This is the second part of my piece on Raymond Louis Wilder (1896–1982) and his philosophical, historical and anthropological account of mathematics. I suggest that the reader refer back to the introduction to my ‘Raymond L. Wilder’s Anthropology of Mathematics: Platonism and Applied Mathematics’for Wilder’s biographical details.

Introduction

R.L. Wilder called for mathematics to be analysed by the social sciences. He 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”.

Moreover, 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.”

What follows is primarily a commentary on Wilder’s well-known book Evolution of Mathematical Concepts: An Elementary Study; which was written in 1968.

R.L. Wilder’s General Account of Mathematics

R.L. Wilder offered us a constructivist and anthropological view of mathematics which is radically at odds with the general Platonic (or at least quasi-Platonic) position.

Wilder himself wrote:

“Mathematics derives its concepts initially from the existing world of reality and uses them as a way of dealing with this reality.”

The traditional view is that it’s indeed the case that mathematics can be used “as a way of dealing with” reality”. However, it’s not usually also said that mathematics actually “derives its concepts [] from the existing world of reality”. Indeed it’s quite hard to see how mathematics can derive its concepts from reality without mathematicians already having at least some mathematical concepts to begin with. That said, if mathematical concepts are derived from reality (if only initially), then it’s no surprise that they can then be applied to — or deal with — that reality.

Now here’s the English cosmologist, theoretical physicist and mathematician John D. Barrow expressing the same position as it was advanced by the 19th century Dutch mathematician Diederik Korteweg:

“Korteweg’s own philosophy of mathematics was a straightforward one. He believed that we had discovered mathematics from the physical world and so its applicability there was just following the stream back to its source.”

Following on from Barrow’s words, there’s an interesting qualification — if it is a qualification — to what Wilder wrote above. Wilder continued by saying thatthis reality

“embraces not just the physical environment, but the cultural — which includes the conceptual — environment” .

In fact Wilder claimed that mathematical concepts “are just as real as guns or butter”. The point being made here is that human (even if deemed to be abstract) concepts don’t run free of the “physical environment”. And this is something that mathematical — or any other kinds of — Platonists won’t accept.

Wilder then explicitly put the anti-Platonist position. He argued that the concepts of mathematics

“were no longer embodiments of an independently existing realms of ideas, having an existence before and after the fact of their discovery, but only of a world of concepts continually under construction and having no existence until conceived in the minds of the mathematicians who created them” .

This is pure mathematical constructivism.

Wilder on Logic, Proof and Set Theory

Wilders stated that “the Greeks brought the notion of proof by logic into mathematics”. That said, many people still associate proof with mathematics.

Now take the well-known logical laws of contradiction and excluded middle.

It wasn’t the case that the law of non-contradiction and the law of excluded middle had logical proofs themselves. Instead, their “trustworthiness” in mathematics (according to Wilder) “was not questioned”. In addition, the mathematical

“conclusions reached by the use of such ‘laws’ were considered absolutely reliable if the premises were”.

Thus the law of contradiction and the excluded middle were essentially used as logical axioms in mathematics. Indeed these laws (or axioms) were deemed to be at the very basis (or foundation) of all mathematics - at least until the late 19th century.

So what about sets?

Wilder told us that it was “nineteenth-century mathematicians [who] introduced set theory into mathematics”. Wilder then provided a constructivist — and “materialist” (his own word) — account of set theory. He went on to saythat set theory

“was derived from experience with the finite collections of the physical and cultural environments”.

This is no surprise if we view sets as simply being collections of their concrete members. That is, if we see sets as being definable exclusively in terms of their members. However, this position leads to problems if it truly is all about “the finite collections of the physical environment”.

In other words, what about the null set and “infinite domains”?

Wilder continued:

“That extension of the classical logic and of set theory to infinite domains might lead to difficulties was not generally anticipated until around 1900, when a number of contradictions were found.”

Because infinite sets led to contradictions, then it was no wonder that the English philosopher Bertrand Russell (at least at one time) argued that sets are nothing over and above the sum of their members. Clearly this makes less sense when we take into consideration infinite sets and the null set. What are the members of such sets? How do we count them? How can an infinity determine a (circumscribed) set at all?

It was because of these problems — i.e., at the end of the 19th century — that commentators began to speak in terms of the “crisis in mathematics”. And, in order to end that crisis, a

“new foundation for the whole of mathematics seemed necessary to meet this crisis, not just a revised formulation of the real number continuum; for all parts of mathematics depended to a greater or lesser extent on logic and set theory” .

This explains the obsessive search for foundations in mathematics in this period; as can be seen in Frege, Russell and many other philosophers, logicians and mathematicians. It also shows us the importance of logic — in the guise of set theory — to mathematics. (Incidentally, similar “crises” have also occurred in philosophy; such as the case with epistemology and its equivalent search for foundations. See epistemological foundationalism.)

