The indispensability of mathematics
If mathematics is indispensably part of our best scientific theories, does the scientific confirmation of those theories provide a reason to accept the existence claims of mathematics?
Please enjoy this free extended excerpt from Lectures on the Philosophy of Mathematics, published with MIT Press 2021, an introduction to the philosophy of mathematics with an approach often grounded in mathematics and motivated organically by mathematical inquiry and practice. This book was used as the basis of my lecture series on the philosophy of mathematics at Oxford University.
Indispensability of mathematics
What philosophical conclusion can we make from the fact that mathematical tools and vocabulary seem to lie at the very core of nearly every contemporary scientific theory? How remarkable that at every physical scale, from the microscopic to the cosmic, our best scientific theories are thoroughly mathematical. Why should this be? The laws of Newtonian physics are expressed in universal differential equations that explain the interaction of forces and motion, unifying our understanding of diverse physical phenomena, from the harmonic oscillations of a mass on a spring to the planetary motions of the heavenly bodies; our best theory of electromagnetism posits unseen electrical and magnetic fields that surround us all, enveloping the Earth; relativity theory explains the nature of space and time with exotic mathematical geometries; quantum mechanics uses Hilbert spaces; string theory uses still more abstract mathematical objects; and all the experimental sciences, including the social sciences, make fundamental use of mathematical statistics. Physicist Paul Dirac (1963) describes the situation like this:
It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better.
(Footnote: One should not misunderstand Dirac's views on God, however, for he also said, “If we are honest—and scientists have to be—we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination.” Dirac, quoted by Werner Heisenberg (1971).)
Thus, mathematics appears to be indispensable for physics and other sciences.
On the basis of this, Hilary Putnam and Willard Van Orman Quine mount the indispensability argument for mathematical realism, arguing that we ought to have an ontological commitment to the objects that are indispensably part of our best scientific theories (see Putnam (1971) for a classic presentation). Just as a scientist finds grounds for the existence of unseen microscopic organisms on the basis of the well supported germ theory of disease, even in the absence of direct observations of those organisms, and just as a scientist might commit to the atomic theory of matter or the existence of electrons or a molten iron core in the Earth, or black holes, or wave functions, even when the evidence for them in our well supported theories is indirect, then similarly, according to the indispensability argument, we should find grounds for the existence of the abstract mathematical objects that appear in our best theories. Quine emphasizes a view of confirmational wholism, by which theories are confirmed only as a whole. If mathematical claims are an indispensable part of the theory, then they are part of what is confirmed.
Science without numbers
Attacking the indispensability argument at its heart, Hartry H. Field (1980) argues that the truth of mathematics is not actually indispensable for science. Defending a nominalist approach to mathematics, he argues that we do not require the actual existence of these abstract mathematical objects in order to undertake a successful scientific analysis.
Rather, Field points out that there is a kind of logical error underlying the indispensability argument. Namely, even if the mathematical theories are indispensable to the scientific analysis, this is not a reason to suppose that the mathematical theories are actually true. It would be sufficient, for example, if the mathematical claims formed merely a conservative extension} of the scientific theory. Specifically, as mentioned in chapter 1, a theory S is conservative over another theory T that it extends with respect to a class of assertions ℒ if, whenever S proves an ℒ assertion, this assertion is already provable in T. In other words, the stronger theory S tells us no new ℒ facts that we could not already know on the basis of T. This does not mean that the theory S is useless or unhelpful, however, for perhaps S unifies our knowledge somehow or is more explanatory or makes reasoning easier, even if ultimately, no new ℒ assertions will be proved in S.
In the case of the indispensability argument, our scientific theory S describes the nature of the physical world, but this theory includes mathematical claims making existence assertions about various mathematical objects. Let N be the nominalist fragment of the scientific theory, omitting the mathematical claims. If the full theory S were conservative over N concerning assertions about the physical world, then we could safely use the full theory S to make deductions about the physical world, whether or not the mathematical claims it makes are true. In the mathematically augmented theory, the scientist may safely reason as if the mathematical part of the theory were true, without needing to commit to the truth of those additional mathematical assertions.
