Against Set Theory (2005) [pdf] | Hacker News Reader

Against Set Theory (2005) [pdf]

ontology.buffalo.edu

55 pointsdanielam49 comments

A mathematician's (or, at least, this mathematician's) instinctive response to a paper titled "Against set theory" is to think that it must be the work of a crank (not to disparage the work of reputable mathematicians exploring alternative foundations—but I think that most of them know that, to earn their ideas a receptive foundation, they had better focus at least on what they are for rather than what they are against), but it should probably be noted that this is against set theory from a philosopher's point of view, not a mathematician's. (With which I can't quibble, though neither can I agree; I am no philosopher, and the fact that mathematics can be used fruitfully in philosophy doesn't mean that it always should.) Indeed, at a skim, the complaint seems to be much more about the mathematisation of philosophy broadly speaking, rather than about the encroachment of set theory in particular.

(I also take issue with the claim on p. 3 that Cauchy was doing only unconscious set theory. It is true that he came before what we might call Cantor's formalisation of the subject, but I think he probably thought in something much closer to a modern "naïvely set theoretic" way about mathematics than almost all o his predecessors.)

Introducing mathematical notation into philosophy gives, not the firm foundation, but some firm foundation, upon which to argue. Otherwise a whole lot of shite is spewed and no progress is made.

In its are sentences of long words that serve to emote, cajole and frankly baffle rather than enlighten. There's some of it here but it's not the worst. What I can't accept is stuff like

" has been to persuade many philosophers that the rich panoply of entities the world throws at us can be reduced to individuals and sets of various sorts, for example sets as properties, sets of ordered tuples as relations, sets of possible worlds as propositions, and so on and so forth. It is hard to know where to start in revealing the scope of the damage caused to ontology by the thoughtless or supposedly scientifically economic reduction of various entities to sets"

To model something you need to simplify it. What's he suggesting instead?

Other oddities " Richmond Thomason notes that Montague saw grammar as a branch of mathematics and not (as in Chomsky) of psychology". Pretty sure Chomsky's hierarchy of grammars, types 3 to 0, are considered by him as mathematical, not psychological.

Not my area but I'm not convinced it's worth digging into this paper.

(edit)

Further down we get to the bullshit. The emotive crap: "tiresome to continue citing further absurdities in philosophy resulting from the over-zealous application" so instead we replace defined set theoretic terms with english, delights such as, replacing <x,y> with x followed by y; x and y in that order. Ha ha ha now explain what logical implication means without being formal. (edit: and then show me some symbol manipulation using it, without the symbols) Not happening.

I do understand the terminology is intimidating, and there's much to be said for annotating the set-theoretic with plain english as he suggests but to replace it... that would be a massive step backwards.

> To model something you need to simplify it. What's he suggesting instead?

But is set theory "simple"? The author mentions several ways in which it is the opposite of simple.

Also - if you need to model _something_, that doesn't mean you should try to model _everything_ with the same kind of model.

Could you point to those ways? I guess I missed them.

> Other oddities " Richmond Thomason notes that Montague saw grammar as a branch of mathematics and not (as in Chomsky) of psychology". Pretty sure Chomsky's hierarchy of grammars, types 3 to 0, are considered by him as mathematical, not psychological.

I would suggest looking up Chomsky and mentalism. This is a core feature of his theory. Early in his career Chomsky did have interesting things to say about formal language theory and you're right that Chomsky would not consider regular languages or the grammar of Python, for instance, to be psychological but this isn't the grammar that the author is concerned with here.

Fortunately, this is about the (mis)use of set theory in philosophy and not mathematics. Why set theory, though? Since it is being gradually replaced by category theory as a more modern, practically better, and a more powerful foundation of mathematics, I expected that philosophers would be more interested in (mis)using category theory these days.

My hope is type theory will be the future adopted foundational system as it allows easier computer-checked proofs and a marriage of mathematics and computation. Category theory can then be formalized within type theory.

Intuitionistic logic and intuitionistic type theory follow essentially the same set of rules, so how does type theory allow for a "marriage of mathematics and computation" any more than logic, which everyone already uses?

