The role of Geometry in the interpretation of Observations
While observing objects in reality, we automatically construct an abstract framework used to frame the picture we make abstractly. We observe objects but they usually move, so we have to consider places, say, some abstraction of emptiness where an object can fit. This is a somewhat problematic concept! What ensures that some empty looking “space” we cannot observe (since it is empty) can contain some object we want to move there?
First we have to believe that when nothing is observed, that “space” is empty and free (later we may discover it was reserved for an unseen entity!). Then we have to believe that space is connected and one can move from one place to another via intermediate places. When an object is in a place, everything in the object also finds a place, but how dense is the object and how do places compare. It turns out that on the set of places, we should have some operations like the intersection, [1. What is common to both places and the union? 2. What is in one place or the other?]
In mathematics, the corresponding notion to such structure is a topology – the structure and the theory describing the site [the place structure of some space]. In the topology of space there are a priori no numbers involved. To see where these fit in, we have to define points. A place can usually be obtained as the union of two smaller places, that is it decomposes into smaller places.
Problem: Is there an indecomposable place?
Well, in abstract topology that could be easily obtained but we are looking at reality and it “looks like” we can always embed a smaller place in to a place. Nevertheless the idea of atoms was already known to the ancient Greeks, so is there an atom of place? This led to the invention of the concept of a “point” thought of as a place with zero size hence no dimension. Obviously, points do not exist in reality, anything observed as existing (and we will define that concept in Section 3 about time) in a place has a size, except effects and forces like gravity that act in a not precisely defined locality. Then we make an important assumption, every place is consisting of points, and thus it is a disjoint union of points! If only a finite number of points is considered, that only spans a place with size zero (later in Measure Theory the theoretical background for such facts is constructed). So to describe places in reality, we have to think abstractly of subsets of space consisting of infinitely many points. Sets with finitely many points are not really observable to us because these have no size (measure), space itself is then the union of all places and also consists of infinitely many points.
Now in Mathematics, there are many different infinities (countable ,non-countable, the continuum,…), so in reality how big is infinity for the number of points? In Physics, the main interest (perhaps this might be debated) is to measure by numbers so the notion of Algebraic geometry comes to the rescue. René Descartes invented the notion of algebraic geometry (analytical geometry) by associating coordinates to a point, this is done by choosing an axis system and representing a point in space by its projection on the axes associating an n-tuple of numbers to the point (where n is the dimension of space).
Which numbers are used to define a point on an axis with an origin (the intersection of the axes used)?
You may choose but there is a nice candidate. The ancient Greek believed geometry was a matter of ratio (hence fractions of integer numbers), so the field of rational numbers seems to be a first candidate. However, for approximations one would like that a converging series (one where the differences between consequent numbers becomes always strictly smaller after some steps) of numbers is also a number. Yet from Pythagoras theorem the square root of 2 was discovered and one can easily proof that that is not a fraction of integer numbers using elementary arithmetic and division properties (try this ,see if you can compete with people 2000 years ago!), yet you can write a series by taking the decimal form of that square root up to n digits and letting n grow. So you discover the real numbers and the completeness of the field of these numbers (i.e. every converging sequence of real numbers defines again a real number). Those real numbers are by their completeness suitable for practical approximation, however, note that a computer can never write down the value of an infinite decimal expression, so it can never write the number pi in numbers.
The Physicists are happy, they have the mathematical tool of Algebraic Geometry, later also Differential Geometry that allows calculations and approximation, so they want to use this for describing reality and that works surprisingly well, certainly as far as prediction of observations is concerned. But the price of using coordinates is that you have to treat objects from reality as if they are points. In mechanics that is often done by replacing rather irregular shaped objects by their gravity centre! In electricity one integrates often over surfaces of some shape, thus assuming the regular smooth division of some electrical load over all points of the surface, and so on. Obviously that is not reality where only finitely many events exist in the universe (the most common assumption nowadays, anyway it is too hard to deal with infinite many effects, finite but too big is already a serious unsolvable problem!).
The notion of atom and point has become so common that everybody thinks in these terms about reality, if you talk to somebody unprepared about nature and how we observe, almost always, they will talk in term of places and points (and lines and other derived concepts). Even it was scientifically proved that some geometric insight is in the human genes by tests on 3 months old babies who could not have learnt that from anyone in that time.
The description of real objects thus uses non-existing concepts like points and further abstract geometry, non-existing numbers like real numbers to form properties not necessarily having a background existing in reality. These abstract properties form the abstract onion in layers we view as the interpretation or image of the object (or other phenomenon). Whether that abstract image is a good approximation can only be tested via predictions of behaviour and measurement (again via the model made of the behaviour via observation), so it is largely a self- realizing prophesy but it is acceptable until a better approximation is needed and the model has to be changed.
Abstract images resulting from the logic in our cognitive system are not the reality they model, a trivial statement that may not be forgotten however!
So the question remains, what is the geometry of “real space” in reality, not in our observations?
We should construct a formal model less depending on non-existing abstractions like points, lines etc…. Moreover, the notion of measurement has to be scrutinized. The ancient Greeks wanted to measure everything with rational numbers but they discovered (and kept it as the Pythagorean secret for some time) the irrationality of the square root of 2. Now, the Physicists believe you can measure everything by real numbers (perhaps also allowing complex numbers if the square root of -1 has to be allowed). For anything to measure, you just define a unit (that may come from the mathematical formula expressing the item in other already known items) and then every amount of it is a real number times that unit. This assumes without proof or reason that the item to be measured is a one dimensional line over the reals and moreover that the scale is going to be totally ordered. To assume that for every measurement in the universe, without even any evidence, except the obvious fact that it is an approximation and perhaps not so bad at first sight, is a strange leap of faith. Look at temperature, it cannot go below the absolute zero, say about -273 °, that is not really using the real numbers in a normal way.
It is clear from the foregoing that we need to develop a geometry with places being in reality and not consisting of points, thus we give up for the moment the possibility to measure with real numbers (later I will point out how to reintroduce some measuring in the non-commutative geometry of reality. This real geometry should be related to the usual geometry used in the observed reality as explained above, this can be done mathematically and in fact it fits in viewing the reality as a deformed process (the micro-process of Section 1 of the learning process about observed reality, the latter may be identified to the usual space-time manifold), hence reality is a deformation of observed reality, not conversely since we do not know reality we can indeed not deform it and know the result!
The abstract framework given by classical geometry (algebraic, differential…) provides an abstract embedding of reality in a geometric space that allows calculations with real numbers, thus yielding some useful approximations of reality like relativity theory,sl string theory, quantum mechanics,…. The approximation is only as good as the modern testing devices allowed, it cannot be identified to reality. To get a better approximation we have to define a real geometry of reality in terms of abstract concepts that allow the real situation as a specialization, so notions have to be generic enough to include existing structure, for example: places cannot be union of infinitely many points since points do not exist and moreover we want to keep the possibility for a finite universe open.
Professor Emeritus Fred Van Oystaeyen [department of Mathematics and Computer Science, University of Antwerp, Belgium]