Upon reflection, I don’t think much of my previous resolution of Richards’ Paradox, which was as follows: Such finite strings of words as specify real numbers between 0 and 1 can be listed in order of increasing number of symbols used in the description and then, within each length of description, in alphabetical order. Given that list, one may seem able to define a real number between 0 and 1 and not in that list using a diagonal procedure: The digit in its nth decimal place is the final digit of d + 1, where d is the digit in the nth decimal place of the nth number on the list. Richards’ paradox is that such a number has such a finite description (so it is in our list, whence that procedure is contradictory, where it is, so there is no such number, in our list, whence that procedure is not contradictory, and so on).
......My previous resolution was just to note that our list includes anything that anyone could possible say (in finitely many words) that would specify a real number between 0 and 1. So in order to specify a number via that list, any (finitely describable) diagonal procedure must explicitly exclude its own entries in the list. But that now seems patently inadequate. Given such a list, we have its diagonal, and so a real number not on that list is clearly indicated. And we can say that it is. What we say cannot be in the list, but that just means that such a list—of all the finite specifications of such real numbers—is impossible. I would not be surprised if it was impossible, because words are typically a bit vague and variable in meaning. But it does seem odd that we can deduce that they must be fuzzy or incomplete, which is why I am not very fond of that resolution either. All in all, I find Richards’ paradox very puzzling.
Friday, November 21, 2008
Monday, November 17, 2008
How We Reason
Either Jane is kneeling by the fire and she is looking at the TV,Most individuals say, “yes”, see Walsh, C., and Johnson-Laird, P.N. (2004: Co-reference and reasoning, Memory and Cognition 32, 96–106). Given the first premise, they think of two possibilities: in one, the first conjunction is true; and in the other, the second conjunction is true. They overlook that when the second conjunction is true, the first conjunction is false, and that one way in which it can be false is when only its first clause is true, i.e., Jane is kneeling by the fire but not looking at the TV. Hence, the correct answer to the question is: “no”.
or else Mark is standing at the window and he is peering into the garden.
Jane is kneeling by the fire. Does it follow that she is looking at the TV?
The above is from ‘How We Reason: A view from Psychology’ in The Reasoner 2(3), 4–5.
......Philip Johnson-Laird’s book How We Reason is out in paperback next month.
That was an example of a fallacy; there’s a nice list of fallacies in The Reasoner 2(5), 7–8, incidentally.
Thursday, November 13, 2008
Berry's Paradox
The natural numbers are one, two, three and so forth. They can grouped by the number of syllables it takes to say them: {one, two, three, ...}, {seven, thirteen, fourteen, ...}, {eleven, seventeen, twenty-one, ...}, ... . So we can easily find, for example, the smallest natural number that cannot be given to us, in that way (by saying its English name), in fewer than three syllables; it is eleven.
......Let us now allow any method of verbally specifying a natural number—e.g. “two hundred and thirty-four times fifty thousand and fifty” (saving us six syllables on simply saying its name)—and consider the following specification:
(B)......The smallest natural number that cannot be specified in less than twenty-five syllables.
Berry’s paradox (which is over a hundred years old) is that (B) is itself only twenty-four syllables long. It seems paradoxical because we expect (B) to specify a number, in at least twenty-five syllables (much as earlier we got eleven, in three syllables), and yet whichever number it is has therefore been given to us by (B) in less than twenty-five syllables.
......However, since we have allowed any method of verbally specifying a natural number, we have allowed that I might take any number and say it at a certain time and date and then refer to it (and hence specify it) in less than twenty-five syllables as, for example, “The number mentioned by me between two and two thirty on the first of June two thousand and eight.” It is quite indeterminate which numbers one might do that for, and hence the number specified by (B) is also indeterminate.
......And if we don’t generalise completely, but instead limit to some extent the ways in which the numbers considered by (B) may be specified, then (B) will either give us some specific number with no problem (much as the specification above gave us eleven), or else it will be rather more obviously self-contradictory (much as Richard's paradox is), so that we would hardly expect it to give us a number (whence it should stop appearing paradoxical).
