Preface

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"All Cantor's set theory is built on a sand [...]. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered. [...] Point set topology is a disease from which the human race will soon recover." [Quoted from here.]

And, here is Gauss (arguably the greatest mathematician of any century):

"I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction." [Quoted from here (this links to a PDF).]
 

Kronecker:

"I don’t know what predominates in Cantor's theory -- philosophy or theology, but I am sure that there is no mathematics there." [Quoted from here (this links to a PDF).]

Weyl:

"...classical logic was abstracted from the mathematics of finite sets and their subsets.... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory...." [Quoted from here (this links to a PDF).]
 

"Cantor's set theory as a whole is a pathological incident in the History of Mathematics and a moral crime against future generations (of meta-mathematicians and AST-people)". [Quoted from here.]

[AST = Axiomatic Set Theory.]

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Read, R. (No Date), 'Gödel's Theorem Over-Interpreted: There Is No Such Thing As De Re Self-Reference'.

Wildberger, N. (No Date), 'Set Theory: Should You Believe?' [This links to a PDF. The first few pages of this paper have been reproduced in the Appendix.]

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Logic And Language Website

A friend has just sent me a link to a site that is partly devoted to exposing the serious logical and mathematical flaws in Gödel's Incompleteness Theorem. The site also contains links to on-line papers that argue to the same end.

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Set Theory: Should You Believe?

Professor N J Wildberger

School of Mathematics, University of New South Wales

"I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction." (Gauss)

"I don’t know what predominates in Cantor’s theory -- philosophy or theology, but I am sure that there is no mathematics there." (Kronecker)

"...classical logic was abstracted from the mathematics of finite sets and their subsets.... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory..." (Weyl)

Modern Mathematics As Religion

Modern mathematics doesn't make complete sense. The unfortunate consequences include difficulty in deciding what to teach and how to teach it, many papers that are logically flawed, the challenge of recruiting young people to the subject, and an unfortunate teetering on the brink of irrelevance.

If mathematics made complete sense it would be a lot easier to teach, and a lot easier to learn. Using flawed and ambiguous concepts, hiding confusions and circular reasoning, pulling theorems out of thin air to be justified 'later' (i.e. never) and relying on appeals to authority don't help young people, they make things more difficult for them.

If mathematics made complete sense there would be higher standards of rigour, with fewer but better books and papers published. That might make it easier for ordinary researchers to be confident of a small but meaningful contribution. If mathematics made complete sense then the physicists wouldn't have to thrash around quite so wildly for the right mathematical theories for quantum field theory and string theory. Mathematics that makes complete sense tends to parallel the real world and be highly relevant to it, while mathematics that doesn't make complete sense rarely ever hits the nail right on the head, although it can still be very useful.

So where exactly are the logical problems? The troubles stem from the consistent refusal by the Academy to get serious about the foundational aspects of the subject, and are augmented by the twentieth centuries' whole hearted and largely uncritical embrace of Set Theory.

Most of the problems with the foundational aspects arise from mathematicians' erroneous belief that they properly understand the content of public school and high school mathematics, and that further clarification and codification is largely unnecessary. Most (but not all) of the difficulties of Set Theory arise from the insistence that there exist 'infinite sets', and that it is the job of mathematics to study them and use them.

In perpetuating these notions, modern mathematics takes on many of the aspects of a religion. It has its essential creed -- namely Set Theory, and its unquestioned assumptions, namely that mathematics is based on 'Axioms', in particular the Zermelo-Fraenkel 'Axioms of Set Theory'. It has its anointed priesthood, the logicians, who specialize in studying the foundations of mathematics, a supposedly deep and difficult subject that requires years of devotion to master. Other mathematicians learn to invoke the official mantras when questioned by outsiders, but have only a hazy view about how the elementary aspects of the subject hang together logically.

Training of the young is like that in secret societies -- immersion in the cult involves intensive undergraduate memorization of the standard thoughts before they are properly understood, so that comprehension often follows belief instead of the other (more healthy) way around. A long and often painful graduate school apprenticeship keeps the cadet busy jumping through the many required hoops, discourages critical thought about the foundations of the subject, but then gradually yields to the gentle acceptance and support of the brotherhood. The ever-present demons of inadequacy, failure and banishment are however never far from view, ensuring that most stay on the well-trodden path.

