Non-philosophers I meet sometimes ask: do I think mathematical facts are invented or discovered? IMO, this is a weird question - and not one that comes up much in the phil math literature- because the contrast between invention and discovery is not very well defined. For example, did Alexander Gram Bell *invent* the telephone, or did he *discover* that putting components together in a certain way would build a telephone? Intuitively, one might say both.
Maybe what people mean to be asking by this question is just this: do mathematicians bring new mathematical objects into existence, or do they discover already existing objects? For, paradigmatic cases of invention typically do involve creating a new physical object, while paradigmatic cases of discovery involves visiting an already existing physical object. So e.g. Columbus discovered America (because it already existed and he went to visit it) whereas Bell invented the telephone, by physically creating the first prototype.
However, the contrast between invention and discovery can't really just track the distinction between cases where a new object is made vs. not. This is because making a new thing isn't required for invention *or* discovery. Consider an imaginary scenario where Bell just thought up a plan for a telephone, and told someone else who physically constructed the first one years later. Bell would still have invented at telephone, if he though up the plan and then worked out from known principles that the plan would work, but never made one.
While we are talking about invention and discovery, I think there's a third notion -artistic creation (e.g. what happens when someone composes a story or a poem)- which bears an interesting relationship to mathematical discovery. When a writer writes a story, they are putting down a sequences of sentences which already exists as an abstract object.
I mean, suppose that the story teller composes a story today. If a linguist said yesterday 'no intelligible sequence of English sentences has property P', the and the sequence or sentence which the story teller writes down today has property P, then then the linguist's claim yesterday was false. The domain of potential counterexamples to linguistics claims today, already contains all sequences of English sentences which literary ingenuity could ever devise. Note also that to compose a story or poem doesn't require writing it down anywhere, (the person in the Borges story who has time stop so he can finish writing a poem before he gets shot still counts as creating the poem). For this reason the task of literary "creation" doesn't really seem to involve creating anything, (neither a physical artifact, nor an abstract string of sentences), but rather directing your attention to an abstract object that already exists - carefully sorting out which string of sentences will combine certain varied and subtle properties in the right way.
Now, if I'm right about this- the creativity of a poet or novelist doesn't need to involve creating any new object, but rather amounts to discovering a pre-existing string of sentences which has a certain property - this suggests a potential confusion about the relationship between mathematical creativity and ontology. Arguably, mathematical creativity is much like literary creativity. But, if mathematical creativity is like literary creativity, it does not follow from this that the mathematician creates the mathematical objects he describes, or that he creates anything else. For (if the above is right) literary creativity isn't a matter of bringing new objects into being, but rather a matter of discovering, amid the combinatorial explosion of possible sequences of English sentences, one that has a certain special features.
Saturday, July 24, 2010
Why Math and Morals Aren't Companions in Guilt
Intuitively, many people feel that epistemic worries about moral facts (if there are moral facts, how to explain why our moral intuitions should be even even remotely correct about them?) are WAY more serious than epistemic worries about mathematical facts (if there are mathematical facts, how to explain why our mathematical intuitions should be even even remotely correct about them?). But is there really a difference here?
Well, here's one thing that I think does make a difference: mathematical claims about number theory have direct and specific consequences for stuff that we can check by logic and/or scientific observation.
-what will happens whenever a person or a computer to successfully applies a certain syntactic alogorithm
-how many apples-or-oranges do you have when you have n apples and m oranges (cf Frege for why this is a logical fact)
This matters because, plausibly, the need to get these concrete applications right likely prevents our beliefs about number theory from getting too off the wall - whereas, our moral intuitions have no such multitude of consequences which are directly checkable by logic and observation.
Well, here's one thing that I think does make a difference: mathematical claims about number theory have direct and specific consequences for stuff that we can check by logic and/or scientific observation.
-what will happens whenever a person or a computer to successfully applies a certain syntactic alogorithm
-how many apples-or-oranges do you have when you have n apples and m oranges (cf Frege for why this is a logical fact)
This matters because, plausibly, the need to get these concrete applications right likely prevents our beliefs about number theory from getting too off the wall - whereas, our moral intuitions have no such multitude of consequences which are directly checkable by logic and observation.