......Let us now allow any method of verbally specifying a natural number—e.g. “two hundred and thirty-four times fifty thousand and fifty” (saving us six syllables on simply saying its name)—and consider the following specification:
(B)......The smallest natural number that cannot be specified in less than twenty-five syllables.
Berry’s paradox (which is over a hundred years old) is that (B) is itself only twenty-four syllables long. It seems paradoxical because we expect (B) to specify a number, in at least twenty-five syllables (much as earlier we got eleven, in three syllables), and yet whichever number it is has therefore been given to us by (B) in less than twenty-five syllables.
......However, since we have allowed any method of verbally specifying a natural number, we have allowed that I might take any number and say it at a certain time and date and then refer to it (and hence specify it) in less than twenty-five syllables as, for example, “The number mentioned by me between two and two thirty on the first of June two thousand and eight.” It is quite indeterminate which numbers one might do that for, and hence the number specified by (B) is also indeterminate.
......And if we don’t generalise completely, but instead limit to some extent the ways in which the numbers considered by (B) may be specified, then (B) will either give us some specific number with no problem (much as the specification above gave us eleven), or else it will be rather more obviously self-contradictory (much as Richard's paradox is), so that we would hardly expect it to give us a number (whence it should stop appearing paradoxical).
The Possibility of Free Will
Consider a real object in the world around you, e.g. a brown chair. Maybe the chair is really made of atoms, but if so then that underlying chair is not so much brown as capable of reflecting photons in certain ways. And since there is only one chair not two, out there in the real world—where you can see that the brown chair is—so there is no atomic chair. But of course, we need not become Idealists for that reason.
......Why should there not be many different but equally sound ways of regarding things? That there appears to be a puzzle may just be due to our being in the world that we are thinking about. So we might expect greater puzzles when thinking about ourselves, due to our being them identically. Therefore the following argument—that we couldn’t have morally significant free wills—shouldn’t convince us that we don’t.
......The argument is that, whatever a free choice between at least two alternatives—say, X and Y—is, either something beyond one’s power to choose determines that choice or else nothing does. One chooses, say, X; but why? If some reason for choosing X appealed to one, then something in one’s nature must have been predisposed to be so appealed to (and if that thing was chosen, then the regress just goes one step back, to why one so chose), but if nothing does then one’s choice was made randomly, irrationally, irresponsibly and so forth.
......Why should there not be many different but equally sound ways of regarding things? That there appears to be a puzzle may just be due to our being in the world that we are thinking about. So we might expect greater puzzles when thinking about ourselves, due to our being them identically. Therefore the following argument—that we couldn’t have morally significant free wills—shouldn’t convince us that we don’t.
......The argument is that, whatever a free choice between at least two alternatives—say, X and Y—is, either something beyond one’s power to choose determines that choice or else nothing does. One chooses, say, X; but why? If some reason for choosing X appealed to one, then something in one’s nature must have been predisposed to be so appealed to (and if that thing was chosen, then the regress just goes one step back, to why one so chose), but if nothing does then one’s choice was made randomly, irrationally, irresponsibly and so forth.
Wednesday, November 12, 2008
The Action of Free Will
Materialism, in its most plausible forms (e.g. property dualism, cf. this old crosspost), implies that something like micro-psychokinesis should be observable, via the likelihood of Gaia as a self-aware wielder of such of its parts as us, self-aware and language (and other tool) using as we are; because if we are purely material, if matter is such that amongst its properties it includes those that give rise to us as we are—much as sunlight is such that amongst its properties it includes those that allow lasers to blast rocks to smithereens—then it is surely indicated, by our existence, that a more complex and unitary structure such as the Earth’s ecosphere would be more like the goddess Gaia than, say, a car or a crystal.
