Coming up with a systematic way to paraphrase sentences involving some wacky new term W in biology, sociology, psychology, or art criticism, into sentences that are just a logical product of claims about sets, mereological sums of other more commonplace objects, and preserves all our intuitive reasoning about W, is useful in three ways.
a) New Applications for Old Knowledge: Getting a method of paraphrase lets us bring our logical/set theoretic/substantive knowledge about the terms used in the analysis to bear on the new term in question. If the facts about the Ws parallel the facts about sets of such and such kind, set theory may have interesting implications for facts about the Ws.
b) Avoiding Adding Terms Which are "Incoherent" or "Have False Presuppositions": Getting a method of paraphrase may let us prove the consistency and conservativity of reasoning about the wacky new entities. For example if you analyze 'x is bachelor' as 'x is unmarried & x is a man', and then only accept informal reasoning about bachelors that can be reconstructed using this analysis, then it is clear that adding the term 'bachelor' and doing this informal reasoning will not allow you to derive contradiction, or any other new consequences. So, adding informal reasoning about bachelors will do no harm. Here the proof theory (any proof of P which uses the term "bachelor" could be turned into one that doesn't) is so obvious that it's easy not to notice. But the mathematical issues involved in showing consistency and/or conservativity of adopting some term (together with analytic feeling reasoning that goes with that term) can become more interesting when conceptual analysis only provides an *implicit* or recursive definition of the term.
This is valuable, to the extent that you are worried a purported new concept may be 'incoherent' (in the sense that intuitive, analytic feeling reasoning about it literally lets you prove contradiction) or may have bad 'presuppositions' (in the sense that that intuitive, analytic feeling, reasoning using the term allows one to derive new propositions not using that term, which are false)
c) Teaching: Obviously getting a method for paraphrase sentences involving new terminology in terms of old terminology provides a way of teaching the new terminology to people who already understand the old terminology.
Note that none of these purposes require that conceptual analyses be unique. Different analyses of claims about, say, the imaginary numbers, in terms of set theory can each serve this purpose equally well. Nor do these uses for conceptual analysis require that one make any claim about the metaphysical status of the objects in question. It's useful to know you can reconstruct all intuitively acceptable reasoning about the imaginary numbers in terms of intuitively acceptable reasoning about sets, even if you don't want to claim that the imaginary numbers ARE sets or anything like that. Nor, lastly, do they require that the analyses have some kind of psychological reality - that when you are thinking about imaginary numbers you are really somehow implicitly (subconsciously?) considering one or the other paraphrase in terms of sets.
[Hidden agenda: Even if it turns out that Occam's razor doesn't apply to positing special sciences objects like livers, species, trade deficits and languages, so there's no need to look for paraphrases which would allow us to *deny* that such "extra" objects exist, finding Quinean-style parapharases will still be illuminating and useful for other reasons. So we philosophers won't be talking ourselves out of a job :). Also, to the extent that you feel like something substantial is going on when one looks for Quinean methods of paraphrase, this may be because these paraphrases illuminate the structure of our intuitive reasoning about Ws, and let us relate the W facts to facts about objects we understand better - not because there is a serious question about whether the Ws really exist.]
Wednesday, May 19, 2010
Tuesday, May 18, 2010
A Depressing Theory of Ceterus Paribus Clauses
We want to say "sugarcubes dissolve in water, ceterus paribus", but what does that mean? Philosophical analysis of the phrase ceterus paribus has proved surprisingly difficult. For example, the quoted sentence doesn't mean that all or most pieces of sugar that actually will be dropped into water will dissolve.
Here's a depressing proposal for how ceterus paribus clauses work. We have a substantive (implicit) theory of what "the normal cases" are like, which is based on human daily life and maybe some random traditions too. We use this when evaluating ceterus paribus sentences to choose which way of making the target sentence true to consider. So, for example, 'ceterus paribus' clauses get filled in so that "dropped eggs break, ceterus parbus" is true, because people tend to hang out in places near the surface of the earth, which don't have thick rugs, so it's part of our substantive theory of what's "normal" that when something is dropped there's a hard surface below it (as opposed to a thick rug, or the empty expanse of space).
Here's a depressing proposal for how ceterus paribus clauses work. We have a substantive (implicit) theory of what "the normal cases" are like, which is based on human daily life and maybe some random traditions too. We use this when evaluating ceterus paribus sentences to choose which way of making the target sentence true to consider. So, for example, 'ceterus paribus' clauses get filled in so that "dropped eggs break, ceterus parbus" is true, because people tend to hang out in places near the surface of the earth, which don't have thick rugs, so it's part of our substantive theory of what's "normal" that when something is dropped there's a hard surface below it (as opposed to a thick rug, or the empty expanse of space).
Sunday, May 9, 2010
Miniature Phil Math
Almost everyone agrees that our mathematical talk is practically helpful. Unlike astrology, doing math helps us build bridges. But how is math practically helpful? And does the way in which talking about numbers is practically helpful give us any reason to think numbers actually exist?
