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It is generally assumed that mathematics is necessary, such that any mathematical theorem is necessarily true. This can be read as a de dicto necessity such that for any mathematical proposition p, []p is true. Call this the rule of mathematical necessitation (from p, infer []p; abbreviate this MN).

But this doesn't commit us to hold that mathematical objects have their properties necessarily. For instance, consider the statement "0 isn't the successor of any number". This is true in Peano Arithmetic (in fact, it is generally an axiom) and so by MN we have "necessarily, 0 isn't the successor of any number". From this, the de re claim that 0 has the property "not being the successor of any number" necessarily or essentially doesn't follow.

In fact, if we think that the 0 that appears in modular (or, so-called "clockwork") arithmetics is the same as the natural number 0 then we have a straightforward counterexample to the claim that "not being the successor of any number" is an essential property of 0. In mod 4 arithmetic, for instance, 0 will be the successor of 3. Even if we expand our focus only to the integers, 0 will be the successor of -1 and so "not being the successor of any number" will not be truly predicable of 0.

It is only within the natural numbers (and similar systems/subsystems) that 0 will have this property. In a sense, then, 0 has this property contingently.

It might still be, however, that "necessarily 0 is not the successor of any number" is true because the models under consideration are being tacitly restricted to models which satisfy the Peano Axioms, and in all of those models "0 is not the successor of any number" will be true. In other words, "necessarily 0 is not the successor of any number" is true when read as "in all models of PA, 0 is not the successor of any number".

Is it right to think of this as a case where a mathematical object has a mathematical property contingently?

Are there other interesting cases of this (preferably cases not dealing with numbers, since those are the ones I find easiest to generate)?

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    Sure. 2^57,885,161 − 1 is the largest known prime. That's true today, will eventually be false. – Janet Williams Mar 21 '13 at 21:21
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    @JanetWilliams I don't consider that to be a "mathematical property". I had in mind properties that would be ascribed in a formal system. Properties like "being my favorite number" or "being the largest known prime" don't count as mathematical in the relevant sense. – Dennis Mar 21 '13 at 21:55
  • @NieldeBeaudrap - I'm hoping you might weigh in on this. I can't think of anyone better suited to answering this question. – Ryder Mar 22 '13 at 11:59
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It depends on how you describe a mathematical object, and what you call "a single" mathematical object.

Contigency in Axioms

The example of zero being a successor of another number is a good one; I use this myself quite frequently to explain some of the mathematics I work on to my non-mathematically-inclined friends. Sometimes it pays to assert that 6+1 = 0; and sometimes it doesn't.

When do we distinguish one mathematical object from another? If we operate within a formal system, we may choose to adopt the premise that 0 has no predecessors (giving rise to the non-negative integers) or not (giving rise instead to a group, which may be the integers, or perhaps an additive cyclic group). If we suspend any axiom determining which, then both systems are models for the resulting set of axioms. What does contingency mean, apart from having significantly different possible models for the axioms?

Is zero a different object in the two models? If you believe that mathematical objects obtain their 'identity' through their relationships with others, you might say so. But in the axiomatic theory, there is only the one name for the object in the different possible models; as if we could identify Charlemagne-equivalents in two different alternative histories involving different conditions for the collapse of Alexandrian Greece. You might say that perhaps that the axiomatic system has only one object, whose properties are underdetermined. But of course, without being able to prove whether or not the system is consistent (or whether or not the model being described is the group of order one), we cannot say whether or not 0 = 1 holds in the axiomatic system; that's a pretty important relationship to be able to ascertain, and yet the best we can do is to suppose that it does not hold (and hope that we're right). So in fact any consistent system is underdetermined in this way. One can then make a good argument that there is one object, some of whose properties are contingent on further axioms.

As a deeper example, one may consider the continuum hypothesis, in the guise of the question: are all subsets of the real numbers countable, or equipotent to the real numbers themselves? If you restrict your axioms to the usual suspects of ZF±C, (Zermelo-Frankl with or without Choice), there simply is no answer. What does one mean by 'contingent', here — do we mean "depending on information that we don't have"? Independence from axiom systems seems like a bit of a hollow way of obtaining that dependence, but it would fit the bill. Of course, we usually mean by 'contingent' something which depends on something beyond our control, such as a random process or even some sort of diabolical adversary. Perhaps you could argue for the contingency of the Continuum Hypothesis if you believe in some manner of One True Set Theory; the truth-value of the Continuum Hypothesis would be definite but unknown to you, and dependent on 'factors' as yet unknown. If you were a formalist who believed that there might in some aesthetic sense be One True Set Theory (or perhaps A Small Handful of True Set Theories), some of these contingent factors might boil down to arbitrary decisions such as those surrounding whether or not Pluto is "a planet".