Because conventional mathematics, whether based on set theory or some categorical alternative, has (usually classical) logic as a background theory. In type theory, that's all you need. It is a theory of computation that has a "built-in" logic. And there's more than just intuitionism. There are type theories that model non-constructive/classical logic and are yet still computable programming languages. So we can write computable code and prove properties about that code in the same theory.

> There are type theories that model classical logic and are yet still computable programming languages.

Thanks, but this seems absurd to me. Please elaborate!

I'm not an expert, so please take the following with the grain of salt.

One of the examples of computation formalisms providing the classical logic at the type level is the Parigot's [λμ-calculus](https://en.wikipedia.org/wiki/Lambda-mu_calculus). λμ-calculus adds μ-abstraction to the classical λ-calculus.

μ-abstraction resembles the notion of continuation. With that addition Peirce's law of classical logic becomes deductible without any modifications (e.g. double-negation translation). The program that proves Peirce's law is just call/cc function.

This topic seems to be an ongoing research, so you may find lots of articles in public internet space. Many interesting introductory articles are among the advanced materials of [this course](https://www.cs.ru.nl/~freek/courses/tt-2011/), including the original Parigot's paper.

There are a number of different type theory models for classical logic. The first to my knowledge was a model that uses the call-with-current-continuation (call/cc) control flow operator. While this does model classical logic, some say it is not particularly useful as a practical programming construct, which is perhaps why it does not seem to be adopted by any major proof assistant. More recently I've seen a few papers showing that type theoretic models of concurrency can also model classical logic, e.g. see "Classical Proofs as Parallel Programs" < https://arxiv.org/abs/1809.03094 >. The original idea that the Curry-Howard correspondence only applies to intuitionist logic is wrong.

Ever heard of Ludwig Wittgenstein, the guy who popularized truth tables? Check out his little known work "on the foundations of mathematics." beautiful stuff.

-- an analytic philosopher

Yes, I believe undergrads(!) cover this in Honors Advanced Truth Tables at UChicago.

Well,

I don't think it really matters whether he's talking set theory or category theory because the way he talks about set theory implies to me he'd have similar problems with category theory.

For example, he describes the result of Godel and Kohen, who together showed the continuum hypothesis is independent of ZFC, as being a failure/problem of ZFC. It's not. That you have significant theories that are independent of a given axiom system is a product of Godel's theorem. Modern mathematics operates with the assumption that most theories are true, false or unprovable.

But the situation also isn't specific to set theory. Any formal system is going to be subject to this "problem"

The thing is this state of affairs isn't appealing to the intuition. Godel himself was unhappy with his result. But it is basically inevitable if you get to the level of rigor of a formal system.

>But the situation also isn't specific to set theory. Any formal system is going to be subject to this "problem"

I think that part of the author's point can be read generously as, "no formal system can satisfy philosophy's pseudo-religious search for truth evident in itself."

Well, that's a problem for the pseudo-religious search, not for mathematics.

In their attempt to remain relevant, (amateur) philosophers will take anything and run with it, sometimes totally incorrectly.

There's a special place in face-palm hell for philosophers who've argued that Godel's Incompleteness Theorems place a fundamental limit on human understanding.

> Since it is being gradually replaced by category theory as a more modern, practically better, and a more powerful foundation of mathematics

i don't see this. set theory is very practical and is a good thing to use to build things up in a granular manner. category theory is useful to connect and relate things. they're at two different ends of the spectrum. category theory isn't "replacing" anything.