In this tiny essay I will propose a theory of how the practice of talking as if there were numbers is helpful. Then, I will say that we can appeal to numbers to explain how this practice is helpful, though there are also other correct explanations for this phenomenon which do not commit themselves to numbers. I will conclude by turning to the question of whether there are numbers. On the basis of the previous section I will propose that we do not *need* to posit the existence of numbers to explain the practical usefulness of our mathematical talk. However, we have another reason to believe in numbers which is the following: We want to make statements like "the number of cupcakes doubles every day" true (under certain circumstances), and the pattern of inferences we make with this sentences is quantificational. But this (being describable by some true sentences associated with a existential pattern of inferences) is the only thing that the many different kinds of non-mathematical objects which intuitively exist have in common.
1. How talking about abstracta like numbers is helpful
Talking about abstract objects, like numbers, is helpful because it lets us economically hypothesize patterns 'in the world around us' as well as patterns that might be described as artifacts of language (patterns in which distinct descriptions are logically or otherwise necessarily equivalent). We can say one sentence (about numbers) that will cause people to be willing to infer infinitely many different sentences that aren't about numbers.
For example, suppose I say: "The number of cupcakes doubles every day" This is a claim that quantifies over numbers and days, in the sense that we might represent it as "Ad An if d is a day, and n is a number, then there are n cupcakes on d there are 2n cupcakes on the day after d. "
Hearing this single sentence will lead my listeners to accept many different statements that do not quantify over cupcakes:
"if Ex7 cupcakes today Ex14 cupcakes tomorrow."
"if Ex8 cupcakes today Ex16 cupcakes tomorrow."
"if Ex7 cupcakes tomorrow Ex14 cupcakes the day after tomorrow."
2. What role do abstract objects play in explaining why talk of abstract objects is helpful?
Now we can ask: what role do various objects play in explaining the success of this talk? We might explain the helpfulness of my statement by saying that it is helpful because it...
- lets us track and predict what cupcakes there are and will be
- lets us track *the pattern in* what cupcakes there are and will be
- lets us track and predict how *the doubling function* relates *numbers*, and then predict what cupcakes there will be when, by relating this to facts about the behavior of the doubling function.
It seems to me that all of these are intuitively decent explanations. I take it that what we have here is a typical phenomenon where the same phenomenon (a war) can be explained by accounts that quantify over various different objects (countries vs. people vs. atoms). However, not much would be lost if we just stuck to giving the first explanation, which does not involve any mention of abstract objects.
3. Are there numbers? A good and bad reason for believing in numbers.
If this story about how math is practically helpful is right, should we believe that there really are objects of the kind talked about in these explanations e.g. patterns in the provenance in cupcakes, or numbers and a doubling function?
I don't think there is an *inference to the best explanation* for the existence of patterns in the provenance of cupcakes, or numbers from the helpfulness of this talk. It's not the case that we *need* to posit abstract objects called "patterns in the provenance of cupcakes" or "numbers" to explain how saying the thing described above could help people cope with the cupcakes around them.
Instead, I think it's reasonable to believe in numbers because we have an intuitively true sentence ("the number of cupcakes doubles every day") which allows a existential pattern of inferences - and playing this logical role is all there is to being an object.
The idea here is that when we look at the variety of different "objects" in the world e.g. electrons, magnetic fields goats, holes, waves, contracts, countries, these different kinds of talk don't seem to have much in common with regard to their relation to the physical world. What they do have in common is the pattern of inferences we make between sentences between them. In each case we accept sentences, such that the inferences with these sentences in are elegantly captured (in first order logic) by something of the form "Ex Fx". Now it turns out that talking about numbers and the doubling function shares this same feature.
In this tiny essay I will propose a theory of how the practice of talking as if there were numbers is helpful. Then, I will say that we can appeal to numbers to explain how this practice is helpful, though there are also other correct explanations for this phenomenon which do not commit themselves to numbers. I will conclude by turning to the question of whether there are numbers. On the basis of the previous section I will propose that we do not *need* to posit the existence of numbers to explain the practical usefulness of our mathematical talk. However, we have another reason to believe in numbers which is the following: We want to make statements like "the number of cupcakes doubles every day" true (under certain circumstances), and the pattern of inferences we make with this sentences is quantificational. But this (being describable by some true sentences associated with a existential pattern of inferences) is the only thing that the many different kinds of non-mathematical objects which intuitively exist have in common.
1. How talking about abstracta like numbers is helpful
Talking about abstract objects, like numbers, is helpful because it lets us economically hypothesize patterns 'in the world around us' as well as patterns that might be described as artifacts of language (patterns in which distinct descriptions are logically or otherwise necessarily equivalent). We can say one sentence (about numbers) that will cause people to be willing to infer infinitely many different sentences that aren't about numbers.
For example, suppose I say: "The number of cupcakes doubles every day" This is a claim that quantifies over numbers and days, in the sense that we might represent it as "Ad An if d is a day, and n is a number, then there are n cupcakes on d there are 2n cupcakes on the day after d. "
Hearing this single sentence will lead my listeners to accept many different statements that do not quantify over cupcakes:
"if Ex7 cupcakes today Ex14 cupcakes tomorrow."
