Monday, September 29, 2008

Theodicies, nice but unnecessary?

Can the atheistic force of the evidential problem of evil be countered by noting that, were there a good God, there would be some true theodicy?
......Atheists can hardly complain (coherently) that such a response does not take the problem seriously enough if, while using their standards to judge hypothetical Gods, and while having the power to reduce the amount of serious suffering in the world (as they usually do, to some extent), they do not use it to that end because they do not regard that problem as sufficiently serious. (They can complain incoherently, and justify such incoherence on the grounds that they do not claim to be more than evolved apes; but then, why bother to justify it?)
......Theists do see evil as a problem though, and not just as a problem that God can help them to deal with. Even theists may wonder, when bad things happen to them, whether that is because they are bad people, or if bad things can happen to good people (they may find that in either case there is some problem with God being what they would call ‘good’). Still, they can always believe that there must be some explanation (even if we could never understand it) since they do believe in a good God; they can always be Sceptical Theists, much as atheists can be Promissory Materialists (or go Mysterian) when faced with the problem of how awareness could possibly arise from within an entirely material universe.
......There must, similarly (they may think), be some explanation of why God does not tell them what that explanation is (cf. how evolved apes would not be expected to know much beyond the ordinary phenomenal world), and very probably the same explanation (as on my preferred theodicy). Having said that, there is an important role for a theodicy in an evidential argument for theism, or in some similarly rational justification for a particular theology (as with my theodicy and Open theism) and hence for a particular metaphysics (as with Open theism and Presentism).
......Such arguments may not be necessary to justify theism but even so, we might be morally obliged to give them if we can. God would surely prefer to tell us why we must suffer the evils of this world, as Rowe has recently argued (via the analogy of God with a parent taking her child to the doctor or dentist). According to my preferred theodicy, God would prefer to let us find out such things for ourselves—if we can (and we can, on that theodicy, since that theodicy)—because such causal and epistemic distance is what we asked for and he agreed to (whence his obligation not to tell us) when he decided to create the Earth (as well as Heaven).
......I therefore disagree with those theists who say that we should not give a theodicy on the grounds that the Bible asks (rhetorically) “Shall the thing formed say to him that formed it, Why has thou made me thus?” Why should she not, I wonder, since her form is that of a rational agent with problems? While I agree that we are in no position to judge God (as I begin this post by observing) because good is by definition (according to divine command metaethics, which are plausible if there is a God) whatever God wants, giving a theodicy is not a matter of judging, or even of apologising for God, but of trying to understand Creation.

