On Quantifying Over Everything

Consider the following argument against quantifying over everything.

"It can't be possible to quantify over everything, because if you did, there would have to be a set, your domain of quantification, which contained all objects as elements. However, this set would have to have to contain all the sets. But there can be no set of all sets, by Russell's paradox argument."

I claim it's unsound for the following reason:

We presumably can quantify over all the sets (e.g. when stating the axioms of set theory). So, if (as this argument assumes) quantifying over some objects required the existence of a set containing all the objects quantified over, we would already have a set containing all the sets, hence Russell's paradox and contradiction.

Thus, meaningfully making an assertion about all objects of a certain kind does NOT require that there's a set containing exactly these objects.

---

BONUS RANT: Why would one even think that where there is quantification there must be a set that's the domain of quantification? Because of getting over-excited about model theory I bet. [warning: wildly programmatic + underdeveloped claims to follow]

Model theory is just a branch of mathematics which studies systematic patterns relating what mathematical objects exist and and what statements are always/never true. It's not some kind of Tractarian voo-doo that `explains how it's possible for us to make claims about the world'. Nor do sets (e.g. countermodels) somehow actively pitch in and prevent claims like "Every dog has a bone" from expressing necessary truths!

Is "Set" Vague?

The (normal) intuitive conception of the hierarchy of sets is roughly this:

The hierarchy starts with the empty set, at the bottom. Then, above every collection of sets at one stage, there's a successor stage containing all possible collections made entirely from elements produced at that stage. And, above every such infinite run of successor stages, there's a limit stage, which has no predecessor, but contains all possible sub-collections whose elements have already been generated at some stage below.

But how far up does this hierarchy of sets go? Is there a fact of the matter, or does our conception not determine this?

The conception/our intuitions about sets doesn't directly tell us when to stop. For any stages we suppose we are looking at, it always seems to make sense to think of new collections that contain only sets generated by that point (e.g. the collection of all things so far generated). Of the sets generated by any collection of stages we can ask:
- Does the proposed next stage/limit stage of these stages really make sense? Are there really such collections?
- If so, are the collections generated at this stage still sets?

A textbook will tell you that at some point the things generated by the process above DO make sense, but DON'T count as sets. So, for example, there is a collection of all sets, but (on pain of paradox) this is not itself a set, but only a class.

However,
a) This just pushes the philosophical question back to classes: is there a point at which there stop being classes? Are there something else (classes2) which have the same relation to sets as sets do to classes? [One of my advisors calls this the "neopolitan" view of set theory]


b) We don't have any idea of WHEN the things generated by the process above are supposed to stop counting as sets.

Note that the issue with b) is not just that we don't know whether sets of a certain size exist. There are lots of things about math we don't know, and (imo) could never know. Rather, the uneasy feeling is that our conception doesn't "determine" an answer to this question in the following much stronger sense:

There could be two collections of mathematical objects with different structures, each of which equally well satisfies our intuitive conception of set.

For, consider the hierarchy of classes (note: all sets are classes). There might be two different ways of painting the hierarchy to say at what point the items in it stop counting as sets. Our intuitive conception just seems to generate the hierarchy of classes, not to say when things in it stop being sets!

In contrast, in the case of the numbers, I might not know whether there are infinitely many twin primes, but any two objects satisfying the intuitive, second order, characterization of the numbers would have to have the same structure (and hence make all the same statements of arithmetic true).

Thus, our intuitive conception of set seems to be hopelessly vague about where the sets end. Hence, even if you are a realist about mathematical objects, we seem forced to understand set theory as making claims about features shared by everything that satisfies the intuitive conception of set, rather than as making claims about a unique object.

Questions:

1. If you buy the reasoning in the main body of this post, does it give an advantage to modal fictionalism? e.g. the modal fictionalist might say: "You already need to agree with us that doing set theory is a matter of reasoning about what objects satisfying the intuitive conception of set would have to be like. What does incurring extra commitment to the actual existence of mathematical objects (as opposed to their mere possibility) do for you?".

