Thursday, September 6, 2007

Anti-Quinean Hoaxers?

This NYT article quotes a letter in which Quine tells his friend’s kid about an arithmetic trick.

It is interesting to hear about the private life of the great man, but I think the comment section is even more interesting: rather than proving to themselves via normal elementary school math why the trick *must* work for all positive integers many posters seem to just try a bunch of cases. e.g. “The algorithm does not work for 29 x 31.” “In fact, it does not work for the number 29 at all. Why is that?” Then, sometimes they forget to take into account the first column (which causes trouble only when the first number you are trying to multiply is odd) so these cases seem to ‘falsify’ the claim, from which they conclude that the algorithm has “holes”. They even guess (apparently purely inductively) what the holes might be e.g. ‘all pairs of numbers starting with 29’. Other responders disagree by going through their observations of the particular instance in question. Then the same thing happens with another pair of numbers!

Is this just a case of decaffeinated blog posting or a bit of sly performance art which envisages what the state of mathematics would be like if we *did* form and revise mathematical beliefs in the same way as other more traditionally empirical parts of our web of belief?

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Sunday, August 19, 2007

realist Pen Maddy, waffles

I just bought "Organic Vanilla Mini-Waffles 8 sets of 4 waffles". Hopefully the ur-elements are in the same box. Read more!

Monday, June 25, 2007

from L.W.'s notes on Culture and Value

"If it is true that Mahler’s music is worthless, as I believe to be the case, then the question is what he ought to have done with his talent. For obviously it took a set of very rare talents to produce this bad music. Should he, say, have written his symphonies and then burned them? Or should he have done violence to himself and not written them? Should he have written them and realized that they were worthless? But how could he have realized that?"

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Friday, June 22, 2007

incompatible complaints?

A traditional objection to empiricism is that you couldn’t learn math empirically because math doesn’t make empirically testable predictions. On the other hand a traditional objection to formalism is that it would be surprising if a mere formal game had the scientific applications which math does. So prima facie it seems like on the one hand we are saying that the problem with empiricism is that math *doesn’t* make empirically testable predictions while on the other hand the problem with formalism is that math *does* seem to make such predictions and these predictions turn out right. So it seems like these traditional objections can’t both work.

One natural/obvious thing to say here would be that the applications problem for formalism is not that math itself makes empirical predictions (as the empiricist would like) but that it can be combined with a lot of empirical stuff to make predictions. But if the “applicability of math” just consists in the fact that mathematical statements can be usefully combined with a lot of empirically discovered stuff to make correct predictions this doesn’t seem like much of a problem for the formalist. Given two complex systems like the game of producing mathematical proofs on the one hand and the physical world on the other hand it would be more surprising if we *didn’t* find some parallels between the two. Newton et al just noticed the bits of math that seem to match certain transactions in the physical world. For all we know you *could* do just the same with any sufficiently complex formal game.

Specifically, the feature of math which (immediately seem to) pose a problem for the formalist are the *direct* applications – the claim that when there are 2Fs and 2Gs there are 4 F-or-Gs and predictions about what a computer will do so long as its electricity behaves in a certain way, or whether you will literally be able to tile a certain floor with certain physical features and certain tiles or whether a someone doing a certain kind of formal manipulations will get a certain string. It’s not surprising that some connection can be found between a complex formal game and what goes on in the external world, but here the ‘arbitrary formal manipulations’ of mathematical practice seem to call the shots and directly make predictions –and then these predictions turn out to be correct. If one allows that the formalism of math has direct predictive applications then it does indeed seem miraculous that the particular mathematical game which we ended up with makes the right predictions.

But if math does have these *direct* predictive applications (if this program halts you should expect to find a mouse in the wiring, no one will ever manage to cover a flat floor with tiles in those shapes) which turn out to be correct when tested then how can we make the other traditional objection that ‘you can’t learn math empirically because math doesn’t make empirical predictions’?

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Thursday, May 17, 2007 will

this really speaks for itself, i think Read more!

Sunday, May 6, 2007

Win Sharon’s Money #4

Quine, Backsliding

It’s been a while since I have come to a philosophical question such that knowing the answer to it was worth $5 to me but here we have the latest Win Sharon’s Money with a prize of $5 to the first answer that convinces me (I will send out an email saying so) or what I think is the best one if none of them convince me. This one is about defending a very famous position of Quine's, so it should be on the easy side…

If you are feeling public feel free to post your answers as comments here on the blog, otherwise email me:

So, as happens with alarming frequency whenever I think about stuff in the neighborhood, my grasp of Classic Quine has gotten unstuck. What did it this time was reading a bit of Davidson, in case that helps you see where this is question is coming from (not to imply that Davidson would agree with this point, as I understand it he wouldn’t, which is what got me thinking)…

The question:

Why can’t you take into account facts about whether a person says S in situations where it would be appropriate/normal to say that P as well as those about whether they assent to S in situations where P in deciding whether or not to attribute someone the belief that P? (If you did it, seems like this would cut down on a lot, though probably not all, indeterminacy)

Specifically, what if our ideas about when it is appropriate/normal to say P are sometimes not just some kind of empirical/sociological knowledge about what 21st century western humans like to say but are part of our very conception of what it is to mean that P.

