Peter Smith's Blog, page 134

Slides for more introductory logic lectures

I've somewhat belatedly put online slides for the last six of last term's intro logic lectures. Lectures 11–13 introduce propositional trees, and lectures 14–16 introduce the language QL. To be honest, you would do much better to read my book: but since slides for previous lectures have been downloaded a surprising number of times, here's the remainder.

TTP, 2. Introduction: Options and Weir's way forward

Faced with the Benacerrafian challenge, what are the options? Weir mentions a few; but he doesn't give anything like a systematic map of the various possible ways forward. It might be helpful if I do something to fill the gap.

One way of beginning to organize (some) positions in the philosophy of mathematics is to consider how they answer the following sequence of questions. Start with these two:

Are '3 is prime' and 'the Klein four-group is the smallest non-cyclic group', for example, (unqualifiedly) true?
Are '3 is prime' and 'the Klein four-group is the smallest non-cyclic group', for example, to be construed — as far as their 'logical grammar' is concerned — as the surface form suggests (on the same plan as e.g. 'Alan is clever' and 'the tallest student is the smartest philosopher')?

The platonist answers 'yes' to both. A naive formalist and one stripe of fictionalist, will get off the bus at the first stop and answer 'no' to (1) —  the former because there is no genuine content to be true, the latter because the content is (supposedly) a platonist fantasy. Another, more conciliatory stripe of fictionalist can answer 'yes' to (1) but 'no' to (2), since she doesn't take '3 is prime' at face value but re-construes it as short for 'in the arithmetic fiction, 3 is prime' or some such.

Eliminative and modal structuralists will also answer 'yes' to (1) and 'no' to (2), this time construing the mathematical claims as quantified conditional claims about non-mathematical things (schematically: anything, or anything in any possible world, that satisfies certain structural conditions will satisfy some other conditions). It is actually none too clear how structuralism helps us epistemologically, and when given a modal twist it's not clear either how it helps us ontologically. But that's quite another story.

Suppose, however, we answer 'yes' to (1) and (2). Then we are committed to saying there are prime numbers and there are non-cyclic groups, etc. (for it is true that 3 is prime, and — construed as surface form suggests — that implies there are prime numbers). Next question:

Is there a distinction to be drawn between saying there are prime numbers (as an unqualified truth of mathematics, construed at face value) and saying THERE ARE prime numbers? – where 'THERE ARE' indicates a metaphysically committing existence-claim, one which aims to represent how things stand with 'denizens of the mind-independent, discourse-independent world' (following Weir in borrowing Terence Horgan's words and Putnam's typographical device)

According to one central tradition, there is no such distinction to be drawn: thus Quine on the univocality of 'exists'.

The Wright/Hale brand of neo-Fregean logicism likewise rejects the alleged distinction. Their opponents are puzzled by the Wright/Hale argument for platonism on the cheap. For the idea is that, once we answer (1) and (2) positively (and just a little more), i.e. once we agree that '3 is prime' is true and that '3' walks, swims and quacks like a singular term, then we are committed to '3' being a successfully referring expression, and so committed to its referent, which (on modest and plausible further assumptions) has to be an abstract object; so there indeed exists a first odd prime which is an abstract object. Opponents think this is too quick as an argument for full-blooded platonism because they think there is a gap to negotiate between the likes of 'there exists a first odd prime number' and 'THERE EXISTS a first odd prime number'. Drawing on early Dummettian themes (which have Fregean and Wittgensteinian roots), the neo-logicist platonist denies there is a gap to be bridged.

Much recent metaphysics, however, sides with Wright and Hale's opponents (wrong-headedly maybe, but that's where the troops are marching). Thus Ted Sider can write 'There is a growing consensus that taking ontology seriously requires making some sort of distinction between ordinary and ontological understandings of existential claims' (that's from his paper 'Against Parthood'). From this perspective, the claim would be that we must indeed distinguish granting the unqualified truth of mathematics, construed at face value, from being committed to a full-blooded PLATONISM which makes genuinely ontological claims. It is one thing to claim that prime numbers exists, speaking with the mathematicians, and another thing to claim that THEY EXIST 'in the fundamental sense' (as Sider likes to say) when speaking with the ontologists.

