When I say that "the more practically benign a system of proto-mathematics is, the more likely it is to count as expressing largely true claims about some domain of objects", I realize that this sounds horribly woolly. People naturally ask me: but how practically benign does a practice have to be to guarantee that it succeeds in talking about some object? (not to mention: how do you measure `largely true'?)
Why Be Woolly
Here's a classic example of a claim that isn't woolly:
H: Any logically consistent system of mathematical beliefs counts as expressing truths about some suitable domain of objects.
We can see H is false because it implies that if I believe ZFC+{X} and you believe ZFC+{~X} where X is some statement about number theory independent of ZFC, since we both have logically consistent systems of belief, we will both be right - just talking about different objects.
But what goes wrong?
Note the problem isn't that there aren't enough mathematical objects (if we just have sets every first order consistent theory has a model). Rather (I claim) it's because actual people will use words in the mathematical theory like 'finite' or 'smallest' or 'number' which have meaning that goes beyond their role in this first order logical stipulation.
When we both say that by the "numbers" we mean (among other things) the smallest collection containing 0 and closed under successor, smallest (intuitively) means the same thing for both of us, so it is NOT correct to then interpret each of us as talking about whatever larger non-standard model makes our claim true.
Hence our informal use of the words like "smallest" or "all possible collections" imposes constraints on interpreting us which go beyond the first order logical content of our mathematical statements.
If you buy this, here's why you should be wooly. In general there will be some vagueness with regard to how wrong you can be about Madagascar, Christmas or to use Quine's famous example, atoms, and still count as talking about these things. Once a theory is sufficiently wrong it can be a tossup whether to say that the objects in question are real, and the person is wrong about them, or that there are no such objects. But this is exactly what we face with regard to mathematical objects as well! We have an amorphous informal practice, and a norm that people count as referring to whatever the most natural object is that best satisfies their methods of reasoning about these putative objects, provided there is one that matches suitably well.
There's no bright line about how wrong your various formal and informal beliefs about some putative object can be while you still still count as referring - for the same reason in math as in physics or history.
Hence, I don't try to draw one, and that's why I'm woolly on this issue, and why you should be to!
All we can say generally is: Mathematicians can posit new objects, and the more logically consistent their reasoning about these objects is, and the less their intuitions about consequences of reasoning about these objects lead to false conclusions about other things, the more likely it is that they will count as expressing largely true claims about some suitable piece of the mathematical universe.
p.s.
The other non-woolly alternative is to give a list: a mathematician counts as referring if they have x beliefs (which are true of the integers), y beliefs (true of the reals), z which are true of imaginary numbers, w for quaterinians, v for sets, k for arrows ... and thats all the mathematical objects that it is metaphysically possible to think about! But surely this is insane.
Showing posts with label thesis. Show all posts
Showing posts with label thesis. Show all posts
Thursday, March 11, 2010
Saturday, November 14, 2009
Practical Helpfulness: Why Care?
Readers of the last two posts may well be wondering why I'm going on so much about the "practical helpfulness" of mathematics.
One thing is, I wish I had a better name for it than "practical helpfulness", so maybe someone will suggest one :).
More seriously, I think the fact that our mathematical methods are (in effect) constantly making predictions about themselves, and other kinds of a priori reasoning - not to mention combining with our methods of observation to yield predictions that observation alone would not have yielded (see the computer example) has two important consequences.
Firstly, it shows that our reasoning about math is NOT the kind of thing you are likely to get just by making a series of arbitrary stipulations and sticking to them. All our different kinds of a priori reasoning (methods for counting abstract objects, logical inference, arithmetic, intuitive principles of number theory, set theoretic reasoning that has consequences for number theory) fit together in an incredibly intricate way. Each method of reasoning has myriad opportunities to yield consequences that would lead us to form false expectations about the results of applying some other method. And yet, this almost never happens!
Thus, there's a question about how we could have managed to get methods of armchair reasoning that fit together so beautifully. Some would posit a benevolent god, designing our minds to reason only in ways that are truth-preserving and hence coherent in this sense. But I think a process of free creativity to come up with new methods of a priori reasoning, plus Quinean/Millian revision when these new elements did raise false expectations, can do the job. This brings us to the second point.
Secondly, if we think about all these intended internal and external applications as forming part of our conception of which mathematical objects we mean when we talk about e.g. the numbers, then Qunian/Millian revision when applications go wrong will amount to a kind of reliable feedback mechanism, maintaining and improving the fit between what we say about "the numbers" and what's actually true of those-mathematical objects-whose-structure-mirrors-the-modal-facts-about-how-many-objects-there-are-when-there-are-n-Fs-and-m-(distinct)-Gs etc.
