Why math-fans really love set theory

Turns out, math fans dig set theory for almost exactly the same reason that some Christian fundamentalists absolutely hate it — all that messy uncertainty, which is either an affront to the idea of intelligent design or really, really sexy and fascinating, depending on your outlook.

At Nautilus, which is currently hosting an entire issue on topic of uncertainty, math professor Ayalur Krishnan writes about an idea in set theory that he calls "The Deepest Uncertainty". This is the Continuum Hypothesis — an idea that, paradoxically, can be proven to be unprovable and proven to be something you can't disprove. (And, with that, I've just typed the word "proven" so many times that it has lost all meaning in my brain.)

The uncertainty surrounding the Continuum Hypothesis is unique and important because it is nested deep within the structure of mathematics itself. This raises profound issues concerning the philosophy of science and the axiomatic method. Mathematics has been shown to be “unreasonably effective” in describing the universe. So it is natural to wonder whether the uncertainties inherent to mathematics translate into inherent uncertainties about the way the universe functions. Is there a fundamental capriciousness to the basic laws of the universe? Is it possible that there are different universes where mathematical facts are rendered differently? Until the Continuum Hypothesis is resolved, one might be tempted to conclude that there are.

Read the full story, which explains what set theory and the Continuum Hypothesis actually are. I could that here, but then this link would end up being as long as the story it's trying to link you to. Ahhhh, set theory.

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1. Shay Guy says:

You can’t prove that any given set is pink. You can’t prove that it isn’t pink, either. This is because color is simply not part of set theory, unless you make up a special definition for it.

I see the continuum hypothesis as being the same way. It’s just more counterintuitive, because infinity is counterintuitive. (Disclaimer: I don’t actually know how either proof on the provability of the CH works.)

1. Skyler Nelson says:

In part, what’s counterintuitive is that we expect there to be an answer, but there’s a sense in which there just really isn’t.  You might ask if lots of things are pink – electrons, ideas, sets.  In general it may not even make much sense to ask, but you can still classify those things as “not pink” even if there’s no other color that they are.  Maybe there’s an answer there, and maybe there’s not but we don’t really expect one to be given to us by the axioms – color isn’t part of our normal set theory talk, so we have to bring in outside predicates to deal with it.

But existence is damn sure part of our normal set theory talk, and insofar as we expect our sets to be determined by the axioms, we expect there to really be an answer as to which sets exist.

It’s funny – the way we talk about mathematics makes it easy to slip into a kind of Platonism.  We imagine our theories and their axioms as simply describing a domain of pre-existent objects and allowing us to prove things about them.  We like to think of them as having this independent existence and that all the facts about them are set in stone in some way or another.  But there’s a sense in which what we’ve learned is that you can never quite set everything in stone; some facts are just never determined.  It’s strange to find out that there’s just no fact of the matter when it comes to a system you’re working with – you thought set theory gave you a relatively concrete domain of objects, but your set theory hasn’t decided if certain perfectly reasonable sets exist or not.

For those that take the sort of Platonistic talk of mathematics seriously, there might really be an answer about whether CH holds, but this just makes our axiomatization look too weak to capture important features of the theory of sets – in fact, it makes *every* axiomatization look weak.  It’s getting used to this other viewpoint where things are only definite up to proof that takes a bit of effort for a lot of people to wrap their heads around.

2. Skyler Nelson says:

Set theory is soooo last century.  Real math nerds are into category theory now.  Scratch that, we’re into homotopy type theory/univalent foundations!

In all seriousness, there’s lots of stuff going on in set theory, but more than being an interesting domain in its own right, it’s mostly just a necessary tool for doing a lot of higher math.  It’s a kind of basic language out of which mathematical objects can be constructed.

Beyond this role, set theory was intended to be a kind of unifying framework that aided interdisciplinary work – since the 60s, category theory (and now higher category theory) has sort of slowly filling that role.  There are certain things that are much more natural from a category theoretic perspective.  The best one sentence description of the difference that I’ve heard is this: when it comes to mathematical structures, set theory is good for answering the question “How is it made?” while category theory is good for answering the question “How does it work?”