Instances of this same pattern have arisen entirely in mathematics. Consider the early use of the complex numbers in mathematics, before the nature of imaginary numbers was well understood. In those early days, skeptical mathematicians would sometimes use the so-called imaginary numbers, using expressions like 1 + √(-5), even when they looked upon these expressions as meaningless, because in the end, the imaginary parts of their calculations would sometimes cancel out and they would find the desired real number solution to their equation. It must have been mystifying to see calculations proceeding through the land of nonsense, manipulating those imaginary numbers with ordinary algebra, and yet somehow working out in the end to a real number solution that could be verified independently of the complex numbers. My point here is that even if a mathematician did not commit to the actual existence of the complex numbers, the theory of complex numbers was conservative over their theory of real numbers, so far as assertions about the real numbers are concerned. So even a skeptical mathematician could safely reason as if imaginary numbers actually existed.
Field is arguing similarly for applications of mathematics in general. Ultimately, it does not matter, according to Field, whether the mathematical claims made as part of a scientific theory are actually true or false, provided that the theory is conservative over the nonmathematical part of the theory, so far as physical assertions are concerned, for in this case, the scientist can safely reason as if the mathematical claims were true.
Impressively, Field attempts to show how one can cast various scientific theories without any reference to mathematical objects, replacing the usual theories with nominalized versions, which lack a commitment to the existence of mathematical objects. He provides a nominalist account of Newtonian spacetime and of the Newtonian gravitational theory. One basic idea, for example, is to use physical arrangements as stand-ins for mathematical quantities. One may represent an arbitrary real number, for example, by the possible separations of two particles in space, and then refer to that number in effect by referring to the possible locations of those particles. Thus, one avoids the need for abstract objects.
Critics of Field point out that although he has strived to eliminate numbers and other abstract mathematical objects, nevertheless his ontology is rich, filled with spatiotemporal regions and other objects that can be seen as abstract. Shall we take physics to be committed to these abstract objects? Also, the nominalized theory is cumbersome and therefore less useful for explanation and insight in physics—isn't this relevant for indispensability?
Fictionalism
Fictionalism is the position in the philosophy of mathematics, according to which mathematical existence assertions are not literally true, but rather are a convenient fiction, useful for a purpose, such as the applications of mathematics in science. According to fictionalism, statements in mathematics are similar in status to statements about fictional events. An arithmetic assertion p, for example, can be interpreted as the statement, “According to the theory of arithmetic, p.” It is just as one might say, “According to the story by Beatrix (1906), Jeremy Fisher enjoys punting.” In his nominalization program for science, Field is essentially defending a fictionalist account of mathematics in science. Even if the mathematical claims are not literally true, the scientist can reason as if they were true.
I find it interesting to notice how the fictionalist position might seem to lead one to nonclassical logic in mathematics. Let us suppose that in the story of Jeremy Fisher, the cost of his punt is not discussed; and now consider the statement, “Jeremy Fisher paid more than two shillings for his punt.” It would be wrong to say, “According to the story by Beatrix Potter, Jeremy Fisher paid more than two shillings for his punt.” But it would also be wrong to say, “According to the story by Beatrix Potter, Jeremy Fisher did not pay more than two shillings for his punt.” The story simply has nothing to say on the matter. So the story asserts neither p nor ¬p. Is this a violation of the law of excluded middle?
As I see it, no, this is not what it means to deny the law of excluded middle. While one asserts neither p nor ¬p, still, one asserts p ∨ ¬p, since according to the story by Beatrix Potter, we may reasonably suppose that either Jeremy Fisher did pay more than two shillings for his punt or he did not, since Jeremy Fisher's world is presented as obeying such ordinary logic. Another way to see this point is to consider an incomplete theory T in classical logic. Since the theory is incomplete, there is a statement p not settled by the theory, and so in the theory, we do not assert p and we do not assert ¬p, and yet we do assert p ∨ ¬p, and we have not denied the law of excluded middle. Fictional accounts are essentially like incomplete theories, which do not require one to abandon classical logic.
