“Accessible to students and relevant to specialists, this remarkable book by a prominent educator offers a unique perspective on the evolutionary development of mathematics. Rather than conducting a survey of the history or philosophy of mathematics, Raymond L. Wilder envisions mathematics as a broad cultural phenomenon. His treatment examines and illustrates how such concepts as number and length were affected by historic and social events.”
As for R.L. Wilder himself:
“Raymond L. Wilder received both his Bachelor's and Master's degrees from Brown University, while the University of Texas granted him a Ph.D. He was instructor at the University of Texas from 1921-24, and Assistant Professor at Ohio State University for the following tow years. In 1925 he was appointed Assistant Professor at the University of Michigan, where he has been successively Associate Professor, Research Professor and Professor Emeritus. He served as President of the American Mathematical Society and the Mathematical Association of America.” (From the 1968 Transworld Publishers Ltd. edition.)
i) The Applications of Pure Mathematics
ii) Platonic Mathematics vs. Applied Mathematics
iii) Wilder’s Materialist Account of Mathematics
iv) Mathematics & Reality
v) Mathematics & Truth
vi) Logic, Proof & Set Theory
vii) Kronecker, Intuitionism & the Law of Excluded Middle
viii) The Foundations of Mathematics & Completed Infinities
While studying philosophy one often hears it said that philosophy is of no practical value and that it's a waste of time. Sometimes that's also said of higher mathematics. And it's also said of basic mathematics. Though, as in science, theoretical research has often led to practical advances and applications outside mathematics, in both science itself and technology. R.L. Wilder cites various examples of this from maths and science:The Applications of Pure Mathematics
“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…)...” (181)
We can see that even higher mathematics has an effect on the hollowed ‘Real World’ we hear so much about. As the writer puts it:
“… it 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.” (185)
One wonders why the writer singles out radio and air travel particularly. We can say that, in a sense, maths is the distillation (or abstractification) of radio, air travel and much else in that, like physics, it captures what is truly important and fundamental to these real situations.
We can see the close relation (or, indeed, unity) of physics and maths in the history of classical mechanics:
“When the basic postulates of classical mechanics were established by Galileo and Newton… Classical mechanics was thus regarded as a branch of applied mathematics.” (186)
However, “as a result of the theory of relativity, we know that the classical postulates do not correspond to physical reality” (186). The writer therefore concludes that because of this change to classical mechanics, it should no longer be seen s a branch of applied mathematics. It must be seen, instead, “as an abstract doctrine pertaining to pure mathematics” (186). This simply seems to mean that applied mathematics must be applicable, evidently, to "physical reality"; whereas pure mathematics need not be. However, classical mechanics as a branch of pure mathematics has nevertheless survived despite being overthrown by the theory of relativity. Thus, in this instance, classical mechanics, qua pure mathematics, has an independence from application and indeed the nature of the physical world.
We shouldn't, however, get too fixated on the terms ‘applied’ and ‘pure’; at least not when we take into account the history of mathematics.
“… what 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’.” (186)
Let’s put that this way. Maths can't contradict the world. It can only confirm its basic form (as it were); just as Wittgenstein's Tractatus logic attempted to do. Wilder cites some examples here:
“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.” (186)
To state the obvious. It can be seen from the above that mathematics - in its many forms - has as many ‘technological’ applications as physics. However, as I said earlier, these applications may not be apparent to either the mathematicians themselves or to anyone else for quite some time. (One just wonders, for example, how set theory crosses over "to production and distribution problems" or how modern algebra crosses over into electronics.)
Platonic Mathematics vs. Applied Mathematics
Strangely enough, the Platonic and practical approach to mathematics could be found at the same time in ancient Greece:
- “… mathematics was considered to be, on the one hand, at attempt to describe the forms, quantitative and geometric, that one finds in the environment…” (186)
- Mathematics was “a description of an ideal world of concepts existing over and above the so-called real world” (186).
Wilder’s Materialist Account of Mathematics
Wilder offers us what may be called a conventionalist - or even an anthropological - view of mathematics which is radically at odds with that of the Platonic one:
“Mathematics derives its concepts initially from the existing world of reality and uses them as a way of dealing with this reality…” (187)
The traditional view was that it's indeed the case that mathematics can be used “as a way of dealing with… reality” (187); though not that it “derived its concepts… from the existing world of reality”(187). In any case, it's quite hard to see how mathematics can derive its concepts from reality without, in a sense at least, already having at least some of those concepts. In addition, if mathematical concepts are derived from reality, if only ‘initially’, it's therefore not surprising that they can then be applied to - or deal with - that reality.
