Hipparchia's Paradox

The most famous cynic philosopher was Diogenes of Sinope, who lived in an old wine jar and told Alexander the Great to get out of his light. But he wasn’t the only cynic; there was a whole bunch of them. The second or third most famous cynic was Hipparchia. (The third or second was Crates, Hipparchia’s husband.) Hipparchia doesn’t seem to have written much if anything, as tended to be the way with the cynics, but history has recorded at least one of her arguments, via an anecdote about an exchange she had with some jackass called Theodorus at a party one time. Here’s how Diogenes Laertius (not to be confused with Diogenes the jar-dweller) tells it:

Theodorus, the notorious atheist, was also present [at Lysimachus’s party], and she posed the following sophism to him. ‘Anything Theodorus is allowed, Hipparchia should be allowed to do also. Now if Theodorus hits himself he commits no crime. Neither does Hipparchia do wrong, then, in hitting Theodorus.’ At a loss to refute the argument, Theodorus tried separating her from the source of her brashness, the Cynic double cloak. Hipparchia, however, showed no signs of a woman’s alarm or timidity. Later he quoted at her lines from The Bacchae of Euripides: ‘Is this she who abandoned the web and women’s work?’ ‘Yes,’ Hipparchia promptly came back, ‘it is I’. But don’t suppose for a moment that I regret the time I spend improving my mind instead of squatting by a loom.’ [Lives of the Ancient Philosophers 6: 96-8; pp45-6 in Dobbin]

I’ve quoted the context as well as just the argument, the alternative being to quote it out of context. I think it’s pretty clear that Hipparchia is the winner of this story, although it’s possible the reality of the situation was pretty unpleasant for everyone concerned. But having acknowledged the context, I’d like to think a bit about the argument in isolation. Here’s the argument laid out neatly:

• Anything Theodorus is allowed, Hipparchia should be allowed to do also.
• If Theodorus hits himself he commits no crime.
• So neither does Hipparchia do wrong in hitting Theodorus.

The first premise is about universalizability: morality is supposed to apply equally to everyone. It’s a bit less clear what the theoretical basis of the second premise is. It seems like a part of most people’s common sense morality that if someone wants to hit themselves then that’s their own business, and while it might be inadvisable, it isn’t immoral. Common sense morality changes from place to place, but I guess this is part of it that my society has in common with Hipparchia’s. You could explain the truth of the second premise in various ways, some of which will mean qualifying or restricting it, and I think that how exactly we explain it will affect how the paradox gets resolved. The conclusion is meant to be absurd, showing that something is wrong with either the premises or the inference.

I think the most obvious way to try to resolve the paradox is to interpret the permission in the second premise as being explained by a general permission for people to hit themselves, rather than a general permission to hit Theodorus. The action that Theodorus is allowed to do is hitting oneself, not hitting Theodorus. Hipparchia is allowed to do the action hitting oneself too, so universalizability is saved.

There’s a problem with this, though: Theodorus is also allowed to do hitting Theodorus. He’d better be, because if an action is immoral under some description, then it’s immoral. This means there is something he’s allowed to do and Hipparchia isn’t, and so universalizability isn’t saved. Universalizability isn’t the idea that some of morality applies equally to everyone; it’s the idea that all of morality applies equally to everyone. Now, I don’t mean to be disingenuous. I’m not saying that Hipparchia’s paradox shows that universalizability is bunk; I’m just saying there’s more work to do. I don’t think there can be much doubt that it somehow matters that the description of the action as hitting oneself applies to Theodorus’s action and not Hipparchia’s. It just doesn’t resolve the paradox completely, and it’s perhaps more of a restatement of the paradox than anything else. Sometimes a restatement of a paradox is more or less all you need, but in this case I don’t think the restatement is enough.

Here’s another line of attack. Maybe on any given occasion it really is only OK for Theodorus to hit Theodorus if it’s OK for Hipparchia to hit Theodorus. The difference is that occasions when he hits himself will be those rare occasions when he wants to be hit, whereas occasions when she hits him are likely to be occasions when he doesn’t want to be hit. (And also he won’t hit himself harder than he wants to be hit.) This kind of reasoning is behind some anti-paternalist thinking in political philosophy. The classic anti-paternalist work is On Liberty, which was published under John Stuart Mill’s name but was probably coauthored with Harriet Taylor Mill, if you take its dedication literally. (It’s possible the Mills were the greatest philosophical power couple since Hipparchia and Crates. I can’t think of a greater one in the roughly 2150 years betweeen them, although perhaps you can, and perhaps there’s an obvious one I’m missing. [UPDATE: A friend pointed out I forgot Abelard and Heloise.]) They argued that the state shouldn’t be interfering with you if you’re not doing anyone else any harm. Here they are:

The object of this Essay is to assert one very simple principle, as entitled to govern absolutely the dealings of society with the individual in the way of compulsion and control, whether the means used be physical force in the form of legal penalties, or the moral coercion of public opinion. That principle is, that the sole end for which mankind are warranted, individually or collectively, in interfering with the liberty of action of any of their number, is self-protection. That the only purpose for which power can be rightfully exercised over any member of a civilised community, against his will, is to prevent harm to others. His own good, either physical or moral, is not a sufficient warrant. [On Liberty: p17]

People disagree over how far you can reconcile this with the consequentialism you find in Utilitarianism, but if you’re trying to reconcile them it usually goes roughly as follows. People will do things that have good consequences for themselves, so if their actions don’t have bad consequences for anyone else then they don’t have bad consequences for anyone. Given consequentialism, that means the actions aren’t bad. That means the state shouldn’t be interfering with them. It’s a bit of a Swiss cheese of an argument, and I think it remains so even if you’re properly doing justice to it, but I also think they were on to something important.

