## Saturday, June 30, 2007

### No Credence

The end of a very wet June (not yet so weird as the butterflies last January though), but hopefully I'll be spending less time at the computer next month... Anyway, today (via TAR) I discovered Williamson's recent paper (in Analysis) in which he showed just how problematic is the assumption of regularity: that a credence (or subjective probability, or degree of belief, or of acceptance) of 1 implies absolute (not just everyday) certainty; that, conversely, if your credence in some proposition is 0 then you believe that it is impossible. But surely that assumption is clearly fallacious, like assuming that 0 length of a line means no part of a line, whereas points seem to be possible; so I'm wondering why it's so common: Is it something to do with the meaning of subjective probability?

## Friday, June 29, 2007

### Carnivals

Yesterday's Carnival of Physics contained lots of philosophically interesting stuff; and today's Carnival of Mathematics includes a couple of interesting views of the totality of the natural numbers, not only mine (as a realistic potential infinity) but also via categorification. (And speaking of ancient philosophical topics, there is a very funny digest of the Nicomachean Ethics.)

## Thursday, June 28, 2007

### Memo on Zero

Mention of Relative Identity (and fractional objects) reminds me of Kessler's Millian approach to the natural numbers: basically (as far as I can recall) the number zero is thought of as a relationship between a property (e.g. being an actual aardvark) and some stuff (e.g. what's now visible). Although I'm not a Millian, that strikes me as a pretty realistic place to start from (and seems prima facie to go with that relative identity approach in logic) because something like a set-theoretical lasso has to capture stuff and turn it into units...?

## Wednesday, June 27, 2007

### The Empty Set?

Standard mathematics is based upon an empty set (a set with no members), which is identified as the whole number 0. But while there is only one whole number 0 (the number of things that we have when we have nothing), there are many set theories. There is only one standard theory, but standards change (while 0 does not) and anyway, even were one particular theory especially elegant, coherent and useful, that would not make it true (cf. ideal fluids and Newtonian astrophysics). So the question arises, what could sets be if not formal? Well, what is a chess set? It is 32 chess pieces (and possibly also a checkerboard) of the right kind. If one of them is missing, then that is not a chess set, although it is still a collection of chess pieces—what logicians mean by ‘set’ is something like a collection.
......So, what are collections? Well, what is a stamp collection? It is a collection of stamps—all of the stamps collected by, owned by some individual. It does not include the stamp albums, just the stamps; and if it is sold then it remains the same collection of stamps, belonging to someone else. And if the collector had to sell most of his stamps, and had only one left, then his collection would consist of just that stamp—his collection would be that stamp.
......Michael Potter would presumably say (judging by his 2004 “Set Theory and its Philosophy,” page 22) that while that collection would have one member, that stamp would have a value, a weight, a design, but no member, but I think that it would only be misleading to say that it had one member. Our way of talking about a collection’s members is taken from the paradigm case of a plurality, and there the collection has members in the way that a composite object has component parts; and it is not exactly wrong to say of a simple object that it has only one (improper) part. (Similarly a particular photon has a particular energy, but nonetheless that photon is that energy; and a person both has and is his or her personality, his or her mind, etc.)
......Furthermore if our collector got rid of his last remaining stamp, then although he might have an empty stamp album, he would surely no longer have a stamp collection. And yet there is (supposed to be) an empty set, so a set is not just a collection of things—it is something more, but what? That something more is presumably like that album, but less specific; it is presumably something that all collections have as a matter of logic (rather than physics, or convention etc.).
......Now we have, in logic, classes of things (e.g. the class of men, the class of men now in this city, the class of classes), and we have empty classes, e.g. the class of aunts that are now in this room, and the class of ants that are now in this room. In one sense (intensional) those are two classes, with different modal properties (e.g. had I left the window open, ants might have got in), but in another sense (extensional) they are the same because they do not have different members. And that latter sense is the one we want because there is only one whole number 0. But surely the extensional class is just the collection of the class’s members (and we have already rejected collections), so the empty set is not a logical class. But I cannot think of anything else that sets could be, if not formal.