Nevertheless, there's an interesting qualification - if it is indeed a qualification - to what Wilder perceives to be this reality. He writes that this reality “embraces not just the physical environment, but the cultural – which includes the conceptual – environment” (187). In fact concepts “are just as real as guns or butter”. If reality includes the conceptual, then Wilder’s position on mathematics could be said to include the Platonic position as well as the materialist conception of mathematics. The point will be, I conclude, that the conceptual doesn't run free of the "physical environment". This is something, evidently, that Plato wouldn't have accepted.
Wilder then explicitly puts the anti-Platonic or constructivist position. He says 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” (187).
This sounds like pure constructivism or intuitionism. Wilder even uses the phrase "concepts under construction" to get his point across. Considering the sociological (or even the "materialist") slant of this book, it's no wonder that Frank Ramsey talked of mathematical constructivism as “that Bolshevik menace”. (Having said that, this crudely sociological or materialist account of mathematics is almost as unappealing as pure Platonism.)
The ironic thing about the constructivist theory is that rather than abiding by the dictates of reality (as it were), mathematics is free to journey wherever it likes. That's because mathematics, according to Wilder, is a human construction and each mathematical concept is an individual mental construction.Maths & Reality
Wilder sees this new mathematical freedom in terms of developments which came to fruition in the 19th century. He writes:
“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. 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.” (187)
This freedom from "the world of experience” (or from physical reality) may make one think in terms of a Platonic conception of mathematics. However, if one is a Platonist, one must be equally committed to Plato’s ideal world which would inevitably “limit [one’s] discoveries”. It's no wonder that philosophers have said that Platonic conception of mathematics is implicitly - or even explicitly - committed to a correspondence theory 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. As one can see, this is just as much of a limitation as making one’s mathematics abide by the dictates of experience or of physical reality. It was no wonder, then, that some mathematicians rejected the infinite, never mind Georg Cantor’s infinite infinities. They did so because there is no actual infinite in the physical world or in the world of experience.
Mathematics & Truth
What is the role of truth in mathematics? What do mathematicians take truth to be?
Wilder writes 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” (190).
Does he mean that the theories themselves aren't strictly speaking true? Or does he mean that the propositions found within modern algebra and geometry aren't strictly speaking true? Perhaps this is just a fact about his phrasing. Or perhaps theories aren't true in the way that individual equations or propositions are true.
In any case, what make things ‘true’ is that they're a “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 are the logical consequences of particular axioms. This is why the late Wittgenstein preferred the word ‘correct’ instead of ‘true’.
Again, are we talking about the theories of geometry and algebra or their particular propositions? Perhaps this doesn't matter. Similarly, are we talking about algebra and non-Euclidean geometry, for example, or their individual theorems or propositions?
According to Wilder, we're definitely denying the status of truth to Euclidean and non-Euclidean geometries:
“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.” (190)
However, Wilder does make an exception to this. He writes:
“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.” (190).
One must ask what is the vital difference between Euclidean, non-Euclidean geometry, algebra and geometry and what Wilder calls “the natural number system and its extensions”. It seems, though Wilder doesn't say this directly, that truth is the issue for the natural number system and its extensions. Is that simply because this system uses - or refers to – numbers; whereas the others don't (at least not completely)?
As I have just said, Wilder doesn't use the word ‘truth’ in this context. He talks instead of “the absolute character of [the] conclusions’ of the natural number system and its extensions”.
Logic, Proof & Set Theory
Many people associate proof with mathematics; though it is, or it was, a notion in logic (just as set theory is a logical theory). Historically speaking, “the Greeks brought the notion of proof by logic into mathematics” (191). It wasn't really a case that the Aristotelian laws of contradiction and of the law of excluded middle had no logical proofs; but that their ‘trustworthiness’ in mathematics “was not questioned” and the mathematical “conclusions reached by the use of such 'laws' were considered absolutely reliable if the premises were” (191). Thus it seems that in mathematics Aristotle’s laws of contradiction and the excluded middle were, and are, used as axioms in mathematics. Indeed isn’t it the case that these laws, or axioms, are at the very basis or foundation of all mathematics, at least until the 20th century when we had ‘alternative’ or ‘deviant’ logics?
I mentioned set theory earlier. It was “nineteenth-century mathematicians [who] introduced set theory into mathematics” (191). Wilder then goes on to provide a constructivist - or even materialist - account of set theory.
He writes that set theory “was derived from experience with the finite collections of the physical and cultural environments” (191). This is no surprise if we view sets as simply collections of their concrete members. That is, if we see them as defined exclusively in terms of their membership. Though this materialist view leads to problems if it's all about “the finite collections of the physical environment”. What about the null set and ‘infinite domains’?
