"The raw result of experience
may be expressed by the relation
A = B, B = C, A < C
which may be taken as the formula
for the physical continuum."
HENRI POINCARE.

fuzzy geometry
TIM POSTON

The refined result of experience may also be expressed by Poincare's expression above (taking ¹ instead of < if we want more than one dimension), since not merely our sensations but our instruments are limited in the fineness of their discrimination, and limited essentially - not merely by the coarseness of their design and manufacture, which might be corrected. Although this fact is connected with the Heiseriberg Uncertainty Principle, it does not follow from it in a straightforward fashion; though the principle predicts uncertainty in combined measurements of position and momentum, it permits arbitrarily fine discrimination in either at the expense of a correspondingly large irreducible error in the other. However, it is easily seen as follows:-

Suppose we wish to measure, in centimetres, to fifty places of decimals. Physically, all length measurement ultimately involves comparison with a wavelength, and to get that fiftieth decimal place with a minimal expenditure of energy we need something with wavelength around 10^-50cm (see e.g. (2)). And we need at least one quantum of it (remember, everything comes in quanta except - in ye olde quantum mechanics - the space and time it happens in) and we need that quantum to react with what we are measuring. Now.'react with' means to exchange some energy with. But a single quantum with this wavelength would have energy equivalent to the total annihilation of 20,000,000 metric tons of matter; an H-bomb annihilates a few micrograms. So 10^-50 cm is not a length: it is a catastrophe.

A fortiori, 'arbitrarily short' lengths are ridiculous, and a fortissimi the distinction between a rational and an irrational number is physically unreasonable if not downright irrational. This distinction - very significant in the theory of dynamical systems - demands not merely arbitrarily fine measurement, which is bad enough, but an explicitly infinite degree of accuracy. All very jolly as mathematics, but as PHYSICS?

So, let's drop these infinities.

But, says Prodnose, both the twin gods Continuity and Differentiation are meaningless without 'arbitrarily short' lengths.

The differential of f at x is defined as the number approached arbitrarily closely by f(x+h) - f(x)/ h as h is made arbitrarily small, and continuity in the case of the real functions boils down to taking the limit of any sequence to the limit of the images of the points of the sequence where for p to be a limit means precisely that the points of the sequence all come within any selected 'arbitrarily small' distance of p after a finite number of them have been discarded.

So scrap the differential calculus.

Scrap continuity (and hence topology).

Scrap the ghostly real numbers themselves, which are defined anyway as the collection of limits of rational number sequences.

Quite a programme. Quite a lot of toys gone down the memory hole.

What have we left for describing the universe? Differential equations, after all, seem rather fundamental to physics. However, George Boole (the inventor of symbolic logic) showed last century (1) that difference equations, where d/dx is replaced by D_X (where D_X f at a is defined as f(a+h) - f(a)/h for some standard difference h; no going to the limit) often have solutions that are approximate solutions to the corresponding differential equation. Ever since, they have been used increasingly - especially since computers have been available to do the donkeywork - precisely to find approximate solutions to particular differential equations that people believed described something. (They are well adapted to computers since all those built so far are finite, like differencing, not infinite, like differentiating.) But what actually happens in a physical system is essentially approximate too, by the arguments above (or at least we can never prove otherwise), so the idea that an 'exact' solution is better than an approximate one results from faith in a differential universe; rather dubious metaphysics if an answer as close to the physical facts as you can measure emerges from a finite computation never mentioning limits at all.

So let us take the basic observation that points/electrons/lengths sufficiently close together are indistinguishable (electrons in principle - not merely experimentally - become indistinguishable when close enough, even in the present theory) and start from there. I will state the basic definitions that are useful, and then give examples.

DEFINITIONS AND EXAMPLES

A fuzzy space is a set X with a fuzzy: a symmetric. reflexive relation denoted * on X (symmetric and reflexive means that if x*y then y*x, and that x*x always). If x*y we say x is indistinguishable from y.

