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For, if such is the case, the latter geometry can easily be replaced by a discrete version in any physical theory that makes use of this particular mathematical background. Straightforward as the task may seem, there are however at least two ways in which the concept of approximation can be understood.
In the sections that follow an overview will be presented of some of the various attempts that fall under a or b. However, before embarking on this journey, several caveats have to be mentioned. Are they logicians, mathematicians, computer scientists, physicists or philosophers to list the five most frequently occurring cases? Do they want to solve a mere technical, a physical or a philosophical problem? Are they worried about foundational aspects or is the object of their research to further develop existing theories?
It is worthwhile to go into some more detail for each of the five types of author s mentioned to illustrate these questions. Logicians are often interested in displaying the underlying logical structure of a theory, physical or mathematical, and in exploring whether or not there are alternatives, usually by changing the underlying logical principles. One could imagine a geometry based not on classical logic, but, e. Often the goal is to find a complete classification of all possibilities. This approach implies that the logician working on and developing discrete models, does not necessarily believe that these models are correct or true in some sense.
They merely help to understand better what classical geometry is. A perfect illustration of such an approach is the work on spatial logic, see Aiello et al. The authors compare their approach to the work done in temporal logic see the entry on temporal logic in this encyclopedia. One example: if time is linear in the future, then this property can be expressed as follows. In an entirely similar way, to construct such a language is what spatial logic wants to achieve for geometry and is thus related to the proposals that we will discuss in section 3. A mathematician might be looking at or studying a discrete or finite counterpart of an existing theory just to see, e.
This in itself is interesting from the perspective of so-called reverse mathematics. The core question is to find out what is necessarily required to prove certain theorems? See, e. Proofs that also hold in a discrete geometry are thus independent from any assumption about discreteness or continuity.
One might however go deeper into the foundations of mathematics and study finite geometries from a foundational perspective. One such approach is strict finitism although sometimes the terms ultra-finitism or ultra-intuitionism are used as well that is not meant as a subtheory of other foundational theories but as an alternative on its own. It shares with the many forms of constructivism the fundamental view that mathematical objects and concepts have to be accessible to the mathematician in terms of constructions that can be executed or performed.
Most constructivists allow for the potentially infinite, i. Strict finitism wants to go one step further and argues that an indefinite outcome is not be accepted as an outcome, since, as all computational resources are finite, it could very well be that these resources have been used up before the outcome has been reached. As might be expected, strict finitism is not a popular view in the philosophy of mathematics. Nevertheless a number of proposals have been put forward. A history and an account of the actual though now somewhat dated state of affairs can be found in Welti In section 2 more will be said about such proposals.
In the computer sciences the theories and proposals that have been put forward are of a quite different nature than the logical and mathematical ones, although they do inspire one another. The problem one faces here is precisely to set up a translation from a classical geometrical, analogous model to a model whereof the domain usually consists of the finite set of pixels or cells that make up the computer screen. The obvious drawback from the perspective of this entry is that nearly all these models assume the classical infinite model in the background and, hence, do not have a proper foundation of their own—a situation quite analogous to numerical analysis that relies on classical analysis for proving the correctness of the procedures.
Most attention is paid to the problem of proving correspondences between the original and the discrete model to make sure that the image obtained is, in certain respects, faithful to the original.
Continuous geometry. | Institute of Mathematics
A simple mathematical example concerns the number of holes in a 3-dimensional Euclidean surface. One wants to be sure that every hole that shows up in the digital picture does indeed correspond to a hole in the original mathematical object. See Kulpa and, more recently, Danielsson for some nice examples.
Note also that these theories should not be confused with computer programs that have the ability to reason about geometrical objects. This is part of the research area of automated reasoning—see Chou et al. As is commonly known, one of the hot topics in physics is the search for the unification of quantum field theory and general relativity theory.
As is equally well-known, the hardest problem to solve is how to deal with space-time. Quantum field theory requires space and time as a background, whereas in general relativity the structure of space-time is largely determined by the masses and the energy present. Most of these models, speculative as some of them may be at the present moment, turn out to be discrete and hence these proposals, in contradistinction, e. From the historical perspective, it must be added that on and off some physicist tried to find out what discrete counterparts of existing classical physical theories could look like.
Usually the philosophical underpinnings of such an attempt tend to be rather idiosyncratic. In section 2 one such example will be presented. Typically such attempts did not create a major stir, they quickly disappeared into the background, but nevertheless they do contain some interesting and relevant ideas. In a rather straightforward sense, all of the above involves philosophers as well. Discussions about logical systems, about foundational mathematical theories, about Zeno paradoxes, about supertasks, about what a model and a representation are, …, are typically topics that belong to the domain of the philosophers.
Suppose that there are excellent arguments from an epistemological or ontological perspective, claiming that the world should be considered discrete, then these arguments can support the search for such a discrete worldview, including the elaboration of a discrete geometry.
Even if from the mathematical point of view, the theory looks rather clumsy or difficult to work with, nevertheless, because of the philosophical considerations, it has to be so. Finally, they also pay attention to the historical side of the matter. As said, these five groups are the most important ones so completeness has not been demonstrated and neither has mutual exclusiveness been shown.
This short overview only meant to list the different intentions, motivations, purposes and methodologies of the parties involved. The first question to settle is what the classical theory will be. As most of the work that has been done has been limited to the plane, this presentation will also be restricted to that particular case in most proposals the extension to higher dimensional geometries is considered to be completely straightforward. But that is not sufficient, for there are different routes to follow as to the presentation of plane geometry.
