I should be reading about Aristotle because I have to write a paper on Metaphysics Z this week. Instead, I’ve been sucked into the vortex of math blogs and in particular this useful post on categorification, from which I learned quite a bit. I haven’t got much to say about the gist of the post, except to note with interest this passage in which it seems that an eliminativist reading of Hume’s Principle is suggested:
“What? Madness!”, you say, and it’s easy to say that when you’re holding the category in your hands. The dirty little secret is that we do this all the time without saying so explicitly, and we teach all our children to do the same thing. The term we tell them is “counting”, and the more technical term is cardinality. We can set up a bijection between two finite sets exactly when they have the same number of elements. Identifying such sets amounts to forgetting everything about them except how many elements they have. The tautology “3=3″ expands to, “the number of elements in
is the same as the number of elements in
,” which really means, “there is a bijection between
So, he paraphrases “the number of elements in
I’ve heard it said that scientists are all realists on weekdays and empiricists on weekends (when they bump into philosophers at cocktail parties). I’ve thought that much the same could be said of mathematicians, that perhaps they are weekday Platonists/abstractionist and weekend formalists (or maybe eliminative structuralists). As the post continues Armstrong speaks of numbers in a rather abstractionist way (e.g., speaking of “the natural numbers”), but I wonder if the author might cotton to further “expansions” of mathematical language along the lines suggested by the expansion of the tautology “3=3″.
jrshipley
7 responses so far ↓
John Armstrong // December 10, 2008 at 10:29 am
Well, to be fair I speak of “the” natural numbers because I defined them earlier, and I had another post somewhere in which I dealt with the weirdness of the definite article there. The natural numbers are defined as any structure satisfying the Peano axioms, which (thanks to the inductive axiom) guarantee that any two such structures are isomorphic (with respect to all the things the axioms talk about, at any rate). So if you’re just speaking with the terms the axioms provide for you, it doesn’t really matter which model you use. As a standard example, there’s no difference as far as the Peano axioms are concerned between the Von Neumann and the Church numerals (two standard constructions given a model of set theory).
And yes, I would most definitely give at least conceptual priority to the RHS of Hume’s principle, because this principle is part and parcel of defining the notion of “number” in the first place. “Number” is rooted in “number of things”, and the semantics of abstract numbers must be defined in terms of the semantics of collections of objects. “Equality of numbers” must be defined, and the definition is that two cardinal numbers are equal if the sets they refer to are in bijection.
Turning it around, we can build up the category of (finite) sets as I did, and then “decategorify” it by identifying isomorphic objects (sets in bijection). We replace equivalence by equality, and consider the set of equivalence classes. Then it turns out we can show that these equivalence classes satisfy the Peano axioms (with some other structures boiled down from the category level), and thus form a model of the natural number structure. Thus the cardinal numbers — defined as isomorphism classes of sets — act like the natural numbers, justifying the use of the Peano axioms (and the structure they define) to talk about (an abstract version of) numbers of things.
And yes, I would like more such expansions. That’s what the whole program of “categorification” is about. Every mathematical equation is potentially an equivalence in disguise, and there’s gold in unearthing the equivalences behind familiar principles.