Saturday, September 27, 2008

Not from Presentism to Theism

The following is the abstract of Alan Rhoda’s forthcoming “Presentism, Truthmakers, and God
The truthmaker objection to presentism (the view that only what exists now exists simpliciter) is that it lacks sufficient metaphysical resources to ground truths about the past. In this paper I identify five constraints that an adequate presentist response must satisfy. In light of these constraints, I examine and reject responses by Bigelow, Keller, Crisp, and Bourne. Consideration of how these responses fail, however, points toward a proposal that works, one that posits God’s memories as truthmakers for truths about the past. I conclude that presentists have, in the truthmaker objection, considerable incentive to endorse theism.
As a rational continuant, my starting position is (naturally) Presentism. But I wonder, why can’t the reason why, for example, it’s true that I drank that coffee (the one now warming me) be that “I am drinking this coffee” was then (when I drank it) corresponding (in the right way) with reality? In the changing present (that is all that is) those words (involving reference to the present me) have so corresponded, so it seems to me that that they have could be a property that grounds the truth of such related words as “I did drink that coffee” via natural linguistic rules—that that property could continue to be associated with such words much as I continue to be associated with those ‘I’s.
......Such an account could cohere pretty well with common sense when it says—in its informal way—that my drinking that coffee when I did is what grounds the truth of “I did drink that coffee,” since we don’t normally mention obvious linguistic rules when giving ordinary explanations. Of course, that may well not be the correct account; but there are probably quite a few more possibilities yet to be refuted decisively. And if, for example (and firstly), Bigelow’s suggestion—that the world as a whole has past-tensed properties that can ground truths about the past—could be defended under atheism (or similarly, if one of the others could), then Alan’s argument could become (as below) an argument for atheism from Presentism, were (secondly) God’s memories unsatisfying truthmakers.
......Regarding the latter, might God give some creature the freedom to be unobserved for a bit? That does not seem to be impossible. But if so, if God did that then some of such a creature’s acts won’t enter into God’s memories; whence God’s memories could hardly be the truthmakers for truths about the past. And if not, if God’s power can’t go that far then we have, in that, a new reason to doubt that God could give his creatures the sort of totally free wills that require the unreality of the future. Theists would therefore lose one of their main reasons for being Presentists in the first place; and there are those who argue from theism to Eternalism, who might agree with Alan’s conclusion and add that if Presentism implies theism (which implies Eternalism) then so much the worse for Presentism. So even if the truthmakers we want are God's memories, maybe we don’t get to theism unless Eternalism leads there (which remains to be shown).
......Regarding the former, were atheism true nomological necessities would be safe from being over-ridden by God; and it might well be that any world like this (made of this stuff, and having this form) must have begun in a Big Bang (assuming that this one did so begin), which would take care of sceptical scenarios such as Russell’s (the world popped into existence 5 minutes ago). And it is surely possible that a truth about the future might be grounded in a present range of propensities (surely Presentists think so), so why should a truth about the past not be grounded in a present state that must—given present natural laws (constantly specified by the stuff of the world)—have developed from a state so described?
......In short, I think that Alan’s conclusion is inadequately supported by his arguments. Maybe the existing accounts of truthmakers are all inadequate, but why should that mean anything other than that we have, as yet, been insufficiently clever? Alan does not say. And those theorists criticised by Alan could have had ideas that did not work perfectly (if Alan is right; if they don’t) then maybe Alan’s idea won’t pan out either. Why should we think that they will? Alan does not say.
......After all, do you (by analogy) have a reason to endorse atheism because of the problem of evil? Or is it rather than if you are already an atheist then you will see that problem as one of the reasons why your position is a reasonable (an intellectually comfortable) one? Conversely (and more appositely if one is an agnostic Presentist), do the problems facing non-dualistic accounts of mind and matter give us any reason to endorse theism (substantial dualism being more reasonable given an underlying monotheism) rather than, for example, that we have not yet thought of everything, or (more pessimistically) that human concepts just cannot stretch to such explanations, or to endorse panpsychism (and so forth)?
......So there are reasons to be sceptical of any such argument to theism. Much as Alan asks three questions of Bigelow’s account (to motivate rejecting it), so one could question how much one should conclude on the basis of there being such (currently) unanswered questions.

Wednesday, September 24, 2008

What Maths is All About

According to van Benthem (a logician) and Dijkgraaf (a physicist), and I tend to agree (as an amateur scientist), a natural division of mathematics is the following list (pp. 42-3 of Five Questions):

......Symmetry, Invariance and Language
......Counting, with Numbers and Language
......Computation and Complexity
......Paradoxes and Meta-theorems
......Games and Information Dynamics
......Prediction and Dynamical Systems

