Tuesday, December 7, 2010

Here are two ways to think about mathematics (among possibly others). (1) As a discipline: an institution with some degree of political organization, a history, etc. (2) As an intellectual virtue: a habit of the cognitive faculty which consists in the enhanced potential to carry out certain cognitive acts. (I say “enhanced potential” because a faculty in itself is already a potential.)

It is in sense (2) that mathematics can be said to have objects. A cognitive act in general is a representation, and a representation, if at all successful (if not, e.g., self-contradictory) has some at least possible being as its object. If there is such a thing as mathematics at all in sense (2), then there must be something which characterizes the objects of just those acts for which it is the enhanced potential.

(Note I’m using “object” in the correct, relative sense: an object is the object of something. According to idealists of various stripes, every being, or at least every contingent being, must be the potential object of some cognitive faculty. But for just that reason it’s a bad idea, especially if, like me, you’re an idealist of some stripe or other, to use “object” as a synonym for “being.”)

It would be wrong to define the intellectual virtue (2) solely in terms of the institution (1). We should want to say that every normal human, past some fairly young age, possesses (knows) at least some mathematics. On the other hand, the people who play the leading role in institution (1) — that is, professional mathematicians — owe their position, on the whole, to being very good at something. There may be some disciplines where success depends, not on any cognitive ability, but rather, for example, on political or social skills. But if you know any even moderately good mathematicians you will know for sure that their skills don’t mostly lie in that direction. We should want the intellectual virtue (2), which we all share to some extent, to be the one successful mathematicians must have to an unusual extent.

But this puts constraints on the objects of mathematics. For example: although some may think that mathematicians are particularly good at counting to very high numbers, that is not actually the case. Hence, the cognitive act of representing a given (high) finite cardinality is not as such an act of the virtue (2). It follows that a given natural number simply as such (unless perhaps it is very small) is not an object of mathematics.

13 comments:

  1. You: "...though some may think that mathematicians are particularly good at counting to very high numbers, that is not actually the case. Hence, the cognitive act of representing a given (high) finite cardinality is not as such an act of the virtue (2). It follows that a given natural number simply as such (unless perhaps it is very small) is not an object of mathematics."
    Me: Sorry, but this is just a non-sequitur. Just because Mat cannot *count*, it does not follow that he cannot *represent* a given (high) finite cardinality. He can: "4567329", for example is such a representation and it is one that can be cognized. Therefore, cognitively vituous Mat can represent any natural number and these are te proper objects of his discipline.

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  2. Bruno: The point is not that a mathematician can't represent a large number, but rather that a mathematician as such isn't better than anyone else at doing so. (I take it that the act of representing a cardinality is basically an act of counting, or at least is essentially related to that. But even if that's wrong, I think the main point --- that the mathematician is not better at such acts of representation --- remains correct.)

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  3. What about the popular notion that mathematics is a structure or set of structures? That doesn't seem to fit into (1) or (2).

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  4. That *mathematics* is a structure, or that the objects of mathematics, as such, are structures?

    If someone does say the former I take that to be a somewhat awkward way of expressing the latter.


    Maybe I should add: I think structuralism must be either a form of realism or a form of nominalism about the objects of mathematics. It might mean that some things are mathematical qua things, i.e. that mathematics has real objects, and that those things (those real objects) are structures. Or it might mean that no thing is as such the object of mathematics, but that things are rather objects of mathematics qua structured. (By "thing" I mean the transcendental, "res": that which has realitas.) I intended what I say here as neutral between realism and nominalism generally speaking (about the objects of mathematics).

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  5. So, the important question for you is: what are mathematicians especially good at?

    Suppose that what they are specially good at is representing abstract structures (including numbers) in multiple ways.

    Wouldn't it follow that the objects of maths are abstract structures (including numbers)?

    I think its at least plausible that this is what good mathematicians are especially good at: not representing big numbers, but reoresenting things in lots of ways.

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  6. wringe: For sure mathematicians are good at representing something in lots of ways, but that's still kind of vague.

    For sure also numbers in some way are objects of mathematics. The question is whether a natural number *as such* is an object of mathematics, and I'm arguing that it's not (although, say, the natural numbers as a whole — the set of natural numbers, I want to say, but maybe that's already saying too much — may be such an object).

    However, your claim might be that a good mathematician is good even at representing a particular number, say 4567329, in many different ways. I'm not sure I believe that. But it does raise a larger issue: maybe I've described an intellectual virtue too narrowly, as if there were only one type of excellence possible here. That sounds plausible, I need to think about it.

    "Abstract" I don't know what to do with, in general. Can say more about what you mean by it in this context?

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  7. "There is no unique set of objects that are the numbers. Number theory is the elaboration of the properties of _all_ structures of the order type of the numbers. The number words do not have single referents. Furthermore, the reason identification of numbers with objects works wholesale but fails utterly object by object is the fact that the theory is elaborating an abstract structure and not the properties of independent individuals, any one of which could be characterized without reference to its relations to the rest." Benacerraf, What Numbers Could Not Be, in the Benacerraf & Putnam volume pp. 291-292.

