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)
Showing posts with label charles parsons. Show all posts
Showing posts with label charles parsons. Show all posts
Wednesday, April 14, 2010
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.
Subscribe to:
Comments (Atom)