And the following, from Feferman's Answers (pp. 131-2) to those Five Questions (on PoM), is, I think, very true.
Discovery in mathematics is one of the highest exercises of creative intelligence. But confirmation of mathematical discoveries requires rigorous calculation and demonstration, and in this respect mathematics is logical at its core. Moreover, mathematics is progressive, it builds on what came before. Thus, since there can be no infinite regress, from the point of view of logic mathematics must rest ultimately on some sort of axiomatic foundations. While mathematicians may accept this in principle, there is a sharp dichotomy between the logicians’ conception of mathematics and that of the practicing mathematician. The latter pays little or no attention to logical or foundational axioms, even if he or she subscribes to some overall foundational viewpoint such as that of axiomatic set theory. And in fact, the logical picture of mathematics bears little relation to the logical structure of mathematics as it works out in practice. The use of certain basic structures like the natural numbers and the real numbers (and of structures built directly from them like the integers, rationals and complex numbers) is ubiquitous, and there is constant appeal to such principles as proof by induction and definition by recursion on the naturals and of the lub principle for the real numbers. But these are not viewed from an axiomatic point of view, e.g. from that of the Peano Axioms for the naturals. The essential difference is that the language of PA is limited to a fixed vocabulary, whereas induction and recursion can be applied in any subject in which natural numbers play some sort of role. For example, the operation x^n is defined in any (multiplicative) semi-group for every element x and natural number n, and its properties are proved by induction on n. So even where the practicing mathematician invokes the basic axioms of the natural numbers, that is done without restriction to a fixed vocabulary. According to the current set-theoretical point of view, all such concepts that the mathematician might want to use in addition to those expressed in PA are defined in the language of ZFC, so we need only look no further in order to give full logical scope to what underlies daily mathematics. It seems however, that if we accept the language of set theory we ought to accept notions not defined in that language, such as the notion of truth in the set-theoretical universe. Moreover there are informal outlying notions that have mathematical coherence, but are not (as given) defined within set theory. [... Feferman proposes] an informal framework to account for mathematical practice and its actual and future possible applications in a more direct way than through the use of the various formal systems currently dominating logical work. This is work in progress, as an extension of my earlier work on unfolding of open-ended schematic systems. An essential new feature is the introduction of a quite general underlying “proto-mathematical” framework for operations and properties; that allows for the interaction of basic schematic systems like those for the natural numbers, real numbers, and subsets of any domain.
Feferman is an antiplatonist, but his points appeal to me nonetheless (other promising logics were mentioned in that book, e.g. by Hintikka, Visser, Weir and Zalta).
......By contrast, although Hellman favours "an objective, broadly "realist" view of mathematics" he fails to notice how widespread open quantification is in real mathematics. So he tries to distinguish (p. 162) between "absolutely every object" and "anything we would ever come to recognise as an object" (i.e. between 'all' and 'any'). Now, I liked the way he nicely listed (pp. 158-160) our collective failure to justify the axiom of infinity, but not his distinguishing between quantifying over a given set of ordinals, and over all ordinals (p. 163):
The cases are, after all, entirely different. In the former, we are given a set and then asked to consider all subsets of it, or all functions defined on it with values in some other given set (e.g. {0, 1}). That is, these are limited notions, "already restricted" as it were. Moreover, they are commonplace in mathematical practice. [...] But "all sets" or "all ordinals" in some putatively absolute sense are supposed to be entirely unlimited and unrestricted, and they raise suspicions in much the same way "absolutely all objects" does—or should, at any rate! Not surprisingly, they are quite foreign to ordinary mathematical practice.
But in ordinary maths 5 + 7 = 12 basically means any five things plus another seven things are twelve things, and those things can be anything whatsoever, e.g. possibilities, feelings, even proper classes. And "for all" just means "for any" in ordinary maths, of course. Even when we move to ordinals (starting from 1), the finite ordinals are given by a rule (keep adding 1), and the ordinals need another (include limit ordinals), but other sets of ordinals need yet another (stop somewhere), so Hellman's distinction is not really that natural.
......A more obvious distinction is between finite sequences and those simply infinite endless sequences that might well be expected to be indefinitely extensible, in a way slightly analogous to, but basically quite unlike (since arising from endlessness), vagueness (due to predicates not being well defined) or fuzziness (due to objects being not logical). And as he had already shown, we have no reason to assume that they are not indefinitely extensible (and there is also this reason to assume they are).