Now, we can think of Sider et al. as mounting an attack from the right wing on the Quine/neo-Fregean rejection of a special kind of philosophical discourse about what exists: the troops are mustered under the banner 'bring back old-style metaphysics!' (Sider: 'I think that fundamental ontology is what ontologists have been after all along'). But there is a line of attack from the left wing too. Consider, for example, Simon Blackburn's quasi-realism about morals, modalities, laws and chances. Blackburn is no friend of heavy-duty metaphysics. But the thought is that certain kinds of discourse aren't representational but serve quite different purposes, e.g. to project our moral attitudes or subjective degrees on belief onto the world (and a story is then told about why a discourse apt for doing that should to a large extent retain the same logical shape of representational discourse). So, speaking inside our moral discourse, there indeed are virtues (courage is one): but as far as the make-up of the world on which we are projecting our attitudes goes, virtues do not EXIST out there. From the left, then, it might be suggested that perhaps mathematics is like morals, at least in this respect: talking inside mathematical discourse, we can truly say e.g. that there are infinitely many primes; but mathematical discourse is not representational, and as far as the make-up of the world goes – and here we are switching to representational discourse – THERE ARE NO prime numbers.

To put it crudely, then, we can discern two routes to distinguishing 'there are prime numbers' as a mathematical claim and 'THERE ARE prime numbers' as a claim about what there really is. From the right, we drive a wedge by treating 'THERE ARE' as special, to be glossed 'there are in the fundamental, ontological, sense' (whatever that exactly is). From the left, we drive a wedge by treating mathematical discourse as special, as not in the ordinary business of making claims purporting to represent what there is.

And now we've joined up with Weir's discussion. He answers 'yes' to all three of our questions. A fourth then remains outstanding:

Given there is a distinction between saying that there are prime numbers and saying THERE ARE prime numbers, is the latter stronger claim also true?

If you say 'yes' to that, then you are buying into a version of platonism that does indeed look epistemically particularly troubling (in a worse shape, at any rate, than for the gap-denying neo-logicist position; for what can get us over the claimed gap between the ordinary mathematical claim and the ontologically committing claim)? Weir thinks this position is hopeless. Hence he answers 'no' to (4). Hence he endorses claims like this: There are infinitely many primes but THERE ARE no prime numbers. (p. 8 )

But this isn't because he is, as it were, coming from the right, deploying a special 'ontological understanding of existence claims'. Rather, he is coming more from the Blackburnian left: his 'THERE ARE' is ordinary existence talk in ordinary representational discourse, and the claim is that 'there are infinitely many primes', as a mathematical claim, belongs to a different kind of discourse.

OK, what kind of discourse is that? "The mode of assertion of such judgements, I will say, is formal, not representational". And what does 'formal' mean here? Well, part of the story is hinted at by the claim that the formal, inside-mathematics, assertion that there are infinitely many primes is made true by "the existence of proofs of strings which express the infinitude of the primes" (p. 7). Of course, that raises at least as many questions as it answers. There are hints in the rest of the Introduction about how this initially somewhat startling claim is to be rounded out and defended in the rest of the book. But they are much too quick to be usefully commented on here; so I think it will be better to say no more here but take them up as the full story unfolds.

Still, we now have an initial specification of Weir's location in the space of possible positions. His line is going to be that, as a mathematical claim, it is true that are an infinite number of primes: and this truth isn't to be secured by reconstruing the claim in some fictionalist, structuralist or other way. But a mathematical claim is one thing, and a representational claim about how things are in the world is another thing. And the gap is to be opened up, not by inflating talk of what EXISTS into a special kind of ontological talk, but by seeing mathematical discourse (like moral discourse) as playing a non-representational role (or dare I say: as making moves in a different language game?). That much indeed sounds not unattractive. The question is going to be whether the nature of this non-representational game can be illuminatingly glossed in formalist terms.

Truth Through Proof, 2. Introduction: Options and Weir's way forward

Faced with the Benacerrafian challenge, what are the options? Weir mentions a few; but he doesn't give anything like a systematic map of the various possible ways forward. It might be helpful if I do something to fill the gap.

