One can reduce natural numbers to finite sets (usually, reduce $\mathbb{N}$ to $\omega$.) One can reduce integers to equivalence classes of pairs of natural numbers. For example, identify $\mathbb{Z}$ with the set $\{[(n, m)]: n, m \in \mathbb{N}\}$, where $[(n, m)]$ is the equivalence class of pairs $(k, p)$ such that $n+p = k+m$. The intuitive idea is that the ordered pair $(n,m)$ represents the integer $n-m$. But there are infinitely many such; so we take the equivalence class of all. By a similar method (this time with $\times$ instead of $+$), we get the rationals, $\mathbb{Q}$. One can reduce real numbers to equivalence classes of Cauchy sequences of rationals, or to Dedekind cuts of rationals. One can reduce the ordered pair $(x,y)$ to some set, say $\{\{x\}, \{x, y\}\}$. One can reduce ordered pairs of numbers (and, more generally, finite sequences of numbers) to numbers. See what I did? - I conflated ordered $n$-tuples with finite sequences. No matter: we can reduce either to the other. And relations can be reduced to sets of ordered $n$-tuples and functions to special relations satisfying a uniqueness condition on one argument place. One can reduce ordinals to transitive $\epsilon$-well-founded sets and reduce cardinals to ordinals.

Even so, I don't think numbers

*are*sets; I don't think real numbers

*are*equivalence classes of Cauchy sequences; etc. These reductions are examples of interpretations or encodings - successful ones, but generally non-unique. This is the message of Paul Benacerraf's famous 1965 article, "What Numbers Could Not Be" (

*Philosophical Review*74). Given the non-uniqueness of these reductions, one can perhaps try and argue that one is special or distinguished, but this is quite implausible (despite the convenience of identifying $\mathbb{N}$ with $\omega$.) Or one can argue that there are no numbers, integers, ordered pairs, etc., and all there are are sets or classes (and so we consider mathematical theories to be convenient definitional extensions of standard set theory, while agreeing that these extensions are not unique). That seems to be Quine's view. Or one can think of these special mathematical entities as

*sui generis*. Or one can think of them structurally (e.g., to be a real number is to be a node in the abstract real number structure). This is Shapiro's and Resnik's view. Or one can be a nominalist (there are no natural numbers, rationals, reals, sets, functions, etc.). This is Field's view.

(Constructivism, as an ontological claim (e.g., Heyting), seems indefensible: if numbers are "mental constructions", then either there are infinitely many mental constructions (one for $0$, one for $1$, and so on) or only finite many numbers. The former contradicts empirical fact and the second contradicts mathematical fact. Perhaps numbers are "possible" mental constructions, in the mind of God? Well, then one might as well just admit that they are not constructions after all. Constructivism, as an epistemological claim, concerning which proofs are legitimate, is not affected by this argument.)

The views are: sets are basic and there's a special reduction of mathematical entities to sets (implausible, I think); there are only sets and classes and all other mathematical entities are introduced by convenient but non-unique definitions (Quine); different kinds of mathematical entities are

*sui generis*; mathematical entities are nodes/places/positions in abstract structures (Shapiro); mathematical entities are mental constructions (Heyting); there aren't any mathematical entities (Goodman, Field).

Of these, I think my preferred view is the

*sui generis*one. Nominalism and constructivism are inconsistent with science. Quine's view is too austere. Structuralism is a bit too weird (the "nodes" in the domain of an abstract structure have an odd status, but maybe structuralism is

*sui generis*-ism in disguise).

So, the natural numbers are just what they are, not another thing, which brings me to Bishop Butler:

If the observation be true, it follows, that self-love and benevolence, virtue and interest, are not to be opposed, but only to be distinguished from each other; in the same way as virtue and any other particular affection, love of arts, suppose, are to be distinguished. Every thing is what it is, and not another thing. (Bishop Joseph Butler 1726,Ok, so that's not mathematics, I admit; but, as Tom Lehrer once joked, the idea's the important thing.Fifteen Sermons Preached at the Rolls Chapel. Preface.)

I'm not a constructivist, but 'indefensible' presents a challenge that can't be refused.

