I contribute my own perspective in the spirit of Harvey Friedman's invitation "...I am trying to get a dialog going on the FOM and in these other forums as to "what foundations of mathematics are, ought to be, and what purpose they serve"." http://www.cs.nyu.edu/pipermail/fom/2016-April/019724.html
I propose to distinguish implicit math (which our minds intuit and interpret) and explicit math (which is expressed as written symbols). I apply insights from implicit math to yield results in explicit math, specifically, interpretations of the -1 simplex, the Gaussian binomial coefficients, the generation of the regular polytopes An, Bn, Cn, Dn, as well as ideas as to how we might think about F1, the field with one element. These results may or may not be new but I hope to persuade you that both explicit math and implicit math may benefit from a mutual conversation. I offer a question of how to define meaningful q-analogues for Bn, Cn, Dn? And how to define the Euler characteristic? I also suggest the development of a set theory based on finite fields of characteristic q such that q is interpreted as infinity. And I'm curious how to interpret Dynkin diagrams, especially with branching paths, and what the generation of polytopes An, Bn, Cn, Dn might say as to why there are four classical Lie algebras and groups.
The distinct advantages of explicit and implicit math
Harvey Friedman has noted: "Among the greatest mathematical events of all time (different than greatest mathematics of all time) are results of the form: this or that mathematical question cannot be proved or refuted using accepted mathematical standards for rigorous proofs." Referring, in particular, to Goedel's Incompleteness Theorems. And he continues, "We need a general purpose foundation for mathematics where mathematicians readily see that ANYTHING that is done throughout the whole of rigorous mathematics is OBVIOUSLY incorporated in that general purpose foundation for mathematics." http://www.cs.nyu.edu/pipermail/fom/2016-April/019727.html
Taking his point more broadly, and his word "cannot" more broadly, some of the greatest mathematics is that which axiomatizes conditions so rigorously that it is able to prove what CANNOT be done:
All statements to be taken under the relevant conditions. Similarly, great results include the classification of Platonic solids, Lie groups, simple groups, and so on, where it is shown that one CANNOT construct additional objects.
These are all magnificent accomplishments of explicit mathematics. Moreover, millions of people are able to use explicit math to work together, to write, read and check rigorous thousand-page proofs and even computer-assisted proofs. Furthermore, Harvey Friedman notes that this all depends on a robust consistency, a lack of contradiction. This grows in impressiveness along with the mathematical community.
The distinction between explicit and implicit math comes up in Harvey Friedman's exchanges with John Baez. John writes about the roles of explicitly stated rules and implicitly active mathematical taste: "However, some rules are more interesting than others, and it's one of the highest expressions of mathematical taste to be able to choose interesting rules to work with." http://www.cs.nyu.edu/pipermail/fom/2016-April/019664.html
I suppose there are even fewer university courses, if any, on "mathematical taste" than on "foundations of mathematics". But this unspoken notion is at play in a range of mathematical discoveries which I think are just as significant but open up what mathematicians CAN do. These include "introducing" and "playing" with:
"Sound" axiomatic systems typically arose decades after much experimentation led by mathematical inquiry and taste. Implicit math, if developed as a science, would seek and find principles for play, inquiry, taste and so on.
Furthermore, implicit math can consider questions that make sense in a dynamic cognitive environment but not in a static document. In the human mind, the logical assumptions can change. Logic can be dynamic. Indeed, the human mind can analyze counterfactuals, can consider logical alternatives. We can even take the state of contradiction as a starting point and consider how a noncontradictory system might evolve from it. This type of "experienced" logic might yield a novel intepretation of Goedel's Incompleteness Theorem as regards what it means for one who actually lives in an unfolding system. Explicit math might usefully model part of this but I think not absolutely all of it.
My own interest is to know everything and apply that knowledge usefully. I aspire to model the limits of our mind. We experience our lives internally, not externally. I feel that I have been able to model much of that internal life. In order to share that, I want to show that it relates usefully to math and physics. I have had some success with some aspects of implicit math. But, in my experience, mathematicians avoid considering implicit math. I am thus sharing some initial results which I think are relevant for explicit math.