A map f from X to Y (fuzzy spaces) is fuzmic if x*y implies that f(x)*f(y).

Two maps f, g are indistinguishable if x*y implies f(x)*g(y). (Notice that this means that a map is fuzmic if and only if it is indistinguishable from itself. Nice.)

The preceding definition gives a fuzzy on the set Y^X of fuzmic maps from X to Y, and the resulting fuzzy space is the function space. We have a nice theorem that if T is another fuzzy space (Y)^XT.is exactly Y^XxT called the exponential theorem, which is true in topology only under strong restrictions, or in general via bizarre expedients.

(There is a lot more material of this 'like but unlike general topology' kind, with resemblances and tricky deviations. For the topologist I will mention, e.g.: homology and homotopy groups are defined, with analogous properties to the topological case; the homotopy extension property is so rare as to be useless; fibrations are more in evidence - the fuzzy Hopf map is hanging in my window; the analogue of a local homeomorphism must be a covering; there is a Brouwer-type approximately-fixed-point theorem. For the non-topologist I would need more space than MANIFOLD can spare me to go into that area. It is all done in mind-battering detail in (7).)

Examples of fuzzy spaces are legion. Take the points in your field of vision, and the fuzzy of being unable to distinguish them. Take the set of all sounds, and a similar fuzzy. Take the set of all pictures on a screen, with again a visual fuzzy, and the map from time to this set defined by projecting a film. At indistinguishable moments you see indistinguishable pictures, and the result is perception of continuous motion. Mathematicians have spoilt the word "continuous" by identifying it with the bizarre definition discussed above, so, as the map Time --> Pictures in fuzmic, it is better to say that what we perceive is fuzmic motion.

Another useful example of a fuzzy space is the atoms of this page with the relation of being less than one centimetre apart. (So points closer than 1 cm. are to be thought of as indistinguishable.) Let us label as A and B, say, two points of this space:

Now the other points I have indicated give a chain of hops less than one centimetre and hence from point to indistinguishable point, which gets from A to B in 28 hops. Call such a chain a path in the fuzzy spaces and the number of hope its length. Then define the hop distance d(A, B) between A and B as the shortest length possible for a path between them. This obviously.obeys the usual rules for a distance function ( d(A, B) = d(B, A), d(A, B) = 0 if and only if A = B and d(A, B) + d(B, C) ³ d(A, C)), and moreover in our example it is plainly the original distance in centimetres rounded up to the next whole number. In other words, from the local (small-scale) indistinguishability alone we can recover approximately the global (large-scale) geometry we had to begin with.

From just installing the assumption that the centimetre is our limit of accuracy, as a piece of mathematical structure we get a description of the complete geometrical properties of the space, up to that same limit of accuracy.

Now if we take not the atoms of a page, but the collection of physically definable positions of something as small as we can measure ("points,of space".for short, since these are what Euclid idealised into his geornetrical points), and 10^-13cm. instead of 1 cm, the above discussion of the geometry again applies, but with the extra wrinkle that many physicists believe that we can't get more accurate than this distance anyway (see (5)). So we get this time a description of the complete geometrical properties of space itself, UP TO THE BASIC ACCURACY OF ALL MEASUREMENTS, and hence a description as complete as can be tested, which simply means 'complete' for all purposes except metaphysics. And it is a great deal simpler than the present description built up from the real numbers with its nondenumerable infinity and weird theorems (e.g. you can take the unit ball in three dimensions and divide it into five pieces such that by rigidly moving them around individually you can assemble a ball twice that size; or - with more pieces - a ball a million times the size, or a millionth. I wouldn't believe it if it hadn't been proved (4) and I still don't). So why not suppose that space is in fact fuzzy rather than Euclidean in structure, and instead of saying "points are indistinguishable if they are less than l0^-13cm. apart" say "1 cm. is about 10¹³ hops". We can then even suppose that space has only a finite number of points in a given volume witout losing anything testable from the geometry. Of course, to suppose that space is in fact anything at all is contrary to the Way. But this absurd activity is what Physics in all about.

