Thursday, April 21, 2016

Three Projects Involving Dispensing With Mathematical Objects



One of the many unfortunate things about academic fashions is that when a popular project goes out of fashion, superficially similar-looking projects which don't face the same difficulties can be tarred with the same brush.

Many people (myself included) feel cautious pessimism about formulating a satisfying nominalist paraphrase of contemporary scientific theories [one major issue is how to make formulate something like probability claims without invoking abstract-seeming events or propositions]. But we wouldn't want to over generalize.

 In this post I'm going to suggest three different motives for seeking to systematically paraphrase our best scientific theories (and many other true and false ordinary claims) in a way that dispenses with quantification over mathematical objects, and note that the requirements for success in the first (ex-fashionable) project are notably laxer in some ways than the requirements for the others.

[Note: I don't mean to endorse any of the projects below, but I think that 3 and Burali-Forti based versions of 2 are at least interesting.]

Three Motivations for Paraphrasing Mathematical Objects Out of Physics
  1. General rejection of abstracta: You deny the existence of mathematical objects because you think allowing any abstract objects are bad. (this is the classic motive)
  2. Explaining special features of mathematical practice by rejection of mathematical objects: You deny the existence of mathematical objects because you think that not taking mathematical existence claims at face value is allows for the best account of certain special features of pure mathematical practice, (e.g., by mathematicians’ apparent freedom to choose what objects to talk in terms of/disinterest in mathematical questions that don’t effect interpretability strength, or by the Burali-Forti paradoxes in higher set theory) not to take apparent quantification over mathematical objects at face value. 
  3. Grounding math in logic/bringing out a claimed special relationship between math and logic: You may allow the existence of mathematical objects, but you’re moved by the close relationship between an intuitive modal notion of coherence/semantic consistency/logical possibility and pure mathematics to seek some kind of shared grounding and think that the coherence/logical possibility notion looks to be the more fundamental. As a result, it seems promising to seek a kind of "factoring" story, which systematically grounds all pure mathematics in facts about logical possibility, and all applied mathematics in some combo of logical possibility and intuitively non-mathematical facts.


 Distinguishing these motivations/projects matters, because what you are trying to do influences what ingredients are kosher for use in your paraphrases: 

If you have the first motivation (defending general nominalism), you need to avoid quantifying over any other abstracta, including: platonic objects called masses or lengths which don’t come with any automatic relationship to numbers, corporations, marriages, sentences in natural languages, metaphysically possible worlds, and (perhaps) propensities. 

But if you have the second motivation (defending mathematical-nominalism-adopted-to-explain-special-features-of-mathematical-practice), then quantification over abstracta which don’t have the relevant special feature, (e.g., objects in domains where we don't think appear to have massive freedom to choose what objects to talk in terms of or objects which don't give rise to a version of the burali-forti paradox), is fine. 

And if you have the third motivation, then quantification over any objects which aren’t intuitively mathematical-- or aren't mathematical in whatever way you are claiming requires a special relationship to coherence/semantic consistency/logical possibility -- is OK.



Tuesday, April 19, 2016

Hello World (again)!

As you can see, I haven't posted to this blog for ages.

I've been busy a) enduring the horrors of the job market b) getting a sweet 5 year postdoc (I still can't express how luck and grateful I feel) c) finishing a stack of old papers and d) writing a zillion page monograph to answer a minor technical question about Potentialism and logical possibility which my advisers asked in grad school (and then rewriting all the proofs 3+ times because a grumpy mathematician friend didn't think the prose was clear or concise enough!).

But now that I have time to focus on new research, I'm thinking it might be fun start blogging again. I'm certainly touched by the number of lurkers who still turn up to check this blog out.

Let's see how things go!

Monday, April 18, 2016

Art and the Examined Life

1.
It's a common thought that works of art can (somehow!) imaginatively suggest different ways of approaching and experiencing the objective world around us. For example, a character in Rebecca Goldsmith's curious novel The Mind Body Problem* says

``The interesting thing about art is your being presented with another's point of view, looking out at the world from his perspective, seeing the dreaminess of Renoir's world the clarity of Vermeer's, the solemnity of Rembrandt's, the starkness of Wyreth's.''

Some philosophers and literary scholars have suggested that engaging with such works of art is valued/valuable because it helps us understand and sympathize with other possible points of view (whether these really were occupied by specific artists or not) -- and might thereby make us nicer to people. Art experiences might make us more inclined to try to be nice because we are more sympathetic to certain people, and they might make us better at actually doing nice things because we understand these people better.

