I'm also reasonably comfortable with basic ring theory and the manipulation of ideals and modules. I'd like to go up to Artin's reciprocity theorem and at least be able to discuss it at the conceptual level. That means I should understand at least that much, and that will be the subject of my next post or two here.

I'm going to assume some level of comfort with rings and modules. In particular, the set of ideals has sums, products and the obvious -action

The story of class field theory begins when we ask what division could mean in the context of ideals. Everything takes place in a integral domain, where we can build a field of fractions for our ring, . A fractional ideal ought to be a submodule . We need more, though, because we also want to relate relate to via the multiplication, so we require that there is some for which . Note that this holds trivially in case is finitely generated.

It is easy to see that the multiplication of integral ideals extends to the fractions, and this gives us an associative operation with the unit . It is not the case that every fractional ideal has an inverse, but we do have a sort of "best approximation"

Then the product is trivially contained in . Moreover, if anything more were included that condition would fail, so if has an inverse in the semigroup, is it.

We also have the notion of a principle ideal , generated by a single element. This generalizes to fractional ideals as well, now allowing to range over the field . Clearly, every principle ideal is invertible, with inverse . We immediately have the notion of "principle fractional domain", where all fractional ideals are principle. This would certainly imply that itself was a principle ideal domain, but the converse turns out to be true as well, since two principle ideals and are equal iff for some unit .

We finally get to the notion of class fields by taking the quotient of all fractional ideals by the subgroup principles. In general we only get back another monoid; if we want to get a class field group, we have to pass to the more restricted notion of a Dedekind domain. That will be the topic of my next post.

This describes Moebius transformations, which are maps of the complex plane which have the form

for fixed . This video really emphasizes how anything having to do with the complex numbers is really about rotation.

www.youtube.com/watch?v=JX3VmDgiFnY

I don't think we have a plug-in for embedded video yet. Maybe I'll work on that later. Until then, enjoy the link. [UPDATE: We now have embedded youtube functionality, as can be seen. -- Ed]

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.

A significant theme at all levels is the promise of technology to flatten barriers and cut costs in education. At the same time, using these new methods to their fullest will require modifying fundamental attitudes about the role of educators and the purpose of institutions. There is no doubt that many of my prescriptions will sound counter-intuitive in a modern world where education seems so important. I urge you to resist the fallacy of considering proposed changes against what the current system promises, rather than what it delivers.

I believe that the crucial failing of our modern educational system is a failure to admit that elementary school has a two-fold purpose. In a modern economy, school's function as a daycare is just as important as its role as an educational institution. If we take this notion seriously, the ramifications are far-reaching.

First and most importantly, there is no longer any call for forcing twelve-year-olds to sit at a desk seven hours a day. Much of my interest in educational practice comes from sheer indignation that mine was not carried out better, and I had an excellent education by most metrics. Once we accept that school is as much about minding kids as teaching them, it becomes acceptable to allow them to run around and play for a few hours every day. These pictures of a young test-taker speak more eloquently than I ever could:

Of course, school is also about education, but today's curriculum is crammed full of information that the average student neither wants nor needs to know. In place of this, I believe we need a basic (very basic) universal curriculum, nationally defined, which is required of all students for "graduation" from elementary school (no yearly assessment). This basic curriculum would consist of only those skills which are actually necessary for day-to-day life: reading, writing, basic arithmetic, perhaps basic computer competency and a few others. Most importantly, this test is not intended to set a high standard, but rather to assess basic competency; this should be something that most 4th or 5th graders could pass handily.

What then of the other subjects: science, literature, history? Without a doubt these should still be offered, but they should no longer be required of all students. There is certainly the worry of producing students without even basic knowledge in these crucial subjects but, truth be told, the same is true today.

I don't want to pass this off as an idle concern, but we should weigh it against the presumptive benefits for students who remain in classes (and teachers, as well). Without forcing unwilling students to sit through classes, we should expect that misbehavior and disruptions should drop significantly; this would also allow teachers to throw out students that are a perpetual disruption in their class.

In many cases, we can then aim for a higher standard for those students who remain. Students who are engaged and want to learn do so faster and easier; we should also be able to cut out much of the tedium and repetition which results from trying to force ideas into unlistening ears. I have little doubt that the top quarter of motivated students could learn a typical 5-6 year arithmetic curriculum in under two.

Nor does the curriculum need to be completely unguided. I would envision a system of dependencies. Knowledge builds on itself, and key components like arithmetic and reading are necessary for everything from science to music to woodshop. The pressure of peer groups will encourage adequacy in basic areas, as well. Once everyone your age has learned to read, there is a strong incentive for you to do so as well. This is especially true in an environment where knowledge is viewed as a commodity rather than a prescription.

Where does technology come in? The most immediate change which we can expect (I hope) is a relaxation in the crushing grip of textbook publishers like Houghton Mifflin and Prentice-Hall. We are very nearly at a point where the physical book becomes irrelevant, and at that point we no longer need to pay for historical ideas and methods which belong to all of us. More generally, as we allow students greater freedom in following their own interests at school, it is inevitable that those interests will outstrip the knowledge, the capabilities and (when different for every student) the time of their instructors. Digital texts will have to take up the slack. Fortunately, digital texts offer a much more powerful means of expression than static books; I can hardly guess at the diversity of these new methods, although I have some ideas about the presentation of mathematical ideas (that will wait for another day). To sum up the basic idea in a pithy statement:

Although a computer may never teach as well as a good teacher, we can certainly design them to teach better than a bad teacher, and there are plenty of those.

That describes the basis of my position. Most importantly, give kids more freedom to do and explore what they want while they're at school. We don't have to take them off the leash all together, but we need to let them run around a bit. While we're at it, let's only expect kids to know those things that we do; that doesn't mean we have to give up on them learning anything more, it just means we shouldn't tell them what that 'anything' has to be. From there, our biggest challenge is to make good on our promise to teach them 'anything.' Technology and free, open information is the best, most efficient way for us to do that.

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)

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?