A classic example of paternalism is seatbelt laws. Idealizing a bit, the set-up is this: by not wearing a seatbelt you’re not putting anyone at risk but yourself. But by having laws demanding people wear seatbelts, you can save lives. Let’s consider a couple of things a libertarian might have to say about this:

• “If I value my life so much and my convenience so little that the small chance that wearing a seatbelt will save my life is worth the inconvenience of wearing one, then I will wear a seatbelt.”
• “The only person who stands to get hurt here is me, and I’m fine with it. Mind your own business.”

The first is a simple consequentialist argument: we don’t have to worry about people not wearing seatbelts in situations where the expected consequences are negative. (It also takes the relative value of someone’s life and convenience to be the relative value they themselves assign to them, but maybe that’s not so silly at least in the case of most adults.) The second libertarian response is harder to categorize. It can still be made out as consequentialist in a way, but it says that people are allowed to waive consideration of negative consequences to themselves. The first objection, where it applies, flows straightforwardly from a simple consequentialism that says the right thing to do is the thing with the best consequences. The second applies more generally, but it says that sometimes it’s OK to do the thing that doesn’t have the best consequences. If we’re allowing people to waive consideration of consequences to themselves in the moral evaluation of their own actions, this raises questions about what other kinds of waivers are allowed:

• Can I waive consideration of consequences to myself in the moral evaluation of someone else’s actions?
• Can I do this on an action-by-action basis, or at least a person-by-person basis, or do I have to waive it for all people or all actions if I waive it for one?
• Can I waive consideration of some but not all negative consequences to myself?
• Can I waive consideration of bad things happening to me even if someone else cares about me and so these would also be negative consequences to them?
• Are there ever situations where someone can waive consideration of a negative consequence to someone other than themselves?

None of these seem to me like they have obvious answers, with the possible exception of the last one, even if we grant that people can waive consideration of harm to themselves in the moral evaluation of their own actions. I expect some readers will think some of the answers are fairly obvious (and that the last one is obviously obvious), or will at least have views on some of the questions, perhaps based on the literatures which presumably exist on each of them. To be clear, I’m not saying that a consequentialism with a self-sacrifice caveat can’t be made coherent. You could say that an action is permissible iff it either maximizes expected utility or has an expected utility for other people at least as high as the expected utility for other people of some permissible action. That seems to get the right results. The problem I have is that if waivers are a thing, then there are other waivers we might want to include in our theory as well, and after a while our theory might end up not looking much like consequentialism at all.

One way to avoid these questions is to deny that people can waive consideration of themselves in the first place. But then Hipparchia’s paradox comes back, at least a little. The problem with this simple consequentialist response to the paradox is that people don’t always do what’s best for them. Unless we supplement the response somehow, it will mean that whenever Theodorus hits himself and it isn’t what’s best for him, he is doing something wrong after all. (At least when he had enough information to work out that it probably wouldn’t be best for him.) Is this what we want to say?

I can sort of see how some people might want to bite this bullet. If you’re an agent-neutral consequentialist, then you think that the only information relevant to whether an action is wrong or not is how good its consequences are. Who did the action isn’t relevant. So this kind of consequentialist should say that Theodorus hitting himself really is immoral whenever it’s inadvisable. If someone gets on their high horse with you about how you’re not doing what’s best for yourself, they actually do have the moral high ground. Perhaps this is right. But it’s weird.

I don’t really feel like I’ve got very far with this. But my main aim was to present the argument as something worth thinking about, because I do think it’s worth thinking about. I’ll close by presenting another argument, which is also a Swiss cheese of an argument, but which I’m also worried might be on to something.

• Hipparchia’s paradox shows that fully agent-neutral consequentialism is absurd.
• The only promising arguments for consequentialism are arguments for fully agent-neutral consequentialism.
• So there are no good arguments for consequentialism.

References

• Dobbin, R. 2012: Anecdotes of the Cynics, selected and translated by Robert Dobbin. Penguin Random House.
• Mill, J. S. 1859/2011: On Liberty, Project Gutenberg ebook #34901, http://www.gutenberg.org/files/34901/34901-h/34901-h.htm
• Mill, J. S. 1863/2004: Utilitarianism, Project Gutenberg ebook #11224, http://www.gutenberg.org/files/11224/11224-h/11224-h.htm

Comparing Size Without (Much) Set Theory

At the end of my last post, I said that I’d like to know whether it’s possible to make sense of there being more Xs than Ys when there are uncountably many of each, without using set theory. I’m not a proper mathematician, as I expect will become painfully apparent to any proper mathematicians reading this, but I’ve tried to hack something together that might sort of work. It uses plural quantification, which George Boolos (1984) has argued isn’t set theory in disguise. It does use some actual set theory too. But hopefully it’s a start.