Consider the example –from W I think, though I don’t remember where - of a person who is trained to say the words ‘I think that…’ before every assertion. A person who was taught language in this way wouldn’t be a person who refused to make claims about anything which extended beyond their own epistemic state (wouldn’t you agree?). Rather, using the words ‘I think that’ in this uniform way would deprive them of their usual meaning. So this person’s sentence ‘I think it is raining’ would be much closer in meaning to my sentence ‘It is raining’ than to my homophonic sentence ‘I think it is raining’.

I think this example shows us (among other things) that the evidence relevant to whether a given person’s sentence S means ‘I think that it is raining’ goes beyond their *assenting* to S in the right circumstances. For, the conditions under which one should *assent* to ‘I think it is raining’ and ‘It is raining’ are the same. However the conditions under which one should *assert* ‘It is raining’ and ‘I think it is raining’ are quite different (we use the latter to signal respect and the existence of disagreement or hint that there is a certain kind of relevant justification which one doesn’t have). Now I claim that there is a difference between meaning ‘It is raining’ by your sentence and meaning ‘I think it is raining’ and we are rationally required to attribute the former state and not the latter one to the person described in the paragraph above.

If one allows that these notions of appropriate assertion (as opposed to merely assent) as being part of what is necessary for a person to mean P by their sentence S then it seems like a lot of traditional indeterminacy disappears. So, for example, there are definite (though relatively limited) conditions under which it is appropriate to assert ‘here is an undetached rabbit part’ or ‘the spatial complement of a rabbit is presently avoiding this spot’ which differ from those under which it is appropriate to assert ‘here is a rabbit’. Thus one might think that more detailed consideration of the way that a person’s linguistic behavior constrains their meaning (it’s not just a matter of assenting to what’s true but of asserting what’s appropriate) removes a lot of apparent indeterminacy of reference.

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Thursday, April 5, 2007

Fun with Quotes

Back when I was TFing for QR22 I used to pass the time by making up puzzles. At some point, I came up with some quote puzzles (of the standard use/mention type) which I proposed to Peter as extra credit problems. He thought we had best not use them, which in retrospect was a wise decision. Anyway, no reason they should go to waste, so here they are!

To answer these puzzles, all you have to do is put a certain number of quotes into the given sentence to transform it into a truth. To give you a feel for the locutions I use, the following sentence has two quotes in it, the first of which is an opening quote, and the second of which is a closing quote.

“Boston” names Boston

Puzzle (1) below is my attempt to create a slightly more challenging limerick puzzle than the well known one given in most logic classes which features the sentence about Boston above. One of (2) - (4) throws self-reference into the mix. One of the nice things about puzzle (4) is that (so far as I know) it has a unique solution. (6) is inspired by part of Dave Gray's recent Eminees presentation. I might post hints in the comments. Enjoy!

(1) A few quotes placed right help construe
the sense of this jumbled word stew:
James names names names James
names names names James names
names names James names names, which is true.

(2) This sentence has the quoted expression this in it, but there is no instance of it unquoted.

(3) This sentence has exactly two instances of in it.

(4) The number of quotes in the sentence on this page beginning with the words the number of quotes on this page is three, and moreover, they are all opening quotes occurring before the first letter t in it.

(5) This sentence uses but has no mention of the word akimbo.

(6) Names name names name name.

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Saturday, March 24, 2007

A little Davidsonian Wisdom

Nothing terribly deep, but something worth reminding ourselves of every now and then:

"Plato, Aristotle, Descartes, Spinoza, Leibniz, Hume, and Kant, to pick a few winners, recognized no lines between metaphysics, epistemology, moral philosophy, psychology, philosophy of language, and the history of philosophy, and neither would we if our universities and colleges [and departmental workshops] didn't often compel us to think of ourselves and our colleagues as belonging in one or another field." (from "Aristotle's Action" p. 291) Read more!

Monday, March 19, 2007

Agreement, Political Authority and Procrastination

Political philosophy is not my area either, but I wanted to add a post here to keep up the momentum of this blog and since I’ve been thinking about political philosophy in relation to my teaching obligations, here are some undeveloped thoughts which I could be easily persuaded to abandon.

