Theorems from Biology

16 07 2008

Thanks to Ars Mathematica, I found an interesting article (.pdf) demonstrating some recent theorems in pure mathematics that emerged primarily from doing research arising from biology (construction and inference of phylogenetic trees in particular). Unfortunately, I couldn’t really work through the math enough to really understand any of the theorems.

But really, it doesn’t seem terribly surprising that interesting theorems arise when one considers sufficiently interesting applications of mathematics. These theorems don’t appear to be like the one I mentioned earlier, where the result could be very strongly supported by scientific argument – they’re the more standard results of applied mathematics that mathematicians came up with and turned out to be applicable. But the work was still motivated by the science.

I don’t see any particularly strong argument here that biology will be more productive in this way than physics has been, but it seems to me that any field that applies sufficiently interesting mathematics will lead to the development of new mathematics that eventually turns out to be interesting for purely mathematical reasons as well. Philosophy led to mathematical logic, and the Gödel results in particular, not to mention the results of social choice theory (which I suppose could equally be attributed to economics or political science or a variety of other social scientific enterprises). I don’t know about much mathematics that can be attributed to chemistry, but I did see a very interesting lecture at last year’s Canada/USA Mathcamp about levels of topological classification relevant for describing chirality of molecules (I don’t know whether this has spurred interesting new areas of topology).

I suppose the main problem is just that the paradigm of applied mathematics that many people have (or at least I do) is differential equations. Once we see that so much of mathematics is applied, or can be applied, and that the applications very often lead to interesting new methods of development that occasionally lead to substantial insights within mathematics itself, it should be clear that whatever the status of Hardy’s claims that the best mathematics is essentially pure, there’s no reason for mathematics to cut itself off from the other sciences, or even to seek to remain pure at all times.

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Putnam on Indispensability

29 10 2007

I mostly wrote this post three weeks ago, when Hilary Putnam was at Berkeley to give the three Townsend lectures, but I didn’t get around to finalizing it until just now.

Putnam gave three talks, which were all quite interesting. The first discussed his own type of realism, and tried to counteract false impressions people have had of his views changing drastically with regards to this question over time. (He admitted that they had changed at a few points, but several other apparent changes were just due to some poor choice of terminology on his part. In particular, he had used the phrases “internal realism” and “scientific realism” in a few different ways by accident.)

The second discussed the Lucas/Penrose arguments trying to use Gödel’s theorems to undercut mechanism about the mind – he claimed that although these arguments couldn’t really work, there is an interesting modification that should undercut Chomsky’s thesis that there is a shared innate “scientific competence” that underlies the notion of scientific justification. If there were such an algorithm, then we could never recognize that algorithm as the true one – but since this thesis has to be an idealization (because of the infinities involved) it doesn’t make much sense to argue that our competence is represented by a Turing machine without our being able to say which one. However, I wonder if there’s still some sense to be made of the question – perhaps the idealization says that our scientific competence is given by some arithmetically definable set. But in that case, it is an interesting question whether the best such set is itself Turing computable (even though we can’t know which Turing machine gives it), or more complicated.

The third discussed his picture of “mathematics as modality”. What I found most interesting about this talk was his discussion of the so-called “Quine-Putnam indispensability argument”. He pointed out that when he gave the argument, it had a very different form and conclusion from Quine’s – while Quine used it as an argument for “realism in ontology” about mathematics (that is, that mathematical objects actually exist), Putnam claimed merely that it established “realism in truth-value”. Putnam’s argument was intended to show that there is a serious tension between scientific realism and verificationism about mathematics, on which mathematical claims don’t have truth-values unless they can be proven or disproven. The scientific realist says that there are some sets of equations (say, equations of motion, or wave-equations) that correctly describe the physical world. Now, if these equations have enough complexity (I suppose a three-body system under Newtonian gravitation suffices), then given the truth of certain initial conditions, there will be some interval (a,b) and some time t such that the question of whether one of the objects is in that interval or not at that time is undecidable. But if there is a fact of the matter for the physical claim, and the equations correctly describe the physical world, then there must be a fact of the matter about the mathematical claim, despite its being undecidable.