## Tuesday, June 26, 2007

### Potential Infinity?

Clearly whole numbers are objective abstract objects, whose properties are independent of our theories (judging by the obvious objectivity of, for example, these three words). And the natural numbers clearly comprise, collectively, a definite totality (which is therefore another thing, one collection) because a very simple, finite rule (of adding 1 to the previous number, starting with 1) completely determines what they are. Consequently their totality has qualities that are in many ways finitesque (i.e. it has properties that are paradigmatically possessed by the finite collections that we more easily envisage). We can, for example, consider how there are as many rational numbers as natural numbers, by considering a bijection between those two totalities.
......Nonetheless such simply infinite collections differ from finite collections in going on and on, in being generated by their finite rules endlessly. So although mathematical realists (such as myself) do think that they form actual collections (objective structures), if their properties happen not to be as finitesque as is commonly believed (which must be discovered, not stipulated), then their totality will be more like the potential infinity of the constructivists, in many ways, than the actual infinity of the set theorists. The underlying cause of such infinitesque behaviour (if it exists) would not be that our numbers are mental constructions (which is the usual reason for thinking of the natural numbers as potentially infinite) but that they are given by an endless reiteration. That obvious fact about the natural numbers (which all can agree upon) might naturally lead some collections of natural numbers (some sub-collections of that definite totality) to be relatively indefinite—it might, or it might not (and for a realist, such obscure facts must be discovered, not stipulated).
......If we were repeatedly tossing a coin forever, with heads meaning membership (of N in the collection, when obtained on the Nth toss) and tails non-membership, then clearly we would never obtain a definite sub-collection—and whether or not we would in principle be able to obtain one (e.g. by tossing ever faster, or by simultaneously tossing aleph-null coins, etc.) is an open question. It is generally assumed (e.g. by most mathematicians) that we could, but why? Even if set theory is consistent, that would not make it true; and there are indications that we could not (e.g. my forthcoming paper). That the potentially infinite does not depend upon a metaphysically prior constructivism is also indicated by the infinitesque behaviour of the proper classes that inevitably accompany the usual sets.
......But I shall end with a devilish detail, another way in which my notion of potential infinity departs from the norm. I see no compelling reason why lines should not exist (as objective structures) and be full of points, but then a point on a line might determine (in conjunction with two points labelled ‘0’ and ‘1’) an endless sequence of binary places. Then even a random point would determine a sub-collection of the natural numbers. Still, even then the totality of such sub-collections would not exist (so saving my resolution of Cantor’s paradox) because even such pre-determined sequences would be relatively incomplete. They would not be unities in their own right but would depend, for their definition, upon their generating points—in themselves they just go on and on, endlessly, without the intrinsic definition that the existence of their totality would require (cf. Russell's paradox). That is, each generating point would contain far more information (in its precise position relative to the points 0 and 1) than would fit into such a potentially infinite sequence (and while those points would of course form a totality, a line, that would not be a set of points, but rather as described in my last paper).

## Saturday, June 23, 2007

### Tidy Physics?

Recent comments on a recent post by Peter were (inevitably) inappropriate, so I thought I'd better post something here, regarding the causal closure of the physical world. That closure seems to be widely presumed (e.g. by Searle, and Chalmers, and those I've read less recently), presumably because of (what is said to be) overwhelming evidence (although some physicists disagree), but I'm wondering why? I doubt that it is simply a matter of a lot of evidence: cf. how not so long ago there was apparently overwhelming evidence that space was Euclidean (e.g. the huge explanatory success of the extremely neat Newtonian physics of the day), so much so that Kant (and Poincare etc.) could reasonably say that it was a matter of logic, not empirical evidence (rather paradoxically); and of course, modern physics (and the associated philosophy) is hardly so tidy.
......What there was (we now know) was a lot of evidence that space was approximately Euclidean (as we already knew) and about where to look for the non-Euclidean stuff (for Einstein). Similarly a (substantial) dualist about mind and body (such as myself) might expect non-closure to show up only in the details of the mind-brain interaction (e.g. via an explanatory gap remaining even with a completely detailed theory of the brain, or via observations of exceptional neurones directly and therefore unlikely, or via theoretically related phenomena beyond the brain, and so forth). The "overwhelming evidence" cited in defence of closure seems to lie far from where that is likely to be.

