Infinite Powers: How Calculus Reveals the Secrets of the Universe by Steven H. Strogatz
My rating: 3 of 5 stars

The Language of God

(view spoiler)[ [Prefatory apology: my comments below are a bit rushed since I’m off for spinal surgery. As the great Oates said, “I may be some time.” I do hope not as long as he.] (hide spoiler)]

Steven Strogatz seems conflicted. He presents mathematical calculus as a description of reality and within a paragraph or two recants and calls it a useful fiction. He claims that it reveals the hidden structure of the universe; yet he admits that its fundamental presumptions contradict the findings of modern physics. He is proud of the strictly logical development of mathematical inference; but he recognises that the invention of calculus was not the result of mathematical rationality. He is a philosophical Platonist who is sympathetic to the idea of the existence of numbers quite apart from our use of them. But he is also pragmatically Aristotelian in his insistence that his devotion is not because numbers are the Truth but because they Work.

Strogatz is not embarrassed by these contradictions because his mission is to generate enthusiasm for his subject among non-mathematicians. He is willing to say whatever it takes to create interest and understanding about the usefulness, beauty, and profound insights of his branch of mathematics. Infinite Powers Reads like a sort of investment prospectus for Calculus Inc. It’s purpose is to sell, not to criticise, judge or evaluate. What we are meant to buy is not just the fact that calculus has been an immense human triumph historically, but also that it is an emblem of a culture that deserves to be more widely spread. The book is, in short, a gospel which seeks to spread the good news of real rationality.

I sympathise with Strogatz’s appreciation of mathematics in general and calculus specifically. They are obviously useful, beautiful for those who take the time to study them, and they enforce a discipline of reasoning which is unparalleled in any other intellectual endeavour. It is this last with which I have issue with Strogatz, however. Gospel-writing evangelists are notoriously unreliable when it comes to what they mean by rationality. I understand that he wants to present “the world according to calculus.” But he is constantly slipping in to the ‘world as calculus,’ which is a very different matter. In doing so he distorts both calculus and the world outside of calculus.

Infinite Powers lays out a simple and, on the face of it, plausible epistemological theory: “Calculus is an imaginary realm of symbols and logic; nature is an actual realm of forces and phenomena. Yet somehow, if the translation from reality into symbols is done artfully enough, the logic of calculus can use one real-world truth to generate another.” The implicit claim is that the universe possesses characteristics and properties that can be revealed through the relationships of the symbols and the logical connections among these symbols within the calculus itself. This is an apparently uncontroversial claim which he makes buoyantly and triumphantly.

The claim is also wrong and demonstrably so. The ‘realm’ of calculus is a language, the characteristics of which have no referent at all in the realms outside of calculus. The most obvious of these is the essential presumption of calculus that the world it deals with is continuous. That is, that every force, event, and movement is describable to an infinite level of precision. This presumption is indeed warranted when dealing with what mathematicians call the ‘number line’ which is infinitely dense - because it is defined as such. This means that there is always another number between any two numbers, no matter how small those numbers are (this is the reason Zeno can’t reach his wall when thinking mathematically). And many of these numbers are ‘real’ in calculus but have no existence elsewhere.

Strogatz recognises that this is a problem. The world outside of mathematics is not continuous, nor does it contain distances of indeterminate length (like those involving a multiple of π). Quantum science, for example, is based on the presumption that everything about the world - matter, energy, even space - is ‘lumpy’ and cannot be infinitely divided. The world is discontinuous. More accurately the world outside of calculus as described by physicists contradicts the presumptions of calculus. Strogatz dismisses this with a wave and calls continuity a useful fiction. And that it may be, but like Newton’s gravitational action at a distance and Democritus’s solid atoms, it is a fiction which has no certain existence outside the world of language, and is disputed by alternative scientific fictions.

This applies as well to all other mathematical ‘phenomena,’ from infinitely thin lines to infinitesimal points, to n-dimensional surfaces. These are aspects of the syntax of mathematical language. They exist because they are defined within and by the language itself. No one would claim that these things have a referent outside mathematics. Like continuity in calculus, they are all useful fictions. That is, mathematics, including the mathematics of calculus, have no reliable connections to anything non-mathematical. In other words mathematical symbols and relations cannot ‘represent’ the world much less reveal it.

But there is also a related issue of which Strogatz seems unaware. Probably the best way to expose this issue is by using another artificial language, the Law. As in any language, legal terms are defined in relation to each other. These terms define various legitimate ‘actors’ as well as things upon which they can act. In Roman Law, for example, there are two ‘persons’ specified - the emperor (a unique individual) and the paterfamilias (a class of individuals). The paterfamilias is further defined as a male head of household. The terms ‘male,’ ‘head,’ and ‘household’ in the way of legalese are then further defined. And so on, potentially ad infinitum, but practically until sufficient precision has been reached so that it can be determined if a party involved in a law suit is a paterfamilias or not by a normally intelligent judge.

In other words ‘paterfamilias’ is a sort of legal mask (the Greek root of the word person) or, perhaps more plainly, a categorical box in which a litigant is placed and which determines how he will be treated in law. It may not be obvious, but this categorisation is not in any way a ‘property’ of the litigant. Neither are the various other criteria - maleness, familial seniority, or indeed familial status - which underlie the definition of paterfamilias. All of these are the consequence of record-keeping not existential characteristics. They are ‘facts’ because they are recorded somewhere by someone (or testified to by someone) and are presented to the court. The documentation defines the litigant as a person or it doesn’t.