Some mathematicians find set theory a bit annoying as a foundational theory in certain regards. There’s ‘too much structure’ in a certain sense that can allow stupid distinctions and hide important ones.  There have been some recent and very interesting attempts to replace it. The most promising (from my perspective) is the whole univalent foundations program.  The idea is to get rid of all the excess structure and have mathematical objects and domains be only describable in terms of distinctions that actually matter (rather than the too-fine-grained distinctions that come up in set theory).  It’s all rather fascinating, but I haven’t been totally convinced they’ve created a workable meta-language yet.  Time will tell.

1. MonkeyBoy says:

Yes, Category Theory and HoTT can in some sense be regarded as alternatives to set theory or a “more natural” framework.

But neither brings any resolution  to the Continuum Hypothesis, so why did you bring them up?

1. Skyler Nelson says:

The title of the article is “Why math-fans really love set theory,” and the CH is used as an example to answer the implicit question.  I bring it up to (gently) dispute the notion that math fans really love set theory – in fact, the trend in the community is away from “we love set theory” and towards “we find set theory cumbersome in certain important ways”.

1. Boundegar says:

As a math fan, I love set theory. The set is therefore not empty. QED.

1. Skyler Nelson says:

I made no claim to the effect that there were no math fans who love set theory – just that the title implied a kind of generalization that folks who care for precision (e.g. math fans) ought to be uncomfortable with.  An analogous situation would be an article titled “Why geeks love Apple products” – surely some do, but it’s worth pointing out that this is not a universal opinion and in fact the present trend is in the opposite direction.

Second, I should mention that I intended no real criticism with my comments.  I mostly took the thread as a chance to give a little bit of background information about foundational stuff that most people who are just getting into higher math don’t usually hear too much about.  I knew about set theory for a pretty long time before I learned anything about category theory, and the latter is what really got me into higher math.  Exposing future math geeks to interesting topics is worth going a little off topic.

3. MonkeyBoy says:

One might think that constructive/intuitive math might resolve the Continuum problem.  (in such maths “P or not P” is not true – you can’t prove P by just making it up and then by proving “not P” to be false conclude that P must be true)

However such maths don’t seem to be a magic bullet.

4. Gilbert Wham says:

I’m not a Math fan. Mathematics however…

1. Felton / Moderator says:

Mathematics fanatics?

5. Gendun says:

No hyphen necessary for “math fans”.

6. Thomas Andrews says:

Nothing about the Continuum Hypothesis should affect religious people. Cantor’s idea of different sizes of infinity upset religious people at first, but that was due to deep confusion about what he meant. The fact that all sufficiently complex axiom systems are incomplete can actually be a comfort to religious people – if God is in the unknowable, then this shows there are always unknowables.
Is not surprising, really, that axiom systems are incomplete. People for ages tried to prove Euclid’s fifth postulate from the other four, since it was seen as complex, but it was shown in the 18th or 19th century that the fifth postulate was undecidable from the other four. (Euclid’s fifth, also, was about infinity, too, btw – what does it mean that two lines never intersect, after all?)

1. BillStewart2012 says:

Nothing about any kind of math bothers sensible religious people, but it doesn’t take much to upset some of the crankier Bob Jones types.

But I don’t think their objection is based on anything that deep.  I went to elementary school in the 1960s, so I got to learn arithmetic the old-fashioned way and then learn it over again through New Math, including set theory, and my guess is that somebody’s kid brought home their New Math homework and mom&dad didn’t have a clue what it meant, and they decided that New Math in general and Set Theory in particular teaches rebellious children to challenge their parents’ authority by thinking they know things their parents don’t*.

It’s possible that they also blame Bertrand Russell for some of it, or would if they could understand him.  And set theory gets into paradoxes about whether the barber is a member of the set of people that don’t shave themselves or not, and paradox bothers people like that.

(*My parents didn’t have that kind of problem; my mom borrowed my 7th grade math book to prep for the GREs when she went back to college, since set theory hadn’t been something she’d learned in high school or as an undergrad French major.)

