Mike Resnik’s book *Mathematics as a Science of Patterns* gives a picture of mathematics that I generally agree with. He takes something like a Quinean position, saying that mathematics, logic, science, and observation are all together in our web of belief, and the process of confirmation is always global. Any statement in the web is in principle revisable. However, we still have the intuition that particular experiments test particular components of our theory individually, because a change in that component wouldn’t reverberate as much through our web of belief. We are free to make the larger changes if further experiments eventually suggest that it would be correct, but the changes involved would often be so drastic as to be effectively ruled out. Thus, we get the illusion of non-holism in our practice of confirmation.

Resnik denies that mathematics is in some absolute sense *a priori* – instead, it is just more general than any science and therefore less susceptible to refutation, due to the greater effects any change on it would have in all our other beliefs. “The relative apriority of mathematics is thus due to its role as the most global theory science uses rather than to some purely logical considerations that shield it from experiential refutation. The same good sense counsels us to use Euclidean rescues to save our mathematical hypotheses from empirical refutation, that is, to save them by holding that a putative physical application failed to exhibit a structure appropriate to the mathematics in question.” (p. 173) He calls such a move a “Euclidean rescue” by analogy with the case of Euclidean geometry – when Einstein and others gave evidence showing that physical space did not obey Euclidean geometry, geometry was reinterpreted as being about abstract points and lines, ratehr than physical ones as had always been presupposed. Every non-Euclidean geometry contains within it models of Euclidean geometry, so we can save the truth of Euclid’s theorems by saying that he was talking about these models, rather than actual space.

However, it’s not clear if his notion of a Euclidean rescue really allows us to shield our web of belief from the repercussions of a change. Let’s consider an example. Say at some hypothetical future point in time, people have been led to adopt some axiom A of number theory that goes beyond PA (since PA is relatively weak, this might even be some consequence of ZFC, but it might not be). Because this axiom will be taken to be true of our concept of number, it will end up having an impact on various complicated computations in the natural sciences. But then let’s say that our notion of physics has also progressed to the extent that we have theoretical justification for saying that a particular machine can carry out a hypercomputation. (That is, it can carry out steps in exponentially decreasing amounts of time, so that it can go through an entire sequence of omega steps in a finite amount of time.) We can then use such a machine to check a simple universally quantified statement of number theory, let’s say C, and assume that C is a consequence of PA+A. If the machine tells us that C is false, then we can either say that our physical theory was wrong in saying that the machine accurately models hypercomputation, or reject PA+A as our correct number theory, or possible revise logic in some way so that C is no longer a consequence of PA+A.

Clearly, the last option is going to be unpalatable, because this will have repercussions throughout our entire web of belief. Ordinarily, we would use this experiment to reject our physical theory (as we reject the theory that a calculator is built in a certain way, if it tells us that 3+5=7), but in this case it seems that the justification for A is likely to be much weaker than that for PA, and possibly even weaker than that for our physical theory. In that case, the right thing to do is reject A.

Resnik suggests that then, for purposes of minimal mutilation of our overall theory, we should perform a Euclidean rescue on PA+A, and say that it’s not meant to be a theory of the actual notion of number, but just some other structure (say, “schmumber”). That’s fine as far as it goes, but it doesn’t prevent us from having to take back many of the other empirical consequences of A that we derived earlier – all of those were derived from A’s role *as an axiom about number*, not schmumber. So performing the Euclidean rescue doesn’t save us from having to revise large parts of our web of belief. Of course, revising the physical theory might be just as bad (especially if it’s a fairly far-reaching theory, and the number-theoretic statement is of relatively limited application), so we’ll have to make drastic changes of this sort no matter what choice we make. But I’m just suggesting that Resnik is wrong to say that the Euclidean rescue can prevent most of this revision.

“Because one can always save a consistent branch of mathematics via a Euclidean rescue (and we have assumed that our mathematicians have excluded this), for them to reject the axioms of ZFC+A would be to take them to be inconsistent!” (p. 134) He’s right that we probably won’t want to say just because of this calculation that PA+A is inconsistent. But saying that it is consistent but false is not the same as attempting a Euclidean rescue – we don’t automatically have a structure that realizes the axioms, the way we actually did in the Euclidean case as a subset of physical space.