One thing is, I wish I had a better name for it than "practical helpfulness", so maybe someone will suggest one :).
More seriously, I think the fact that our mathematical methods are (in effect) constantly making predictions about themselves, and other kinds of a priori reasoning - not to mention combining with our methods of observation to yield predictions that observation alone would not have yielded (see the computer example) has two important consequences.
Firstly, it shows that our reasoning about math is NOT the kind of thing you are likely to get just by making a series of arbitrary stipulations and sticking to them. All our different kinds of a priori reasoning (methods for counting abstract objects, logical inference, arithmetic, intuitive principles of number theory, set theoretic reasoning that has consequences for number theory) fit together in an incredibly intricate way. Each method of reasoning has myriad opportunities to yield consequences that would lead us to form false expectations about the results of applying some other method. And yet, this almost never happens!
Thus, there's a question about how we could have managed to get methods of armchair reasoning that fit together so beautifully. Some would posit a benevolent god, designing our minds to reason only in ways that are truth-preserving and hence coherent in this sense. But I think a process of free creativity to come up with new methods of a priori reasoning, plus Quinean/Millian revision when these new elements did raise false expectations, can do the job. This brings us to the second point.
Secondly, if we think about all these intended internal and external applications as forming part of our conception of which mathematical objects we mean when we talk about e.g. the numbers, then Qunian/Millian revision when applications go wrong will amount to a kind of reliable feedback mechanism, maintaining and improving the fit between what we say about "the numbers" and what's actually true of those-mathematical objects-whose-structure-mirrors-the-modal-facts-about-how-many-objects-there-are-when-there-are-n-Fs-and-m-(distinct)-Gs etc.
How (Our Beliefs About) Math are Practically Helpful
All philosophers of math will agree that people do something they call "math", and that this activity is practically helpful, in a certain sense. This is often put pretty loosely by saying `Math helps us build bridges that stand up'. But I think we can say something much clearer than that. Here goes:
Our grasp of math (such as it is) has at least three aspects:
- We can follow proofs. You will accept certain kinds of transitions from one mathematical sentence to another, (or between mathematical sentences and non-mathematical ones) when these are suggested to you.
- We can come up with proofs. You have a certain probability of coming up with chains of inference like this on your own.
- Proofs can create expectations in us. Accepting certain sentences makes you disposed to react with surprise and dismay should you come to accept other sentences. e.g. if you accept "n is prime" you will react with surprise and dismay to a situation where you are also inclined to accept "n has p, q, and r as factors".
Now, the sense in which our mathematical practices are helpful is this:
First, our reasoning about math fits into our overall web of beliefs in such a way as to create additional expectations. Here's what I have in mind: Fix a situation. People in that situation who realize their dispositions to make/accept mathematical inferences arrive in a state where they will be surprised by more things than those in the same situation who don't.
For example, plonk a bunch of people down in front of a bowl of red and yellow lentils. Make each person count the red lentils and the yellow lentils. Now give them some tasty sandwiches and half an hour. Some of the people will add the two numbers. Others will just eat their sandwitches. Now, note that the people ho did the math have formed extra expectations, in the following sense. If we now have our subjects count the lentils all together, the people who did the sum will be surprised if they get anything but one particular number, whereas those who didn't do the math will only be surprised if they get anything outside of a certain given range.
Secondly, the extra expectations raised by doing math are very very often correct. When doing mathematical reasoning about your situation puts you in a state where (now) you'd be surprised if a certain observation/reasoning yields anything but P, applying this process tends to yield P. (This is especially true if we weight the satisfaction/dissatisfaction of strong expectations more heavily). Thus, composing a process of mathematical reasoning M with some other reasoning processes A yields nearly always yields correct expectations about the result of going through a different process B, if it yields any expectations at all.
And finally, this is (potentially) helpful, because it means not only do we acquire the disposition to be surprised if B yields something different, but any further inferences/actions which would get triggered by doing B happen immediately after doing A and M without having to wait for B to take place. For example, in the case from the previous post: if we imagine that all of our sample population have inductively associated counting 1567 lentils in total with having enough to make soup, the people who did the addition after counting the lentils separately, start cooking earlier than those who did something else instead.