Georg Cantor, and apparently David Hume before him, came up with a rule for comparing the sizes of infinite collections. If the Xs and the Ys can be paired off one-one, then there are the same number of each. If the Xs can be paired of one-one with some of the Ys, there are at least as many Ys as Xs. In set theory, you can use this idea to make a nice precise open formula expressing that a set x is at least as big as a set y, in terms of there being another set z that represents this one-one pairing. The usual way is to make it a set of ordered pairs with one member from each of x and y, having previously said what it is for a set to count as an ordered pair.

Since this set-theoretic version of “at least as big as” relies on there being a set in the model to represent the correspondence whenever there is such a correspondence, you can sometimes get models that don’t give the results about which sets are bigger than which that you intuitively might think they ought to. That’s how you end up with things like Skolem’s paradox, which is the puzzle of how set theories that say (under their intended interpretations) that there are uncountably large sets can have models with only countably many things in the domain. We can sort of ignore this here, although if you know a lot more than I do about Skolem's paradox it may help to keep it in mind.

Suppose I want to do this pairing thing without set theory. One thing I could do is take “at least as many” as primitive, so I’ve got a predicate Xs ⪰ Ys, which takes plural terms on both sides, and is true just when there are at least as many Xs as Ys. That’s not really legitimate for this project though, because the good standing of the concept is exactly what we’re trying to establish. What I’ll suggest is that we use just enough set theory to make comparisons of size, but not all the extra stuff that leads to indeterminacies in which model we're talking about and whether or not the continuum hypothesis is true in it.

What do we need to define “Xs ⪰ Ys”? We’ve already got plural quantification, outsourcing the defence of its set-theoretic innocence to Boolos, as is traditional. What I’m suggesting is adding just ordered pairs of things which aren’t themselves ordered pairs, and then saying that there at least as many Xs as Ys whenever there are some ordered pairs representing a one-one correspondence between some of the Xs and all of the Ys. And then you throw away the ordered pairs again, since ordered pairs are fictional and all.

So, you start off with the model M you’re interested in, and you want to extend it to a model M+ with Xs ⪰ Ys defined in it. To do that, you take another model N which is the same except you add in a bunch of ordered pairs. Whenever there are one or two things in the domain of M, there are the ordered pairs of them in N. None of the ordered pairs are duplicated, and there’s nothing else in N. Then you can define Xs ⪰ Ys as being true in M+ iff it’s true in N that whenever there are some ordered pairs Ps such that nothing is the first of more than one p in Ps, and nothing is the second of more than one p in Ps, and all the firsts of a p in Ps are in Xs, and all the seconds of a p in Ps are in Ys, and all the Ys are the second of some p in Ps.

I’ve tried to be careful not to introduce any general set-theoretic stuff in the definitions, except for the ordered pairs. The idea is that given a model M of plural logic without ⪰, we can always pin down a unique model N, and then we can define a new model M+ of plural logic with ⪰ in terms of M and N. The M+ models constructed in this way are the admissible models for plural logic with ⪰. The way this definition goes is supposed to be unaffected by what is and isn’t true about the universe of sets out there, if it even is out there, and in particular it’s unaffected by the truth or otherwise of the set-theoretic version of the continuum hypothesis. This means we should be able to express the non-set-theoretic version of the continuum hypothesis that I talked about in the last post, purely in terms of plural logic and without leaving any hostages to set theory.

A potential source of problems is that the model theory for plural logic, just like the model theory for most things, tends to be given in terms of set theory. Can you avoid that, and just give it in terms of plural logic? I sort of expect you could, perhaps with a little extra stuff but way short of full set theory, although I’m not sure whether this is something anyone has taken it upon themselves to do. The idea would be that instead of saying things like “a model M is an ordered pair <D, V> where D is a set of objects and V is a valuation function”, you say “a model M is defined by some things the Ds, which are its domain, and…”. (This is the point at which it becomes difficult.) If set theory does turn out to be indispensible to the model theory, then there will always be a suspicion that the definitions are hostage to set theory. It’s a little bit like the problem of doing the model theory for non-classical logics in classical logic, or giving a model theory for variable domains modal logic without committing yourself to a possibilist ontology. I don’t really want to get into this debate because in debates like this there’s always a danger you’ll find Tim Williamson on the other side.

So, I’m not going to present a non-set-theoretic semantics for plural logic, and I’m also not going to defend the set-theoretic innocence of plural logic with a set-theoretic semantics. But when I try to formalize the method for constructing the M+s, I’ll try to mention sets as little as possible. In particular, the ordered pairs in the domain of the intermediate model won’t actually be ordered pairs. But the domains will be sets, and the extensions will be sets of ordered n-tuples of objects and/or sets, the way you’d normally do it if you weren’t worried about set theory. The idea is that if plural logic can be set-theoretically innocent unless the subject matter happens to be sets, then this construction is set-theoretically innocent too. The model theory helps us clarify what we're saying, but you still only have to commit to the entities in the domains.

Here’s the syntax of the language L. It doesn’t have ⪰ in it yet; adding that will make L+.