But  (to my knowledge) there's no clear empirical or folk-psychological case that having deep artistic experiences does make people significantly nicer. And I'm inclined to be skeptical.  There's a funny line in C.S. Lewis somewhere about how any inclination to think that art makes people more virtuous will be dispelled by asking an English professor to think about their colleagues. (Lewis was an English professor at Oxford and Cambridge).

2.
I'm gonna suggest a different idea about why we might highly value art for its power to evoke a different way of looking at things (in addition to valuing it as a source of pleasure, a tool for distraction etc.).

Maybe art experiences are valued for expanding our knowledge of how it would be psychologically possible for us to approach the world (including what adopting such approaches would feel like from the inside). In doing so, they help us a) live an examined life and b) choose how to live by expanding our sense what the space of (psychologically accessible) options is like.

For great works of art seem to reveal the possibility of ways of approaching and experiencing life [noticing things, finding projects appealing, reacting emotionally] which it would have otherwise been very hard for us to first personally imagine (imagine Jane Austen reading Nietzsche, or Nietzsche reading Jane Austen!). Like Hume's first taste of pineapple (and unlike his first experience of a missing shade of blue) such art experiences expand our knowledge of what kind of experiences it is possible to have.

This has two benefits.

First, art can help us adopt a life we want to live, in approximately the same way that travel or visiting different social scenes does -- by making us aware of regions within the space of possible approaches to life [i.e. the space of options which are at least sufficiently psychologically possible for us to take up that we can imaginatively simulate them/enter into them for a while] is -- hence potentially aware of attractive options which were hitherto overlooking.

Second, I suspect that merely getting this kind of knowledge of psychological/phenomenological possibility space (whether it ever gives us practical benefits or not) contributes more to `living an examined life' (in whatever rough intuitive sense that seems desirable) than many big pieces of awesome philosophical knowledge would (e.g. knowledge of the right solution to the liar paradox).

*[The main character is a philosophically-trained mathematics groupie!]

Tuesday, February 7, 2012

Kripke's Paderewski and Frege's Common Coin

Maybe this was already obvious to everyone but ...

Kripke's Paderewski example can be modified to refute both the attractive principle Frege's Common Coin, below, and contemporary weakenings of that principle which exempt indexical and demonstrative sentences, or require that all speakers be normally linguistically component.

Frege's Common Coin:
a) When two sincere speakers "disagree over" a sentence*, there is a single proposition expressed by this sentence in this context which one believes and the other does not believe,(and indeed believes the negation of).
b) When two sincere speakers "agree about" a sentence, there is a single proposition expressed by this sentence in this context which they both believe.

*[I realize this is an awkward locution, but I just mean the intuitive kind of disagreement which occurs when I say "snow is red" and you say "snow isn't red" but doesn't occur when I say "I'm tired" and you say "I'm not tired". It sounds far more natural to say `disagree over a proposition' but we will see that it is actually not clear whether these scenarios involve disagreement over a proposition.]

Consider the following drama involving Pierre, a man in a thought experiment of Kripkie's who knows the musical statesman Paderewski in two different ways, and doesn't realize that Paderewski the pianist is Paderewski the anarchist. Suppose all the following utterances are sincere,
Act 1 (musical evening)
pierre:"Paderewski is tall" p1
alice:"yes, Paderewski is tall" p2
Act 2 (on the street)
alice:"Paderewski is tall" p3
bob:"yes" p4
Act 3 (political rally)
bob:"Paderewski is tall" p5
pierre:"no, he's not!" let p6 be the proposition that Pierre *denies*

Frege's Common Coin tells us that there are propositions p1…pn which speakers express attitudes towards in all the different phases of our play, that p1=p2, p3=p4, p5=p6 and that Pierre believes p1 and does not believe p6.

But this is a very bad thing to say: By the fact that a person like Alice or Bob who has a single grip on Paderewski presumably says the same thing by asserting this sentence on the street vs. at a political rally or a musical evening p2=p3 and p4=p5.
By transitivity of identity p1=p6.
Thus Pierre believes p1 and does not believe p1. Contradiction.

[If you are worried about the fact that Pierre is unusually ignorant for his society, and hence may not count as "normally linguistically compent" substitute in the name "John". Now cases like the one above will turn out to be so ubiquitous that denying that Pierre, Alice and Bob know enough to have linguistic competency would imply that no one ever has linguistic competency for names.]