......Similarly theism, in its most plausible form (e.g. as indicated by the most plausible theodicy), indicates that something akin to micro-psychokinesis would occur within living brains, if not elsewhere, as the soul-brain interaction. Reports of such things as micro-psychokinesis are therefore most interesting philosophically, because their empirical details should have—or so one might expect upon reflection upon what we know pretty well nowadays—the potential to discriminate between the most plausible materialisms (not, e.g., Humean supervenience) and theisms (not, e.g., Islamist fundamentalism). It is therefore sociologically interesting that there is so little professional interest in making rigorously objective observations of such things, even though there are reports by scientists of such things.
......How many other ways are there, whereby 'collapse' and 'no-collapse' interpretations of Quantum mechanics could be distinguished (the true from the false) empirically? If there was micro-psychokinesis then minds would not just be occupying slices through a world described by the wave function, since in that latter case the external events would have to seem random to us. A possible reason why there is a lack of professional scientific interest in such experiments is that 'collapse' interpretations seem to need an observer external to the entire physical universe, e.g. a God, and many scientists prefer to presume that there is no such being. They would say that since there is no such being, so 'collapse' interpretations are false, and hence there is no micro-psychokinesis to look for. Really a rather unscientific attitude (hardly letting the world itself tell you what is true of it).
......Similarly theism, in its most plausible form (e.g. as indicated by the most plausible theodicy), indicates that something akin to micro-psychokinesis would occur within living brains, if not elsewhere, as the soul-brain interaction. Reports of such things as micro-psychokinesis are therefore most interesting philosophically, because their empirical details should have—or so one might expect upon reflection upon what we know pretty well nowadays—the potential to discriminate between the most plausible materialisms (not, e.g., Humean supervenience) and theisms (not, e.g., Islamist fundamentalism). It is therefore sociologically interesting that there is so little professional interest in making rigorously objective observations of such things, even though there are reports by scientists of such things.
......How many other ways are there, whereby 'collapse' and 'no-collapse' interpretations of Quantum mechanics could be distinguished (the true from the false) empirically? If there was micro-psychokinesis then minds would not just be occupying slices through a world described by the wave function, since in that latter case the external events would have to seem random to us. A possible reason why there is a lack of professional scientific interest in such experiments is that 'collapse' interpretations seem to need an observer external to the entire physical universe, e.g. a God, and many scientists prefer to presume that there is no such being. They would say that since there is no such being, so 'collapse' interpretations are false, and hence there is no micro-psychokinesis to look for. Really a rather unscientific attitude (hardly letting the world itself tell you what is true of it).
Wednesday, November 05, 2008
Theism implies Open theism
According to theism there is a God who has, for example, the most understanding that anyone could possibly have. Open theism is the thesis that God’s future is to some extent open. It is not that God exists within time, but that time—or rather, changeability—is another of God’s attributes. The temporal dimension is certainly implicit in much of our ordinary talk of ordinary things (changeable continuants), but it is only our imperfect, quasi-spatial reification of change. Changeability itself originates with God’s power to change (e.g. to choose to create contingent continuants like us) should he wish to. The following shows that changeability is indeed a power, rather than a liability.
......It is one of several arguments I produced in response to Mawson’s recent argument that, since a temporal God would not know all about the future, if we had free will, whereas a timeless God would, and since God is maximally knowledgeable, so God is timeless. It is based on the observation that if God could be timeless—if a timeless divinity could create a world of people like us, while being above and beyond our personal and physical temporalities (or ways of being changeable)—then surely an everlasting (or Open theistic) God could have created such a world in a single moment of his relatively transcendental time. He would just have been creating things whose temporalities differed that much from his own, just as a timeless divinity would have been doing.
......Now, there are lots of possible worlds, which God would know all about even if he did not actually create all of them. Not creating some of them would hardly be a failure of omnipotence, as being unable to create them would. So suppose that God is everlasting and that he has chosen not to instantaneously create such a world as ours would be were God timeless, but has instead made it as Open theists believe it is. If God was timeless he could not do that, because he would have to know all about the future of any world that he could possibly create. So an everlasting God knows about (and is able to create) all the possible worlds of a timeless God, and more besides.