• Singular names a, b, c, etc
• Singular variables x, y, z etc
• Plural names C, D, E, etc
• Plural variables X, Y, Z etc
• Predicates P, Q, R etc, which can be any finite number of places ≥ 1, and which can be singular or plural in each position.
• A binary “one of” predicate <, singular in the first position and plural in the second.
• A binary “among” predicate ⊑, plural in both positions.
• A binary identity predicate =, singular in both positions. (Plural identity can be defined in terms of = and < in the normal way if need be. (Our language L can't express many-one identities, even though regular readers will recall that I think some many-one identities are true.)
• Atomic wffs composed out of predicates and names or variables in the normal way.
• Compound wffs composed from wffs and & and ¬ in the normal way, with other connectives defined as normal.
• Quantifiers ∃ and ∀. ∃! is defined as a unique-existence quantifier in the normal way.
• If φ is a wff and v is a singular or plural variable, ∃vφ and ∀vφ are wffs.

Now the semantics:

• A model M is an ordered pair <D, V> where D is a set of objects and V is a function on members of L.
• An assignment A is a function from singular variables to members of D and from plural variables to non-empty subsets of D.
• If t is a singular name, VA(t) = V(t) ∈ D.
• If t is a singular variable, VA(t) = A(t) ∈ D.
• If t is a plural name, VA(t) = V(t) ⊆ D, and must be non-empty
• If t is a plural variable, VA(t) = A(t) ⊆ D, and must be non-empty
• If P is an n-place predicate, V(P) is a set of n-tuples <o1, o2, …, on>, where oi ∈ D when P is singular in the ith place, while oi is a non-empty subset of D when P is plural in the ith place.
• V(<) is the set of ordered pairs <x, y> where y is a subset of D and x is in y.
• V(⊑) is the set of ordered pairs <x, y> where y is a subset of D and x is a subset of y.
• V(=) is the set of ordered pairs <x, x> where x is in D.
• If P is an n-place predicate and t1 … tn are terms, then VA(Pt1...tn) = T if <VA(t1), …, VA(tn)> ∈ V(P), and  VA(Pt1...tn) = F otherwise.
• The values for & and ¬ are assigned truth-functionally in the normal way.
• VA(∃vφ) = T iff there is an assignment B which differs from A at most in the value for v, such that VB(φ) = T, and VA(∃vφ) = F otherwise.
• VA(∀vφ) = T iff all assignments B which differ from A at most in the value for v are such that VB(φ) = T, and VA(∀vφ) = F otherwise.
• M(φ) = T iff VA(φ) = T for all assignments A, and M(φ) = F otherwise.
• Σ ⊨ φ iff for every model M such that M(ψ) = T for all ψ ∈ Σ, M(φ) = T as well.

This is the basic logic. It isn’t supposed to be original. It’s supposed to be unoriginal, because if it was original I’d be in danger of having to defend its set-theoretic innocence myself, instead of outsourcing the job to Boolos. (I'm not sure if Boolos himself is the first person to formalize the model theory along these lines, but other people in the tradition use a set-theoretic model theory and lean on Boolos for the case for ontological innocence. I think they do, anyway. If I'm honest it's a long time since I read Boolos's paper. I think he says something pretty persuasive about how when you eat a bowl of Cheerios you're eating the Cheerios, not a set of Cheerios.) I felt it was important to write it down so you could see just how much set theory is involved, and what it's doing.

Now we construct a model M+ = <D, U> from a given model M = <D, V>. We start by constructing a model N = <E, W>.

• E = D ∪ G, where G is the set of objects representing ordered pairs of members of D.
• D and G are disjoint.
• Introduce two binary predicates P1 and P2 which are undefined in M. These are singular in both positions. You can think of N as a model of an expanded language L*.
• For every object x in D, there is one object y in G such that <x, y> is in W(P1) and W(P2).
• For every two objects x and y in D, there is exactly one object z in G such that <x, z> is in W(P1) and <y, z> is in W(P2).
• Nothing else is in G.
• For every object x in G, <y, x> is in W(P1) and <y, z> is in W(P2) for only one y and only one z.
• Nothing else is in W(P1) or W(P2).
• Let A be an assignment on D, and let B be an assignment on E that extends A.
• Now we define the extension of ⪰ in M+, that is U(⪰), in terms of the assignments A relative to which W evaluates an open sentence with two free plural variables X and Y as true.
• Let φ be ∃Z[∀y(y<Y → ∃!z[z<Z & P1yz]) & ∀z(z<Z → ∃x[x<X & P2xz & ∀w[(w<Z & P2xw) → w = z]])]
• U(⪰) = {<s, t>: WA(φ) = T for some assignment A such that A(X) = s and A(Y) = t}
• In words, φ is meant to mean “There are some things [stand-ins for ordered pairs] such that every Y is the first of exactly one of them, and each of them has a distinct X as its second.”
• Now we can say that a model of L+ is admissible iff it is the model M+ for some admissible model M of L.