The truth is, at least since the early 1960s category theory has been a much, much more powerful tool in many branches of modern mathematical research than set theory. (By the way, it can be an enlightening exercise, to try and restate the simple notions about sets purely in terms of maps - notions like 'B is a subset of A', 'C is a union of A and B', 'map f is injective/surjective', 'C is a Cartesian product of A and B', etc. Highly recommended!) Category theory is very deep and easily subsumes much, if not all, of set theory. The difference in power is staggering. Sets in general are now considered just one of many categories and play a subordinate role. I agree that set theory is more useful when studying math at an elementary level, but the sooner one starts getting used to categorical ways of reasoning, the better one's chances to make progress in understanding modern mathematics.

you're missing the argument i made. i didn't say category theory isn't useful or powerful. i said it isn't replacing set theory in practice.

you can make it through the lion's share of a ph.d. in math without using category theory, but you'll use set theory ubiquitously up until you start doing categorical things. of course category theory is a powerful relator. but it is awkward for when simple sets do just fine. loring tu's manifolds book is a good example of this. set theory is used throughout with categorical concepts sprinkled about to show there's power of relation there.

and that's what i said. set theory is helpful as a brick. category theory is helpful as an architect.

"Sign up to download"? No. Just no.

See the link posted by ivan_ah (https://news.ycombinator.com/item?id=23207024 ): http://ontology.buffalo.edu/04/AgainstSetTheory.pdf .

That annoys me too. However, you can easily read the full text below that button - it's all there.

I've only made it partway through this so far, but footnotes 4 through 7 make me think I already don't understand the author, since he seems to be presenting these as self-evidently absurd sentences.

Well, I think the idea that one can buy a set from a fruiterer is somewhat absurd. Suppose a fruiterer has only three apples left in stock – let us name them Apple1, Apple2, Apple3. Could I buy the set {Apple1, Apple2, Apple3} from the fruiterer? Could I instead buy the set {{Apple1, Apple2}, {Apple1, Apple3}}? How would these two purchases differ?

Part of the motivation of mereology is that, overall, it maps better than set theory to everyday life. One can come up with some really simple examples where set theory matches "common sense" well, but for more complex examples that breaks down. Many set theory texts try to justify set theory based on those simple examples while ignoring the more complex ones, and ignoring the alternative of mereology which claims to handle those more complex cases in a way which better respects common sense.

> Well, I think the idea that one can buy a set from a fruitier is somewhat absurd. Suppose a fruitier has only three apples left in stock – let us name them Apple1, Apple2, Apple3. Could I buy the set {Apple1, Apple2, Apple3} from the fruitier? Could I instead buy the set {{Apple1, Apple2}, {Apple1, Apple3}}? How would these two purchases differ?

Is it possible to buy two apples or you can only buy “this” apple and “that” apple? How would these two purchases differ?

I would say depends on how you define the structure and operationalize the problem. In your case the straightforward way would be to allow only for the first variant to make sense.

> Many set theory texts try to justify set theory based on those simple examples while ignoring the more complex ones, and ignoring the alternative of mereology which claims to handle those more complex cases in a way which better respects common sense.

I am not sure what you mean by “the more complex ones”, but the advancements in the set theory allowed to develop the measure theory, which is the basis of rigorous probability theory and statistics.

> I am not sure what you mean by “the more complex ones”

What I mean is that when you introduce nested sets (sets of sets, sets of sets of sets), and then allow repetition of elements in nested sets, it isn't clear what that means (if anything) if you think of sets as groups of physical objects. Mereology avoids this particular issue.

> the advancements in the set theory allowed to develop the measure theory, which is the basis of rigorous probability theory and statistics.

Well, there are two different questions about set theory (1) is it an accurate model of everyday human thinking about grouping objects? (2) is it useful as a foundation for developing various useful mathematical theories? A lot of defenders of set theory assume the answer to both questions is "Yes", or even fail to clearly keep the two questions clearly distinguished. The correct answers could well be "No" and "Yes".

> it isn't clear what that means (if anything) if you think of sets as groups of physical objects.

Isn't this similar to the idea that you can count apples or birds: one, two, three, four, etc, but there's nothing countably physical about infinity, or even arbitrarily large finite numbers. That doesn't impede the usefulness of infinity as an abstraction of counting, in a similar way to the fact that sets are useful as an abstraction of treating groups of things as a thing (and, in fact, we have to treat groups of things as a thing -- most of the physical things we are familiar with have fuzzy boundaries and definitions, because they're all groupings of smaller things).

