36 Physicist Paul Davies asserts that it “is generally agreed” among scientists that the Laws of Nature have four characteristics: they are universal, they are absolute, they are eternal and they are omnipotent (“all powerful,” meaning that they cannot be avoided or evaded). The Mind of God, pp.82–3. He observes that these qualities “that were formerly attributed to the God from which [the Laws] were once supposed to have come.” Id., p.82. Davies suggests that the question that does divide scientists is whether the Laws are somehow real (that they exist independently of the physical world) or are simply things discovered by scientists. Id., pp.83–4.
37 And, certainly, theoretical work does go on exploring the possible consequences to our understanding of cosmology, for example, if one permits (or assumes) changes in the laws or the constants. See, e.g., Smolin, Time Reborn; Joao Mangueijo, Faster Than the Speed of Light (2003).
38 Lord Rees explains that if one takes a few meters as a normal distance for man, that distance would have to be increased by twenty five factors of ten to reach the observable limits of the Universe. Just Six Numbers, pp.5–6. From a meter distance, our smallest measurable size (using electron microscopes and particle accelerators) would be about seventeen negative factors of ten smaller. Physicists speculate that the smallest structures of nature, like the proposed superstrings, would be smaller by another seventeen negative factors of ten. Id., pp.6–7.
39 The example of the developments leading to Newton’s theories is relatively easy to understand. The change from an Earth-centered to a Sun-centered then a no-centered worldview is widely understood and accepted in modern society. There was a paradigm shift that was far broader than the scientific theories directly involved. It is likely that certain societal and cultural developments were useful, if not necessary, to the acceptance of the new theories and, perhaps, even to the conception of them. What about Einstein and his theories of relativity? At a superficially appealing level, one can note that societal developments concurrent to the scientific theory included the breakdown of established orders and universal truths and the emergence of cultural and moral relativity. Are these all elements of a paradigm shift? Can one say that the Special Theory of Relativity is a cousin to moral relativity and then to political correctness? I think not. The relativity of the theories in physics does not in any way suggest that there are no universal truths or bases from making judgments of the correctness or value of various positions. Indeed, as Lord Rees has observed, “It’s a pity, in retrospect, that he called his theory ‘relativity’. Its essence is that the local laws are just the same in different frames of reference. ‘Theory of invariance’ might have been a more apt choice, and would have staunched the misleading analogies with relativism in human contexts.” From Here to Infinity, p.137. Nonetheless, there is certainly some truth to the observation that our values and views of the world are being challenged on many fronts. I do not attribute that phenomenon to Albert Einstein. I think that it reflects the sense of tumult and uncertainty that will accompany significant shifts in views of the world, whether revolutionary in the Kuhnian sense or more accretive and cumulative.
The Nature of Mathematics
Mathematics is encountered everywhere in today’s world. Its usefulness (even, indispensability) as a tool in applications from almost all fields is beyond dispute. The big question that I address here is: why is it that mathematics is so useful? Or, in other words, what is the relationship between mathematics and the physical world? The question is a methodological one, similar to the types of issues that have already been introduced.
I do not think that the deeper philosophical questions about the foundations of mathematics need generally concern us, such as whether mathematics reflects something that is in some sense real or is only a creation of the mind; whether mathematics should be limited to concepts that have physical analogs; whether mathematics is the study of logic or structure or something else or whether all of mathematics can be captured in a single comprehensive and consistent structure (such as set theory). See, e.g., “Philosophy of Mathematics” (revised 2 May 2012), Stanford Encyclopedia of Philosophy; Ian Hacking, Why Is There Philosophy of Mathematics at All? (2014); Mark Colyvan, An Introduction to the Philosophy of Mathematics (2012).
Perhaps, the more accurate assertion is that the issues to be explored here do not require us to take a position on a comprehensive philosophy of mathematics or on many of the issues that have been the subject of debate among those who have attempted to formulate or critique philosophies of mathematics. However, the discussion will suggest that some of the dilemmas encountered may be the result of attempting to fit fundamentally different things into a single category.
I do think it useful, therefore, to state explicitly that the term mathematics refers to a rather wide range of intellectual methods and activities that can look quite different and can serve very different methodological functions in science. Think of just the basic academic subjects: arithmetic, geometry, algebra, trigonometry, and calculus. Many of the symbols used are different, the types of statements presented look different and the uses of the various disciplines are different; yet, one recognizes intuitively that there is something fundamentally similar among all of these activities.
Arithmetic and geometry are recognized as highly useful tools for dealing with the world around us, whether in our daily errands, our work or our play. Indeed, many of the applications of the more elementary branches of mathematics as a tool seem almost second nature. Counting, adding, subtracting, multiplying are functions that tie directly to our actual experience of the things of the external world. Geometry also appears to be an idealized conceptualization of relationships that exist and can be observed in the physical world, such as the volume of a cube or the area of a circle. So, we could say that these branches of mathematics may have been initially developed in order to represent the relationships that existed.1
Yet, science makes use of many forms of mathematics and does so in many different ways.
Statistical analyses and related quantitative methods are used to organize data and to extract information from it, such as correlations between or among variables reflected in the data. This branch of mathematics, including probability theory, has found wide application in biology and the social sciences, as well as in astronomy and other natural sciences. Morris Kline, Mathematics for the Nonmathematician (1967), p.499 (hereinafter “Mathematics”).
In addition, there has been considerable use of “game theory” models in economics and in evolutionary biology. See, e.g., Martin A. Nowak (with Roger Highfield), SuperCooperators (2011). Many modern sciences make use of large simulations in which values for certain variables are placed into sets of equations and operations are performed hundreds, thousands or millions of times by computer.
A particularly intriguing use of mathematical simulations is to perform hypothetical tests for potential causality. The idea is to make discreet changes in specific assumptions, run the model and then see how the resulting scenario compares to our observations of the existing world. If the constructed results resemble what has in fact happened, one has some support for the validity of the new assumption. See, e.g., Jonathan Webb, “Evolution ‘favours bigger sea creatures,’” BBC News, Science & Environment, 19 February 2015 (a study of marine fossils suggests that marine life have in general gotten large since the Cambrian period; the question asked was whether the larger body size could have been the result of evolution in which larger size had adaptational value; in some of the computer simulations, the researchers gave larger size a beneficial effect; when the program with that adjustment was run, the tendency was for the marine life to get larger as appears to have happened).2
The point here is simply that examples of the mathematics in these various uses, if set forth on the page, would look very different one from another, both in form and function.
What is mathematics?
Given the diversity of mathematical techniques, one might want to ask exactly what