The philosophy of applied mathematics

told a guest at a recent party that I use mathematics to try to understand migraines. She thought that I ask migraine sufferers to do mental arithmetic to alleviate their symptoms. Of course, what I really do is use mathematics to understand the biological causes of migraines.

My work is possible because of a stunning fact we often overlook: the world can be understood mathematically. The party goer's misconception reminds us that this fact is not obvious. In this article I want to discuss a big question: "why can maths be used to describe the world?", or to extend it more provocatively, "why is applied maths even possible?" To do so we need to review the long history of the general philosophy of mathematics — what I will loosely call metamaths.

What is applied maths?

A stunning fact: the world can be understood mathematically.

Before we go any further, we should be clear on what we mean by applied mathematics. I will borrow a definition given by an important applied mathematician of the 20th and 21st centuries, Tim Pedley, the GI Taylor Professor of Fluid Mechanics at the University of Cambridge. In his Presidential Address to the Institute of Mathematics and its Applications in 2004, he said "Applying mathematics means using a mathematical technique to derive an answer to a question posed from outside mathematics." This definition is deliberately broad — including everything from counting change to climate change — and the very possibility of such a broad definition is part of the mystery we are discussing.

The question of why mathematics is so applicable is arguably more important than any other question you might ask about the nature of mathematics. Firstly, because applied mathematics is mathematics, it raises all the same issues as those traditionally arising in metamaths. Secondly, being applied, it raises some of the issues addressed in the philosophy of science. I suspect that the case could be made for our big question being in fact the big question in the philosophy of science and mathematics. However, let us now turn to the history of metamaths: what has been said about mathematics, its nature and its applicability?

Metamathematics

The long history of mathematics generally lacks a distinction between pure and applied maths. Yet in the modern era of mathematics over, say, the last two centuries, there has been an almost exclusive focus on a philosophy of pure mathematics. In particular, emphasis has been given to the so-called foundations of mathematics — what is it that gives mathematical statements truth? Metamathematicians interested in foundations are commonly grouped into four camps.

Formalists, such as David Hilbert, view mathematics as being founded on a combination of set theory and logic (see Searching for the missing truth), and to some extent view the process of doing mathematics as an essentially meaningless shuffling of symbols according to certain prescribed rules.

Logicists see mathematics as being an extension of logic. The arch-logicists Bertrand Russell and Alfred North Whitehead famously took hundreds of pages to prove (logically) that one plus one equals two.

Intuitionists are exemplified by LEJ Brouwer, a man about whom it has been said that "he wouldn't believe that it was raining or not until he looked out of the window" (according to Donald Knuth ). This quote satirises one of the central intuitionist ideas, the rejection of the law of the excluded middle. This commonly accepted law says that a statement (such as "it is raining") is either true or false, even if we don't yet know which one it is. By contrast, intuitionists believe that unless you have either conclusively proved the statement or constructed a counter example, it has no objective truth value. (For an introduction to intuitionism read Constructive mathematics.)

Plato and Aristotle as depicted in Raphael's fresco The school of Athens.

Moreover, intuitionists put a strict limit on the notions of infinity they accept. They believe that mathematics is entirely a product of the human mind, which they postulate to be only capable of grasping infinity as an extension of an algorithmic one-two-three kind of process. As a result, they only admit enumerable operations into their proofs, that is, operations that can be described using the natural numbers.

Finally, Platonists, members of the oldest of the four camps, believe in an external reality or existence of numbers and the other objects of mathematics. For a platonist such as Kurt Gödel, mathematics exists without the human mind, possibly without the physical universe, but there is a mysterious link between the mental world of humans and the platonic realm of mathematics.

It is disputed which of these four alternatives — if any — serves as the foundation of mathematics. It might seem like such rarefied discussions have nothing to do with the question of applicability, but it has been argued that this uncertainty over foundations has influenced the very practice of applying mathematics. In The loss of certainty, Morris Kline wrote in 1980 that "The crises and conflicts over what sound mathematics is have also discouraged the application of mathematical methodology to many areas of our culture such as philosophy, political science, ethics, and aesthetics [...] The Age of Reason is gone." Thankfully, mathematics is now beginning to be applied to these areas, but we have learned an important historical lesson: there is to the choice of applications of mathematics a sociological dimension sensitive to metamathematical problems.