To summarize:
Doing math is practically helpful in the sense that spending time doing math raises extra expectations (relative to spending that time eating sandwiches) about the results of certain other processes, and these expectations are generally correct. Thus, mathematical reasoning constitutes a reliable shortcut, leading us to take whatever actions would be triggered by going through some other process B without actually going through B.
NOTE: I don't mean to suggest that this is all there is to math, or that math is somehow *merely* instrumental. I'm just trying to concretely state some data about the successful "applications" of math, which I think everyone will agree to.
Our grasp of math (such as it is) has at least three aspects:
- We can follow proofs. You will accept certain kinds of transitions from one mathematical sentence to another, (or between mathematical sentences and non-mathematical ones) when these are suggested to you.
- We can come up with proofs. You have a certain probability of coming up with chains of inference like this on your own.
- Proofs can create expectations in us. Accepting certain sentences makes you disposed to react with surprise and dismay should you come to accept other sentences. e.g. if you accept "n is prime" you will react with surprise and dismay to a situation where you are also inclined to accept "n has p, q, and r as factors".
Now, the sense in which our mathematical practices are helpful is this:
First, our reasoning about math fits into our overall web of beliefs in such a way as to create additional expectations. Here's what I have in mind: Fix a situation. People in that situation who realize their dispositions to make/accept mathematical inferences arrive in a state where they will be surprised by more things than those in the same situation who don't.
For example, plonk a bunch of people down in front of a bowl of red and yellow lentils. Make each person count the red lentils and the yellow lentils. Now give them some tasty sandwiches and half an hour. Some of the people will add the two numbers. Others will just eat their sandwitches. Now, note that the people ho did the math have formed extra expectations, in the following sense. If we now have our subjects count the lentils all together, the people who did the sum will be surprised if they get anything but one particular number, whereas those who didn't do the math will only be surprised if they get anything outside of a certain given range.
Secondly, the extra expectations raised by doing math are very very often correct. When doing mathematical reasoning about your situation puts you in a state where (now) you'd be surprised if a certain observation/reasoning yields anything but P, applying this process tends to yield P. (This is especially true if we weight the satisfaction/dissatisfaction of strong expectations more heavily). Thus, composing a process of mathematical reasoning M with some other reasoning processes A yields nearly always yields correct expectations about the result of going through a different process B, if it yields any expectations at all.
And finally, this is (potentially) helpful, because it means not only do we acquire the disposition to be surprised if B yields something different, but any further inferences/actions which would get triggered by doing B happen immediately after doing A and M without having to wait for B to take place. For example, in the case from the previous post: if we imagine that all of our sample population have inductively associated counting 1567 lentils in total with having enough to make soup, the people who did the addition after counting the lentils separately, start cooking earlier than those who did something else instead.
To summarize:
Doing math is practically helpful in the sense that spending time doing math raises extra expectations (relative to spending that time eating sandwiches) about the results of certain other processes, and these expectations are generally correct. Thus, mathematical reasoning constitutes a reliable shortcut, leading us to take whatever actions would be triggered by going through some other process B without actually going through B.
NOTE: I don't mean to suggest that this is all there is to math, or that math is somehow *merely* instrumental. I'm just trying to concretely state some data about the successful "applications" of math, which I think everyone will agree to.
Thursday, October 22, 2009
Why I am not Carrie Jenkins
Carrie Jenkins' 2009 book Grounding Concepts: an Empirical Basis for Arithmetical Knowledge, proposes a theory that has a lot in common with my thesis project.
Both of us:
- want to give a naturalistic account of mathematical knowledge
- in particular, want to explain how humans can have managed get "good" combination of inference patterns that count as thinking true things about some domain of mathematical objects/having a coherent conception of what those objects must be like, rather than "bad", 'tonk' like patterns of reasoning.
-appeal to causal interactions with the world, to explain how we wind up with such combinations of inference dispositions.
BUT there are some important differences. Here's why (I claim) my view is better.
Jenkins' theory:
Jenkins winds up positing a whole bunch of controversial, and perhaps under-explained philosophical notions to account for how experience gives us good inference dispositions. She proposes that:
Experience has non-conceptual content which grounds our acquisition of concepts so as to help us form coherent ones. Then when we have a coherent concept of something like the numbers, we inspect it to see what what must be true of the numbers and reason correctly about them.
-The idea that there's non-conceptual content is a controversial point in philosophy of perception.
-The idea that experience can "ground" concept acquisition without playing a justificatory role in the conclusions drawn is not at all clear. What is this not-justificatory, but presumably not just causal relationship of grounding supposed to be? (Kant's notion of a posteriori concepts seems relevant, but that's none-too clear either).
