Thursday, May 31, 2007

Pink Floyd

A white-hot diamond
alights on liquid oxygen—
facets in a cloud
of flamboyancy.

Wednesday, May 30, 2007

The Porpoise of Life

...dreams of swimming through the seas of language... I weave these words into the World Wide Web, but why? To catch the attention of the wise, perhaps (I’ll be hosting the Carnival in August), or to clear out an overcrowding head, ready for the start of my serious studies (I’ll have to start posting on Continuity soon), or just to clarify my thoughts (which always surprises me): E.g. I’ve used the word ‘agnostic’ (about p) to mean something like knowing (or believing) that one neither knows p nor knows not-p, but it seems (from an interminable debate about kinds of atheism on the ScienceBlogs) that it can also mean knowing (or believing) that one cannot ever know p or not-p. It makes me wonder what I’ve been saying (and whether possible worlds can help to clarify that). Anyway I ought to leave myself a memo about the Philosophy Carnival in August (so start thinking now!), which will favour stuff related, in some way, to the Philosophy of Mathematics (and of Sciences, Logics, mathematical philosophy and so forth), whatever such words mean. When I use them, I have various ideas about what they are likely to mean, in the back of my head, and one such posit is that they mean what their most authoritative users take them to mean, which is confusing for we who find such authors especially obscure. Anyway speaking of possible worlds, what if water is actually the platonistic shadow of a spirit-dragon; is that not possible (for all we know for sure)? But then ‘water’ might refer to Mathematics in some other possible world, if such was the shadow of that self-same spirit-dragon there (much as ‘Bob’ might refer to a rabbit, if Bob reincarnates as a bunny). So, XYZ might be Mathematics?!

Tuesday, May 29, 2007

What is possible?

Presumably for something to be possible, e.g. epistemically possible (i.e. subjectively consistent with our other beliefs) or nomologically possible (i.e. objectively consistent with the laws of nature), it must be at least logically possible. So then, what is logical possibility? It cannot be mere consistency, for consistency is always with something. Perhaps it is self-consistency, i.e. the absence of self-contradiction, but what is that?
...... To begin with common sense, G E Moore thought that the idea that he was only dreaming, when he was actually lecturing, contradicted what he knew of himself, but was it then logically impossible that he was dreaming? After all, dreams might be very vivid, and even Putnam’s argument would not make it logically impossible for him to have just fallen asleep. And (on another topic that I’ve waffled about recently) the idea that I am nothing more than biochemical complexity, that seems to contradict what I know of myself, but I don’t imagine that I’d be allowed to call the theory of evolution logically impossible.
...... I’ve seen logical possibility associated with conceivability, e.g. a round square is inconceivable because it appears to be self-contradictory. But surely the laws that actually govern nature might be inconceivable by us (even in principle), and yet we would want them to be logically possible. And from mathematics, a typical real number is by itself too complex to be conceivable, but we think of it as conceivable because the real number line (which includes it) is conceivable, if not by you then by someone suitably authoritative. (And we allow that formal real numbers might be logically impossible, if inconsistencies lurk within the formalism.)
...... Suppose that there are aliens out there who know a lot more than we do about geometry. And suppose (if you can) that they tell us (one day) that there are square squares (in our geometries, with the usual meanings for those words) and round squares (in some others, similarly). Prima facie those conceivable aliens could, conceivably, be right about that (or so we could think, not unreasonably, if that happened, because after all, our experts have been wrong about elementary geometry, in the recent past, for all that it seemed to many of them that they could not possibly get such things wrong). But are round squares logically possible?
...... Anyway, does the discovery of a contradiction necessarily amount to the discovery of logical impossibility (and hence of non-existence)? I don’t think so because it might instead be that our predicates have been found to be less definite than we had thought (if their subject clearly exists). So even when that does not seem to be the case, how sure may we justifiably be that that was not possible? (I’m thinking here of the superficial similarity between a wave-particle, which is logically possible, and a round square, which is not.)
...... After all, Descartes’ demon (the rationalist’s friend) does not seem to be inconceivable or self-contradictory, and such a demon might possibly hypnotise us, so that not only do we make errors, when we reflect upon some of our predicates, we also believe ourselves to be incapable of such errors (much as we rationalists like to think we are, sometimes).
...... So, to return to common sense (i.e. to delete most of the above waffle, and post the rest), I’m left thinking that logical possibility must be, if anything, consistency within some given system of logic. Since we have many (inconsistent) logics, we would therefore have many (inconsistent) sorts of logical possibility. But presumably we want there to be one special one, that our system of possible worlds (with which we analyse our subjunctive conditionals), built upon our preferred system of logic, is attempting to model accurately; so I’m back to my original question (with no clear view of how to attempt to answer it), what is that?

Saturday, May 26, 2007

More Weird Ideas

Speaking of other worlds, it occurs to me that since I’ve assumed the falsity of the many worlds interpretation (MWI) of quantum mechanics in my forthcoming paper, hence I ought to develop something more like an argument against it—rather than the vague antipathy towards it that I've expressed at On Philosophy (for many-minds) and Mormon Metaphysics (against MWI). To begin with, either materialism is true or it ain’t; and if it is then I’m essentially my brain. According to MWI, I’d be continually budding into infinitely many others who had (until then) been me, which is what I intuitively reject. Maybe I can’t say for sure that that isn’t what it’s like to be me, because I can’t really imagine what that would be like, but prima facie that contradicts what I know for sure about myself (via my direct acquaintance with myself). And of course, were materialism false, the mind-brain interaction would need to be understood, and so in that case MWI would be much less attractive anyway. Not much of an argument so far, so I guess I ought to try to imagine what it would be like to be constantly splitting into infinitely many people... (Maybe not!)

Friday, May 25, 2007


Speaking of indicatives, "If A then C" might be asserted because of a belief in the corresponding subjunctive, "Were A the case, C would be the case." What would make us believe that? Probably a belief in some underlying law (of nature) with such consequences. A common way of analysing subjunctives uses possible worlds, and I'm not at all sure about why I don't like them (as they seem suited to many normal uses of subjunctives, e.g. when the apposite law is probabilistically physical), or how I would rather analyse subjunctives (so this is yet another fishing post), but this is certainly an important philosophical topic (so I hope to post more in due course) because analytic philosophy is all about the logical analysis of various hypotheses, about discovering what would be the case under various assumptions (?) But therefore an obvious problem with any sort of possible worlds is that we are naturally going to want to consider what would be the case were there no such possible worlds.

