************************
There is a recognition among many mathematicians of a pressing need for a powerful organizational scheme for mathematics. There is general recognition that mathematics today has gotten out of control, in that no one seems to be able to get a "good grasp" of mathematics as a whole.
Many mathematicians chalk this up to the mere volume of (interesting) material being produced, and are very skeptical of powerful organizational schemes.
However, some others, including me, take a different point of view.
We believe in the idea of a powerful organizational scheme for mathematics, that would put the myriad advances into some clear general picture that is fully intelligible to the general mathematical community - and largely to other scholarly communities as well.
I know of only two serious attempts to provide such a powerful organization scheme for mathematics as a whole. I cite the first as a joke, and the second as more serious. (However, Dieudonne?)
Obviously 1 is a monstrosity, but still provides an obvious service to the mathematical community.
Certainly 2 has some admirable qualities, and there is no question that Mac Lane was trying to address this desperate need. This is highly admirable.
However, 2 is nowhere near systematic enough to really do what we want. It is too descriptive, and is missing key powerful organizational schemes.
Our mathematics textbooks don't even provide the appropriate foundational organizations of limited areas of mathematics. The best of them aim at reasonably efficient presentations of a hodge podge of topics that are known to represent the "tools of the trade" for the relevant area. To be sure, the best of these textbooks provide essential services to the profession, but do they discuss how to formulate productive research programs, or how to weigh the relative interest and importance of various mathematical developments? I do not think so.
I am optimistic about the possibility of a suitably powerful organizational scheme for mathematics. One reason is this: my belief that I operate under such a scheme for the focused area of foundations of mathematics, which I have effectively used for decades. I use it to formulate research programs, and evaluate relative importance in f.o.m.
There is the serious question of whether such a scheme that works for f.o.m. could work for mathematics as a whole. Perhaps mathematics is different in essential ways from f.o.m.
I doubt that mathematics is different from f.o.m. in this essential way. However, one thing is absolutely clear.
I am clearly not in a position to push through a powerful organizational scheme for mathematics as a whole to any kind of advanced state on my own. If this is going to be truly successful, it will inevitably have to be a seriously collaborative effort.
######################################
Let me make a bold beginning. We wish to "generate" all of "fundamental" mathematics starting with a very few "basic" concepts, using a very few basic "intellectual operations".
DISCLAIMER: I am in no position to prematurely inject rigor into this from the beginning.
By generate, we mean that we start with the basic concepts, and apply the intellectual operations, yielding more ideas, definitions, and (statements of) theorems, and then applying the intellectual operations again, again yielding more ideas, definitions, and (statements of) theorems.
In the setup we propose, sometimes these intellectual operations also yield at least the essential ideas behind the proofs of the theorems so generated, but not always. We emphasize the production of definitions and statements of theorems.
DIGRESSION: In a later posting, I want to talk more specifically about the project of producing a "foundational exposition" of mathematics, which includes ideas, definitions, and statements of theorems, but specifically not proofs of theorems. END.
ONE BASIC CONCEPT. The measurement of "items". This is the assignment of a
"quantity" to a variety of entities. We usually like to divide this concept
into some obvious embodiments that are immediately clear to everyone.
a. The area of a region in two dimensions.
b. The volume of a region in three dimensions.
c. The number of elements of a finite set.
Now obviously if we are doing *foundations* in the usual sense meant in f.o.m., we cannot be throwing ideas like this around at the very beginning.
But f.o.m. is about foundations in a different sense. I.e., formulations of primitive concepts and fundamental principles and rules of inference, etc.
But here we are talking about something quite different. We are trying to get at the fundamental intuitions that drive a fundamental organizational scheme for mathematics.
FIRST BASIC INTELLECTUAL OPERATION. Formulation of basic principles about concepts that have been generated. Ideally, establishing that these basic principles are complete in that they determine a unique mathematical embodiment of the concept. (Existence and uniqueness theorems). These formulations may require the identification of additional concepts, and some development of them. Such identification and development is considered an output of this intellectual operation. This would generally be their principal foundational justification.