7. jon_anon says:

The article is a bit shoddy. This isn’t quite right: “It remains possible
that new, as yet unknown, axioms will show the Hypothesis to be true or false.” (It also contradicts the statement that CH has been shown to be undeterminable from the usual bases of Set Theory.) If we add axioms to any of the foundations, we will *create* a mathematics in which CH is true, or is false–we won’t be showing the hypothesis to be true or false. Essentially, adding different axioms could split mathematics into one field where CH is true and another where it is false. This says nothing about the real world or the naturalness of the conjecture: what Godel and Erdos proved was that neither the assertion or the negation of CH can be shown to be inconsistent with current foundational axioms (and hence, neither is inconsistent with anything that follows from those axioms.)

1. steve849 says:

Interesting comment. Maybe you should write a guest piece on CH, for non-math-experts such as me.

8. Lloyd Cogliandro says:

“Is there a fundamental capriciousness to the basic laws of the universe?”
Hail Eris! All hail Discordia!

9. GawainLavers says:

Correspondences can be used to compare the sizes of much larger collections than six goats—including infinite collections. The rule is that, if a correspondence exists between two collections, then they have the same size. If not, then one must be bigger. For example, the collection of all natural numbers {1,2,3,4,…} contains the collection of all multiples of five {5,10,15,20,…}. At first glance, this seems to indicate that the collection of natural numbers is larger than the collection of multiples of five. But in fact they are equal in size: every natural number can be paired uniquely with a multiple of five such that no number in either collection remains unpaired. One such correspondence would involve the number 1 pairing with 5, 2 with 10, and so on.

I’m sure I’d have gotten this question out of my system if I’d taken a few more math courses, but I have a problem completely getting my head around this.

I get if we pair 1 with 5, 2 with 10, etc. that we never run out of numbers in category A (goats) to tie to category B (trees).  But what if I choose another kind of pairing?  What if I choose to pair 5 with 5, 10 with 10, etc.  I also tie a goat to every tree, but now for every tree I also have 4 extra goats that got passed over.  It seems like the equivalence only works for an arbitrary method of correspondence.

1. Boundegar says:

Yes, it’s confusing. The problem comes from thinking of “infinity” as if it was a really big number. Then you end up with ideas like “infinity plus one,” and everything gets discombobulated. Infinity is not a number, and your pluses and timses don’t work on it.

You think that’s bad? Try mapping the real numbers into the points on a plane.

10. Ito Kagehisa says:

Chesterton said “Poets do not go mad; but chess-players do.
Mathematicians go mad, and cashiers; but creative artists very seldom. I
am not, as will be seen, in any sense attacking logic: I only say that
this danger does lie in logic, not in imagination.”  David Foster Wallace replied that stories of mad, suicidal mathematicians are overemphasized, especially in regards to Cantor and the other set theorists – and then, after writing a truly brilliant book on the subject, went on to kill himself.

1. BillStewart2012 says:

Chesterton must not have paid attention to van Gogh.

Or Kurt Cobain and the rest of the 27 Club. Of course, he died before any of them became famous.

11. miasm says:

I’m so pervasive that I’m still in the set of everything minus me.

1. Ito Kagehisa says:

You must be my barber!

1. miasm says:

Because the thought experiment is so ludicrous my suggested solution is actually for the barber to shave himself accidentally.
If you can remove any agency on his part the logical structure can continue to eat itself; because, lets face it, the proposition is fucking nuts.
Ah yes, what *does* it mean to have shaved ones self?

12. Joseph Francis says:

Seems to put a monkey in those ‘Science Delusion’ people’s wrenches.

13. Preston Sturges says:

Are they still teaching set theory to 3rd graders to prepare them for graduate school?

14. “Either an affront to the idea of intelligent design or really, really sexy and fascinating”

Or both!

15. I understand hating math because it’s hard but not for religious reasons. Even God thinks that’s dumb.

My very favourite thing about math is that in the UK they say “maths” but here in the US we just say “math”. I like “maths” better. Also, shrimps.

1. nemryn says:

Conversely, it’s ‘sports’ in the US but ‘sport’ in the UK. I’m not sure what the significance of this is.

16. anansi133 says:

There’s always been something about Godel’s theorem that scratches a deep itch in my soul. Likewise it’s satisfying to find he worked on the continuum hypothesis. Far from having my spirituality be challenged by this stuff, this only seems to confirm it.

My hunch is that if we could look directly into our own blind spot and *see* something, this is the kind of work that would get us there.