That’s the proposal formalized. There’s a lot of set theory in the formalization, and indeed there’s so much that you could be forgiven for forgetting that I was trying to avoid set theory at all. But I was trying to avoid set theory. There are two things set theory is doing there. One is to construct the models of plural logic. I already said I wasn’t going to try finding a non-set-theoretic model theory for plural logic. The other thing is a very weak set theory that adds something equivalent to ordered pairs to the domains of the intermediate models (N in the construction), but the ordered pairs don't themselves form further ordered pairs. How should we interpret this? I think the most principled way for a fictionalist about sets like me is to interpret those models as representing a fiction. The fiction says that every one or two objects that aren’t themselves ordered pairs form one or two ordered pairs respectively. (So for every non-pair x there’s <x, x> and for every non-pair x and non-pair y there are <x, y> and <y, x>.) When it’s true in the ordered pairs fiction that there are some ordered pairs representing a one-one correspondence between some of the Xs and all the Ys, it’s true in reality that there are at least as many Xs as Ys.

The point of using the ordered-pairs fiction instead of the full-ZFC fiction is that the ordered-pairs fiction specifies a single fully determinate model N, given a model M for reality-minus-size-comparison-facts. You then use this to get a model M+ for reality-including-size-comparison-facts. Full ZFC doesn’t specify a single model, and the different models may have different one-one correspondences in them, which will give you different size-comparison facts. The models themselves are set-theoretic objects. I’m not sure how much of a problem that is. I think the kind of answer I’d like to give is along the lines people give for variable domains modal logic: we already understand plural logic, and the use of this model theory is just supposed to precisify which particular thing that we already understand we’re talking about. Someone who thinks you can’t understand plural logic without set theory won’t buy that, and those are the people I’m referring to Boolos.

Maybe a promising way around this would be to construct a model theory for plural logic along the same lines as the ordered-pairs fiction itself. There’s a whole lot of ZFC not being used in the model theory, so maybe you could have a much lower-powered fiction which could still do the job but didn’t have the underspecification you get with ZFC. I only have a vague idea of how that might go though, and there could be straightforward reasons why it wouldn’t work.

In closing I’d like to make it clear what my ambitions are. I claimed in my last post that we could understand a version of the continuum hypothesis independently of set theory. The continuum hypothesis without sets, or CHWS, is a statement about how many real numbers there are. To make sense of CHWS without using sets, we need to understand how there can be more Xs than Ys when there are uncountably many of each. Normally we do that using sets. I’ve been trying to show how we might do it while avoiding full ZFC and its indeterminacies, using only plural logic and the much lower-powered and more determinate fiction of ordered pairs. I’m trying to show that we can understand CHWS as something with a determinate answer, even if we’re fictionalists about sets. I’m not trying to offer any reason for optimism that we could ever settle CHWS. And if I had to guess, I’d say we probably never will.

**********************

• Boolos, George (1984). To be is to be a value of a variable (or to be some values of some variables). Journal of Philosophy 81 (8):430-449.

The Continuum Hypothesis Without Set Theory

Regular readers may recall that a while ago I posted a fallacious proof of the continuum hypothesis. They may also recall that a bit more recently I tried unsuccessfully to understand forcing. Forcing is a technique for building models of set theories, which Paul Cohen used to show that the continuum hypothesis doesn’t follow from the ZFC axioms. (The ZFC axioms are the Zermelo-Fraenkel axiomatization of set theory plus the axiom of choice, and I’m told that most normal maths that can be proved at all can in principle be cast in terms of set theory and then its set-theoretic version can be proved from these axioms, if you feel the need to do that.) Anyway, in spite of the evidence that I’m not very good at it, I’ve been thinking about the continuum hypothesis again.

The continuum hypothesis, for those of you who don’t know but are still reading, is the proposition that the number of real numbers is the second smallest infinite number. It’s been proved that the number of integers is the smallest infinite number, that the number of real numbers is bigger, and that there is a second smallest infinite number. Georg Cantor conjectured that the number of real numbers was the second smallest, Kurt Gödel proved that the continuum hypothesis was consistent with the ZFC axioms, and Paul Cohen proved that it didn’t follow from them. So we’ve established that the ZFC axioms, even if true, don’t settle the question. And in the time since Cohen finished the independence proof, set theorists have learned a great deal more about which axiom systems do and don’t settle the continuum hypothesis, which models it is and isn’t true in, and the relationships between them. (I don’t understand this work myself: in order to understand it I’d have to understand how forcing works, and I’m sorry to report that I still don’t.)

Now, some people respond to this situation by saying that the continuum hypothesis (CH from now on) is indeterminate. This might be because they think that the notion of a set corresponding to ZFC doesn’t pin down a particular model, and CH is true in some and not in others. It might be because they think there are multiple set-theoretic universes out there, and CH is true in (or of) some of them but not others. It might be because they think mathematical truth basically amounts to provability, and we know that neither CH nor its negation is provable. Or perhaps they think the indeterminacy comes in somewhere else, for example as fundamental metaphysical indeterminacy.

Some people are set-theoretic Platonists, who think there’s a real, determinate universe of sets out there, and that even though we don’t know enough about it yet to prove that CH is true or false in it, CH is nonetheless true or false in it. To settle the question we would have to learn things about the set-theoretic universe that don’t follow logically from what we already know about it. How exactly we’d go about learning something like that is a vexed question, but (so the story goes) we have already managed to learn some things about it that don’t follow from nothing, so perhaps we could pull off the trick again. I’ve heard something like this view attributed to Gödel, though I’m not sure how fair a reflection of his views what I heard was, or how badly I’ve garbled what I heard.