## Thursday, June 21, 2007

Given some things, we have not only those things but also that collection (the plurality as another unity, the many as one), so the question arises, is a collection (so to speak) with just one member also a further thing, over and above its only member? Intuitively it is not, but it is often (as below) tidier mathematically to say that it is, and then (if we also outlaw self-membership, and potentially infinite collections, etc.) then our collections are sets (and their members are elements). Cantor’s paradox concerned his sets.
......Another question is, when does one collection (a whole number of things) have as many members as another? Cantor realised that it was when the members of one collection correlate one-to-one with those of the other, when they have the same cardinal number of members. The most basic numbers are the natural numbers, 1, 2, 3 and so forth (adding 1 ad infinitum), so how many of them are there? The classical answer was that, since they are generated endlessly (are potentially infinite) they have no number, but Cantor’s answer was that there are aleph-null of them, where aleph-null was the first of his transfinite cardinal numbers.
......Similarly there are aleph-null rational numbers (since they correlate one-to-one with the natural numbers) but more real numbers—there are as many endless sequences of 0s and 1s as there are real numbers (which we may think of as endless decimal or binary expansions), and hence as many subsets of the set of natural numbers (since we may think of each 1 as signifying membership of that set, and each 0 non-membership), and hence there are more real numbers, as follows.
......Counting a singleton (a set with a single element) as different to its element tidies up the mathematics of the set of all the subsets (the power-set) of a given set. E.g. if S = {a, b, c} then its power-set, P(S) = {{a, b, c}, {a, b}, {b, c}, {a, c}, {a}, {b}, {c}, {}}. In general, if the size of S is # (e.g. 3) then the size of P(S) is 2 to the power of # (e.g. 8). Cantor showed that the power-set of any set (even one of transfinite size) is bigger than that set by showing that there could not be any bijection between that set and its power-set (via his diagonal argument), so that since, for each element of that set, there is a singleton that is an element of its power-set, hence the power-set is bigger.
......Cantor’s paradox concerned the set of all the sets. Although it would be the biggest set (by its definition), its power-set would be bigger (by the diagonal argument), whence there is no such set. And although that seems no worse than Russell’s paradox, consider the pure sets. They are built out of (by taking collections of (collections of, etc.)) empty sets, where the empty set does not have a single element (e.g. {} above). Within any particular set theory, the unique existence of the empty set, as one thing, accords with the intuition that a singleton is a different thing to its element (although I would rather begin with a pair of definite things). But that there can be no set of all the pure sets is worse because we cannot resolve this paradox by pointing to the intrinsic fuzziness of (e.g.) the totality of the possible red chairs.
......The totality of all the pure sets is a pure proper class, a class being a collection of all and only those things satisfying some definite predicate, and a proper one being one that is not a set. Mathematicians who are interested in sets tend to regard mathematics as set theory, and so they can ignore proper classes, but the philosophical puzzle remains, motivating us to develop a plausible theory of classes—to point to the definitive property that sets have, and that classes lack. To begin with, if each pure set exists, as a definite thing, then all of them will, their totality existing as some sort of collection. And clearly, the totality of the pure sets and the totality of the cardinal numbers are two proper classes, and each has as many members as the other.
......In short, whatever intuitions led us to transfinite mathematics are likely to lead us towards trans-set mathematics, and inconsistency. In particular, if all the pure sets exist then not only would that totality seem to exist, each collection of pure sets would seem to exist already, as a sub-collection of that totality, whose power-class would therefore also seem to exist—much as the (standard full) power-set of the natural numbers seems to exist. So note that most of the subsets of the natural numbers (which collectively make that power-set larger than aleph-null) correspond to random sequences of 0s and 1s. Although the totality of the natural numbers is clearly a definite collection, a random sequence of 0s and 1s, which is like the probable result of endlessly tossing a fair coin, may well not be a definite thing, because its elements are not determined by a finite rule.
......If the natural numbers are like that (a potential infinity) then, collections being collections of things rather than of fuzzy stuff, numerical collections (at least) avoid Cantor’s paradox—there is only really a paradox when one has (with Cantor) the intuition that the natural numbers form an actual infinity. For Cantor, each potential infinity presupposed an actual infinity (since the former was for him like a variable ranging over the latter) so he thought of proper classes as being actual infinities that were nonetheless inconsistent, a view that is unattractive to most philosophers (who would conclude from an inconsistency the falsity of one or more presuppositions).
......So the problem is, if it is not the endlessness of the simply infinite sequence that makes some infinite collections (those whose members are not determined by any finite rule) unfit to be members of other collections, then what is it? (Note that even if the natural numbers are potentially infinite, their totality may nonetheless be regarded one thing, with a definite number of elements—the same as the number of the rational numbers, via the standard bijection between those two collections—although the devil is in the details).