Putnam took care to point out that this only establishes that there is a fact of the matter about the mathematics, and not that the mathematical objects actually exist. Also, he took care to point out that he was not trying to make an epistemological point – nothing in this argument establishes that this is how we know the truth-values of mathematical claims, just that there must be such truth values. Since Quine concluded that mathematical claims must be about something (because of his picture of ontology as being given by the existential commitments of the best theory of the world), he also concluded that this is how we know which mathematical axioms are true.





Set Theory and String Theory

29 10 2006

One remark that Penelope Maddy makes several times in Naturalism in Mathematics, is that if the indispensability argument was really important in justifying mathematics, then set theorists should be looking to debates over quantum gravity to settle questions of new axioms. Since this doesn’t seem to be happening, she infers that the indispensability argument can’t play the role Quine and Putnam (and perhaps her earlier book?) argued that it does.

However, since the awarding of the Fields Medal to the physicist Ed Witten in 1990, it’s not totally clear that Maddy is right about this. Set theorists certainly don’t pay much attention to string theory and related theories, but other mathematicians in low-dimensional topology and algebraic geometry seem to. I don’t know much about the details, but from what I understand, physicists have conjectured some deep and interesting connections between seemingly disparate areas of mathematics, in order to explain (or predict?) particular physical phenomena. These connections have rarely been rigorously proved, but they have stimulated mathematical research both in pursuing the analogies and attempting to prove them. Although the mathematicians often find the physicists’ work frustratingly imprecise and non-rigorous, once the analogies and connections have been suggested by physicists, mathematicians get very interested as well.

If hypothetically, one of these connections was to turn out to be independent of ZFC, I could imagine that there would at least be a certain camp among mathematicians that would take this as evidence for whatever large cardinal (or other) principle was needed to prove the connection. Set theorists themselves haven’t paid too much attention to these issues, because the interesting connections are in mathematical areas traditionally considered quite distant from set theory. Instead, they have traditionally looked at intra-set-theoretic considerations to justify large cardinals. But if it became plausible that some of these other debates would turn out to be connected, I’m sure they would start paying attention to the physics research, contrary to what Maddy suggests.





Maddy on Indispensability?

13 06 2006

I was discussing indispensability arguments at the bar this evening with some of the philosophers in Canberra, and an interesting question came up regarding Penelope Maddy’s position on them. I unfortunately don’t have access to a copy of her book Naturalism in Mathematics right now, so I’m half doing this as a reminder to myself to check it out when I get a chance (or in case someone who knows better than me reads this and decides to comment to clear things up).

Anyway, my recollection of her position (in her naturalist phase) is roughly as follows. Quine has pointed out that natural science is a powerful and progressing body of knowledge that has helped us build a tremendous amount of understanding. Therefore, we should adopt the methods of its practitioners (or at least, the methods they follow when doing their best work, not necessarily the methods they say they adopt) when we want to find out what’s really going on fundamentally in the world. Maddy points out that mathematics is also such a body of knowledge, and that when Quine applied the methods of the natural sciences, he ended up with a much weaker theory than mathematicians (or at least, set theorists) want. Therefore, she suggests that when we talk about mathematics, we should adopt the methods of mathematicians – the needs of scientists are neither necessary nor sufficient (nor, perhaps, even relevant) for answering questions about whether various mathematical claims are true.

Of course, the methods of mathematics are fairly restrictive and straightforward, so we can’t even say anything about many supposed ontological questions about mathematics (like whether numbers really exist), and about basically all epistemic questions about mathematics (like how we come to have knowledge about numbers). As a result, these questions are effectively meaningless, because there is no way to answer them. So Maddy’s naturalism is a sort of third way, distinct from both realism and nominalism.

