Mathematics can be utilized to explore various possible characteristics of a theory and make “predictions” of results that would occur if those characteristics obtained in the real world. In other words, the theories we construct about relationships in the world might lend themselves to mathematical representation. If so, we can use the power and versatility of mathematics to investigate those relationships and to make predictions about events, as well as to gain insights into the relationships.
Wigner’s question again
But, is there something more? Can we actually answer Wigner’s question: why is it that we find that many branches of mathematics have real world applications?
One might approach the matter with a more limited question. Why is it the case that one branch of mathematics so often can be applied to or is useful to another branch of mathematics? See Hacking, Why Is There Philosophy of Mathematics at All?, pp.3, 8–11. Geometry has been applied to algebra, and vice versa. Sometimes a form of proof in one branch can be applied to create a proof in another branch. The possible answers to this question include the assertion that (i) there exists a mathematics that is true, parts of which are reflected in various branches;(ii) it is mere coincidence; or (iii) the relationships are really more of analogies between the branches than applications of one to the other. Id., pp.11–4, 16–21.10
A similarly curious phenomenon occurs where it is “discovered” that a branch of mathematics just happens to seem to reflect other relationships that exist in the physical world. An interesting example is that of Boolean algebra,11 which happens to capture the dynamics of two-phase or on/off circuitry. It happens that one can specify a result that one wishes to achieve through a series of switches, perform computations with Boolean algebra to design a simplified circuitry and find that the new design performs the desired function.
Why? Or, perhaps the question is, how?
The point for here is that this example and many others like it suggest that man has not always “forced” his own constructs on to nature (or nature into his own constructs) but, at least sometimes, has "found" an apparent good fit. Interestingly, Boole created the algebra in 1854, but it was not until 1937 that a young graduate student named Claude Shannon wrote a master’s thesis in which he combined Boolean logic with electrical engineering, demonstrating that the logic could be implemented electronically. “Thus is borne the electronic ‘logic gate’—and soon enough, the [computer] processor.” Brian Christian, The Most Human Human (2011), pp.49–9.
Whatever the answer, it is clear that mathematics has led to scientific discoveries and scientific insights have led to new mathematics. There are numerous examples of what seems to be an interactive relationship between mathematics and the physical sciences. Shing-Tung Yau has described the human process leading to his proof of the Calabi conjecture and the subsequent developments in string theory that flowed from or followed that proof. See Shing-Tung Yau and Steve Nadis, The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions (2010).
Professor Barrow describes instances where scientific exploration found existing mathematics available to fill the scientists’ needs and examples where the work of scientists identified areas of new mathematics that were needed for the science. These needs in some cases were filled by mathematicians and in some cases are still awaiting solutions. Theories of Everything, pp.188–193. For example, when Max Born worked to explore the apparently new mathematical system that Heisenberg had utilized for the multiplication of two lists of quantities, representing frequencies and amplitudes in creating a system of representing the behavior of electrons in an atom, he discovered that a whole branch of mathematics had already been developed that would serve for this new quantum mechanics—matrix algebra. David Lindley, Uncertainty (2007), pp.123, 113–4.
At the same time, it is also true that much of mathematics has no scientific application, at least not yet. Wigner, “The Unreasonable Effectiveness,” p.7.
Are mathematical propositions true?
The question then arises: Are the propositions derived by mathematics necessarily true as a factual matter (assuming that the original axioms or propositions were factually accurate)? In other words, does the world conform to the logic of mathematics? This is the point at which I want to suggest a distinction (discussed further below) between mathematics derived (or derivable) from observations of the physical world and mathematics that has been constructed based upon concepts that are not part of our physical experiences (infinity, complex numbers, etc.).
There is the possibility that man has been able to extract causal relationships from observable experience and incorporate the logic of those relationships in the resulting mathematical system. To the extent that the observations have indeed captured the “logic” of the underlying physical relationships, then the mathematics should enable us to extrapolate to phenomena yet to be observed and to make predictions with accuracy. The notion here is not that the constructed theory necessarily corresponds to reality in fact, but that the structure of the relationships corresponds to what does happen—i.e., our theory is incorrect as an explanation but the relationships hypothesized in the theory do occur in fact.
In contrast, if the mathematics are structured around concepts of purely of human invention, then it would seem that we might find some coincidences of correspondence to the physical world and many examples of mathematics as simply intellectual exercises.
Problems of Infinity
Charles Seife has written: “Zero…is infinity’s twin. …The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity.” Zero, p.2. I think that the statement is a serious oversimplification, and a misleading one at that.12
But, it is correct to suggest that these concepts give rise to problems in scientific applications. I think it is useful to explore why. As we shall see, the situations are quite different with respect to zero and to infinity. And, the differences are important.
The construct of zero
Mathematics certainly had problems with zero,13 but at least zero was a concept that could be related to the actual experience of counting (sort of, anyway). Contrary to some popular perceptions, one does not need zero for purposes of counting, i.e., the absence of any As (“no As”) does not have to be characterized as “zero As” in the sense that the presence of A and A is clearly “2 As.”
On the other hand, the use of zero as a "place holder" in larger numbers does make intuitive sense. For example, a number 100 represents one hundreds, zero tens and zero ones. It is clearly a convenient – indeed, a highly useful – convention. In fact,the first use of zero, Seife says, was not as a number but as a place holder to distinguish 1 from 10 or 100. Zero.,pp.26–53. See also, Deutsch, The Beginning of Infinity, p.131. The use of zero as another number is attributed to mathematicians in India. Seife, Zero, pp.66–71; Deutsch, The Beginning of Infinity, p.131.14
Zero may not have direct obvious application in daily life, but it is not in concept really different from the other cardinal numbers. See Whitehead, An Introduction to Mathematics, p.63. While there may not have been many pressing applications for the addition or subtraction of zero from a positive quantity, the concept does not seem foreign to our physical experiences. Even the notion of zero times a positive number (say, zero groups of five apples) has an intuitive sense.
The mathematical solution for zero has been effective in application to the physical world, with one important exception—division. The concept of division by zero seems of no use in our daily activities. As mathematics developed, however, it was deemed that a positive number divided