-Finally, what is concept inspection, (presumably you don't literally visit the 3rd realm and see the concepts) and how is it supposed to work? Jenkins admits that this is an open question for further research.
My theory:
In contrast, my view gives a naturalistic account of mathematical knowledge that doesn't need any of this controversial philosophical machinery. I propose that:
Note that we don't need to posit any mysterious faculty of concept-inspection, or any controversial non-conceptual experience. All I appeal to is perfectly ordinary processes. People go from one sentence to another in a way that feels natural them (whether or not they are so fortunate as to be working with coherent concepts like +, rather than doing reasoning like Frege did about extensions) And when this natural-feeling reasoning leads to a surprise, they revise.
[Well, perhaps I'm also committed to the view that innate stuff about the brain makes some ways of revising more likely than others, and certain initial inference-dispositions more likely than others, in a way that doesn't make us always prefer theories that are totally hopeless at matching future experience. But you already need something like this even to explain how rats can learn that pushing a lever releases food, so I don't think this is very controversial.]
Both of us:
- want to give a naturalistic account of mathematical knowledge
- in particular, want to explain how humans can have managed get "good" combination of inference patterns that count as thinking true things about some domain of mathematical objects/having a coherent conception of what those objects must be like, rather than "bad", 'tonk' like patterns of reasoning.
-appeal to causal interactions with the world, to explain how we wind up with such combinations of inference dispositions.
BUT there are some important differences. Here's why (I claim) my view is better.
Jenkins' theory:
Jenkins winds up positing a whole bunch of controversial, and perhaps under-explained philosophical notions to account for how experience gives us good inference dispositions. She proposes that:
Experience has non-conceptual content which grounds our acquisition of concepts so as to help us form coherent ones. Then when we have a coherent concept of something like the numbers, we inspect it to see what what must be true of the numbers and reason correctly about them.
-The idea that there's non-conceptual content is a controversial point in philosophy of perception.
-The idea that experience can "ground" concept acquisition without playing a justificatory role in the conclusions drawn is not at all clear. What is this not-justificatory, but presumably not just causal relationship of grounding supposed to be? (Kant's notion of a posteriori concepts seems relevant, but that's none-too clear either).
-Finally, what is concept inspection, (presumably you don't literally visit the 3rd realm and see the concepts) and how is it supposed to work? Jenkins admits that this is an open question for further research.
My theory:
In contrast, my view gives a naturalistic account of mathematical knowledge that doesn't need any of this controversial philosophical machinery. I propose that:
People are disposed to go from seeing things, to saying things, to being surprised if we then see other things, in certain ways. When these inference dispositions lead us to be surprised, we tend to modify them.
Thus, it's not surprising that we should have wound up with the kind of combination of arithmetical inference dispositions + observational practices + ways of applying arithmetic to the actual world, which makes our expected applications of arithmetic work out.
For example: insofar as we had a conceptions of the numbers which included the expectation that facts about sums should mirror logical facts in a certain way, it's not surprising that we would up also believing the kinds of other claims about sums, which make the intended applications to logic work out (e.g. believing 2+2=4 not 2+2=5).
Note that we don't need to posit any mysterious faculty of concept-inspection, or any controversial non-conceptual experience. All I appeal to is perfectly ordinary processes. People go from one sentence to another in a way that feels natural them (whether or not they are so fortunate as to be working with coherent concepts like +, rather than doing reasoning like Frege did about extensions) And when this natural-feeling reasoning leads to a surprise, they revise.
[Well, perhaps I'm also committed to the view that innate stuff about the brain makes some ways of revising more likely than others, and certain initial inference-dispositions more likely than others, in a way that doesn't make us always prefer theories that are totally hopeless at matching future experience. But you already need something like this even to explain how rats can learn that pushing a lever releases food, so I don't think this is very controversial.]
Tuesday, October 20, 2009
Mathematical Concepts and Learning From Experience
I've been reading Susan Carey's new book on the development of concepts, which features a lot of interesting stuff about the development of children's reasoning about number. The last two chapters are philosophical though, and bring up an important point, which it had not occurred to me needed to be stressed:
Learning from experience need not take the form of someone explicitly forming a hypothesis, and then letting experience falsify it/doing induction to conclude the hypothesis is true.
If this were all that experience could do, it would be hopeless to appeal to it to help explain how we could get mathematical knowledge. For, plausibly, you only count as having the concept of number, once you are willing to make certain kinds of applications of facts about the numbers, reason about the numbers largely correctly etc. So, by the time that experience could falsify hypotheses containing the mature concept of number, you would already have to have lots of mathematical knowledge.