ReplyDeleteFirstly, just because there are infinitely many numbers doesn't mean a constructivist has to propose infinitely many actions by which they are constructed - perhaps the whole structure of the natural numbers is constructed in one act (or a finite sequence of acts), but the individual numbers are parts of the structure in such a way that they can all be said to have been constructed together through this process? This might make the hypothetical constructivist in question to be a structuralist as well (constructuralist?), but they’ll be one with a distinctive position (a distinctively constructivist one - their constructivism won‘t be drowned in their structuralism).

Secondly, even if the acts/constructions in question needn’t be actually completed ones (in the way that painting the Eiffel Tower blue is an action that could be done, but not the outcome of anyone’s real action to date), and the constructivist must give an account of possible actions which will look like some other theory of possible things (probably Platonism, but maybe even something like fictionalism), the fact that these ideal or fictional things are ideal or fictional constructions, rather than any other kind of ideal or fictional thing, is still a substantive claim. Wouldn’t a second hypothetical constructivist who said e.g. ’Yes, numbers are fictions, but specifically fictional constructions, and here’s how the fiction of constructions works: ...’ be saying something interesting? It seems to me that if the fictional things we’re studying are constructions and fictionalists spend all their time talking about some guy called Holmes or whatever, they might be missing important details, because Holmes isn’t a construction. (Well, the character is a construction, but in the fiction, no-one built Holmes.)

OK, I seem to have made my hypothetical constructivists into hybrids with other positions. Although, the second one need not take any position on those possible constructions - they could just say ’Whatever nature they have as merely possible things, they nonetheless have the nature of constructions’, and perhaps also give details as to what this means for Platonists, fictionalist, etc. in detail. I also haven’t really defended constructivism so much as hypothesised people who could, and I suspect if a real constructivist were to turn up, they’d just attack your argument in some existing constructivist way. Oh well, I’ve had fun at least.

Sorry, same anon. Since the above isn’t really directed at your own position (though I don’t think it’s irrelevant, or I wouldn’t leave it here), I’ll try a question: are the non-negative integers identical with the natural numbers? And secondly, do you think the question makes sense? I think it does for at least e.g. structuralists - they can ask whether the structure of the integers contains that of the natural numbers as a part or if it is indivisible, and if only the non-negative integers are that part, or whether e.g. all the integers bigger than 6, or even all the integers less than 6, or even all the odd numbers less than 6, are also the natural numbers (since there are such natural isomorphisms with the natural numbers, though here structuralism and reductionism can blur into each other a bit).

ReplyDeleteIf the natural numbers’ being ‘sui generis’ is meant to rule out their being the same kind of thing even as otherwise similar mathematical objects like the integers (as opposed to being a different kind of thing from sets), that sounds a bit off to me. Surely they’re in some sense the same kind of object as the integers. And if they are, then I think it’s in that sense that philosophers ask what kind of thing they are. Absolutely everything is of its own kind in the sense of being itself and nothing else, that’s just what your quote says, but I think to philosophers who are asking what kind of thing numbers could possibly be, ‘numbers, of course’ isn’t going to be a satisfying answer (even if it turns out to be the only one possible).

Thanks, anon! (Glad you couldn't resist; it's Saturday and not much on the telly.)

ReplyDeleteThe argument I gave is basically an argument that having each number $n$ as a particular mental construction requires a supertask, while also insisting that our minds can't do those. Maybe it would require infinite amount of energy (this would require supposing there is a finite lower bound on the energy content of a mental construction: which seems right).

The first suggestion you consider is to suppose that the whole structure, (N, 0, S), say, is "constructed" tout court by some mental act. (David Miller once made a similar suggestion, comparing it to giving birth). Yes, that becomes some sort of structuralist. But it does leaves inner working of the infinite construction rather mysterious, unless it simply means

define(e.g., either explicitly, or implicitly, by some categorical axiom system).(In mathematical parlance, one does often say "construct" just to mean "define" or "prove to exist" - but these definitions and proofs, understood as being tokened in certain human actions, are small and finite.)

The second idea involves going either modal or fictional. But I'd protest that neither fictionalism nor modal nominalism is constructivism. The fictionalist has their fiction - usual some specific (finite) piece of text. Say, some specific Conan Doyle story. Since the fictionalist applies an error theory to this story, there simply aren't any referents to account for. So, it isn't a view according to which there are numbers, and they're mental constructions. Rather it's a view that there aren't numbers, there is only talk of numbers. One could go mentally modal - talking of "possible mental constructions", but I think this requires postulating supertasks again.

Thanks for the question. The short answer is. Not sure.