One of the best known (at least to philosophers) quests for the foundations of mathematics can be found in the work of Gottlob Frege. According to Wilder, Frege

“insisted that number and all of mathematics can be grounded in logic — a doctrine sometimes called ‘the logicist thesis’”.

From what’s been said about set theory’s importance to 19th-century mathematics (indeed, all mathematics), it isn’t surprising that Frege attempted to reduce mathematics — beginning with arithmetic — to logic. Wilder also tells us that Giuseppe Peano attempted something similar when he “refined and utilised the axiomatic method to achieve a basis for mathematics”.

A decade or so later, it was “largely under the influence of the works of Frege and Peano that the work of Russell and Whitehead was fashioned”. This too was alogicist programme. More technically, in the Principia Mathematica “an attempt was made to derive mathematics from self-evident universal (‘tautological’) logical truths”.

Despite the above, it’s not the case that logical tautologies (or logical truths) are required to be “self-evident” or even evident in nature. (These notions belong to what philosophers call psychologism.) What they need to be is necessarily true. As Ludwig Wittgenstein later put it, tautologies don’t even allow the possibility of their own falsehood. Thus their necessary truth is a result of their form, not of their content (see here).

I’ve just mentioned that tautologies needn’t be self-evident in nature. This lack of self-evidence contributed to the problems which irked the logicist programme. Wilder went on to say that as this foundationalist

“work proceeded to the higher realms of mathematical abstraction, it became necessary to introduce axioms that could hardly be admitted as constituting ‘self-evident logical truths’…”

Again, this raises the following question.

What did these philosophers, logicians and mathematicians mean by the term “self-evident”?

The problem here is that what’s self-evident to a mathematician may not be self-evident to a layperson. What’s more, what’s self-evident to the higher-level mathematician may not be self-evident to the lower-level mathematician. Indeed if a logical truth (or axiom) is necessarily true, then why do we need the added property self-evidence at all? And can a logical truth (or equation) simply (as it were) become self-evident after the mathematician (or logician) has worked on it for some time? That said, would this work on self-evidence constitute some kind of contradiction (or negation) of something’s being self-evident?

Kronecker, Intuitionism & the Law of Excluded Middle

According to Wilder, the German mathematician Leopold Kronecker fiercely derided Georg Cantor’s infinities. That was mainly because Kronecker believed that mathematics

“was a construction based on the natural numbers, which, in turn, were an outgrowth of man’s ‘intuition’”.

So not only do we have a reference to “construction” here: we also have a reference to “intuition”.

We can now ask why the natural numbers are so special and why they too aren’t constructed. In addition, what did Kronecker mean by the word “intuition”? (This term was initially taken from Immanuel Kant’s philosophy of mathematics — see here.)

Wilder then went into more detail as to what it was that Kronecker believed. He stated that Kronecker

“avoided all use of numbers that could not be constructed (as can, for instance, fractions like 2/3) from natural numbers”.

We now need to ask what exactly is meant by the words “constructed from natural numbers”. This — at least partly — means that numbers are constructed by use of operations such as +, x, etc. In any case, Kronecker

“asserted that numbers like [x] for example, simply do not ‘exist’, since there are apparently no ways of constructing them from natural numbers”

We can say, then, that Kronecker was a constructivist — but only about certainnumbers.

I mentioned Kronecker’s acceptance of natural numbers earlier and he accepted them because they are… well, natural.

Wilder went on to say that “[v]irtually no one agreed with” Kronecker on these issues.

Kronecker can also be seen as a kind of (proto)intuitionist. Wilder said that his “thesis was reaffirmed (in modified form) by Brouwer”. Wilder continued:

“[L]ogic that had been introduced into mathematics by the Greeks was tossed overboard, except for what could be salvaged through use of the constructive methods of intuitionism.”

What, exactly, was thrown overboard? Perhaps, most importantly, the

“use of the law of the excluded middle, so important in reductio ad absurdum proofs, was no longer permissible except for finite sets”.

Interestingly enough, Wilder claimed that the law of excluded middle is acceptable to the intuitionist. However, this was the case only when that law was applied to finite sets — which is, of course, a big exception! Wilder did go on to tell us why the law of excluded middle is applicable to finite sets when he saidthat for

“any finite set of natural numbers, it was permissible to assert that either at least one of the numbers was even or none was even” .

For intuitionism, what is important is how a mathematical statement is proven. That is, there must

“exist an elementary constructive was of demonstrating such use of the law of the excluded middle, namely by examining the numbers one by one!”.

So here we have an explicit example of intuitionist construction — “examining the numbers one by one”. That is: Is the number 1 even? No. Is the number 2 even? Yes. Okay. That means that there is at least one even number in this set of two natural numbers.

So what is it about infinite sets that renders the law of excluded middle non-applicable?

We can say that

“the same assertion about an arbitrary infinite set of natural numbers could not be made” .