Thursday, May 24, 2007


Speaking of common sense, it strikes me that the indicative conditional is obviously suppositional (subsection 4.3 of Edgington’s Conditionals), i.e. that “C, if A” asserts “C” iff A. Suppose I say: “The cat will sit on the mat.” Then (whatever my motives for doing so, however likely or unlikely it seemed, and however likely or unlikely it actually was), surely what I say will be literally true iff the cat sits on the mat. Similarly, if I say: “If the mat is not removed, the cat will sit on it,” then what I say will be true iff the cat sits on the mat that has not been removed.
...... If the mat is removed (for whatever reason) then my conditional prediction, while not false, failed to come true. What could make it true, given that the mat is actually removed? That the cat would have sat on it, had it not been removed? But that would only have made the corresponding subjunctive conditional true. Now, it may well have been a belief in that subjunctive (or counterfactual) conditional that made me assert the indicative conditional (about actuality), but it was the latter that I asserted, not the former. Of course, it may not have been—and indeed, the subjunctive might be known to be false (via the counterfactual probabilities) while both the antecedent and the consequent of the indicative are correctly guessed to be true.
...... The major alternative (to such a suppositional interpretation) seems to be the material conditional, but I don’t see why indicative conditionals with false antecedents should have truth-values. Are there any good reasons for thinking that they should? Even Edgington (third paragraph of subsection 2.1) calls it uncontroversial that “If that is a square, it has four sides” is true even when said of a triangle, but in fact what is clearly true is just “All squares have four sides.” Anyway, one problem for the material interpretation (first paragraph of subsection 2.5) is that according to it “It is not the case that if it is a triangle, it has four sides” ought to be false when said of a non-triangle. And while I suppose that if our logic is set-theoretical then the material interpretation is tidier, conversely if that interpretation is indeed false then that is indicative of the falsity of set theory.
...... And what about “If you disagree with me, I’ll hit you;” isn’t there some intuitive pull to the thought that I have, after all, made an assertion here? Well, the former is certainly an assertive speech-act—a threat—but what was actually said? To find that out consider that, because of the threat, I hide my disagreement and so avoid being hit—then the antecedent is true (I do disagree) but the consequent is false (I am not hit) and so the conditional is false, and yet the assertive speech-act is nonetheless successful (I do not show my disagreement). And although in a sense I complied because I believed what was said, what I believed was that if I had shown disagreement I would have been hit.
...... Mind you, that "If you disagree with me, I'll hit you" was probably intended subjunctively anyway, to mean the same as the counterfactual "Had you disagreed with me, I would have hit you" uttered later. Still, the speaker might have been (possibly correctly) a determinist, who thought (correctly) that I would not disagree, given his threat, and then there would have been no possibility of my having disagreed with him... Although clearly indicatives have the truth-conditions of material conditionals in possible world semantics, since to say if A then C is to say that the A-worlds are all in the C-worlds, which is just to say that no A-worlds are also not-C-worlds.

Wednesday, May 23, 2007

Surprisingly Common Sense

Because of my common sense, I know, as I type this, that I have hands; but presumably a disembodied, and therefore handless, brain-in-a-vat might think so too, as a result of its experiences being so arranged that they resembled mine, down to the smallest detail—so, perhaps I am such a brain-in-a-vat? My problem is that if I do not know that that is very unlikely (or impossible)—and how could I examine its likelihood?—then I can hardly justify my belief that I have hands; so I would not know (for sure) that my hands exist. That is a problem because one naturally feels that one’s confidence in such everyday propositions is fully justified, somehow.
...... But therefore we have G E Moore’s common sense refutation of such Cartesian scepticism (and Humean scepticism can be treated similarly, e.g. see my A Pair of ‘Sceptical’s): All such stupefying scenarios can be known to be impossible because one really can know that one’s hands (knees, feet etc.) certainly do exist. That is an almost complete refutation—only a scientific question remains. Sceptical scenarios are seemingly rational possibilities that are not made sufficiently unlikely by the empirical evidence, whence even when we are not tempted by the scepticism, they naturally raise such scientific questions as: How do we know that we have hands?
...... Insofar as we think of our knowledge of the external world as being constructed indirectly, from nerve-signals coming from sensory organs external to our brains, it is quite mysterious (in view of such sceptical scenarios) how our natural confidence in our knowledge of the ordinary things around could be justified. Nonetheless the foundations of our scientific knowledge are surely our common sense certainties (e.g. as we perform experiments), so we cannot really question them, we can only seek their scientific justification. Now, scientific questions require scientific answers and of course, as a philosopher of mathematics I don’t have a scientific answer, but what I can do is speculate.
...... The scientific answer would surely be some theory devised to account for scientific observations of the relevant phenomena, so we might ask ourselves: What would be relevant here? Well, in what ways might we become directly acquainted with things? A clue comes, it seems to me, from our need to explain the direct mind-brain interaction (since something else that is common sense is Cartesian dualism), because quantum mechanics is likely to be a key component of that explanation. If so, then (wavefunctions being spread out) we might expect direct knowledge of external objects to be possible too, and so the relevant phenomena could well include the micro-psychokinetic (the ‘micro’ means that the mind interacts directly with chance phenomena in the external world, i.e. not levitation).
...... Of course, common sense also tells us that the paranormal is not worth investigating. After all, such things are investigated, by reputable sceptics and by believers, and nothing ever comes of it. But then, a socio-economic explanation for that might involve that word ‘reputable’—I don’t want to delve into such issues here because they are intrinsically complicated (as political issues are). I’ll just observe that if I’m right (and why not?) then the paranormal would, insofar as it occurs, be the extremity of something quite normal (much as superfluids are), varying in the usual way of such biological things (and insofar as it does not occur—e.g. poltergeists—its plausibility could be similarly explained, i.e. via the ubiquity of something that is both obscure and like the fiction). After all, the empirical evidence is not inconsistent with lesser degrees of the so-called sixth sense being surprisingly common (e.g. consider the sense of being stared at, which would probably have been selected for if our underlying physics does allow it; cf. this recent research).
...... Just as Humean scepticism motivates our taking seriously what is common sense, that there are laws of nature (indicated by the correlations due to them), by verging upon a reductio of their non-existence (of there being only the correlations), so Cartesian scepticism motivates our taking seriously what is common sense, that we are directly acquainted with such natural kinds as hands (e.g. via a process akin to the mind-brain interaction), by verging upon a reductio of its non-existence (of our having only an indirect acquaintance with our own hands).