A challenge for explicit math: The field with one element
In defending classical set thory (ZFC=Zermelo-Fraenkel axioms along with the Axiom of Choice), Harvey Friedman asked if there was any issue in mathematics which it is not addressing successfully but another foundations might. http://www.cs.nyu.edu/pipermail/fom/2016-April/019656.html
Currently, there is a mathematical object which has attracted attention but is not yet grounded in set theory and perhaps might never be. I am thinking of F1, the field with one element, which Jacques Tits proposed in 1956 as he introduced his theory of buildings in algebraic geometry. https://en.wikipedia.org/wiki/Field_with_one_element There are many areas of math where such a field suggests itself: https://ncatlab.org/nlab/show/field+with+one+element Conferences have been devoted to this object. It appears as #14 in Richard Stanley's survey of most intricate and beautiful mathematical structures. http://mathoverflow.net/questions/49151/most-intricate-and-most-beautiful-structures-in-mathematics However, this object does not exist! There is no finite field in which 0 and 1 would be the same number. That is, as of yet, there is no settled way to make sense of such a field-like object, which suggests itself in so many areas of math.
Here is a discussion https://golem.ph.utexas.edu/category/2007/04/the_field_with_one_element.html of a 586 page paper where Nikolai Durov describes it as "the free algebraic monad generated by one constant or the universal generalized ring with zero". http://arxiv.org/abs/0704.2030
I present my own approach to show it is simple but fruitful.
Examples of implicit math: A "mind game" and opposites.
I will start by giving an example of reasoning that occurs in implicit math. Here is a "mind game" which serves to define the concepts "one", "all" and "many":
Search for constancy. Either you find "one" example of constancy or "all" is constantly unconstant. But in searching thus, you need to know that in each case, what you select is the same as what you inspect, that is, "many" are constant.
Thus we define "singularly constant", "universally constant" and "multiply constant". This kind of reasoning is standard in recursive function theory. Note that it did not require any explicit symbols. Sure, I used words, but it is the concepts that matter. It could be thought without any words at all. It is the kind of reasoning that I imagine takes place even in the womb. This "mind game" is a participatory example of how to define primitives without appealing to even simpler primitives. They are defined relative to each other through an activity which evokes them. And they are actually defined more precisely, I think, then they are ever defined in set theory. The concept "one", as defined above, supposes that a search algorithm halts, and the concept "all" supposes that it does not. Immanuel Kant tried to derive these same three concepts as "categories" by appealing to the form of a logical implication, which however, others never took up. I simply want to suggest that such reasoning is possible, and that such metaphysical, pre-mathematical thinking can ground implicit math and ultimately even explicit math, physics and other domains of understanding.
Now I want to point out three different kinds of "opposites" that are relevant for implicit math and can be observed in explicit math as well.
By "implicit opposites", I mean that I may be presented with two choices which I cannot distinguish in any way except that one is not the other. We may call them "unlabelled opposites". Of course, this is difficult to talk about because we end up introducing distinctions. Still, there is a sense in which we can be absolutely indifferent as to "clockwise" and "counterclockwise". In math, there are two square roots of -1 and we have no way to distinguish them. We call one of them i, but which one? Do you call the same one i that I do? It shouldn't matter. And this makes clear to me, as I never understood before, why complex conjugation is so important, because that is simply reminding us that on some level it shouldn't matter which is which. So the opposition here between the two square roots is mathematically implicit, in our mind.
By "explicit opposites", I mean that the two choices are expressed and one is primary, "unmarked", and the other is secondary, "marked". Linguistically, we may have the opposites "happy" and "unhappy", and the derivation "unhappy" is marked with the negation "un". Mathematically, we have 1 and -1. Generally, in my thinking, it's not so much the actual symbols that are most important but rather the way we think about them. Be that as it may, mathematically, there are natural reasons to distinguish 1 and -1, for example, 1 times itself is 1 whereas -1 times itself is 1. Indeed, it is reasonable for 1 to be unmarked and -1 to be marked. However, I think it is very unhelpful that the square roots of -1 are written as i and -i rather than, say, i and j. Implicit, unlabelled opposites are being presented as explicit, unmarked-marked opposites. I have a Ph.D. in math and yet only recently did I realize that I have a completely baseless prejudice by which I think of i as more basic than -i. It is as if I thought that "clockwise" was more clocklike than "counterclockwise" or "lefthandness" was sinister and "righthandness" was righteous.