Possible Moral: If we want to take propositions to be the objects of belief then, contra Frege's Common Coin, each sentence must be associated with (something like) a class of different propositions which someone could sincerely assert that sentence in virtue of believing.

Friday, December 9, 2011

The Sheffer stroke, and the blandest antirealism ever

In college my metaphysics prof said realism was the doctrine that there is a complete true description of the world, though it might take infinitely many sentences to express this description. Do the following incredibly bland considerations about different truthfunctional connectives commit me to "antirealism"?

(1) If you want thinking about propositions to play a role in psychology, then you need to individuate propositions narrowly enough that truthfunctionally equivalent sentences express different propositions.

(2) If you individuate propositions as narrowly as required by (1) then tautologies involving the sheffer stroke will be different propositions than any corresponding sentences which use more standard truthfunctional connectives.

(3) The same argument goes for all the other infinitely many n place truthfunctional connectives.

Conclusion: no language with finitely many basic connectives can express all true propositions. No finite language can be used to give a complete true description of the world. In particular if the a language L has only n place truthfunctional connectives then there will be tautologies using n+1 place truthfunctional connectives that cannot be expressed in L.

"Just what I mean by the term"

Whether or not you think there are philosophically interesting facts about analyticity, normal people do respond to certain a) epistemic challenges and b) calls for scientific/philosophical explanation by saying 'that's just what I mean by the term'.

So I think any philosopher who doesn't want to accept massive error about justification and argument owes some account of what is going on here. What does this expression do?

Maybe one thing the claim 'x is just part of what I mean by the term' maybe does is diffuse the expectation that there is an interesting natural kind or scientific fact in the neighborhood. Here's what I mean...

Suppose I had a made a map with mountain ranges labeled and you said, how come all mountains have a slope of at least 1 percent (cf. wikipedia definition). If you said "how do you know that mountains all have a slope of less than 1percent?" or "why do all mountains have a slope of less than 1 percent" I might say `that's just what I mean by the term mountain'.

I *think* hearing this might helpfully lead us to rule out possibilities like the following: 99% of paradigm mountains are caused by a certain geological process, and this process always produces things that have a steep slope, and then reliably maintain this steep slope through the process of erosion. If there was some more natural kind in the neighborhood of "mountain" then there would be a less trivial answer to questions like 'why do all mountains have at least a 2% slope?' and 'how do you know that all mountains have at least a 2% slope.

Tuesday, November 1, 2011

Pictureability and Definiteness in Mathematics

A fairly popular position in philosophy of mathematics is the following: there are definite right answers to questions about arithmetic, but not to questions about set theory (especially not to questions like the Continnum Hypothesis whose answers can be changed by forcing), because we have a clear notion of what the numbers are supposed to be like but not what the sets are supposed to be like.

This claim about clarity is sometimes supported by appeal to notions of mental picturing. The idea would be that we can picture all the numbers, but that we can't picture things like `all the subsets' of the numbers, or the hierarchy of sets as an item that contains, at each level, all subsets of the levels below. But I want to ask: In what sense can we picture all the numbers? In what sense CANT we picture the sets?

All the ways I can think of of picturing all the numbers involve some kind of "..." e.g. you might picture all the numbers by having a mental picture like the physical picture I have drawn below:

| || ||| |||| ....

When we use this picture to conceive of the structure of the numbers, we are employing a highly non-trivial method of picturing, which lets us represent the state of there being a countable infinity of numbers by considering some finite collection of drawings like the above.

And people certainly do *in some sense* draw pictures of the hierarchy of sets. At the beginning of a set theory class (and at the beginning of my college analysis class) the professor will frequently draw up a V on the board. In a way that is a picture of the heirarchy of sets. And, presumably, one could mentally picture the heirarchy of sets in the same.

For this reason I am (a little) skeptical about justifying the claim that we have a definite idea about the sets but not the numbers by appeal to considerations about what we can mentally picture. We can picture the numbers if we are willing to let "..." represent the idea of all finite successors to a stroke symbol. We can picture the hierarchy of sets if we are willing to let the increasing girth of the V represent the idea that each level of the hierarchy of sets contains all possible subsets of the level below. What (mathematical) objects one can picture depends on what methods of picturing we will accept, as allowing one to genuinely conceive of a situation by entertaining a mental picture.

At the very least, I think that if someone wants to argue that claims about numbers are definite and claims about sets are not on the basis of appeals to mental pictures, they owe us some kind of story or explanation for why one method of picturing is OK and the other is not.