jrshipley // December 10, 2008 at 10:59 am
Thanks for the reply John. I am interested in the philosophical implications of category theory precisely for the reasons you suggest and your blath has been really helpful as I tentatively begin to grasp some of the basics. Hume’s Principle has been used by abstractionists/neologicists/neoFregeans (they go by many names) recently to prove the Peano Axioms, a program launched by Crispin Wright’s Frege’s Conception of Numbers as Objects in the early 80s. HP plus second order logic proves the Peano Axioms. So, I think that there are a number of philosophers of mathematics that would object to using the Peano Axioms to define number in the way you suggest in your reply. Hume’s Principle defines number and is conceptually prior to the Peano Axioms, the contention would go, and that the numbers so defined satisfy the Peano Axioms is an interesting and desirable property for them to have. The idea is roughly that HP is a plausibly analytic/logical truth that contextually defines the term forming operator “the number of Fs” (where F is a sortal concept). According to this approach, the numbers are those abstract objects that make true Hume’s Principle and they can be shown to satisfy the Peano axioms. As you note, many, many structures in set theory also satisfy the Peano Axioms. According to the abstractionists, as I understand their view, this may be an interesting fact in addition to the fact that the numbers do. I’m actually more sympathetic to the spirit of your comments than to the abstractionist approach. The abstractionists view mathematical theorems as statements about abstract objects, reference to which is constituted by abstraction principles (principles analogous to HP). I take your comments to be in the spirit of construing mathematics as moreso about structures than about objects, which I am sympathetic to for generally antiplatonist reasons. But there is much more to be said philosophically to make this go through. After all, a structuralist with eliminativist inclinations doesn’t want to be saddled with his own abstracta: viz. structures!
John Armstrong // December 10, 2008 at 11:53 am
Well, I know the Universal Philosophical Refutation will undercut me, but I have to say that those philosophers have it precisely backwards. The Peano axioms are not to be proven because they’re not a theorem. They’re a definition, and the structure they define (the natural numbers) would (or “exists” if you don’t want to get into ontological questions quite yet) even if the cardinal numbers didn’t satisfy them.
Compare to the definition of the real numbers (follow my links and you’ll eventually get to it). We may be willing to grant “existence” to arbitrarily large collections of objects for the purposes of discussing cardinal numbers, but we have precisely zero hard evidence that anything is infinitely divisible. In fact, it may be that space and time are granular on some extremely fine scale, and so no continuum actually exists.
And yet the formal definition of the real numbers still makes sense. The structure defined by its axioms hangs together, and we can reason about it. The whole of the calculus is based on this structure, whether or not it “really” describes the physical space which inspired it.
What is to be proven is whether or not a given real-world system (like finite collections of actual objects) satisfies some abstract collection of axioms. You can’t prove Peano’s axioms, but you can prove that the “numbers” we use to measure the sizes of collections of objects satisfy them.
So where does HP come into it? HP is simply an assertion that categorical isomorphism in the category of (finite) sets (that is, bijection) is a natural notion of equivalence, and that the equivalence classes of this relation (the cardinal numbers) are natural objects of study. That this is an interesting concept is well-known to category theorists today, but it wasn’t always. In a historical view, HP was a landmark because it was stated before we had the proper categorical language in which to express it, and it’s a prototypical example of analogous statements in other categories.
As for abstractionists and “the numbers”, I defy them to tell the difference between the Church and Von Neumann numerals on purely number-theoretic grounds (as defined by the Peano axioms). They simply can’t do it, because the two structures are isomorphic as models of the axiom system.
And here (in my view) is the clincher: philosophy of mathematics is ostensibly about mathematics, which is what mathematicians do. And mathematicians (outside some logicians and basic set theory classes) don’t talk about the Von Neumann numerals. What do we do? We talk about “the natural numbers” and we use the functions and predicated provided to us by their axioms.
It’s an apt analogy to consider programming in, say, C++. When I write a program, I don’t consider at all what system it will be implemented on. I can take the same source code and compile it on an Intel 586-processor-based machine running Windows 95, or I can compile it on a DEC Alpha running some flavor of Linux, and each time I’ll get the same — an isomorphic — program. The point is that when writing the program I don’t care about the underlying machine code. The only thing I care about is the high-level semantics of the C++ programming language.
Similarly, mathematicians discussing the natural numbers do not care whether they’re being modelled by the Church or the Von Neumann numerals. As far as the semantics of their “mathematics language” (here, the Peano axioms) are concerned, each model will result in the same — in isomorphic — behavior.