Saturday, July 17, 2010
Epistemology verses Foundations in Philosophy of Math
The epistemology of math task: Get a true theory of what under what circumstances a person counts as knowing something. Or, at least, square our beliefs about what people have or lack knowledge of what particular mathematical beliefs, with general beliefs about what’s required for knowledge (e.g. causal contact.
The foundations of math task: extend our mathematical knowledge.
I claim that making this distinction matters a lot, because:
Arguments that are helpful for foundations of math are (in themselves) useless for the epistemology task. Suppose we have a working derivation D of certain facts of arithmetic from logic. And suppose we have a perfectly adequate, intuition-matching story about what it takes to count as knowing the relevant logical facts.
This still does not allow us to account for current knowledge of arithmetic (i.e. reconcile our theory of knowledge with the intuition that people now know things about arithmetic). This is because - in general - it is not enough for S to know that P, for P to be true, S to believe that P, and P to be derive*able* from things which S knows. In general, the subject S needs to have some kind of access to the derivation. The mere fact that I believe that P, and P can be proved from other things that I know, hardly suffices to establish that what I have counts as knowledge. If a lawyer is asked to show that some contractor knew that a bridge was safe, it doesn’t suffice to show that one *could* derive from laws of physics and facts about the blueprint which the contractor knew that the bridge was safe - we also need to suppose that the contractor did derive it, or get testimony from someone who derived it or the like.
Hence, a foundational argument which derives (say) one body of mathematics from premises that are more certain is not directly relevant to the general epistemological project.
Conversely, an accurate epistemology of mathematics can be almost perfectly useless to the task of setting some shaky region of mathematical theory on firmer foundations. For example, one classic account of knowledge is reliablism. If we modify reliablism so as to apply non-trivially to mathematics (following suggestions by Linnebo and Field) we get the idea that someone has knowledge if they have a true belief which is reliable in the sense that: they accept a sentence which expresses p, and if that sentence had not expressed a truth, they would not have accepted it. This is a perfectly decent candidate for a general account of mathematical knowledge. But note that, even supposing that it is right, it does nothing to help satisfy foundational desires for, say, more secure foundation for the axiom of choice. If someone has foundational worries about the axiom of choice, they have worries about whether it is true. They might express these worries by saying ‘how do you know that the axiom of choice holds?’ but the emphasis here is on truth, not on knowledge. It would be silly to respond by saying that we know AC because AC is true, and we have reliable beliefs (as defined above) to that effect. What the foundation-seeker really wants is to know whether AC. They want to acquire knowledge about whether AC, not get a general theory of what it would take to count as knowing AC.
So, I have been trying to argue that it’s important to make a distinction between the epistemological project of trying to come up with a general theory of when someone knows something about math, and the foundational project of trying to make it the case that we know more things about math, by supplementing inadequate arguments with additional arguments that appeal to premises which are already known. The one focuses on the most bland an uncontroversial cases of mathematical knowledge, and tries to reconcile our other beliefs about the nature of knowledge with our particular judgments about this case. The other seeks out the most controversial regions of mathematical claims, and seeks to secure knowledge for us about these claims, by connecting them to claims that are more securely known. Enticing answers to one project can easily seem to frustratingly miss the point for someone who is interested in the other, as shown in the examples above. Hence it’s important to make the distinction.
However, this is not to say that there’s no relationship between the epistemological and foundational projects. Thinking about big picture issues about justification in general, can influence your judgments about particular cases. A kind of trivial example of this is intuitions about what you can take for granted, while still counting as being justified. Just off the top of one’s head, it can seem attractive to say that someone doesn’t count as knowing that P if all they can give is a circular justification for P, an infinite regress of justifications, or a justification that comes to a halt at a certain point. But when you consider these three options together and notice that they exhaust all the possibilities, you will likely be inclined to give up the principle that someone who can only give a justification of one of these kinds must thereby not count as having knowledge. So, if two realists about AC are attempting to provide and evaluate firmer foundations for AC, it may be helpful for them to general questions about what’s required for knowledge and justification – to make sure that their evaluation of the evidence in this case, doesn’t depend on assumptions about justification which turn out to be incoherent or conflict with what they take to be sufficient evidence more generally.
