In fact, one could posit another, different sort of relationship between logic and truth. To the extent that logic and mathematics reflect the inherent characteristics of the human mind, they may not be truly objective nor universal nor, in that sense, independent of the physical world (that is, of the human mind). Apparent consistency between mathematical models and our understanding of the external world could simply be the result of both types of models being human creations. From this perspective, mathematics may not actually be “objective and concrete,” meaning the same to other intelligent life (if such aliens exist) as to human beings. Cf., e.g., Martin A. Nowak, SuperCooperators, p.2 (“universal logic acting on universal rules”).
Wigner’s famous question
The question is often asked why it is that we find that many branches of mathematics have real world applications. Nobel-prize winning physicist Eugene P. Wigner gave a famous lecture in 1959 entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” Communications of Pure and Applied Mathematics, Vol. XIII (1960), pp.1–14. The title seems to have captured the philosophical puzzle in an imaginative way, since the lecture is frequently cited, and there are numerous articles published utilizing easily recognizable variations on Wigner’s title. I do not intend below to discuss all of the matters illuminated in that lecture, but I simply note here that Wigner did not, in my view, present a very satisfactory answer to the question he asked. Yet, many scientists since have limited their discussion of the issue to a citation to that lecture, as if it held the dispositive answer.6
Of course, as already suggested, it may be a mistake to talk about mathematics as if it were all the same thing epistemologically. One might observe that arithmetic, at least in its simple form, reflects one aspect of material things, captured in counting, addition and subtraction. Kant argued that arithmetic reflected the structure of time, based upon counting which takes place in time. Id., p.7. Geometry reflects the structure of space. Id. One might speculate that each of these branches of mathematics developed simply as an accurate representation of relationships that exist in various aspects of the real world. It is a curious fact, then, that algebra (derived from arithmetic) can be successfully applied to geometry and geometry to algebra, each being used to simplify or solve problems arising in the other. See id., pp.20–21.7 Perhaps, this applicability evidences some underlying logic in the nature of reality. However, as we have come to learn, Euclidean geometry is not in fact a totally accurate representation of space.
As we have noted, mathematics is essentially an elaborate exercise of human reasoning or logic. See, e.g, Wigner, “The Unreasonable Effectiveness,” at pp.2–3. The historians of science can debate, and maybe establish, the extent to which more sophisticated mathematics developed through an analysis of real world applications, such as counting things or describing the physical features of our three dimensional space, or as an independent, logical exercise that to the surprise of the creators has some real world application. Undoubtedly, there has been some of each. See, e.g., Kline, Mathematics, pp.30–7, 51–4; Wigner, “The Unreasonable Effectiveness,” pp.2–3.
In any event, the questions I want to highlight here are:
First, why is it that the phenomena in the physical world seem to be capable of description or modeling through mathematics; and
Second, could one conclude that any logically consistent mathematical model has a physical analog in some universe.
The underlying question, reworded, is why does human logic seem able to capture natural phenomena? (“Nature speaks in equations. It is an odd coincidence. The rules of mathematics were built around counting sheep and surveying property, yet these very rules govern the way the universe works.” Charles Seife, Zero: The Biography of a Dangerous Idea (2000), p.117.8
As previously noted, it might be suggested that mathematics and human reasoning are the results of evolution and that, as such, we would expect them to reflect reality. But, as also observed, such an argument would only apply to aspects of the world around us that have been directly relevant to our survival and reproduction. Modern science, and its mathematical models, go far, far beyond such matters, into realms of knowledge that have only recent or expected future relevance to our physical existence and other realms of knowledge that have no currently foreseeable material utility.
Indeed, Cambridge paleobiologist Simon Conway Morris has asserted that “being a product of evolution gives no warrant at all that what we perceive as rationality, and indeed one that science and mathematics employ with almost dizzying success, has as its basis anything more than sheer whimsy.” “Darwin was right. Up to a point,” The Guardian, 12 February 2009. For the reasons just expressed, I think that he overstates his case here. But, he goes on, suggesting an argument that many scientists strongly resist: “If, however, the universe is actually the product of a rational Mind and evolution is simply the search engine that in leading to sentience and consciousness allows us to discover the fundamental architecture of the universe—a point many mathematicians intuitively sense when they speak of The Unreasonable Effectiveness of mathematics—then things not only start to make much better sense, but they are also much more interesting.” Id.
Mathematics and Logic
At this stage, it may be useful to take another look at logic. Let’s use a well-known example: “All men are mortal. A is a man. Therefore, A is mortal.” The conclusion seems beyond dispute. In fact, it is hard to imagine how it could not be true. But, what does true mean in this context? Suppose that we discover an apparent man B who appears to be immortal because he has substantially outlived any prior example of a man. There are two clear possibilities. Obviously, B may eventually turn out to be mortal—that is, B may die. Alternatively, we may conclude that B is not actually a man. Indeed, it may be that mortality is part of the definition of “man.” In other words, if B is not mortal, then he is not a “man” (he may be an “angel” or a “god”). But, when worded this way, the logic is not very satisfying.
There are other possibilities. We might find an ambiguity in the meaning of mortal or, at least, in the empirical observation of mortality (such as the length of life). Another possibility is in the meaning of the word “all.” If it has a probabilistic meaning, then occasional counter-examples are compatible.
In any event, we would hope that our logical argument could go beyond merely asserting, just in different words, a definition that we have adopted. To have some meat in the theory, we would want the definition of man to be in some sense independent of the assertion that man is mortal. Clearly, it cannot be completely independent. There could be aspects of man, defined without explicit reference to mortality, that would still mean that man was in fact necessarily mortal.
Example: One plus one
Take a simple arithmetical example. One plus one equals two. How could it be otherwise? If the proposition means no more than if you have one item and obtain another item, then you have two items; then it seems necessarily true—that is, true by reason of the meanings of the terms. By that, I do not mean just a matter of mere definition, but as a result of the structure of the proposition.
To illustrate, are there situations where one plus one does not “equal” two?
If the word “equals” in the question is limited to its definition in logic or mathematics, then the answer must be “no” because of how the term “equals” is defined. In contrast, if “equals” is taken to mean “result in” or “yield,” then there are some other interesting possible answers. For example, one male plus one female could turn out to result in three (or four or five, etc.). Or, one particle of matter plus