Monday, May 21, 2007


So, a second week of OPC2—last week I found Williams’ paper helpful, clarifying for me some of what Wittgenstein was up to at the beginning of Philosophical Investigations. I know too little to engage with that debate, but it strikes me prima facie that (in the builders’ language-game) builder A’s “Slab!” (spoken to builder B) must have been the expression of a thought that we might adequately describe as “Bring me a slab!” or “Make it so that B brings A a slab.” Certainly A must have wanted a slab, and must have thought that saying “Slab!” to B would help to satisfy that want—otherwise, there could hardly have been a language-game. Cf. two robots, 1 and 2, with 1 beaming a pulse of light (of some sort) towards 2 whenever 1 detects an insufficiency of slabs, and with the detected pulse causing 2 to take a slab to 1. Could such a transmission of photons be a linguistic act? Maybe, in the sense of computer languages, but how about if a billiard-ball collides with another, so that the momentum of the first is transferred to the other—is that a linguistic act? And if so, what is not? So, underlying A’s command “Slab!” there must have been (as a basic linguistic component) a thought that we might adequately describe as A entertaining the epistemic possibility that B could (actually, but in other cases fictionally, or theoretically) bring A a slab (no thoughts, no language, is how I see it—I mean, language is a medium for the communication of our thoughts), which is a relation of intentional objects (?)

Saturday, May 19, 2007

Such bandwidth!

My series of posts inspired by DawkinsThe God Delusion finally ends (I really ought to be thinking more about logic) with his rather comic view of God. Someone once said, that when people told him that they did not believe in God, he asked them which God they did not believe in, and it strikes me that I would not believe in a God described by the following two points, which Dawkins was adding to an earlier response (the possibility of hallucinations etc.) to claims that personal experiences of God cannot be scientifically refuted:

"First, that if God really did communicate with humans that fact would emphatically not lie outside science. God comes bursting through from whatever other-worldly domain is his natural abode, crashing through into our world where his messages can be intercepted by human brains – and that phenomenon has nothing to do with science? Second, a God who is capable of sending intelligible signals to millions of people simultaneously, and of receiving messages from all of them simultaneously, cannot be, whatever else he might be, simple. Such bandwidth! God may not have a brain made of neurones, or a CPU made of silicon, but if he has the powers attributed to him he must have something far more elaborately and non-randomly constructed than the largest brain or the largest computer we know."
(Dawkins 2006: 154)

Dawkins’ God seems here to be like the leader of some alien invaders from a parallel universe, because surely the Creator of this world would not need to “burst” or “crash” into it—no more than a human author would have to magically appear inside her characters’ fictional world in order to influence them. Of course, fictional characters are not alive, like we are, but that was only an analogy, and the key word is ‘Creator’ (if we have one, this universe is more her/his world than it is our world).
...... Her/his messages might bypass our brains (designed as they would have been for our dealings with this material world) and be received via our consciences (in some immaterial way), or they might be relayed via angels (for whom this world might be like the seas are to us, i.e. not so much unnatural—are houses natural?—as just relatively uncomfortable), or they might indeed arise within our brains, but since those would be brains that s/he made (or is making), so such information would hardly have to travel more than the zero distance from the tip of her/his metaphorical pen (which would be quite unlike any part of Creation) to the metaphorical paper (Creation).
...... No analogy is perfect, so consider another. In Flatland, mathematical creatures of both 2 and 3 dimensions occupy the same space, the former in a plane through which the latter might move as easily as they move through any of the planes of space. E.g. a cube passing through Flatland might seem to appear from nowhere, first as a small equilateral triangle, growing and changing into a rotating hexagon before reverting to a shrinking triangle and disappearing again—or as some other sequence of 2-dimensional shapes. It might be quite mysterious, to the dimensionally challenged Flatlanders, how the different possible sequences could be the same thing, but nonetheless Flatland is just part of the natural abode of the cube.
...... Now, perhaps elves would burst into this world upon the backs of unicorns, from their natural abode (Elfland), were they not fictional, but the Creator would be more like something that was to this world (Creation) as this world is to such fictional realms, were s/he to exist. And whatever the Creator is composed of (if anything), it would hardly be like the stuff of this world, whence we can hardly conclude that s/he must have any kind of elaborately and non-randomly constructed information-processor (although of course s/he might have, e.g. a host of angels).
...... Dawkins’ second point relates to scientific refutation because his argument for the improbability of God rests on our need to explain a complex God—but why should the Creator not be simple? Cf. how an author need not contain anything like her own complicated plots (just the insight, imagination and inspiration to construct such things, with pen and paper); cf. how a sphere is not a complicated Dr. Triangle/Mr. Hexagon character (within Flatland). Anyway, the theory of evolution having shown us one way in which complex (physical) structures might arise from simple (physical) beginnings, the assumption that the Creator could not be simple is an odd one for Dawkins to make.
...... Anyway, whether the Creator is complex (in certain respects, and from our point of view) or simple, there is hardly any necessity to her/him being non-randomly constructed. If s/he had an origin (and why should s/he?), it would be quite mysterious (why should it not be?), so maybe s/he arose randomly (and if we don’t know out of what sort of stuff, how can we say if it is unlikely?) and maybe s/he was constructed (why not an infinite sequence of Creators?). (But there should be time enough for Theology later, if there is any point to doing any.)

Friday, May 18, 2007


135 years ago today, something along the lines of Russell first cried. Therefore there was crying. “But here is the problem: a cry, which cannot be called a description, which is more primitive than any description, for all that serves as a description of the inner life.” (Wittgenstein 1953, IIix) Funnier is Calculus Cat, some of these, and these, and Lump of Clay and Thad Guy, and this and that, and xkcd...

Thursday, May 17, 2007


As kingfishers catch fire, dragonflies draw flame;
... As tumbled over rim in roundy wells
... Stones ring; like each tucked string tells, each hung bell’s
Bow swung finds tongue to fling out broad its name;
Each mortal thing does one thing and the same:
... Deals out that being indoors each one dwells;
... Selves—goes itself; myself it speaks and spells,
Crying What I do is me: for that I came.

That's one of Hopkins' poems (undated).