I’ve got a certain amount of sympathy with this kind of view in principle, but I don’t agree with it. I think sets are made up, and that makes me a fictionalist about sets. I don’t think that integers and real numbers are made up, and my attitude towards them probably makes me a Platonist about those. I sympathize with the Platonist view about CH in principle because I’d be happy to take this attitude towards mathematical entities I didn’t think were made up. But it’s not my view because I do think sets are made up. In fact, the more I hear about sets, the more made up they sound.

Sometimes we talk about things being true or false in fictions, and I think that’s a reasonable way to talk. It’s true in the relevant fictions that Miss Marple solves crimes, for example. I think that there are a few fictions relating to set theory, and that we have pretty tight rules for establishing what is and isn’t true in several of these fictions. (Much tighter rules than we have for Miss Marple.) As is the way with fictions, sometimes the rules don’t settle what’s true in the fictions, and maybe sometimes nothing settles it at all. We tend to be more easygoing about indeterminacy in fiction than about indeterminacy in reality, and that’s probably fair. If you give a set-theoretic statement of CH, that will tend to be true in some of these fictions, false in others, and indeterminate in others. So you might think that a fictionalist about CH should think this is all there is to the truth or otherwise of CH.

Well, that’s not what I think. Take another look at what I said CH was, earlier in the post:

"The proposition that the number of real numbers is the second smallest infinite number."

The eagle-eyed and literal-minded among you will notice that this doesn’t mention sets. It mentions numbers. Since in this instance we don’t need to worry about uninstantiated numbers, we can cast CH without mentioning sets, or even mentioning infinite numbers:

"There aren’t any things such that there are more of them than there are integers but fewer of them than there are real numbers."

We’ll call this CHWS, for Continuum Hypothesis Without Sets. I think that when we ask whether the continuum hypothesis is true, this is the question we’re ultimately interested in, at least under the assumption that the integers and real numbers exist. But I expect that some set theorists will like to think of the set-theoretic formulation of CH as the continuum hypothesis, and that’s fine. I’ll try to sidestep the terminological issue by giving it a slightly different name: CHWS. In deference to these imagined set theorists, from now on I’ll use CH for their set-theoretic version.

I said earlier that I was a realist, and probably a Platonist, about integers and real numbers. That’s more or less true, in that it’s my working hypothesis even if I’m not fully committed to it. As I see it, if you’ve got some things, then it makes sense to talk about how many of them there are, and whether the number of them is infinite, and whether there are more of them than there are of some other things. And none of this presupposes the existence of sets. When you say there are more even numbers than prime numbers between 1 and 100 you’re saying something about numbers, not about sets. We’re able to give a precise explication of these notions in set-theoretic terms, and this explication serves for most purposes, but you might think that one lesson of the independence of CH from ZFC is that ZFC set theory can’t help us answer the question about how many real numbers there are. But if I’m a Platonist about real numbers, which I probably am, and I don’t think that there’s indeterminacy in how many of them there are, which I don’t, then CHWS can still have an answer. It’s just that set theory alone can’t help us find it. Perhaps sometimes when you ask whether there are more Xs than Ys this doesn’t always have a determinate answer, and so CHWS could be indeterminate for that reason. But I don’t see that anything we’ve learned about set theory compels us to think that. I think it’s a very strange idea, although the subject matter seems sufficiently strange that I should be open to the possibility, and so I am. But being open to the possibility that CHWS is indeterminate is very different from thinking it actually is indeterminate. And if I had to guess, I’d say it probably isn’t.

So in summary, here are the things I said I think:

• Sets are made up, so I’m a fictionalist about sets.
• Integers and real numbers aren’t made up: I’m a Platonist about those.
• We can make sense of questions about whether or not there are more Xs than Ys without understanding them in terms of sets.
• Sometimes those questions have determinate answers, and maybe they always do.
• The continuum hypothesis can be cast as a question about how many real numbers there are, without reference to sets.
• This question may have a determinate answer, and nothing about set theory gives us much reason to think it doesn’t.

As I understand it, this isn’t a popular combination of views, and I suppose it might even be provably incoherent. Perhaps there are people who could sensibly be confident that they could prove that it’s incoherent in an afternoon. But I’ve held these views as working hypotheses for a while now, and although I don’t think I’ve converted anyone, I also haven’t noticed them leading to any problems in the general pursuit of truth, so to speak. For somebody who has the information and interests I have, they make a livable position, and livability is a source of evidence in philosophy. But I do worry that I don’t understand the issues well enough to have earned the right to hold this combination of views. Maybe my taking a position on this stuff at all is pure hubris, and my foolish, incoherent position is a result of that hubris. Or maybe I’m more or less right, and my reasons for arriving at the position are sensible enough that being right is some kind of epistemic achievement. (Or maybe I’m entitled to a view, but am nonetheless wrong.) On the one hand the Enlightenment ideal is supposed to involve having confidence in your own rational capacities and making sense of things for yourself, instead of taking things on trust from people you’re told are authorities. On the other hand, that’s exactly how people end up posting flat-earth videos on Youtube.