There’s also something misleading, it seems to me, about calling it “naturalism”. She develops it on analogy with Quinean naturalism, but it has important differences. In particular, it says that there is a body of knowledge that is not continuous with the natural sciences, namely mathematics! On at least some ways of putting Quinean naturalism, this is exactly what he wants to reject! (Of course, the alternative bodies of knowledge he was thinking about were things like “first philosophy”, rather than mathematics.)

But now I wonder – since Maddy (as I understand her) accepts something very much like Quinean naturalism about the physical sciences (when considered separately from mathematics), what does she have to say about traditional indispensability arguments? She obviously doesn’t think that they give one reason to believe that numbers and sets really exist (at one point she says something like, “if science can’t criticize, it also can’t support mathematical claims”). However, it seems that if entities of these sorts really are indispensable in doing natural science, then don’t we have scientific reason to say they actually exist, even if not mathematical reason to say so? Just as science makes us say there are electrons and quarks and genes and ions, it also seems to make us say that there are numbers and functions and the like, because all of these entities appear in our best theories. Maybe this is no reason from the mathematical point of view, but don’t we end up in effect with a reason to believe in physical objects with all the properties of numbers and functions and sets and the like? (Of course, these are quite unusual physical objects that have no spatiotemporal location and no causal properties, but science has already told us about strange particles that have no identity conditions and multiple positions (like electrons) or strange causal isolations (like black holes), so the “mathematical” entities mentioned in the theories could be seen as just even stranger physical objects, if Maddy won’t accept them as mathematical objects.)

I’m actually fairly sympathetic to this position – if I believed in the actual indispensability of mathematics, then I would grant mathematical entities exactly this kind of physical existence. But I’m also inclined to think that most people would regard this as a reductio of any position if it made one say that mathematical entities had such existence. Especially if the point of the theory was to remove mathematical existence claims from special philosophical consideration.

But maybe I’m just misreading one or more parts of the theory.





APA Blogging: Dorr, Bennett

31 12 2005

Cian Dorr‘s talk, “Of Numbers and Electrons” on Thursday morning made me realize that we’ve got a lot of the same metaphysical goals. The point of his talk was to show that a weakened (and therefore tractable) version of Hartry Field’s program will be able to support realism about theoretical entities of physics and anti-realism about mathematical entities. The scientific anti-realist might suggest a theory like the following:
BAD: As far as observable matters are concerned, it is just as if T
where T is our actual scientific theory, that talks about electrons and other unobservables. However, almost everyone agrees that such a theory is bad (hence the name Dorr has given it). The mathematical realist then claims that the mathematical anti-realist would have to give a theory like:
AS-IF: As far as the concrete world is concerned, it is just as if T
where T is our actual scientific theory, that talks about functions and numbers and other abstract entities. Dorr proposes an alternative.
Read the rest of this entry »





Disjunctive Justifications for Mathematics

7 11 2005

One of Penelope Maddy’s main reasons for objecting to the indispensability argument in her book Naturalism in Mathematics is because it seems to make mathematics too contingent – as she says, if indispensability were our grounds for believing in the axioms, then set theorists arguing about large cardinal axioms should be paying attention to quantum gravity and other cutting-edge physics to see what sorts of math are indispensable for it. And more importantly, she thinks that Quine has shown that indispensability arguments only get us ZFC and not the further large cardinal axioms – and that in fact we are limited to V=L, which is incompatible with most of the larger axioms that set theorists emphatically want us to adopt (and which she thinks we have good mathematical reason to adopt).

However, it seems that a Fieldian nominalist has an easier time justifying our mathematical practice, if the program can ever be made to succeed. The goal is to show that mathematics is actually dispensable (and thus the entities it appears to talk about don’t actually exist) using the Fieldian strategy of giving an attractive nominalistic physical theory that the platonistic theory conservatively extends. If this can be done, it undercuts mathematics in one sense, by saying that it is not literally true. But it supports it in another (perhaps more important?) sense, by showing that it’s a perfectly useful way to talk that (while not itself true) will help us get to the truth more easily in the domains we’re actually concerned with, namely the physical.