SECOND BASIC INTELLECTUAL OPERATION. Identification and exploitation of generality. Often we see that Theorems hold not only for the objects and concepts intended, but also for any system of objects and concepts obeying certain simple properties. This is our principal tool for justifying algebra (in the sense of modern mathematics).
DISCLAIMER. Of course, I have no premature interest in attempting to be
complete in what I am doing in this posting. So let us assume that we have
justified the following.
i. The natural numbers as counts of finite sets.
ii. Addition of natural numbers in connection with counting disjoint unions
of finite sets.
iii. Multiplication of natural numbers in connection with counting Cartesian
products of finite sets.
iv. The integers for many reasons, including measurement of change from one
natural number to another.
v. Extension of addition and multiplication to integers for many reasons,
including uniqueness. (Although here this might have to be combined with an
explanation of the fundamental character of the ring laws and ordered ring
laws also).
vi. So we arrive at the ordered ring of integers.
vii. Proportianality. This leads to the extension of the ordered ring of
integers by the ordered field of rationals.
viii. Completion. This leads to the extension of the ordered field of
rationals by the ordered field of reals.
ix. Exponentiation on natural numbers. From counting the number of sequences
of a given length from a finite set of given size. This gives 0^0 = 1, and
for n > 0, 0^n = 0.
We now come to two more delicate issues:
So here we at least get into algebraic laws, infinite series, rotations/angles, simple differential equations, continuity, etcetera. Is, and why is, infinite series, fundamental?
But all of this so far is still at a far more elementary level than what is needed to be developed in order to even test the idea that we are on to any kind of truly powerful organizational scheme for mathematics.
Things seem to get more substantive when we start talking about measuring "figures" in two and three dimensions. Even before this, there needs to be an intense discussion of the Euclidean plane (and higher dimensions). Here there are all sorts of informal geometric notions such as colinearity and angles, that need to be treated foundationally. One must show at every single stage that the embodiment in mathematics is entirely unique.
We arrive at the measurement of lengths and areas of "figures" in the Euclidean plane (and higher), first by means of finite additivity. Finite additivity, for basic figures alone, is sufficient to fix areas of figures if they are nice enough figures. (Here one could strive to justify semialgebraic sets, or perhaps more, like semianalytic sets?). For more general "figures", one must invoke countable additivity. There seems to be a lot of serious mathematics involved in sorting out just when finite additivity enough, and just when countable additivity must be invoked, and even when countable additivity is not good enough (this would involve highly pathological sets unavailable without the axiom of choice).
An extremely key idea when we get to measurement of lengths and areas, is
the consideration of transformations that are to preserve lengths and areas.
We are rather quickly plunged into permutation groups, automorphism groups,
and by default, some group theory (second basic
intellectual operation).
4. What is (are) the right way(s) to justify linear transformations on
Euclidean spaces?
Note that serious linear algebra - where determinants make their appearance - are forced on us once linear transformations appear. I.e., linear transformations are volume preserving up to a constant factor, and this constant is the absolute value of the determinant. This introduces the determinant (up to sign change) as the stretching factor. The determinant itself is the almost unique linear operator whose absolute value is this stretching factor. And what is the significance of this sign? Etcetera. Linearity of the determinant falls under: fundamental properties of the determinant, with the idea that obtaining fundamental, simple, clarifying laws of concepts already introduced either falls under our second basic intellectual operation, or needs to be specifically added to our list of basic intellectual operations. Orientation, alternating groups, etc.
What about topology? Maybe we need (?)
SECOND BASIC CONCEPT. Shapes, including time indexed transformations of one figure to another.
TO BE CONTINUED.
PS: Another key ingredient in this approach is:
the return to original purposes.
All kinds of original purposes went unrealized because the area was too new to have generated various powerful techniques. But in the course of developing these powerful techniques, communities seldom return (in any systematic way) to the original purposes which now can be fulfilled. Instead, the communities become enamored with the latest razzle-dazzle for its own sake, independent of original purposes. This seems to be the *critical error* in business-as-usual.
A suitable foundational organization of mathematics promises to put an end to this business-as-usual.
Harvey Friedman