Elegant Design

I was out walking today, and I noticed how beautiful the wild flowers were. Nature has presumably selected such structures because they attract pollen-transporting insects, which were simultaneously selected because they associated the flowers’ colours, or their ultraviolet shines, with food, or something along those lines. So why, I wonder, do we find flowers attractive? A story might have been told, of how those who did not find flowers beautiful did not pick them, and so were stung less often and had more honey (etc.), had we evolved to find flowers unattractive, which we did not. Similarly, one could have told a story of how we came to find raw potatoes beautiful, were they beautiful. Now, we do find fruit attractive, and so such stories are indeed told about fruit, but therefore the beauty of the wild flowers is all the more puzzling—why do we not find, for example, dead animals attractive? (Did those who found raw flesh as attractive as the furry creatures that contain it succumb more frequently to food poisoning, while those who discovered fire and cooking survived?) So, I’m wondering (as in all my posts, what others may know) about the evolution of our appreciation of art. I guess that the natural selection of chemical structures might in all likelihood produce beautiful physical objects, but I’m puzzled by how it could produce people who find them beautiful. (E.g. rooks are clearly elegant, to pick up an example from my previous post, but why do we find elegance aesthetically attractive? We admire aerodynamic shapes in nature, and have done so in our own designs, from stylised cave-paintings to futuristic furniture, but we are not birds, so why?) The thing is, when fans of ‘Intelligent Design’ accuse materialistic scientists of giving us a bleak, soulless world-view, many of those scientists respond by pointing out that it was the majesty, the beauty of nature that attracted them to science in the first place. But it seems, prima facie, quite unlikely that evolved matter should find such things beautiful—what could the evolutionary point of aesthetic attraction have been? Beautiful clothes need not be practical, or even sexy, and there is more to art than the cuteness that we find in our young.

Tuesday, May 15, 2007


Although, I admit, I desire,
Occasionally, some backtalk
From the mute sky, I can’t honestly complain:
A certain minor light may still
Leap incandescent

Out of kitchen table or chair
That's from Plath's Black Rook in Rainy Weather, 1956

Monday, May 14, 2007

Online Philosophy

I've just noticed OPC2, to look at later (whence this brief, memoesque post).

Faith unfit, in Fair weather

Having recently blogged about hope, to charity and faith. As I don’t yet know enough about theology to write much (or to keep my speculations nicely brief, unfortunately), I’ll post this under 'Science' and consider charity (e.g. donations, mercy etc.) and faith (e.g. traditions, loyalty etc.) as having evolved because social animals sharing the genes for such traits came to dominate our gene pool. If some such materialistic explanation is correct, then the sort of critical reasoning that we ought (I think) to favour over faith would presumably have evolved too, and would therefore seem to be, at heart, pragmatic, whence it might well be no more reliable, beyond our everyday uses of it, than our innate intuitions about, e.g., space-time. So I wonder about the limitations of materialistic justifications of our favouring of science over faith. Let us say that some materialistic Darwinian, D (not Dawkins, as the literalists ought to try to refute him), prefers science to faith, while some literalistic Christian, C (not me, as I’m a natural heretic), prefers faith to science. D thinks that C is wrong, but why?
...... Even D values authoritative testimony and teamwork within the sciences. Of course, C’s faith is getting in D’s way, and D does not like that. Nor does D like C’s sort of charity, which moves money that could be going to science towards religious interests. All that is quite coherent, but it might also seem a little selfish. Still, D is a scientist, pursuing collective truths, and so D does think that science is a good thing for everyone. But then, C thinks that faith is a good thing for everyone too. And C’s faith, like D’s science, is coherent on its own terms. So, is there anything more objective that D can say, against C’s faith? Paradoxically, what C and D prefer is (according to D) just a matter of how C and D evolved, so they are pretty much on a par at the base level of existence (according to D). If D fears that most people have so evolved that they are in danger of preferring faith to science, then what can D say about why they should prefer science?
...... Well, they probably already want useful truths (although they may well not want to get them for themselves), and so D could say that science is more likely to get them to useful truths. But of course C would disagree and although, for an example of the ensuing difficulties, D might mention in his defence that scientific sorts of evidence are what naturally sway juries, not protestations of faith, nonetheless whenever juries are swayed by faith D would say that they should not be, and so C could quite fairly say that in response to D’s defence. So, it begins to seem as if D is after all just grunting in disapproval at C. D can say that science is more useful than faith, that science is a better route to the objective truth than faith is, but not only can C say the same about faith, can D even explain why utility and truth are so important (let alone why a philosophically interesting truth is more important than a socially useful lie)?
...... The problem is that even if sciences are more useful than faiths, since when did a species have to do something useful when it did not want to? If humans end up being largely like C then they will value being that way; they may die out, but what species won’t. And if humans end up being largely like D then they will value being that way. And they might die out even more quickly, since the fruits of science do seem to present the immediate danger of extinction (while faith just involved millennia of tribal warfare). D could say that such things are irrelevant to whether or not materialism is true—(a species does not have to value its own long-term survival, according to D, it is just that it exists because it did survive, whence whatever it wants probably did have survival value)—but can she say any more than that about why truth is important?
...... D will probably, eventually, justify her interest in truth much as she explains it, i.e. on the grounds of its likely utility, so her problem (as I see it) is that, by the lights of her materialism, both C and D exist because their genes have competed successfully in the past. That is, D’s own theory puts D’s innate preferences on a par with those of C at the most basic level; and it even seems to imply that both of them are similarly intrinsically irrational. Since D cannot (without insincerity) appeal to anything more fundamental than shared preferences, arising through our common evolution, things look bleak for D because the evidence (that the culture war, naturally polarising between C and D, is not going her way) indicates their insufficiency.
...... Consider a lot of monkeys on typewriters, and let us suppose that one of them happens upon the printed truth. They ought to be out eating bananas of course, but there is a mechanism on each typewriter whereby they get bananas by tapping the keys. Now, each mechanism is switched off when its monkey types out a truth (of some specified sort). Consequently although our monkey has chanced upon a truth, it fails to produce viable offspring (through a too-early shortage of bananas). Many of the others will not so fail, but will rather become ancestors of the monkeys of the future. Of course, D does not say that our monkey should have produced viable offspring (or even that it should have wanted to), only that whatever it did want (i.e. bananas) would probably have been such as to lead to it having viable offspring. (Well, they did all want bananas, and eventually they all died.)

