A Staples IntroductionShopping at Staples (today 9/29/2022) I had a glance at the Public Speaking tutorial laminated card, and it said your Introduction should be made up of: Attention Getter. Significance. Credibility. Thesis. Shall we try it?
Attention GetterFor me, the story started when in 1978 with NSF support I spent a summer studying David Hilbert's axiomatic approach to geometry under Edmund Deaton at San Diego State University. Thanks to both of them, and to you American taxpayers for paying the taxes that once better funded the National Science Foundation! I was an ubergeek at the time, in Hindi-English they'd say I was the math topper, not just in my school but in our league of many schools in a vast swath of southern/central california which held regular math competitions, which I normally won, at least senior year. I would outscore Barstow High, the whole school team, by myself. Strangely enough, I was suffering from low self-esteem at the time, which made me highly motivated. Counterattack my own misery by doing well in math. So I was eager to go to math summer school. You know that concept of the Big Man On Campus, whose life peaks in high school, and goes downhill after that? That was me.What I signed up for, it turns out, was to study the "Moore Method", of proving things for yourself, in this case in geometry. We used the axioms-to-theorems approach of the great Hilbert, as filtered through the greatest American math teacher R.L. Moore. Moore was a colleague of the adviser of my own teacher, Deaton. Their idea was, prove it yourself, and that's how you learn. Simple idea, hard to implement, but with lots of value. I can see why the NSF funded it, and I feel so privileged to have had a summer to dive deep with that approach. How amazingly lucky! No BS, it was like, Thinking School. The highlight of my summer was proving the Midpoint Theorem: For any two points A and B, there is a third point C on the line AB such that AC is congruent to CB. Don't ask me why it was so hard, but I had fifteen pages of notes written in ink. (I always wrote in ink, partly from an arrogance of errorfree writing, but mainly for the synesthesia. Pencils are like fingernails on chalkboards: ugh.) It took two whole class days, and about 10 chalkboards, to present the proof from axioms to conclusion. Once Dr. Deaton said Hey you can combine those two steps by symmetry. Other than that, I talked and wrote, and talked and wrote, and the whole classroom listened silently the whole time, and at the end, it was true. Pinnacle of my math career. Truth be told, it did not make me a scholar. It did make me unafraid of thinking about things that might be hard. Mathematical gumption, or mathematical presumption, perhaps, is the gift. It's what mathematicians have. Okay enough with the history and self-congratulation, and please forgive me that theme. Our story passes on to quite recently, after many years with that experience in the back of my head. I thought I'd look at those axioms once again. I got in touch with Dr. Deaton, now retired, got a copy of the axioms, and I started reading. And I discovered something deep and amazing, that I would like to share with you. Did I get your attention? Great, that was the attention getter bit.
SignificanceAh, significance. I think this is pretty significant. I'm delighted and excited by it. Why?Humans, like us, think we understand space. Indeed geometry is the mother knowledge of mathematics and logic, going back to Pythagoras and Euclid. My own view is, perhaps even worms intuit or understand space -- perhaps in much the same way as we do (as I argue here) -- and perhaps spatial awareness is the organizing basis of math, logic, and information-integration, and consciousness itself, and their ability to be amazingly logical and mathematical. Like, if you have space represented in your brain, and you experience stuff as spatial, or at least part of you experiences stuff as spatial, then you can read out the logic of stuff right from your spatial representation. Just like Mr. Venn taught us, any spatial arrangement of stuff drives a compelling logical understanding of that stuff. Of course you can do spatial reasoning, but that also gives you distance and count and number and all of math in there. That's the reason that the math don't lie, and that's why logic is incontrovertible, and how we evolved to be the kind of creatures that can think mathematical thoughts.But it seems our organismal thinking capacity, which emerged from the incidents of our evolutionary history, has certain arbitrary limitations, in terms of the richness of the spaces we can think about and intuit naturally. Our retinal maps are only 2 dimensional, after all, and with stereoscopic merger we get a sort of 2+ dimensional view, more like a stack of cards than a full, rotatable 3D space. Add animalian travel, going along through it. Add proprioception, sensing our own bodies multidimensionally. From these we seem to have a pretty good sense of, at least a piece of, locally centered 3D space. That's what we're built with. So we are not exactly super flexible about different numbers of dimension. As you know, most of us have quite a hard time above 3 or 4 dimensions. Einstein and Reimann blew our minds saying time is the fourth dimension, since most of us can only barely imagine a fourth dimension by thinking about three dimensions evolving through time. In a way it is as if the time dimension were, like, all around us, while we can only try to look side-to-side, to try to see it. We also have a problem, a sort of congenital disability, thinking about zero dimensional or at least negative dimensional objects. My teacher Dr. Deaton told me, that smart folks have said, such things are uninteresting -- they don't exist! Do negative numbers exist? Or infinitesimal points? Or imaginary numbers? No. Do perfect abstractions like numbers exist, apart from the things we can actually count? No. We concoct them and name them and can write expressions using them in the same way that language does not have to be real or true or free from paradox.