    I don't necessarily agree with this -- just because the "numbers are objects" view fails object by object doesn't mean we must jump to "abstract structures". But Benacerraf's essay has been pretty influential, so I wondered whether you disagree with his view?

    I think you and Benacerraf may agree that a natural number is not an object, but disagree about why.

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  8. Whoa, I didn't say "a natural number is not an object": I said "a natural number is not, as such, an object *of mathematics*."

    I believe Benacerraf is using "object" absolutely, as a synonym for "thing." But as I explained in the post, I'm not using it that way. Of course a natural number is the object of some cognitive act. That is completely consistent with nominalism about numbers: that is, the view that there is no such thing (res) as a number (in fact, it's needed to make any sense of nominalism).

    What I'm claiming is that the cognitive act(s) in question (the act(s) of representing that number) is/are not act(s) of the cognitive virtue, mathematics (in sense (2)). I don't think Benacerraf is addressing that question at all.

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  9. In case it's not obvious: I'm trying to get to a point, so to speak, behind both the usual ontological/epistemological puzzles of Analytic philosophy of mathematics and the Mark Steiner problem about applicability, from which I can see both as distinct but related.

    Also, a point far enough back, so to speak, that contemporary philosophy of mathematics can be discussed together with Natorp, Hegel, Kant, and pre-Kantian thought about mathematics (and/or about quantity and shape).

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  10. I'm a bit confused, but I think clearing up the confusion may clarify your point for me. Above you say that "Of course a natural number is the object of some cognitive act. That is completely consistent with nominalism about numbers: that is, the view that there is no such thing (res) as a number".

    But in the post, you say "A cognitive act in general is a representation, and a representation, if at all successful (if not, e.g., self-contradictory) has some at least possible being as its object." Are you using "being" in some other sense here? Or is "at least possible" the significant modifier here? Is the idea that natural numbers are "at least possible" beings, and in that sense they can be the objects of cognitive acts?

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  11. I'm very sympathetic to the overall argument here, incidentally, I just wonder about the upshot. It gets back to wringe's point: what, then, are the proper objects of maths?

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  12. Lydia: A representation must have a possible being as its object, and every being is a res, but a representation needn't determine the being which is its object via a determination which is predicated of that being in quid, that is, by a determination of that being as a res.

    See Thomas's explanation of the meaning of res as a convertible transcendental:

    For there is not found anything said affirmatively and absolutely which can be accepted in every being, except for its essence, according to which it is said to be. And this is how the name "thing" is imposed, which (according to Avicenna in the beginning of [his] Metaphysics) differs in this [way] from "being": that "being" is taken from the act of being, but the name "thing" expresses quiddity or essence of [a] being. (De ver. 1.1 c.)

    The object of a representation is a thing only if the representation represents some being via its quiddity.

    For example: suppose you think that all beings are bodies. You could say: to represent a number is to represent a being as in a certain way divisible into (spatial) parts. Call these potential parts "units." Then you could say: to represent a being as a number is to represent it as in a certain way a mereological sum
    of units.

    Then since no body is essentially a sum of units in one way as opposed to another, no being is essentially a number: a name like "two" is never predicated in quid. There are beings correctly represented as two (the cognitive act of representing two has possible and/or actual beings as its object), but there is no being for which the answer to the question, quid est, is (in whole or in part) "two." (This is called "nominalism" because, on this view, although "two" functions grammatically as a name, it isn't the name of anything --- of any thing. It's a "pure name.")

    (As I understand it, this is equivalent to what Quine would call nominalism about numbers — roughly, a view on which we do not need numbers as instances in inferences of universal instatiation, i.e. on which they are not "values of a bound variable" — given, and only given, various presuppositions peculiar to Quine.)

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  13. Ah, I see. I have neglected my Aquinas studies of late, so the precise sense of his nominalism wasn't immediately evident!

    If you're still interested in discussing this: Now that you've explained it, I think that nominalism of the sort you describe and structuralism of the sort I've described are answering two different questions. For instance, as I (somewhat vaguely) remember his view, Shapiro talks about structuralism in the sense that numbers are **whatever** structures there are that can be ordered serially, defined by recursion, and support proofs by induction. As I understand it, this way of thinking about numeric series goes back to Hilbert (and Bernays) and then Benacerraf. This kind of view makes the assumption of what Hilbert and Bernays called "existential axiomatics": that numerical objects exist and are representable, and that the processes of ordering, recursion, etc. are executable on them. Wilfried Sieg has written an excellent new paper on this.

    But the nominalism you describe is not answering the question of how we can demonstrate that numbers exist, as objects that can be ordered, defined by recursion etc. Instead, it's answering the question of what sort of beings individual numbers could be: are they sums of units, or are they names for beings, and so forth.

    There is a possible point of commonality, though. Hilbert is famous for arguing that Frege's pocket watch can be a geometrical point, or that in principle any set of objects -- {pepper shaker, Iphone, cat} -- can be a numeric series if it has the right ordering properties. This seems to possibly agree with the view you describe, it's just that Hilbert's view, anyway, goes further than nominalism in giving numeric series a *kind* of (assumed) existence.

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