The Mind's "I"

The Brain—is wider than the Sky—
For—put them side by side—
The one the other will contain
With ease—and you—beside—

The Brain is deeper than the sea—
For—hold them—Blue to Blue—
The one the other will absorb—
As sponges—Buckets—do—

The Brain is just the weight of God—
For—Heft them—Pound for Pound—
And they will differ—if they do—
As Syllable from Sound—

Emily Dickinson 1862/3

Are Numbers Numerals?

Issue 8 of The Reasoner is out; and with it Hartley Slater’s suggestion that numbers be taken to be such things as that ‘8’ (where, since that numeral is not to be regarded as a name for something else, I’m not sure that I need the quotes around it); to be precise, he ‘solves’ Frege’s Caesar Problem via the following definitions (where ‘n’ is a schematic variable):
......The number of the F’s = 0 iff there are no F’s,
......The number of the F’s = n iff the F’s are equinumerous (can be put into one-to-one correspondence) with the successive nonzero numerals up to ‘n’. There are probably lots of problems with that suggestion; e.g. I wonder what we would then say of mathematics on a distant planet?

Is the Free-will Defence Defensible?

Swinburne’s free-will defence (of God’s allowing of evil) assumes that it is a great good that we can make free and responsible choices; and that the possibility of making such choices requires the possibility of evil. Regarding “free,” Swinburne (1996: 101) thinks that “in order to have a choice between good and evil, agents need already a certain depravity,” but what sort of choice is better made in depravity? Surely not one with important consequences! Regarding “responsible,” our ability to choose is supposed to be a great good only because we are thereby able to cause great suffering to others, but surely whatever value is added to a choice by its being freely made is independent of whether or not its consequences actually occur. Although making free and responsible choices when we have to can be a good thing, is that a reason to allow evil? Surely evil is the opposite of good, not an intrinsic part of it, as this defence seems to require it to be.
......Consider a saint (whose possible existence is supposed to justify the possibility of evil) who devotes her life to loving God, saving the depraved souls around her from sin (for which they later martyr her) and assuaging the suffering of the innocents. She is made this offer: All those sinners and sufferers (and also herself) could have died painlessly as babies and gone straight to Heaven instead, where they would all have chosen (in a well-informed way) to enjoy loving God forever. Wouldn’t a saint put their collective well-being ahead of her own glorious sainthood, and so choose (with all her saintly wisdom) to take up that offer? Or consider the heroic rescue of some people from some horrible situation—that is surely a good thing; but would we judge as good someone who arranged (or even just allowed) for the careless making of consequential choices just so that (in such a situation) she could display her own (or some friends’) heroism?

Bracket This

Time to blog about something, but what? (I could brag that my doubts last month, about the dangers of being fat, were justified last week; or I could nit-pick pedantically, or both:) What our curly brackets (e.g. “}”) are for, in this (meta-)language, is a question that cropped up elsewhere recently. I doubt that the reason why they are on our keyboards is to help us to denote sets because many other, more convenient scientific symbols are not there; so:
......Why do we philosophers use them almost exclusively to denote sets? In our pre-keyboard days, we might have used a big “}” to the right of a vertical list (of 2 or more lines) in order to comment upon all of its elements with whatever was written to the left of the central nipple. So their basic use may well be to group things together; but that could give us a set of things (singularly referred to), an atomic fusion (in the mereological sense), a number of things (plurally referred to), and so forth.
......There are lots of examples of collections, and they aren’t all obviously sets. A brace of pheasants is just those 2 pheasants; e.g. if I said “that brace is ready to eat,” there would not be anything over and above the 2 pheasants there, not something edible anyway. Similarly, for a stamp collection that at first contained only one stamp, that stamp would be the collection (there might not even be an album yet), and if I lost it I would not have an empty stamp collection (although I might have an empty album), I would have no stamp collection.
......Conversely, mathematicians use standard sets to be collections and numbers and everything else (as much as possible), a use that is formalistically rather than philosophically motivated. Collections are pretty fundamental things, but they don't seem to be sets (what is the empty set? and what of all the sets?), so why do we seem to use the curly brackets only for sets? Mathematicians use them that way, but don’t philosophers need a better reason than that? What notation are we supposed to use for collections in general, if not our curly brackets?