2. An alternative would be to reject the textbook view, and say EVERYTHING generated by the process above is a set. Hence, you couldn't talk about a class of sets. Would this be a problem?

3. [look up] Is it possible that all initial segments of the hierarchy of classes that reach up to a certain point are isomorphic? (I mean, the mere existence of one-to-one, membership preserving, function that's into but not onto -the identity- doesn't immediately guarantee that some *other*, more clever, function that IS an isomorphism)
[maybe you can prove this is not poss by using the fact that one initial segment would have extra ordinals, and this iso could be used to define an iso between ordinals of different sizes which is impossible]

4. Is there some weirdness about the idea that collections in general (whether they be sets, or classes) eventually give out - so there's no collection of all collections.

We could say there are sets, classes, classes2, classes 3 and so forth. This lets us say there's a class of all sets, and a class2 of all classes etc. But as far as collections in general we must admit that there's no collection of all collections, on pain of contradiction via Russell's paradox.

Well, I don't personally find this that problematic. It's a surprising fact about collections maybe, but mathematics often yields surprising results.

Relations vs. Sets of Ordered Pairs

(Normally in math) a relation is defined to be a sets of ordered pairs.

But the `elementhood' relation between sets can't, itself, be a set of ordered pairs - since there can't be a set which contains each ordered pair of sets such that x is an element of y. [From the existence of such a set you could use the axiom of collection in ZF to derive the existence of a set of all sets, and hence the Russell set and contradiction.]

Therefore, not all relations (in the ordinary sense) are sets of ordered pairs (i.e. relations in the mathematical sense).

Thin Realism #2

Hmm on further reflection, `thin realism' is just Lumpism.

So see the essay above for why Lumpism is right and all that seductive stuff about the world having a logical structure/existence claims having a special epistemological status is wrong.

"Thin Realism" - what could it be?

I always want to say "I think there are numbers - but I understand existence in a thin logical sense". But I feel kindof dishonest saying this. It's too much like the sleazy "Of course P - but I don't mean that in any deep philosophical way" which happens when Wittgensteinians get lazy.

So here are some actual concrete ways in which I differ from other platonists (i.e. other people who believe there are mathematical objects).

1. I don't think we need to posit numbers to explain how there can be unknowable mathematical facts.

2. I think fictionalism/if-then-sim is is perfectly coherent. We could have had a mathematical practice which was completely based around mathematical properties, and studying their relaitons to one another e.g.: `Insofar as anything heirarchy-of-sets-izes, it's mathematically necessary that it satisfies the contiuum hypothesis'.

And here's an attempt to say what having only "a thin logical notion of existence" means:

When we ask what objects exist, this is equivalent to asking what sentences with a given logical form (Ex) Fx are true. So far, this is just Quinean orthodoxy.

But now the question is: what makes a given sentence (say, of English) have a certain logical form?

Now, I think having existential form is just a matter of what inferences can be made with that sentence, and what other -contrasting- sentences are in the language. We cook up various logical categories in order to best represent, and exploit, patterns in which inferences are truth preserving. Furthermore, there's noting special about objects, and object expressions. Each component of a sentence (be it concept-word, object-word, connective or opporator) makes a systematic contribution to the truth conditions of the sentences it figures in (i.e. the class of possible situations where the sentence is true).

On this view, choices about the logical form of a sentence wind up not being very deep - the question is just what's the most elegant way to capture certain inference relations.

In contrast, (I propose) having a "thick" notion of objecthood and existence, means thinking that there IS something more than elegant summary of inference relations at stake when we decide how to cut sentences up into concepts and objects. For example, you might think

1. It's easy to learn statements which don't imply that any objects exist (all bachelors are unmarried), whereas learning statements that do imply the existence of at least one object (there are some bachelors) is harder.