......God being maximally knowledgeable (and maximally powerful), we should be Open theists, at least according to Mawson’s methodology (and given the validity of such possible-worlds-talk).
......It is perhaps more clear that we should not conclude that God is timeless just because he could then be completely knowledgeable (and powerful) in respect of ourselves. That is because 100% of a little could be much less than 1% of a lot. To see that even more clearly, let RoboGod be an infinite computer that can create arbitrarily complex virtual beings, about which it would know (insofar as computers can know) everything, and over which it would have complete control. One might think that one might be such a creature (e.g. because Functionalism is conceivable) but even so, RoboGod might not know as much as (and is clearly less powerful than) someone who could create such a computer in the first place.
......It is one of several arguments I produced in response to Mawson’s recent argument that, since a temporal God would not know all about the future, if we had free will, whereas a timeless God would, and since God is maximally knowledgeable, so God is timeless. It is based on the observation that if God could be timeless—if a timeless divinity could create a world of people like us, while being above and beyond our personal and physical temporalities (or ways of being changeable)—then surely an everlasting (or Open theistic) God could have created such a world in a single moment of his relatively transcendental time. He would just have been creating things whose temporalities differed that much from his own, just as a timeless divinity would have been doing.
......Now, there are lots of possible worlds, which God would know all about even if he did not actually create all of them. Not creating some of them would hardly be a failure of omnipotence, as being unable to create them would. So suppose that God is everlasting and that he has chosen not to instantaneously create such a world as ours would be were God timeless, but has instead made it as Open theists believe it is. If God was timeless he could not do that, because he would have to know all about the future of any world that he could possibly create. So an everlasting God knows about (and is able to create) all the possible worlds of a timeless God, and more besides.
......God being maximally knowledgeable (and maximally powerful), we should be Open theists, at least according to Mawson’s methodology (and given the validity of such possible-worlds-talk).
......It is perhaps more clear that we should not conclude that God is timeless just because he could then be completely knowledgeable (and powerful) in respect of ourselves. That is because 100% of a little could be much less than 1% of a lot. To see that even more clearly, let RoboGod be an infinite computer that can create arbitrarily complex virtual beings, about which it would know (insofar as computers can know) everything, and over which it would have complete control. One might think that one might be such a creature (e.g. because Functionalism is conceivable) but even so, RoboGod might not know as much as (and is clearly less powerful than) someone who could create such a computer in the first place.
Friday, October 31, 2008
Being there
She couldn't just tell him to go away, or ignore him. That would have been so much easier for her. But he was a seventeen-year-old human being with feelings; just like everyone else he had never asked to be born – no matter the strange nature of his birth. He deserved to be treated with consideration and respect.That's from The Temporal Void (p. 721), but I'm wondering about the implicit implication: had we asked to be born, would we not deserve consideration and/or respect? Or is it that nothing can ask to exist, since there is nothing anything can do before it exists, and everything deserves consideration? (I wonder because I think I can show that it is likely that we did ask to be born, if not for whatever bad luck has befallen us.) And what if someone did ask to be born (e.g. Jesus, maybe), would they not deserve consideration (e.g. if they fell victim to malice)?
Saturday, October 11, 2008
Tales of the Unexpected
In the middle of his exhilarating exploration of science and the imagination, Richard Holmes takes us up with the first balloonists soaring from earth in the 1780s. They had expected to find out about the sky. Instead, what they saw was the earth: "A giant organism, mysteriously patterned and unfolding, like a living creature." Their new view of fields and roads, rivers and hills spurred the map makers, while their flight also stirred an interest in meteorology and the formation of clouds. Holmes compares his awed balloonists to the astronauts of the 1960s looking back at the "single blue planet" they had left behind. Each jolt in perception makes us see the familiar map of our lives differently and revaluate our place in the universe.So begins Jenny Uglow's review (in today's Guardian) of Richard Holmes' "The Age of Wonder" (a new book on Romantic scientists), next week's Book of the Week.