...... Now, both C and D want the truth, and both of them can justify wanting it on the grounds of expected rewards that are relatively big. They both think that they are approaching it, but what value do they put upon getting a truth, just insofar as it is a truth? They both prefer their method (faith or science) to the other, even in those cases when it is the other that chances (as they would see it) upon the truth. And they do both value the truth for its own sake, to some extent, more so than many people, e.g. those who prefer love, and faith motivated by love (and who would happily let a popularity contest decide the political issue between C and D). In particular, D values truth (since she evolved that way, in her opinion) but when she looks at things more objectively—say at that poor monkey—she can see little value in it having printed out a truth (or, indeed, in it getting bananas, or producing viable offspring).
...... D might point out that our search for the truth (via the scientific method) would bring us a high likelihood of rewards (that we are not like that poor monkey), but that is surely not quite so obvious when we are fighting and losing culture wars. While C could justify martyrdom for the cause of truth quite happily, D can only justify an extreme interest in truth on the relatively irrational grounds that that is just the way she is (a justification that would justify all other extremisms equally well, of course, for all that D could think of hers as being, since hers, special). Should D be a fair weather friend of truth? If so then perhaps she ought to be insincere about that, in order to protect her wider support.

...... Suppose (for the sake of argument) that D will not produce viable offspring, but will be killed, whereas C will produce viable offspring, since he is (let us say) going to win the culture war. Perhaps the human gene pool is just naturally moving in that direction, at this time. If D knew that, the wider public might ask her, would she still value objective truth, the way a scientist should? And if she did, could she see herself (in so doing) as objectively anything other than an evolutionary dead-end, with values that turned out to have little survival-value, in a world where life was all about survival?

Saturday, May 12, 2007

Not so unlikely after all

Wolfgang notes this fake zebra. I've noticed that this blog's settings were all wrong, so I've changed them, and its colours too. It's rainy in England at the moment, but I'm not going to post properly, I'm going to read Philosophical Investigations (finally).

Friday, May 11, 2007

Ariel's ayre

Where the bee sucks. there suck I:
In a cowslip's bell I lie;
There I couch when owls do cry.
On the bat's back I do fly
After summer merrily.
Merrily, merrily shall I live now
Under the blossom that hangs on the bough.

That's from Shakespeare's The Tempest, 1611.

Words Whirled

......In the beginning
........... God created
............ the heaven
and the earth.

......And the earth was without form, and void:
............ and darkness was upon the face
............ of the deep.
......And the Spirit of God moved
............ upon the face
............ of the waters.

And God said, Let there be light:
...... and there was
...... light.

...... And God saw the light, that it was good:
............ and God divided the light
............ from the darkness.

...... And the Lord God formed man of the dust of the ground,
............ and breathed into his nostrils the breath of life;
............ and man became a living soul.

...... And the Lord God said, Behold, the man
............ is become as one of us, to know good and evil:
............ and now, lest he put forth his hand,
.................. and take also of the tree of life,
.................. and eat, and live for ever:

............ And the Lord said unto Cain,
.................. Why art thou wroth?
............ and why is thy countenance fallen?

.................. And God said unto Moses, I am
...... that I am
: and he said, Thus shalt thou say unto the children of Israel,
.................. I Am hath sent me unto you.
............. Two and twenty years old
....... was Ahaziah when he began to reign;
....... and he reigned one year in Jerusalem.

..And his mother's name was Athaliah, the daughter of Omri king of Israel.
............. Forty and two years old
....... was Ahaziah when he began to reign,
....... and he reigned one year in Jerusalem.
..His mother's name also was Athaliah the daughter of Omri.
...... Where wast thou when I laid the foundations
............ of the earth? declare if
............ thou hast understanding.
...... Who hath laid the measures
............ thereof, if thou knowest?
............ or who hath stretched the line upon it?
Whereupon are the foundations
...... thereof fastened? or who
...... laid the corner stone thereof;
When the morning stars sang together,
...... and all the sons of God
...... shouted for joy?
Hast thou entered into the springs
...... of the sea? or hast thou walked
...... in the search of the depth?
Have the gates of death been opened
...... unto thee? or hast thou seen
...... the doors of the shadow of death?
Hast thou perceived the breadth
...... of the earth? declare if
...... thou knowest it all.
Who hath put wisdom
...... in the inward parts? or who hath given
...... understanding to the heart?
Who can number
...... the clouds in wisdom?
...... or who can stay the bottles of heaven,
When the dust
...... groweth into hardness,
...... and the clods cleave fast together?
Wilt thou hunt the prey
...... for the lion? or fill the appetite
...... of the young lions,
When they couch in their dens,
...... and abide in the covert
...... to lie in wait?

Who provideth
...... for the raven his
...... food? when his young ones cry
............ unto God, they wander
................. for lack of meat.

My God,
...... my God, why hast thou forsaken
...... me? why art thou so far from helping
............ me, and from the words of
............ my roaring?
They gaped upon
.................. me with their mouths, as a ravening
.................. and a roaring lion. As the hart
........................ panteth after the water brooks, so
........................ panteth my soul after thee, O God.
......................................... My soul thirsteth

..........................................for God, for the living God: when
........................... shall I come and appear before God?

My tears have been my meat
...... day and night, while they continually say unto me Where
is thy God? I will say unto God my rock, Why
....... hast thou forgotten
why go I mourning
.................... because of
the oppression of the enemy?
..................... For thou hadst cast
.......... me into the deep, in the midst of the seas;
........................ and the floods compassed

.................... me about: all thy billows and
........................... thy waves passed over

.............................. me.
.......................................... Then said
.............................. the Lord, Doest
............. thou well to be angry?

...... And the rain descended,
............ and the floods came,
............ and the winds blew,
.................. and beat upon that house;
.................. and it fell: and great was
........................ the fall of it.

.................. And in the fourth watch of the night
........................ Jesus went unto them,
........................ walking on the sea.
............ And when the disciples saw him walking on the sea
.................. they were troubled, saying, It is a spirit;
.................. and they cried out for fear.
...... And he said, Come.
............ And when Peter was come down
............ out of the ship, he walked on the water,
.................. to go to Jesus. But when he saw
.................. the wind boisterous,
........................ he was afraid; and beginning
........................ to sink, he cried, saying,
.............................. Lord, save me.
............ And immediately Jesus stretched
.................. forth his hand, and caught him,
.................. and said unto him, O thou of little faith,
........................ wherefore didst thou doubt?
...... And I say also unto thee, That thou art Peter,
............ and upon this rock I will build my church;
............ and the gates of hell shall not prevail against it.
........For nation shall rise against nation,
. and kingdom against kingdom: and there shall be famines,
... and pestilences, and earthquakes, in divers places.
. All these are the beginning of sorrows.
........ Then shall they deliver you up to be afflicted, and shall kill you:
. and ye shall be hated of all nations for my name's sake.
... And then shall many be offended, and shall betray one another,
. and shall hate one another.
........ And many false prophets shall rise, and shall deceive many.
. And because iniquity shall abound, the love of many shall wax cold.
... But he that shall endure unto the end, the same shall be saved.
. Heaven and earth shall pass away, but my words shall not pass away.
Then saith Jesus unto them, All ye shall be offended
...... because of me this night: for it is written,
...... I will smite the shepherd, and the sheep of the flock
............ shall be scattered abroad.
............ But after I am risen again,
.................. I will go before you into Galilee.