"if Ex8 cupcakes today Ex16 cupcakes tomorrow."
"if Ex7 cupcakes tomorrow Ex14 cupcakes the day after tomorrow."
2. What role do abstract objects play in explaining why talk of abstract objects is helpful?
Now we can ask: what role do various objects play in explaining the success of this talk? We might explain the helpfulness of my statement by saying that it is helpful because it...
- lets us track and predict what cupcakes there are and will be
- lets us track *the pattern in* what cupcakes there are and will be
- lets us track and predict how *the doubling function* relates *numbers*, and then predict what cupcakes there will be when, by relating this to facts about the behavior of the doubling function.
It seems to me that all of these are intuitively decent explanations. I take it that what we have here is a typical phenomenon where the same phenomenon (a war) can be explained by accounts that quantify over various different objects (countries vs. people vs. atoms). However, not much would be lost if we just stuck to giving the first explanation, which does not involve any mention of abstract objects.
3. Are there numbers? A good and bad reason for believing in numbers.
If this story about how math is practically helpful is right, should we believe that there really are objects of the kind talked about in these explanations e.g. patterns in the provenance in cupcakes, or numbers and a doubling function?
I don't think there is an *inference to the best explanation* for the existence of patterns in the provenance of cupcakes, or numbers from the helpfulness of this talk. It's not the case that we *need* to posit abstract objects called "patterns in the provenance of cupcakes" or "numbers" to explain how saying the thing described above could help people cope with the cupcakes around them.
Instead, I think it's reasonable to believe in numbers because we have an intuitively true sentence ("the number of cupcakes doubles every day") which allows a existential pattern of inferences - and playing this logical role is all there is to being an object.
The idea here is that when we look at the variety of different "objects" in the world e.g. electrons, magnetic fields goats, holes, waves, contracts, countries, these different kinds of talk don't seem to have much in common with regard to their relation to the physical world. What they do have in common is the pattern of inferences we make between sentences between them. In each case we accept sentences, such that the inferences with these sentences in are elegantly captured (in first order logic) by something of the form "Ex Fx". Now it turns out that talking about numbers and the doubling function shares this same feature.
Wednesday, April 14, 2010
Contrast w/ Tait "The Platonism of Mathematics"
Both my view (Lumpist Platonism) and Tait's might be considered unusual or quirky versions of platonism. Platonism (in phil math) is the view that mathematical objects exist.
I think that the world is fundamentally (something like) a space-time manifold [as opposed to a set of facts, or a set of objects and relations], and that all statements are true or false in virtue of how the manifold is. This includes statements about objects, and different statements about objects will correspond to very different claims about the state of the manifold (e.g. saying that there's a table vs that there's a whirlpool vs a trade deficit vs. a marriage contract vs. a number or string of symbols or a proposition). So facts obtain, and objects and relations exist, in virtue of how the physical stuff of the world is configured, not vice versa. Necessary truths (like all statements of pure math) correspond to the trivial claim about the state of the manifold (one that doesn't rule out any possible configurations).
Tait, as I understand him, thinks that mathematical sentences show that objects exist by constructing suitable objects. He writes "A proof is a presentation or construction of an object: A is true when there is an object of type A and we prove A by constructing such an object."
Both of these views contrast with what you might call a "two worlds" version of platonism. On this view: in addition to whatever objects exist in virtue of the physical stuff of the world comporting itself a certain way, there is also an "extra" component of reality. So far as I understand the force of the word "extra" here, the point of saying that there's an extra component of reality is this: An infinite and putatively exhaustive description of the world given purely in the language of microphysics e.g. (this point has that property, this point has that property etc.) would be missing out on the existence of sets, *in some stronger sense then the sense than in which it would be missing out on rabbits and trade deficits*.
Tait and I also agree that sentences are the right place to start when considering how semantics relates to metaphysics and ontology. For a sentence to be meaningful you just need the whole sentence to somehow make a claim about the world. Thinking about particular words in the sentence as having favored relations with particular chunks of matter will help in some cases but not others.
However, I disagree with Tait on some really important points:
Firstly, I don't really understand what he means by construction. The best sense I can make of the idea of constructing mathematical objects (how can you bring an abstract object into being?) is that it's something like the way I can create a) a marriage contract with another person by signing things the courthouse, or b) the set with ur-elements {Sharon's mullet} by giving myself an ill-judged haircut and thereby bringing a particular mullet-token into being, and hence it's corresponding singleton. But if this is what he had in mind, then...
a) it has the (at the very least) wildly counterintuitive to say that there wasn't a number between 3 and 5 before someone wrote down a proof inscription.
b) quantification in math works very weirdly and differently from quantification in general. For, since people have only written finitely many proofs there will be some number - say 347892-, such that no one has inscribed a proof of "3457892 has a sucessor". On the other hand, we certainly have inscribed proofs of "Ax if x is a natural number then x has a successor". So it would seem that the general statement is true. But the instance is (at the moment) false.