The foundations of math task: extend our mathematical knowledge.
I claim that making this distinction matters a lot, because:
Arguments that are helpful for foundations of math are (in themselves) useless for the epistemology task. Suppose we have a working derivation D of certain facts of arithmetic from logic. And suppose we have a perfectly adequate, intuition-matching story about what it takes to count as knowing the relevant logical facts.
This still does not allow us to account for current knowledge of arithmetic (i.e. reconcile our theory of knowledge with the intuition that people now know things about arithmetic). This is because - in general - it is not enough for S to know that P, for P to be true, S to believe that P, and P to be derive*able* from things which S knows. In general, the subject S needs to have some kind of access to the derivation. The mere fact that I believe that P, and P can be proved from other things that I know, hardly suffices to establish that what I have counts as knowledge. If a lawyer is asked to show that some contractor knew that a bridge was safe, it doesn’t suffice to show that one *could* derive from laws of physics and facts about the blueprint which the contractor knew that the bridge was safe - we also need to suppose that the contractor did derive it, or get testimony from someone who derived it or the like.
Hence, a foundational argument which derives (say) one body of mathematics from premises that are more certain is not directly relevant to the general epistemological project.
Conversely, an accurate epistemology of mathematics can be almost perfectly useless to the task of setting some shaky region of mathematical theory on firmer foundations. For example, one classic account of knowledge is reliablism. If we modify reliablism so as to apply non-trivially to mathematics (following suggestions by Linnebo and Field) we get the idea that someone has knowledge if they have a true belief which is reliable in the sense that: they accept a sentence which expresses p, and if that sentence had not expressed a truth, they would not have accepted it. This is a perfectly decent candidate for a general account of mathematical knowledge. But note that, even supposing that it is right, it does nothing to help satisfy foundational desires for, say, more secure foundation for the axiom of choice. If someone has foundational worries about the axiom of choice, they have worries about whether it is true. They might express these worries by saying ‘how do you know that the axiom of choice holds?’ but the emphasis here is on truth, not on knowledge. It would be silly to respond by saying that we know AC because AC is true, and we have reliable beliefs (as defined above) to that effect. What the foundation-seeker really wants is to know whether AC. They want to acquire knowledge about whether AC, not get a general theory of what it would take to count as knowing AC.
So, I have been trying to argue that it’s important to make a distinction between the epistemological project of trying to come up with a general theory of when someone knows something about math, and the foundational project of trying to make it the case that we know more things about math, by supplementing inadequate arguments with additional arguments that appeal to premises which are already known. The one focuses on the most bland an uncontroversial cases of mathematical knowledge, and tries to reconcile our other beliefs about the nature of knowledge with our particular judgments about this case. The other seeks out the most controversial regions of mathematical claims, and seeks to secure knowledge for us about these claims, by connecting them to claims that are more securely known. Enticing answers to one project can easily seem to frustratingly miss the point for someone who is interested in the other, as shown in the examples above. Hence it’s important to make the distinction.
However, this is not to say that there’s no relationship between the epistemological and foundational projects. Thinking about big picture issues about justification in general, can influence your judgments about particular cases. A kind of trivial example of this is intuitions about what you can take for granted, while still counting as being justified. Just off the top of one’s head, it can seem attractive to say that someone doesn’t count as knowing that P if all they can give is a circular justification for P, an infinite regress of justifications, or a justification that comes to a halt at a certain point. But when you consider these three options together and notice that they exhaust all the possibilities, you will likely be inclined to give up the principle that someone who can only give a justification of one of these kinds must thereby not count as having knowledge. So, if two realists about AC are attempting to provide and evaluate firmer foundations for AC, it may be helpful for them to general questions about what’s required for knowledge and justification – to make sure that their evaluation of the evidence in this case, doesn’t depend on assumptions about justification which turn out to be incoherent or conflict with what they take to be sufficient evidence more generally.
Wednesday, July 7, 2010
Are mathematical truths "substantive"?
One thing that that has caused me great puzzlement (in the past few years), is the question of whether math tells us anything 'substantive'. I want to suggest that our intuitive notion of "substantiveness" combines two distinct notions, which come apart in this case.