What does applicability say about the foundations of maths?

The logical next step for the metamathematician who bothers to think about the applicability of mathematics would be to ask what each of the four foundational views has to say about our big question. Discussions along this line have been written by a number of mathematicians and scientists, such as Roger Penrose in the book The road to reality, or Paul Davies in his book The mind of god.
I would like to take a different path here by reversing the "logical" next step: I want to ask "what does the applicability of mathematics have to say about the foundations of mathematics?" In asking this question I take for granted that there is no serious disagreement about whether mathematics is applicable: the entire edifice of modern science and technology, depending heavily as it does on the mathematisation of nature, bears witness to this fact.

So what can a formalist say to explain the applicability of mathematics? If mathematics really is nothing other than the shuffling of mathematical symbols in the world's longest running and most multiplayer game, then why should it describe the world? What privileges the game of maths to describe the world rather than any other game? Remember, the formalist must answer from within the formalist worldview, so no Plato-like appeals to a deeper meaning of maths or hidden connection to the physical world is allowed. For similar reasons, the logicists are left floundering, for if they say "well, perhaps the universe is an embodiment of logic", then they are tacitly assuming the existence of a Platonic realm of logic which can be embodied. This turns logicism into a mere branch of platonism, which, as we will shall see below, comes with its own grave problems. Thus for both formalists and non-platonist logicists the very existence of applicable mathematics poses a problem apparently fatal to their position.

Neither logicism nor formalism is widely believed any more, despite the cliché that mathematicians are platonists during the week and formalists at the weekend. Both perspectives fell out of favour for reasons other than the potentially fatal one about the applicability of mathematics, reasons largely connected with the work of Gödel, Thoralf Skolem, and others. (See Gödel and the limits of logic.)

Is the world inherently mathematical or is maths a construct of the human mind?

The third proposed foundation, intuitionism, never really garnered much support in the first place. To this day, it is muttered about in dark tones by most working mathematicians, if it is considered at all. What is seen as a highly restricted toolkit for proofs and a bizarre notion of limbo, in which a statement is neither true nor false until a proof has been constructed one way or the other, make this viewpoint unattractive to many mathematicians.

However, the central idea of the enumerable nature of processes in the universe appears to be deduced from reality. The physical world, at least as we humans perceive it, seems to consist of countable things and any infinity we might encounter is a result of extending a counting process. In this way, perhaps intuitionism is derived from reality, from the apparently at-most-countably infinite physical world. It appears that intuitionism offers a neat answer to the question of the applicability of mathematics: it is applicable because it is derived from the world.

However, this answer may fall apart on closer inspection. For a start, there is much in modern mathematical physics, including for example quantum theory, which requires notions of infinity beyond the enumerable. These aspect may therefore lie forever beyond the explicatory power of intuitionistic mathematics.

There is one modern idea which could benefit from the finitist logic of the intuitionists: so-called digital physics. It holds that the Universe is akin to a giant computer. The fundamental particles, for example, are described by the quantum state they happen to be in at a given moment, just as the bit from computer science is defined by its value of 0 or 1. Just like a computer, the Universe is based on information about states and its evolution could in theory be simulated by a giant computer. Hence the digital physics motto, "It from bit".

But this world view too fails to be truly intuitionistic and seems to sneak in some platonic ideas. The bit of information theory seemingly posits a platonic existence of information from which the physical world is derived.

But more fundamentally, intuitionism has no answer to the question of why non-intuitionistic mathematics is applicable. It may well be that a non-intuitionistic mathematical theorem is only applicable to the natural world when an intuitionistic proof of the same theorem also exists, but this has not been established. Moreover, although intuitionistic maths may seem as if it is derived from the real world, it is not clear that the objects of the human mind need faithfully represent the objects of the physical Universe. Mental representations have been selected for over evolutionary time, not for their fidelity, but for the advantage they gave our forebears in their struggles to survive and to mate.

