Posted by Today we’ve got a bit of a treat. I’ve been holding off on this for a while, because I wanted to do it justice. This isn’t the typical wankish crackpottery, but rather a deep and interesting bit of crackpottery. A reader sent me a link to a website of a mathematics professor, N. J. Wildberger, at the University of New South Wales, which contains a long, elegant screed against the evils of set theory, titled “Set Theory: Should You Believe?”
It’s an interesting article – and I don’t mean that sarcastically. It’s over the top, to the point of extreme silliness in places, but the basic idea of it is not entirely unreasonable. Back at the beginnings of set theory, when Cantor was first establishing his ideas, there was a lot of opposition to it. In
my opinion, the most interesting and credible critics of set theory were the constructivists. Put briefly, constructivists believe that all valid math is based on constructing things. If something exists, you
can show a concrete instance of it. If you can describe it, but you can’t build it, then
it’s just an artifact of logic.
Some of that opposition continues to this day, and it’s not just the domain of nuts. There are
serious mathematicians who’ve questioned the meaningfulness of some of the artifacts of modern
set-theory based mathematics. Just to give one prominent example, Greg Chaitin has given lectures in which he discusses the idea that the real numbers aren’t real: they’re just logical artifacts which can never actually occur in the real world, and rationals are the only real real numbers. (I don’t think that Greg really believes that – just that he thinks it’s an interesting idea to consider. He’s far
too entranced with set theory. But he clearly considers it valid enough to be worth thinking about
and talking about.)
Professor Wildberger is very much in the constructivist school of thought. That doesn’t mean
that he’s right, and in fact, I think he makes some dreadful errors in his article. But it’s
worth the look. His basic point is a good one: that the way we teach math really stinks. His
explanation of why it stinks is, I think, pretty far off base. Here’s the gist of
his fundamental thesis:
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.
The problem with what comes later is apparent from just this much. Professor Wildberger believes
that we’re starting from the wrong foundations for math – and that the problems of mathematical
education come not just from poor teaching, but from the fact that the teachers are building on
a flawed foundation.
I think he’s pretty off-base with this. I remember my high school math teachers. Most of them didn’t know diddly-squat about set theory. They didn’t know or care about the axioms of set theory, or the idea of infinite sets. They didn’t know how real numbers are formally defined. It didn’t matter to them,
or to the material they were teaching. That fact is a big part of Wildberger’s point: that teachers at that level leave out, or don’t understand foundations. But that’s not why the lousy teachers were lousy teachers. The great math teachers that I had also didn’t know any of the deep foundations. They didn’t need to.
Teaching a subject like math doesn’t really start from foundations. In fact, teaching any really complex subject doesn’t start from foundations. Foundations are left until after you understand a bit. My fellow SBer Chad Orzel has said that part of teaching introductory subjects is lying to your students. You start by telling them about approximations to the truth, which you know are wrong in the details. In physics, you start off with basic newtonian physics, immutable masses. You don’t start off with relativity and quantum phenomenon – the students simply don’t yet have the intellectual tools to deal with that. You start off with the simple approximation, and then teach from there; then once the students have some grasp on the basics, you move on to start introducing the subtleties. And even there, you continue with your white lies. Again in physics, you move on to teach relativity – but you don’t start with the problems of the incompatibility of quantum physics with relativity; you start by just teach relativity. Once the students grasp that, then you can make the next step towards the foundation, and talk about the interactions between relativity and quantum phenomena.
We do the same thing with language. We don’t start teaching a student a language by presenting them with the foundational grammar of the language. We start by lying. We tell them that “a sentence looks like this”, and teaching them to use that basic structure with some basic vocabulary. Then we move on to show
them how the structure can vary, by introducing new clauses, tenses, senses, voices, etc.
So I disagree with him right from the beginning. But I admit that his idea has some validity; I don’t
think he’s right, but I think that the argument can be made by a reasonable person, and that’s entirely
possible that he’s right and I’m wrong.
What happens next is where he starts to go off the rails. He’s clearly a strict constructivist, and
as a result, has a deep, visceral dislike of set theory, and all things related to it. The problem is that
he takes his hate of set theory to such a point that he basically argues for discarding logic in math.
Skipping past his diatribe about how most math professors don’t understand fundamental math (which I find
to be a rather shallow bit of hand-waving snobbery), we get to the meat of his argument about set
theory:
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.