Maybe the paradox asserted implicitly by negatives is merely made evident by imaginaries. Just a passing thought, triggered by noticing that the Sqrt of -1 is a reexpression of the unobserveable number -1.But do we find them interesting? Certainly. So let's not be prevented from thinking about it by other people's limitations. Now, suppose we were indeed able to develop a new way of thinking about geometries and spaces and dimensionalty, we might actually overturn the basis of everything. What you thought you knew, might not be so. What you thought could not be, might be, or is. And perhaps from the new perspective, as I would assert, the different mathematical sciences of knowledge and of space might become integrated just a little bit more. That would the significance, the sweep of things, if you are willing to bear through with my argument, which is given below.
CredibilityBut hold on, my Staples' Public Speaking card says I should establish credibility before making my argument. After all, Tom Veatch is a mere retired plumber, you might say, returning to math in an advanced, perhaps demented, age. Who does he think he is? Oh, but I have a great answer for that, which is simply my lineage of teachers. Oh my goodness! The world-wide tree of math teachers is well documented, and produces this, for me:
Ed Deaton became my higher-math professor at the National Science Foundation Student Science Training Program in 1978, where we used the "Moore Method" to make a special study of geometrical proof, particularly focussed on the axioms derived from the work of "Hilbert, often considered the greatest mathematician of the 20th century". From Dr. Deaton, I receive such great good fortune! Through him I have studied under this lineage of historical geniuses from the obscure Regiomontanus to Copernicus, Leibniz, Bernoulli, Euler, Laplace, Lagrange, and Poisson, on top of my American lineage, which even touches the greatest American math teacher R. L. Moore. These include numbers 4, 7, and 10 on the Top 10 list of the greatest mathematicians in history not even including Hilbert, #9. The card said, establish credibility, so there it is. I find myself somewhat uncomfortable going on like this, to tell the truth. Really, one shouldn't have to do such a thing: credibility ought to come from the truth of what one says, irrespective of pedigree. But people do seem to like it, and I do want you to properly consider what I'm saying, so I submit.
ThesisFinally, it's time for my thesis. I will be brief.
TL;DR: When Geometries of some dimension, like planes of dimension 2 for example, intersect with each other, the result is a Geometry of lower dimension, in this case a line of dimension 1. On the other hand, Geometries can project, or combine with each other, into higher dimensional geometries, as for example two (crossing) 1-dimensional lines project an entire 2-dimensional plane. But intersection and projection are perfectly general; they aren't limited to relationships between geometries of certain specific dimensions. They can be used to increase dimension as high as you want, 2 to 3 to 4 to, well, keep on going, why not? And they can be used to lower dimension as low as you want, 4 to 3 to 2 to 1 to, watch it, 0 and -1 and -2, et cetera. Not only is it possible but Ockham demands negative dimensionality, since it falls out from the more economical statement of the axioms. What does it mean? We will consider that here, after first demonstrating it true.Our introduction is complete. It's time to turn hand waving into proof, or at least a deeper persuasion. Let us begin, then.
