Raymond Louis Wilder was born 1896 and died in 1982. He was an American mathematician who specialised in algebraic topology and the theory of manifolds. Wilder was professor at the University of Texas, Ohio State University and at the University of Michigan. He was also vice president of the American Mathematical Society and its president from 1955 to 1956. (He was the Society’s Josiah Willard Gibbs Lecturer in 1969.) From 1965 to 1966, Wilder was the president of the Mathematical Association of America. (This association awarded him its Distinguished Service Medal in 1973.) Wilder was elected to the American National Academy of Sciences in 1963.
Evolution of Mathematical Concepts
Raymond L. Wilder called for mathematics to be analysed by the social sciences. He therefore predated elements of George Lakoff and Rafael E. Núñez’s book Where Mathematics Comes From; which was published only twenty years ago (in 2000). Wilder himself suggested that we should
“study mathematics as a human artefact, as a natural phenomenon subject to empirical observation and scientific analysis, and, in particular, as a cultural phenomenon understandable in anthropological terms”.
More relevantly to this piece, Wilder wrote the following words:
“The major difference between mathematics and the other sciences, natural and social, is that whereas the latter are directly restricted in their purview by environmental phenomena of a physical or social nature, mathematics is subject only indirectly to such limitations.”
In terms of the word “Platonism” in the title above, Wilder went on to say that
“Plato conceived of an ideal universe in which resided perfect models [however] the only reality mathematical concepts have is as cultural elements or artefacts”.
Finally, the following is primarily a commentary on Wilder’s well-known book Evolution of Mathematical Concepts: An Elementary Study; which was written in 1968. It tackles only specific ideas in that book.
Platonic Mathematics vs. Applied Mathematics
R.L. Wilder informed his readers that both “the Platonic” and practical approaches to mathematics could be found at one and the same time in ancient Greece. He wrote:
“[M]athematics was considered to be an attempt to describe the forms, quantitative and geometric, that one finds in the environment.”
On the other hand, mathematics was also seen as a
“description of an ideal world of concepts existing over and above the so-called real world”.
These two approaches weren’t always in conflict — even if for Plato himself they were indeed in conflict.
So it’s possible that there wouldn’t have been a pure — or Platonic — mathematics if it weren’t for prior mathematics “describing the forms — quantitative and geometric, that one finds in the environment”. On the other hand, it might have been the other way around. That is, there might not have been a practical mathematics without a prior Platonic — or Pythagorean — mathematics which described “an ideal world of concepts existing over and above the so-called real world”. More likely, however, perhaps both pursuits always existed in tandem — even if particular mathematicians or philosophers chose one approach or the other.
If we bring all that up to date.
Theoretical research (as in physics) in mathematics has often led to practical advances and applications in both science and technology. Wilder cites various examples of this in the following:
“The cases of Faraday and his researches in electricity and magnetism (making possible the electric motor) and of Clerk Maxwell and his equations (revealing the existence of radio waves) are classical instances. There are matched by the history of mathematical logic — the utmost in abstraction, one might say — and its ultimate importance in the computing industry (von Neumann, was originally a worker in the foundations of mathematics…).”
We can even see that pure mathematics has an effect on the hallowed “real world” we hear so much about. As Wilder put it:
“[I]t appears that no matter how abstract and seemingly removed from physical reality mathematics may become, it works — it can be applied either directly or indirectly to ‘real’ situations — as witness radio, air travel, and the like, none of which would have been possible without mathematics.”
One may initially wonder why Wilder singled out radio and air travel particularly. He probably did so because mathematics is the (as it were) distillation (or, to use an ugly word, abstractification) of radio and air travel in that it captures what is truly important and fundamental (from a scientific and technological point of view) in these concrete cases.
We can also see the close relation (or, indeed, unity) of physics and mathematics in the history of classical mechanics. Wilder wrote:
“When the basic postulates of classical mechanics were established by Galileo and Newton  classical mechanics was regarded as a branch of applied mathematics.”
Wilder then stated that “as a result of the theory of relativity, we know that the classical postulates do not correspond to physical reality”. Wilder made this conclusion because this (as it were) qualification of classical mechanics made it the case that it could no longer be seen as a branch of applied mathematics. He believed that it must be seen, instead, “as an abstract doctrine pertaining to pure mathematics”.
All this simply means that applied mathematics must be, well, applicable to “physical reality”; whereas pure mathematics needn’t be. Nonetheless, classical mechanics — as a branch of pure mathematics — has survived. Yet classical mechanics can also be said to have survived as a purely physical theory — despite being added to (not “overthrown”!) by the theory of relativity. The English mathematical physicist and mathematician Roger Penrose, for example, argues that classical mechanics retains its status in physics — though only “as a limit”. He wrote:
“Current physics ideas will survive as limiting behavior, in the same sense that Newtonian mechanics survives relativity. Relativity modifies Newtonian mechanics, but it doesn’t really supplant it. Newtonian mechanics is still there as a limit. In the same sense, quantum theory, as we now use it, and classical physics, which includes Einstein’s general theory, are limits of some theory we don’t yet have.”
On Wilder’s reading (as stated), on the other hand, classical mechanics — qua pure mathematics — now has an independence from application and indeed from the nature of the physical world.
Despite all the above, we shouldn’t get too fixated on the prefixes “applied” and “pure”; at least not when we take into account the history of mathematics. Wilder himself wrote:
“[W]hat is considered ‘applied’ mathematics today may… become ‘pure’ mathematics tomorrow. And, at any given moment in time, there is no clear distinction between what is ‘pure’ and what is ‘applied’. Even the ‘purest’ of mathematics may suddenly find ‘application’.”