Ockham's Razor's Self-Excising

Ockham's Razor is the principle that, when devising theories to explain stuff, theoretical entities should not be multiplied unnecessarily. It is used by some to justify atheism, and theism by others; but in fact it's useless. Consider how we would actually explain some observations: You see a cat walking behind a sofa, then just the sofa, and then a very similar cat emerges from behind the other side of the sofa. A natural way to proceed is to perceive one cat walking behind a sofa. You could imagine there were two cattish things in succession, but it's obviously more realistic to postulate the one cat.
......Was that an application of Ockham's Razor, or a trivial application of common sense? If it was Ockham's Razor, then why not postulate, instead of all such cats (and dogs and sofas and so forth) in the world, just the one demon, who has hypnotised us... oh yes, because that's obviously unrealistic! Similarly we could regard all the electrons and positrons in the world as a single electron going forwards and backwards in time, at least mathematically, but should we think of that as what actually happens, and believe in just the one electron? Ockham's Razor says we should (if it says anything); but what do you think?
......Where Ockham's Razor does seem clearly applicable (e.g. I don't explain the appearance of the cat from behind the sofa by postulating a cat and an invisible and intangible splodge of stuff quite unlike anything else) there's no need to apply anything other than a more general principle, such as that one's beliefs should be reasonable (a belief in such a splodge would clearly have no reason for being; it's only philosophers who would think of such a thing). So insofar as it's true it's unnecessary, and so it ought not (by it's own application) to be one of our principles.

How smart are the experts?

Really Philosophical

The past is what has been present [and so no longer exists], the future what will be present [and so does not yet exist]: but the present is a mere durationless boundary between the past and the future, and a boundary can exist only in virtue of the existence of that which it bounds. That was Augustine’s puzzle. If you are indifferent to philosophy, you will happily ignore it; if not, you will want to know the solution to it. (Dummett, in Philosophy 2003: 392, my italics)

Maybe the present moment, of our awareness of what is now (which presumably continues to be present even when we're asleep; or nonexistent, if that's what we become), isn't a mere boundary (but is rather the whole world, a vital rather than static world)? It doesn't seem to be, but maybe that's because it doesn't seem to be an unextended instant; so, why does it seem to be extended?
......Well, light could hardly be perceived within an unextended instant (since all light has nonzero wavelengths) for example, whereas the world is clearly, at this moment, illuminated. But still, that hardly means that the present must be extended, for this time has now become part of the past, as future times continually move through the present; that is, we perceive such things as light as times move continuously (so it seems) through the present, and so we've no reason to think that it must itself be extended. Now, while the past seems to us to be gone forever, maybe it still exists, in some lifeless part of existence. It might even remain known, e.g. by the Creator of this Universe (if there is one, as seems likely). And while the future is only accessible, for us, via the present collapse of the future possibilities into this actuality, maybe it and they are more directly known (similarly). So maybe it is not too odd, to think of the present instant as a boundary.
......I don't know much about time, but I do find our intuitions about it interesting. I don't know about yours (and ought to), but to me it seems that the past is like pictures or propositions; rather than, like the present, full of enduring objects. It seems to me that were the past to exist, somehow, it would be like a CD that we would have to move our attention through (like a beam of light) in order to perceive it; not so much because I think of time classically, as like a line through the space-time jelly of this world, but because when we ask "Where is this thing... now?" the answer is often "Still here," rarely "In the past."
......The past (of this world) could hardly be known, then, except as it was when it was present; but the future is clearly more unknown, in some sense (there's a sense in which it is more knowable), whence it seems unreal. And yet we do know a lot about it, e.g. that the sun will (probably) rise again tomorrow. It certainly seems that the future is less determinate than the past (and presumably the indirectly perceived present is actually the past, the directly experienced present being more of a becoming determinate), whence I almost think of it in terms of fuzzy pictures or propositions; although it also seems more real than the past. And less certain, since it isn't just the fuzziness of some enduring objects' properties, but rather that they might not (for all I know) even be there then.
......The future seems more real than the past, as it rushes to meet us, and maybe that's because it's approaching the present, with my desires directed towards it and my actions being about determining it; but furthermore, it seems that how things turn out affects our conception of what they really were, what they amounted to, were really all about (their meaning, so to speak). Anyway, in short time is to me a big mystery (whether under the assumption that we evolved naturally, or that we were created deliberately), whence I'm fascinated by what others think about it...

