I’ve just put a draft of one such translation up on my web page. It is an excerpt from the 1894 edition of the Dirichlet-Dedekind Vorlesungen über Zahlentheorie. The selected portion focuses on the structure of the lattice of subfields of , and it spells out how Galois theory can be construed in the field-theoretic framework laid out by Dedekind (i.e. the modern take on Galois theory).

Any comments on the draft are welcome.

Here are a few of the points which I think distinguish this approach:

Beneath the fold I will hint at a few of these points, and give some small detail for the first. Although the details are by no means worked out, I am certain that most of our methods for the real numbers (e.g., calculus, topological groups, measure theory) will carry over in a natural way. A significant exception may be  descriptive set theory (and other theories of the fine logical detail of ); in fact it seems to me that my notion of the continuum doesn’t even have a well-defined cardinality. Rather than a failing of this approach, I see this as a sign that set theory has injected unnecessary baggage into our theory of the number system.

These ideas are mostly idle musings, not in active development, and I’m releasing them to the wind. If anyone who happens by wants to develop these ideas, go ahead. You can take them for yourself (with an attribution) or I would be happy to work on a joint project. Otherwise, I’ll continue a haphazard development in these pages from time to time.

Let’s start by recalling the set-up for my model. The underlying machinery of the model will be the nonstandard theory of the natural numbers. A very nice introduction to these ideas can be found in the fourth chapter of Edward Nelson’s book Radically Elementary Probability Theory. Roughly speaking, a nonstandard model of has two nested models of the natural numbers. The raw ingredients are a pair of nested sets , an element and a bijective successor function which also acts as a successor . We call the elements of the standard elements; this is an important property, so we’ll give it a special name: . All the rest of the elements are non-standard. Corresponding to these two different types, we have  two notions of induction. “External” (true) induction says that the standard elements form a model for the natural numbers:

The internal induction axiom will apply to all of ; we want to say that and the compound propositions which include it are the only ones which are not inductible relative to the entire set. Thus for any proposition which does not involve

What all this means is that the standard numbers make up an initial segment of , and both sets act like the natural numbers. I like to think about the standard numbers as corresponding to the actions in our day-to-day life (like counting or dividing) or in our formal system (lengths of derivations), whereas the rest of consists of the huge numbers like astronomical distances or the number of atoms in an everyday object. At a practical level, if I can count to I can count to , but I’ll never be able to count the grains of sand on a beach even though there are a finite number of them. This non-inductible character to our actions is exactly the intuition which non-standard analysis captures.

Since and $0$ are inductible, the entire arithmetic/order theory of comes for free, and we can use all our usual intuitions about these operations. Just to emphasize this, we’ll start referring to the non-standard model with the bold notation: . Now we want to recover the theory of the continuum. To do this, we rescale until the gaps in the natural numbers are too small to see. Specifically, we fix a non-standard number , and we want the number to be our new unit. This doesn’t effect the addition or order operations at all, since these are indifferent to scaling. Multiplication, however, must be modified so that the new unit squares to itself. Using to distinguish this new operation, we would like to say

Of course, division by is not defined in , so really we should put a ceiling or floor operator on this definition.

But wait, the approximation is going to screw up our axioms like associativity and commutativity; sometimes we’ll end up off by one or two. To keep our operations well-behaved, we have to mod out by infinitesimal distances. Non-standard analysis to the rescue! We can define an equivalence relation by

The we can define the half-ray by . In fact, we’ll be focusing on the compact interval

It is a bit of work, which I will leave to another day, to see that addition, multiplication, order and (truncated) subtraction are well-defined on these equivalence classes. Division is a bit trickier, but I would argue that we don’t even really need it; it will be enough to work out division by two, and this is easily accomplished. Moreover, this squares with elevated position that is accorded to dyadic rationals in the theory.

Notice that the definition of involves the predicate. This means that we may need to be careful when using propositions defined from elements of . In some sense this is like taking limits, as it pushes us from a relatively simple domain (the internal theory of the natural numbers) into a more complicated situtation (the nonstandard theory). Because of this, we will usually try to make all our definitions and conduct our calculations before passing from to .

It is worth noting here that the structure I’ve just described cannot be the real numbers to which we are accustomed. To see this, consider the element . We assumed at the beginning that is a non-standard element, so and is not the zero element of . If we try to think about distances in , we must rescale so that . Then

Thus is a true infinitesimal. Moreover, there are lots of others, , all prearranged in a nice arithmetic hierarchy. Indeed, any real number defines a different scale of infinitesimal . These are precisely the “new entities” referred to above the fold. Although I haven’t worked though the details yet, I believe these infinitesimal scales ought to help us provide a more concrete picture of local properties, especially different degrees of differentiability.

I’m going to leave it at that for now. The next time I revisit this topic I’ll say some words about the function theory of my non-standard interval.