So, am I being like a flat-earther? The unpopular part of my position, as I understand it, is that I’m a fictionalist about sets but I think CHWS may still have a determinate truth value. It seems to me that a couple of things differentiate me from the flat-earthers. One is that I’ve studied a lot of philosophy, including some philosophy of maths, and this should give me some protection against making really silly mistakes when thinking about this kind of thing. Another is that members of the academic establishment put quite a lot of effort into debunking flat-earthism, whereas nobody’s putting much effort into debunking the combination of set-theoretic fictionalism and realism about a version of the continuum hypothesis. Although like I say, I wouldn’t be all that surprised if a philosophically informed set-theorist with a free afternoon could do it. If there is a quick debunking to be made, the main pressure point seems to be the notion that we can make comparisons of infinite sizes without dealing with sets. More specifically, there’s the notion that we can make comparisons between uncountably infinite sizes without dealing with sets. Can we? It’s not obvious to me whether we can or not. Maybe that’s the question I need to answer before I’m allowed to have an opinion on this stuff.

Metaphysical Whack-A-Mole

Here’s an argument for the existence of God:

• There’s no contradiction in an omnipotent being spontaneously coming into existence.
• Nothing that was not omnipotent would be able to prevent an omnipotent being coming into existence. (Think of it like a game of metaphysical whack-a-mole.)
• In a given time period, if there is no contradiction in something happening and there is nothing to prevent it from happening then there is a non-zero probability that it will happen.
• So in any given time period at the beginning of which there is no omnipotent being, there is a non-zero probability that an omnipotent being spontaneously comes into existence.
• So over an arbitrarily large time period the probability that an omnipotent being spontaneously comes into existence will be arbitarily close to certainty.
• There has been enough time that we can be practically certain that an omnipotent being has spontaneously come into existence.
• Once an omnipotent being exists, it will see to it that it continues to exist, since it is omnipotent and wants to continue to exist.
• There cannot be more than one omnipotent being, since it follows from their omnipotence that they would be both able and unable to frustrate each other’s intentions, and this is a contradiction.
• If there is exactly one omnipotent being, then that being is God.
• So we can be practically certain that God exists.
• So God exists.

What do you think? Let me know in the comments!

Goat Product

I did a lot of maths at school, and while I’ve got a good memory and do use bits of it here and there, it’s been fifteen years and some of it’s a bit rusty. Partly in an attempt to remedy this, I recently I watched some videos on Youtube about calculus and linear algebra. They’re by someone called Grant Sanderson, whose account is called 3Blue1Brown, and they have a friendly style with lots of neat animations. The idea of the videos is that a lot of people are taught maths in a way that involves a lot of number crunching and rule following but doesn’t get them to understand the underlying concepts, and that this leaves a gap that can be filled by presenting the ideas using visual interpretations. For example, a 2x2 matrix represents a transformation of 2D space that turns every square of size 1 into a parallelogram whose size is the magnitude of the matrix’s determinant. (If it’s negative, that means things are mirrored.) It’s possible I learned this at school, but stressing it and showing a nice animation helps you really understand what the deal is with determinants. They’re cool videos, and if you like maths then I recommend them, although they’re not really designed to teach the topics from scratch. I also recommend this one by the same person, which shows how to give a topological proof of something that doesn’t really seem to be about topology. I watched it a while ago and it really is beautiful.

Anyway, as I was going through the linear algebra videos I wrote down some questions I still had so I could find answers to them later, and one of them was about the dot product (also known as the scalar product). The dot product of two vectors <a,b> and <x,y> is a dimensionless directionless scalar, equal to ax + by. Why? What’s it measuring? The video on the dot product gave it a couple of interpretations.

One was to project <a,b> onto the line <x,y> is on via the shortest route, and then multiply the length of the new vector by the length of <x,y>. That might be OK for giving a visual heuristic for calculating or estimating the dot product, but for me it didn’t get me much of a handle on what sorts of thing it represents. Multiplying the length of two vectors is something you’d do if they were the sides of a rectangle and you wanted to find its area, but if the vectors are on the same line, where’s the rectangle? And if it’s an area, why is the dot product a scalar? It makes sense for the determinant to be a scalar because it’s the scale factor for areas, but with the projection interpretation of the dot product I couldn’t really see which areas were getting scaled up.

The second interpretation replaced one of the vectors with a 1x2 matrix, and then visualized this as a transformation taking points in 2D space onto a number line embedded in that space. The 1x2 matrix <a,b> will take the point <x,y> in 2D space to the point ax+by on the number line. That’s great, and it shows you the nice duality between the dot product and this kind of transformation, and indeed that was the point of this part of the video. (The video was called “Dot Products And Duality”, after all.) But a 2x1 vector isn’t a 1x2 matrix, and the fact that that taking the dot product is dual to performing a process that outputs a scalar doesn’t really tell you why the dot product itself is a scalar.

There will be good reasons for the dot product to be a scalar, of course, but this video didn’t help me see what they are. I also thought it’d be good to have a better visual interpretation of the dot product, where I could point to something I could see and say “that’s the dot product”, instead of pointing at two things I could see and having to multiply them together to find the dot product, even though the multiplication didn’t correspond to anything in the picture. In the video series on calculus he actually flags up that he doesn’t have a nice picture for the derivative of 2t where we can point to it and say: “See? That part! That is the derivative of 2t!” He says he’d like one, but doesn’t know one. Perhaps there isn’t one. Although perhaps there must be one. I guess different people will have different views about how high our expectations about this sort of thing should be.