Tuesday, October 07, 2008
Richards' Paradox
Finitely definable real numbers are endless sequences of digits (plus a decimal point) that can be specified in a finite number of words, e.g. “two plus two” defines 4.000... (the 3 dots signifying an endless sequence of zeros, as usual). Sounds simple, but there is a paradox here (published by Richards in 1905), which inspired Gödel to worry about the definability of arithmetic truth (and was recently discussed in an open access issue of Philosophy), which are the subject of this week’s In Our Time.
......Such finite strings of words as specify real numbers can be listed in order of increasing number of symbols used in the description and then, within each length of description, in alphabetical order. And given that list one can, it seems, define a number (between 0 and 1 and) not in that list using the diagonal procedure made famous by Cantor: The digit in its nth decimal place is the final digit of 1 plus the digit in the nth decimal place of the nth number on the list. The paradox is that such a number, say R, has a finite description (as above) and so it should be in our list. (Indeed, as with all the finitely definable reals, R will occur in the list infinitely many times, as it can be specified by arbitrarily longer descriptions too.)
......Consider such a description of R as the above, occupying the Nth place on our list of such descriptions (for some particular positive integer N). What does it say that R’s Nth digit is? That R’s Nth digit is different from what it is! So then, that descrpition could hardly be defining a number after all. So it should not be on our list in the first place. But then, if it’s not on our list we should after all be able to get a new number from that diagonal procedure, which therefore means that R should be on our list.
......This paradox is therefore a bit like the paradox of the club for all those who are not in a club. There is, of course, no problem setting up a club for all those who are not in any other club (which was surely what was intended). Similarly a barber could shave everyone else who doesn’t shave himself unremarkably, and we would (were it not for the other set-theoretical paradoxes) be able to have a well-founded set of all the other well-founded sets; or we could have a class of all and only the well-founded sets (which is little more than terminologically different) of course, or a self-membered set of all the sets, and so forth.
......So maybe the description of R above included an implicit exclusion of the Nth description from the diagonal procedure (an exclusion that would be explicit on the Nth line of our list).
......That is counter-intuitive (since the imagined diagonal cuts across the whole list) but any resolution must be, and note that the list included already—whether or not we noticed it—anything that anyone could possibly say (finitely) that would define a definite real number. So why should the diagonal procedure without such an exclusion not fail? Or, to put it another way, insofar as we think that that procedure won’t fail, because all those numbers exist before the procedure, maybe the aforementioned exclusion was implicit (in that intention), for all that it was unnoticed.
......I find it hard to make any realistic sense of (the alternative resolution of Richards’ paradox, the one that inspired Gödel’s syntactical results) a sentence that neither defines a real nor fails to define one. Maybe that does make sense (there is something philosophically obscure about the nature of mathematical truth), but if the description is so unclear that it fails objectively to define a real, how then could it also be failing to fail to define one? Some philosophers mention mathematical creativity in this context, but surely any mathematical object that could be created by us is already described in our original list.
......In my defence (of my sticking with the simple resolution, of R’s description containing, if R exists, an implicit exclusion of the Nth description) I can make realistic sense of the indefinite extensibility of the simply infinite. Consider a geometrical line of points, with two points labelled ‘0’ and ‘1’ (to define a metric) and another point between them. That third point can be surrounded by nested intervals, focussing in upon it, yielding its decimal expansion. Such an expansion may, clearly, be generated endlessly by the point in such a way that it fails to encapsulate all the information contained in that point’s precise position (relative to the points 0 and 1)—for details see here.
......Such finite strings of words as specify real numbers can be listed in order of increasing number of symbols used in the description and then, within each length of description, in alphabetical order. And given that list one can, it seems, define a number (between 0 and 1 and) not in that list using the diagonal procedure made famous by Cantor: The digit in its nth decimal place is the final digit of 1 plus the digit in the nth decimal place of the nth number on the list. The paradox is that such a number, say R, has a finite description (as above) and so it should be in our list. (Indeed, as with all the finitely definable reals, R will occur in the list infinitely many times, as it can be specified by arbitrarily longer descriptions too.)