At the same time mathematical ideas on structuralism, especially as de veloped in category theory, may offer philosophical dividends by helping us to understand the mathematical value judgements neglected by Benacerraf and his successors. This is because the mathematician’s structuralism addresses the “How?” rather than the “What?” of mathematics. Rather than invalidating alternative approaches, structuralists argues that theirs is a better way of doing mathematics. In the case of the ordinal systems in ZFC, Von Neumann’s ordinals provide a better reduction of the natural numbers than Zermelo’s because they require shorter and simpler definitions and proofs. This is because we are able to use the existing structural machinery of set theory (i.e., products, disjoint unions, etc.) both in building the internal theory of N and in the extension
to transfinite ordinals.

The Red Herring of Ontology (right click & save)

By modern logical lights, geometry did not receive a properly rigorous foundation until works such as Hilbert’s informal Grundlagen der Geometrie, and later formal work by the likes of Tarski. To be sure, Euclid certainly leaves certain necessary assumptions unstated (a minor problem which is easily remedied); but Euclid’s methods of proof have also been derided for relying on spatial intuition rather than precisely formulated logical rules. If one is not careful when proving with diagrams, one can easily misstep and use features of a particular diagram that are not general in order to make unwarranted deductions (as in classic “proofs” such as that all triangles are isosceles).

But of course, Euclid does not prove absurdities such as these! As Ken Manders has stressed, Euclidean geometry was a remarkably stable practice for thousands of years, and its practitioners exhibited a certain diagram discipline. One central idea is this:

What makes Euclid’s proofs (as opposed to those of Hilbert, say) diagrammatic has nothing to do with an improper reliance on spatial intuition; rather, it is merely that Euclid’s one-step inferences are of a very different nature than those of Hilbert.

While Euclid’s Elements might appear very far from rigorous when looking at it with systems like Hilbert’s or Tarski’s in mind, the point is that Euclid’s methods of proof can be given a solid formal foundation in and of themselves. (And in this light Euclid’s text is much closer to formalized mathematics than typical informal-yet-rigorous mathematical works, which goes well with the fact that Euclid was considered to be a paragon of mathematical rigor for many centuries.)

I don’t want to get into the nitty gritty details of spelling out the construction of , and I can get to the topic I want to discuss without doing so. As indicated above, we want single inference steps in to match the sort of “diagrammatic” inferences that Euclid makes. Consider:

Proposition I.12: Given a point off of line , construct line through perpendicular to .

Diagram for Proposition I.12

Proof. Let be a point on the other side of from . Let be the circle through with center . Let and be the intersection points of and . By Proposition I.10, let bisect the segment from to . Let be the line through and . By Proposition I.8, , hence and are perpendicular.

Inferences like the highlighted one above are the sort of step where we might worry that we could misuse a diagram to reach unwarranted conclusions (though this case is rather obviously benign). Our particular instantiation of our diagram has such intersection points, but we have given no reasoning to indicate why it must, i.e. why this isn’t merely an artifact of the particular way we drew it. A Hilbertian or Tarskian proof would provide a chain of logical reasoning in order to arrive at the existence of these intersection points. That is all well and good, but we end up with very different proofs than Euclid’s of course. In fact, in too we will have such chains of reasoning that look nothing like moves Euclid would explicitly carry out, but we have these only on a “background” or “internal” level. The proof given above (N.B. this has nothing to do with the picture) actually is the proof of Proposition I.12. (I led you to believe I was giving Euclid’s proof; one can check that ours has come out essentially identical to Euclid’s.)

In order for to properly license these kinds of diagrammatic inferences, but not go too far, there is a formalized notion of “direct diagrammatic consequence” in our paper. Next time, I will talk about some of the interesting proof-theoretic points surrounding its design.

We’re all probably familiar with the usual constructions of the real numbers via either cauchy sequences or dedekind cuts. Add to this the theory of limits and we have a foundation sufficient for all the calculus and analysis that we need to do. However, these definitions don’t exactly correspond to our basic intuitions; they’re actually the product of a long struggle to eliminate the infinitesimals of Newton and Leibniz.

Nonstandard analysis (NSA) offers a rigorous approach to these more intuitive methods; let’s briefly recall a sketch of one way the story can go. Using set theoretic techniques (ultraproducts) we embed our existing model into  nonstandard model   in such a way that

for formulas built from the symbols and constants above (which includes <).

However we can show that there exists an element smaller than every positive standard real number, and we make this observation the basis of our calculus. The problem with this approach, in my opinion, is that we must first build something which is already good enough to do calculus and then add an extra layer of structure. Instead, I think that we should try to build our continuum directly from a nonstandard model of .

This is approach will be easier to develop from a different nonstandard point of view. The important aspects of the semantic description above are also captured by a syntactic approach, where we add a new predicate ““. This predicate holds of all the usual numbers , , , , but not of the infinites and infinitesimals mentioned above. We must be careful, though, because we are not allowed to form sets using predicates which are built from , so that some reasonable-looking definitions like are actually nonsense.