Secondly, Tait doesn't seem to allow that quantified statements of arithmetic (like, say, the Godel sentences for various formal systems) already have truth values now. He seems to think we are free to choose which kinds of proofs to construct (i.e. what formal system to adopt). And then he says that "the incompleteness of formal systems such as elementary number theory can be proved by induction, is best seen as an incompleteness with respect to what can be expressed in the system rather than with the rules of inference." And he points out that by extending the language (and adding suitable instances of the induction schema) you can prove the Godel sentence (and con) for this system.
But when I wonder about e.g. con(PA+X) [it's pretty hard to wonder about con(PA) imho] or con(ZFC), I'm not just wondering whether I could extend my formal system in such a way as to allow these sentences (or their negation) to be derived. Obviously, I could start making derivations (and hence constructing objects, for Tait) in any formal system I want. Nor am I pondering what kind of lifestyle choice to adopt in the future. Rather, I think that *right now*, I understand what it means to ask whether there's a proof of 0=1 from ZFC. And this is what I want to know. Is this sentence provable in that formal system or not? Is there such a proof or not?To the extent that we can ever be sure that we really understand something, and are asking a sharply meaningful question, this is it! [I think this may be why my advisor PK disagrees with Tait too]
Overall, I'm tempted to suspect that Tait is getting into bed with unattractive antirealism because he wants to avoid an epistemological problem. He sees how (/doesn't worry about how) you could know that something exists if you are able to bring it into existence (construct it). Such knowledge is sometimes called "maker's knowledge". And then he wants to say what mathematical knowledge is, in such a way that all mathematical knowledge turns out to be accessible in this way - which leads to weird consequences about large numbers, and unknown arithmetical facts.
In contrast, if you use the ...ahem... magic of Sharon's thesis, to provide a general naturalistic mechanism for how physical creatures could have gotten a faculty of reliable rational insight into abstract mathematical/logical truths :) , then you don't have to do any of this fancy (and potentially distorting) footwork.
I think that the world is fundamentally (something like) a space-time manifold [as opposed to a set of facts, or a set of objects and relations], and that all statements are true or false in virtue of how the manifold is. This includes statements about objects, and different statements about objects will correspond to very different claims about the state of the manifold (e.g. saying that there's a table vs that there's a whirlpool vs a trade deficit vs. a marriage contract vs. a number or string of symbols or a proposition). So facts obtain, and objects and relations exist, in virtue of how the physical stuff of the world is configured, not vice versa. Necessary truths (like all statements of pure math) correspond to the trivial claim about the state of the manifold (one that doesn't rule out any possible configurations).
Tait, as I understand him, thinks that mathematical sentences show that objects exist by constructing suitable objects. He writes "A proof is a presentation or construction of an object: A is true when there is an object of type A and we prove A by constructing such an object."
Both of these views contrast with what you might call a "two worlds" version of platonism. On this view: in addition to whatever objects exist in virtue of the physical stuff of the world comporting itself a certain way, there is also an "extra" component of reality. So far as I understand the force of the word "extra" here, the point of saying that there's an extra component of reality is this: An infinite and putatively exhaustive description of the world given purely in the language of microphysics e.g. (this point has that property, this point has that property etc.) would be missing out on the existence of sets, *in some stronger sense then the sense than in which it would be missing out on rabbits and trade deficits*.
Tait and I also agree that sentences are the right place to start when considering how semantics relates to metaphysics and ontology. For a sentence to be meaningful you just need the whole sentence to somehow make a claim about the world. Thinking about particular words in the sentence as having favored relations with particular chunks of matter will help in some cases but not others.
However, I disagree with Tait on some really important points:
Firstly, I don't really understand what he means by construction. The best sense I can make of the idea of constructing mathematical objects (how can you bring an abstract object into being?) is that it's something like the way I can create a) a marriage contract with another person by signing things the courthouse, or b) the set with ur-elements {Sharon's mullet} by giving myself an ill-judged haircut and thereby bringing a particular mullet-token into being, and hence it's corresponding singleton. But if this is what he had in mind, then...
a) it has the (at the very least) wildly counterintuitive to say that there wasn't a number between 3 and 5 before someone wrote down a proof inscription.
b) quantification in math works very weirdly and differently from quantification in general. For, since people have only written finitely many proofs there will be some number - say 347892-, such that no one has inscribed a proof of "3457892 has a sucessor". On the other hand, we certainly have inscribed proofs of "Ax if x is a natural number then x has a successor". So it would seem that the general statement is true. But the instance is (at the moment) false.
Secondly, Tait doesn't seem to allow that quantified statements of arithmetic (like, say, the Godel sentences for various formal systems) already have truth values now. He seems to think we are free to choose which kinds of proofs to construct (i.e. what formal system to adopt). And then he says that "the incompleteness of formal systems such as elementary number theory can be proved by induction, is best seen as an incompleteness with respect to what can be expressed in the system rather than with the rules of inference." And he points out that by extending the language (and adding suitable instances of the induction schema) you can prove the Godel sentence (and con) for this system.