......Consider such a description of R as the above, occupying the Nth place on our list of such descriptions (for some particular positive integer N). What does it say that R’s Nth digit is? That R’s Nth digit is different from what it is! So then, that descrpition could hardly be defining a number after all. So it should not be on our list in the first place. But then, if it’s not on our list we should after all be able to get a new number from that diagonal procedure, which therefore means that R should be on our list.
......This paradox is therefore a bit like the paradox of the club for all those who are not in a club. There is, of course, no problem setting up a club for all those who are not in any other club (which was surely what was intended). Similarly a barber could shave everyone else who doesn’t shave himself unremarkably, and we would (were it not for the other set-theoretical paradoxes) be able to have a well-founded set of all the other well-founded sets; or we could have a class of all and only the well-founded sets (which is little more than terminologically different) of course, or a self-membered set of all the sets, and so forth.
......So maybe the description of R above included an implicit exclusion of the Nth description from the diagonal procedure (an exclusion that would be explicit on the Nth line of our list).
......That is counter-intuitive (since the imagined diagonal cuts across the whole list) but any resolution must be, and note that the list included already—whether or not we noticed it—anything that anyone could possibly say (finitely) that would define a definite real number. So why should the diagonal procedure without such an exclusion not fail? Or, to put it another way, insofar as we think that that procedure won’t fail, because all those numbers exist before the procedure, maybe the aforementioned exclusion was implicit (in that intention), for all that it was unnoticed.
......I find it hard to make any realistic sense of (the alternative resolution of Richards’ paradox, the one that inspired Gödel’s syntactical results) a sentence that neither defines a real nor fails to define one. Maybe that does make sense (there is something philosophically obscure about the nature of mathematical truth), but if the description is so unclear that it fails objectively to define a real, how then could it also be failing to fail to define one? Some philosophers mention mathematical creativity in this context, but surely any mathematical object that could be created by us is already described in our original list.
......In my defence (of my sticking with the simple resolution, of R’s description containing, if R exists, an implicit exclusion of the Nth description) I can make realistic sense of the indefinite extensibility of the simply infinite. Consider a geometrical line of points, with two points labelled ‘0’ and ‘1’ (to define a metric) and another point between them. That third point can be surrounded by nested intervals, focussing in upon it, yielding its decimal expansion. Such an expansion may, clearly, be generated endlessly by the point in such a way that it fails to encapsulate all the information contained in that point’s precise position (relative to the points 0 and 1)—for details see here.
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.
......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”
......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.
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
......Order
......Proof
......Computation and Complexity
......Paradoxes and Meta-theorems
......Probability
......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.
......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):
......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).
......Symmetry, Invariance and Language
......Counting, with Numbers and Language
......Order
......Proof
......Computation and Complexity
......Paradoxes and Meta-theorems
......Probability
......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.
(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?
......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.)
......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
Banach-Tarski
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.
......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.
Tuesday, August 12, 2008
Philosophy of Mathematics
It's now 10 years since my interest in mathematics became an interest in the philosophy of mathematics, but I'm not much closer to knowing why mathematicians standardly assume an axiom of infinity (which is described in the second comment below)—not the historical or sociological reasons, but what philosophical (or amateur-scientific) reasons those applying standard mathematics have, for thinking that an axiom of infinity is true.
......That they have found few problems with that axiom in a hundred years, for example, is an answer in a different ball-park—cf. how medieval astronomers found few problems with assuming (what seems clear) that the stars circle (and the planets epicircle) a fixed Earth. Ultimately the interesting question was does the Earth turn? And while our being able to imagine a star in each cubic lightyear of some infinite space is at least a metaphysical reason (e.g. see Benardete's "Infinity: An essay in metaphysics"), mathematicians have naturally been moving away from their previous reliance upon such geometrical intuitions (towards greater rigour).