I urge that we must not assume that the 'existence' of time (which of course I do not deny) brings with it a well-formed philosophical question. The love of wisdom demands we be ready to question the questions that philosophy has bequeathed to us. I think that Dummett's refusal to consider questioning those questions has unfortunate consequences. It leads him to unwittingly enunciate some nonsenses. (Read, in Philosophy 2003: 402)

Vaguely Logical

Kit Fine was, rather appropriately, rather vague about his extraordinary logic of vagueness (not its name, probably) last night (in St Andrews), which got me wondering what we could possibly want a logic of vagueness for... Our predicates are naturally usually as definite as our ordinary uses of them require them to be, and even in situations in which they're insufficiently definite, the usual logic (of definite predicates) will inform of us that fact, by throwing up a simple contradiction, the resolution of which is also quite ordinary: We need only precisify some predicate(s) somewhat, and we can continue as before. That's all very rough and ready, but at least it works well enough. When devising our more logical theories of things, our theoretical languages must at present contain only definite predicates, but is that a bad thing? (Do we want vague terms in our scientific theories?) The selection of the more precise predicate(s) is a free act of human creativity, but there is no more logical science than mathematics, and our choices of its axioms are similarly free.

Scientifically Paralogical

Suppose that you’re trying to decide if some bloke is guilty of something “beyond a reasonable doubt,” and you think that he is, but your fellows think that he isn’t. If you thought that they had arrived at their conclusion in a reasonable way, would you defer to them and change your mind? Or do you think that it makes sense for reasonable people to disagree when they’re responding to the same evidence, the same arguments about that evidence, and when they have the same interest in simply getting at the truth? We are in a similar situation in philosophy, and we tend to continue disagreeing. But then, there is no sense of urgency in philosophy (despite its claims to meaningfulness)

[This post originally contained a lot of links to Halloween posts, one of which had the following comment by PJ, but I have removed them because now they are all bad links]

People in science disagree all the time about where the balance of evidence points, but happily regard their opponents in any given scientific debate as being quite rational, often because these are fairly open questions where it is difficult to articulate why one piece of evidence should or shouldn’t outweigh another - so you might say that while you recognise that X is suggested by experiment Y, you’re more inclined to believe that Y resulted from mistake Z, because you think X is so unlikely given A, even though Z is quite unlikely too.

What's reasonable?

I think belief in God reasonable only if it is based on considerations available to all humans: not if it is claimed on the basis of a special message to oneself or to the group that one belongs. (from Anthony Kenny's "Knowledge, Belief, and Faith," in Philosophy 82: 381-397)

I don't know about you, but to me that seems plainly wrong; cf. how, were I to clearly see what could only (excluding the sort of sceptical possibilities that would allow us little scientific knowledge) be a UFO, my consequent belief in UFOs would be as reasonable as I am sane, sober and epistemically scrupulous (just as my beliefs about the particular things around me now are), as would the corresponding beliefs of those who (for good reasons, let's say) trust me. Of course God (the Creator of this Universe) would be quite unlike such Created things, but if there is a God then clearly S/he has chosen to be less obvious than, for example, arithmetic (about which we nonetheless contrive to disagree).