Legally we are our documents - birth certificate, driving licence, passport, etc. This is so even in a much more technologically advanced society than ancient Rome, in fact more so because of the technology. For example, it might be argued that DNA analysis now can determine sex, and therefore maleness, without the need for other records. But DNA analysis is a report about a chemical reaction that involves a number of stages and many procedural guarantees to ensure the ‘integrity’ of the result. These are all documents, that is to say, language-based evidence. In fact this modern evidence is even more embedded in language than a quick peek under the toga and a nod by a court official in a Roman tribunal (note that the nod is also language, just a bit more concise than required in the modern legal system).

We are our documents for legal purposes. Or, more accurately, the language that is associated with our proper name, determines which category we are assigned within the legal system. These documents are certainly not ‘us’ in an existential sense. We do not necessarily have the properties recorded about us on our driving license - blue eyes, brown hair, glasses. Neither do these descriptive characteristics exhaust the ways in which we could be described. They are also categories into which we are placed by the system. But they nevertheless constitute our identity.

This is why ‘identity-theft’ is such an increasing problem. Defining documents can be mistaken, forged, mis-laid, duplicated, or mixed up with others. Proving such a crime or mistake requires producing yet more documents and showing that these documents are somehow superior to those produced by the criminal or incompetent clerk. The more steps in the process of document-production, the more complex this proof becomes. This complexity is itself the result not of some existential mystery - ‘I’ may be standing physically in front of a judge and jury trying a case - but of the issues of language involved: error, fraud, intent (also a linguistic issue), etc.

The point of this legal digression is to make two things clear. First, the world of language, including the language of mathematics, is a very different world from that of ‘what happens’ or ‘events’ or what we casually call reality. Legal definition and classification is a kind of measurement. It is a less precise kind of measurement than that of science or engineering but in principle it is the same process. A recording of events is assigned to a place within the definitional framework at hand. In the law this framework is a status such as a ‘person.’ In science and engineering it is usually a position on a numerical scale, a metric. The measurement involved is never a property of the thing measured. Rather, the thing measured becomes a property of the category or the metric.

So, second, when Strogatz writes about “artfully translating” between reality and mathematics, he is playing to an audience for whom this seems plausible only because it hasn’t had to think seriously about this kind of claim before. He doesn’t want to confuse potential devotees. But as has been said before by philosophers “Things happen; everything else is literature.”

Scientific measurement is as literary as a novel by Jane Austen. The two may be written in different languages but unless ‘what happens’ to the scientist or to the novelist is recorded, that is finds its way into language, it does not exist. It is merely hearsay in both law and science; it has no evidential value. And once ‘what happens’ becomes language, it is part of its artificial world in which it will be compared, connected, and judged by other documents, recordings and measurements contained in that world.

This isolation of the world of language from the world of not-language is not news. Plato’s metaphor of the cave within which shadows are cast by a flame behind us we cannot see is an ancient expression of the condition we are in. Immanuel Kant’s permanently elusive thing-in-itself is the classic Enlightenment statement of the same issue. And 20th century philosophers like Heidegger and Wittgenstein have clearly shown how language itself, all language, is implicated in what is our unique human condition.

None of these thinkers would have thought of denying that something called reality exists (although a certain 18th century Irish philosopher, Bishop Berkeley, did), or that we experience it more or less accurately (mostly less as it turns out). Their assertion is merely that reality cannot be captured reliably in language, that the world of language is entirely separate from the world of not-language. Or as the American Pragmatist philosopher put it so concisely: Reality “is surface all the way down.” That surface is the language we use to talk about it.

Remarkably perhaps, it is the parallel existence of the worlds of language and not-language that is the source of the power of language in its various dialects from mathematics to books to music. Because language is purely symbolic and purely definitional, that is self-contained, it is relieved entirely of constraints in either physical resources or imagination. It is a human creation rather than a ‘naturally’ occurring thing. We know everything about it since we created it. So we can do with it what we like.

Strogatz is entirely correct about this. But language, including mathematical calculus ‘works’ not because it describes the way the universe is ‘in reality.’ It works because it permits our species to think, to argue, to make judgements, and to learn how to do things progressively over generations. Language frees us from reality. This is our strength and perhaps our ultimate weakness as we spread a dense cloud of language-based consciousness along with its social vices and its overwhelming technology into every part of the planet.

It is this last point which Strogatz’s mathematical boosterism fails to note. This is what gets up my nose. Calculus is a powerful tool, a foundational technology, with a fascinating history. So is the hydrogen bomb. Strogatz wants us to believe that “there seems to be something like a code to the universe, an operating system that animates everything from moment to moment and place to place. Calculus taps into this order and expresses.” This is scientism not science. Scientism is an ideology which claims that there is a privileged way of understanding the world. It is a species of thought which operates under the more general heading of idolatry, the claim to know the essence of reality. What Strogatz is really selling is not calculus as a beautiful, useful, historically interesting tool, but as an inside track to the divine, the language of God (vying of course with Hebrew, Arabic, and Latin).

I’m not buying.

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