......Not that standard mathematicians need any very good metaphysical reasons for finding most interesting those models of arithmetic that contain such an axiom (e.g. there are benefits to having a common language, and there was originally the epistemic possibility to explore) but if they don't have any such reasons then physicists and metaphysicians should perhaps not presume that some such axiom is true. And note that the question arises given that numbers are objective (that proof aims at, but is not to be identified with, truth), so that it is hardly an answer to have included Constructive mathematics within academia.
......That they have found few problems with that axiom in a hundred years, for example, is an answer in a different ball-park—cf. how medieval astronomers found few problems with assuming (what seems clear) that the stars circle (and the planets epicircle) a fixed Earth. Ultimately the interesting question was does the Earth turn? And while our being able to imagine a star in each cubic lightyear of some infinite space is at least a metaphysical reason (e.g. see Benardete's "Infinity: An essay in metaphysics"), mathematicians have naturally been moving away from their previous reliance upon such geometrical intuitions (towards greater rigour).
......Not that standard mathematicians need any very good metaphysical reasons for finding most interesting those models of arithmetic that contain such an axiom (e.g. there are benefits to having a common language, and there was originally the epistemic possibility to explore) but if they don't have any such reasons then physicists and metaphysicians should perhaps not presume that some such axiom is true. And note that the question arises given that numbers are objective (that proof aims at, but is not to be identified with, truth), so that it is hardly an answer to have included Constructive mathematics within academia.
Tuesday, August 05, 2008
Liars, Divine Liars, and Semantics
Let ‘L’ name the sentence, “This sentence is not expressing a truth.” L seems to be saying that L is not being used to say anything true. If so then, if L is expressing a truth then L is not expressing a truth, whence L is not expressing a truth. But then, L seems to have been expressing a truth—that L is not expressing a truth—after all. In short, L is a paradoxical Liar sentence.
......Letting ‘N’ denote (for brevity) the property of not expressing a truth, L seems to say that L is N, but hardly straightforwardly. If we let T be “L is N” then I can say that maybe L only seems to say that L is N because of its resemblance to T. Note that if L is expressing a truth—if it is not N—then L is N (paradoxically), but if T is not N then it only follows that L is N (relatively straightforwardly).
......On this (fairly popular and) Traditional Resolution, L is a special sort of nonsense (and hence not true), its apparent sense being due to its resemblance to T (which is simply true). A related paradox is therefore Moore’s. Let G be the sentence “George believes that S is certainly P, but S is not P.” G may be true, but if George utters G there is something odd about it (even with “George believes” replaced by “I believe”), and were George omniscient he could only say G by lying.
......There are many other ways to resolve the Liar (e.g. Kripke’s, e.g. see subsection 3.2 of the recent SEP entry on Self-Reference), but this is certainly one possibility. And why should we expect there to be a unique resolution? Natural linguistic entities are usually only partially (and fuzzily) defined—maybe such incompleteness facilitates their flexibility?—and Liar sentences are relatively artificial, so it may well be that different resolutions yield different (but equally legitimate) extensions of our languages.
......Atheistic uses of Divine Liars (as follows) have therefore begged the question (so far as I can see). Let ‘DL’ name the sentence “God knows that this sentence is N.” If DL is not N—if it is expressing a truth—then God (an omniscient being, here) exists, and DL is N. So DL is N. But if DL is N then God should know that DL is N, which is what DL seems to be saying; so the atheist is tempted to conclude that God does not exist.
......A theist, on the other hand, could regard DL as some evidence that such appearances are deceptive, at least when it comes to sentences like the Liar (and to a lesser extent with Moore’s paradox), and an intelligent agnostic would be unable to make much of DL in the absence of compelling reasons to regard a different way of resolving such sentences (e.g. Kripke's) as the only correct one (not just the conventional one, as that would certainly beg the philosophical question).
......Incidentally it is usual to use non-well-founded names to define Liars (as Grim defined his Divine Liar, see second comment below), e.g. F = “F is N,” which creates additional problems for this approach (e.g. saying “F is N” is then to speak nonsense), so note that the atheist would additionally need to justify such an extension of the natural process of naming, to reap any benefit from such problems. (Prima facie it would be wise to analyse Liars and such naming seperately.)