Similar considerations apply to measuring short intervals of time as with short distances of space, and the idea of time as a fuzzy continuum is equally well motivated by perception; just as a newspaper photo builds a visually continuous line out of visually indistinguishable dots at one moment, a film builds visually continuous motion out of visually indistinguishable pictures at visually indistinguishable moments. .

STATIC PHYSICS

We can describe more than just lengths (and hence for instance the principle of the lever) admirably in this fuzzy description of space. For instance Stoke's theorem holds, and Dirichlet's principle that a certain integral - here replaced by a sum - which represents the energy in a distribution of potential achieves its minimum for the unique function P which is harmonic (that is, the Laplacian is zero) is true as in traditional fashion. (Physicists will recognize these as the basic tools in the theory of static electric and magnetic fields.) The latter was first proved in the 1920s (3) but since the statement only mentioned the very special fuzzy space consisting of the lattice of points with integer coordinates in three dimensions (though the proof did not in fact use its special character), and since no one could suppose that space is as rectangular as that, it was taken as essentially a contribution to the numerical approximate solution of differential equations, rather than directly to mathematical physics.

Poincare's philosophical view was firmly that the physical continuum is a fuzzy space (6), though in his lifetime (1854-1912) the mathematical tools for investigating it were not around if he did not invent them for himself. (Many of the concepts that apply very fruitfully are descended from his topological work.) If someone had related the difference equation results of the 1920s to Poincare's thoughts, who knows? Perhaps the description of spacetime would have been quantized almost as soon as that of the physics occuring in it.

DYNAMIC PHYSICS

Of course, what I have outlined above is a discrete description of space only, not of space-time. Basically, it is a co-ordinate-free approach to elliptic difference equations in particular the analogues to the Laplace equation

and the Poisson equation

The fuzzy description of space-time must involve coordinate-free hyperbolic difference equations, in particular the analogue to the wave equation

Now in this case the behaviour of the finite difference wave equation models that of the differential equation perfectly, not just approximately, and equations of this type have been the most successful models of physical behaviour (e.g. the Maxwell equations written relativistically - their clearest form - and the Schroedinger equation) so the ground looks promising. But the D'Alembertian

² = Ѳ - d²₂/dt²

is not a straightforward generalization of the Laplacian, and some very deep questions are involved. The nature of causality in particular is tied up with the operator ², which distinguishes between space-like and timelike (and thus causal) directions. A space-time is a much richer structure than a space, and one regrettably ignored by pure mathematicians, who write in the only language I really understand (essentially a sort of highly abstract picture, once you get into it. There is something much more visual/geometrical - and thus in my view more physical - about the mathematicians' approach than in the endless indices of physics.) One thing clear already is that the coordinate-free discrete equations will be impossible to frame non-relativistically, so that its proper formulation would be a bridge between the only uneasily related theories of quantum mechanics and relativity. Which, in the words of the ancient lore of Atlantis, is a nice trick if you can do it.

Of Light and Caterpillars

So space-time presents problems. But the idea of space-time, rather than space and time separately, is a farly new idea, so let us suppose that fuzzy geometry had been adopted as a way of looking at things rather earlier. Suppose it to have been invented not by me (whose contribution has been the development of the theory) nor by Chris Zeeman (who invented it for describing the brain (8), and used it to prove the possibility of seven-dimensional vision and the existence of sleep) nor even by Poincare (who also invented it (6), and used it to prove that vision is three-dimensional, but was too busy founding topology and campaigning, for a decimal clock to develop it mathematically) but by Sir Isaac Newton, and consider first the motion of a particle in space.