## Monday, June 18, 2007

### What His Wife Said

In the beginning,
Eurynome, the Goddess
of All Things, rose naked
......from Chaos, but
found nothing substantial for her feet to rest upon,
and therefore divided the sea from the sky,
dancing lonely upon its waves.
......She danced towards the south,
and the wind set in motion behind her
seemed something new and apart
with which to begin a work of creation. Wheeling about,
......she caught hold of this North Wind,
rubbed it between her hands, and behold!
the great serpent Ophion. Eurynome danced
to warm herself, wildly and more wildly,
......until Ophion, grown lustful, coiled about
those divine limbs and was moved to couple with her.
...
Eurynome and Ophion made their home
upon Mount Olympus, where he vexed her
......by claiming to be the author of the universe.
Forthwith she bruised his head with her heel,
kicked out his teeth, and banished him
to the dark caves below the earth.

That’s from GravesThe Greek Myths, 1960

## Saturday, June 16, 2007

The basic concept in mathematics is the concept of a plurality, a number of things. Prima facie each plurality, being one plurality, is also a unity, one thing. E.g. when we think of the extension of the concept of a plurality, i.e. the totality of all the pluralities, we are clearly regarding each plurality as one thing, one member of that totality. That totality must have, as one of its members, itself (since it is a plurality), but clearly many pluralities are not (in that way) self-membered. So now consider the totality of all the non-self-membered pluralities—is that totality a member of itself? Russell’s paradox is that if it is then (by its definition) it should not be, and if it is not then (similarly) it ought to be.
......By analogy with the Grelling-Nelson paradox, perhaps we should have considered the totality of all the other non-self-membered pluralities—but then what of the plurality whose members are that totality and that totality’s members? This paradox is not so easily resolved. Still, the Grelling-Nelson paradox may nonetheless indicate the best resolution. Think of the ordinary pluralities, of cats and dogs, and tables and chairs—whilst any such plurality will be some definite number of definite things, the totality of all the pluralities would be unlikely to be so well-behaved, because even a merely possible object is a thing of some kind, and would therefore be a member of various pluralities.
......E.g. even though some chairs are definitely red, we cannot consider, not as a definite plurality, the totality of all the definitely red possible chairs, because of the fairly obvious borderline cases, such as we are not thinking of when we think of something that is definitely a red chair. Clearly, in order to justify our initial belief, in the totality of all the pluralities, we would need to rule out even the possibility of continuous transitions between individual things (such as a single white cloud, in an otherwise blue sky) and stuff that is too fuzzy to be one or more things, to rule that out for all possible kinds of thing. So, it may well be that Russell’s paradox just shows that such a belief cannot be justified. (2nd Aug: More of the same posted.)

## Thursday, June 14, 2007

### Dusty Stars

An awesome picture of a dusty galaxy quite like our own (and this and this are also like ours, whle other dusty galaxies are this and this, and the prettiest galaxy is this and this).