But when I wonder about e.g. con(PA+X) [it's pretty hard to wonder about con(PA) imho] or con(ZFC), I'm not just wondering whether I could extend my formal system in such a way as to allow these sentences (or their negation) to be derived. Obviously, I could start making derivations (and hence constructing objects, for Tait) in any formal system I want. Nor am I pondering what kind of lifestyle choice to adopt in the future. Rather, I think that *right now*, I understand what it means to ask whether there's a proof of 0=1 from ZFC. And this is what I want to know. Is this sentence provable in that formal system or not? Is there such a proof or not?To the extent that we can ever be sure that we really understand something, and are asking a sharply meaningful question, this is it! [I think this may be why my advisor PK disagrees with Tait too]
Overall, I'm tempted to suspect that Tait is getting into bed with unattractive antirealism because he wants to avoid an epistemological problem. He sees how (/doesn't worry about how) you could know that something exists if you are able to bring it into existence (construct it). Such knowledge is sometimes called "maker's knowledge". And then he wants to say what mathematical knowledge is, in such a way that all mathematical knowledge turns out to be accessible in this way - which leads to weird consequences about large numbers, and unknown arithmetical facts.
In contrast, if you use the ...ahem... magic of Sharon's thesis, to provide a general naturalistic mechanism for how physical creatures could have gotten a faculty of reliable rational insight into abstract mathematical/logical truths :) , then you don't have to do any of this fancy (and potentially distorting) footwork.
Parsons and Intuitability
I've just been summarizing CH1 of Charles Parsons' Mathematical Thought and it's Objects. It set me thinking that Parsons is oddly concerned with whether you can "see"/percieve/intuit mathematical objects. I say oddly, because IMO what matters for assuaging worries about the weirdness of mathematical objects or the weirdness of our knowing about them (which seems to be part of his aim) isn't whether we can strictly speaking *see*/perceive/intuit abstracta, but rather a) whether positing abstracta isn't a violation of Occam's razor and b) how there can be enough of a connection between mathematical facts and our dispositions to form beliefs about them, for what we have to count as knowledge.
I mean: even in the empirical case, questions about what we can see, as opposed to merely inferring from what we see are super murky. Who knows whether you can "see" that the light is on vs. that the electricity is back on vs. that Jones succeeded at his task etc. as opposed to inferring them or justifiably and reliably forming true beliefs about these subjects)? What matters (for the epistemology worry b) is just that there needs to be some suitable and clear reliable mechanism at work leading you to form true beliefs on these subjects - as there obviously is in the empirical case of the light. Once we see how this reliable mechanism could work, it's (in my opinion) a matter of indifference whether you want to describe this mechanism as seeing the light and then immediately and unconsciously but justifiably inferring that the electricity is back on vs. directly seeing that the electricity is back on.
And the same goes for knowledge of mathematical objects. What we'd like is something that was like perception in the sense that it provided an unproblematic mechanism whereby we could get the relevant kind.Once we have that in place, we can say whatever we like about whether someone staring at a piece of paper can see/percieve/intuit that there's a proof of SS0+S0=SSS0 in PA, or a palindrome containing the word 'adam' vs. merely reliably and justifiably infer these statements from the concrete object that they do see. The million dollar question is how we manage to do this putative seeing/inferring correctly.
Similarly, if someone thinks that construing math as stating truths about genuine abstract objects is a violation of Occam's razor, (as per objection a) they aren't going to be impressed by claims to "see" the abstract object (a string) in the concrete object (a series of inkmarks). When the Platonist stares at the sheet of paper and says they are seeing that there's a proof SS0+S0=SSS0, the Fictionalst will say that they are seeing that there would have to be a proof in the relevant mathematical fiction, and the modalist will say you are seeing that a certain proof is possible.
My point here is not to knock Parson's interest in the relationship between concrete things you can see and abstract mathematical objects. Hearing him talk about this connection was a decisive inspiration for my own view, and I think it's absolutely crucial to think about the concrete physical processes going on when we form and revise mathematical beliefs, if you want to understand how creatures like us could know about (or even think about) something as abstract as math. But I would claim that the key point about string inscriptions isn't what they represent/allow us to intuit (can you stare through the string inscription to the string itself?, can you at least see that a certain string exists?), but (as it were) what you take these inscriptions to represent, i.e. how you are willing to form and revise your beliefs about other things, like strings as abstract objects, in response to seeing them. This is what starts to give us traction in linking up our dispositions to form mathematical beliefs to mathematical facts, to answer challenge (b). (IMO answering challenge (a) requires something else entirely, namely Lumpism, but more about that in the next post)
I mean: even in the empirical case, questions about what we can see, as opposed to merely inferring from what we see are super murky. Who knows whether you can "see" that the light is on vs. that the electricity is back on vs. that Jones succeeded at his task etc. as opposed to inferring them or justifiably and reliably forming true beliefs about these subjects)? What matters (for the epistemology worry b) is just that there needs to be some suitable and clear reliable mechanism at work leading you to form true beliefs on these subjects - as there obviously is in the empirical case of the light. Once we see how this reliable mechanism could work, it's (in my opinion) a matter of indifference whether you want to describe this mechanism as seeing the light and then immediately and unconsciously but justifiably inferring that the electricity is back on vs. directly seeing that the electricity is back on.
