Scientific Proof of Mathematical Truths?
Dear Dr. Craig,
The existence of mathematical laws is included in your list of reasonable yet scientifically unprovable assumptions that we tend to make about reality. But aren't we proving these laws scientifically every time we use them to successfully calculate physical quantities?
Although proof of the mathematical laws would not tell us why they are here, where they came from, or why they nature adheres to them, we might say that mathematical laws are an accurate conceptualization of the way the universe behaves, or equally that the universe embodies mathematical laws.
Your question caught my attention, Colin, because the claim that mathematical truths are confirmed by the evidence for our best scientific theories plays a crucial role in the most widely discussed argument for the existence of abstract mathematical entities (like sets, for example), namely, the so-called Indispensability Argument of Harvard’s late W. V. O. Quine, with which I’ve been lately preoccupied.
To set the stage for Colin’s question: people like Peter Atkins, Jerry Coyne, and a number of other scientistic, new atheist types have endorsed, explicitly or implicitly, a criterion of rationality which states that we should believe only that which can be scientifically proven. To believe any statement without scientific evidence for that statement is irrational. In response to this claim, two objections are typically raised: first, the criterion is too restrictive; and second, the criterion is self-defeating. In support of the first objection, it’s easy to give examples of truths which we all accept and are perfectly rational in so doing but which cannot be proven scientifically. One example would be truths of mathematics. Since these are presupposed by science, science cannot prove these truths without reasoning in a circle.
Quine, however, held to a view called Confirmational Holism. This is the view that empirical confirmation of the truth of our best scientific theories extends to every single statement of those theories. In Quine’s view the statements of scientific theories are not subject to confirmation (or disconfirmation) when they are taken in isolation but rather only as parts of whole theories. It is the theory as a whole which is subject to testing, and its component statements enjoy confirmation or suffer disconfirmation insofar as they share in the confirmation or disconfirmation of the whole. One can test individual statements only by deciding to hold fast the other statements of a theory. Since mathematical statements are an ineradicable part of science, it follows that they, like purely empirical statements, share in the confirmation enjoyed by the theory of which they are a part. Thus, mathematical statements are empirically confirmed by the evidence supporting a theory. This seems to be exactly what you’re suggesting, Colin.
Unfortunately, Quine’s Confirmational Holism is a highly implausible and therefore widely rejected doctrine. Elliott Sober has convincingly exposed its weaknesses, charging that “The confirmation relation that holism invokes is bizarre.”1 Sober importantly distinguishes distributive holism from non-distributive holism. Quine endorses distributive holism, according to which it is not merely a theory as a whole which enjoys confirmation or suffers disconfirmation, but its individual statements as parts of the whole: in virtue of the confirmation of the theory as a whole, each of its several statements is confirmed.
Distributive holism is a strange doctrine, since confirmation does not seem to be distributive in the way the doctrine envisions. How is it that the confirmation which a theory as a whole enjoys gets distributed to every one its several parts? Sober reminds us that it is fallacious to infer that because an observation O confirms a hypothesis H and H entails some statement S, therefore O confirms S. (Let O = the playing card is red; H = the card is the 7 of hearts; and S = the card is a 7.) Sober thinks that this fallacious inference underlies distributive Confirmational Holism, for apart from it all one has is a non-distributive holism, according to which the confirmation or disconfirmation of a whole theory is not distributed to its component parts, and hence, to its mathematical statements.
Furthermore, a property of simple examples of confirmation is symmetry: observation O confirms hypothesis H just in case not-O would disconfirm H. Yet the statements of pure mathematics never suffer disconfirmation from different observational outcomes of theory testing. The same calculus that is used in relativity theory, for example, was used in Newtonian theory and did not share in the disconfirmation of the latter. Since pure mathematical statements do not suffer disconfirmation but are common to all theories, neither can they be confirmed by observational evidence.
Sober emphasizes that to reject holism is not to adopt the positivistic alternative of testing isolated hypotheses. Confirmation/disconfirmation relations are really three-place relations: a hypothesis H is confirmed by an observation O relative to background assumptions A. The shared background assumptions of competing hypotheses are not tested by the observations and therefore are not confirmed/disconfirmed along with H. Now the mathematical statements of science, precisely by being assumed by every scientific theory, belong to the background assumptions of those theories. The empirical confirmation of those theories therefore does not extend to mathematical statements. It follows, then, that the statements of pure mathematics which underlie scientific theories are not tested when these theories are tested and so do not enjoy confirmation as a result of the theory’s confirmation.
You’re quite right that we may still ask why nature adheres to or embodies mathematics. On this score, I think that the theist enjoys a considerable advantage over the naturalist, whether one be a realist about mathematical objects or an anti-realist. As Mary Leng points out in her recent Mathematics and Reality, for the non-theistic realist, the fact that physical reality behaves in line with the dictates of non-causal mathematical entities is just “a happy coincidence.” 2 But the theistic realist can argue that God has fashioned the world on the structure of the mathematical objects. The anti-realist might say that mathematical principles “are an accurate conceptualization of the way the universe behaves,” so that there is no happy coincidence. Well and good, but what remains wanting on atheistic anti-realism is an explanation why the physical world exhibits so complex and stunning a mathematical structure in the first place. The theistic anti-realist, on the other hand, can maintain that God has constructed the world on the fictional blueprint conceived by Him.
1 Elliott Sober, “Quine I: Quine’s Two Dogmas,” Proceedings of the Aristotelian Society Supplementary Volume 74 (2000): 264; cf. idem, “Mathematics and Indispensability,” Philosophical Review 102 (1993): 35-57; idem, “Evolution without Naturalism,” Oxford Studies in Philosophy of Religion 3 (forthcoming).
2 Mary Leng, Mathematics and Reality (Oxford: Oxford University Press, 2010), p.239.