## Wednesday, June 13, 2007

The meaning of ‘heterological’ is most easily shown by examples, e.g. since ‘long’ is not a long word, ‘long’ is heterological, whereas since ‘short’ is short, ‘short’ is not heterological, and similarly ‘monosyllabic’ is heterological because it is not monosyllabic, and ‘polysyllabic’ is not heterological—but what about ‘heterological’? If it is heterological then (by its intuitive definition) it is not, whereas if it is not then (similarly) it is.
......This paradox was designed to shed light on Russell’s paradox (of the collection of all the collections that are not members of themselves) and indeed, the best way to see its triviality is to consider the idea of a club for all and only those who don’t belong to any other club. Whilst such a club is perfectly feasible, if the word ‘other’ were missing from its definition it would become paradoxical unless nobody was not in some club.
......Similarly the intuitively heterological adjectives would just be the other adjectives that don’t describe themselves. After all, while it is quite clear what ‘ugly’ being an ugly word would amount to, we have no similarly intuitive grasp of how ‘heterological’ could be heterological. (Furthermore, how heterological are ‘ugly,’ ‘round,’ ‘necessary,’ etc.? Unlike ‘collection,’ ‘heterological’ does not even seem to name a definite concept.)

## Monday, June 11, 2007

......Let A = ‘C, if A,’ where C is a contradiction. E.g. “If this statement is true then 0 = 1.” If indicative conditionals were material (if, from a falsity, anything followed) we would be able to get C even from a false A and then, since we’d also get C from a true A (as that is just what A says), we would have C, rather paradoxically; but are they? If they are suppositional (if ‘C, if A’ asserts nothing if A is not true) then we could only deduce (correctly) that since if A were true then C would be true, hence A is not true. Still, the subjunctive version remains puzzling, as follows.
......Let S = ‘If S were true, something that is not true would be true.’ Much as before, S is not true because if it were true then (its antecedent being true) it would be true that something that is not true is true (which is a contradiction). But S therefore seems to be saying that if something (i.e. S) that is not true were true, then something that is not true would be true, which appears to be correct. So although S is certainly not true, it therefore appears to be true. Still, since that appearance must be deceptive, we might resolve this paradox along the lines of the Liar paradoxes, as follows.
......Let L = ‘L is not true.’ By the definition of ‘not,’ L is either true or not true, and if L is true then L is (as it says) not true, whence L is not true. But since we know that L is not true, L therefore appears (paradoxically) to be saying something true. The best explanation for that seems (in my opinion) to be that, whilst another expression (i.e. not L) could be sensibly used to assert that L is not true (as that did), L cannot itself say that because it is nonsense. So perhaps the best explanation for why S does not appear to be nonsense is that it resembles (very closely) a true expression?

## Saturday, June 09, 2007

### Skull Star

When the stars threw down their spears,
And watered heaven with their tears,
Did he smile his work to see?
Did he who made the Lamb make thee?
That's from Blake's The Tyger, 1794