Hilbert, Redundancy, OckhamThe great Hilbert axiomatized geometry as follows:
Thoughtful as Hilbert is, he doesn't seem to have focussed on economy of ideas. It seems simpler would be better. Trivial examples include:
A Simplified AlternativeAllow me therefore to offer a more minimalist axiomatization, indeed one which could extend to higher numbers than three, or indeed to lower numbers than zero:
G.2 captures I.3, I.8, while G.3 captures 1.1, 1.2, 1.4, 1.5, 1.6, 1.7. For two distinct-but-intersecting D-dimensional geometries their intersection is a unique D-1 dimensional space, and they project a unique D+1 dimensional space. Thus distinct lines (D=1) intersect if at all in a unique point (D=0); distinct planes (D=2) intersect if at all in a unique line (D=1); presumably we can say "etc., etc. for higher dimensions". Intersecting geometries have a full sub-geometry in their intersection (1.7) so that the intersection of planes is a line, not a point (etc., etc.) Professor Harry Goheen pointed out in his Forward to Hilbert's Foundations of Geometry, Hilbert showed not only a "model for the axioms of geometry" but also proved that "any [other] model is isomorphic to [his]. It may therefore be left as an exercise for the reader to add to the above Geometric Axioms what is isomorphic to Hilbert's other Axioms of Order, Congruence, Parallels and Continuity. This axiomatization, like Hilbert's, was first drafted constructively, or procedurally, in a form similar to computer code whereby one starts at zero, and moves incrementally up through the dimensions. But then I thought, let's try to build it functionally, so that the relations between higher and lower dimensions can be specified, and a whole infinite structure simultaneously given its laws, so that a person or computer thinking about it could travel not just upward and not just up to three, but farther, and also downward, and not just down to zero, but farther.
Projection for example could be defined using a Geometry plus a Point outside it, as Hilbert did, but that approach might not work with negative dimensions, and if we want to consider dimensionality in the abstract, then getting away from particular dimensionalities like D=0 Points, would seem a step in the right direction. Perhaps G.1 could be done away with, considering this recurrence is more local and uses only adjacent-dimension Geometries. Occam's Razor, the force of economy of thought, has led me to propose G.1-3 as replacements for I.1-8, but also leads to a surprising outcome.
ConsequencesThis axiomatization offers the possibility, indeed requires, for it to work, that there exist Geometries of negative Dimension (which some might consider a priori as a reason to discard the whole, but for the sake of argument, let us proceed: I don't think we have actually contradicted ourselves yet).What is the intersection between points? Naively one could say that if they are the same point, the intersection is the union, they are not distinct; and if they are distinct, not the same point, then there is no intersection, or the intersection is empty. But in the current view, a point is just an N-dimensional Geometry where N=0, and (following G.3), Geometries of yet-lower dimension, intersecting yet distinct from each other, somehow Project the next higher-dimensional Geometry. So given two distinct-but-intersecting D= -1 dimension Geometries, they would Geometrically Combine (G.3) meaning they Geometrically Project a D= -1+1=0 Geometry, namely a point. If we take this for granted as obviously true for D=0's (points) projecting D=1's (lines), and for D=1's(lines) projecting D=2's (planes), and D=2's(planes) projecting D=3's(spaces) and imaginably into higher dimensions of time etc. (see footnote 1, below), then why shouldn't we be open to thinking it might be true with D= -1's projecting D=0's? And the same argument goes for intersections, going from D=3 down to D=0. I have forced myself to be open to persuasion on this, sorry for the pun, point! Let us say it can, then. So then let us comfortably consider that a point indeed does not, cannot, intersect as a point with another distinct point (otherwise they would be the same point rather than distinct points), so far so good, but I note, rather less comfortably, that this might mean that G.3 could not apply to project a line from these two points. See my issue? (The conditions in G.3, for projecting higher dimensions and intersecting in lower dimensions, are the same conditions, so if we want to project lines from points, we have to also meet the conditions and be able to Intersect points to make D= -1-dimensional Geometries). Perhaps we can imagine, then, a Geometry in which points intersect one another in negative-dimensional sub-Geometries of Dimension D= -1 where there is a D= -1-dimensional overlap between two D=0-dimensional points, and in such a case two points, distinct, but now in a lower dimension intersecting, D=0 dimensional Geometries, could therefore be considered using G.3 to Geometrically Project a D=1-dimensional (line) Geometry. So negative dimensionality rescues G.3, enables us to go up as well as down from D=0 points and thus to project lines from points, and in short we can go ahead and believe that one generalization applies to all the dimensions up and down. So one might say Reality Intersects Theory only for dimensions 0 to 3, plus or minus 1 (we need D=0-1 so that Projection can apply to something to bring us D=0 Points, and D=3+1 is needed so that Projection of 3D spaces is tolerated too, as G.3 requires).