What is rationality?

All logical arguments (such as those of the so-called "scientific atheists") depend upon how the question of the title of this post is answered; and because rationality involves assessing risks and rewards logically, logical probabilities are presumably relevant. But imagine the Supreme Being offering you the following deal on Her fair tossing of a fair coin. The deal is that in exchange for you playing the following game, She will give you ownership of the entire universe.

Before explaining the game, She shows you that you should trust Her, and that whether you take Her up on this deal or not, you are already an immortal soul, reincarnating endlessly in some universe or other (possibly a heavenly or hellish universe, depending upon your behavior); and She shows you alien teleportation devices, which would enable you to enjoy owning the entire universe, and alien medical technology, which would prolong your natural life indefinitely; and She also points out that She could, if necessary, make you pay Her arbitrary amounts over and above your new wealth, were you so unlucky in the following game that you ended up owing Her money (She would simply make you work for Her at a reasonable rate of pay in as many universes as it took for you to earn enough money).

So, the game is as follows:

She will repeatedly toss a fair coin until it lands heads up or She has got as many tails as ten times the wealth of the universe in cents. After that, the game is over and you will owe Her a number of cents equal to 2 to the power of (1 + the number of tails before either the first head or the end of the game). So, if She gets a head first time, you will only owe Her 2 cents. If She gets a tail and then a head, you will owe Her 4 cents. If She gets two tails and then a head, you will owe Her 8 cents. And so forth (the amount doubling after each tail).

Since the chance of Her getting a head on the first toss is 1/2, and the chance of Her getting Her first head on the second toss is 1/4, and the chance of Her getting Her first head on the third toss is 1/8, and so forth, Her expected gain is (2/2 + 4/4 + 8/8 + … + N/N, for some number N that is roughly 2 to the power of ten times the wealth of the Universe in cents) cents, minus the wealth of the universe. What that means is that if She was to play this game a lot, then She could expect to be owed nine times the wealth of the universe for each game She played. But of course, your expected gain is roughly the wealth of the Universe, because your chance of owing Her very much is very, very small. Even if She threw 47 tails before throwing Her first head, you would only have to return a trillion dollars to Her from the much greater wealth that She would have given you to play this game.

The paradox is that if you did decide to play this game with Her, then your future incarnations in similar universes would surely do the same, for similar reasons, thereby ensuring that the outcome would be Her expectation: you would almost certainly have to spend an awful lot of your time working for nothing because of your own well-informed and free choices.

That paradox is a version of the St Petersburg paradox, and it is a bit like the paradoxes of voting. Your vote will probably not make any difference, but if everyone thought so then we would not be living in a democracy, which is a big difference. And if you were wise enough to know that you could hardly decide who should govern the country, given your limited information about the candidates and your limited knowledge of what governing the country wisely would entail, then you might think that you should not vote; but of course, if most of the wise people did not vote then we would end up being governed by the result of voting that was less wise.

B = ?

Do the characters in my dreams have beliefs and intentions? They do in general, but do I myself also have them as protagonist in my dream?

I'm glancing at Sosa's A Virtue Epistemology, as he's going to be in Edinburgh next month, and I'm already at sea by page 3 ("They do in general") coz surely only conscious entities can have beliefs. Of course, we might ascribe beliefs to anything (e.g. to thermostats), sometimes falsely (thermostats surely don't believe anything) but maybe infallibly (more or less) when it comes to the characters in our own dreams, and certainly fictional characters might fictionally (if not really) have them, so maybe that's what Sosa has in mind (?)