Despite all their differences, there appears to be a surface similarity between Hobbes’ and Socrates’ views on the nature of political obligation: both seem to hold that one’s obligation to obey the law stems from an agreement one has freely made (or would hypothetically make). For Hobbes, you contract with your fellow (future) countrymen to transfer your rights to a powerful sovereign and thereby incur an obligation to obey the law. For Socrates – according to one of the various arguments hinted at in the Crito – you agree to obey the laws of the state by choosing to live in it. Socrates could’ve moved away from Athens but didn’t and thereby incurred an obligation to obey its laws. Unlike Hobbes’ view, Socrates’ isn’t exactly a general theory about political obligation; Hobbes’ theory is mean to apply across the board whereas Socrates’ argument applies only in cases where the agent freely chose to remain in (or emigrate to) the state in question in full knowledge of the fact that doing so would put him under political obligation.

But though both views seem to ground political obligation in an agreement, the mechanism by which the agreement gives rise to obligations is very different in each case. For Hobbes, our agreement sets up a powerful sovereign who in turn makes it self-interestedly rational for each of us to obey the law. (On this reading it sounds rather odd to speak of an obligation to obey the law; the sovereign’s subjects may have decisive reason to obey the law but it sounds odd to my ears to call it an "obligation".) For Socrates, the agreement gives rise to obligation more directly; he doesn’t spell this out, but presumably the agreement works like any other sort of promise: if I promise you that I’ll do X, I have thereby incurred an obligation to do so.

Assuming the interpretations of Hobbes and Socrates I’ve just sketched are (close to) correct, I’ve come to think that neither view succeeds in grounding the obligation to obey the law (or the corresponding rights of rulers to command) in anything plausibly thought of as an agreement.

Socrates’ argument actually presupposes the notions of political authority and obligation thus does not succeed in accounting for them. Socrates imagines that the Laws of Athens tell him to either leave the city or to obey their commands. But for his choice here to be morally transformative in the way he thinks it is (i.e. if it is to yield obligations towards the Laws), the Laws must already possess political authority. If I stop you on the street and tell you to leave the city or pay me $50, I haven’t really succeeded in doing anything other than to utter some powerless words (and perhaps to puzzle or annoy you). I certainly haven’t made it the case that your staying in the city constitutes an agreement to pay me $50. What do the Laws of Athens have that I don’t such that Socrates’ choice to remain in Athens does give rise to specific obligations to obey the laws whereas your choice to remain in the city doesn’t give me a claim against you for $50? I’m inclined to say that it is political authority. Socrates’ decision to remain in the city cannot be what confers political authority on the Laws since the decision is only morally transformative in the way he supposes if the Laws already have that authority.

(I’m muddling here but I can’t quite see my way clearly. Even if we assume that the Laws have the authority to require Socrates to choose between exile and obedience, before he actually makes his choice to remain in the City there doesn’t seem to be any sense in which he has an obligation to obey the law. I’ve been speaking as if political authority and the obligation to obey the law are mirror notions but that’s probably not the case.)

Hobbes is easier since it is really rather straightforward that what does the work of generating reasons to obey the law is not our agreement with our fellow countrymen, but rather the sovereign’s power to punish lawbreaking. There is a sense in which the agreement is a necessary condition of our overriding reasons for obedience but Hobbes’ social contract is not morally transformative the way promises, agreements and freely entered contracts are normally thought to be. Read more!

Saturday, March 17, 2007

When is ‘seeming to see’ enough?

There are a lot of different cases where people claim to have an experience which amounts to simply seeing that some P is the case. Chess players ‘see’ the weakness of a pawn structure, potters ‘see’ that a certain pot will crack when fired, people having religious experiences suddenly ‘see’ that god is real and cares about them, intro math students ‘see’ that you can’t put 4 puppies into 3 boxes without putting more than one puppy into some box, and nearly all ordinary people can ‘see’ that you can’t pick a lock with a banana and, of course we can see that we have hands. This raises a some natural questions: how much justification do these experiences-as-if-of-seeing-that-p provide? And, are there natural divisions in the list of examples I just gave, or do they all have the same epistemic status?

At present I am torn between two opinions about the epistemic status of these ‘seeming to see’ experiences. The simple view would be to say that any such experience where it feels like you are sensing that P provides prima facie warrant for believing that P. The case of religious experience gives me some qualms though, and one can cook up even more implausible cases. Some people claim that they can feel via a sense of forboding in their heart that their twin or loved one is in trouble. Or, imagine looking northwards at the clouds in the direction of Canada and ‘feeling a great disturbance in the force’ as it were, which seems to let you feel that something terrible is happening in a certain small town in Canada.