- mathematical truths DONT rule out any physically or even metaphysically possible states of the world. (This is just another way of putting the truism that mathematical truths are necessary, hence compatible with every metaphysically possible world. I like putting things this way, because it doesn't suggest that necessary mathematical truths arise from something (mathematical objects?) causally blocking any person that tries to being both more than three feet long and less than two feet long)
- mathematical truths DO combine with our background beliefs to lead us to form expectations we wouldn't have formed otherwise (e,g. about the results of future counting procedures, about the programs)
Presumably you admit that these are at least nominally different properties. But you might still wonder *how* these two things could come apart. How could knowing any proposition be useful, if this proposition didn't rule out any possible states of the world? Here's what I think the answer to that is in a nutshell:
Some mathematical facts (i.e. facts which are derivable from math and logic alone) which are useful because they tell us that whenever one description of the world holds, then so does another (e.g. anything that accelerates from standstill at this rate for this amount of time travels that distance, anything that's less than two feet long isn't three feed long.)
And here's the answer in more detail.
- mathematical truths DONT rule out any physically or even metaphysically possible states of the world. (This is just another way of putting the truism that mathematical truths are necessary, hence compatible with every metaphysically possible world. I like putting things this way, because it doesn't suggest that necessary mathematical truths arise from something (mathematical objects?) causally blocking any person that tries to being both more than three feet long and less than two feet long)
- mathematical truths DO combine with our background beliefs to lead us to form expectations we wouldn't have formed otherwise (e,g. about the results of future counting procedures, about the programs)
Presumably you admit that these are at least nominally different properties. But you might still wonder *how* these two things could come apart. How could knowing any proposition be useful, if this proposition didn't rule out any possible states of the world? Here's what I think the answer to that is in a nutshell:
Some mathematical facts (i.e. facts which are derivable from math and logic alone) which are useful because they tell us that whenever one description of the world holds, then so does another (e.g. anything that accelerates from standstill at this rate for this amount of time travels that distance, anything that's less than two feet long isn't three feed long.)
And here's the answer in more detail.
Monday, June 28, 2010
FOL as the language for science
Maybe I'm missing something here...
Quine suggests that we adopt first order logic as the language for science. But, first order logic can't capture the notion of 'finitely many Fs'. It can only express the claim that there are n Fs for some particular n. Yet, we do understand the notion of finite, and use it in reasoning (e.g. if there are finitely many people at Alice's party, there is one person such that no one is taller than him) and potentially in science. Hence, we should not adopt first order logic as the language for science.
[The standard way to try to get around this, is by talking about relations to abstract objects like the numbers (There are finitely many Fs if there's a 1-1 map from the set of things that are F to the some set theoretic surrogate for the numbers). This would give you the right extension, if your scientific hypothesis could say that something had the structure of the numbers. But first order logic can only state axioms, like PA which don't completely pin down the structure of the numbers. Any first order axioms which you use to characterize the numbers will have non-standard models. This is Putnam's point in his celebrated model theoretic argument against realism. So, if you take this strategy, rather than saying that there are finitely many people at Alice's party, you can only say that the number of people is equinumerous items that satisfy a certain collection of first order axioms. And this does not rule out non-standard models.]
Quine suggests that we adopt first order logic as the language for science. But, first order logic can't capture the notion of 'finitely many Fs'. It can only express the claim that there are n Fs for some particular n. Yet, we do understand the notion of finite, and use it in reasoning (e.g. if there are finitely many people at Alice's party, there is one person such that no one is taller than him) and potentially in science. Hence, we should not adopt first order logic as the language for science.
[The standard way to try to get around this, is by talking about relations to abstract objects like the numbers (There are finitely many Fs if there's a 1-1 map from the set of things that are F to the some set theoretic surrogate for the numbers). This would give you the right extension, if your scientific hypothesis could say that something had the structure of the numbers. But first order logic can only state axioms, like PA which don't completely pin down the structure of the numbers. Any first order axioms which you use to characterize the numbers will have non-standard models. This is Putnam's point in his celebrated model theoretic argument against realism. So, if you take this strategy, rather than saying that there are finitely many people at Alice's party, you can only say that the number of people is equinumerous items that satisfy a certain collection of first order axioms. And this does not rule out non-standard models.]