If we declare that the Continuum Hypothesis is false, do there suddenly exist for us objects, whose existence were previously only contingent? Is the power set of the real numbers — an analytically defined construct — a different object depending on whether or not we assert the Continuum Hypothesis to be true? This does not seem a particularly constructive way to consider things; but tastes vary.

Contingency apart from axioms

Consider for example random graph theory. In their seminal paper on the subject Paul Erdős and Alfred Rényi (Erdős+Rényi 1960, "On the evolution of random graphs") consider a process where one connects n abstract 'points' or 'vertices' in pairs by edges, selecting up to some number N of pairs to connect, uniformly at random and in a random order. The description of "what is going on" is presented in highly suggestive time-dependent language. (The very idea of the fact that there is something which is "going on", as opposed to just simply being statically the case, is already a hint of this.) The word 'evolution' in the title is meant literally, for example: they speak of the graph changing with time, of connected components "melting" into one another, and so forth.

Of course, this is no different in principle than how we talk about the position of a random walker with time, or even deterministically about the position of a particle moving smoothly according to Newton's Laws. Some mathematical process is described in which one of the parameters describes time; and then we are inclined to talk about what's true of the particle at particular points in time. Is this a series of facts about the object? Or is it a collection of propositions, of which only one is 'true', depending on 'the time'? Making the distinction is a question of semantics, which can either be helpful or pointless: the leverage mathematicians have gotten out of naïve mathematical Platonism certainly suggests that thinking of things as being absolute and static gives some advantage in certain scenarios, but obviously Erdős and Rényi got more mileage out of thinking of their graphs growing with time, maturing as edges are added.

Something even more provocative happens for N = n(n-1)/4 (so that each pair of points is connected with probability 1/2), and take the limit as n goes to infinity. This is known as "the" infinite random graph; and the reason why it is called "the" infinite random graph is that although different pairs of points are connected depending on which edges are chosen to be in the graph, with probability 1 the outcome will be a particular graph up to relabeling of the names of the points. That is, the outcome is just as much the same graph as all circles of fixed radius 1 are in some sense representations of "the same circle". And so it turns out that a random process, infinitely extended, gives rise to what one might call a deterministic outcome. Yet there are other infinite graphs which you could consider which are not the same as "the" infinite random graph: but the probability of realising them by this process is zero, because it would require a conspiracy of infinitely many events which have some probability of failing. Is there anything contingent about the structure of this graph, then? Much is known about it (see the linked article above), but it comes about as if by an inevitable accumulation of accidents. And even though it is "the" infinite random graph, whether or not any particular realization of it (in the construction process) joins two particular points by an edge is still a random event with probability 1/2. We are speaking essentially of a single mathematical object whose representation is contingent.

Summary

Whether there is anything actually contingent going on depends in part on your philosophy of mathematics, on your interpretation of probability or of the idea of 'contingency', or perhaps merely on what the most convenient way to speak about the subject is.

  • Excellent answer. I like especially the addition of the section on "contingency in axioms". – Dennis Mar 22 '13 at 18:16
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    @Dennis: I belatedly realised that I should address the specific type of issue that you raise in your question, so to actually address it. :-) – Niel de Beaudrap Mar 22 '13 at 18:57
  • I've been writing a paper on a topic related to this question and will probably use some of your examples. Even if I don't I think it would be a nice thing (if you're fine with it) to include you in the acknowledgments since your answer helped me in developing this paper. I don't really want to cite a stackexchange question in a paper that will likely become a chapter of my dissertation and (hopefully) be published, so I just wanted to know if your handle was your real name and whether you'd have a problem with me extending the courtesy of listing you in the acknowledgments. – Dennis Apr 10 '13 at 0:39
  • (I also wrote the same in chat and tried to tag you in it, but I don't know if tagging works in chat and I do know that no one checks chat anyway :)). – Dennis Apr 10 '13 at 0:39
  • @Dennis: I'm not sure that it's a bad thing to cite Stackexchange, but then I've seen CS papers cite "personal communication"; standards may vary from field to field. But yes, that is my name, and I don't mind if you use my examples. :-) – Niel de Beaudrap Apr 10 '13 at 0:56
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I think that you are correct, every mathematical object has a tacit context within which it is being considered. How about geometry, a triangle in euclidean, hyperbolic & elliptic spaces are different. Presumably this can be generalised to manifolds with a metric (patches are locally euclidean, hyperbolic & elliptic). This can't be standard smooth manifolds since they are by definition locally euclidean.

One thing that interests me about this is whether a mathematical object in a context can be considered to be within another context, so that you have for want of a better name a second-order contingency.