Wilder then cited some further examples in the following:
“A problem of great importance to an electrical industry, which had failed of solution by its own engineers, has been solved by using methods of set-theoretic topology. Topics in matrix theory, topology, and set theory have been applied to production and distribution problems; abstract concepts of modern algebra find application in electronics; and mathematical logic is applied to the theory of automata and computing machines.”
To state the obvious. It can be seen from the above that mathematics — in its many forms — has had many technological applications. Nonetheless, these applications were often not apparent to either the mathematicians themselves or to anyone else at the time the various mathematical areas were created. (A layperson may now wonder how set theory crosses over “to production and distribution problems” or how modern algebra crosses over into electronics.)
Let me add an extra philosophical point here.
Mathematics can’t contradict the world. It can only confirm its basic (as it were) form. (Just as Wittgenstein’s logic — in his Tractatus — attempted to capture “the form of the world”.) More accurately, mathematics can’t contradict the world; though elements of mathematics may not have any role when it comes to describing the world — at least not at present! To cite Roger Penrose again, he gives various actual examples of this:
“Cantor’s theory of the infinite is one noteworthy example  extraordinary little of it seems to have relevance to the workings of the physical world as we know it. [See the discussion of singularities at the very end of this piece.] The same issue arises in relation to  Gödel’s famous incompleteness theorem. Also, there are the wide-ranging and deep ideas of category theory that have yet seen rather little connection with physics.”
Mathematics & Truth
What is the role of truth in mathematics?
What do mathematicians take truth to be?
Wilder himself informed us that
“most mathematicians of prominence concur in the doctrine that modern algebraic and geometric theories are true only in the sense that they are logical consequences of the axioms that form their bases”.
Did Wilder mean that these mathematical theories aren’t — strictly speaking — true? Or did he mean that the theorems (or propositions) found within modern algebra and geometry aren’t — strictly speaking — true? In other words, perhaps mathematical theories aren’t true in the same way in which individual theorems (or propositions) are true. Alternatively, perhaps neither mathematical theories nor individual theorems can be taken to be true.
According to Wilder, what make things true in mathematics is that they’re the “logical consequences of the axioms that form their bases”. Clearly other philosophical notions of truth (such as the correspondence theory) aren’t applicable to things which are true simply because they’re the logical consequences of particular axioms. (This is partly why the — late — Wittgenstein preferred the word “correct” rather than the word “true” — see here.) In any case, Wilder definitely denied the honorific true to Euclidean and non-Euclidean geometries. He wrote:
“No mathematicians who is familiar with the modern situation in mathematics will argue for the ‘truth’ of either Euclidean or non-Euclidean geometry, for example.”
However, Wilder did make an exception to this when he continued with these words:
“But in the case of those parts of mathematics that depend on the natural number system and its extensions, as well as on logical derivation therefrom — and this ultimately includes a good part of mathematics — there are those who argue for the absolute character of their conclusions.”
As stated, Wilder didn’t use the word “true” (or “truth”) in this context. He talked, instead, of the
“absolute character of [the] conclusions of the natural number system and its extensions”.
One must now ask what the vital difference is between Euclidean, non-Euclidean geometry, algebra and geometry and what Wilder calls “the natural number system and its extensions”. It seems (though Wilder didn’t say this explicitly) that truth is relevant for the natural number system and its extensions.
Mathematics & Reality
R.L. Wilder also discussed mathematical constructivism within these contexts. This movement is very relevant when it comes to discussing the relation between mathematics and the world.
The ironic thing about mathematical constructivism is that it doesn’t (or didn’t) believe that mathematics must abide by the (as it were) dictates of reality. Instead, it sees mathematics as being free to journey wherever it likes. According to Wilder, this is because mathematics is a human construction and each mathematical concept is itself an individual mental construction.
Wilder saw the rise of this new mathematical freedom in the context of developments which came to fruition in the 19th century. He wrote:
“Following the 19th century developments, the mathematical world came to feel that it was no longer restrained by the world of reality, but that it could create mathematical concepts without the restrictions that might be imposed by either the world of experience or an ideal world to whose nature one was committed to limited discoveries.”
Wilder then put the pure mathematician’s position when he continued with these words:
“One is reminded of the mathematician who, disgusted by the uses to which a backward and laggard world was putting scientific concepts, explained, ‘Thank God that there is no danger of my work ever being put to practical use!’ He was giving expression to that kind of ‘freedom’ that the mathematical world came to feel during the past century.”
This freedom from “the world of experience” (or from physical reality) may make one think in terms of a Platonic conception of mathematics. That said, if one is a Platonist, then one must be equally committed to Plato’s ideal world. Yet this is also a world and it too must inevitably “limit [one’s] discoveries”. So it’s no wonder that that the Platonic conception of mathematics is implicitly — or even explicitly — committed to a correspondence theory of truth for both numbers and equations. That is, mathematicians must be both committed to — and make their numbers and equations correspond to — the abstract mathematical entities in Plato’s ideal world. And surely this is just as much of a limitation as making one’s mathematics abide by the dictates of physical reality (or the dictates of experience).
It was no surprise, then, that some mathematicians (or at least some metamathematicians or philosophers of mathematics) rejected the infinite. They did so because they emphasised the point that there are no actual infinities in the physical world (or in the world of experience). Indeed, in the 20th century, it was seen that the laws of physics “break down” when it came to the ostensible infinities found at black holes and other singularities. (This occurs when mass is believed to have an infinite density or when spacetime has an infinite curvature.) More clearly, the ostensible infinities of physical singularities are the mathematical result of problematic and incomplete physical theories.