I'm currently preparing the final version of my book manuscipt and being puzzled anew by some passages in Stewart Shapiro's book Philosophy of Mathematics: Structure and Ontology. In particular, I've never seen - and still am not seeing - how to reconcile these claims:
(1) [A] structure is ... determined ... by the relations among the places. ... [T]he correct use of the language determines what the relations are. (p. 137, emphases in the original)
(2) Through successful language use, we structure the objective subject matter. (p. 137)
(3) [T]he way the universe is divided into structures and objects - of all kinds - depends on our linguistic resources. (p. 161)
with this one:
(4) The natural-number structure has objective existence and facts about it are not of our making. (p. 137)
If anyone knows of a charitable interpretation, I would be very interested to hear it.
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7 comments:
I agree that there is something odd going on here. In fact, it seems that the problems here are exactly parallel to those encountered in some discussions of externalism in the philosophy and mind in combination with realism. By this I mean the following. The externalist holds that at least some of the aspects of the subject's mental states are determined by external factors. so what is going on in the subject is determined by what's outside ("objective factors", if you wish.) But then some people want to combine this with some version of antirealism, holding that the world (at least in some of its aspects) is determined by the workings of the mind. Here, we should be worried - could we really have it both ways? (Putnam's Reason Truth and History tries to pull of this combination, but he not the only one doing this.) It seems that Shapiro is doing the very same thing.
In the externalism-realism case, we could at least try to solve the conflict by holding that the mind determines some aspects of the world, and in turn is determined by some other aspects, but it seems that this option is not available for Shapiro. This is not very helpful, but I am just registering my agreement with your bafflement - I have had the same problems understanding the view.
(1) [A] structure is ... determined ... by the relations among the places. ... [T]he correct use of the language determines what the relations are. (p. 137, emphases in the original)
(2) Through successful language use, we structure the objective subject matter. (p. 137)
(3) [T]he way the universe is divided into structures and objects - of all kinds - depends on our linguistic resources. (p. 161)
with this one:
(4) The natural-number structure has objective existence and facts about it are not of our making. (p. 137)
Some possibilities: (a) there are constraints on what counts as correct language use that are independent of the rules that we make; (b) once the rules are fixed, we, as individuals, cannot change them; (c) objectivity is thought of as stemming from the satisfaction of a counterfactual test, where this is not inconsistent with the absence of a different kind of mind independence (as you've discussed elsewhere); (d) (This one from Luca Incurvati, sitting next to me in the Grad centre) Shapiro means that what is structured is objective, though the structure itself is not, and he just expresses this pretty badly.
But I have little confidence in any of these.
Hi Carrie,
some suggestions:
(1) Maybe if you appeal to the idea of "metaphysical serendipity" (I think I once used that term in a draft of my PhD thesis but somehow not in the final version.....) you can "somehow" explain the compatibility. Roughly the idea is, structuring, etc. of the world is done by language use and as a matter of metaphysical luck the world actually conforms to the way we structure it. So the facts and the number-structure do exist mind-independently while at the same time it is true that the 'way the universe is divided' is a matter of our use of language.
Also note that there are some potentially weak readings of (2) and (3):
(2) Stewart uses the term '"successful" language use'. If successful means that the use has to be true to the facts, then (2) is not very involving. And maybe relieve the tension with (4).
(3) Also, (3) merely say that "the way the universe is divided _depends_ on linguistic resources." Again, there is a near trivial reading of it. Namely, it depends on our language how we describe the universe. Surely that seems true. So, I'm not sure what, in (3) the dependency relation amounts to and furthermore, what 'the way the universe is divided' really is supposed to involve. It might just mean, as suggested above, "giving a description of the world"...
Anyway, maybe that is too charitable but at least some thought on that
Hope all is well
Philip
Hi
It is indeed very puzzling, the only reading I can think of (though I'm not sure it is coherent..and in any case, it is certainly not very charitable)is taking (4), once he defined structures as language-dependent,as stating that natural-numbers are a kind of "special" structure.
That even though it is a structure, and as such it depends on our correct use of language, that is not the end of the story. That there is an "extra"-objective element (as it is said in a previous comment, it might be read as saying that structures are not independent but what is structured is)
This objective element will prevent us from falling into anti-realism about numbers and, at the same time, accepting the definition of structure given, it might be easier to explain mathematical knowledge and avoid problems with indeterminism.
Now, I have no clue how this extra objective element of natural numbers could be understood (apart from simply adopting a traditional -and kind of radical- platonist view, and in this case, I don't see what would be the use of structures)
So really, even though I cannot think of a more charitable reading, I don't think he had this one in mind. I read the book long time ago..I guess I should re-read it!
maria
*by the way, it is the first time I send a comment, very nice blog!
first, make Shapiro provide a necessary argument for the law of contradiction itself, or the law of the excluded middle. Or you can have a go at it. Then explain how verbs are translated into magick logic language.
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