One of the very first attempts dates back to the late 40s, early 50s and will therefore be presented here as an exemplar in the sense that it has both all the positive qualities required as well as the oddities that seem to go together with such attempts. More specifically, it concerns the work of Paul Kustaanheimo in partial collaboration with G. Next a recent proposal will be discussed, along totally different lines, of Patrick Suppes and a somewhat older proposal of Ludwik Silberstein, where the geometry is directly imbedded in a physical theory, special relativity theory to be precise.
The concluding section of this part deals with some specific problems and tentative solutions. What does a Hilbert-type axiomatisation look like? The first thing one has to do is to fix a formal language. Usually one chooses first-order predicate logic with identity, i. The restriction to first-order logic means that only variables can be quantified over. Without going into details, it should be remarked that a more expressive language can be chosen, e. Once a language has been chosen, the next problem is to determine the primitive terms of the language.
For plane Euclidean geometry, these are points and lines, although sometimes lines are defined as particular sets of points. Next the basic predicates have to be selected. There exist a number of different axiomatisations at the present moment. Note that it is not necessary that all of them occur in an axiomatisation. As an example, if lines are not introduced as primitive terms, then usually there is no incidence relation. The next step is the introduction of a set of axioms to determine certain properties of the above mentioned relations.
As an example, if the axiomatisation uses the incidence relation, then the typical axioms for that relation are:. Finally, one looks for an interpretation or a model of the axiomatisation. This means that we search for a meaning of the primitive terms, such as points and lines, of the functions if any and of the predicates in such a way that the axioms become true statements relative to the interpretation.
Although we often have a particular interpretation in mind when we develop an axiomatisation, it does not exclude the possibility of the existence of rather unexpected models. In a sense finitist models rely on this very possibility as the next paragraph shows. Paul Kustaanheimo was a member of a group of mathematicians based at Helsinki, who were all interested in some form of finite geometry. The most prominent members were G. Kustaanheimo, and R. The origin of their inspiration is to be found in the work of J. Their approach has not known any continuation, one exception being Reisler and Smith Of course, since we humans can only manipulate finite objects in finite ways, a discrete geometry must result.
A standard model for the classical axiomatic theory of Euclidean geometry consists of the cartesian product of the real numbers with itself. Or, as it is usually formulated, a point in the plane is mapped onto a couple of real numbers, its coordinates. The real numbers have the mathematical structure of an infinite field. But finite fields exist as well. So why not replace the infinite real number field with a finite field, a so-called Galois field?
The best result one could obtain would be that every finite Galois field satisfies most of the axioms of Euclidean geometry. That however is not the case. In short, one cannot claim that any finite field will do, but only some and for that matter only part of it.
A finite model is not merely a scaled-down version of an infinite model. Very often a different structure appears. As an analogy take the infinite set of natural numbers. This is classically not possible. So one finds additional structure. Thus perhaps the prime numbers do have a significance. But still the question remains: is this a new sort of Pythagoreanism?
Two elements are important from the strict finitist perspective. Secondly, all models considered will be finite, because no matter what constructions are performed, the starting point will always be a finite set of points. A diagrammatic representation makes clear what is happening:. Of course, merely listing a set of constructions is not sufficient to talk about a geometrical theory, so it has to be shown, as it is indeed done by Suppes, that a formal-axiomatic treatment is possible.
In addition a representation theorem is proven such that points are attributed rational coordinates. Two important comments must be made. First, it remains to be shown that this elementary geometrical theory can be expanded all the way into a full-fledged geometrical theory that can be considered a plausible alternative to classical geometry. Suppes himself seems quite confident as he writes:. Secondly, the focus on constructions opens up a novel way to deal with the problem of the distance function.
We do not need a general distance function, but, for each separate case, we have to be able to attribute coordinates to the points present in the diagram and nothing more. In section 2. First, however, a quite different approach from the physical side. In Silberstein proposes a fairly straightforward discrete theory. In the short booklet that brings together the five lectures on this theme, Silberstein restricts himself to one spatial and one time parameter. Although he acknowledges the problem 15 of higher dimensions, he does not deal with it. So the distance problem becomes rather trivial since on a line, the discrete distance function and the Euclidean distance function coincide.
His proposal is elementary in the sense that the smallest distance, viz.
Analogs for derivatives are defined, differential equations are replaced by difference equations, an analog in terms of finite differences is derived of the Taylor series and most of classical physics can be imitated. It is worth mentioning that the lectures include a rough calculation of the size of the chronons, i. He further applies the discrete spacetime framework to special relativity and here too, an analog is found. Quite interesting in this approach is the fact that additional conditions appear that are not needed in the classical case.
Here is one illustration. Special relativity theory relies on the expression, here restricted to one spatial dimension, viz. This last condition is a pure consequence of the fact that we are thinking in a discrete way, using integers. In this section three specific problems will be discussed that need to be solved if any proposal for a discrete geometry is to be taken seriously: the distance function problem, the dimension problem, the anisotropy problem, and the identification problem.
The distance function problem. Perspectivity by Decomposition Pages Chapter V.
Equivalence of Perspectivity and Projectivity Pages Chapter VI. Properties of the Equivalence Classes Pages Chapter VII. Dimensionality Pages Part II Chapter I. Theory of Regular Rings Pages Order of a Lattice and of a Regular Ring Pages Isomorphism Theorems Pages Definition of L-Numbers; Multiplication Pages Addition of L-Numbers Pages Chapter VIII.
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