And the same goes for knowledge of mathematical objects. What we'd like is something that was like perception in the sense that it provided an unproblematic mechanism whereby we could get the relevant kind.Once we have that in place, we can say whatever we like about whether someone staring at a piece of paper can see/percieve/intuit that there's a proof of SS0+S0=SSS0 in PA, or a palindrome containing the word 'adam' vs. merely reliably and justifiably infer these statements from the concrete object that they do see. The million dollar question is how we manage to do this putative seeing/inferring correctly.
Similarly, if someone thinks that construing math as stating truths about genuine abstract objects is a violation of Occam's razor, (as per objection a) they aren't going to be impressed by claims to "see" the abstract object (a string) in the concrete object (a series of inkmarks). When the Platonist stares at the sheet of paper and says they are seeing that there's a proof SS0+S0=SSS0, the Fictionalst will say that they are seeing that there would have to be a proof in the relevant mathematical fiction, and the modalist will say you are seeing that a certain proof is possible.
My point here is not to knock Parson's interest in the relationship between concrete things you can see and abstract mathematical objects. Hearing him talk about this connection was a decisive inspiration for my own view, and I think it's absolutely crucial to think about the concrete physical processes going on when we form and revise mathematical beliefs, if you want to understand how creatures like us could know about (or even think about) something as abstract as math. But I would claim that the key point about string inscriptions isn't what they represent/allow us to intuit (can you stare through the string inscription to the string itself?, can you at least see that a certain string exists?), but (as it were) what you take these inscriptions to represent, i.e. how you are willing to form and revise your beliefs about other things, like strings as abstract objects, in response to seeing them. This is what starts to give us traction in linking up our dispositions to form mathematical beliefs to mathematical facts, to answer challenge (b). (IMO answering challenge (a) requires something else entirely, namely Lumpism, but more about that in the next post)
Parsons Mathematical Thought and its Objects CH1 summary
No one I've talked to is really sure what's going on. Especially me. But here's my current best guess. Maybe the magic powers of saying something wrong on the internet will help us work our way incrementally to a better interpretation.
1. Abstract objects defined + generic worries about them
Mathematical objects would be abstract objects = acausal, not located in space and time.
Worry: They aren't perceptable, if perceiving something requires locating it. Maybe this suggests there are no such things?
Response:
- electrons don't seem to be directly perceptable either, but they exist
- if we say that mathematical objects don't exist then we will have to explain why talking as if they did is so helpful for science.
- it's not clear whether we can avoid quantifying over abstract objects, hence (if we accept Quine's criterion) saying that they do exist.
2-3 What is an object?
It's hard to answer the question 'what is an object?' since unlike with gorillas we can't point out a contrast class of things that aren't objects.
the right answer: logical role
Philosophers usually ask 'what's an object?' in the context of trying to figure out how language can relate to the world - how we can talk about objects. For these purposes we can define being an object in terms of logical role: objects are what we talk about by using singular terms (e.g. 'Bob' in Bob is happy= Happy(Bob)) and quantification (e.g. 'Ex x happy').
other conceptions of objects/requirements philosophers have had for objects...
i. actuality/causal powers
Digression about Kant: general notion of object vs. "Wirklichkeit"
Kant invented the phrase 'concept of an object in general'. Kant's "categories" are concepts of an object in general. He is conflicted about whether these categories have to be perceivable by the senses [and hence whether "the concept of an object in general" would allow abstract objects?]
a) the categories are supposed to be derivable from logic and general considerations that don't take into account anything specific about the kind of object involved.
b) applying the categories is only supposed to generate knowledge when combined with stuff from the senses (namely: " the manifold given in sensory intuition")
Kant and Frege seem to have a notion of the actual = "wirklich" which only applies to objects you can causally interact with
Kant clearly accepts mathematical objects in some sense, but it's not clear whether he somehow thinks they are merely possible.
Idea: Many people find abstract objects spooky because they assume that they would have to be Wirklich, or something like it. The merely logical conception of object above doesn't require any such thing. So maybe mathematical objects exist in the logical sense i.e. we can state truths using singular terms for them and using quantifiers, but they are somehow not Wirklich.
ii. intuitability
Kant digression:
You use intuition to discover whether things could fall under it. [presumably round square would be an example of a putative concept that doesn't pass this test.]
geometric figures = forms of empirical objects
We can learn about them using intuition.
Perhaps it's an requirement that all objects are 'intuitable'?
defining intuitable
We will use intuition to mean a kind of perception that could apply to physical objects or abstract objects. We can distinguish
- having an intuition of an object, like perceiving an object (e.g. 'I intuit the equilateral triangle')
- having an intuition that some proposition about the object holds (e.g. 'I intuit that the interior angles of the equilateral triangle add up to 180')
Some issues:
-Should we require that one can have intuition *of* the object, rather than merely intuiting some suitable proposition about it? (call this strong intuitability) Or is it enough if you have an intuition of concrete objects that represent abstract objects, like the sequence of strokes Kant appeals to in his proof that 7+5=12? (call such a representation a quasi-concrete representation)
-On what sense does need to be possible to intuit something for that something to count as intuit*able*, and hence satisfy the requirement?