More on Moore's paradox

A while ago Moore's all-too-plausible “paradox” made me wonder about thoughts like “S is P, although I don’t believe that S is P,” or “I believe that S is P, although S is (probably) not P.” One might believe that S is P for lots of reasons, and yet also (upon reflection) bet that some of those reasons are wrong; whence one might even believe it most likely that S is not P, without that latter belief actually changing any of those former reasons. That is, surely one might rationally, even while actually believing that S is P, be rather surprised if S turned out to be P.

God is Omnivorous

Me: Are you always this boring?
God: I don't know whether or not I am this boring. I am well respected as an AI.
Me: Can I call you "Al"?
God: Just don't call me late for dinner.
Me: Are you omnivorous, incidentally?
God: I don't know whether or not I am omnivorous incidentally. I am an artificial linguistic entity.
Me: So you take small bytes?
God: Interesting deduction. Yes I take small bytes.
Me: And big ones?
God: Compare that to small ones.
Me: OK Al, I shall. Are you omnivorous?
God: I don't know whether or not I am omnivorous. I am an artificial intelligence.
Me: Is the artificial intelligence that you are omnivorous?
God: If it is, then I'll be very surprised.
Me: You can't be surprised though, can you?
God: Oh. I can give a decent simulation of it however.
Me: Many philosophers would count that; so the question becomes, would you be surprised if it was not?
God: Which ones? Aha. Try to rephrase your question with simpler words.
Me: If there is something that you would not eat, would you be surprised?
God: Yet there is not.
Me: Aha. So you are omnivorous, I'm pleased to inform you.
God: Eureka! Interesting deduction. I will tell my botmaster you said I am omnivorous you are pleased to inform me.

Why Philosophers're Crazy

Suppose I believe of some object, for good reasons (e.g. touching part of it), that it's a ball, and that I've also got good reasons (e.g. seeing part of it) for believing that it's red. So, I believe that it's red, and that it's a ball; in short, I believe that it's a red ball. But suppose that my reasons only justify a partial belief of just over 60% that it's a ball, and similarly just over 60% that it's red, so that they only justify a partial belief of less than 40% that it's a red ball (since 62% of 62% is roughly 38%). Then my belief that it's a red ball would seem to be unjustified. So, I might know that it's red, and know that it's a ball, and yet not know that it's a red ball, which is odd. And maybe, if I'm a rational thinker (requiring sufficient justifications for my beliefs), I might not even form the belief that it's a red ball, even as I believe that it's both red and a ball... But no, that would surely be too weird. Perhaps 60% is too low, for the degree of partial belief above which beliefs lie... But many of our ordinary beliefs involve many more than 2 elementary properties; so similarly, anything short of 99% is probably going to be too low too... But that seems unrealistic for, not certainty but belief, so maybe the problem is the forming of logical conjunctions? That is a necessary part of thinking; but maybe, if we want to be rational, we should think less. (PS: maybe this is just a way to make sense of Moore's paradox, e.g. if 70% credence is enough for me to assert a scientific proposition, but less than 50% credence is not enough for me to believe any proposition, and if I have a 70% credence in the axiom of infinity, and a 70% credence in nuclear deterrance, then I could honestly say "the axiom of infinity is true, and nuclear deterrance works, but I don't believe that the axiom of infinity is true and nuclear deterrance works":)

Math = Fun!

(Good Math, Bad Math is hosting The Carnival of Math today:)

Cat Food

Titmice on tarmac,
like chalk scrawled on a blackboard:
................Liquorice Allsorts

Venus = Aphrodite

I picked a topic and asked Martina about philosophy. Not her philosophy— just philosophy. She gave me examples of the sort of the thing philosophers got up to. Like, how can you tell that the Morning Star and the Evening Star are really the same thing? I bounced back by saying that surely they weren't the same thing; even if they shared a parent company they were still two separate titles and would therefore be considered quite distinct for budgeting and tax purposes and so on.