One way of beginning to organize (some) positions in the philosophy of mathematics is to consider how they answer the following sequence of questions. Start with these two:

Are '3 is prime' and 'the Klein four-group is the smallest non-cyclic group', for example, (unqualifiedly) true?
Are '3 is prime' and 'the Klein four-group is the smallest non-cyclic group', for example, to be construed — as far as their 'logical grammar' is concerned — as the surface form suggests (on the same plan as e.g. 'Alan is clever' and 'the tallest student is the smartest philosopher')?

The platonist answers 'yes' to both. A naive formalist and one stripe of fictionalist, will get off the bus at the first stop and answer 'no' to (1) —  the former because there is no genuine content to be true, the latter because the content is (supposedly) a platonist fantasy. Another, more conciliatory stripe of fictionalist can answer 'yes' to (1) but 'no' to (2), since she doesn't take '3 is prime' at face value but re-construes it as short for 'in the arithmetic fiction, 3 is prime' or some such.

Eliminative and modal structuralists will also answer 'yes' to (1) and 'no' to (2), this time construing the mathematical claims as quantified conditional claims about non-mathematical things (schematically: anything, or anything in any possible world, that satisfies certain structural conditions will satisfy some other conditions). It is actually none too clear how structuralism helps us epistemologically, and when given a modal twist it's not clear either how it helps us ontologically. But that's quite another story.

Suppose, however, we answer 'yes' to (1) and (2). Then we are committed to saying there are prime numbers and there are non-cyclic groups, etc. (for it is true that 3 is prime, and — construed as surface form suggests — that implies there are prime numbers). Next question:

Is there a distinction to be drawn between saying there are prime numbers (as an unqualified truth of mathematics, construed at face value) and saying THERE ARE prime numbers? – where 'THERE ARE' indicates a metaphysically committing existence-claim, one which aims to represent how things stand with 'denizens of the mind-independent, discourse-independent world' (following Weir in borrowing Terence Horgan's words and Putnam's typographical device)

According to one central tradition, there is no such distinction to be drawn: thus Quine on the univocality of 'exists'.

The Wright/Hale brand of neo-Fregean logicism likewise rejects the alleged distinction. Their opponents are puzzled by the Wright/Hale argument for platonism on the cheap. For the idea is that, once we answer (1) and (2) positively (and just a little more), i.e. once we agree that '3 is prime' is true and that '3' walks, swims and quacks like a singular term, then we are committed to '3' being a successfully referring expression, and so committed to its referent, which (on modest and plausible further assumptions) has to be an abstract object; so there indeed exists a first odd prime which is an abstract object. Opponents think this is too quick as an argument for full-blooded platonism because they think there is a gap to negotiate between the likes of 'there exists a first odd prime number' and 'THERE EXISTS a first odd prime number'. Drawing on early Dummettian themes (which have Fregean and Wittgensteinian roots), the neo-logicist platonist denies there is a gap to be bridged.

Much recent metaphysics, however, sides with Wright and Hale's opponents (wrong-headedly maybe, but that's where the troops are marching). Thus Ted Sider can write 'There is a growing consensus that taking ontology seriously requires making some sort of distinction between ordinary and ontological understandings of existential claims' (that's from his paper 'Against Parthood'). From this perspective, the claim would be that we must indeed distinguish granting the unqualified truth of mathematics, construed at face value, from being committed to a full-blooded PLATONISM which makes genuinely ontological claims. It is one thing to claim that prime numbers exists, speaking with the mathematicians, and another thing to claim that THEY EXIST 'in the fundamental sense' (as Sider likes to say) when speaking with the ontologists.

Now, we can think of Sider et al. as mounting an attack from the right wing on the Quine/neo-Fregean rejection of a special kind of philosophical discourse about what exists: the troops are mustered under the banner 'bring back old-style metaphysics!' (Sider: 'I think that fundamental ontology is what ontologists have been after all along'). But there is a rejection from the left wing too. Consider, for example, Simon Blackburn's quasi-realism about morals, modalities, laws and chances. Blackburn is no friend of heavy-duty metaphysics. But the thought is that certain kinds of discourse aren't representational but serve quite different purposes, e.g. to project our moral attitudes or subjective degrees on belief onto the world (and a story is then told about why a discourse apt for doing that should to a large extent retain the same logical shape of representational discourse). So, speaking inside our moral discourse, there indeed are virtues (courage is one): but as far as the make-up of the world on which we are projecting our attitudes goes, virtues do not EXIST out there. From the left, then, it might be suggested that perhaps mathematics is like morals, at least in this respect: talking inside mathematical discourse, we can truly say e.g. that there are infinitely many primes; but mathematical discourse is not representational, and as far as the make-up of the world goes – and here we are switching to representational discourse – THERE ARE NO prime numbers.