Thus, the problem for the Quinean realist justification is that we need to show that the entities quantified over in mathematics are indispensable for our scientific theories in order to justify our mathematical talk. The problem for Fieldian nominalist justification is that we need to show that these same entities are dispensable in our scientific theories. Thus, Maddy rejects both attempts at justification, which are based on reading the indispensability argument in opposite directions, and instead suggests that mathematics needs no external justification, just as most naturalist philosophers think about science as a totality. However, I think that Maddy’s move is unnecessary, and that we may even be able to put together these two justifications to show why!

Combining Platonistic and Nominalistic Justifications

First, note that the important trouble steps in these approaches are nicely complementary. However, they aren’t negations of one another. In one case, we need to show that the only (nice) physical theory that explains all our data is a theory that includes mathematics. In the other case, we need to show that there is a nice theory that explains all the data and includes the mathematics, and there is also a nice theory that explains all the data that doesn’t include the mathematics. So to get my proposed disjunctive justification off the ground, we’ll need to show that there is a nice physical theory including the relevant mathematics that explains all our data. Fortunately, for any theory up to the strength of ZFC, we’ve got such a theory. (In another post I’ll mention why I think going further up isn’t such a problem. For now, I think it’s good enough to observe that as long as we think the large cardinal axioms are at least consistent, then we can extend our theory to one including all of them.)

So let E be the set of all our relevant evidence, let M be the relevant mathematical theory, and T+M be our nice theory that currently explains all of E. At this point there are two possibilities – either every nice theory that explains E includes M, or there is some alternate theory T’ that doesn’t include M that explains E equally well. (For now let’s assume that T’ is a nominalistic theory referring only to entities also referred to in T+M.) In the first case, the indispensability argument suggests to us that M is in fact true, and thus refers to a class of existing entities, and is therefore a justifiable part of our scientific discourse. (Never mind that those entities may well be acausal, atemporal, or whatever – they’re indispensable for our science, so we know about them just as we know about quarks.) In the second case, the indispensability argument suggests that M is in fact false, and there are no entities of the type it refers to. However, following Field, we can still use M in our scientific reasoning, because it is part of T+M, which is just as good a way of making predictions about E as T’ is. In either case, M is justified as part of our scientific reasoning, so Maddy needn’t be concerned.

Some Generalizations

If E is a complete theory, then both T’ and T+M will be conservative extensions of it, so we’ll be in exactly the situation Field takes himself to have given us for Newtonian gravitation. Of course, E is our set of actual observations, so it won’t be complete, but there’s a sense in which this doesn’t matter. Alternative scientific theories don’t have to agree with our current ones in every prediction – they just have to be equally good at explaining our data. (In fact, they don’t even necessarily have to be equally good at all of it – if one theory does a better job of explaining some data, and the other theory does a better job on a different set, then both might be useful theories.) So in a sense, Field might be aiming too high when he aims for conservativity of mathematical theories over nominalistic ones. All he needs is something more like empirical and explanatory adequacy. I think he comes around to a position like this in his 1985 “On Conservativeness and Incompleteness” where he suggests that it might be ok for the nominalistic theory to miss out on some translations of Gödel sentences – these are unlikely to appear in the data, so they aren’t a good reason to decide between two theories that differ severely in their ontological virtues.

Now, let’s note that once we know that T+M exists, we don’t need to know anything about what T’ is, or even whether it exists. In the ideal situation (which Field approximately gives us for Newtonian gravitation and calculus, and also sketches for a very general theory of counting medium-sized dry goods and the natural numbers) we know exactly what T’ is, and that T+M is a conservative extension of it. But even if we don’t know it to be conservative, we’re justified in using M either way. If we even know it not to be conservative, we may then be able to empirically test which theory’s predictions are correct – but until then, they both explain our current data equally well. But even if we don’t know whether such a T’ exists, we’re justified by the existence of T+M in using M.