.................. Peter answered and said unto him,
............ Though all men shall be offended because of thee,
.................. yet will I never be offended.

............ Jesus said unto him, Verily I say unto thee,
...... That this night, before the cock crow, thou shalt deny me thrice.

.................. Peter said unto him,
........... Though I should die with thee, yet will I not deny thee.
...... Likewise also said all the disciples.
......... But all this was done,
............ that the scriptures of the prophets
......... might be fulfilled.
...... Then all the disciples forsook him, and fled.

...... But Peter followed him afar off
............ unto the high priest's palace, and went in,
.................. and sat with the servants, to see the end.
...... But he denied before them all, saying, I know not what thou sayest.
.................. And again he denied with an oath, I do not know the man.
...... Then began he to curse and to swear, saying, I know not the man.
.................. And immediately the cock crew.
.................. And Peter remembered the word of Jesus,
........................ which said unto him, Before the cock crow,
........................ thou shalt deny me thrice.

.................. And he went out, and wept bitterly.

.................. And about the ninth hour Jesus cried
........................ with a loud voice saying, My God,
........................ my God, why hast thou forsaken me?
He is not the God of the dead,
. but the God of the living:
. ye therefore do greatly err.
............ And when the sixth hour was come,
.................. there was darkness over the whole land
.................. until the ninth hour.
............ And at the ninth hour
.................. Jesus cried with a loud voice, saying, My God,
.................. my God, why hast thou forsaken me?
For he is not a God of the dead,
. but of the living:
. for all live unto him.
...... And it was about the sixth hour,
............ and there was a darkness over all the earth
............ until the ninth hour.

...... And the sun was darkened,
............ and the veil of the temple
............ was rent in the midst.

...... And when Jesus had cried with a loud voice, he said,
................ Father, into thy hands I commend my spirit:
................... and having said thus, he gave up the ghost.
In the beginning was the Word,
. and the Word was with God,
. and the Word was God.

Now Jacob's well was there. Jesus therefore,

. being wearied with his journey, sat thus on the well:
. and it was about the sixth hour. There
...... cometh a woman of Samaria to draw water:

. Jesus saith unto her, Give me to drink. Then
. saith the woman of Samaria unto him,

...... How is it that thou, being a Jew, askest drink of me,
. which am a woman of Samaria? for the Jews
. have no dealings with the Samaritans.
...... Jesus answered and said unto her,

. If thou knewest the gift of God, and who it is that saith to thee,
. Give me to drink; thou wouldest have asked of him,
...... and he would have given thee living water.
The woman saith unto him, Sir, thou hast nothing to draw with,

. and the well is deep: from whence then hast thou that living water?
. Art thou greater than our father Jacob, which gave us the well,
. and drank thereof himself, and his children, and his cattle?
...... Jesus answered and said unto her,

Whosoever drinketh of this water shall thirst again:
. But whosoever drinketh of the water that I shall give him
. shall never thirst; but the water that I shall give him shall be
. in him a well of water springing up into everlasting life.
.................. God is a Spirit:
........................ and they that worship him must worship him
.................. in spirit and in truth.
............ From that time many of
.................. his disciples went back, and walked no more
........................ with him.
.............................. After this, Jesus
...... knowing that all things were now accomplished,
............ that the scripture might be fulfilled, saith,
................................................................. I thirst.
....... But the day of the Lord will come
as a thief in the night; in which the heavens shall pass away
....... with a great noise,
and the elements shall
melt with fervent heat,
the earth also
....... and the works that are therein shall be burned.
Nevertheless we,
according to his promise, look for new heavens and a new earth,
....... wherein dwelleth righteousness.
And I saw a new heaven and a new earth:
....... for the first heaven and the first earth were passed away;

............. and there was no more sea.
The Bible (1611 trans., my emphasis): Genesis 1:1-4; 2:7; 3:22; 4:6; Exodus 3:14; 2Kings 8:26; 2Chronicles 22:2; Job 38:4-7, 16-18, 36-41; Psalms 22:1, 13; 42:1-3, 9; Jonah 2:3; 4:4; Matthew 7:27; 14:25-26, 29-31; 16:18; 24:7-13, 35; 26:31-35, 56, 58, 70, 72, 74-75; 27:46; Mark 12:27; 15:33-34; Luke 20:38; 23:44-46; John 1:1; 4:6-7, 9-14, 24; 6:66; 19:28; 2Peter 3:10, 13; and Revelation 21: 1.

Is my Red your Blue?

Prima facie such a question is unanswerable in principle, because we cannot look at things through others’ minds. And yet, suppose there is at least one telepath in this Universe who can, and who could tell us the answer. Or suppose that God exists (unlikely as that may seem) and that He shows us the answer. Or, if it could be shown that the nature of perceived colours is (probably) due entirely to their use by evolved brains, then we might learn enough empirically to (scientifically) know the answer. And so forth. All very speculative, but I’ve noticed that whereas orange is quite like red (e.g. orange dyes often look red), green is relatively unlike blue. So if there are those who find green and blue to be more similar than orange and red, then perhaps their red is my blue. That is, an increased likelihood of their red being my blue might be deduced, especially if they also find red rather than blue to be more like grey. If your red was my blue then you might think that dark yellow was orange rather than green (whereas I see that as a golden green). (And ain’t it odd that adding blue to yellow is like adding grey to yellow, whereas adding blue to white makes it look whiter?)

Thursday, May 10, 2007

A Platonistic Dialogue

A blind man, Davft Lewni, is travelling in the land of the one-eyed, where he meets Ria Liszt. As our story begins he is trying to sell her his new invention.

RL: But it’s just a stick. And it’s not even the right colour for my outfit.

DL: It is indeed said to be a stick, and not incorrectly; just a stick and yet I may, nonetheless, use it to find out how far things are from me (this actual self that is here) by assuming that it is the same stick each time I use it, and via calculations whose details we may naturally presume must exist. But to begin with, it seems reasonable to suppose that there is space and time out here (for all sorts of reasons that I presume you know of). Now, I don’t know about you, but I find that things distant in space or time are never much of a problem. And with this stick I can calculate whether there is anything of any importance in any spatial direction from here.