>What I mean is that when you introduce nested sets (sets of sets, sets of sets of sets), and then allow repetition of elements in nested sets, it isn't clear what that means (if anything) if you think of sets as groups of physical objects. Mereology avoids this particular issue.

It avoids it by cost of being useless for any serious mathematical endeavor (mereology is essentially a complete Boolean algebra without a zero element). Yes, probably compactness and differentiability is not something clear and easy to grasp, especially when you have groups of physical objects in mind, but they are very useful constructs grounded in modern set theory.

I can't comment on how useful set theory or mereology is in reagrds to classical Western ontology, but I have no reason to doubt that many people in humanities may abuse and misuse mathematical and scientific apparatus.

>(1) is it an accurate model of everyday human thinking about grouping objects? (2) is it useful as a foundation for developing various useful mathematical theories? A lot of defenders of set theory assume the answer to both questions is "Yes", or even fail to clearly keep the two questions clearly distinguished. The correct answers could well be "No" and "Yes".

I think what you perceive as "failing to clearly keep the two questions clearly distinguished" may simply be a misinterpration of deep disinterest in the question, I think most mathematicians wouldn't perceive the question of "how everyday human thinking works" to be in the realm of math. Trying to tie in math with metaphysics fell out of favor since times of Gödel.

> I think what you perceive as "failing to clearly keep the two questions clearly distinguished" may simply be a misinterpration of deep disinterest in the question, I think most mathematicians wouldn't perceive the question of "how everyday human thinking works" to be in the realm of math.

Well, how "everyday thinking works" is very much related to the realm of math in my mind. Most commonly classical logic is selected as a foundation for mathematics, although there has been some work done on alternative foundations (most significantly constructivism/intuitionism, although there are less notable projects trying to build out mathematics on yet other foundations.) But, classical logic is commonly criticised by philosophers as being a poor model of everyday human thought, as represented by issues such as the paradoxes of material implication. So, alternative logics get proposed which attempt to answer those criticisms – for example, relevant/relevance logic. The study of these alternative logical formalisms is itself part of mathematics (mathematical logic, proof theory, etc), and the question of whether any of those alternative logical formalisms can be used as a foundation to develop other parts of mathematics (such as analysis) is an interesting mathematical question. And, added to all that, studying alternative logics is also interesting from the viewpoint of their possible practical applications in computer science (in fields such as automated reasoning.)

On the topic of set theory, while most mathematics assumes ZFC, there has been a lot of work on alternative set theories [1]. For each of these alternatives, we can ask (a) the philosophical question of whether it does a better job of modelling naïve human thought than ZFC does; (b) the metamathematical question of how easy it is to build out the rest of mathematics on that foundation; (c) the question of whether the theory has any useful applications in other fields such as computer science. All three questions are interesting, and they are all interconnected.

[1] https://en.wikipedia.org/wiki/Alternative_set_theory

The real issue is not any set theory, but another one: do sets exist? If they exist, they can function as causal antecedents. However, it is unintuitive to think of sets as causal antecedents. This is why great mathematicians postulate another world for numbers, sets, abstract objects, etc: Platonic world. That way, you can get rid of unintuitive-ness in the world we live in.

You'd think he'd rail against Alian Badiou but he doesn't.

Myself I refuse to accept the axiom of choice and I think Steve Wolfram should grow some balls and reject it too.

If you reject the axiom of choice, what are your thoughts on Zorn's lemma and the well ordering theorem (other than the observation of "they too are false" via equivalency)?

The ultrafilter attempt to bypass Arrow's Impossibility Theorem demonstrates how Zorn sends you up the creek without a paddle.

You can postulate such an object exists but you cannot realize it, so it doesn't translate to praxis. (e.g. you can't use the ultrafilter to decide an election)

That which can be constructed or described in a finite number of bits is more real than the phony numbers that Cantor justified. (e.g. Feigenbaum's constant is more real than any one of those real numbers that classical analysts try to bracket but never catch)

I got my honorable discharge from grad school and part of the climb in mathematical physics is reading some paper from 1957 that looked promising but after a close read you learn they got it wrong at page 47 and you have to figure it out yourself because you can't find the answers in the literature. You find out that the median scientific paper is wrong the hard way.