Saturday, September 20, 2008

Curry's Paradox

A sentence is true insofar as it describes reality, ordinarily (and adequately enough here), so consider the following sentence:
(C)......If this sentence is true then so is the sentence (S).
Suppose that (C) is true. We would have, not only that (C) was true but also—from (C)’s definition—that from (C) being true we would have the truth of (S), so we would have, consequently, that (S) is true. That is, if (C) is, as we supposed, true then (S) is true. But that means—from (C)’s definition—that (C) is true.
......It seems that, logically, (C) is true, which is a paradox because (C) cannot be true, of course, because (S) was arbitrary, and some sentences are certainly false. If there was (as there seems) nothing wrong with our logical steps, then there was something wrong with our original sentence. And note that (C) says (of itself) that contradictions—e.g. (S) = “2 + 2 = 5”—follow from its truth, so essentially it says (of itself) that it is not true. That is, (C) is a Liar-style sentence, and the traditional resolution of such sentences is that they are senseless, a bit like such nonsense as “I met a man who wasn’t there” (can be).
......The propositional content of a Liar sentence such as “This sentence is not true,” which I shall call ‘(L),’ is essentially the same as that of the clearly empty “Disobey this command!” Such an utterance from one’s superior would naturally prompt the thought: What command? And note that just as (L) can seem true—since it seems to say that (L) is not true, and (L) is senseless so it is not—paradoxically because it is not, so similarly (C) can seem true: If (C) is false then it is false that from a falsehood—that (C) is true—we might deduce a contradiction—via all sentences being true—whereas such a deduction is plausibly alright.

Thursday, September 18, 2008

How wrong is lying?

Some (e.g. Alexander Pruss) say that lying is always wrong, but I wonder. (The question arises in the context of teaching, where you have to teach what is to be taught, not what you yourself believe, and where the naturally sociological aspects of teaching can be counter-intuitive, as with the recent Michael Reiss stuff.)
......The familiar counter-example is the knock on the door in the dead of night. It’s the Nazis, come to ask you if there are any Jews hiding in your attic. There are (say) and if you don’t say anything, or if you say anything they don’t like, then they’ll investigate further. Convincing, I find; but I also suspect that the Nazis might not count. Perhaps they’ve left the linguistic community within which lying is wrong, by their actions, and joined the ranks of the dangerous animals. (Language-use is a pretty complicated business, I find.) So suppose you’re a doctor.
......Your patient is fatally ill, with no known cure. Still, if she thought there was a cure, there might be a placebo effect. So you might lie to her, e.g. tell her that there is a new drug being tested. She could join its trial (you might tell her), with a 50% chance of getting a placebo. You cannot tell her any more details (you might tell her) in the interests of scientific objectivity (and in her own interests, naturally). There need be no real trust betrayed here, because people might (say) know that you’re scrupulously honest in general, that you would only lie in this sort of case.
......Suppose your patient knows you might be lying, but doesn’t know that you are. (That would hardly affect the placebo effect because, in a real drug trial, she would know there was a good chance of not getting the drug, and would not know how good the drug would be even if she got it.) Is lying in such a case wrong (as it must be if lying is always wrong)? E.g., is it the lesser of two evils? But if so then what is the other, greater evil? Letting nature take its course when there is nothing (that is morally acceptable) to be done about it, presumably; but if so, what’s wrong with that?
......Of course, you (the doctor) could get the same (or maybe a better) result without lying, e.g. by giving your patient a homeopathic remedy; but the same question will arise: If you are peddling such remedies, is it wrong for you to lie as part of a system that enables doctors to avoid lying? You need not be lying when you say that homeopathey works (since it works insofar as placebos work), but you would have to lie at some point unless you were very naive (dangerously so, since you claim to be selling medicine), so why not let the professionals take care of such things directly?