## Thursday, June 07, 2007

### The Surprise Exam

Whereas Moore’s paradox hardly seems paradoxical at all (no more so than how, although I may not, at some time, be saying anything, I cannot say so truthfully at that time), I’ve found the Surprise Exam to be very puzzling (although experts say that the former is much deeper than the latter). We are to imagine that a good teacher wants her class to revise their lessons, so she decides to set them an exam for the following week, but she does not want them to cram their heads with facts the night before the exam, so she tells them that it will be a surprise exam—they won’t know which of those five days it will be on until she plonks the paper on their desks. One pupil works out that there is therefore no need for him to revise, such an exam being impossible, as follows. The exam cannot be on Friday because if it were then on Thursday evening the class would know that it was the following day. So the exam must be on or before Thursday. But therefore Thursday is similarly ruled out (since Friday has been ruled out the class could know on Wednesday evening that the exam was the following day). And so on, going backwards through the week like that until even Monday is ruled out.
......But of course he was wrong because the exam was on Thursday, and it surprised him most of all because he did not expect it on any day that week! And similarly, anyone else who on Wednesday, for example, worked out that it would definitely be on Thursday because Friday was (as the last possible day) ruled out, would have obtained an inappropriately justified true belief (i.e. would not really have known) that it would be on Thursday, because for such a reason Thursday (as the new last possible day) should also have been ruled out. Now, the earlier in the week the exam is, the less expectation the class could justifiably have, the day before the exam, of it being on the following day, but the reason why the induction failed was that the exam could even have been on Friday (without the teacher being wrong). Suppose it got to Thursday evening, with as yet no exam. Those who thought that therefore it would probably be the next day would probably have overlooked the possibility of the teacher being wrong about there being an exam at all, that week, rather than wrong about it being a surprise—they could only know that it was on Friday if they knew that their teacher was not wrong about it being that week, but they would have less reason to believe what she said about the exam the more they believed that it was likely to be the next day.
......That what the teacher said turns out to be correct even with the exam on Friday, because of it being apparently wrong in some respect, is certainly odd, but hardly contradictory. And so the induction fails, and so the exam could also (less oddly) be on Thursday, or even earlier (even less oddly). It is certainly odd though—e.g. suppose that one pupil trusted the teacher so much that he had complete faith in what the teacher said about the exam (not unjustifiably, because this teacher is indeed sufficiently trustworthy). Prima facie he might know that what she said was correct (as indeed it was, as usual), but then if we also suppose that she had set the exam for Friday then he could, the previous evening, have known that the exam would be on Friday whilst also knowing that he did not know which day it would be on!
......(Added June 8:) Having slept on it (and there's a mystery, that good thinking can get done while we're unconscious!), this seems less puzzling. Since the teacher expressed herself that way (saying that the pupils would not know when the exam would be until the paper was on their desks) she probably had in mind a day early in the week (as otherwise it would have made more sense to say just that she would not tell them which day it was going to be). In the unlikely event of her intentions being frustrated, so that the exam had to be on Friday, the pupils could hardly (given only the above information) be justifiably sure that there would be an exam at all that week. So, the lazy pupil's induction could not get started (the exam might be on Friday, for all that it is probably not intended to be), and the truth of the faithful pupil's belief (that the exam would be the next day) would have been too much a matter of chance to be justified (?)

## Wednesday, June 06, 2007

Another post inspired by my recent reading of Philosophical Investigations (specifically IIx): Moore's Paradox is that although it may be true both that S is P, and that you don’t believe that S is P, whence it might not seem to be self-contradictory to think: “S is P, although I don’t believe that S is P,” nonetheless that thought is self-contradictory. That is not much of a paradox (if we do not have an implausible theory of mind), no more paradoxical than that we might accidentally tell the truth by lying; but still, I wonder if we might sometimes, quite reasonably, assert such thoughts. E.g. upon seeing a spider, an arachnophobic might say, “I know that Spiders are not a serious threat to me. I know that… but still I believe that they are.” Now, maybe that does not count (as it is irrational) but still, I might more rationally say, “Our wills are not ever really free, I know, now that you’ve convinced me to believe in determinism, but still, when I think about such things, as I’ve been doing, I know that I presuppose that I’m free to make up my mind; that is, I actually believe that my will is, in some respects, free,” or more simply “Our wills are not free, but I believe that mine is.” Or suppose that I've learnt that the Battle of Hastings was in 1066, and suppose that I’m being orally tested on my knowledge of History. Still, suppose that the Battle was actually in 1067 (that the history books are slightly wrong about that date), and further suppose that I’m a bit psychic (which may not be logically impossible), so that I have a sudden sense of certainty that the Battle was actually in 1067. Then I might think, or say, “The Battle of Hastings was in 1067, although I know that it wasn’t...” That is, there might be many ways in which we believe the propositions that we express publicly…