It all comes back to those darned nasty infinite sets!
Most of math embroiled mathematicians with logical difficulties. The diatribe that I skipped over
included a rant about polynomials: finding roots of polynomials caused problems in math – from the
arithmetic difficulties, to logical and philosophical difficulties – and those arguments raged for
centuries. Professor Wildberger clearly doesn’t have any problem with the idea of
polynomial roots, nor, I suspect does he have problems with complex numbers, which resolved
that problem – but which met with profound opposition from the precursors of the constructivists: “If your imaginary numbers are real, show me 3+2i apples!”
This really is the heart of his argument. As we’ll see, his fundamental problem is the idea of
the infinite set. He does not believe that infinite sets exist in any meaningful way – because
you can’t build one. And since set theory is so dependent on the idea of infinite sets, it must
be completely bogus. And since modern math is taught using set theory as its fundamental
basis, that means that everything that math students are taught it wrong: mathematicians
do not understand math, because what they think they understand is based on a fundamental
falsehood.
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’.
See what I mean? The infinite sets are the problem. You can’t create an infinite set – they only exist
as a result of logic. They’re not real. And since they’re not real, you can’t trust anything
that you can prove from them. He goes on to try to continue to build his case, and cast more doubt
on the concept:
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.”
This is where he starts to get sloppy. Yes, if you’re not very particular about your definition of
sets, and in particular, infinite sets, you can run into trouble. But as Gödel showed, you don’t need
sets to get into trouble. Any time in math, if you’re not very careful, you can get into trouble. Pick one bogus axiom, and everything you’ve built from it collapses like a house of cards. Set theory isn’t
unique in this way. If you’re not careful when you pick your axioms of geometry, you might get the result
that there’s no such thing as a parallel line. If you’re not careful when you pick your axioms describing
numbers, and you might end up with x+y != y+x.
But Professor Wildberger wants to focus on set theory alone, and claim that this problem
is uniquely one of infinite sets. He can’t show that there’s anything wrong with it, other
that repeating the same old constructivist arguments: “you can’t create a collection of an infinite number of things, so infinite sets don’t exist”. But that’s an old, mostly discredited argument. So
the professor needs to try to build it up, to show that it’s been improperly discredited. So, sadly, he pulls out an old creationist trick: the appeal to authority, and almost a
quote mine: “Look – Russell and Whitehead showed that it’s a mess!” Now, R&W didn’t
conclude that set theory was a hopeless mess. They just accepted the sad fact that
Gödel had showed that their attempt to create the complete, perfect mathematics was
doomed to failure. That doesn’t mean that there’s a problem with set theory; just that there’s
a limit to formal reasoning. That’s not a surprise to any modern mathematician (or even just
a math geek like me). We’ve grown up with Gödel’s limits.
showed “
But Professor Wildberger thinks it’s far more important than that:
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’.
Again, he quotemines, to try to create an impression that great minds agree with him.
And then, it gets sad. Set theory, like any mathematical theory, is built on a set of axioms.
Since the axioms work, and allow proofs of things he doesn’t like, Wildberger concludes that the
axiomatic approach is totally, fatally flawed, and goes on a rant about the evils of axiom. Axioms aren’t math, they’re religion! We need to get rid of those stinking rotten axioms that are causing all of
the grief!
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.
This leads into a presentation of the axioms of set theory, presented in a way that tries to make
them look as silly as possible – so that he can then follow it with a critique of how silly
they are. The problem is, it’s all silliness.
All completely clear? This sorry list of assertions is, according to the majority of mathematicians, the proper foundation for set theory and modern mathematics! Incredible!
See, we’re supposed to be very impressed that the axioms, stated blankly, without explanation,
do not form an obvious basis for math. They’re not easy to really understand, so they must
be ridiculous. And they’ve got the horrible “Axiom of infinity”!
The `Axioms’ are first of all unintelligible unless you are already a trained mathematician. Perhaps you disagree? Then I suggest an experiment—inflict this list on a random sample of educated non-mathematicians and see if they buy—or even understand—any of it. However even to a mathematician it should be obvious that these statements are awash with difficulties. What is a property? What is a parameter? What is a function? What is a family of sets? Where is the explanation of what all the symbols mean, if indeed they have any meaning? How many further assumptions are hidden behind the syntax and logical conventions assumed by these postulates?