Monday, September 08, 2008

Cantor's Paradox Again

The positive integers (1, 2, 3, ...) are the products of endlessly reiterating the addition of 1, starting with 1. Given a common sense arithmetical realism, either they exist (so to speak) altogether like stars coexisting in space, forming a transfinite set—a quasi-spatial collection that is a definite thing in its own right—or else they are more accurately envisaged quasi-temporally, being never a completed collection (or set) but rather being (in Mill’s words) indefinitely extensible. It has become standard to suppose the former, so let us do that, for now.
......For any set there is also its power-set, the set whose members are the subsets (the parts) of the original set. And each set is (cardinally) smaller than its power-set because not only is the former a subset of the latter, there is no way to pair the members of the former with the members of the latter (the proof of that impossibility is basically Cantor’s diagonal argument, which uses axioms that are realistic enough). Reiterating the power-set operation, starting with the set of positive integers, yields an endless sequence of transfinite sets, whose (cardinal) sizes are the Beths. Now, if there was a set of all the other sets, its power-set would contain more sets, which is absurd, whence there is no such set. The sets—and similarly (although the argument is longer) the Beths—therefore form a different sort of collection, a proper class.
......That result is Cantor’s Paradox (which I last blogged about over a year ago). Each integer follows from some finite reiteration of their defining (and clearly definite) algorithm, so it exists, so to speak; they all do, whence one might expect that they form a set. And since that set exists (quasi-spatially), so each proper part of it exists (as a different collection), whence the power-set whose size is the next Beth exists; and so on. Each Beth results from a definite transfinite reiteration, so they all exist. Cantor’s paradox is so-called because realistic intuitions that the integers form a set tend to give the wrong answer for proper classes such as the Beths.
......Consequently those intuitions are not to be trusted; and indeed, there is also an empirical argument—from an equally realistic interpretation of quantum-mechanical probabilities—to the indefinite extensibility of applicable arithmetic. Mathematicians have for the most part preferred to reject such realisms, rather than (formalistic) set theory, presumably because of the predominant Naturalism within academic Science. But Cantor was himself a realist, and not afraid of theism, and might have rejected his set theory before either. (His way of keeping all three was to go paraconsistent when it came to theism, but that did not really solve the problem for arithmetical realism.)

Tuesday, September 02, 2008

Lost Souls

Martha Jones, ex-time traveller and now working as a doctor for a UN task force, has been called to CERN where they're about to activate the Large Hadron Collider. Once activated, the Collider will fire beams of protons together recreating conditions a billionth of a second after the Big Bang - and potentially allowing the human race a greater insight into what the Universe is made of.

But so much could go wrong - it could open a gateway to a parallel dimension, or create a black hole - and now voices from the past are calling out to people and scientists have started to disappear... Where have the missing scientists gone? What is the secret of the glowing man? What is lurking in the underground tunnel? And do the dead ever really stay dead?

Lost Souls is on Radio 4 on Wednesday 10 September at 2.15pm. A report by EONR said there was "no conceivable danger"

But there have been fears about the possibility of a mini-black hole - produced in the collider - swelling so that it gobbles up the Earth... Critics have previously raised concerns that the production of weird hypothetical particles called strangelets in the LHC could trigger the mass conversion of nuclei in ordinary atoms into more strange matter - transforming the Earth into a hot, dead lump.
The vacuum around a mini-black hole is a quantum-mechanical vacuum, full of short-lived virtual particle-antiparticle pairs. So it is almost bound to swallow one of those quanta (of the lowest energy level), which amounts to the creation of an actual particle or antiparticle. And mass-energy being conserved, that means a net loss of mass-energy to the mini-black hole, making it even smaller, etc. It would be better to call it a mini-white hole.
......Still, a black-hole is a crease in space-time, so who knows about the lost souls? There is a nice sci-fi trilogy about souls flooding back into the land of the living (and space-faring), following a Reality Dysfunction (involving an alien investigating spatial distortions in an energistic vacuum)—a nice blend of zombies, gangsters and astronauts, I thought.

Monday, September 01, 2008


Following on from my last post, I’ve noticed that those introducing standard set theory often say it’s a good foundation for maths because it gave answers to questions that mathematicians were asking, a hundred years ago (even if it can’t show them to be the correct ones), e.g. the problem of integrating a curve that takes the value of 1 for irrationals between 0 and 1, and otherwise takes the value 0, to which standard measure theory gives the answer of 1. But note that such problems only arise within this kind of foundation. If continua are absolutely continuous (e.g. they contain 1/0 points) and simple infinities are indefinitely extensible, for example, then no such problems could arise because only a finite number of exceptional points could be given.
......Furthermore standard set theory throws up questions about measures that it certainly cannot answer. The pieces of the decomposed sphere of the Banach-Tarski paradox, for example, cannot be given any measures consistently—not even measures of 0. That paradox is usually 'resolved' by something like the assertion that it is really just an aspect of the proof of a theorem that certain sets of points cannot be given certain kinds of measure. (That assertion is usually followed by a question about why one would expect all sets of points to have measures, in view of such examples of counter-intuitive behaviour with infinite sets as Hilbert’s Hotel.) It’s usually been forgotten, by that stage of the exposition, why set theory was being entertained in the first place.