Is Math Logic?
Is mathematics just a branch of logic? This is the first question many people ask about philosophy of math (sometimes with a vague idea that a) it would solve some kind of metaphysical or epistemological problems if math were logic or b) it's been proved that math isn't logic). Well, unsurprisingly, the answer depends on what you mean by 'logic'. Here are some different senses of the word 'logic' that one might have in mind.
1. first order logic
2. fully general principles of good reasoning
3. a collection of fully general principles which a person could in principle learn all of, and apply
4. principles of good reasoning that aren't ontologically committal
5. principles of good reasoning that no sane person could doubt
The sense in which it has been proved that math isn't logic is (to put things as briefly as possible) this: You can't program a computer to spit out all and only the truths of number theory.
This fact directly tells us that the mathematical truths are not all logical truths, if we understand "logic" in sense #1 - since we *can* program a computer to list off all the truths of first order logic. And it also tells us that the mathematical truths aren't all logical truths in sense #3 or #5 either - if we are willing to make the plausible assumption that human reasoning can be well modeled in this respect by some computer program. For if all human reasoning can be captured by a program, then so can all human reasoning from some starting finite collection of humanly applicable principles, and so can the portion of human reasoning that no sane person could doubt (to the extent that this is well defined).
However, if by "logic" you just mean #2 -fully general principles of reasoning that would be generally valid (whether or not one could pack all of these principles into some finite human brian)- then we have no reason to think that math isn't logic. We expect the kinds of logical and inductive reasoning we use in number theory (e.g. mathematical induction) to work for other things (especially for things like time, which we take to have the same structure as the numbers). If Jim didn't have a bike on day 1, and if, for each subsequent day he could only get a bike if he had already had a bike on the previous day, then Jim never gets a bike. If there are finitely many people at Jane's party, there is one person such that no one is taller than them. The laws of addition are the same whether you are counting gingerbread men and lemon bars, or primes and composite numbers. And this doesn't just apply to principles of mathematical reasoning which we actually accept. We also expect any *unknown* truths about the numbers (as the smallest collection containing 0 and closed under a transitive, antisymmetric relation like successor) to be mirrored by corresponding truths about any other collection of objects which contain some other starter element and are as few as possible while being closed under a transitive, antisymmetric relation (be this a collection of infinitely many rocks, or a collection of some other abstracta like the range of possible strings containing only the letter "A"). Hence, it is plausible that every sentence about numbers is an instance of a generally valid sentence form containing only worlds like "smallest", "collection", "antisymmetric" "finite" etc - and every mathematical truth is a logical truth in this regard.
Finally, if by "logic" you mean #4- ontologically *committal* good reasoning, the answer depends on a deep question in meta-ontology. For, it is well known that standard mathematics can be reduced to set theory, which in turn can be reduced to second order logic. But what are the ontological commitments of second order logic?
People have very different intuitions about whether we should say that there really are objects (call them sets with ur-elements or classes) corresponding to "EX" statements in second order logic. Does the claim that "Some of the people Jane invited to her party admire only each other, so if all and only these people accept, she will have a very smug party" assert the existence of objects called collections? More generally: the quantification over classes in second-order logic ontologically committal? Statements like the one above certainly seem to be meaningful. And, it turns out not to be possible to paraphrase away the mention of something like a set or class, in the sentence above, using only the tools of standard first order logic. This reveals a sense in which we treat reasoning about abstracta like classes (or, equivalently for these purposes, sets with ur-elements), very similarly to ordinary objects in our logical reasoning about them. But is this enough to show that second order logical is ontologically committal (and hence not logic at all, according to meaning #4)?
