a break in the line to suggest yet others, series of lines or cross-hatching to represent shadows, and so on.  Splotches of ink can also represent shadows, though they could be used instead to represent blood or holes or bumps or any number of other things.  The artist might also draw in an illustrative style or a cartoony style, do a tight rendering or a loose sketch, and so on.  Now, a skilled artist might produce a likeness of a person, object, or scene that is so close that we might find it useful for identifying the person or thing, predict how it will look from different angles, and so on.  All the same, some features of the drawing will reflect only the mode of representation rather than the thing represented, and we could be led into serious fallacies and errors if we failed to keep this in mind – for example, if we thought that the thing represented really had a black outline around it, or if we concluded that there must be some interesting relationship between shadows, blood, and holes in things, on the grounds that they all looked like black splotches in the drawing. 

Similarly, since even the most useful system of symbols will have features that reflect the natures of the symbols and the system of rules governing them rather than objective reality, we need to be careful lest we assume that there must in objective reality be something corresponding to a given element of the system.

The example of economics

In economics, Pilkington points out, one starts with abstractions like INCOME, CONSUMPTION, INVESTMENT, GOVERNMENT EXPENDITURE, EXPORTS, IMPORTS, SAVINGS and TAXES.  These already tend to run together very different phenomena.  For example, Pilkington says, CONSUMPTION can cover things as diverse as “the purchase of this book… dishwashers, bananas, underpants, blueprints for perpetual motion machines from dubious internet websites and cat-food” (p. 99).  INCOME might include Medicare payments that don’t actually go to households but rather directly to healthcare providers.  And so on. 

These abstractions are then replaced with algebraic symbols, such as Y, C, I, G, X, M, S and T, respectively, in the case of the examples cited.  These enter into formulae such as (to borrow Pilkington’s examples) the national income identity:

Y º C + I + G + (X – M)

or a formula representing the relationship between income and savings and taxes:

Y º C + S + T

Such equations can then be substituted into one another, which, after cancelling, yields:

I + G + X º S + T + M

Now, Pilkington notes, we are at this point really manipulating abstractions from abstractions – what I have, again, called meta-abstractions – and thus are even farther from concrete economic reality than the abstract concepts we started out with are.  That need not be a bad thing, but one must constantly keep in mind the limits of what can be captured in such representations and the way that the results of our manipulation of formulae might reflect the method of representation rather than empirical reality.  Pilkington writes:

Such algebraic abstraction is extremely useful, but it can also be used to throw up dust and allow people to engage in sophistical nonsense.  An awful lot of microeconomics is precisely this: nice, neat, formal abstractions that have almost a zero-degree of homogeneity with the real world.  They are far, far from Kant and his circular plate.  Rather, they tend to be made-up stories with no real empirical content.  They are, in that sense, fantasy constructions. (p. 100)

Borrowing a point from fellow economist Tony Lawson, Pilkington also points out that the method of mathematical modelling is such that the model is made into a closed system, with the result that the reality modeled is represented in a deterministic way.  This may be appropriate in physics, Pilkington says, but not when representing human action.  To treat economic behavior as if it were deterministic, even just for purposes of the model, is a clear case of reading what is really just a feature of the abstract method of representation into the reality represented, rather than reading it out of that concrete reality.

(I would add, though, that even in physics we should regard such models as simplifying abstractions, for reasons of the sort raised by Nancy Cartwright.  Of course, these days people are, in light of quantum mechanics, happy to allow that nature does not operate in a strictly deterministic way.  But even in the old days, the contrary supposition was not really justified.  The determinism was read out of the mathematical models and into nature, not out of nature itself.  Again, see Aristotle’s Revenge for discussion of such issues.) 

The bias in science

Now, here is a deep irony.  Mathematics is, quite rightly, widely regarded as a paradigm of objective and disinterested knowledge.  And its heavy use in modern science is a major reason why science too has a reputation for objectivity and disinterestedness.  But there is a fallacy hidden in the implicit inference.  For the fact that a mathematical technique is of itself free of bias simply does not entail that its application to some aspect of concrete reality is also free of bias.  And indeed it is not free of it.  When we apply mathematical models to objective reality, we are always making the tendentious assumption that there is nothing more to reality than is captured by the model – or at least nothing more to it that is relevant to the purposes for which we are constructing the model.  Of course, that assumption might be defensible and correct; but then again, it might not be.  Either way, it is an assumption, and one that is extrinsic to science itself.  It is a philosophical assumption about science, an assumption that reflects further philosophical assumptions about what nature is like and about what the best techniques are for studying it.  As E. A. Burtt noted in his classic book, the tendency of many modern scientists has been to make an entire metaphysics out of what is really just a method. 