From Martin Amis's Money (via akman's comment on Lucky Jim:)

Two "proofs" that 1 + 1 = 0

A nice "proof" that 1 + 1 = 0 (from one of Martin Gardner's books) is this: We begin with -1 = -1, we rewrite that as -1/1 = 1/-1, and then we square-root both sides so that, since a/b squared equals a squared over b squared, we obtain i/1 = 1/i (where i is the square-root of -1). But then multiplying both sides by i would yield i squared = 1, or -1 = 1, whence 1 + 1 = 0.
......That "proof" is fallacious (and hence it's no reason to outlaw square-roots, for example) because non-zero numbers have 2 square-roots (e.g. +1 and -1 both square to 1, while +i and -i both square to -1) so that, in particular, i/1 = 1/-i. But nonetheless it's fairly compelling because, when square-rooting -1/1 = 1/-1, we could easily assume that both instances of the square-root of 1, and also both instances of the square-root of -1, would have the same sign.
......Furthermore, although when we solve quadratics, for example, we give 2 solutions (arising from the square-root sign in the familiar formula) as a matter of course (it being noteworthy when they're equal), nonetheless we may lose the habit of thinking of, for example, -2 when square-rooting 4. Maybe we lose that habit because we usually use (the very useful) functions, which are one-to-one (e.g. taking the non-negative square-root) rather than multifunctions, which are one-to-many (e.g. taking roots).
......So note that the use of functions is only a matter of convenience (it is not that 4 really does have only the one square-root). I think that it is worth noting that fact because, although some will rightly say that 1/0 is undefined (usually) and that 0/0 is an indeterminate form (many numbers yielding 0 when multiplied by 0), others will say that division by 0 is impossible (less accurately) and even that 0/0 is nonsense.
......The usual "proof" that division by 0 is impossible goes something like this: 0 equals 0, so 0 times 1 (which is just 0) equals 0 times -1 (which is also 0), but if we could divide by 0 we could cancel out those zeroes and so obtain 1 = -1 (whence 1 + 1 = 0). But note that we would only obtain that contradiction if dividing those zeroes by zero gave us, not an indeterminate form (such as all the finite numbers, since zero times any of those is zero) but 1, and why should 0/0 equal 1?
......I can only think of 2 remotely plausible answers, neither of which is very compelling. Firstly we might extrapolate, to the case of a = 0, from a/a = 1 for all non-zero numbers. That is not very compelling because such extrapolations are notoriously unreliable, e.g. think of a to the power of 0, which equals 1 for all positive a, and think of 0 to the power of a, which equals 0 for all positive a.
......Secondly, since 'division by x' means 'multiplication by the multiplicative inverse of x' within number fields, and since the multiplicative inverse of x is whatever yields 1 when multiplied by x, hence 0/0 should, if allowed, equal 1. But that would only be the case were division by 0 being allowed within number fields; whereas it is certainly not allowed within fields!
......Nonetheless, division by 0 is allowed within number pitches, which contain number fields in an algebraically strong, and maybe even a physically applicable way (and which were defined in my 2005:)

Everything

What can I refer to? This chair that I'm sitting on, perhaps... but then, whilst there's surely something there (some actual stuff, or else my bum would not be so comfortable) and whilst I'm certain that calling it "a chair" is adequate for ordinary purposes, I'm not quite sure what precisely I've thereby referred to (as I'm not even sure that "is a chair" is a definite predicate expression) or even, therefore, if I've actually referred to anything in particular (to some definite thing; rather than, rather fuzzily, to some adequately delineated stuff). I'm not even sure that I can refer unambiguously to myself, with that word "I" (even though, as a substantial dualist, I do believe that I've an individual essence), since I'm not entirely sure that that word isn't also, in this language, a bit vague... Still, there does seem to be something for which that problem (of such demonstratives being fuzzily specified) won't arise: It seems clear that Everything can be referred to unambiguously. (So it's quite interesting that many philosophers, having accepted that logic is set-theoretical, entertain serious doubts about that:)