Why is that? It’s because we can never know what surprises an infinite set contains or will (as it were) throw up later.

For example, we may count a billion billion numbers of an infinite set and find that, so far, it doesn’t instantiate property x. However, that doesn’t mean that it may not display x somewhere further up the line. Thus we can’t apply the law of excluded middle (i.e., either p or not-p) to this particular case.

So what is positive about intuitionism? What does it (or did it) attempt to achieve?

Wilder went on to say that the

“great advantage of intuitionistic philosophy was its freedom from contradiction — limitation to constructive methods guaranteed this”.

The idea here is that if one is constructing everything (as it were) by hand , then one can’t make mistakes. In addition, there’s no room for speculation or conjecture in intuitionism — and that’s why it is (supposedly) guaranteed to be free from contradictions. In other words, all such contradictions will be nipped in the bud.

Of course intuitionism does have its defects. Wilder did say that its

“fatal defect was that it could not derive, using only constructivist methods, a major portion of the concepts that were regarded as being among the greatest mathematical achievements of the modern era” .

Perhaps many of these great mathematical concepts were the result of speculationor mathematical creativity; which seem to have no place in intuitionism. Not only that: Wilder saw this period of intuitionism “as as attempt to stem the flow of mathematical evolution — a kind of cultural resistance”. Most of this can be blamed on the then obsessive desire to root out “the threat of contradiction”. And, according to Wilder, that threat didn’t need “such drastic action as intuitionism demanded”.

Again, this seemed to be a sacrifice of speculation, conjecture and mathematical creativity on the part of intuitionism.

Despite these defects and criticisms, it was still the case — according to Wilder at least — that intuitionism “had a great and seemingly beneficial influence”. For example, a “number of prominent mathematicians shared in some, or all, of its tenets — for example, H. Poincaré and H. Weyl”.

To sum up.

Despite intuitionism’s radical nature and its rejection of the law of excluded middle (at least for infinite sets), Wilder nevertheless finished off by saying that

“its doctrine of constructivity was found to be adaptable to numerous situations within the framework of conventional mathematical theory”.

Completed Infinities

More technically, Wilder expressed his constructivist, anthropological and psychological position on mathematics by elaborating on the notion of a completed infinite. The idea of a completed infinite — at least initially — seems like a blatant and direct contradiction. How can any infinite set be completed (or complete)? If such a set is completed, then surely it’s not infinite.

So what, exactly, did Wilder say on the issue of the completed infinite? This:

“For example, an infinite decimal is not something that ‘just goes on and on without end’. It is to be conceived as a completed infinite, just as one conceived of the totality of natural numbers as a completed infinity.”

So Wilder actually gave us two examples of completed infinities — the “infinite decimal” and “the totality of natural numbers”. However, he didn’t really go into detail as to what such things actually are or what the words “completed infinite” mean. That said, the following words hint at an explanation (though they don’t help the non-mathematicians very much). Wilder wrote:

“Symbolically, it may be considered a second-order symbolism, in that it is not susceptible to complete perception, but is only conceptually perceivable.”

One may have a vague idea of what Wilder meant by the words directly above. Perhaps he was referring to a kind of “direct insight” (or intuition) into the nature of completed infinities. That is, completed infinities are “conceptually perceivable” (philosophers today would say conceivable); though they can’t literally — or even metaphorically — be seen.

Conclusion

From the retrospective point of 2021, the late-19th-century and early-20th-century obsession with the foundations of mathematics may seem strange. It may seem even stranger if we realise what the end result of this obsession was - at least according to Wilder - the following:

“The most powerful symbolic tools and his powers of abstraction and generalisation have failed the mathematicians in so far as ‘explaining’ what mathematics is, or in providing a secure ‘foundation’ and absolutely rigorous methods.”

It’s quite remarkable that Wilder claimed that modern mathematicians failed to explain what mathematics is considering the fact that even the layman would have a good go at the job. So there are two further questions:

Why can’t mathematicians explain what mathematics is? Why is this task so difficult?

Wilder argued that it was largely Kurt Gödel’s theorems which stopped mathematicians from “providing a secure foundation”; as well as from finding “absolutely rigorous methods”. Yet, from what Wilder wrote next, it seems as if mathematics not having any (secure) foundations (or not being free from all contradictions) might not have been such a bad thing. More precisely, Wilder went on to say that

“perfect rigour and absolute freedom from contradictions in mathematics are no more to be expected than are final and exact explanations of natural or social phenomena”.

Of course in science we don’t always — or ever — have “exact explanations of natural and social phenomena” and such things aren’t even expected. So is this really the case for mathematics as well? Perhaps this conclusion, on Wilder’s part, is simply a result of his constructivist and anthropological position on the practice and history of mathematics.

So to sum up with a single statement from Wilder..

It’s no surprise that Wilder’s general position was what it was if he believed that

“the only reality mathematical concepts have is as cultural elements or artefacts”.