Instead, experience helps us correct and hone our mathematical reasoning all through the process of "developing a concept". How can this be?
Well, firstly, think about the way students are normally introduced to the concept of set. No one makes a hypothesis that there are sets, nor do math profs attempt to define sets in other terms. Rather the professor just demonstrates various ways of reasoning about sets, ways of using these claims to solve other mathematical problems etc. and gets the students to practice. Given this, the student's usage and intuitions conform more and more to standard claims about the sets, and eventually they count as having the concept of set.
I propose (and I think Carey would agree) that the original development of many concepts in mathematics works similarly, only with trial and experience playing the role of the teacher.
You start out not having the concept, and try various usages. Here, however, rather than having a professor to imitate, you just have your general creativity/trial and error/analogical reasoning to suggest ways of reasoning about "the X"s and then an ability to check whatever kinds of consequences and applications you expect at a given time. Often this kind of creative trying and analogical reasoning will turn out to fail in some way, such as leading to contrdiction, or underspecifying something important. But then you can correct it. Inconsistent reasoning about limits in the 19th century and sets in the early 20th would be examples of the former. And the kind of process of refinment of the notion of polygon in Lakotosh's Proofs and Refutations would be an example of the latter.
We try out various patterns of reasoning about the world (e.g. calling certain things Xs, trying to apply the analogue of good reasoning about one domain to another) -with perhaps a nudge from brain structures subject to evolution effecting which patterns we are likely to try- and experience corrects these inference patterns until they cohere enough that we count as genuinely having some new concept. And note that no conscious scientific reasoning must be assumed to start this process, all we need some disposition to go from seeing things to making noises to doing things, together with a playful/random/creative inclination to try extending those dispositions in various ways!
p.s. I haven't emphasized this point the past, because I think questions like 'when exactly does someone start having the concept of X?', don't generally cut psychology or metaphysics at their joints. I mean: when exactly did people start having the modern conception of atom? The interesting facts are surely facts about when people started accepting this or that idea atoms "atoms", or reasoning about "atoms" in this or that way. Coming up with a decision about exactly what amount of agreement with us is necessary for people to count as having the same concept is a matter of arbitrary boundary setting.
But I realize now that ignoring the whole issue of concepts can be confusing. So let me just say:
When I say mathematical knowledge is a joint product of mathematically shapted problems in nature, correction by experience, the wideness of the realm of mathematical facts and the relationship between use and meaning, "Correction by experience" doesn't just mean what happens when hypotheses consciously proposed by people who already count as having all the right mathematical concepts get refuted. Rather, "correction by experience" includes what happens when you are inclined to reason some way, you get to an unexpected conclusion, and then subsequently become disposed to draw slightly different inferences/feel less confident when engaging in some of the processes that lead you there. You might or might not count as revising some hypothesis, phrased in terms of fully coherent concepts, when you do this.
p.p.s. The idea that experience helps us form coherent mathematical concepts, (while not figuring in the justification of our beliefs) is also a central theme in Carrie Jenkins' 2009 Grounding Concepts: an empirical basis for arithmetical knowledge.
Learning from experience need not take the form of someone explicitly forming a hypothesis, and then letting experience falsify it/doing induction to conclude the hypothesis is true.
If this were all that experience could do, it would be hopeless to appeal to it to help explain how we could get mathematical knowledge. For, plausibly, you only count as having the concept of number, once you are willing to make certain kinds of applications of facts about the numbers, reason about the numbers largely correctly etc. So, by the time that experience could falsify hypotheses containing the mature concept of number, you would already have to have lots of mathematical knowledge.
Instead, experience helps us correct and hone our mathematical reasoning all through the process of "developing a concept". How can this be?
Well, firstly, think about the way students are normally introduced to the concept of set. No one makes a hypothesis that there are sets, nor do math profs attempt to define sets in other terms. Rather the professor just demonstrates various ways of reasoning about sets, ways of using these claims to solve other mathematical problems etc. and gets the students to practice. Given this, the student's usage and intuitions conform more and more to standard claims about the sets, and eventually they count as having the concept of set.
I propose (and I think Carey would agree) that the original development of many concepts in mathematics works similarly, only with trial and experience playing the role of the teacher.