ReplyDeleteI'd like to have inclusions:

$\mathbb{N} \subset \mathbb{Z} \subset \dots \subset \mathbb{C}$.

Then, the natural number 0 is the integer 0.

(The Benacerraf argument focuses on non-unique identification of numbers with sets of various kinds.) But this would, perhaps, violate the spirit of sui generis-ism.

On the other hand, it seems that structuralism has to reject the identification of the natural number 0 and the integer 0: for the former is a node in the natural number structure while the latter is a node in a non-isomorphic structure, and presumably domains of distinct abstract structures have to be disjoint.

If constructing the numbers as individual objects is something like drawing more and more tally marks or counting on a very large number of fingers (and I can’t think of any other way of doing it, though I wonder how you’re meant to ‘construct’ 0 using tally marks or finger counting), then I agree that constructing all of them would require completing a supertask. I would, like you, be surprised if humans could do this. I’m not sure what you mean when you say that postulating an infinity of possible mental constructions involves also postulating supertasks, though. Each of those possible mental constructions is finitely completable (though perhaps some are very long indeed, which is probably just as bad for human calculators as if they were infinite - I don’t know how we could determine how long they would have to take, as it doesn‘t seem a determinate enough question to answer mathematically).

ReplyDeleteAs to constructing the whole structure at once, I don’t know how to do it. I’m not a constructivist, after all. I’m not sure what to start with (surely there must be something you construct the natural numbers from - if they come out of nowhere, that’s not a construction). When reductionists start with sets, they usually build the naturals first and work from there. Could it ever be OK to start with the real numbers and construct the naturals from them? I’m not sure how to do that in a way not involving supertasks; you can maybe invoke the integers as the group the acts on something real in some particular way - could that constitute constructing them all at once? - then pick one element to be 0, define arithmetic and isolate the natural numbers.

On fictionalism, I was proposing, perhaps stupidly, that whether the thing that is claimed to not really exist is a non-existent construction as opposed to a non-existent something else might make a difference. It’s not quite the same as pointing out that even though Holmes is fictional, it’s important that he is a fictional detective rather than a fictional bus driver, because if the corresponding fiction in our case is the theory of the natural numbers, the constructivistic fictionalist doesn’t want to say that it is a truth in that theory that the numbers are constructions. It’s more like someone being a fictionalist about deities, so they think there aren’t any but that some claims about them are especially appropriate (in a way that mimics being true), and being specifically a monotheist fictionalist rather than a polytheist one. Even though they think there aren’t any, they think there isn’t one rather than that there aren’t 7. (OK, obviously if there isn’t one, there aren’t 7, but the claim that there are 7 isn’t even relevant, or at least isn’t appropriate, whereas the claim that there is one is relevant, appropriate, and false.) But in our case, what are appropriate or not to fictionalists are claims of arithmetic, not philosophical claims like ‘numbers are constructions’, so perhaps I’m getting my levels muddled.

Hi - many interesting points there. On getting the natural numbers from the reals, there is a difficulty based on a famous technical result (due to Tarski), concerning the first-order theory of real numbers (technIcally, RCF, for "real-closed field").

ReplyDeleteRCF is a complete theory. (Every sentence in the language of RCF is either a theorem or refutable.)

This means that Peano arithmetic, which is essentially undecidable, is not interpretable in RCF. One cannot therefore (at least not in the first-order theory) define a special subset of reals and prove that they satisfy the desired properties of successor, addition, multiplication and induction. One can define 0, 1, 2, etc., "one-by-one", as: 0, (0+1), (0+1+1), etc. But one can't get "x is an ancestor of 0 under successor". To overcome this, one needs to quantify over sets of reals (adding a comprehension principle). This will give a theory which is incomplete, and can define "x is a natural number". So, if one wants to start with the (first-order theory of the) reals, one needs to add sets of reals in order to get the natural numbers.

I think the difficulty of combining (ontological) constructivism and fictionalism is that they disagree about not only what the numbers, etc., are, but crucially about whether there even are such things.

ReplyDelete(i) Constructivism: there are numbers, but they're "mental constructions" in some sense. (A view like this is articulated by Brouwer's student Arend Heyting, for example.)

but

(ii) Fictionalism: there are no numbers, but there is talk about numbers. (The sort of fictionalist view I am thinking of is Field's Science Without Numbers, and more recently, Mary Leng's Mathematics and Reality.)