I propose that the key issue here concerns how closely ontology is tied to inferential role. Both advocates and deniers of abstract objects will agree that many of the same syntactic patterns of inference that are good for sentences containing "donkey" and sentences containing "set". But what exactly does this tell us about ontology? If you think about ontological questions as being questions about what the logical role of an expression in a given language, this tells you something very decisive. On the other hand, if you think about ontology can swing somewhat free of the inferential roles of sentences in languages (so an expression can have an object-like inferential role without naming an object), it's open to you in principle to say that - however similar their logical role- second order quantifiers are not ontologically committal. On this view, claims about sets with ur-elements are just ways to make very sophisticated claims (generally claims that could not otherwise be finitely expressed) "about" the behavior and relationship between ur-elements, and true claims about pure sets (i.e. sets that can be built up just from the empty set) are true in a way that does not involve any particular relationship to any objects, but can illuminate the necessary relationships between different expressions about classes that do have ur-elements. [At the moment I prefer the former view, that quantification in second order logic is ontologically commital, but this is a subtle issue]
Thus, to summarize, it is fully possible to say - even after Godel- that math is the study of "logic" in the sense of generally valid patterns of reasoning. However, if you say this, you must then admit that "logic" is not finitely axiomatizable, and there are logical truths which are not provable from the obvious via obvious steps (indeed, plausibly ones which we can never know about). Note that to make this claim one need not give up on the idea that logical arguments proceed from the obvious via obvious steps. For, if you take this route you can (and probably will want to) distinguish the human practice of giving logical arguments, from the collection of logical truths. You can say: only some of the logical truths seem obvious to us, and only some of the logically-truth-preserving inferences seem obviously compelling to us. We make logical arguments by putting these inferences together to get new results which are also logical truths. But (what Incompleteness shows) is that not all logical truths can be gotten from the ones that we know about. You can even claim that mathematical truths are logical in the further sense of not being ontologically committal, if you allow (contrary to the usual close association between objecthood and logical role) that the set quantifiers in second order logic are not ontologically committal.
1. first order logic
2. fully general principles of good reasoning
3. a collection of fully general principles which a person could in principle learn all of, and apply
4. principles of good reasoning that aren't ontologically committal
5. principles of good reasoning that no sane person could doubt
The sense in which it has been proved that math isn't logic is (to put things as briefly as possible) this: You can't program a computer to spit out all and only the truths of number theory.
This fact directly tells us that the mathematical truths are not all logical truths, if we understand "logic" in sense #1 - since we *can* program a computer to list off all the truths of first order logic. And it also tells us that the mathematical truths aren't all logical truths in sense #3 or #5 either - if we are willing to make the plausible assumption that human reasoning can be well modeled in this respect by some computer program. For if all human reasoning can be captured by a program, then so can all human reasoning from some starting finite collection of humanly applicable principles, and so can the portion of human reasoning that no sane person could doubt (to the extent that this is well defined).
However, if by "logic" you just mean #2 -fully general principles of reasoning that would be generally valid (whether or not one could pack all of these principles into some finite human brian)- then we have no reason to think that math isn't logic. We expect the kinds of logical and inductive reasoning we use in number theory (e.g. mathematical induction) to work for other things (especially for things like time, which we take to have the same structure as the numbers). If Jim didn't have a bike on day 1, and if, for each subsequent day he could only get a bike if he had already had a bike on the previous day, then Jim never gets a bike. If there are finitely many people at Jane's party, there is one person such that no one is taller than them. The laws of addition are the same whether you are counting gingerbread men and lemon bars, or primes and composite numbers. And this doesn't just apply to principles of mathematical reasoning which we actually accept. We also expect any *unknown* truths about the numbers (as the smallest collection containing 0 and closed under a transitive, antisymmetric relation like successor) to be mirrored by corresponding truths about any other collection of objects which contain some other starter element and are as few as possible while being closed under a transitive, antisymmetric relation (be this a collection of infinitely many rocks, or a collection of some other abstracta like the range of possible strings containing only the letter "A"). Hence, it is plausible that every sentence about numbers is an instance of a generally valid sentence form containing only worlds like "smallest", "collection", "antisymmetric" "finite" etc - and every mathematical truth is a logical truth in this regard.
Finally, if by "logic" you mean #4- ontologically *committal* good reasoning, the answer depends on a deep question in meta-ontology. For, it is well known that standard mathematics can be reduced to set theory, which in turn can be reduced to second order logic. But what are the ontological commitments of second order logic?