What does it mean?The mental projection of higher dimensional Geometries than the third dimension, for example, on analogy with a time dimension or with higher-dimensional ordered n-tuples of vector spaces are of course part of learning higher math.But what it might mean to be negative-dimensional, somehow connecting points outside of space and time, may be a Zen exercise for the intuitive, but perhaps formally no more so than to imagine "point" objects of zero spatial extent. In any case it can be no objection to negative-dimensional Geometries that there is nothing, or less than nothing, there, since there is nothing in a point, either. One possible analogy for negative-dimensional geometries might be taken from the intersecting of a line(D1) with a plane(D2) in a point(D0): imagine a sort of interpenetrating space with a locally conical set of rays coming from a D= -1 de-point located out there somewhere, sending rays to every actual D=0 point in the universe, and thus rather conically sending rays to all the points around here, for example, and having those rays pick out and define actual points in actual space only upon intersecting with the higher-dimensional space of Reality. The analogy seems imperfect since lines-intersecting-planes-in-points is a D1/D2/D0 relationship while the analogy proposes imagining a D-1/D3/D0 relationship. This analogy might tempt a physicist to consider whether Gravity, which interconnects all space through some mysterious fabric ungoverned by the normal laws of space and time, is a Geometry of negative dimension. But no, subGeometries are smaller, so I would consider Gravity the other way around, as a phenomenon expressed within the Geometric Intersection of some higher-dimension Geometries. Another imaginable view on dimensionality is as a sliding window on information dimensions.
Taking as primary the perspective of the analyst, the universe may be considered as an unbounded set of informational dimensions, and the sorting and grouping of dimensions considered as subject to the interest and focus of the person looking at them, but not necessarily assuming that the list of dimensions under consideration starts at a beginning, but rather that there's a bunch before, some that we're thinking about now, and a bunch after, since after all everything is in this universe of information and why we would assume our attentional focus happens to pick out the first seems presumptuous and arbitrary. So we're looking at some subset, in the middle of an ordered list of dimensions. A fact or scenario, i, in such a universe might be an infinite cartesian vector we could represent as (...,xi, yi, zi, ...), incidentally seeming to solidify the cloud of vagueness with names for some three of the dimensions in the list. We're just looking in the middle of the list somewhere. Now suppose we decide to arrange things so up/down, forward/backward, and left/right are adjacent, but there are some other parameters for each situation under consideration in the preceding and following positions or dimensions. Okay, then a point is picked out by setting x,y,z to particular values; setting two of those dimensions to constant or independent values picks out a line (making the third a linear function of the other two instead of a constant, would make the line non-orthogonal in the particular basis being used); setting one to a constant picks out a plane; etc. In the conceptualization of any set of continuous dimensions, one could sort them in one direction or another, and one could start in the middle of a list, for example. Then starting from some arbitrary index within the list, if we pick the next three values, we can envision them as distributed in a 3D space. Well, equally we could go backwards in the list and talk about the values preceding those selected by that arbitrary index. Which dimensions are those, when they precede our arbitrary starting point? In such a system, dimensionality is rather like attentional focus, in that we may say the dimensions we are looking at now correspond to a 3D space, but we might shift our attentional focus to a different set, a re-ordered set, or if an ordering is externally imposed, then a set preceding in the order the ones we were looking at a moment ago, and these would be the negative dimensions of our informational space. Along these lines, the dimensions of a given 3D Euclidean space may be considered as based on viewing as constant the contextual information or preceding dimensions, for example, viewing time or any other analytic precondition as having a contextually-specified value and being thus the preceding or lower dimension (D=-1) in an ordering of the dimensions that places the properly spatial dimensions as dimensions 0, 1, and 2. In such a view, time is a negative dimension, not the fourth dimension, but it may be stressed that these Geometries, Dimensions, and Orderings are all analytical conveniences taken up in one view or another but not exactly representing the Truth out there, which is a matter more of Physics than this abstract Geometry. So G.1 and G.2 which assert a kind of Reality on which to base G.3, could be internally dispensed with, and supplied by external considerations: the analyst who decides what are the dimensions of interest, how to group them together, which to focus on, etc., may be depended upon to supply the initial Point of G.1 and distinct-yet-intersecting Other Geometries of G.2. The job of abstract geometry's 8 Incidence Axioms are here reduced to a single statement, G.3, and the job of applied mathematicians is to separately put forth objects of their own understanding and interest, to which G.3 then applies, in order to think Geometrically about their problem.