To put it crudely, then, we can discern two routes to distinguishing 'there are prime numbers' as a mathematical claim and 'THERE ARE prime numbers' as a claim about what there really is. From the right, we drive a wedge by treating 'THERE ARE' as special, to be glossed 'there are in the fundamental, ontological, sense' (whatever that exactly is). From the left, we drive a wedge by treating mathematical discourse as special, as not in the ordinary business of making claims purporting to represent what there is.

And now we've joined up with Weir's discussion. He answers 'yes' to all three of our questions. A fourth then remains outstanding:

Given there is a distinction between saying that there are prime numbers and saying THERE ARE prime numbers, is the latter stronger claim also true?

If you say 'yes' to that, then you are buying into a version of platonism that does indeed look epistemically particularly troubling (worse, at any rate, than for the gap-free neo-logicist position; for what can get us over the claimed gap between the ordinary mathematical claim and the ontologically committing claim)? Weir thinks this position is hopeless. Hence he answers 'no' to (4). Hence he endorses claims like this: There are infinitely many primes but THERE ARE no prime numbers. (p. 8 )

But this isn't because he is, as it were, coming from the right, deploying a special 'ontological understanding of existence claims'. Rather, he is coming more from the Blackburnian left: his 'THERE ARE' is ordinary existence talk in ordinary representational discourse, and the claim is that 'there are infinitely many primes', as a mathematical claim, belongs to a different kind of discourse.

OK, what kind of discourse is that? "The mode of assertion of such judgements, I will say, is formal, not representational". And what does 'formal' mean here? Well, part of the story is hinted at by the claim that the formal, inside-mathematics, assertion that there are infinitely many primes is made true by "the existence of proofs of strings which express the infinitude of the primes" (p. 7). Of course, that raises at least as many questions as it answers. There are hints in the rest of the Introduction about how this initially somewhat startling claim is to be rounded out and defended in the rest of the book. But they are much too quick to be usefully commented on here; so I think it will be better to say no more here but take them up as the full story unfolds.

Still, we now have an initial location of Weir in the space of possible positions. His line is going to be that, as a mathematical claim, it is true that are an infinite number of primes: and this truth isn't to be secured by reconstruing claim in some fictionalist, structuralist or other way. But a mathematical claim is one thing, and a representational claim about how things are in the world is another thing. And the gap is to be opened up not by inflating talk of what EXISTS into a special kind of ontological talk, but by seeing mathematical discourse (like moral discourse) as playing a non-representational role (dare I say, as making moves in a different language game?). And that much indeed sounds not unattractive. The question is going to be whether the nature of this non- representational game can be illuminatingly glossed in formalist terms.

TTP, 1. Introduction: Platonism vs 'naturalized epistemology'

Let me begin by setting the scene, embroidering only a little on Weir's opening pages.

Consider then the following claims, ordinarily regarded as mathematical truths:

3 is prime.
The Klein four-group is the smallest non-cyclic group.
There is an uncountably infinite set of nested subsets of Q, the set of rationals.

On the surface, (1) looks structurally very similar to 'Alan is clever'. The latter is surely about some entity, namely Alan, and is true because that entity has the property attributed. Likewise, we might initially be pretty tempted to say, (1) also is about something, namely the number three, and is true because  that thing has the property attributed. Similarly (2) is about something else, the Klein four-group. And (3) is about the set of rationals, and claims there is a further thing, an uncountably infinite family of sets of rationals nested inside each other.

So what  kind of things are three, the Klein group, the set of rationals? Not things we can see or kick, but non-concrete things, surely — i.e. abstract objects. And given the standard view that (1), (2) and (3) don't just happen to be true but are necessarily true, it would seem that these abstract objects must be necessary existents.