His statement that a Euclidean rescue is possible for any consistent theory presumes Gödel’s completeness theorem, which guarantees that every consistent theory has a model. If ZFC is true (or a certain fragment of it at least) then we have the completeness theorem. In fact, even if ZFC is false as a set of statements about sets (the way four dimensional Euclidean geometry is false as a set of statements about actual space-time), but we have performed a Euclidean rescue on it to say that it talks about some domain other than the “actual sets”, then we can use the rescued completeness theorem to give us a domain (not necessarily among the actual sets) for any consistent theory we want to talk about.

But this means that if some piece of empirical evidence were to challenge ZFC (or some principle of the weaker system that proves the completeness theorem), we wouldn’t immediately be justified in performing a Euclidean rescue on it, the way we can be for another theory in the context of ZFC. We would either have to be rejecting it in favor of some theory that proves ZFC has a model, or else we would have to have other reason to believe that a structure supporting the rescue exists.

One reason Resnik might presuppose all of ZFC is that it seems to be necessary for an adequate (Tarskian) account of first-order logical validity. But in the very difficult section 8.3 of his book, he endorses something like a relativist view about logic. (Not that the *truth* of logical sentences is relative, but merely their status as *logical* truths, rather than other sorts, is relative.) I’ll have to read this section again more closely, and I’d be glad if any readers could help clarify it for me. But at any rate, it doesn’t seem clear to me that a role in explicating logic would make ZFC indispensable on Resnik’s account, so it’s not clear how he gets ZFC off the ground to perform his Euclidean rescues later on. (Unless he just means that *for us* (ie, people who believe in ZFC) a Euclidean rescue is always possible for mathematical theories.)

The only way a Euclidean rescue can prevent large-scale mutilation of our web of belief is if we can find a structure closely related to the original one, for which the theory actually is true. But the only way to be sure we can always provide a Euclidean rescue seems to be through the completeness theorem, which doesn’t guarantee that the new structure is at all related to the old one. So his two uses of Euclidean rescues seem to work at cross purposes with one another.

Peter(08:42:14) :Kenny, you write:

“He calls such a move a “Euclidean rescue” by analogy with the case of Euclidean geometry – when Einstein and others gave evidence showing that physical space did not obey Euclidean geometry, geometry was reinterpreted as being about abstract points and lines, ratehr than physical ones as had always been presupposed.”I’ve not read Resnik’s book, so I’m not sure if my comment is a criticism of his argument or of your summary. But the historical record is other than what this statement says. Hilbert published his axiomatic treatment of geometry (in which geometry was about abstract entities) in 1899, before Einstein’s re-conception of physical space as non-Euclidean from 1905.

The historical impetus to the re-interpretation of geometry as being about abstract entities was not the failure of physical space to conform to the axioms of Euclidean geometry, but was the presence of multiple competing axiom systems for geometry (Euclidean and non-Euclidean). These were discovered in the early 19th century and discussed widely among mathematicians from the mid-19th century. In other words, the revision of the beliefs of mathematicians about the nature of geometry was entirely due to evidence from mathematics, and not due to any evidence from the real-world.

Kenny(17:50:29) :That’s an interesting point – I didn’t think about the history of it, but of course you’re right. I don’t think it affects Resnik’s arguments too much, because all that matters is that such a move is possible. Whether it was the existence of alternative geometries or the actual physical data that made people re-evaluate what Euclidean geometry was supposed to mean seems somewhat irrelevant. He does seem to think that we regard it all as “true” though, in a sense entailing that structures of the appropriate sort exist, whether or not they’re physical as we originally thought they would be.

Peter(04:22:09) :Kenny — To verify my claims above, I just read Jeremy Gray’s nice book, which confirms my argument about the development of non-Euclidean geometry in mathematics being independent of work in physics. However, Hilbert was not the first mathematician to propose an axiomatic treatment of geometry independent of any physical interpretations. That honour belongs to Mario Pieri, who published his axiom systems in 1895 and 1897-8.

@BOOK{gray:book04,

author = “Jeremy Gray”,

title = “Janos Bolyai: Non-Euclidean Geometry and the Nature of Space”,

publisher = “Burndy Library”,

year = “2004”,

address = “Cambridge, MA, USA”}