Set theory doesn’t exist on its own. Wildberger is trying to suggest that it does. But it really doesn’t. Set theory works in conjuction with first order predicate logic. You can build all
of math using those two – but take away sets, and the logic doesn’t have a model; take away the logic,
and you can’t reason about sets. What Wildberger is doing is pretending that the places where set theory
depends on logic are faults in set theory. The axioms are statements of predicate logic (with the
exception of the axiom of subsets, which is actually an axiom schema – a second-order statement- parametric on a predicate.) The rest of it is all built on logic. But Wildberger pretends that it
isn’t – that the fact that “property”, “function”, “parameter” are all ill-defined terms, because
they aren’t defined in set theory. But they are defined in predicate logic – the necessary companion of set theory.
And then we get back to his favorite hobby-horse:
And Axiom 6: There is an infinite set!? How in heavens did this one sneak in here? One of the whole points of Russell’s critique is that one must be extremely careful about what the words `infinite set’ denote. One might as well declare that: There is an all-seeing Leprechaun! or There is an unstoppable mouse!
Oh, the horror! The infinite set! We can’t have that! And again, we see the pseudo-quote mine of
Russell – the implication that Russell said that you can’t have infinite sets. But Russell didn’t
say that. And Wildberger uses a form of the axiom of infinity that I’ve never seen before. The way that Wildberger presents it is “There exists an infinite set”. The way that it’s normally presented is
“∃N: ∅∈N ∧ (∀x : x∈N ⇒ x∪{x}∈N)”. In other words, the
usual presentation of the axiom of infinity doesn’t just say that there’s an infinite set; it defines
a specific infinite set with specific properties according to the other axioms. It’s downright
constructivist, in fact.
There’s a reason that Wildberger chose to present it in that dreadful way: because if he presented it in its standard form, then his rant following it wouldn’t have worked. He’s trying to say that the axioms are invalid, because they don’t specify what an “infinite set” really means, and that since Russell showed that unless you’re very careful of how you define the concept, that infinite sets cause trouble. But the real axioms do carefully define the infinite set in a way that works – and that shows us how
to create the set of natural numbers.
Wildberger clearly understands this problem – which leads into the next section of his article, in which he takes on the entire concept of axioms in mathematics.
Occasionally logicians inquire as to whether the current `Axioms’ need to be changed further, or augmented. The more fundamental question—whether mathematics requires any Axioms —is not up for discussion. That would be like trying to get the high priests on the island of Okineyab to consider not whether the Divine Ompah’s Holy Phoenix has twelve or thirteen colours in her tail (a fascinating question on which entire tomes have been written), but rather whether the Divine Ompah exists at all. Ask that question, and icy stares are what you have to expect, then it’s off to the dungeons, mate, for a bit of retraining.
Mathematics does not require `Axioms’. The job of a pure mathematician is not to build some elaborate castle in the sky, and to proclaim that it stands up on the strength of some arbitrarily chosen assumptions. The job is to investigate the mathematical reality of the world in which we live. For this, no assumptions are necessary. Careful observation is necessary, clear definitions are necessary, and correct use of language and logic are necessary. But at no point does one need to start invoking the existence of objects or procedures that we cannot see, specify, or implement.
This is where the constructivism because bleedingly obvious, and where we see how he misuses the
word “axiom” to build his case. Logic is fine to him – so long as it is never used to describe something that we cannot create a concrete physical instantiation of.
He wants to replace axioms by definitions, and as near as I can tell, the difference between
axioms and definitions is that definitions are, according to him, absolutely concrete. Things
like the axiom of infinity, which describe something in terms of logic which you can’t build
out of matter, are unacceptable: they’re axioms, not definitions. In his own words:
The difficulty with the current reliance on `Axioms’ arises from a grammatical confusion, along with the perceived need to have some (any) way to continue certain ambiguous practices that analysts historically have liked to make. People use the term `Axiom’ when often they really mean definition. Thus the `axioms’ of group theory are in fact just definitions. We say exactly what we mean by a group, that’s all. There are no assumptions anywhere. At no point do we or should we say, `Now that we have defined an abstract group, let’s assume they exist’. Either we can demonstrate they exist by constructing some, or the theory becomes vacuous. Similarly there is no need for `Axioms of Field Theory’, or `Axioms of Set theory’, or `Axioms’ for any other branch of mathematics—or for mathematics itself!
