The human mind and the physical world
In the end, however, it is physical reality, not mathematics, that determines what is true in fact. “[T]he mistake is to confuse an abstract attribute with a physical one of the same name. Since it is possible to prove theorems about the mathematical attribute, which have the state of absolutely necessary truths, one is the misled into assuming that one possesses a priori knowledge about what the laws of physics must say about the physical attribute.” Deutsch, The Beginning of Infinity, p.183. Thus, mathematics may not always reflect the existing physical relationships.
In addition, Deutsch argues that the conclusions of Hilbert, Turing and Gödel about the limits of computing or proving mathematical statements22 are the results of the existing laws of physics, reflected in the operations of our brains and our computers. “Different physical laws would make different things infinite, different things computable, different truths—both mathematical and scientific—knowable. …[I]f the laws of physics were in fact different from what we currently think they are, then so might be the set of mathematical truths that we would then be able to prove, and so might the operations that would be available to prove them with. The laws of physics as we know them happen to afford a privilege status to such operations as “not”, “and” and “or”, acting on individual bits of information… .” Id., p.186.
Deutsch asserts “a computation of a proof is a physical process in which objects such as computers or brains physically model or instantiate abstract entities like numbers or equations, and mimic their properties. …It works because we use such theories only in situations where we have good explanations saying that the relevant physical variables in those objects do indeed instantiate those abstract properties.” Id., p.188. Thus, mathematics will not necessarily provide good predictions, let alone good explanations, of the physical world. In addition, there may be significant aspects of our reality that are not susceptible to mathematical modeling, that are “non-computable.” “So there is something special—infinitely special, it seems—about the laws of physics as we actually find them [to date], something exceptionally computation-friendly, prediction-friendly and explanation-friendly.” Id., p.189.
Deductive theories again
In addition to the use of mathematical logic to explore the implications of the axioms of a deductive theory, there is “the interesting possibility that mathematical consistency might be used to guide us, along with experimental observation, to the laws that describe physical reality.” Cox and Forshaw, Why Does E=mc2?, p.25. In other words, in the development of a theory, one might find ideas by considering the assumption that the correct theory will be internally consistent from a mathematical standpoint. Thus, one can do theory construction by a combination of inductive empirical reasoning and abstract mathematical reasoning, adding to the equations suggested by the data factors suggested by the mathematics. Of course, the prudent course is to conclude that the propositions are, at least, worth considering and then look for experimental methods to attempt to support or disprove the conclusion. One would still want to test the resulting theory against observation.
To investigate this question a bit further, let us take another look at deductive theories. Such theories, as noted, will have a set of axioms or assumptions and rules for manipulation. In some cases, one might be able to test the assumptions to determine whether they are empirically sound. If they are, then the results of the theory (the predictions) would be expected also to be empirically sound if the rules of manipulation (the mathematics) correspond to the laws of nature, that is, to the actual causal relationships at work. In that type of theory, the “validity” of the mathematics is effectively an empirical question. In contrast, one might have a deductive theory where the axioms are either not verifiable or are not really empirical, that is, they are either merely assumptions or are essentially definitions. There will also be the rules of manipulation or the mathematics. For such a theory to have scientific meaning, there will need to be certain variables that can have empirical content. So, essentially, one inputs certain observations and the theory generates a predication of certain other potential observations. If the theory “works,” one will find that the predicted observations occur.
The point I want to make is that in this type of deductive theory, the assumptions or definitions need not have empirical content (or be “true”) and the mathematics need not mimic the actual causal relationships at work; the theory just gives the right answers. All that one can say about the mathematics is that it works.
The methods of Newton and Galileo
This characterization of a theory might sound strange or contrived, but it is essentially a description of Newtonian or classical mechanics and is an apt characterization of much of modern science.23 Modern science has been attributed the development of a new scientific method that, with a bit of overstatement, has been described as “fashioned almost entirely by Galileo Galilei” in the early seventeenth century. Kline, Mathematics, p.284. See, id., pp.284–290, 337–51. Galileo concluded that matter (shape and size) and motion were phenomena of the natural world that existed independent of man’s perception, so they were suitable subjects for science. Id., p.284. He was interested in quantitative relationships based upon experimentation and observations (rather than introspection and contemplation). So, he focused on concepts that were quantifiable or measurable, like mass, speed, time, distance, etc.—concepts fundamentally different than the qualities that were the focus of Aristotelian philosophy, like fluidity, potentiality and purpose. Id., p.288. And Galileo believed that the quantitative relationships in Nature could be expressed mathematically.
Galileo’s interest was essentially descriptive. The question was not why but how. Thus, mathematical formulas were perfectly suited to the pursuit: “formulas do not explain; they describe.” Id. “[T]hrough Galileo … the connection between math and the physical world became solidified.” Mazur, The Motion Paradox, p.7. See also, id., pp.62–5. Of course, it was Newton, building on the work of Galileo and other Italian scientists of the fifteenth and sixteenth centuries, who put together the pieces and created the science of Dynamics. See, e.g., Whitehead, An Introduction to Mathematics, pp.30–31.
Newton similarly believed that mathematics underlay the design of the natural world, and he believed that the quantitative principles of that design should be explored through observation and experimentation. He also focused on matter and motion. To the work of Galileo, Newton added the Second Law of Motion and the more general law of gravity. Kline, Mathematics, pp.337, 359. The result was a set of principles that enabled the description of all motion of matter. Like Galileo, Newton utilized axioms and, through mathematics, deduced the laws of motion. The important point here is the nature of the resulting science, which is mathematics with great predictive and descriptive power, but no explanation of the underlying phenomenon itself.
A hostage to mathematics?
It is often observed that contemporary science is almost all mathematics. See, e.g., Barrow, Theories of Everything, p.174 (“Modern science is founded almost entirely upon mathematics.”); Kline, Mathematics, p.361 (“…our best knowledge of the physical world is mathematical knowledge”), p.555 (“[i]n all of these and in other significant and powerful bodies of science, mathematics, as we now know, is the method of construction, the framework, and indeed the essence”); Wigner, “The Unreasonable Effectiveness,” p.1. In fact, that is an overstatement, for several reasons. For example, it is not true of modern biology. However, as discussed further below, it does seem accurately to reflect the state of modern physics, including particle physics and cosmology.24
We have found that many physical phenomena are susceptible to mathematical modeling. It is certainly not obvious that it had to be so. The apparent congruence between mathematics and the physical world is characterized by Cox and Forshaw as “one of the deepest and in some ways most mysterious insights into the workings of modern science. Physical objects out there in the real world behave in