In this project, I will investigate existing proposals for a finite, discrete geometry and reinterpret infinite geometry in a finite manner. I will argue why and how the infinite and continuous prove not to be indispensable. As we all learned in secondary school, the number of points on a straight line is infinite, in two senses: a line has to fit on the blackboard, but it can in principle be extended in both directions. And even a line segment has an infinite amount of points on it, for between every two points, we can identify a third one. But on the other hand, we conduct our reasoning in a finite brain, in a finite amount of time. When we consider concrete geometric objects, we only obtain a finite collection of points by “instantiating” a few of them with a dot. And even if we want to instantiate “all” of them, we only have finite resources at our disposal. If we revert to a computer to conduct the reasoning, we encounter exactly the same limitations. And after all this, we apply this geometry to a finite physical reality. There is a tension between the seemingly obviousness of infinity in geometry and our failure to truly grasp its meaning and use. Therefore, in this project, I investigate why, where and how the infinite can be avoided in geometry. My research should appeal to a wide range of people, from the philosopher concerned about our understanding of the infinite to the physicist interested in the potential of a discrete and finite description of reality.
|Effective start/end date
|1/10/16 → 14/09/18
Flemish discipline codes in use since 2023
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