The theory/metatheory distinction
In a robust sense, fictionalism is a retreat from the object theory into the metatheory. Let us make the theory/metatheory distinction. In mathematics, the object theory is the theory describing the mathematical subject matter that the theory is about. The metatheory, in contrast, places the object theory itself under mathematical analysis and looks into meta-theoretic issues concerning it such as provability and consistency. In the object theory—take ZFC set theory, for instance—one asserts that there are sets of all kinds, including well-orders of the real numbers and diverse uncountable sets of vast cardinality, while in the metatheory, typically with only comparatively weak arithmetic resources suitable for managing the theory, one might make none of those existence claims outright, and instead one asserts like the fictionalist merely that “According to the theory ZFC, there are well-orders of the real numbers and uncountable sets of such-and-such vast cardinality.” In this way, fictionalism amounts exactly to the metamathematical move. For this reason, there are affinities between fictionalism and formalism (see chapter 7), for the formalist also retreats into the metatheory, taking himself or herself not to have asserted the existence of infinite sets, for example, but asserting instead merely that, according to the theory at hand, there are infinite sets.
Continue reading more about this topic in the book:
Lectures on the Philosophy of Mathematics, MIT Press 2021
There is an aspect of the indispensability question that has always struck me as conspicuously absent from most discourse. Discussion of indispensability often proceeds as if the complete body of mathematical and scientific knowledge of the time were handed down from on high in some great volume. Had this been so, it would certainly seem baffling! But the actual history is nothing like this, mathematical and scientific understanding developed over millennia. To boot, much of contemporary mathematics would probably not be recognizable as mathematics at all by the ancients. It seems to me that the development of scientific theories is one of increasingly honing and making precise one's notions. It took great effort (by Galileo, Descartes, Newton, among others) to clarify the once poetic notions of motion, force, change, etc. The tools of calculus were no doubt conceived with the basic ideas of Newtonian mechanics in mind.
To my way of thinking, this kind of ever increasing "precisification" of one's concepts is necessary to the scientific enterprise. If the goal is understanding the physical world via some kind of explanatory theory (not a priori a mathematical one necessarily), then we are (most strongly) persuaded by one theory or another on the basis of its verifiably correct predictions. But the world is subtle and complicated [citation needed] and to make unambiguous verifiable predictions requires unambiguous conceptions in the first place. It seems to me that through this honing process by which one's concepts become more precise and rigorous, they *take on the character of mathematics*. This is one way that a subject or idea becomes a mathematical one (even if it was perhaps not so already). This phenomenon is not limited to physics. For ages, the notion of infinity was viewed as purely philosophical or poetic (not meant pejoratively). But in making various conceptions of the infinite precise (limits, infinite cardinalities, the ordinal numbers, etc.), they *became* mathematical notions which have in turn vastly sharpened the philosophical ideas.
Perhaps some combination of abstraction and rigour (and nothing else) is the defining character of mathematics. As I have suggested above, I believe that rigour is ultimately necessary for scientific reasoning and understanding. The role of abstraction is murkier maybe but, from experience, abstraction is often a key part of successful *explanations* (an enormous list of ultimately correct specific predictions would hardly count as an "explanation" of the physical phenomena). On this way of thinking, the mathematical flavour, on the one hand, of our most successful physical theories is as inevitable as their physical character, on the other; rigour and abstraction are part of the answer to the question "What is mathematics?" in the first place (maybe) and they are desirable features of our scientific theories.
I am certainly no expert and I would welcome scholarly references that engage with this line of thought. How is it that new mathematics emerges (or how do we come to understand what's already out there in the Platonic realm, if you prefer) and *might the answer to this question also hold the key to the role of mathematics in our scientific theories*?