Now how does this apply the the natural numbers? Start by assuming the basics of . Now adjoin the predicate; most of the axioms translate directly because of the transfer principle. This allows us to prove, for instance, that

Thus we can think of as a non-inductible predicate. This is particularly useful for capturing a notion of indeterminately bounded computation. For example, if I can count to , I can probably count to . However, I obviously can’t count to any number; NSA gives us a means of saying that there is a small (practical) infinity of counting numbers contained in the much larger infinity of all numbers.

*     *     *

So how do we connect this up with the continuum? We want to exploit an analogy which says that is like , because both carry linear orders which are bounded on one side and unbounded on the other. The problem, of course, is that one order is dense while the other is discrete. However, if we think of standard numbers as infinitesimals then this discreteness does not manifest at the macro-scale; say

Then implies that . Thus any nonstandard model of has a dense linear order lurking inside; this is just the quotient of the original order by the eqivalence relation

The next issue is how we ought to scale this correspondence. It’s obvious that corresponds to , and that must correspond to some nonstandard natural. But which one? Any choice will lead to a sufficient theory, but remember, we are interested in aesthetics here. My suggestion is that we fix a nonstandard and define (any other base will do just as well). In this theory, the dyadic rationals take pride of place over ; these are simply the numbers for standard .

The algebraic and order relations are almost trivial to define in this framework. Addition in is exactly the same as that in , as is . Multiplication, on the other hand must be scaled so that . Since we expect multiplication to be linear, this forces us to define

We must include the floor operation because may not be divisible by ; fortunately, this discrepancy washes out at the large scale. All of the arithmetic facts that we like about (commutativity, associativity, distributivity, etc.) follow immediately from the corresponding laws in , although some may be fuzzy in the sense that they hold only up to -equivalence.

I think I’m going to leave it here for the moment. I hope everyone will at least agree that this is a satisfactory definition for the continuum. In my next post I’ll come back to this topic and say why I think this is a preferable method. Any guesses?

My old advisor Peter Koellner, in a paper on pluralism in mathematics, raises the kind of interesting point (based on a simple observation) that I’m often disappointed didn’t occur to me already. The main issue Koellner is considering is the problem of selection of mathematical theories, analogous to the problem of selection in physics. For physics, the problem is to select from among classes of empirically equivalent theories; for math, it is to select from among various interpretability degrees, i.e. equivalence classes under the relation of mutual interpretability. (Don’t worry about the precise definition of that for now.) The point I have in mind is an aside Koellner makes.

He writes:

The structure of the hierarchy of interpretability is more disorderly than one might expect—it forms a distributive lattice that is neither linearly ordered nor well-founded. This is shown via the construction of non-standard theories via coding techniques. Remarkably, however, when one restricts to the natural theories that occur in mathematical practice the theories are well-behaved—they are well-ordered under interpretability. (28)

(NB: The single best source I know for such material is Lindström’s Aspects of Incompleteness, where he investigates the structure of interpretability degrees for theories containing arithmetic.) Koellner has in mind the hierarchy of large cardinal axioms—e.g. inaccessibles, Mahlos, measurables, Woodins, and so on—for extending . Similarly, in the context of second-order arithmetic instead of set theory, there are the extensive results in reverse mathematics showing that many theorems of ordinary mathematics fall directly into the well-ordered progression of subsystems , , , , - and so on. (Simpson’s Subsystems of Second Order Arithmetic is the standard reference.)

As Koellner points out, there is some imprecision in the notion of what counts as a “large cardinal” axiom. But setting that aside, we can just talk of the finitely many instances of axioms that currently go by the name. Or in the context of we can talk of the finitely many subsystems in the list above. The fact is that “natural” theories tend to fall into the corresponding well-orderings of interpretability. But now there is the notion which would be interesting to examine and perhaps try to make somewhat precise, that of a natural theory.

Let’s focus for the moment on subsystems of , taking as a base theory. We know that theorems from countable algebra and separable analysis tend to fall into the hierarchy of subsystems listed above, that is, along one particular chain from to in the lattice of interpretability degrees. Here are a couple of questions I would ask; they are rather vague, and I do not know what form satisfactory answers would take.

Question 1: What is so special about this particular chain? Why does ordinary mathematics lie on it? (How could we characterize this in a precise manner?)

Question 2: Would ordinary mathematics be different (and in what ways) if it, say, lived on a different chain? Could one argue that it necessarily resides where it does, in some interesting sense of the word?

Again, these are pretty vague, but I could imagine rigorous logico-mathematical work that would inform a consideration of these philosophical questions and others like them.

In sum, while I’ve long been aware of the fact that “natural” theories tend to fall along the well-ordering of large cardinal axioms (or, within , the well-studied subsystems), and also of the rather wild structure of the lattice of interpretability degrees, for some reason I never thought about examining the inherent tension there. I agree with Koellner that this is “a mystery that calls for clarification” (28).