RL: Good for you, you big Lewni.

DL: Thank you, but I am not so great (not compared with Hume). But to return to the job in hand, I'm sure that you too could use such a device, from where you are (it being reasonable for me to suppose not only that such a place exists, but that you do too, of course (at least while I'm speaking to you))... As for your earlier objection, about the wrongness of my stick’s colouration, I must confess that I find your supposition of colouration quite nonsensical. You can hardly show me a colour (as you must admit) and so I cannot take them seriously, as scientific possibilities, until you explain how such a reification of a such fuzzily delineated class of sensations could possibly exist. Please do not insult my intelligence by asking me to believe in colours without giving me a complete (and completely logical) theory of them.

RL: But I am sure that, since I do have one good eye, I shall not really need to mess about with your stick, which is actually rather cumbersome and complicated, for a stick.

DL: Ah, but it is just such presumption that would lead you into danger, in infinitely many possible worlds, were you without the assistance of such a device as this, my stick, which you might call 'white' if you wish, as it has in itself, I'm sure, no colour. You presume that your eye can be used to see with, but if so then you might indeed see colours, which is not only absurd, prima facie, but unnecessary, e.g. I do not feel the need to see colours. [Etc. ad nauseum]

Wednesday, May 09, 2007

The Bounds of Love

Today's post will be a bit long, but it's my first serious written foray into theology, and I know too little (and have too much else to do) to attempt a longer piece elsewhere. Where to begin? Well, I guess that facts are aspects of reality, but are also the elements of our mental maps of reality—what makes a fact objectively true is not that one’s mental map is a perfect representation, but that the reality that is being modelled sufficiently well by that map objectively exists. Similarly beauty is in the eye of the beholder (as is colour, etc.) whilst also being a property of the beheld—and perhaps what makes some perception of a beloved’s loveliness (of moral goodness, etc.) a veridical perception is the existence of God (linguistically male below, for convenience). Now, that would of course depend upon the details of such perceptions; but let us suppose that someone might somehow have come by a justified belief in the existence of this Universe’s Creator, much as we come by our beliefs in the existence of this Universe. After all, the divine creation of a man who naturally believes in his Creator’s existence might also give that belief some justification. So, can we say much about this Universe’s possible Creator?
......Well, something like omniscience would seem to be indicated, since an author will know all there is to know about her works, on which she is the authority. But the question arises, must the Creator know now all that we will actually do? Such would seem to contradict our directly known freedoms, few and limited as they are. A recent argument in Alanyzer shows that from plausible assumptions it follows that the Creator also exists within time. And after all, there being an act of Creation in the first place would seem to presuppose something like temporality on the part of the Creator. Open or Process Theism is not so much a limitation as a positive freedom—if the Creator did not exist within something like time (with future possibilities, and past actualities) then there would have been no free choice to Create, so there could have been no real motivation for Creation, which would make it all pretty pointless (and this argument for a Creator was the absurdity of his absence). There are problems with Open Theism, e.g. how are we to understand either the time before God, or else God’s infinite past. But time is a great mystery anyway (about which I shall post eventually), so let us move on.
......There are many conceivable motivations, and the real one may well be inconceivable, but prima facie quantum mechanics seems like a good way to put reasoning beings into a world of prescribed possibilities. Living beings exist as they do—being aware of changes and making choices—because they exist during the present collapse of the future possibilities into the past actuality that we perceive all around us. Why would that be a useful thing to do? Well, maybe the Creator knows that He does not know what He does not know, and in particular He does not know for sure whether or not there are others of His kind. How could He? (Maybe He could, maybe not; this is only a conceivable possibility.) So, maybe this Universe is a means of finding out more. One problem with that hypothesis could be that God’s infinite past becomes more of a problem—we have to imagine God not being motivated, for an infinite time, to find out what He is now so interested in. But still, perhaps there were infinitely many other things to find out about (and infinity is another great mystery anyway), so let us press on.
......Fortunately the infamous problem of evil is not a problem for this possibility. Maybe we did choose to be born (clichés notwithstanding). Maybe in a previous, more Heavenly existence (following our Creation as minds) we were asked if we wanted to participate in such an investigation, and we bravely said yes. Maybe the point of it all is therefore (paradoxically) the facilitation of the presence of other gods. We just don’t know; there are just so many possibilities. But therefore we can see each such possibility, not so much as a reason to believe in that motivation, as a reason for not being presumptuous. If the point of our existence in this Universe were the facilitation of the presence of (or more generally, the finding out more about the possibility of) other gods, then such activities as paganism and polytheism might even cohere with the Creator’s intention. That is, monotheism and multiculturalism need not conflict.
......That possibility may well seem too odd, but after all, why is there so little direct evidence for the Creator’s existence? Presumably Heaven is full of such evidence, so why were we not Created there in the first place? If the point of life is to be with and to love our Creator, then why are we here? Furthermore various people have noticed how lazy the Creator seems to have been, from (all the evidence for) natural laws to natural selection, whereas with a motivation like this one He would actually require a minimal support for minds (one that facilitated complicated structures without specifying too much about them). In short, the possibility of fleeting evils may have been the acceptable price of a useful space for free will, within an orderly, lawful cosmos; a price that we may even have chosen to pay, before we were born here (those of us who are not divinely animated, or machines—and perhaps we should not presume without good reason that none of the atheists are machines). Our ignorance (if we are honest) of the precise nature of evil may just be part of the whole point of our existence.
......Would it be so bad, that our existence had some point to it, beyond hopes of being the Creator's lap-dogs in Heaven? Perhaps in Heaven things are lovelier and more perfect, but perhaps also divine plans are laid there, plans to bravely find out more about the unknown. (And note that I'm not suggesting that we could find out about such things, only that our lives may have some ultimate point to them.) Anyway, in short, far from there being no way to make sense of what little data we have, there are actually too many live possibilities for us to be too dogmatic about, e.g., the unlikelihood of a Creator (about which I shall post more eventually).