Anyway, I made a note of the question, and when I got to the end of the videos I still didn’t have an answer, so I googled things like “visual interpretation of dot product” and “why is the dot product a scalar”. And you know what? Nobody else seemed to have a decent answer either! I mean, I didn’t look very hard, but there seems to be a fair bit of mystification out there, both from people asking the question and people trying to answer it. On this Quora page about the question, one person said that’s just how it’s defined, which isn’t helpful, and another said it was the distance between points the two vectors point at, which isn’t true on any interpretation I could work out. On another page I found more helpful someone said it was like a booster ramp in Mario Kart, which boosts you best if you drive over it in the direction of the ramp, and doesn’t boost you so well if you drive over it obliquely. I quite liked that one, but it has some problems. One is that if you drive over it backwards it still boosts you forwards, so there’s nothing much that seems to correspond to negative dot products. The other problem is that it’s not obvious what corresponds to multiplication here. It seems to me that the ramp adds a certain amount of velocity, rather than multiplying a vector from the ramp by the vector given by your initial velocity. Also the result of going on a ramp seems more like a vector than a scalar. I think these three issues are related. I did still find it helpful, and maybe I just don’t have a good enough feel for how the booster ramps work (although I have played the game quite a lot), but I wanted something better. He had one involving the angles of solar panels too but he didn’t like that as much as the Mario Kart one, and I didn’t really either. (I couldn’t see how it produced negative dot products, for example.)

So anyway, I had a go at coming up with my own visualization, and I found one that I think works pretty well. You know how people share those pictures of goats standing on almost vertical surfaces, seemingly defying gravity? Well, imagine a goat standing on one of those. It could be a very steep one like the cliffs in the pictures, or it could be shallower like a hillside. Basically any inclined plane will do. We can represent the slope with a vector u, for upslope. The vector points up the slope and its magnitude is the gradient of the slope in that direction. <0,0> corresponds to a horizontal plane, which is fine, but no vector corresponds to a vertical one, which is just as well because even a goat can’t stand on a vertical plane. Now the goat sets off along the surface in a straight line, with its horizontal velocity given by a vector v. It may be easiest to picture v if you look down on the goat from above. Now the dot product is the rate of change in the goat’s elevation. Positive means the goat is going up, and negative means the goat is going down. It’s a scalar, not a vector, because it’s measuring how fast the goat is moving up or down, not measuring how fast it’s moving and which direction it’s moving. (Well, it’s measuring whether the goat is going up or down, but even scalars distinguish a positive and a negative direction. It’s not measuring the direction in three dimensions, or even two.)

The sums seem to work out. Think of u = <a,b> as giving the gradient a when moving north and the gradient b when going east, and think of v = <x,y> as giving the velocity’s northern component x and eastern component y. Negative gradients are downhill, and negative velocity north or east is movement south or west. So if it goes x north and then y east it will go a*x up and then b*y up.

It gives straightforward visual interpretations to some properties of the dot product too. To get maximum height for its horizontal displacement, the goat goes straight uphill, which means u and v have the same direction. Change in elevation for a given horizontal dispacement is proportional to gradient, and change in elevation for a given gradient is proportional to horizontal displacement. It’s clear when the dot product will be positive, negative or zero, depending on whether it’s moving uphill, downhill or along a contour. It’s a good visualization of how a difference in the angle between u and v makes less of a difference to the dot product when the angle is small than when it’s big. It’s also clear that if u = <0,0> the plane is flat and the goat won’t go up or down however much it moves horizonatally, and it’s clear that if v = <0,0> the goat is stationary so it won’t go up or down however steep the slope is.

Finally, we can think about how different vectors u can multiply by the same vector v to give the same dot product. To do that, imagine that the plane is one side of a V-shaped valley whose edges and base are horizontal, and the goat wants to get to the top or the bottom, depending on whether the dot product is positive or negative. The dot product represents the height to the top or bottom of the valley, and u represents the slope of the valley. Now imagine you’re looking at the goat from above, and it wants to get to the top. There are lots of ways it could get to its destination. It can run straight uphill, and then it won’t have gone so far horizontally. Or it can set off almost at right angles to the slope, and then it’ll go a very long way horizontally before it gets to the top. Or it can do something in between. But it can’t go at right angles or an obtuse angle to the slope, or it’ll never get to the top. And there’s also a minimum horizontal distance it has to go. Viewing from above, you can draw a little circle around the goat and see that if the circle is small enough everything inside it will still be in the valley.

So there’s my visualization for the dot product. I like it! Commutativity holds, as it must, but it doesn’t immediately drop out of the setup the way it might if u and v represented more similar sorts of things. I won’t try making a virtue of that. But I think it’s OK, and I like it better than the other ones I found, and assuming I’ve not made mistakes with it, I think other people would like it too. I’m sure there are other adequate ones out there, but not everyone is aware of them, and coming up with these things yourself is all part of the learning process anyway. I didn’t really get what the deal was with the dot product, and now I think I do. The dot product is the goat product.