“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.” (191)
If infinite sets lead to contradictions, it's no wonder that Bertrand Russell (at least at one time) argued that sets are nothing but the sum of their members taken collectively. Clearly this makes less sense when we take into consideration infinite sets. What are the members of such sets? How do we count them? How can an infinity determine a set at all?
It was because of these contradictions, primarily those found by Russell, that writers speak in terms of the ‘crisis’ in mathematics at the end of the 19th century. In order to end the 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” (192).
This explains the obsessive search for foundations in mathematics in this period; as can be seen in Frege, Russell and many more philosophers, logicians and mathematicians. It also shows us the importance of logic - in the guise of set theory - to mathematics. (Similar crises, incidentally, occurred in philosophy; as was the case in epistemology and its equivalent search for foundations.)
One of the best known, at least to philosophers, quests for foundations in mathematics can be found in the work of Frege.
“insisted that number and all of mathematics can be grounded in logic – a doctrine sometimes called 'the logicist thesis'”(192).
From what has been said about logic and set theory’s importance to 19th century mathematics, and, indeed, all mathematics, it isn't therefore surprising that Frege attempted to reduce mathematics - beginning with arithmetic - to logic. Peano attempted something similar when he “refined and utilised the axiomatic method to achieve a basis for mathematics” (192).
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” (192). This too was a logicist programme. More technically, in the Principia Mathematica “an attempt was made to derive mathematics from self-evident universal ('tautological') logical truths” (192). However, I’m not sure that logical tautologies, or ‘logical truths’, are required to be ‘self-evident’ or even evident in nature. What they need to be is necessarily true. As Wittgenstein put it later, they don't even allow the possibility of their falsehood. Their necessary truth is an effect of their form, not their content.
I've just mentioned that tautologies needn't be self-evident in nature. This lack of self-evidence, as it were, contributed to the problems which irked the logicist programme. As this foundationalist or reductionist
“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'” (192).
This raises the following question. What did these philosophers, logicians and mathematicians mean by ‘self-evident’? After all, what is self-evident to a mathematician may not be self-evident to the layperson. In addition, what is self-evident to the higher mathematician may not be self-evident to the lower-level mathematician. Indeed if a logical truth or axiom is necessarily true, why would we need the added property self-evidence at all? And can an axiom (or equation or logical truth) become, as it were, self-evident after one has worked on it for some time? Or would this constitute some kind of contradiction or negation of something’s being self-evident?
Kronecker, Intuitionism & the Law of Excluded Middle
Leopold Kronecker, the man who fiercely derided Cantor and his infinities, can interestingly enough be seen as a proto-constructivist or intuitionist. He believed that mathematics “was a construction based on the natural numbers, which, in turn, were an outgrowth of man’s 'intuition'” (193). So not only do we have here a reference to ‘construction’, but also one 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 ‘intuition’? Was this term taken from Kant’s philosophy of mathematics?
Wilder goes into more detail as to what it was that Kronecker believed.
Kronecker “avoided all use of numbers that could not be constructed (as can, for instance, fractions like 2/3) from natural numbers” (193), as has already said. We need to ask what exactly he meant by ‘constructed’ from natural numbers. Did it mean by use of the operations +, x and such like? In any case, he “asserted that numbers likep* [problems with symbols], for example, simply do not 'exist', since there are apparently no ways of constructing them from natural numbers” (193). We can say, then, that he was a kind of anti-realist or nominalist about only certain numbers in that he claimed that some numbers, p for instance, do not ‘exist’.
I mentioned his acceptance of natural numbers earlier and perhaps he accepted them because they are, well, natural. In any case, “[v]irtually no one agreed with him” on this issue.
I mentioned that Kronecker can be seen as a kind of proto-intuitionist: Wilder writes that his “thesis was reaffirmed (in modified form) by Brouwer” (193). Intuitionism, according to Wilder, is indeed radical. He writes that the
“logic 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” (193).
What, exactly, was thrown overboard?
For a start, and 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” (193/4).
Interestingly enough, Wilder claims that to the intuitionist the law of excluded middle is acceptable; though only when applied to finite sets – which is a big exception! What is it about infinite sets that render the law of excluded middle non-applicable? In any case, Wilder does go on to say why the law of excluded middle is applicable to finite sets. 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” (194).
I can’t help thinking that this isn't much of a claim! However, what is important, to intuitionism, is how this claim is proven. That is, there
“exists an elementary constructive was of demonstrating such use of the law of the excluded middle, namely by examining the numbers one by one!” (194).