The large international conferences let the fellowship gather together and congratulate themselves on the uniformity and sanity of their world view, though to the rare outsider that sneaks into such events the proceedings no doubt seem characterized by jargon, mutual incomprehensibility and irrelevance to the outside world. The official doctrine is that all views and opinions are valued if they contain truth, and that ultimately only elegance and utility decide what gets studied. The reality is less ennobling -- the usual hierarchical structures reward allegiance, conformity and technical mastery of the doctrines, elevate the interests of the powerful, and discourage dissent.

There is no evil intent or ugly conspiracy here -- the practice is held in place by a mixture of well-meaning effort, inertia and self-interest. We humans have a fondness for believing what those around us do, and a willingness to mold our intellectual constructs to support those hypotheses which justify our habits and make us feel good.

The Problem With Foundations

The reason that mathematics doesn't make complete sense is quite easy to explain when we look at it from the educational side. Mathematicians, like everyone else, begin learning mathematics before kindergarten, with counting and basic shapes. Throughout the public and high school years (K-12) they are exposed to a mishmash of subjects and approaches, which in the better schools or with the better teachers involves learning about numbers, fractions, arithmetic, points, lines, triangles, circles, decimals, percentages, congruences, sets, functions, algebra, polynomials, parabolas, ellipses, hyperbolas, trigonometry, rates of change, probabilities, logarithms, exponentials, quadrilaterals, areas, volumes, vectors and perhaps some calculus. The treatment is non-rigorous, inconsistent and even sloppy. The aim is to get the average student through the material with a few procedures under their belts, not to provide a proper logical framework for those who might have an interest in a scientific or mathematical career.

In the first year of university the student encounters calculus more seriously and some linear algebra, perhaps with some discrete mathematics thrown in. Sometime in their second or third year, a dramatic change happens in the training of aspiring pure mathematicians. They start being introduced to the idea of rigorous thinking and proofs, and gradually become aware that they are not at the peak of intellectual achievement, but just at the foothills of a very onerous climb. Group theory, differential equations, fields, rings, topological spaces, measure theory, operators, complex analysis, special functions, manifolds, Hilbert spaces, posets and lattices -- it all piles up quickly. They learn to think about mathematics less as a jumble of facts to be memorized and algorithms to be mastered, but as a coherent logical structure. Assignment problems increasingly require serious thinking, and soon all but the very best are brain-tired and confused.

Do you suppose the curriculum at this point has time or inclination to return to the material they learnt in public school and high school, and finally organize it properly? When we start to get really picky about logical correctness, doesn't it make sense to go back and ensure that all those subjects that up to now have only been taught in a loose and cavalier fashion get a proper rigorous treatment? Isn't this the appropriate time to finally learn what a number in fact is, why exactly the laws of arithmetic hold, what the correct definitions of a line and a circle are, what we mean by a vector, a function, an area and all the rest? You might think so, but there are two very good reasons why this is nowhere done.

The first reason is that even the professors mostly don't know! They too have gone through a similar indoctrination, and never had to prove that multiplication is associative, for example, or learnt what is the right order of topics in trigonometry. Of course they know how to solve all the problems in elementary school texts, but this is quite different from being able to correct all the logical defects contained there, and give a complete and proper exposition of the material.

The modern mathematician walks around with her head full of the tight logical relationships of the specialized theories she researches, with only a rudimentary understanding of the logical foundations underpinning the entire subject. But the worst part is, she is largely unaware of this inadequacy in her training. She and her colleagues really do believe they profoundly understand elementary mathematics. But a few well-chosen questions reveal that this is not so. Ask them just what a fraction is, or how to properly define an angle, or whether a polynomial is really a function or not, and see what kind of nonuniform rambling emerges! The more elementary the question, the more likely the answer involves a lot of pilosophizing and bluster. The issue of the correct approach to the definition of a fraction is a particularly crucial one to public school education.

Mathematicians like to reassure themselves that foundational questions are resolved by some mumbo-jumbo about 'Axioms' (more on that later) but in reality successful mathematics requires familiarity with a large collection of 'elementary' concepts and underlying linguistic and notational conventions. These are often unwritten, but are part of the training of young people in the subject. For example, an entire essay could be written on the use, implicit and explicit, of ordering and brackets in mathematical statements and equations. Codifying this kind of implicit syntax is a job professional mathematicians are not particularly interested in.

The second reason is that any attempt to lay out elementary mathematics properly would be resisted by both students and educators as not going forward, but backwards. Who wants to spend time worrying about the correct approach to polynomials when Measure theory and the Residue calculus beckon instead? The consequence is that a large amount of elementary mathematics is never properly taught anywhere.