By "mixed opposites", I mean that an implicit opposite is opposed by an explicit opposite. In other words, an undefined notion is opposed by a defined notion. This is especially common when we oppose content and the form or symbol that conveys it. For example, a "nonaction" may be written down as the identity element e. The emptiness of a box may be referenced by the number 0 or the empty set.
I continue by noting why I find the "field with one element" interesting. It must be similar to both 0 and 1, to both the empty set and its empty content. Thus it may relate to my own insights into God and Everything as well as the kinds of opposites that I described above.
The -1 Simplex is the center of a simplex
I will share my exploration of a simple example where the "field with one element" comes up. Namely, the (n,k) entry of the Gaussian binomial coefficients is a q-polynomial which counts the number of k-dimensional subspaces of an n-dimensional vector space over a finite field of characteristic q. When q is set to 1, then we recover the number of subsets of size k of a set of size n. These latter subsets of a set can thus be imagined as 1-dimensional subspaces of an n-dimensional vector space over the (mysterious or nonexistent) finite field of characteristic 1.
In trying to understand this, I learned that the binomial coefficients (n,k) count (among so many other things) the number of k-simplexes within an n-simplex. A 2-simplex is an equilateral triangle, a 3-simplex is a tetrahedron, and there is a nice table of vertices, edges, faces at the Wikipedia article: https://en.wikipedia.org/wiki/Simplex From the article it is striking that the 1-simplex is an edge, a 0-simplex is a point and from the binomial coefficients there ought to be an interpretation for a -1-simplex. There should always be one -1-simplex in any k-simplex. Wikipedia explains that the -1 simplex is sometimes defined as the empty set. I imagined that there should be a more informative interpretation which might be a first step in interpreting F1.
We can consider the n-simplexes and the k-simplexes inside of them to be generated as the terms of the expansion: (chosen + unchosen)^n Thus the -1 simplex can be interpreted as the unique way of choosing no vertices.
My solution is that the -1 simplex is the unique "center" of the k-simplex in question. And that "center" is defined implicitly, that is, we must imagine it with our minds. It is not explicitly indicated as are the vertices and edges. So I think it is a great example of a well-defined mathematical object which, however, is not defined by set theory in any natural, helpful or faithful way. It is very much an implicit object. Let me show you how the simplexes unfold from it!
First we have just the "center".
Next, the implicit center finds expression as an explicit point. They are a mixed opposite. They look like a fish eye. It is seeing zero dimensions.
We can think of the point as a constant and the center as a sliding variable. The center can thus define a dimension. And it can express itself as a point along that dimension. Thus we have a line segment. And it also has a center, I suppose, the same center.
Now we can imagine that center as a variable, let us say, sliding up and down. It is as if we are seeing a vertical triangle from above, so that it looks like a line segment to us. Let the center find expression as a point. Then indeed we have a triangle.
Now the triangle has a center. Again we can imagine it as sliding up and down a dimension. Of course, we can picture it in 2-dimensions or in 3-dimensions as well, but in either case, when the center finds expression as a point, then we have a tetrahedron. (If we are in 2-dimensions, then we could unfold the new faces and imagine the tetrahedron and subsequent k-simplexes as a tiling of the plane.)
And likewise the tetrahedron has a center. And if we draw that center then we can imagine the 4-simplex in 4 dimensions. We can practically see it, the 5 volumes, 10 faces, 10 edges and 5 vertices, how they must be organized! If we like, then to visualize this we can invert the tetrahedrons so that the 4 new tetrahedrons are on the faces of the old one.
This is a wonderful thing, this link between implicit and explicit math. In thinking one center and drawing five points, I feel the solemnity of the center, the spirited gaze of that first point, the companionship in the second point, the humor when our third point opens up like a tent, the beauty of the pyramid that surveys our spatial imagination, and the power in the fifth point at its center which has us transcend the world we know.