Wolfram wants to use computation (e.g. simulation, construction) as a praxis for all intellectual activity so he should privilege that map out of the Borges story over the territory of that deteriorating Empire which it mirrors.

Scientists in 2020 don't calculate in Cantor's phony numbers, but instead with those IEEE floats which never work quite right when you decimalize them.

> Feigenbaum's constant is more real than any one of those real numbers that classical analysts try to bracket but never catch

What do you mean?

Feigenbaum's constant is a number like Pi or e. It is transcendental. Even though you can't write it down with a finite number of digits, you can write down a formula to compute as many digits as you want (if you are ready to boil the oceans, build a Dyson sphere, harness a Quasar) You can give it a name and refer to it directly.

Any formula like that provides a set of brackets, "real" numbers with a finite number of digits (e.g. names) that we can say that the "phony" number is between. We can make the brackets finer and finer, but you can't pick out one in particular.

Thus 3, pi, pi/e + 6, sqrt(pi-e) are more "real" than the the continuum we imagine between them. Being able to name things, for instance, makes it possible to talk about them.

Constructivist mathematics [1] is an approach to math that restricts the universe of discourse to objects that can be explicitly defined. With this restriction, the subset of the real numbers considered consists of the reals that are definable [2], such as those that are computable [3] and constructible [4].

Similarly, the axiom of choice allows for the existence of nondefinable choice functions [5] in certain cases, so is rejected.

Regarding the part about analysis, the field of computable analysis [6] exists to establish analysis on constructivist footing.

[1] https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...

[2] https://en.wikipedia.org/wiki/Definable_real_number

[3] https://en.wikipedia.org/wiki/Computable_number

[4] https://en.wikipedia.org/wiki/Constructible_number

[5] https://en.wikipedia.org/wiki/Choice_function

[6] https://en.wikipedia.org/wiki/Computable_analysis

Better: what do you think of "the product of nonempty sets is nonempty".

I believe this statement only requires the axiom of choice if the number of sets is infinite.

> what distinguishes a from {a}?

Heavens. Stopped there.

It's not a bad question. Set theories answer that question somewhat differently, depending on whether they are 'structural' (like ETCS, SEAR etc.) or membership-based (like ZF(C) or NF).

Here's a nice early 20th century example[0]. Lesniewski's hatred of the singleton (and set theory in general) is legendary. In his case, the motivation was his own nominalism.

[0] https://plato.stanford.edu/entries/lesniewski/

the sentence you quoted is prefixed by "set theory has no natural interpretation.". that's the problem.

if you're trying to talk about herds of sheep, you might decide to represent them with sets – that sounds like what sets are for! but then is a single-sheep-set meaningfully different from an "unwrapped" sheep? how?

in general the article seems to be talking about the issues with using set theory to talk about real-world stuff; it's not questioning the math.

There is an obvious isomorphism between the set of all sheep and the singleton sets of sheep, so practically speaking it shouldn’t be much of a barrier.

Mathematics is about theories, philosophy is about describing reality. Theories that have little to do with reality are philosophically irrelevant, confusing and to be avoided despite being mathematically interesting.

It is at least a good reason to reject set theory for certain theoretical applications. Imagine you're a linguist trying to develop a theory of compositional natural language semantics. You want to have a good theory of what is entailed by a sentence of "Bill ate cheerios." You decide that pluralities are sets because that seems intuitive. Does that mean that when you say that Bill ate cheerios, he might have only eaten a singleton cheerio? That seems like the wrong prediction. It is clear that one cannot simply say that pluralities are sets, they are sets plus certain conditions, at least. This isn't only a philosophical exercise. This is about building a predictive model of natural language meanings where one doesn't want to make spurious predictions. The complexity of using sets to represent plurals, collections and groups quickly ramps up in this domain. This is a practical issue.

Better link: http://ontology.buffalo.edu/04/AgainstSetTheory.pdf

Changed from https://www.academia.edu/7590816/Against_Set_Theory?email_wo.... Thanks!

Reply on news.ycombinator.com