People have very different intuitions about whether we should say that there really are objects (call them sets with ur-elements or classes) corresponding to "EX" statements in second order logic. Does the claim that "Some of the people Jane invited to her party admire only each other, so if all and only these people accept, she will have a very smug party" assert the existence of objects called collections? More generally: the quantification over classes in second-order logic ontologically committal? Statements like the one above certainly seem to be meaningful. And, it turns out not to be possible to paraphrase away the mention of something like a set or class, in the sentence above, using only the tools of standard first order logic. This reveals a sense in which we treat reasoning about abstracta like classes (or, equivalently for these purposes, sets with ur-elements), very similarly to ordinary objects in our logical reasoning about them. But is this enough to show that second order logical is ontologically committal (and hence not logic at all, according to meaning #4)?
I propose that the key issue here concerns how closely ontology is tied to inferential role. Both advocates and deniers of abstract objects will agree that many of the same syntactic patterns of inference that are good for sentences containing "donkey" and sentences containing "set". But what exactly does this tell us about ontology? If you think about ontological questions as being questions about what the logical role of an expression in a given language, this tells you something very decisive. On the other hand, if you think about ontology can swing somewhat free of the inferential roles of sentences in languages (so an expression can have an object-like inferential role without naming an object), it's open to you in principle to say that - however similar their logical role- second order quantifiers are not ontologically committal. On this view, claims about sets with ur-elements are just ways to make very sophisticated claims (generally claims that could not otherwise be finitely expressed) "about" the behavior and relationship between ur-elements, and true claims about pure sets (i.e. sets that can be built up just from the empty set) are true in a way that does not involve any particular relationship to any objects, but can illuminate the necessary relationships between different expressions about classes that do have ur-elements. [At the moment I prefer the former view, that quantification in second order logic is ontologically commital, but this is a subtle issue]
Thus, to summarize, it is fully possible to say - even after Godel- that math is the study of "logic" in the sense of generally valid patterns of reasoning. However, if you say this, you must then admit that "logic" is not finitely axiomatizable, and there are logical truths which are not provable from the obvious via obvious steps (indeed, plausibly ones which we can never know about). Note that to make this claim one need not give up on the idea that logical arguments proceed from the obvious via obvious steps. For, if you take this route you can (and probably will want to) distinguish the human practice of giving logical arguments, from the collection of logical truths. You can say: only some of the logical truths seem obvious to us, and only some of the logically-truth-preserving inferences seem obviously compelling to us. We make logical arguments by putting these inferences together to get new results which are also logical truths. But (what Incompleteness shows) is that not all logical truths can be gotten from the ones that we know about. You can even claim that mathematical truths are logical in the further sense of not being ontologically committal, if you allow (contrary to the usual close association between objecthood and logical role) that the set quantifiers in second order logic are not ontologically committal.
Friday, June 18, 2010
Knowledge and Cannonical Mechanisms
In my first epistemology class in college, the prof encouraged us to look for adequate necessary and sufficient conditions for knowledge by making the following (imo appealing) argument. We expect that there's SOME nice relationship between facts about knowledge and descriptive facts not containing the word knowledge, since our brains seem to be able to go, somehow, from descriptions of a scenario (like the Gettier cases) to claims about whether the person in that scenario has knowledge. However, philosophical attempts to find a nice definition of knowledge in other terms seem to have systematically failed. This suggests that there may be a correct and informative definition of knowledge to be found, but this definition is just too long to be an elegant philosophical hypothesis, but not too long to correspond to what the brain actually does when judging these claims.
So here's what I propose that the true definition of knowledge might look like:
We describe messy physical processes by talking about symple mechanisms, and a notion of what these mechanisms tend to do "ceterus paribus". People agree surprizingly much on which mechanisms approximate what (e.g. how to go from facts about swans to claims about the swan lifestyle, how to divide up actual dispositions to behavior into "behaving normally" vs, "something special happening whereby the ceterus aren't paribus"). One thing that can be so approximated is human belief forming. We think about actual human belief formation by saying that it "ceterus paribus" it approximates combination of various belief forming mechanisms (e.g. logical deduction, looking etc). A reliable beleif forming mechanism is one whose ceterus paribus behavior yields true beliefs.