Now the description of such motion in Newtonian terms consists of giving a position for each moment of time, with the requirement that this be continuous. We've scrapped continuity, however, so we assume instead a fuzzy on space and a fuzzy on time, and require that the particle occupy indistinguishable positions at indistinguishable times, as above with a film. So the description of the motion is a fuzmic function p:

p: TIME --> SPACE
t --> p(t) = position at time t.

Next consider the particle at times t₁, and t₂. We must have the hop distance d(t₁,  t₂) equal to some number n, and thus there exists a path

t1= t0, t1, t2, ......, tn = t₂

of length n from t₁ to t₂, where each t_i is indistiguishable from t_i+1. Then since p is fuzmic we also have a path

p(t₁), p(t₁), p(t₂), ........, p(t₂)

between the positions of the particle at t1 and t2, also of length n. Now this may not be a minimal path, but the minimum cannot be longer (if x is in X x ³ min(X) by definition). So

d(p(t₁), p(t₂ )) £ d(t1, t2 ) always.

In words, between times t₁ and t₂ the particle cannot go more than a distance equal to the difference in time ~ equal not proportional here, because we are using fundamental units; measuring both space and time by the hop distance. (This distance-decreasing property, of course, applies to all fuzmic maps of fuzzy spaces, and means that fuzzy spaces cannot 'stretch' as topological ones do; if topology is india-rubber geometry, fuzzy is chain-mail geometry. This restriction is physically natural: you can't stretch an atom either.) Just as a fuzzy structure gives you a distance function with no extra assumptions, we have assumed only fuzmicity of the motion to get a limiting velocity, with no mention of relativity. And it gives more.

Consider now the motion of an object, not a point mass, in space. For the sake of a simple diagram we will take a one-dimensional space consisting of a row of points with each point indistinguishable from its two nearest neighbours on each side, though our discussion will apply equally to three dimensions, and an object O, five hops long. Now a position of the object consists of a position for each point of it, i.e. a map f: O --> SPACE, which must obviously be fuzmic. (The object must not be broken.)

A description of its motion, then, is a map for each moment such that the maps specifying positions at indistinguishable moments are indistinguishable as maps. Recall the definition of this and consider a 'next position' after f, which is to be indistinguishable from f. The map

will not do, as O₁ and O₂ are indistinguishable but f(O₁) and g(O₂),are not. So g is not within fuzzy of f. It can in fact be reached from f, but only by a path going from f to

Thus O takes six hops to travel a distance that a particle could go in one, and so moves at only a fraction of the speed. If we want it to go faster, we must start pushing a wrinkle through before the first one reaches the front: a representative position will then look like

and it will then go twice as fast. In general, the faster we want to travel, the more wrinkles we must have travelling through it, and the shorter its image in the spaces e.g. at a third of the limiting velocity it must look like:

for instance.

Only via maps of the form

can it travel at the limiting velocity. This is illustrated rather more ornamentally by the caterpillar on the cover. A caterpillar moves by pushing a wrinkle through from the back, and moves faster by pushing more wrinkles through, the hunching up involved making him shorter. (Your usual creepy-crawly, however, is not in such a hurry as Aliice's friend on the mushroom seems to be; we don't need to be told what's in that hookah of his.) Going faster and faster, he moves more and more and tighter and tighter wrinkles go through till he reaches the limiting velocity. At which point he acquires infinite energy and loses his cool.

To summarize, we have
(a) There is a limiting velocity.
(b) Rigid body motion is impossible; motion is essentially a wave phenomenon.
(c) Wave phenomena themselves, such as the compression waves passing through an object, of which the object's own motion is a composite, travel - unlike the object itself - at the limiting velocity. Thus if light itself is a wave phenomenon (not of course Newton's view) it may be expected to move at the limiting velocity; put the other way round, the limit is precisely the speed of light. Light waves are exactly waves of motion passing through and moving the electromagnetic field, and it is precisely in this sense that electric currents in a wire move at the speed of light.
(d) Faster motion in any direction involves contraction in that direction until zero length is reached at the speed of light, than which nothing can go faster. Be it noted that (a), (b), (c), and (d) are all deduced from only the replacement of "continuous" by "fuzmic" in classical mechanics, with no relativistic assumptions.