These are claims in the metatheory, of course, and not the object theory. Both would like to satisfy something like the assertibility of ground level arithmetic claims - but it's important they have very different semantic metatheories. The fictionalist's semantic metatheory is, usually, a standard Tarskian semantics (or maybe deflationary) along with an error theory (existential claims of arithmetic are false). While the ontological constructivist will almost certainly defend some sort of assertibility semantics for arithmetic (thus joining with epistemological constructivism - resulting in the rejection of certain instances of LEM).

I see the problem with starting from the reals. I wasn’t sure how much structure to suggest starting with. Actually, I was thinking more about starting from Euclidean geometry (where there is a notion of ‘construct’ slightly different from just ‘define’, which I hoped would be closer to the kind of construction that constructivists might accept). E.g. if an equilateral triangle is constructed, maybe a constructivist could say that by inspection of it and its rotations, we come to construct arithmetic mod 3 all at once (so each rotation is associated with, though not identified with, a number, and addition with concatenation). That seems adequately finite to me, and reassuringly traditionalist. But is there any finite constructible structure that will give us an infinite arithmetic (other than the real numbers themselves) in the same sort of fashion?

ReplyDeleteThe plan was for a constructivist fictionalist to be able to say ‘There are no numbers, there is acceptable talk of numbers as constructions, any talk of things other than constructions isn’t even arithmetic.’ I’m sure constructivism does posit numbers, but I don’t see it has to. If constructivists really are committed to those semantics, then it does look as if there is a problem, though, because the conditions for asserting that a+b=c or something like that are probably going to require the person making the assertion to have a construction, or possible construction, to hand, and the theory is that there are no such things to be provided. Here is where the level confusion was, I think: epistemological constructivists can’t be error theorists about arithmetical statements themselves, because they don’t think about (some of? - isn’t it OK to talk about the strictly finite statements as true?) them in terms of truth, just acceptability; it’s the acceptability conditions where constructions come in, and I think they could be error theorists here - it’s acceptable to claim a+b=c if you have a supporting claim about a corresponding construction, and the latter claim is truth evaluable, and strictly false (because of fictionalism about possible constructions), but still assertible. This doesn’t help me with respect to ontological constructivists, though, and they’re the ones we’re really interested in. I’m tempted to say that their claims about the constructive nature of numbers are at the same level as the epistemological constructivist’s claims about acceptability, and so can be both truth evaluable and false, but I’m not really sure.

Also, on structuralism, I suppose I was suggesting that there could be a debate over whether distinct structures have to be disjoint. A structuralist who thinks that structures actually contain objects, but that it makes no sense to ask which thing an object is outside of a particular structure would probably have to say the do have to be disjoint. That sounds like Benacerraf’s view, in that one paper at least (and maybe Shapiro’s? - I’m really not sure there), but I don’t think it’s essential to structuralism. Though now I’ve thought about it some more, it does sound a bit unstructuralist to say that the things called ‘0’ in the naturals and the reals are the same object (if they are objects at all).

I thought I posted this earlier, but it's not there, so I'll try again. Hopefully, it won't now appear twice.

ReplyDeleteI see the problem with starting from the reals. I wasn’t sure how much structure to suggest starting with. Actually, I was thinking more about starting from Euclidean geometry (where there is a notion of ‘construct’ slightly different from just ‘define’, which I hoped would be closer to the kind of construction that constructivists might accept). E.g. if an equilateral triangle is constructed, maybe a constructivist could say that by inspection of it and its rotations, we come to construct arithmetic mod 3 all at once (so each rotation is associated with, though not identified with, a number, and addition with concatenation). That seems adequately finite to me, and reassuringly traditionalist. But is there any finite constructible structure that will give us an infinite arithmetic (other than the real numbers themselves) in the same sort of fashion?

