Structural Axioms and Terminological Disputes

When I presented my paper on the role of axioms in mathematics in Berlin, one comment I got from several people (which I’ve also gotten from some mathematician friends) was that the project falls apart in my second footnote. Basically, they want to treat all sets of axioms as structural, rather than allowing for a class of foundational axioms, encompassing something like Peano Arithmetic or possibly ZFC or some replacement. The idea is that any consistent system of axioms describes some mathematical structure (of course, only some of those structures are actually of interest to us), but there isn’t an intended model (like the actual natural numbers, or actual sets) to fall back on to extend these sets in a natural way. The Peano axioms show us some results that are useful when counting, but there isn’t some unique structure beyond this practice that fills in truth values for further claims. So the project of wondering whether we should look for new axioms falls apart. I wasn’t able to really say anything terribly convincing at the time, but I think I’ve got some general ideas now.

About a week later, I was discussing David Chalmers’ talk on terminological disputes with some friends in the department here, and I realized that this is a useful way to motivate the distinction between foundational and structural axioms. Consider any dispute over some axiom – say one mathematician says that in every ring there is an element such that multiplying it by any other element fixes the other element, while another mathematician denies that this is in general true. Consider another dispute – say one mathematician says that in the natural numbers, every element has a successor, while another denies that this is in general true. In the former case, it seems that there’s just a disagreement in terms of what we mean by “ring” – some people build in commutativity and identities, while others don’t. The disagreement is merely terminological. However, the ultrafinitist who denies the successor axiom isn’t engaged in a merely terminological dispute. We both intend to be talking about the counting process. We know what the other means by natural numbers, and say that they’re wrong in terms of what they claim. This dispute is not merely terminological.

The fact that there can be disputes like this, that aren’t about logical consequence and aren’t merely terminological, suggests that the axioms involved are in fact of this other “foundational” type. This doesn’t necessarily mean that they provide a foundation for the rest of mathematics, but just that they play a different sort of role. And this is the role that I am concerned with.

Dimensionless Constants

Over at the n-category cafe, John Baez discusses dimensional analysis in physics, as the beginning of a “collection of wisdom on gnarly issues in physics”. (It sounds like “gnarly issues” means “philosophical issues” – so philosophers of physics should definitely go contribute!)

Many quantities in physics (like the speed of light) inherently carry with them certain “dimensions” (in this case, length divided by time). In general, physicists can solve a lot of problems quickly by noting that it only makes sense to add quantities that have the same dimensions, and that in many situations, a single dimensionless constant will be the only relevant factor. For instance, once you know that the only factors relevant to the length of time it takes for an object to fall to the ground are its initial altitude and the acceleration due to gravity, and one knows that the dimensions for these three quantities are time, length, and length divided by time squared, then one can quickly see that when the gravitational acceleration is fixed, the time it takes to fall is proportional to the square root of the initial altitude.

John Baez points out that this sort of reasoning seems to beg a question, in that we have no clear justification for why dimensionless quantities in our theories are always constant. To put it more perspicuously, there are some constants that appear in our theories (like the speed of light and Newton’s universal gravitational constant) that have dimensions associated with them, and the numerical value therefore depends on the choice of units (say, feet versus meters for length, and seconds versus minutes for time). However, other constants (like the “fine structure constant”, which is the square of the charge on the electron, divided by 4π times the permittivity of the vacuum, Planck’s constant, and the speed of light) are dimensionless, and their numerical values (in this case approximately 1/137) don’t depend on our choice of units. Why do physicists find the latter particularly compelling?

At any rate, I found that this stuff all made more conceptual sense after I had read Hartry Field’s Science Without Numbers a few times. In this book, Field argues that numbers don’t really exist – they (like all mathematical abstractions) are just useful fictions that let us extend our physical theories in ways that make calculations much easier. Just as Hilbert showed how to properly axiomatize geometry in terms of betweenness, distance-congruence, and identity of points and lines without talking about real numbers or coordinates or actual distances or the like, Field shows how to add a few more basic relations on spatial points and axiomatize Newtonian gravitational mechanics without any talk of real numbers or coordinates. When thought of like this, it seems to become more clear why the laws have to be phrased in dimensionless terms (or in terms where the dimensions on opposite sides of an equality sign are equal). All the relevant relations that occur in Field’s theories are something like distance-congruence, which is a relation between four points, that are taken to define two distances in pairs. There just is no relation in the language that considers a pair of points defining a distance and a pair of points defining a duration. All the quantities involved are actual physical things (no dimensionless numbers), and the operations only make sense when the quantities are of the same sort.

I don’t know how much this makes sense of what’s going on, but it seems to me to help clarify some things. (I don’t know if it’s evidence that Field’s view is the right way to see things – I’m tempted to think it is, but that may just be my fictionalist bias.) Another way to clarify these things is in the language of torsors, as a commentor on Baez’ post mentions. (I’ve only heard of torsors once before, on a page by John Baez explaining them. But even then, I was struck by how they seem to relate to the way Field thinks of physics and mathematics.)

Return

I just realized it’s been almost 2 months since I’ve posted here! You may have noticed trouble leaving comments in the last month or so – apparently my host site updated something in their system, and only today did I find the change I needed to make to make it work again.

Anyway, after finishing up at the Canada/USA Mathcamp, I visited some friends in Bellingham and Vancouver, and then had the beginning of the semester to deal with, which all distracted me from blogging for a while. Last week I was in Berlin for a workshop, Towards a New Epistemology of Mathematics, attached to the Gesellschaft für Analytische Philosophie’s large conference. An overview of the workshop by David Corfield is here.

A few talks caught my attention that he didn’t mention, so I’ll briefly mention those here. Tatiana Arrigoni presented some discussion of the candidate set-theoretic axiom V=L, mentioning that although many philosophers of set theory argue that it should be rejected, there are some (like Ronald Jensen) that argue in its favor. Her idea seemed to be that there might be at least two different ideas of set-theoretic intuition that lead to different sets of axioms.

Curtis Franks suggested that although Hilbert is traditionally regarded as a mathematical formalist, some of his early writings suggest that his program was motivated by a sort of naturalism, perhaps in Maddy’s vein. He objected to the intuitionists and others by saying that standard mathematical practice is just obviously justified, because it’s been so successful and hasn’t led to any problems – in a sense, their philosophical arguments are no better than those of the skeptic. However, Hilbert wanted to reformulate the consistency of mathematics as a mathematical (rather than philosophical) question – Gödel just showed that this was impossible.

And in the main conference, Øystein Linnebo suggested that John Burgess’ system inspired by Bernays and Boolos, although it very elegantly derives all of ZFC (and large cardinals up to indescribables) just by means of a fairly straightforward plural quantification system together with something like Cantor’s limitation of size principle, doesn’t provide a much stronger justification than Bernays’ original system. The particular plural logic used here does matter, and a seemingly similarly justified limitation of size principle leads to Russell’s paradox. I’m not convinced that Linnebo undermines Burgess’ system terribly much, but it’s definitely interesting to see how these systems develop.

Anyway, now I should return to more regular posting.