Idea cont. - Maybe mathematical objects are real in the logical sense, and intuitable, but not wirklich/causally effecations...
4. objecthood=having the logical role of an object
We will stick with Quine and Frege and say that the logical criterion (not wirklichkeit or intuitability) is all that's required for objecthood.
Some questions arise if you accept this definition of "object", about how to further spell out the view.
a) Which logic has the property that *its* singular terms and quantifiers correspond to objecthood? Maybe we should allow modal or other intentional notions, and if we do we will get different answers about what objects there are.
b) Maybe there are some entities which aren't objects? (i.e. maybe there's some important ontological category that's wider than objecthood - like some kind of meinogian being)
c) Maybe there are some objects which don't exist? (i.e. maybe there's some important ontological category that's narrower than objecthood - like fictional objects might be said to logically objects, but not really exist)
5-6 are about b and c respectively
7. Quasi-concrete objects
We will call abstract objects quasi concrete if they have a special relationship to certain concrete objects that 'represent' them e.g.
strings of letters --- inscriptions of strings of letters
sense qualities --- experiences of those sense qualities
shapes --- physical things that have that shape
We can look at the physical representatives, and keep in mind individuation criteria for the abstract objects. These individuation criteria say when two different concrete things 'represent' the same abstract one.
Some sets are quasi-concrete: sets with concrete ur-elements are represented by those ur-elements. But pure sets are not quasi concrete.
Overall Conclusion: mathematical objects exist in the logical sense, although they are not Wirklich, and although some of them are not intuitiable even in the weak sense allowed by looking at concrete objects that represent them.
1. Abstract objects defined + generic worries about them
Mathematical objects would be abstract objects = acausal, not located in space and time.
Worry: They aren't perceptable, if perceiving something requires locating it. Maybe this suggests there are no such things?
Response:
- electrons don't seem to be directly perceptable either, but they exist
- if we say that mathematical objects don't exist then we will have to explain why talking as if they did is so helpful for science.
- it's not clear whether we can avoid quantifying over abstract objects, hence (if we accept Quine's criterion) saying that they do exist.
2-3 What is an object?
It's hard to answer the question 'what is an object?' since unlike with gorillas we can't point out a contrast class of things that aren't objects.
the right answer: logical role
Philosophers usually ask 'what's an object?' in the context of trying to figure out how language can relate to the world - how we can talk about objects. For these purposes we can define being an object in terms of logical role: objects are what we talk about by using singular terms (e.g. 'Bob' in Bob is happy= Happy(Bob)) and quantification (e.g. 'Ex x happy').
other conceptions of objects/requirements philosophers have had for objects...
i. actuality/causal powers
Digression about Kant: general notion of object vs. "Wirklichkeit"
Kant invented the phrase 'concept of an object in general'. Kant's "categories" are concepts of an object in general. He is conflicted about whether these categories have to be perceivable by the senses [and hence whether "the concept of an object in general" would allow abstract objects?]
a) the categories are supposed to be derivable from logic and general considerations that don't take into account anything specific about the kind of object involved.
b) applying the categories is only supposed to generate knowledge when combined with stuff from the senses (namely: " the manifold given in sensory intuition")
Kant and Frege seem to have a notion of the actual = "wirklich" which only applies to objects you can causally interact with
Kant clearly accepts mathematical objects in some sense, but it's not clear whether he somehow thinks they are merely possible.
Idea: Many people find abstract objects spooky because they assume that they would have to be Wirklich, or something like it. The merely logical conception of object above doesn't require any such thing. So maybe mathematical objects exist in the logical sense i.e. we can state truths using singular terms for them and using quantifiers, but they are somehow not Wirklich.
ii. intuitability
Kant digression:
You use intuition to discover whether things could fall under it. [presumably round square would be an example of a putative concept that doesn't pass this test.]
geometric figures = forms of empirical objects
We can learn about them using intuition.
Perhaps it's an requirement that all objects are 'intuitable'?
defining intuitable
We will use intuition to mean a kind of perception that could apply to physical objects or abstract objects. We can distinguish
- having an intuition of an object, like perceiving an object (e.g. 'I intuit the equilateral triangle')
- having an intuition that some proposition about the object holds (e.g. 'I intuit that the interior angles of the equilateral triangle add up to 180')
Some issues:
-Should we require that one can have intuition *of* the object, rather than merely intuiting some suitable proposition about it? (call this strong intuitability) Or is it enough if you have an intuition of concrete objects that represent abstract objects, like the sequence of strokes Kant appeals to in his proof that 7+5=12? (call such a representation a quasi-concrete representation)
-On what sense does need to be possible to intuit something for that something to count as intuit*able*, and hence satisfy the requirement?
Idea cont. - Maybe mathematical objects are real in the logical sense, and intuitable, but not wirklich/causally effecations...
4. objecthood=having the logical role of an object
We will stick with Quine and Frege and say that the logical criterion (not wirklichkeit or intuitability) is all that's required for objecthood.