The plan was for a constructivist fictionalist to be able to say ‘There are no numbers, there is acceptable talk of numbers as constructions, any talk of things other than constructions isn’t even arithmetic.’ I’m sure constructivism does posit numbers, but I don’t see it has to. If constructivists really are committed to those semantics, then it does look as if there is a problem, though, because the conditions for asserting that a+b=c or something like that are probably going to require the person making the assertion to have a construction, or possible construction, to hand, and the theory is that there are no such things to be provided. Here is where the level confusion was, I think: epistemological constructivists can’t be error theorists about arithmetical statements themselves, because they don’t think about (some of? - isn’t it OK to talk about the strictly finite statements as true?) them in terms of truth, just acceptability; it’s the acceptability conditions where constructions come in, and I think they could be error theorists here - it’s acceptable to claim a+b=c if you have a supporting claim about a corresponding construction, and the latter claim is truth evaluable, and strictly false (because of fictionalism about possible constructions), but still assertible. This doesn’t help me with respect to ontological constructivists, though, and they’re the ones we’re really interested in. I’m tempted to say that their claims about the constructive nature of numbers are at the same level as the epistemological constructivist’s claims about acceptability, and so can be both truth evaluable and false, but I’m not really sure.

Also, on structuralism, I suppose I was suggesting that there could be a debate over whether distinct structures have to be disjoint. A structuralist who thinks that structures actually contain objects, but that it makes no sense to ask which thing an object is outside of a particular structure would probably have to say the do have to be disjoint. That sounds like Benacerraf’s view, in that one paper at least (and maybe Shapiro’s? - I’m really not sure there), but I don’t think it’s essential to structuralism. Though now I’ve thought about it some more, it does sound a bit un-structuralist to say that the things called ‘0’ in the naturals and the reals are the same object (if they are objects at all).

Hi - the disappearance is caused because by comments being "anonymous", which get caught by Blogger's spam filter. So, when I notice something has been caught, I have to "release" it.

ReplyDeleteThe situation with RCF is the same as with first-order geometry in n dimensions. Call it $EG_1(n)$.

The result is due to Tarski.

$EG_1(n)$ is complete (and decidable).

Any sentence $\phi$ in the language of $EG_1(n)$ is either a theorem or refutable. So, we can't get arithmetic going in this theory. Tarski also gives a representation theorem for this theory:

$\mathcal{A} \models EG_1(n)$

iff

$\mathcal{A}$ has the form

$(\mathcal{F}^n, Bet(\mathcal{F}^n), Cong(\mathcal{F}^n))$,

where $\mathcal{F}$ is a real-closed field. Model-theoretically, first-order geometry is intimately related to the first-order theory of real numbers.

See this wikipedia page on the topic

http://en.wikipedia.org/wiki/Tarski's_axioms

The question about the domain identity for structuralism is the really central problem that arises, I think.

ReplyDeleteIf N is the abstract natural number structure and Z is the abstract integer structure, then it looks like the relation of the domain of former to the domain of the latter cannot be inclusion, for 0 in N seems to be what it is in virtue of the whole surrounding structure - N. And 0 in Z seems to be what it is in virtue of the whole surrounding structure - Z. So it seems that, because these are distinct structures, the 0 in N has to be distinct from the 0 in Z.

This is all rather speculative, though. I think it's probably a consequence of Shapiro's structuralism.

Hopefully, the sui generis approach doesn't face this problem.

Everything is what it is: $\forall x (x = x)$.

ReplyDeleteNothing is another thing: $\forall x \forall y (x \neq y \rightarrow x \neq y)$.

The quote from Bishop Butler -- 'Everything is what it is, and not another thing' -- seems to say merely that identity is reflexive. Reductionists needn't deny that. If numbers are sets, then saying that numbers are sets is just saying that they are what they are, not that they are another thing.

Hi Campbell, yes, you're right. Literally read, it just means what you say. Still, Bishop Butler manages to get his point across ...

ReplyDeleteIt's quite hard to articulate this exactly. I think it's some sort of meta-claim of non-implication in disguise, maybe: "From the possibility of reducing $A$s to $B$s (or the structural similarity of $A$s and $B$s), it doesn't follow that As are Bs".

Hi Jeff and Campbell,

ReplyDeleteLike Jeff, I'm sympathetic to the sui generis view (which I used to call 'unruffled Butlerism').

Campbell's point is a good one. A suggestion: we obviously have some working criteria for deciding whether a purported reduction of mathematical objects is genuinely *reductive* (ignoring the question of adequacy or truth), or just trivial in some sense.

I think what the Butlerian-Benaceraffian really wants to do is to take those reductivity-criteria and re-employ them as falsity-criteria.

To clarify, by 'purported reduction' above I meant reduction of the sort which does entail identity - the sorts of claims discussed in Benacerraf's article. (Reductions in a weaker sense can be true on the Butler-Benacerraf view.)

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ReplyDelete