Theoretical physicist Lee Smolin expresses some frustration with the emphasis placed upon mathematics in the development of scientific theories. He asserts that it is absurd to think that “mathematics is prior to nature. Math in reality comes after nature. It has no generative power. …[I]n mathematics conclusions are forced by logical implication, whereas in nature events are generated by causal processes operating in time. …[L]ogical implications can model aspects of causal processes, but they’re not identical to causal processes. …Logic and mathematics capture aspects of nature, but never the whole of nature.” Time Reborn, p.246. Smolin does not manage actually to address the question raised here of why it is that mathematics seems to be able to “capture aspects of nature” and whether the aspects captured by the mathematics or the aspects that are left out are the things that really matter. Nonetheless, as we shall see in a later chapter, there are a host of examples where I conclude that mathematics has in a sense taken the science hostage and led us in directions that, at least, merit re-examination.
The curiosity is not that certain specific observed relationships can be depicted by a mathematical formula, but that a few of such formulae corresponding to observed physical relationships can be incorporated into a mathematical system that will enable the user to compute other relationships that can then be observed in fact. In other words, whole complex structures in mathematics can have their analog in the physical world.
One could assert that some regularity and causal relationships are necessary features of the physical world, otherwise we would not find a structured Universe. But, that observation does not explain very much. It simply says that if it were not so, this world would not be. Maybe that is all that we can say, but it is not very satisfying nor, one suspects, the whole story.
Thus, we are left with no real answer to the question posed of why mathematics is so useful or to the derivative question of what are the limits, if any, to the use of mathematics to describe the physical world. It seems that we are just left with the fact that relative to much of the physical world, mathematics seems to work. But, to be consistent with what we have said about induction, past success cannot be taken as a guarantee of future success. Moreover, even if mathematics reflects many important relationships in the physical world, it may not capture the relationships of many other aspects of our “natural world” that we seek to understand. There may be very significant areas that may come someday to be called science but that are not capable of representation by equations or other tools of mathematics. Nonetheless, it is easy to note that there may be, or even probably are, limitations to knowledge based upon mathematics. What is hard is to figure out what those limitations are and what could supplement or replace our knowledge based upon mathematics.
Kline concludes his book with these observations: “One should also question the extent to which mathematics really represents the physical world. …It treats those physical concepts which can be represented by numbers or geometrical figures. But physical objects possess other properties as well. …[The use of mathematics] may cause us to look at the world with blinders. …It may be that man has introduced some limited and even artificial concepts and only in this way has managed to institute some order in nature. …[But,] [i]n those domains where it is effective it is all we have; if it is not reality itself, it is the closest to reality we can get.” Mathematics, pp.554, 555.
Concluding comments
Apart from the possibility that there may be many aspects of our world that do not follow the “logic of mathematics,” there are two other cautions I would suggest:
First, the fact that mathematics does not provide actual causal explanations clearly indicates that there may be a more fundamental theory even of the laws of motion and matter.
Second, where science becomes largely mathematics, with only few and tenuous contacts with observable phenomena, there is the risk that it is only an intellectual game that can create an artificial and erroneous belief that we know something when, in fact, we do not.
Thus, it would seem that we need to remain cautious and be ready to test empirically the applicability and accuracy of mathematical predictions continuously as advances in, extensions to or replacements of theories based upon mathematics are proposed. In other words, the assertion that “this is what the mathematics establishes” should be met with some skepticism: the assertion should be viewed as interesting, maybe insightful, but not dispositive.
Endnotes
1 Interestingly, however, the discovery of the Pythagorean Theorem presented the problem of “incommensurables,” numbers that cannot be written as the ratio of two whole numbers (like 1/2, 1/3, 2/5). The diagonal of a triangle with two other sides of the length of 1 would be the square root of two. Such a square root cannot be written as such a ratio or as a finite series of digits. See Joseph Mazur, The Motion Paradox, p.18. Geometry happens to present such incommensurables with some frequency, such as the very important number Pi. These incommensurable numbers, by the way, are now referred to as irrational numbers.
2 The historical study of the fossil record is clearly interesting. The computer simulation is less impressive. Where no advantage is granted for larger size, random variations appear not to lead to size increases; where there are advantages, the advantages tend to affect the evolution of the species. One wonders that computer simulations were necessary to generate that result. And, of course, the observed trends in real life could still be the results of totally random processes.
3 “If the amazing three volumes of Whitehead and Russell’s Principia Mathematica … (1910–13) had wholly succeeded, our seemingly naïve question would have a direct answer. Something is mathematics if it is logic!” Hacking, Why Is There A Philosophy of Mathematics At All?, p.54.
4 Roger Penrose does elsewhere write extensively about the relationships between mathematics and the human mind. E.g., Shadows of the Mind: A Search for the Missing Science of Human Consciousness (1994); The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics (1989). Some of his insights concerning human consciousness are discussed in the later chapter on The Mind. However, he continues to express the view that “major revolutions are required in our physical understanding … [before] much real progress can be made in understanding the actual nature of mental processes.” The Road to Reality, p.21.
5 In an earlier work, Penrose identified what he considered to be three deep mysteries that relate one to another the world of our consciousness, the physical world and the world of mathematical forms. Shadows of the Mind, pp.412–4. The first mystery is why mathematical laws play such significant roles in the behavior of the physical world. The second is how the physical world can give rise to entities that are conscious and able to perceive. The third is that that mental capability is able to develop the mathematical models that seem so accurately to describe the physical world.
6 Not to suggest that there have not been more detailed explorations of the question. See, e.g., Colyvan, An Introduction to the Philosophy of Mathematics, pp.99–117. One suggestion is that the question focuses on the notable successes and fails to note the instances of failure in the application of mathematics. Similarly, it is possible that “we tackle only the physical problems that are amenable to the mathematical methods we have at our disposal.” Id., p.105. These are, at best, only partial answers.
7 This fact may seem particularly strange given that the two branches of mathematics developed historically in different geographical regions and different cultures (geometry from the Greeks and algebra from India through Persia and Islam). Id., p.7.
8 One possibility is that we are living in an elaborate computer simulation constructed using the mathematics that we have been in the process