Some questions arise if you accept this definition of "object", about how to further spell out the view.
a) Which logic has the property that *its* singular terms and quantifiers correspond to objecthood? Maybe we should allow modal or other intentional notions, and if we do we will get different answers about what objects there are.
b) Maybe there are some entities which aren't objects? (i.e. maybe there's some important ontological category that's wider than objecthood - like some kind of meinogian being)
c) Maybe there are some objects which don't exist? (i.e. maybe there's some important ontological category that's narrower than objecthood - like fictional objects might be said to logically objects, but not really exist)
5-6 are about b and c respectively
7. Quasi-concrete objects
We will call abstract objects quasi concrete if they have a special relationship to certain concrete objects that 'represent' them e.g.
strings of letters --- inscriptions of strings of letters
sense qualities --- experiences of those sense qualities
shapes --- physical things that have that shape
We can look at the physical representatives, and keep in mind individuation criteria for the abstract objects. These individuation criteria say when two different concrete things 'represent' the same abstract one.
Some sets are quasi-concrete: sets with concrete ur-elements are represented by those ur-elements. But pure sets are not quasi concrete.
Overall Conclusion: mathematical objects exist in the logical sense, although they are not Wirklich, and although some of them are not intuitiable even in the weak sense allowed by looking at concrete objects that represent them.
Friday, April 9, 2010
Field on Normativity and Logic
In "What is the Normative Role of Logic" Field argues that you can't understand logic descriptively as (eg. the project of studying necessarily truth preserving syntactic manipulations), and so are forced to a more normative conception of logic (logic is the study of how one ought to reason), by the following dilemma.
-classical logics can't state a general truth predicate (if they could, we could inductively argue for the soundness of logic, and hence a consistency proof for logic L in logic L, contra Godel 2)
-non-classical logics which can state a general truth predicate, sometimes fail to preserve truth, in some degenerate cases (in places where good reasoning wouldn't lead you to in the first place).
So (Field says) the only people who can *state* the descriptive criterion for being a logic, deny that logic has to have that property.
But I think there's a gap in this argument: why should you have to be able to state your criterion for what a good logical system is, *in the formal language of that logic*? In particular, why can't the anti-normativitst about logic reply like this:
A. Classical Logic Version:
Logic is the study of formal systems of syntactic manipulation which are truth preserving for various fragments of our language (e.g. english sans any truth predicate, english sans any repeated application of the truth predicate). Practically speaking, this is all we need for almost every purpose except philosophy of logic and truth. And the moral of Tarski-Godel considerations above is that this is all we can get.
Formal, exceptionless, rules for truth-preserving reasoning are great when you can get them (i.e. for limited fragments of our language) but what Field has shown, is that we can't get any such rules that apply to the informal notion of truth (as opposed to the notion of truth-of-a-sentence-in-L, for various restricted L)
Admittedly, taking this route involves giving up the traditional and somewhat attractive Fregean idea that logical principles are fully general, and hence would apply to all possible reasoning, but - at least- this seems way less revisionary than the normative relativism about logic where Field winds up.
B. Non-Classical Logic Version:
It was indeed wrong to say that logic studies patterns of inference that are always truth preserving. Field is right that Logic studies patterns of reasoning that are truth preserving "where it counts". But "where it counts" doesn't mean something normative like 'with regard to premises that one could be justified in believing', but rather, something descriptive like 'with regard to premises that people are likely to every actually accept'.
-classical logics can't state a general truth predicate (if they could, we could inductively argue for the soundness of logic, and hence a consistency proof for logic L in logic L, contra Godel 2)
-non-classical logics which can state a general truth predicate, sometimes fail to preserve truth, in some degenerate cases (in places where good reasoning wouldn't lead you to in the first place).
So (Field says) the only people who can *state* the descriptive criterion for being a logic, deny that logic has to have that property.
But I think there's a gap in this argument: why should you have to be able to state your criterion for what a good logical system is, *in the formal language of that logic*? In particular, why can't the anti-normativitst about logic reply like this:
A. Classical Logic Version:
Logic is the study of formal systems of syntactic manipulation which are truth preserving for various fragments of our language (e.g. english sans any truth predicate, english sans any repeated application of the truth predicate). Practically speaking, this is all we need for almost every purpose except philosophy of logic and truth. And the moral of Tarski-Godel considerations above is that this is all we can get.
Formal, exceptionless, rules for truth-preserving reasoning are great when you can get them (i.e. for limited fragments of our language) but what Field has shown, is that we can't get any such rules that apply to the informal notion of truth (as opposed to the notion of truth-of-a-sentence-in-L, for various restricted L)
Admittedly, taking this route involves giving up the traditional and somewhat attractive Fregean idea that logical principles are fully general, and hence would apply to all possible reasoning, but - at least- this seems way less revisionary than the normative relativism about logic where Field winds up.
B. Non-Classical Logic Version:
It was indeed wrong to say that logic studies patterns of inference that are always truth preserving. Field is right that Logic studies patterns of reasoning that are truth preserving "where it counts". But "where it counts" doesn't mean something normative like 'with regard to premises that one could be justified in believing', but rather, something descriptive like 'with regard to premises that people are likely to every actually accept'.
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