De-GeometriesIs it time for some terminology? So the English morpheme "de-" means "away from", shares /d/ with "dimension" and shares "de" with Dr. Edmund I. Deaton my teacher, and while I don't wish to demean Hilbert whose shoulders we are standing upon here, "de-" also shares initial /d/ with Dilbert, a sort of anti-Hilbert. So perhaps we might call, for convenience of reference, a D= -1 Geometry a de-point, a D= -2 Geometry a de-line, and a D= -3 and D= -4 Geometries de-planes and de-spaces. Maybe unpoints, unlines, and unplanes will prove more popular, but I'll use de- for now. Let us clarify the requirements of G.3 here: We would need two distinct D= -1 de-points Geometrically Intersecting each other (in a D= -2 de-line) in order to Geometrically Project a D=0 point. Similarly two distinct D= -2 de-lines Geometrically Intersect in a D= -3 de-plane and Geometrically Project a D= -1 de-point. The recurrence relationship of G.3 can be followed to send our thinking back to de-spaces, etc. ad infinitum, as far as you may like. Now it is clear that the names have a reversal in them in that de-point vs de-line reverses the relationship of intersection and projection found between point and line: points project lines and are their intersection, while de-lines project de-points and are their intersection. Yes, according to G.3. However! the difference between projection and intersection is primarily that they are opposite in picking out a higher-or-lower dimensionality of the uniquely so-selected adjacent-dimensioned Geometry, but the construction is almost perfectly symmetrical and so it is worth considering whether the polarity we happen to choose first is non-essential. If we started with de-points instead of points by renumbering dimensions as d = -(D+1), and relabelled "intersect" and "project" each as the other, then would the same properties of Geometry still continue to follow throughout? Is there much difference between the world we think we are actually in, with D=0..3, and the de-world with D= -1..-4? If we were in the other, would it make any difference? The only difference is the direction of inclusion when we use the word "distinct", when distinct Geometries intersect, as for example, the only difference between the Fourier transform and its inverse Fourier Transform is a negative sign, or between considering a line with one end in the positive direction and itself the other way around, is that single bit of orientation, even while multiple levels of infinity are packed into any segment thereof. We could swap intersection and union, along with Geometrically Intersect and Project, and have the same world inside out. So we can in imagination turn the dimensional increase and decrease around the other way and get a de-Geometry which looks rather similar to the Geometry. Now I think I have proven:
Footnote 1: We have stretched our minds to think of negative dimensions in which D=0 points might intersect so as to enable G.3 to apply thereby gaining as a theorem the proposition that two points project a line. Let us also briefly consider how two D=3 spaces might intersect to make a plane. Two 3D spaces might conceiveably (a) not intersect at all, or (b) if they did intersect it might take some thought to imagine how. Consider as an example time as a fourth dimension. Then an entire Euclidean 3-D space at one moment in time would have no intersection at all with the same entire Euclidean 3D space at any different moment in time, because nothing in the first is at the time of the second. Thus (a). How then might we construct (b), an intersection of 3D spaces? As a thought experiment imagine something traveling like a photon or a flying camera dolly straight through a 3D universe from one end to the other, so to speak, or at least through our bit of it here, but having something like a 360-degree camera looking out in all directions but only perpendicularly to its path, and able to see or index everything in that perpendicular plane at that instant (disregarding the limited speed of light in this idealization). Then every point in the universe would at some time be scanned by our travelling time-indexing camera, and every point that is scanned or indexed at the same moment is in a plane with all the other such points. Now this collection of planes at different times can be considered to be a 3D space, because the planes are 2D and the travelling camera's location-and-time is a third dimension projecting planes along its infinite flight path. It just has newer planes in one direction, and older planes in the other direction, but is equally infinite, continuous, and geometrically dimensional as any other 3D space one might imagine. So far so good. Now what is the intersection of this traveling space with our usual Euclidian 3D universe -- taken of course at one time, say, now? Of course the intersection is that plane picked out by our travelling idealized camera at the current moment! Footnote 2: A challenge stands here: can all of geometry be derived from an axiomatic treatment built around the general rule of combining geometries, G.3? It is an open question. Footnote 3: I asked Dr. Deaton what he thought of this; he asked some mathematician colleagues; they said this is "uninteresting" because there aren't any existing objects they can imagine that can be thought of using de-points, de-lines, etc. Hence: Footnote 4: After Geoff Szabo emailed me today I gave him an idea of my current way of thinking about this, which has some practical angles to it. Here's what I wrote. |