You start out not having the concept, and try various usages. Here, however, rather than having a professor to imitate, you just have your general creativity/trial and error/analogical reasoning to suggest ways of reasoning about "the X"s and then an ability to check whatever kinds of consequences and applications you expect at a given time. Often this kind of creative trying and analogical reasoning will turn out to fail in some way, such as leading to contrdiction, or underspecifying something important. But then you can correct it. Inconsistent reasoning about limits in the 19th century and sets in the early 20th would be examples of the former. And the kind of process of refinment of the notion of polygon in Lakotosh's Proofs and Refutations would be an example of the latter.
We try out various patterns of reasoning about the world (e.g. calling certain things Xs, trying to apply the analogue of good reasoning about one domain to another) -with perhaps a nudge from brain structures subject to evolution effecting which patterns we are likely to try- and experience corrects these inference patterns until they cohere enough that we count as genuinely having some new concept. And note that no conscious scientific reasoning must be assumed to start this process, all we need some disposition to go from seeing things to making noises to doing things, together with a playful/random/creative inclination to try extending those dispositions in various ways!
p.s. I haven't emphasized this point the past, because I think questions like 'when exactly does someone start having the concept of X?', don't generally cut psychology or metaphysics at their joints. I mean: when exactly did people start having the modern conception of atom? The interesting facts are surely facts about when people started accepting this or that idea atoms "atoms", or reasoning about "atoms" in this or that way. Coming up with a decision about exactly what amount of agreement with us is necessary for people to count as having the same concept is a matter of arbitrary boundary setting.
But I realize now that ignoring the whole issue of concepts can be confusing. So let me just say:
When I say mathematical knowledge is a joint product of mathematically shapted problems in nature, correction by experience, the wideness of the realm of mathematical facts and the relationship between use and meaning, "Correction by experience" doesn't just mean what happens when hypotheses consciously proposed by people who already count as having all the right mathematical concepts get refuted. Rather, "correction by experience" includes what happens when you are inclined to reason some way, you get to an unexpected conclusion, and then subsequently become disposed to draw slightly different inferences/feel less confident when engaging in some of the processes that lead you there. You might or might not count as revising some hypothesis, phrased in terms of fully coherent concepts, when you do this.
p.p.s. The idea that experience helps us form coherent mathematical concepts, (while not figuring in the justification of our beliefs) is also a central theme in Carrie Jenkins' 2009 Grounding Concepts: an empirical basis for arithmetical knowledge.
Friday, October 16, 2009
Empirical adequacy and truth in mathematics
The current weakest link in my thesis is this (IMO): how to connect merely having beliefs about mathematics that help us solve problems, and yield correct applications to concrete situations to having beliefs about mathematics that are reasonably reliable.
Couldn't totally false mathematical theories nonetheless be perfectly correct with regard to their concrete applications?
Also, even if our beliefs would indeed perfectly accurately describe some concrete objects, how can we count as refering to these objects, given that we have no causal contact with them?
My current best answer is this:
Think of human mathematicians as observing certain regularities (e.g. whenever there are 2 male rhymes and 2 female rhymes in a poem there are at least 4 rhymes all together), and then positing mathematical objects "the numbers" whose relationship to one another is supposed to echo these logical facts.
(This is a reasonable comparison because what we actually do is like this, in that we happily make inferences from a proof that "a+b=c" to the expectation that when there are a male rhymes and b female rhymes there are c rhymes all together. We behave as though we know there's this relationship between the numbers and logical facts, so it's not too much of a stretch to compare us to people who actually consciously posit that there is some collection of abstract objects whose features echo the relevant logical facts in this way.)
Now either there are abstract objects or not.
If there aren't abstracta (as the fictionalist thinks), the fact that mathematicians only care about structures makes it plausible to think of them as talking about the fiction in which there are such objects.
Thus, our abstract-object positing mathematicians will count as speaking about the fiction in which there are objects whose features echo the logical facts about addition in the intended way. They will also count as knowing lots of things about what's true in this fiction.
Also, note that insofar as these mathematicians propose new things that "intuitively must be true of the numbers" their intuitions will be disciplined and corrected by the fact that the relevant applications are expected, so there's a systematic force which will keep some degree of match between their claims about this fiction and what's actually true in this fiction.
If there are abstracta, then there are abstract objects with many different structures, in particular structures corresponding to every consistent first order theory (note this is even true if the only mathematical objects there are are sets! the completeness theorem guarantees that there are models of every such theory within the heirarchy sets). So there will be some collection of objects whose features match those expected by our positers (note that the positers only really care about structural features of "the numbers" not whether they are fundamental mathematical objects etc).
