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The Uses and Usefulness of Mathematics

The impressive success of modern technology is a result of the inexplicable fact that the physical world can be simulated by computational models and that greater and greater accuracy has been achieved through more and more sophisticated mathematics. See Penrose, Shadows of the Mind, p.203. It is the “fruitfulness” of mathematics that is, perhaps, the most surprising fact. Mathematics has been used not only to capture the patterns of observed phenomena, but as a guide to the discovery of new theories and models that have then led to the discovery of empirical patterns and relationships that had not previously been noticed. See id., pp.416–7.

This capability may seem particularly surprising if one accepts what has been said so far about mathematics being just one or more deductive system in which the implications of the initial assumptions are all necessarily contained in those assumptions. In that sense, all mathematics is circular. There can be nothing new derived by manipulating the logical system, only different ways of expressing the same thing. So, how can that be such a powerful tool?

Where a set of relationships in the observed world has been defined (or hypothesized) logically as a deductive theory, it may be the case that the logical rules of the theory can be represented mathematically, using existing mathematics or even creating mathematics to fit the situation. The mathematical representation of the theory often provides very significant pragmatic benefits. Mathematics is based upon a system of notation using symbols. The use of symbols in place of words and phrases forces precision and clarity, as well as providing great economy and simplicity in the formulation of propositions or assertions. The simplicity itself is important. As Whitehead observed, “[b]y relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power.” An Introduction to Mathematics, p.59.

Furthermore, the mathematical rules allow extensive manipulation by hand and, even for calculations, by machine (including the computer). Moreover, the process of breaking logical relationships down into many small steps makes it easier to avoid errors in constructing predictions and theories or to check for errors after the fact. These are very significant practical benefits attainable by setting forth theories in mathematical form.

However, it also appears that the use of the mathematics may even enable the development of concepts that have useful analogs for or provide surprising insights consistent with the non-mathematical theory.

Let us be more specific. Physicists Cox and Forshaw assert that “[e]quations are the most powerful tools available to physicists in their quest to understand nature.” Why Does E=mc²?, p.21. An equation sets out on each side of the “equals sign” two formulations that, when reduced to numerical values, will be equal in quantity. Sometimes the equality is “by definition,” in which case the equation can be considered to be a definition of one of the terms. In other cases, the equation expresses an empirical relationship.

It may be hard to tell the two apart in some cases. For example:

Distance (traveled) = Time in transit × Speed, or D = T × S,

where S is the speed of movement. Is that an empirical statement or a definition?

Either way, one can manipulate the formula and derive other relationships that are “necessary,” that is, compelled by or implicit in the original definition. So,

Speed = Distance divided by Time, and

Time (traveled) = Distance divided by Speed. It is apparent that these simple manipulations of the formula generate new statements of the relationship defined that can be useful.

Again, one might pause to ask whether these equations are “factual”—do they make contingent statements about the physical world? You might immediately say “of course.” If one travels for T minutes at S speed, one will in fact travel D distance. Agreed. But, is that a factual statement? Or, is it merely a definition of D (and of T and S) and, as a definition, is true because it is defined so, not because of inherent characteristics of the physical world?

Moving onward, mathematical logic tells us that if you do the same thing to both sides of an equation, then the two sides will still be equal. So, you can add the same value to both sides or subtract the same value, or multiple or divide by the same value, and the equality will still hold. Why is this proposition “true”? Because it is inherent in the definition of “equals.” But, that fact does not make the proposition vacuous.

Indeed, surprisingly, there are many examples where this process of performing the same operation on both sides of an equation, followed by manipulations of the resulting equation according to the established rules, results in a statement (in the form of something “=” something else) that seems to provide new, unexpected information about the physical world. We can generate propositions that were not intuitively obvious before the mathematics was performed, even some that are not intuitively comfortable when we do see them. We will come later to some examples. For now, suffice it to say that this application of simple mathematical methods appears to yield many quite striking results.

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.