But how can we possibly know about such things? Once upon a time (Weir might have reminded us), the thought seemed attractive that we are made in God's image, and — albeit to a limited extent — partake in his rational nature (for Spinoza, indeed, 'the human mind is part of the infinite intellect of God'). And God, the story went, can just rationally see all the truths of mathematics: sharing something of his nature, in a small way we can come to do that too. Thus Salviato, speaking for Galileo in his Dialogue, says that in grasping some parts of arithmetic and geometry, the human intellect 'equals the divine in objective certainty, for here it succeeds in understanding necessity'. And Leibniz writes `minds are … the closest likenesses of the first Being, for they distinctly perceive necessary truths'. (For more on this these, see Ch. 1 of Edward Craig's wonderful The Mind of God and the Works of Man (OUP, 1987), from which the quotations are taken.)

However, this conception of ourselves as approximating to the God-like has lost its grip on us. So, getting back to Weir, given the sort of beings we now think we are, with the sorts of limited and ramshackle cognitive powers with which evolution has provided us to enable us to survive in our small corner of the universe, the question becomes pressing: how come that we can possibly get to know anything about supposedly necessarily existent abstract entities (entities in Plato's heaven, as they say)? How are we supposed to cognitively 'lock on' to such things? Indeed, we might wonder, how do we ever manage even to frame concepts of things apparently so remote from quotidian experience?

Now, note that to find this sort of question pressing it isn't that we already have to bought into the idea that epistemology should be 'naturalized' in some strong sense, or that a Quinean 'naturalized epistemology' exhausts the legitimate parts of what used to be epistemology. And though Weir does talk about mathematical platonism being "put to the test by naturalized epistemology", he officially means no more than that our conception of ourselves as natural agents without God-like powers "imposes a non-trivial test of internal stability" (p. 3) when combined with views like platonism. The problem-setting issue, then, is an entirely familiar one: as Benacerraf frames it in his classic paper, 'a satisfactory account of mathematical truth … must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have'.

So far, then, so good. We have a familiar but still pressing question, and in the next post, I'll say something about various lines we might take in response and indicate how, in rest of his Introduction, Weir situates on the map the position he wants to defend.

But first just a word or two more about Weir's enthusiasm for naturalism. He goes on to write "I accept, as a general methodological maxim, the prescription that one should push a naturalistic approach as far as it will go." (p. 5) And what does that involve? "The methodological naturalist … prescribes that one ought to follow scientific method, at a level of sophistication appropriate to the problem at hand, whenever attempting to find out the truth about anything." Really? If using the methods of science is construed more narrowly as involving the careful rational weighing of evidence to test specific, antecedently formulated, empirical conjectures, etc., then of course this isn't the only way to discover truths. Evolution has thankfully provided us with other quick-and-dirty ways of fast-tracking to the truth reliably enough to avoid fleet-footed predators often enough! While if the methods of science are understood in a more relaxed and embracing way, as whatever goes into the mix as we develop our best overall theory of nature, then (a familiar old-Quinean point) these methods would seem to subsume the methods of (much) mathematics which seem so entangled, and 'methodological naturalism' in itself has no special bite again platonism (over and above Benacerraf's problem, which doesn't depend on the naturalism).

But we really don't want to start off on that debate again, about the appropriate formulation of a possibly-defensible 'naturalism'. So let's re-emphasize the key point that I think Weir would make, which his rhetoric hereabouts could possibly obscure: we don't need to endorse any strong form of naturalism, or have any commitment to naturalized epistemology as the one legitimate residuary legatee of the epistemological tradition, to find troubling the combination of platonism with our conception of ourselves as limited creatures epistemically geared to the sublunary world. That's enough of a problem to get Weir's project going.

Truth Through Proof, 1. Introduction: Platonism vs 'naturalized epistemology'

Let me begin by setting the scene, embroidering only a little on Weir's opening pages.

Consider then the following claims, ordinarily regarded as mathematical truths:

3 is prime.
The Klein four-group is the smallest non-cyclic group.
There is an uncountably infinite set of nested subsets of Q, the set of rationals.