I disagree with his characterization. In fact, I think this is where he stops being a reasonable
constructivist, and starts to verge on crackpottery. There are no axioms of field theory? No axioms of
group theory? How does that work? I really would like to know just how Professor Wildberger does group
theory without working with the set of real numbers – since that’s clearly infinite, and many of the
theorems of group theory require the axiom of choice!
This is nothing but linguistic game-playing: exactly what he accuses his opponents of doing. He’s playing with definitions and terms in a vague and ambiguous way in order to create a false distinction
between the valid logical statements that he likes (“definitions”), and the valid logical statements that he doesn’t like (axioms). The only real difference between the two is his personal opinion of them. Clearly, as someone who does work in group and field theory, he’s using axioms that work on infinite sets, like the set of numbers. But he just declares them to be “definitions” of concrete things, and
since they’re not dirty, nasty, evil “axioms”, they’re OK.
At least he gets honest about admitting why he does this:
We have politely swallowed the standard gobble dee gook of modern set theory from our student days—around the same time that we agreed that there most certainly are a whole host of `uncomputable real numbers’, even if you or I will never get to meet one, and yes, there no doubt is a non-measurable function, despite the fact that no one can tell us what it is, and yes, there surely are non-separable Hilbert spaces, only we can’t specify them all that well, and it surely is possible to dissect a solid unit ball into five pieces, and rearrange them to form a solid ball of radius two.
And yes, all right, the Continuum hypothesis doesn’t really need to be true or false, but is allowed to hover in some no-man’s land, falling one way or the other depending on what you believe. Cohen’s proof of the independence of the Continuum hypothesis from the `Axioms’ should have been the long overdue wake-up call. In ordinary mathematics, statements are either true, false, or they don’t make sense. If you have an elaborate theory of `hierarchies upon hierarchies of infinite sets’, in which you cannot even in principle decide whether there is anything between the first and second `infinity’ on your list, then it’s time to admit that you are no longer doing mathematics.
Yes, infinite set theory creates some strange results. The Banach-Tarski paradox (the thing about the balls that he mentions is a B-T variant) is definitely strange and uncomfortable. That I agree with. B-T bothers me. It’s a nasty beast, which I find it damned hard to wrap my head around. But I don’t reject things just because they’re hard.
Where he gets silly is the “uncomputable numbers” bit. That’s where he’s starting to touch on my
specialty. And sorry, prof, but the uncomputable numbers are absolutely real. The square root of two is
(weakly) non-computable. (By weakly non-computable, I mean that we can compute an approximation of it to any degree of accuracy that we want, but we can never computationally produce the exact number.) ω is well-defined, and thoroughly (strongly) non-computable (although you
can compute the first couple of dozen digits of it). They’re real, they’re a fact of life in
computing. Our comfort or discomfort means nothing. Quantum physics makes me damned uncomfortable too, but I can’t argue that it must be wrong because it makes me queasy.
The part about the Continuum hypothesis is, to me, even sillier. It’s just more of the extreme
constructivist viewpoint, but carried to an extreme that I thought died with the failure of Russell and
Whitehead. Before Gödel and incompleteness, there was a dream that math could be solved,
that we could create a perfect mathematics in which every possible statement was provably true or false,
and where there were no contradictions. Wildberger seems to still be a part of that school, to believe in
that dream. But it doesn’t work. Math has limits: it can’t be both consistent and complete. There are, inevitably, statements that aren’t provable. The Continuum hypothesis is one of those strange places. Personally, I don’t even see it as a problem. Euclid defined the axioms of geometry. He added the
parallel axiom, even though he didn’t think it should be necessary. It turns out that without it,
you can get different results. You can’t get to standard Euclidian geometry without defining the answer to “Given a point p and a line l, How many lines parallel to l pass through p?” as being 1. There’s no reason that it has to be. You get different geometries if you pick different values. But that’s fine! The different geometries are interesting, and have interesting applications. With
respect to the continuum, it’s the same story: there’s no reason that it has to be either true or false. You get a different structure to your universe of sets depending on which value you choose.
Anyway, he continues in this vein for quite a while longer. He also takes some time to attack the idea
of the natural numbers as an infinite set, and the concept of the real numbers – but they’re just more of the same kind of thing that we’ve already seen in the above quoted sections: extreme constructivism,
combined with an intense revulsion for the concept of infinite sets.
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