These results are familiar, of course, but not in the context of Newtonian mechanics. (Some people are less familiar with (b) but it follows from the normal theory of relativity: if a body is 'rigid' the speed of sound in it is infinite - not even lightspeed - which is impossible. The occurence of rigid bodies in books on relativity follows from the normal theory of the psychology of physics text book writers.) They do not amount to a properly relativistic outlook or a physical theory - every observer would agree on all bodies shortening in relation to the same fixed frame. contrary to experiment - but they seem highly suggestive. Exactly what do they suggest?

That is the object of my current research (i.e. I don't know). But fuzzy geometry seems to be rich enough mathematically to have a lot more interesting things to be done with it yet, and the horizon-possibility of a quantized space-time theory is enough to cause me to continue to try to learn some physics. A propos of which, I will end with a heartfelt appeal. At the end of a seminar I gave last autumn I asked if anyone, please, knew what the Laplacian operator 'really is', and a suggestion a few days later put me on the traill that led to the right fuzzy version. If you have a better mental picture of the D'Alembertian than

d2/dx2 + d²/dy² + d²/dz² - d²/dt²

let me know,buddy.

References
(1) A Treatise on the Calculus of Finite Differences, George Boole
A very nice bit of reading, once you get used to the solidly Victorian prose, and remarkably comprehensive as a basic text even now. The bizarre idea of continuity and differentiation, introduced by Newton, was still causing confusion two centuries later, e.g. " ... whether in the proper sense of the term limit we can regard sin x and cos x as tending to the limit 0, when x tends to become infinite.", p.271. Such difficulties have finally been sorted out with the rise of topology, but why worry? Scrap continuity.
(2) Science and Information Theory, Leon Brillouin
A splendid account of the uncertainties inherent in measure-ment, deterniinism, etc., analysing in depth among other things the relation between entropy and information. For a more popular approach, and a very strong blast against infinistic physics, read "Scientific Uncertainty and Information" same author.
(3) Uber die Partielle Differenzengleichungen der Mathematischen Physik, R.Courant, K.Friedrichs und H.Lewy; Matliematische Annalen, 1928 pp. 32-74.
If you want to learn mathematical German, do it by working through this paper. I do not think I have ever read such a masterpiece of mathematical writing. Beautiful material beautifully presented.
(4) On the Congruence of Sets and their Equivalence by Finite Decomposition, W.Sierpinski (Lucknow Univ. Studies)
An interesting, readable little book.
(5) An Analysis of the Structure of Space-time, Roger Penrose, (Birkbeck College, London)
Mainly about spinor methods in cosmology, but the opening section dissects interestingly the possible alternatives to the idea that space-time is a differentiable manifold.
(6) Henri Poincare: (i) Science and Hypothesis (ii) Science and Method (iii) The Value of Science (iv) Last Essays
Essays of the last universal genius of mathematics, covering a great deal of ground. For his discussion of fuzzy spaces see ch. II, (iv)-ch.III,and in greatest detail (iii) ch.III. His definition of dimension is not quite satisfactory (c.f. (7),I.5.3).
(7) Fuzzy Geometry, Tim Poston (Maths.lnst.U. of Warwick)
Covers everything mentioned here, sets up discrete vector fields, forms, etc, defining a Laplacian, generalizing that of (3) by this approach. Some light relief, but full of pernickety detail.
(8) Topology of the Brain and Visual Perception, E.C.Zeeman (in Topology of Three-manifolds, ed K.Fort)
Very entertaining. Imaginative and uninhibited, towards both maths and biology, but somewhat shaky in parts. My introduction to the subject.