On the surface, (1) looks structurally very similar to 'Alan is clever'. The latter is surely about some entity, namely Alan, and is true because that entity has the property attributed. Likewise, we might initially be pretty tempted to say, (1) also is about something, namely the number three, and is true because  that thing has the property attributed. Similarly (2) is about something else, the Klein four-group. And (3) is about the set of rationals, and claims there is a further thing, an uncountably infinite family of sets of rationals nested inside each other.

So what  kind of things are three, the Klein group, the set of rationals? Not things we can see or kick, but non-concrete things, surely — i.e. abstract objects. And given the standard view that (1), (2) and (3) don't just happen to be true but are necessarily true, it would seem that these abstract objects must be necessary existents.

But how can we possibly know about such things? Once upon a time (Weir might have reminded us), the thought seemed attractive that we are made in God's image, and — albeit to a limited extent — partake in his rational nature (for Spinoza, indeed, 'the human mind is part of the infinite intellect of God'). And God, the story went, can just rationally see all the truths of mathematics: sharing something of his nature, in a small way we can come to do that too. Thus Salviato, speaking for Galileo in his Dialogue, says that in grasping some parts of arithmetic and geometry, the human intellect 'equals the divine in objective certainty, for here it succeeds in understanding necessity'. And Leibniz writes `minds are … the closest likenesses of the first Being, for they distinctly perceive necessary truths'. (For more on this these, see Ch. 1 of Edward Craig's wonderful The Mind of God and the Works of Man (OUP, 1987), from which the quotations are taken.)

However, this conception of ourselves as approximating to the God-like has lost its grip on us. So, getting back to Weir, given the sort of beings we now think we are, with the sorts of limited and ramshackle cognitive powers with which evolution has provided us to enable us to survive in our small corner of the universe, the question becomes pressing: how come that we can possibly get to know anything about supposedly necessarily existent abstract entities (entities in Plato's heaven, as they say)? How are we supposed to cognitively 'lock on' to such things? Indeed, we might wonder, how do we ever manage even to frame concepts of things apparently so remote from quotidian experience?

Now, note that to find this sort of question pressing, it isn't that we already have to bought into the idea that epistemology should be 'naturalized' in some strong sense, or that a Quinean 'naturalized epistemology' exhausts the legitimate parts of what used to be epistemology. And though Weir does talk about mathematical platonism being "put to the test by naturalized epistemology", he officially means no more than that our conception of ourselves as natural agents without God-like powers "imposes a non-trivial test of internal stability" (p. 3) when combined with views like platonism. The problem-setting issue, then, is an entirely familiar one: as Benacerraf frames it in his classic paper, 'a satisfactory account of mathematical truth … must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have'.

So far, then, so good. We have a familiar but still pressing question, and in the next post, I'll say something about various lines we might take in response and indicate how, in rest of his Introduction, Weir situates on the map the position he wants to defend.

But first just a word or two more about Weir's enthusiasm for naturalism. He goes on to write "I accept, as a general methodological maxim, the prescription that one should push a naturalistic approach as far as it will go." (p. 5) And what does that involve? "The methodological naturalist … prescribes that one ought to follow scientific method, at a level of sophistication appropriate to the problem at hand, whenever attempting to find out the truth about anything." Really? If using the methods of science is construed more narrowly as involving the careful rational weighing of evidence to test specific, antecedently formulated, empirical conjectures, etc., then of course this isn't the only way to discover truths. Evolution has thankfully provided us with other quick-and-dirty ways of fast-tracking to the truth reliably enough to avoid fleet-footed predators often enough! While if the methods of science are understood in a more relaxed and embracing way, as whatever goes into the mix as we develop our best overall theory of nature, then (a familiar old-Quinean point) these methods would seem to subsume the methods of (much) mathematics which seem so entangled, and 'methodological naturalism' in itself has no special bite again platonism (over and above Benacerraf's problem, which doesn't depend on the naturalism).

But we really don't want to start off on that debate again, about the appropriate formulation of a possibly-defensible 'naturalism'. So let's re-emphasize the key point that I think Weir would make, which his rhetoric hereabouts could possibly obscure: we don't need to endorse any strong form of naturalism, or have any commitment to naturalized epistemology as the one legitimate residuary legatee of the epistemological tradition, to find troubling the combination of platonism with our conception of ourselves as limited creatures epistemically geared to the sublunary world. That's enough of a problem to get Weir's project going.