So that’s an example of intuitionist construction – “examining the numbers one by one”. That is: Is 1 even? No. Is 2 even? Yes. OK. That’s it. There is at least one even number in this set of two natural numbers.
What about infinite sets? We can say that “the same assertion about an arbitrary infinite set of natural numbers could not be made” (194). Why is that? Because we can never know what surprises an infinite set contains or will come up with. For example, we may count a billion billion numbers of an infinite set and find that, so far, it does not have property Æ [problems with symbols]; though that doesn't mean that it may not display Æ somewhere further up the line (so to speak). Thus we can't apply the law of excluded middle (either p or not-p), in this particular instance.
So what is so good about intuitionism? What does it, or did it, attempt to achieve?
Wilder says that the “great advantage of intuitionistic philosophy was its freedom from contradiction – limitation to constructive methods guaranteed this” (194). Perhaps this means that if one is constructing everything by hand (as it were), one cannot make mistakes. In addition, perhaps there's no room for speculation or conjecture in constructivist intuitionism – and that is why it's guaranteed to be free from contradictions because all such contradictions will be nipped in the bud.
Though, of course, intuitionism does have its defects.
“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” (194).
Perhaps these great mathematical concepts are the result of speculation or even mathematical creativity; which have no place in constructivist intuitionism. Not only that: Wilder sees this period of intuitionism “as as attempt to stem the flow of mathematical evolution – a kind of cultural resistance” (194). Perhaps all this can be blamed on the then (?) obsessive desire to route out “the threat of contradiction”. However, “but not such drastic action as intuitionism demanded” (194). Again, was this a sacrifice of speculation, conjecture and mathematical creativity on the part of intuitionism?
Despite these defects and criticisms, it was still the case that intuitionism “had a great and seemingly beneficial influence” (194). For example, a “number of prominent mathematicians shared in some, or all, of its tenets – for example, H. Poincare and H. Weyl”(194). From what I personally know about Poincare, I'm surprised that he was, or was even influenced by, intuitionism with its emphasis on formalism and constructivism. He, for a start, rejected the tautological or analytical view of mathematics and emphasised the importance of creativity in this discipline.
To finish. Despite intuitionism's radical nature and its rejection of the law of excluded middle (at least for infinite sets), Wilder still says that
“its doctrine of constructivity was found to be adaptable to numerous situations within the framework of conventional mathematical theory” (194).
The Foundations of Mathematics & Completed Infinities
From the outside, the 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. According to Wilder, the modern mathematicians with
“his 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’ (197).
It's quite remarkable that Wilder claims that the modern mathematicians have failed to explain what mathematics is considering the fact that even the layman would have a good go at the job.
The question is:
Why can’t they explain what mathematics is? Why is this task so difficult?
Was it Kurt Gödel’s results that stopped mathematics from “providing a secure foundation” as well as “absolutely rigorous methods”? Is it really the case that the search for foundations - as well as for absolutely rigorous methods - is well and truly over, let alone when Wilder wrote these words in 1968?
From what Wilder says next, it seems as if mathematics not having any foundations, or not being free from all contradictions, may not be such a bad thing. More precisely, he writes 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” (197).
Of course, in science we don't have “exact explanations of natural and social phenomena” and such things aren't even ‘expected’ in science. Is this really the case in mathematics as well? Surely not. Perhaps this conclusion, on Wilder’s part, is simply a result of his materialist, sociological or even Marxist position on the practice and history of mathematics. Surely even these positions accept different standards from maths – indeed, they do. Again, it's no surprise that Wilder says what he says if he believes that “the only reality mathematical concepts have is as cultural elements or artefacts” (197). This position seems to go even further than constructivism; though perhaps not as far as the late Wittgenstein.
More technically, Wilder expresses his constructivist, Marxist or sociological position on mathematics by elaborating on the notion of a completed infinite (198). This sounds like a blatant and direct contradiction. How can any infinite be complete or completed? If it's completed, then surely it's not infinite. What, exactly, does Wilder say on this issue of the completed infinite? –
“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.” (198)
Wilder gives us examples of completed infinities, the infinite decimal and “the totality of natural numbers”; though he doesn’t say what such things actually are or what the phrase ‘completed infinite’ means. The following hints at an explanation; though it doesn't help the non-mathematicians much. He writes:
“Symbolically, it may be considered a second-order symbolism, in that it is not susceptible to complete perception, but is only conceptually perceivable.” (198)
One has a vague idea of what Wilder means by the above. Perhaps it's a kind of ‘direct insight’ (as in Husserl?) or intuition into the nature of completed infinities. It can be conceptually perceived; though not seen – literally or even non-literally.