2. The *world* has a logical structure too! - so the most elegant way of cutting up your sentences to capture inference relations might still be wrong, because it fails to respect the logical structure of the world.

[Oh yes, they are kindof seductive. More about why they are wrong later.]

Unknowable Truths without Objects

I believe in mathematical objects, but I think the following appeal to them is dead wrong:

"The existence of mathematical objects is what allows there to be unknowable mathematical truths, whereas there are no unknowable logical or `conceptual' truths."

Corresponding to every unknowable AxFx statement in arithmetic, there's a purely modal statement, that's not ontologically commital, but would let you infer the arithmetical statement and hence must be equally unknowable, namely:

"It is impossible for there to be a machine on an infinite tape which a) acts in such and such such-and-such a physically specified way (here we have we list physical correlates of rules for some Turing machine program that checks every instance of the AxFx statement), and b) stops."

Why I am not Carrie Jenkins

Carrie Jenkins' 2009 book Grounding Concepts: an Empirical Basis for Arithmetical Knowledge, proposes a theory that has a lot in common with my thesis project.

Both of us:
- want to give a naturalistic account of mathematical knowledge

- in particular, want to explain how humans can have managed get "good" combination of inference patterns that count as thinking true things about some domain of mathematical objects/having a coherent conception of what those objects must be like, rather than "bad", 'tonk' like patterns of reasoning.

-appeal to causal interactions with the world, to explain how we wind up with such combinations of inference dispositions.

BUT there are some important differences. Here's why (I claim) my view is better.

Jenkins' theory:

Jenkins winds up positing a whole bunch of controversial, and perhaps under-explained philosophical notions to account for how experience gives us good inference dispositions. She proposes that:

Experience has non-conceptual content which grounds our acquisition of concepts so as to help us form coherent ones. Then when we have a coherent concept of something like the numbers, we inspect it to see what what must be true of the numbers and reason correctly about them.

-The idea that there's non-conceptual content is a controversial point in philosophy of perception.
-The idea that experience can "ground" concept acquisition without playing a justificatory role in the conclusions drawn is not at all clear. What is this not-justificatory, but presumably not just causal relationship of grounding supposed to be? (Kant's notion of a posteriori concepts seems relevant, but that's none-too clear either).
-Finally, what is concept inspection, (presumably you don't literally visit the 3rd realm and see the concepts) and how is it supposed to work? Jenkins admits that this is an open question for further research.

My theory:

In contrast, my view gives a naturalistic account of mathematical knowledge that doesn't need any of this controversial philosophical machinery. I propose that:

People are disposed to go from seeing things, to saying things, to being surprised if we then see other things, in certain ways. When these inference dispositions lead us to be surprised, we tend to modify them.

Thus, it's not surprising that we should have wound up with the kind of combination of arithmetical inference dispositions + observational practices + ways of applying arithmetic to the actual world, which makes our expected applications of arithmetic work out.

For example: insofar as we had a conceptions of the numbers which included the expectation that facts about sums should mirror logical facts in a certain way, it's not surprising that we would up also believing the kinds of other claims about sums, which make the intended applications to logic work out (e.g. believing 2+2=4 not 2+2=5).

Note that we don't need to posit any mysterious faculty of concept-inspection, or any controversial non-conceptual experience. All I appeal to is perfectly ordinary processes. People go from one sentence to another in a way that feels natural them (whether or not they are so fortunate as to be working with coherent concepts like +, rather than doing reasoning like Frege did about extensions) And when this natural-feeling reasoning leads to a surprise, they revise.

[Well, perhaps I'm also committed to the view that innate stuff about the brain makes some ways of revising more likely than others, and certain initial inference-dispositions more likely than others, in a way that doesn't make us always prefer theories that are totally hopeless at matching future experience. But you already need something like this even to explain how rats can learn that pushing a lever releases food, so I don't think this is very controversial.]