Truth Through Proof, 1. Introduction: platonism vs 'naturalized epistemology'

Let me begin by setting the scene, embroidering only a little on Weir's opening pages.

Consider then the following claims, ordinarily regarded as mathematical truths:

3 is prime.
The Klein four-group is the smallest non-cyclic group.
There is an uncountably infinite set of nested subsets of Q, the set of rationals.

On the surface, (1) looks structurally very similar to 'Alan is clever'. The latter is surely about some entity, namely Alan, and is true because that entity has the property attributed. Likewise, we might initially be pretty tempted to say, (1) also is about something, namely the number three, and is true because  that thing has the property attributed. Similarly (2) is about something else, the Klein four-group. And (3) is about the set of rationals, and claims there is a further thing, an uncountably infinite family of sets of rationals nested inside each other.

So what  kind of things are three, the Klein group, the set of rationals? Not things we can see or kick, but non-concrete things, surely — i.e. abstract objects. And given the standard view that (1), (2) and (3) don't just happen to be true but are necessarily true, it would seem that these abstract objects must be necessary existents.

But how can we possibly know about such things? Once upon a time (Weir might have reminded us), the thought seemed attractive that we are made in God's image, and — albeit to a limited extent — partake in his rational nature (for Spinoza, indeed, 'the human mind is part of the infinite intellect of God'). And God, the story went, can just rationally see all the truths of mathematics: sharing something of his nature, in a small way we can come to do that too. Thus Salviato, speaking for Galileo in his Dialogue, says that in grasping some parts of arithmetic and geometry, the human intellect 'equals the divine in objective certainty, for here it succeeds in understanding necessity'. And Leibniz writes `minds are … the closest likenesses of the first Being, for they distinctly perceive necessary truths'. (For more on this these, see Ch. 1 of Edward Craig's wonderful The Mind of God and the Works of Man (OUP, 1987), from which the quotations are taken.)

However, this conception of ourselves as approximating to the God-like has lost its grip on us. So, getting back to Weir, given the sort of beings we now think we are, with the sorts of limited and ramshackle cognitive powers with which evolution has provided us to enable us to survive in our small corner of the universe, the question becomes pressing: how come that we can possibly get to know anything about supposedly necessarily existent abstract entities (entities in Plato's heaven, as they say)? How are we supposed to cognitively 'lock on' to such things? Indeed, we might wonder, how do we ever manage even to frame concepts of things apparently so remote from quotidian experience?

Now, note that to find this sort of question pressing, it isn't that we already have to bought into the idea that epistemology should be 'naturalized' in some strong sense, or that a Quinean 'naturalized epistemology' exhausts the legitimate parts of what used to be epistemology. And though Weir does talk about mathematical platonism being "put to the test by naturalized epistemology", he officially means no more than that our conception of ourselves as natural agents without God-like powers "imposes a non-trivial test of internal stability" (p. 3) when combined with views like platonism. The problem-setting issue, then, is an entirely familiar one: as Benacerraf frames it in his classic paper, 'a satisfactory account of mathematical truth … must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have'.

So far, then, so good. We have a familiar but still pressing question, and in the next post, I'll say something about various lines we might take in response and indicate how, in rest of his Introduction, Weir situates on the map the position he wants to defend.

But first just a word or two more about Weir's enthusiasm for naturalism. He goes on to write "I accept, as a general methodological maxim, the prescription that one should push a naturalistic approach as far as it will go." (p. 5) And what does that involve? "The methodological naturalist … prescribes that one ought to follow scientific method, at a level of sophistication appropriate to the problem at hand, whenever attempting to find out the truth about anything." Really? If using the methods of science is construed more narrowly as involving the careful rational weighing of evidence to test specific, antecedently formulated, empirical conjectures, etc., then of course this isn't the only way to discover truths. Evolution has thankfully provided us with other quick-and-dirty ways of fast-tracking to the truth reliably enough to avoid fleet-footed predators often enough! While if the methods of science are understood in a more relaxed and embracing way, as whatever goes into the mix as we develop our best overall theory of nature, then (a familiar old-Quinean point) these methods would seem to subsume the methods of (much) mathematics which seem so entangled, and 'methodological naturalism' in itself has no special bite again platonism (over and above what is revealed by Benacerraf dilemma which doesn't depend on it).