Now I hesitate to say, given this kind of example, that seeming to see that P provides prima facie justification. It’s pretty clear that in these circumstances a rational person should not immediately believe what their strange experience seems to show them but check (say, by making the appropriate phone calls) whether this experience really does reliably track how their twin is doing or what is going on in Canada. And a supporter of the ‘prima facie warrant’ idea can agree with this. But what about cases where there is no practical possibility of checking – suppose you have these experiences when you are out in the woods, or suppose God tells you as part of your religious experience that you will only get normal empirical evidence for his existence after he is dead [ed: after YOU are dead :)]? Here I am inclined to think that you shouldn’t believe what you seem to see at all, until you have checked the reliability of your experiences as if of seeing – and that once you do this the amount of evidence which your seeming to see provides is proportional to the evidence that you have now accumulated that your experiences of seeming to see are reliable.

But what about the case of sense perception? You can’t check the reliability of your senses against something else, but surely seeming to see that there’s a table in front of you does give you reason to believe it. This leads to the second more complicated theory of the epistemic status of seeming-to-see-that-P.

On this (slightly Peacockian theory) most such experiences give one no reason to believe anything on their own. You are only justified in believing what such experiences seem to tell you if you are also frequently exposed to evidence that confirms the reliability of this supposed perception. So the chess player who ‘seems to see’ that his queenside pawns are weak only has as much reason to believe that the pawns *are* weak as he has evidence that these experiences of his are reliable (so e.g. if he is a chess master he will have strong reason to believe it while if he is infamously bad at chess like myself he will have very little reason to believe this).

BUT (here’s the Peacockian part) in some cases the experience of seeming to see that P is central to, or indeed nearly all there is to our practice of saying that P. In these cases the facts about when we ‘seem to see that P’ largely determine what we mean by P and hence what it would take for P to be true. Specifically, these facts determine the meaning of P in such a way that if we say P whenever we feel like we can ‘see that P’ we are quite likely to be right. So, for example if what tends to give us the experience of ‘seeming to see that there’s a table’ is tables then ‘there’s a table’ means there’s a table, if it is vat state T then ‘there’s a table’ means the vat is in state T and so on a la Putnam on the BIV. Thus, in these very special cases believing that P when you have the experience of seeming to see that P will be a reliable, and maybe even justified method of forming belief.

This proposal has the advantage of giving a motivated way of separating up the list of ‘seeming to see’ experiences I started with in a motivated way. We have other practices which give us an independent grip on what it would be for your twin sister to be in trouble or disaster to be striking in Canada. Thus your feeling of conviction that P remains just that until you have evidence that this sixth sense of yours is reliable. But, on the other hand, in the perceptual case we don’t have this kind of independent grip on the stuff which our five senses seem to show us. Thus your experience of seeming-to-see that there is a table plays a role in determining *what it would mean for there to be* a table there which ensures that you are justified.

So how does this sound? Any takers on the simple proposal or the split (not to say…shudder…disjunctive ;) ) proposal? New proposals? Obvious points in the philosophy of perception which I am missing?

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Conflicting conceptions of what it takes to have knowledge vs. what it takes to have *mathematical* knowledge?

Suppose someone is doing a bunch of really long sums e.g. adding 12 digit numbers, with a blunt pencil and in a hurry. Under these circumstances they are quite likely to make at least one mistake during the course of each of the sums, so (as they learn when they check over their answers) overall they get only about one sum in ten correct. Now after doing this for a while, suppose they do one more sum and, being the confident person they are, they believe that the answer to it is in fact 1789200056911 as their calculations suggest. And suppose that this is, in fact the right answer, and in this case they have been lucky enough not to make any mistakes along the way. Then do you think that they know that whatever + whatever = 1789200056911 or not?/ Is their belief justified?

On the one hand, it seems like they know since they have gone through and been convinced by a correct process of reasoning which entails that this is the right answer. On the other hand, it seems like they don’t know that the answer is that because they usually make so many mistakes that the mere fact of their computing a certain result is very little evidence that that result is correct.

My impulse would be to say that this shows that we two different standards – for mathematical knowledge and for knowledge in general which are clashing in this case. Maybe one can also get a conflict between these standards for knowledge in the opposite direction: setting computers to check the first few billion cases of goldbach’s conjecture (that every number greater than two can be written as the sum of two primes) could eventually give you very strong justification for believing it (and hence perhaps knowledge in the ordinary sense) but it would be strange to say that you know the conjecture was true if you didn’t have a proof.

Also this case seems similar to the familiar lottery example 'do you know that you won't win the lottery, when you have evidence that that your chances of loosing are overwhealmingly good?' so maybe the lottery example is further evidence that our conception of knowledge is fragmented/highly context dependent. Read more!