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 fund the NSF!
Hilbert axiomatized geometry as follows:
Thoughtful as Hilbert is, it seems simpler would be better. For example:
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 lines(D=1) intersect if at all in a unique point(D=0); 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 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. This 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 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). 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, as 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. 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 as being the first three after 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 out 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. Is it time for some terminology? So the English morpheme "de-" means "away from", shares /d/ with "dimension" and shares "de" with Dr. Deaton my teacher, and while I don't wish to be mean to 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, just 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 straight through a 3D universe from one end to the other, but having something like a 360-degree camera looking out in all directions but only perpendicularly to its path. Then every point in the universe would at some time be scanned by our travelling time-indexing camera, and every point that is scanned 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 infinitesimal 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. |