Gaussian Binomial Coefficients define what it means to count
I will now describe the q-analogue of this unfolding process, which will ultimately yield insights into the meaning of the "field with one element" for which q=1. We can use the process above to unify three different combinatorial interpretations of the Gaussian binomial coefficients in terms of distinct:
In playing around with weights q, trying to interpret the terms of the Gaussian binomial coefficients, I realized that:
This can be proven by the recursion relation for Gaussian binomial coefficients: N-choose-K = (N-1)-choose-K + q^N-K * (N-1)-choose-(K-1) In general, this recursion relation says that if we extend our series of "options to choose" with an Nth "option to choose", then: the results of our K choices = the same as before (we already made K choices) and do nothing now + what we made with K-1 choices and make the Kth choice now and give it weight q^N-K In other words: our currently possible solutions = previously possible solutions + newly possible solutions So we only have to consider if the weights make sense for the "newly possible solutions".
We similarly wish to show that the N-choose-K expansion counts, as a polynomial of q, the number of vector subspaces W of V over Fq, where dim W = K. Let us fix a basis of V: e1, e2, ..., eN. Then the "newly possible solutions" are those which make use of V's newest basis element eN. Two such new solutions will be distinct if they extend the subspace in distinct ways, which is to say, if they are distinct modulo the existing subspace. So our subspace's newest basis element should be of the form en + f1*d1 + f2*d2 +... + fN-K*dN-K where d1, ..., dN-K are a basis for the left over subspace of size (N-1)-(K-1). There are no restrictions on the scalars fi and the size of the finite field is q so there are q^N-K possibilities. Note that by this construction every subspace has a basis of the form ei + fi-1*ei-1 + fi-2*ei-2 + ... + f1*e1, which is to say, there is a leading term ei plus so much trash.
Let us now imagine how the center is generating these weights and Pascal's triangle, that is, the series of choices encoded by the coefficients of the expansion (NotChosen + Chosen)^N as N goes to infinity. Each series of choices can be identified with a lattice path in Pascal's triangle. However, these series run in the opposite direction to that by which the center creates Pascal's triangle! The series of choices is made from the top of the triangle to some point within it. Whereas the center keeps adding new vertices and Pascal's triangle keeps growing (let us say, to the left).
In the simplex, the dual of the unique center (where no vertex is chosen) is the unique whole (where all vertices are chosen). Initially, when there are no vertices, the center and the volume are the same. The first vertex is distinct and has weight 1. When the center introduces the Nth vertex, it also creates N-1 new edges, which may be thought of as the new vertices relationships with the N-1 existing vertices by which it may be kept distinct from them. Indeed, the center adds an entire diagonal to Pascal's triangle. The new diagonal has N-1 holes, that is, "diamonds", which the lattice paths go around. In all, there are N-choose-2 holes, N-choose-2 edges, N-choose-2 pairs of vertices. Each hole, each edge represents a pair of vertices. The center gives each pair of vertices a weight q.
The weights come into play whenever A < B but B was chosen and A was not.
I hope soon to send out my essay that I've been writing. I think it might even touch on your "closing the lid" operator. I reinterpret the "demicube" (demihypercube) polytope series Dn as "hemicubes" (halfcubes) where the most opposite corners of the cube have been identified (the cube/sphere has been folded in half... like n-dimensional circle folding?) and so we have spiky Euclidean "coordinate systems" with double edges, with additional double edges linking the tips of all of the coordinate vertices, just as you describe. I just don't know how to call these "trusses"? The point is that we get two different ways of looking at this. On the one hand, we have a simplex that has grown out of the "origin". (Just the angles aren't 60 degrees, they are 90 degrees or 45 degrees.) And because our "origin" could have been any point of the half-n-cube, we get 2^(n-1) versions of these simplexes. So each of these is an "anti-center". On the other hand, we get the big picture of the half-cube and by taking a subset of dimensions we can look at smaller half-cube within that. And from the big picture point of view, it makes no difference which points we chose to fold by. But it is a folded volume, so it is an "anti-volume". So the four series will be:
These correspond to the four families of classical groups / Lie Algebras / Lie groups. That is, they express the symmetries of the above structures in terms of actions. Some day I'll understand...
Now I'm trying to relate what I've done with what John Baez writes about here: http://math.ucr.edu/home/baez/week186.html see letters a, b, c, d
And I'm interested how this all relates to Clifford algebras...