But we really don't want to start off on that debate again, about the appropriate formulation of a possibly-defensible 'naturalism'. So let's re-emphasize the key point that I think Weir would make, which his rhetoric hereabouts could possibly obscure: we don't need to endorse any strong form of naturalism, or have any commitment to naturalized epistemology as the one legitimate residuary legatee of the epistemological tradition, to find troubling the combination of platonism with our conception of ourselves as limited creatures epistemically geared to the sublunary world. That's enough of a problem to get Weir's project going.

Truth Through Proof, 0. Preamble

I am eventually going to be writing a (short) review for  Mind of Alan Weir's new book  Truth Through Proof: A Formalist Foundation for Mathematics (OUP, 2010). The blurb on the publisher's website gives  you an idea what of what the book is about. The clue is in the subtitle — but note, this is a philosophy book, not a technical book in foundational studies.

To help fix my ideas,  I'll be posting a (much longer) series of discussion notes here, as I sporadically work through the book. Given other commitments, however, I'll have to take things pretty slowly over the coming weeks.

As with similar series of postings on other books, I expect that these notes will weave around and about Weir's book (henceforth  TTP) in a more free-ranging way than would be appropriate in a review: but I'll try to make it clear when I'm summarizing TTP, when I'm directly commenting on Weir's views, and when I am striking out more on my own account. And more generally, I'll try to keep things accessible to students, even if that sometimes means including more background explanations than other readers here will need.  All comments as we go along will be very gratefully received!

After the Introduction, Weir's chapters are divided into sections numbered off with roman numerals: so here '3.III' means Chapter 3, §III. Double quotation marks are reserved to signal quotations from TTP (and otherwise unattributed page numbers of course refer to the book too).

OK. With that by way of brisk preamble, let's dive in … tomorrow!

Truth Through Proof, 0: Preamble

I am eventually going to be writing a (short) review for  Mind of Alan Weir's new book  Truth Through Proof: A Formalist Foundation for Mathematics (OUP, 2010). To help fix my ideas,  I'll be writing a (much longer) series of discussion notes here on my blog, as I sporadically work through the book. Given other commitments, however, I'll have to take things pretty slowly over the coming weeks.

As with similar series of postings on other books here, I expect that these notes will weave around and about Weir's book (henceforth  TTP) in a more free-ranging way than would be appropriate in a review: but I'll try to make it clear when I'm summarizing TTP, when I'm directly commenting on Weir's views, and when I am striking out more on my own account. All comments from other readers of the book as we go along will be very gratefully received!

After the Introduction, Weir's chapters are divided into sections numbered off with roman numerals: so here '3.III' means Chapter 3, §III. Double quotation marks are reserved to signal quotations from TTP (and otherwise unattributed page numbers of course refer to the book too).

So with no more preamble, let's dive in … tomorrow!

The Bertrand Russell Chair

That well-known website LeiterLeaks tells the world

Huw Price, Challis Professor of Philosophy at the University of Sydney and a two-time winner of the lucrative Australian Federation Fellowships, has been offered the Bertrand Russell Professorship in Philosophy at Cambridge University (presently held by Simon Blackburn, who will be retiring this year).  Price has written widely in philosophy of science and physics, metaphysics, and philosophy of language.

Hmmm. I thought this was all hush-hush, with things depending on the University's further discussions (in the Cambridge system, the appointing committee makes an election, but then it is up to quite other bodies in the University to come to terms with the hoped-for appointee). So given it's confidential still, I couldn't possibly comment.

Oh well, all right …

Just a bit …

Let's just say it seems to me that if Huw Price does indeed take up the offer, that will be a very good outcome for the Faculty.

And now, back to logic and Alan Weir's book

Ok, back to thinking! Alan Weir's long awaited book Truth Through Proof: A Formalist Foundation for Mathematics has been out a few weeks, and sitting on my desk unread. But now I'm going to be reviewing it for Mind. So I really ought to start reading. Now. And after Christmas, I'll be blogging about it chapter by chapter here. As I said before, it looks as if it should certainly be a fun and provoking read. So watch this space.