Pilkington too complains that scientists tend to make philosophical assumptions about which they are “completely unreflective” (p. 114).  For example, they “have a tendency to fall back on a reactive materialism as their default worldview” but “the details are never worked out – and if we are to be honest, it more so resembles an ideology than a properly reasoned and considered worldview” (pp. 113-14).  He also notes that, even when making grand but undefended philosophical assumptions, “most scientists arguably do not truly understand the philosophical implications of what they their theories tell them because they are so illiterate in the language of philosophy” (p. 114). 

Pilkington observes that though what he calls “lower-level physics” such as “Newtonian mechanics, electromagnetism, thermodynamics, the basic precepts of relativity theory and so on” are relatively free of bias, “speculations about the origins or nature of the universe” are not free of it and often verge on “crossing the boundary into metaphysics” (p. 121).  The former areas of study have a degree of empirical testability that the latter do not, but because they are all lumped together as “physics,” the latter inherits from the former an unearned prestige.

Then, as Pilkington notes, there is the fact that a branch of science like “Newtonian physics is almost completely bereft of politicisation because, say, the theory of gravity really makes no difference to how we as human beings organise our lives and build our societies” (p. 117).  However, when scientific study does have dramatic political implications, the evidence is clear that it is often massively biased.  (Pilkington cites the work of Brian Martin, who took as a case study the debate several decades ago over the effect of supersonic transport aircraft on the ozone layer, and showed how politicized the science on both sides had become.)

Now, consider these four sources of potential bias in science: an excessive confidence in abstract mathematical models; a tendency toward a crudely materialistic analysis; a fallacious attribution of the prestige enjoyed by the directly testable areas of science to the untestable and speculative areas; and a subject matter that has dramatic implications for how we organize our lives and societies.  Have we seen these come together in any recent controversies?   Hmm?

Well, of course we have.  All four have been on display in the defense of the COVID-19 lockdowns, which involved an overreliance on speculative and faulty mathematical models; a fixation on the mechanics of the transmission of the virus while ignoring the psychological damage, destruction of livelihoods, and ruin to education caused by lockdowns; the ridiculous smearing of all skeptics as “science deniers,” as if questioning the lockdown was on all fours with rejecting the Periodic Table of Elements; and considerable politicization, putting vulgar sloganeering in place of dispassionate scientific argumentation and favoring more relaxed measures for political allies such as left-wing protesters.

Of course, opponents of lockdowns can also be influenced by political biases, no less than proponents of lockdowns can be.  That is precisely why, a year ago and early in the pandemic, I was defending the proponents against those who too quickly dismissed the initial lockdown as an unjustifiable infringement on liberty.  However, in the nature of the case it has always been the defenders of lockdowns, and not the opponents, who have to meet the burden of proof.  And as I also argued a year ago, the longer the lockdowns went on, that burden would inevitably become heavier rather than lighter. 

Yet as time went on, lockdown defenders largely acted as if the opposite were the case.  Indeed, they largely conducted themselves disgracefully, exhibiting precisely the reverse of the caution and humility that the four sources of potential bias cited by Pilkington should have made them especially sensitive to.  And at this point it is clear that the most draconian measures were a mistake, inflicting massive harms with no net benefits that could not have been achieved through less extreme measures.  But “mistake” is really far too mild a term for the callous arrogance and manifestly fallacious reasoning of people who imposed on others enormous costs that they mostly avoided themselves.  The dogmatic scientism that motivated this disaster in policy provides an object lesson in how philosophical errors are by no means of mere academic interest, but can have dramatic and indeed catastrophic real-world effects.

Related posts:

Concretizing the abstract

David Foster Wallace on abstraction

Cundy on relativity and the A-theory of time

Color holds and quantum theory

The particle collection that fancied itself a physicist

Think, McFly, think!

Metaphysical taxidermy