Now, how can our positers count as referring to some such objects? Well, as noted above, we have systematic mechanisms of belief revision which kick back and insure that their claims about the numbers must match with logical facts, and hence with the real facts about these collections of suitable abstracta. Just as looking at llamas helps ensure that certain kinds of false beliefs about llamas which you might form would be corrected, applying arithmetic insures that certain kinds of false general beliefs you might form about the numbers would be corrected (those which lead to false consequences about sums).
Thus, we have a situation where people not only have many beliefs that are true about the numbers, and the tendency to make many truth-preserving inferences, but also where these beliefs have a certain amount of modal stability (many kinds of false beliefs would tend to be corrected). Even Fodor thinks that making correct inferences with or is sufficient to allow or to make the right kind of contribution to the truth value of your sentences, so why should the same thing not apply to talk about numbers, given that we now have not only many good inferences but this kind of mechanism of correction which improves the fit between our beliefs about the numbers and the numbers?
You might still worry that there will be so many mathematical objects which have all the features which we expect the numbers to have - how can we count as referring to any one such structure, given that our use fits all of them equally well? And if we don't uniquely pick out a structure, how can our words count as refering and being meaningful? But note that to the extent that our use of the word "the numbers" is somehow ambiguous between e.g. different collections of sets, our use of the word "human bodies" would seem to be equally ambiguous between e.g. open vs. closed sets of spacetime points. So either meaningfully talking about objects is compatible with some amount of ambiguity, or the above kind of reasoning doesn't suffice to establish ambiguity.
Couldn't totally false mathematical theories nonetheless be perfectly correct with regard to their concrete applications?
Also, even if our beliefs would indeed perfectly accurately describe some concrete objects, how can we count as refering to these objects, given that we have no causal contact with them?
My current best answer is this:
Think of human mathematicians as observing certain regularities (e.g. whenever there are 2 male rhymes and 2 female rhymes in a poem there are at least 4 rhymes all together), and then positing mathematical objects "the numbers" whose relationship to one another is supposed to echo these logical facts.
(This is a reasonable comparison because what we actually do is like this, in that we happily make inferences from a proof that "a+b=c" to the expectation that when there are a male rhymes and b female rhymes there are c rhymes all together. We behave as though we know there's this relationship between the numbers and logical facts, so it's not too much of a stretch to compare us to people who actually consciously posit that there is some collection of abstract objects whose features echo the relevant logical facts in this way.)
Now either there are abstract objects or not.
If there aren't abstracta (as the fictionalist thinks), the fact that mathematicians only care about structures makes it plausible to think of them as talking about the fiction in which there are such objects.
Thus, our abstract-object positing mathematicians will count as speaking about the fiction in which there are objects whose features echo the logical facts about addition in the intended way. They will also count as knowing lots of things about what's true in this fiction.
Also, note that insofar as these mathematicians propose new things that "intuitively must be true of the numbers" their intuitions will be disciplined and corrected by the fact that the relevant applications are expected, so there's a systematic force which will keep some degree of match between their claims about this fiction and what's actually true in this fiction.
If there are abstracta, then there are abstract objects with many different structures, in particular structures corresponding to every consistent first order theory (note this is even true if the only mathematical objects there are are sets! the completeness theorem guarantees that there are models of every such theory within the heirarchy sets). So there will be some collection of objects whose features match those expected by our positers (note that the positers only really care about structural features of "the numbers" not whether they are fundamental mathematical objects etc).
Now, how can our positers count as referring to some such objects? Well, as noted above, we have systematic mechanisms of belief revision which kick back and insure that their claims about the numbers must match with logical facts, and hence with the real facts about these collections of suitable abstracta. Just as looking at llamas helps ensure that certain kinds of false beliefs about llamas which you might form would be corrected, applying arithmetic insures that certain kinds of false general beliefs you might form about the numbers would be corrected (those which lead to false consequences about sums).
Thus, we have a situation where people not only have many beliefs that are true about the numbers, and the tendency to make many truth-preserving inferences, but also where these beliefs have a certain amount of modal stability (many kinds of false beliefs would tend to be corrected). Even Fodor thinks that making correct inferences with or is sufficient to allow or to make the right kind of contribution to the truth value of your sentences, so why should the same thing not apply to talk about numbers, given that we now have not only many good inferences but this kind of mechanism of correction which improves the fit between our beliefs about the numbers and the numbers?