The only way we can lose this justification (without simultaneously replacing it by a Fieldian one) is by coming up with some other theory U that does a substantially better job of explaining E, and doesn’t contain M. Since this theory is better than T+M, it would undermine our indispensability justification. But if it doesn’t contain M, then we’d need to show that U+M was a conservative extension of U (and a useful one) in order to get a Fieldian justification.

Field (early in Science Without Numbers) claims to have an argument that mathematical theories are conservative extensions of any physical theory (though not necessarily useful extensions). But even ignoring this claim, I find it hard to imagine that we will find a useful theory to explain the world that neither includes (some substantial fragement of) ZFC nor has a useful conservative extension including it. This is even less plausible if we’re talking about the theory of real-valued functions. And I venture to say that it’ll be impossible to describe the world in a way that wouldn’t be usefully and conservatively supplemented by Peano arithmetic. So whatever the status of our current theories, I think this disjunctive justification will let us use ZFC in good conscience, or at least the theory of real-valued functions, and certainly PA. And once we’ve got these axioms, I think we can get all the way up to where Maddy wants us to be, as I’ll show in a later post.

So Maddy really has no reason to be concerned about indispensability arguments depriving us of mathematics. Field has shown how to convert indispensability refutations into alternative justifications for mathematics, showing why the minor amount of empiricism the indispensability argument brings to mathematics is so utterly invisible to us.





No Gain?

5 10 2005

I’ve been reading Mike Resnik’s book Mathematics as a Science of Patterns and have found a lot of stuff I like in it. He makes the point that if we use the indispensability argument to show that mathematical entities exist, then they shouldn’t be that different from the entities postulated by theoretical physics. I didn’t know much about the particular examples he gives from physics, but I think I would take the argument a little bit in the other direction – if mathematical entities really are indispensable to physical theory, then we might as well take them to be concrete physical objects that just happen to lack a lot of the causal and spatiotemporal properties that other physical objects have.

In addition, the indispensability argument meets the epistemological challenge, because we get epistemic access to the objects by confirming the whole theory to which they are indispensable. I’m not sure if Hartry Field discusses this point much in Science without Numbers, but Burgess and Rosen (in A Subject with no Object) seem to be puzzled by the fact that Field both gives an epistemological challenge and argues for the dispensability of mathematics. They think either one alone should be enough, if successful, and therefore the fact that he has to give both calls each into question. But if the point of the epistemological challenge is merely to show that there is no direct epistemic access to these objects, then we need to establish dispensability to show that the indispensability argument doesn’t give us indirect epistemic access. So both arguments are needed.

However, on page 109, Resnik criticizes Field, saying “whether space-time points are mathematical or physical, abstract or concrete, there will be no real gain in using them to dispense with (other) mathematical objects unless they are more epistemically accessible than the objects they replace.” This sort of objection is related to the ones that say that space-time points really are mathematical objects, and therefore Field hasn’t succeeded in nominalizing anything.

However, I think all of this misses an important point. Although Field says he’s a nominalist, it seems to me that a more important point is that he’s trying to give an internal explanation of everything in physics rather than an external one. (Resnik compares this to the contrast between synthetic geometry (where we only talk about points and lines and such) and analytic geometry (where we refer to real-number coordinates as well) and thus talks about “synthesizing” physics rather than “nominalizing” it.) Whether space-time points are concrete, physical objects or abstract, mathematical ones, and whether we have good epistemic access to them or not, they are somehow much more intrinsic to the physical system than real numbers seem to be. Real numbers are applied in measuring distances, calculating probabilities, stating temperatures, and many other things. These seem to be many different areas of the natural world, and using real numbers to explain all of them seems to involve some sort of “spooky action at a distance” as I discussed several months ago. Field’s reconstruction of Newtonian mechanics is certainly an advance on this front, whether or not it has any metaphysical, epistemological, or nominalistic gains. Thus, Resnik is wrong when he says there is no real gain in using space-time points instead of real numbers.