  • Yes, the second part of your answer is precisely what interests me here. The intuitive picture of the numbers, for instance, is that you go from the naturals to the integers by adding more numbers, i.e., by extending the naturals. But these numbers will have different (sometimes conflicting) properties in the different contexts, as my example shows. The reason I asked this question is because I'm working on a way to handle this sort of "second-order contingency" with counterpart theory. – Dennis Mar 21 '13 at 20:33
  • @dennis: Obviously it can carry on, n-order contingency is possible. The reason why I'm interested is that then meaning of the mathematical object appears to become radically destablised. – Mozibur Ullah Mar 21 '13 at 22:47
  • Oh, I was taking "contexts" here to mean simply "models" (or perhaps, "theories") and taking the second-order contingency to be contingency when evaluated across theories. But now I think I didn't understand what you meant by "second-order contingency". – Dennis Mar 21 '13 at 23:13
  • @Dennis: I hadn't thought of that (models) - but that also seems a good and different way to think about it. – Mozibur Ullah Mar 22 '13 at 0:10
  • "whether a mathematical object in a context can be considered to be within another context"; you might enjoy thinking about that problem in conjunction with the problem of transworld identity. – Dennis Mar 22 '13 at 8:09
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Discussions about the contingency of mathematical properties should definitely consider the existence of Non-standard Models (see the related MathOverflow reference request). The idea is this: in addition to the "standard" model of a mathematical theory, which we might take to be constituted by self-subsistent Mathematical objects, there exist other structures that model the same theory in its entirety (making all its axioms and theorems true) but relative to which we can also specify other objects and properties beyond what can be strictly asserted by the axioms closed under logical consequence.

Here's an example in Arithmetic. "There is some number such that starting from it, we can define a infinitely descending chain of predecessors". On the standard model, this is false, because at some point the chain terminates at zero, which has no predecessors. But not every model of Peano Arithmetic claims that the initial segment of numbers, obtained by starting from zero and closing under succession, covers the entire domain; the existence of alternative models, still validating everything Peano Arithmetic states directly, was proven by Skolem in a paper in 1934.

If the axioms don't say it, can it be true of the objects of the mathematical theory? That depends on whether you're a realist about mathematical objects or not. The Platonist will come down one way or the other, the Constructivist will either suspend judgement or claim indeterminacy, and the Formalist may be happy to posit the existence of many different kinds of "number" systems that differ in what they say.

Of these positions, only the Platonist, I think, can say that there are contingent mathematical properties. For the Constructivist, once we've determined that some axiom system is right, that's it fixed, and for the Formalist, since properties are all strictly tied to systems, there doesn't seem to be any question of the objects themselves having other ways of being. It's the Platonist for whom the possibility of fixing models one way or another makes sense, and for whom a notion of modality with respect to possible mathematical structures might be understood. It's a very different kind of Platonist to the standard theological view of mathematical objects, but it meshes well with Quine's sort of Naturalist Platonism of commitment through deployment in the empirical sciences.

  • Well, you just sort of summarized my paper, haha. Thanks for the input! – Dennis Apr 10 '13 at 13:31
  • Sorry it wasn't more substantial, but I hope the predecessor chain example might be of value. – Paul Ross Apr 10 '13 at 14:05
  • Oh sorry! That wasn't a complaint, it was actually rather reassuring! – Dennis Apr 10 '13 at 15:47
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I partly disagree with Mozibur's answer. I agree that every mathematical object has a tacit context. However, if we change from one context to another we are strictly speaking not talking about the same object, because the meaning of terms involved is given by implicit definition within the mathematical theory as a whole. In the example given, the zero that is not the successor of any number and the zero that is the successor of -1 are two different abstract objects bearing the same name "0".

My answer sounds a bit formalist, though, and I have no idea how to reconcile it with a more Platonist view.

  • I'm actually quite sympathetic with your answer and it need not be construed formalistically. It is actually a quite standard move when talking about the apparent contradictions between euclidean and non-euclidean geometry to index the objects to theories. So, the story would go, in euclidean geometry we're talking about euclidean points, lines, etc. – Dennis Mar 22 '13 at 15:32
  • But I feel some intuitive pull against this move in the case of numbers. It seems like, for instance, if I were to study a finite substructure of the natural numbers (say, including only the elements to 10) I would think I really was studying the natural numbers, but only a portion of them. Likewise, it seems that the intuitive picture of the relationship between the natural numbers, integers, rationals, reals, etc. is that they build on top of one another, filling in the gaps in the number line of the previous theory. So, on this way of thinking about it the natural number 2= integer 2, etc. – Dennis Mar 22 '13 at 15:38
  • I also have some sympathy for this view - however what happens when we're not sure what the context actually is? Think about the Italian school of algebraic geometry where they operated 'intuitively'. When Schemes were discovered people would say, what they were doing was working in schemes, but did not know it. – Mozibur Ullah Mar 22 '13 at 21:01
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    @Mozibur: So according to my view, sometimes when mathematicians operate 'intuitively', unaware of the larger theoretical context, they might not know exactly what objects they are talking about. Is that really such a problem? ;-) – Eric '3ToedSloth' Mar 25 '13 at 10:24

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