## Monday, June 04, 2007

### A Dualist @ Dawn

Summer’s here, so I’ll probably post less often (as, once on the web, I’m stuck here for hours, whereas it’s sunny out), but it occurs to me (halfway through Chalmers’ 1996) that I ought to say what I’ve meant by ‘dualism’ (about mind and brain). Thus far, only what I picked up from reading Descartes, so this short post won’t be cutting-edge (I’ll post a more precise statement of, and proper arguments for, my position when my relevant reading is up-to-date), but it may belatedly clarify my recent Carnival post (and better-informed readers might point me in useful directions as a result of it).
......I’d assumed that if, for example, the evolution of our brains was what caused (not only our mental complexity, but also) our individual subjectivities to arise, i.e. if we could only exist as an aspect of something physical, then even were physics not causally closed (e.g. even if subjective feelings such as pain sometimes caused personal actions) nonetheless the world would be essentially materialistic. So to begin with, I’m surprised that such beliefs (as that one) are dualistic. In Chalmers' terminology, Naturalistic dualism regards physics as causally closed, while Interactionist dualism (e.g. that belief) regards it as open. He is the former, and by his lights I’m the latter, but I see myself as a Substantial dualist, and all of Chalmers' dualisms as Superficial.
......And I don't think that Substantial dualism is opposed to my taking seriously our natural sciences. After all, these do appear to be early days for psychological theorising. Cf. how mysterious time is, for example (for all its obviousness), and how our ideas of what it might be have changed, in various ways, from Newton (and the seeming irrelevance, for our concept of time, of the shift from Ptolemaic astronomy) to Einstein (and beyond); or imagine an AI looking down on some part of America and seeing buildings and cars. It might explain its observations in terms of laws relating those two, but eventually it notices suits moving between them, and so posits that Earthlings are actually those suits. Of course, we are not alienated from our own natures, so we might (even at this dawn of our psychologies) know enough about ourselves to see the prima facie implausibility of all the Superficial dualisms of Chalmers’ 1996.
......For Chalmers (1996: 157-8), Interactionist dualism is no less epiphenomenalistic than Naturalistic dualism: “We can always subtract the phenomenal component from any explanatory account, yielding a purely causal component.” But suppose that minds do sometimes communicate directly (the evidence being currently inconclusive). The non-physicality of the medium might be indicated by the independence of the telepathic correlations from the physical separations of the subjects, for example, but furthermore its non-existence might follow from metaphysical considerations. An explanatory account of such telepathy might appeal to simple psychical laws (e.g. those of a fictional medium, cf. relativity physics with relational space-time) posited to explain all sorts of observed regularities, and if we were to subtract from that the awareness of thoughts we would be left with fatal gaps (in the causal chain that best explained our observations).
......(PS, added July 2) Well, that was a wet month after all. Anyway, it occurs to me to note that even a Substantial dualism (i.e. one that is not epiphenomenalistic, one in which mind does not supervene upon brain) need not be commited to an implausible substance dualism (since that depends upon what is meant by 'substance'). If mind does not supervene upon, does not arise from matter, then it is not so implausible that there is a Creator, that both mind and matter are alike in being divine creations, which would naturally interact (cf. how plausible it is that a metal ball dropped upon a wooden floor, or a sandy beach, should interact with it, should leave a round indentation upon that other material, just because they have so many properties in common, notably a common origin).

## Friday, June 01, 2007

### Good Reasoning

My recently expressed unease with possible worlds was clarified somewhat by the latest issue of The Reasoner (issue 2, the article about counterpossible antecedents (also this reply)): I shalln't delve into details (so this post is mostly a memo to myself) but in mathematics we don't want the vacuous truth of both that if Goldbach's conjecture were false (or true, whichever is not the case) then X would be the case and that if that then not-X would, which we would get on some analyses of subjunctive conditionals, as that case would have no possible worlds. We clearly don't want epistemic possibilities, as the truth and falsity of Goldbach's conjecture are both epistemic possibilities at present, as is constructivism of course, whence I wonder whose metaphysics would yield the literal meaning of 'would'? The background presumptions of the particular communicants would surely give us something too subjective for the literal meaning, but the wider linguistic community is surely going to have too vague an intersection of its various presumptions (which would include those of people like Pythagoras, Cantor, Brouwer and Quine).