Sunday, May 06, 2007

You Read This

That is clearly true. If you are Bob, what is true is that Bob read that title (and if you are Bill, that Bill read that title, and similarly if you are Ben, etc.), because that initial ‘You’ clearly refers to one when one reads it. So now imagine Bob reading ‘You, Bob, are mistaken.’ Bob would presumably take that ‘You’ to be referring to some particular person called ‘Bob’ and so he might wonder if that person was he.
......But why should I not have called (rightly or wrongly) each of the people called ‘Bob’ who reads it mistaken, for wondering which of them was meant? That was certainly my intention and so, pretty much as the initial ‘You’ did, that ‘You’ also refers individually to each element of a set of people (in this case not all those who read it, just those called ‘Bob’). So, that ‘Bob’ (in that sentence) also refers individually to each element of that set. In short, one proper name, on one occasion of its use, may refer to many different things. Note that ‘Bob’ was not being used as a common name, but rather that, for each Bob, that instance of ‘Bob’ would refer to Bob alone upon Bob’s reading of it.
......Now, I’m not entirely sure that that is a problem for a direct reference theory of proper names, but it does strike me as odd. Still, I've not got much beyond Searle's refutation of Kripke's objections to description theories (see
here) before getting confused. So, because of the importance of the philosophy of language for my other philosophical interests, I wonder what others make of such oddities...

Saturday, May 05, 2007

I'm referred to, therefore

I exist! That's what I thought when I saw myself referred to in Philosophy of Logic (within Fletcher's article on Infinity). My first time. Probably of little interest to anyone else, but still. More interesting was the editor's new look at the Liar paradoxes, my view of which is here.

Friday, May 04, 2007

I think, therefore...

I am a genius. I could be anything at all, with half
the chance. But today I am going to change the world.
Something’s world. The cat avoids me. The cat
knows I am a genius, and has hidden itself.
That’s from Duffy’s Education for Leisure, 1985

Thursday, May 03, 2007

Popper's Probabilities

Commenting on Basic Sets, I replied to Torbjörn’s comment (#49) that in most parts of theoretical physics they use frequentist probabilities (in connection with my new argument against standard set theory), as follows—I repeat myself in order to refer to this post in my defence of Popper (sections linked to below).

50 years ago, I noted (in comment #52), Popper noticed that many scientists were not really frequentists, for all that they might call themselves that. While they took certain real numbers in the equations of QM to be probabilities (i.e. they took QM, quantim mechanics, to be saying something that could be tested against observed frequencies) they were not frequentists because, basically, frequentism is either finite or infinite. If it is finite then probabilities cannot be real numbers (see §2 for more reasons). But if it is infinite (as it is usually) then such probabilities actually say nothing about our finite observations—they cannot be tested against observed frequencies because each value for the limit frequency is compatible with any initial values (cf. §3 and §5).

So, most of those scientists who regarded themselves as frequentists, but who were also realists about QM (unlike
Lewis for example), actually believed in single-case propensities (see §4). The propensity bit is the idea that the QM equations model something real, something chancy that is tested via frequencies. The single-case bit is the idea that even a single particle behaves probabilistically, e.g. in 2-slit experiment, and that widely separated things might be causally independent. Note that by QM I mean nothing too technical, just the QM of chemistry, and of 2-slit experiments. Such elementary experiments do appear to be telling us that the underlying stuff of reality is well modelled by Schrödinger's equation. And I take it for granted that there is some such stuff. But of course, I hope that there are some coherent objections to some of that, for me to think about...

Wednesday, May 02, 2007

St. Petersburg Paradox

This paradox (for further details, see here and here) is nearly 300 years old now, but it can still challenge a naive idea of what it is to be rational, despite its simplicity. For a simple version of the paradox, imagine God offering you the following deal, on His fair tossing of a fair coin. You can tell somehow that it is God talking to you, and so you rightly believe all that He tells you.

Firstly God will pay you a certain amount, as below, and then God will repeatedly toss the coin, until it lands heads up. Each time it lands tails up (if it does) you will owe Him a certain amount of money, as follows. For the first tail 2 cents, for the second another 2 cents, but for the third another 4 cents, and so forth, so that for a total of N tails before the first head a total of 2-to-the-power-of-N cents. God helpfully explains what that formula means mathematically. The chance of Him getting a tail on the first toss is 50%, and the chance of Him also getting a tail on the second toss is 25% (since there are four equally likely possibilities, HH, HT, TH and this one, TT) and so forth. So the mathematical expectation is (2/2 + 4/4 + 8/8 + …) cents = (1 + 1 + 1 + …) cents = an infinite number of cents, or dollars (or trillions of dollars).

In view of that fact, God feels that it would not be irrational to offer you the entire wealth of the Universe. In effect, you would become God’s appointed Queen (or King) of the Universe in exchange for you owing Him a number of cents thus determined, by the fair toss of a fair coin. To simplify matters, assume that God has shown you that, whether or not you take him up on this deal, you will live forever in some form or another (e.g. as an immortal soul), and that the wealth of the Universe includes alien medical technology that can prolong your natural life indefinitely, and also teleportation devices (so that you could actually spend all that wealth). Of course, conversely God could (if necessary) make you pay Him arbitrary amounts over and above your new wealth, were you to end up owing Him money (were you that unlucky), by getting you to work for Him at a very reasonable rate of pay, in some relatively pleasant part of Purgatory. But still, there is no trickery involved here (it is God talking, not the Devil). God will not be offended if you do not take the deal, it is only being offered because He suspects that you might wish to take it. Would you?

Well, what is your chance of losing much? It is extremely small, because for you to return as much as 10 dollars, from the vast wealth of the Universe, God would have to throw at least 10 tails in a row. And if God threw less than 47 tails before throwing his first head, which seems almost certain to occur, you would not even have to return a paltry trillion dollars. Your chance of having to work for trillions of years in the afterlife is much, much smaller. Would it be rational to reject such an offer just because of the remote possibility of something that almost certainly won’t happen? I mean, smell the coffee! What would actually happen, if the above scenario were offered, and you took God up on His deal? You would own most of the Universe.

Still, your mathematical expectation (the mean, not the mode) is of an infinite loss, and perhaps rationality should have such a mathematical precision (e.g. via coherent betting quotients). According to such a view, your intuitions about what is rational would, if you took God up on this deal, be letting you down. Nonetheless it does also (as above) seem rational to accept this deal. So, could both options be rational? Incidentally, I don't think that the infinitude of this scenario is the underlying source of the paradox, even though our intuitions are often confounded by infinite scenarios. That is because God's mathematical expectation did not need to be of an infinite gain, only of a gain greater than what He offered you. A paradox would therefore remain even if, after a number of tails equal to, say, twice the wealth of the Universe in cents, further tails would entail no further losses.