But there are two foundational topics that are introduced in the early undergraduate years: infinite set theory and real numbers. Historically these are very controversial topics, fraught with logical difficulties which embroiled mathematicians for decades. The presentation these days is matter of fact -- 'an infinite set is a collection of mathematical objects which isn't finite' and 'a real number is an equivalence class of Cauchy sequences of rational numbers'.

Or some such nonsense. Set theory as presented to young people simply doesn't make sense, and the resultant approach to real numbers is in fact a joke! You heard it correctly -- and I will try to explain shortly. The point here is that these logically dubious topics are slipped into the curriculum in an off-hand way when students are already overworked and awed by all the other material before them. There is not the time to ruminate and discuss the uncertainties of generations gone by. With a slick enough presentation, the whole thing goes down just like any other of the subjects they are struggling to learn. From then on till their retirement years, mathematicians have a busy schedule ahead of them, ensuring that few get around to critically examining the subject matter of their student days.

Infinite Sets

I think we can agree that (finite) set theory is understandable. There are many examples of (finite) sets, we know how to manipulate them effectively, and the theory is useful and powerful (although not as useful and powerful as it should be, but that’s a different story).

So what about an 'infinite set'? Well, to begin with, you should say precisely what the term means. Okay, if you don't, at least someone should. Putting an adjective in front of a noun does not in itself make a mathematical concept. Cantor declared that an 'infinite set' is a set which is not finite. Surely that is unsatisfactory, as Cantor no doubt suspected himself. It's like declaring that an 'all-seeing Leprechaun' is a Leprechaun which can see everything. Or an 'unstoppable mouse' is a mouse which cannot be stopped. These grammatical constructions do not create concepts, except perhaps in a literary or poetic sense. It is not clear that there are any sets that are not finite, just as it is not clear that there are any Leprechauns which can see everything, or that there are mice that cannot be stopped. Certainly in science there is no reason to suppose that 'infinite sets' exist. Are there an infinite number of quarks or electrons in the universe? If physicists had to hazard a guess, I am confident the majority would say: No. But even if there were an infinite number of electrons, it is unreasonable to suppose that you can get an infinite number of them all together as a single 'data object'.

The dubious nature of Cantor's definition was spectacularly demonstrated by the contradictions in 'infinite set theory' discovered by Russell and others around the turn of the twentieth century. Allowing any old 'infinite set' à la Cantor allows you to consider the 'infinite set' of 'all infinite sets', and this leads to a self-referential contradiction. How about the 'infinite sets' of 'all finite sets', or 'all finite groups', or perhaps 'all topological spaces which are homeomorphic to the sphere'? The paradoxes showed that unless you are very particular about the exact meaning of the concept of 'infinite set', the theory collapses. Russell and Whitehead spent decades trying to formulate a clear and sufficiently comprehensive framework for the subject.

Let me remind you that mathematical theories are not in the habit of collapsing. We do not routinely say, "Did you hear that Pseudo-convex cohomology theory collapsed last week? What a shame! Such nice people too."

So did analysts retreat from Cantor's theory in embarrassment? Only for a few years, till Hilbert rallied the troops with his battle-cry "No one shall expel us from the paradise Cantor has created for us!"  To which Wittgenstein responded "If one person can see it as a paradise for mathematicians, why should not another see it as a joke?"

Do modern texts on set theory bend over backwards to say precisely what is and what is not an infinite set? Check it out for yourself -- I cannot say that I have found much evidence of such an attitude, and I have looked. Do those students learning 'infinite set theory' for the first time wade through The Principia? Of course not, that would be too much work for them and their teachers, and would dull that pleasant sense of superiority they feel from having finally 'understood the infinite'.

The bulwark against such criticisms, we are told, is having the appropriate collection of 'Axioms'! It turns out, completely against the insights and deepest intuitions of the greatest mathematicians over thousands of years, that it all comes down to what you believe. Fortunately what we as good modern mathematicians believe has now been encoded and deeply entrenched in the 'Axioms of Zermelo-Fraenkel'. Although there was quite a bit of squabbling about this in the early decades of the last century, nowadays there are only a few skeptics. We mostly attend the same church, dutifully repeat the same incantations, and insure our students do the same....

The rest of the above paper can be accessed here. [This links to a PDF.]

Finally, here's a video where Professor Wildberger explains why he rejects infinite sets:

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