Certain belief forming mechanisms are popular, and remain popular with people even when they undergo lots of reflection. Some of these are cannonical, in the sense that we count them as potential conduits for knowledge. But, if we ever come to believe that some such mechanism is not reliable (jn the sense defined above) we will stop saying that beleifs formed via it count as knowledge. So here's what I think a correct definition of knowledge might look like.
We have, say, 300 cannonical reliable mechanisms for producing knowledge, 200 cannonical reliable mechanisms for raising doubt (100 optional and 100 obligitory), and 200 cannonical reliable mechanisms for assuaging doubt. Call these CRMs. Our definition starts by giving a finite list of all these CRMs.
You know P, if and only if your belief in P was generated by some combination of CRMs for producing knowledge, and you went through CRMs from assuaging doubt corresponding to a) all optional CRMS for doubt raising that you did engage in b) all non-optional CRMs for doubt assuaging that apply to your situation.
Even though this is just a claim about what the form of a correct definition of knowledge would look like, it already has some reasonably testable consequences:
1. That situations where it seems unclear of vague what mechanism best describes a person's behavior (should I think of the student as correctly applying this specific valid inference rule, or fallatiously applying a more general but invalid inference rule?) will also make us feel that it's unclear or vague whether the person in question has knowledge.
2. That we should seem unclear whether to attribute knowledge about when reliable but science fictiony and hence non-cannonized mechanisms are described. For example, most people would say it's OK to take delivarances of the normal 5 senses at face value, without checking them against something else. But what about creatures with a 6th sense that allowed them to reliably read minds, or form true beliefs about arbitrary pi 01 statements of arithmetic (imagine creatures living in a world with the weird physics that allows supertasks, and suppose that they have some gland that has no effect on conscious experience, but whose deliverances reliably check each case). Would they count as knowing if they form beliefs by using these?
So here's what I propose that the true definition of knowledge might look like:
We describe messy physical processes by talking about symple mechanisms, and a notion of what these mechanisms tend to do "ceterus paribus". People agree surprizingly much on which mechanisms approximate what (e.g. how to go from facts about swans to claims about the swan lifestyle, how to divide up actual dispositions to behavior into "behaving normally" vs, "something special happening whereby the ceterus aren't paribus"). One thing that can be so approximated is human belief forming. We think about actual human belief formation by saying that it "ceterus paribus" it approximates combination of various belief forming mechanisms (e.g. logical deduction, looking etc). A reliable beleif forming mechanism is one whose ceterus paribus behavior yields true beliefs.
Certain belief forming mechanisms are popular, and remain popular with people even when they undergo lots of reflection. Some of these are cannonical, in the sense that we count them as potential conduits for knowledge. But, if we ever come to believe that some such mechanism is not reliable (jn the sense defined above) we will stop saying that beleifs formed via it count as knowledge. So here's what I think a correct definition of knowledge might look like.
We have, say, 300 cannonical reliable mechanisms for producing knowledge, 200 cannonical reliable mechanisms for raising doubt (100 optional and 100 obligitory), and 200 cannonical reliable mechanisms for assuaging doubt. Call these CRMs. Our definition starts by giving a finite list of all these CRMs.
You know P, if and only if your belief in P was generated by some combination of CRMs for producing knowledge, and you went through CRMs from assuaging doubt corresponding to a) all optional CRMS for doubt raising that you did engage in b) all non-optional CRMs for doubt assuaging that apply to your situation.
Even though this is just a claim about what the form of a correct definition of knowledge would look like, it already has some reasonably testable consequences:
1. That situations where it seems unclear of vague what mechanism best describes a person's behavior (should I think of the student as correctly applying this specific valid inference rule, or fallatiously applying a more general but invalid inference rule?) will also make us feel that it's unclear or vague whether the person in question has knowledge.
2. That we should seem unclear whether to attribute knowledge about when reliable but science fictiony and hence non-cannonized mechanisms are described. For example, most people would say it's OK to take delivarances of the normal 5 senses at face value, without checking them against something else. But what about creatures with a 6th sense that allowed them to reliably read minds, or form true beliefs about arbitrary pi 01 statements of arithmetic (imagine creatures living in a world with the weird physics that allows supertasks, and suppose that they have some gland that has no effect on conscious experience, but whose deliverances reliably check each case). Would they count as knowing if they form beliefs by using these?
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