Created in the image of mathematics

Formalism and logicism have failed to answer our big question. The jury is out on whether inuitionism might do so, but huge conceptual difficulties remain. What, then, of Platonism?

Galileo Galilei, who believed that the world was written in the language of maths, facing the Roman Inquisition for proclaiming that the Earth moved around the Sun. Painting by Cristiano Banti.

Platonists believe that the physical world is an imperfect shadow of a realm of mathematical objects (and possibly of notions like truth and beauty as well). The physical world emerges, somehow, from this platonic realm, is rooted in it, and therefore objects and relationships between objects in the world shadow those in the platonic realm. The fact that the world is described by mathematics then ceases to be a mystery as it has become an axiom: the world is rooted in a mathematical realm.

But even greater problems then arise: why should the physical realm emerge from and be rooted in the platonic realm? Why should the mental realm emerge from the physical? Why should the mental realm have any direct connection with the platonic? And in what way do any of these questions differ from those surrounding ancient myths of the emergence of the world from the slain bodies of gods or titans, the Buddha-nature of all natural objects, or the Abrahamic notion that we are "created in the image of God"?

Indeed, the belief that we live in a divine Universe and partake in a study of the divine mind by studying mathematics and science has arguably been the longest-running motivation for rational thought, from Pythagoras, through Newton, to many scientists today. "God", in this sense, seems to be neither an object in the space-time world, nor the sum total of objects in that physical world, nor yet an element in the platonic world. Rather, god is something closer to the entirety of the platonic realm. In this way, many of the difficulties outlined above which a platonist faces are identical with those faced by theologians of the Judeo-Christian world — and possibly of other religious or quasi-religious systems.
The secular icon Galileo believed that the "book of the universe" was written in the "language" of mathematics — a platonic statement begging an answer (if not the question) if ever there was one. Even non-religious mathematical scientists today regularly report feelings of awe and wonder at their explorations of what feels like a platonic realm — they don't invent their mathematics, they discover it. Paul Davies goes further in The Mind of God, and highlights the two-way nature of this motivation. Not only may a mathematician be driven to understand mathematics in a bid to glimpse the mind of God (a non-personal God like that of Spinoza or Einstein), but our very ability to access this "key to the universe" suggests some purpose or meaning to our existence.

In fact, the hypothesis that the mathematical structure and physical nature of the universe and our mental access to study both is somehow a part of the mind, being, and body of a "god" is a considerably tidier answer to the questions of the foundation of mathematics and its applicability than those described above. Such a hypothesis, though rarely called such, has been found in a wide variety of religious, cultural, and scientific systems of the past several millenia. It is not natural, however, for a philosopher or scientist to wholeheartedly embrace such a view (even if they may wish to) since it tends to encourage the preservation of mystery rather than the drawing back of the obscuring veil.

Penrose's three-worlds diagram.

Roger Penrose has most lucidly illustrated some of this mystery with a three-worlds diagram. The platonic, physical, and mental worlds are the three in question, and he sketches them as spheres arranged in a triangle. A cone then connects the platonic world with the physical: in its most general form, the diagram shows the narrow end of the cone penetrating the platonic world and the wider part penetrating some of the physical world. This is to show that (at least some of) the physical world is embedded in at least some of the platonic world. A similar cone connects the physical to the mental world: (at least some of) the mental world is embedded in the physical world. Finally, and most mysteriously, the triangle is completed by a cone from the mental to the platonic world: (at least some of) the platonic world is embedded in the mental world. Each cone, each world, remains a mystery.

We seem to have reached the rather depressing impasse in which none of the four proposed foundations of mathematics can cope unambiguously with the question of the applicability of mathematics. But I want you to finish this essay instead with the feeling that this is incredibly good news! The teasing out of the nuances of the big question — why does applied mathematics exist? — is a future project which could yet yield deep insight into the nature of mathematics, the physical universe, and our place within both systems as embodied, meaning-making, pattern-finding systems.



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