You might still worry that there will be so many mathematical objects which have all the features which we expect the numbers to have - how can we count as referring to any one such structure, given that our use fits all of them equally well? And if we don't uniquely pick out a structure, how can our words count as refering and being meaningful? But note that to the extent that our use of the word "the numbers" is somehow ambiguous between e.g. different collections of sets, our use of the word "human bodies" would seem to be equally ambiguous between e.g. open vs. closed sets of spacetime points. So either meaningfully talking about objects is compatible with some amount of ambiguity, or the above kind of reasoning doesn't suffice to establish ambiguity.
Friday, October 9, 2009
kindof a joke: an ad for my solution to the access problem
After a really helpful but sad conversation with KY, I realized that I really haven't done enough to make clear to casual readers just what my thesis project (and paper on the access problem) are trying to do.
This lead to me making the following little advertisement.
addendum:
I just heard that Poincare thought that we evolve and/or prune our beliefs to believe what's advantagious, not what's true. In contrast, my thesis suggests that in evolving/pruning our beliefs to believe what's advantagious, we wind up believing (mostly) the truth about some suitable aspect of objective mathematical reality - but this doesn't make our mathematical beliefs a posteriori!
This lead to me making the following little advertisement.
addendum:
I just heard that Poincare thought that we evolve and/or prune our beliefs to believe what's advantagious, not what's true. In contrast, my thesis suggests that in evolving/pruning our beliefs to believe what's advantagious, we wind up believing (mostly) the truth about some suitable aspect of objective mathematical reality - but this doesn't make our mathematical beliefs a posteriori!
Thursday, October 1, 2009
Stipulation and Easy Mathematical Knowledge
As noted before, I think we get (mature, human) mathematical knowledge by benefiting from caual interactions with the world that lead us to find "coherent" combinations of mathematical statements obvious, and that our acceptance of these coherent stipulations helps determine the meaning of our words in such a way that these stipulations express truths in our language.
But this suggests a question. (Or at least, related views suggested a question to Shapiro and Ebert) Suppose someone accepts ZF and just guesses some elaborate provable truth T, and then stipulates {ZF+T}. Do they count as knowing that T? Doesnt my view commit me to thinking that they do?
The combination of ZF+T is indeed coherent, so I think that people who naturally found T just as obvious as people with mainstream mathematical intuitions find ZF would count as expressing mathematical truths, and indeed knowing that T. (see my paper the Doctoroids for more on this, though I wrote it before seeing the Shapiro).
But what about someone who feels uncertain about whether T, but tries to just stipulate it?
In general, I think, such a person won't count as having knowledge, because they are taking what is (relative to their current state of knowledge) an excessive epistemic risk - and hence they lack justification for their true beliefs. If their current mathematical faculties and other experiences do not give sufficient reason think that adding T to their beliefs would lead to a logically consistent system, they also lack sufficient reason to think that adding T would lead to a system of axioms that correctly describe some realm of mathematical reality. Thus, they are being epistemically irresponcible in adding this axiom.
However, if their current mathematical and other reasoning does suggest (though not prove) that adding T would be consistent, they can be justified in adding T as an axiom (although they may not be justified in assuming that once they have e.g. stipulated the axiom of choice to be true, they are talking about the same mathematical structure as they originally were).
But this suggests a question. (Or at least, related views suggested a question to Shapiro and Ebert) Suppose someone accepts ZF and just guesses some elaborate provable truth T, and then stipulates {ZF+T}. Do they count as knowing that T? Doesnt my view commit me to thinking that they do?
The combination of ZF+T is indeed coherent, so I think that people who naturally found T just as obvious as people with mainstream mathematical intuitions find ZF would count as expressing mathematical truths, and indeed knowing that T. (see my paper the Doctoroids for more on this, though I wrote it before seeing the Shapiro).
But what about someone who feels uncertain about whether T, but tries to just stipulate it?
In general, I think, such a person won't count as having knowledge, because they are taking what is (relative to their current state of knowledge) an excessive epistemic risk - and hence they lack justification for their true beliefs. If their current mathematical faculties and other experiences do not give sufficient reason think that adding T to their beliefs would lead to a logically consistent system, they also lack sufficient reason to think that adding T would lead to a system of axioms that correctly describe some realm of mathematical reality. Thus, they are being epistemically irresponcible in adding this axiom.
However, if their current mathematical and other reasoning does suggest (though not prove) that adding T would be consistent, they can be justified in adding T as an axiom (although they may not be justified in assuming that once they have e.g. stipulated the axiom of choice to be true, they are talking about the same mathematical structure as they originally were).
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