......Letting ‘N’ denote (for brevity) the property of not expressing a truth, L seems to say that L is N, but hardly straightforwardly. If we let T be “L is N” then I can say that maybe L only seems to say that L is N because of its resemblance to T. Note that if L is expressing a truth—if it is not N—then L is N (paradoxically), but if T is not N then it only follows that L is N (relatively straightforwardly).
......On this (fairly popular and) Traditional Resolution, L is a special sort of nonsense (and hence not true), its apparent sense being due to its resemblance to T (which is simply true). A related paradox is therefore Moore’s. Let G be the sentence “George believes that S is certainly P, but S is not P.” G may be true, but if George utters G there is something odd about it (even with “George believes” replaced by “I believe”), and were George omniscient he could only say G by lying.
......There are many other ways to resolve the Liar (e.g. Kripke’s, e.g. see subsection 3.2 of the recent SEP entry on Self-Reference), but this is certainly one possibility. And why should we expect there to be a unique resolution? Natural linguistic entities are usually only partially (and fuzzily) defined—maybe such incompleteness facilitates their flexibility?—and Liar sentences are relatively artificial, so it may well be that different resolutions yield different (but equally legitimate) extensions of our languages.
......Atheistic uses of Divine Liars (as follows) have therefore begged the question (so far as I can see). Let ‘DL’ name the sentence “God knows that this sentence is N.” If DL is not N—if it is expressing a truth—then God (an omniscient being, here) exists, and DL is N. So DL is N. But if DL is N then God should know that DL is N, which is what DL seems to be saying; so the atheist is tempted to conclude that God does not exist.
......A theist, on the other hand, could regard DL as some evidence that such appearances are deceptive, at least when it comes to sentences like the Liar (and to a lesser extent with Moore’s paradox), and an intelligent agnostic would be unable to make much of DL in the absence of compelling reasons to regard a different way of resolving such sentences (e.g. Kripke's) as the only correct one (not just the conventional one, as that would certainly beg the philosophical question).
......Incidentally it is usual to use non-well-founded names to define Liars (as Grim defined his Divine Liar, see second comment below), e.g. F = “F is N,” which creates additional problems for this approach (e.g. saying “F is N” is then to speak nonsense), so note that the atheist would additionally need to justify such an extension of the natural process of naming, to reap any benefit from such problems. (Prima facie it would be wise to analyse Liars and such naming seperately.)
Friday, August 01, 2008
A Platonistic potential infinity?
One would naturally think that, for any time in the future (and similarly, for any point in space, any collection in a hierarchy of collections and so forth), either it is the nth day from now, for some natural number n, or it is infinitely far into the future, so that an infinite future might be divided into the finitely and the infinitely remote, the former region being infinitely many (i.e. aleph-null) days long since otherwise there would be no nth day for some n. But maybe it is not the case that, for any time in the future, it is either the nth day (for some n) or not.
......If it was an nth day, for some n, it would be so objectively; but maybe that property, of being an nth day for some n, is sufficiently like an indefinite property (as a consequence of the endlessness of the natural number sequence, not because of anything fuzzy about units, additions or repetitions, nor because of anything specifically temporal) for it not to follow that an arbitrary future time would either be an nth day (for some n) or not. (Such a possibility is indicated by how the natural assumption, of aleph-null, leaves us with a proper class of such cardinal numbers.)
......If it was an nth day, for some n, it would be so objectively; but maybe that property, of being an nth day for some n, is sufficiently like an indefinite property (as a consequence of the endlessness of the natural number sequence, not because of anything fuzzy about units, additions or repetitions, nor because of anything specifically temporal) for it not to follow that an arbitrary future time would either be an nth day (for some n) or not. (Such a possibility is indicated by how the natural assumption, of aleph-null, leaves us with a proper class of such cardinal numbers.)
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