Math and Philosophy « The Unapologetic Mathematician // December 10, 2008 at 11:59 am
[...] Peano axioms (not to mention “the natural numbers”) mean. Follow along (or jump in!) at if-then knots — an excellent title, in my opinion. Possibly related posts: (automatically [...]
jrshipley // December 10, 2008 at 4:27 pm
You make a number of good points and I agree with most of what you say. For instance, I think you’re absolutely right that it’s an incredibly difficult question for abstractionists to answer external questions about the identity of numbers (e.g., whether they are identical to this or that sequence of sets). Frege noted that HP suffices to tell us when two numbers are identical, but doesn’t suffice to tell us whether Julius Caesar is a number. The under-determination of reference to abstracta by contextual definitions such as HP is a reason why Frege began his logicism with Basic Law V (which is inconsistent due to the Russell paradox), from which he proved Hume’s Principle and subsequently the Peano Axioms.
To pick a point where I have some inclination to disagree with what you say, there is a strong case to be made that at least developmentally HP is prior to the PA in forming cardinal number concepts. After all, this developmental priority is just what you note in the quotation I pulled in the top post of this thread. Also, HP is logically prior in the following sense: that the PA have a model can be proven from HP plus 2nd order logic, in the sense that there is a valid argument showing that numbers as defined by HP satisfy the PA in 2nd order logic. I should clarify at this point. You took issue with the notion that the PA can be “proven”. I agree that that’s loose talk. The point is better put thusly: “the [cardinal] numbers as defined by HP satisfy the Peano Axioms”. I did put it that way in my previous post, but I invited confusion with some loose talk as well. In any case, developmental and logical priority in the preceding senses are indeed not the same as conceptual priority. One can certainly grasp the structure described by the PA without having an inkling of HP (and vice versa).
I happen to think that HP might not be true, since it only has a model in an infinite domain and the actual universe might not have infinitely many objects in it. At this point, the abstractionist will dig in his heels. HP is true, they will insist. It’s analytically true because it’s just what is grasped in forming the concept number. Since the truth of HP requires that there be infinitely many objects, there are infinitely many objects. So, if there aren’t infinitely many physical objects then there must be abstract objects, they would maintain.
There is much, much more to say. As I’ve indicated I’m very sympathetic to your way of looking at things. But consider this. You would press the abstractionist with the “which sequence” question, a version of the Caesar objection. Can’t the same be pressed against the definition of cardinal numbers as equivalence classes under the relation of equinumerocity (which, btw, is basically Frege’s own definition)? After all, don’t we define the term forming operators “the class of __” and “the set of __” (by the relation of coextensiveness) contextually, just as the abstractionists define “the number of __” contextually? And if it is an objection to the latter that the contextual definition does not determine external identity statements, then why shouldn’t it also be an objection to the former?
John Armstrong // December 10, 2008 at 6:06 pm
I don’t think we’re really in much disagreement. My point is that the Peano axioms are in a way unrelated to cardinal numbers. Even if the cardinal numbers didn’t exist, the Peano axioms still would, and they’d still define the same structure. I think that somewhere in here you’re still identifying “natural numbers” and “cardinal numbers”.
As for domains, note that if there are only
objects in the universe then we can consider the category of “sets with fewer than 
elements”. It’s a lot uglier a category, and it lacks a lot of nice properties, but we can still talk about isomorphism in this category, and an analogous notion to HP.
I think your last question ultimately gets into the idea of foundations, and ultimately you have to pick some foundations to just posit. In my case, it’s either set theory or category theory (the latter if you’re really comfortable with Lawvere). The difference, though, is that the claims I’m rebutting aren’t trying to posit the existence of numbers as foundational. They claim to take set theory as foundational and then work with numbers within that context.
notedscholar // December 11, 2008 at 11:22 pm
Dear Misses Jrshipley,
I wouldn’t pay any attention to John Armstrong. He’s an academic obscurantist of the most classical, Orientalist variety.
You’d better place your focus somewhere else.
Don’t say I didn’t warn you.
NS
http://sciencedefeated.wordpress.com/