*How Mathematics Meets the World, by Tim Maudlin* https://fqxi.org/community/forum/topic/2318
Essay Abstract The most obvious explanation for the power of mathematics as the language of physics is that the physical world has the right sort of structure to be represented mathematically. But what this in turn means depends on the mathematical language being used. I first briefly review some of the physical characteristics required in order to unambiguously describe a physical situation using integers, and then take up the much more difficult question of what characteristics are required to describe a situation using geometrical concepts. In the case of geometry, and particularly for the most basic form of geometry— topology—this is not clear. I discuss a new mathematical language for describing geometrical structure called the Theory of Linear Structures. This mathematical language is founded on a different primitive concept than standard topology, on the line rather than the open set. I explain how some other geometrical concepts can be defined in terms of lines, and how in a Relativistic setting time can be understood as the feature of physical reality that generates all geometrical facts. Whereas it is often said that Relativity spatializes time, from the perspective of the Theory of Linear Structures we can see instead that Relativity temporalizes space: all of the geometry flows from temporal structure. The Theory of Linear Structures also provides a mathematical language in which the fact that time is a fundamentally directed structure can be easily represented. https://fqxi.org/data/essay-contest-files/Maudlin_How_Mathematics_Mee.pdf ... The standard answer derives from the standard mathematical tool used to describe the most basic geometrical structure of a space. If that mathematical tool, that mathematical language, is to provide an accurate characterization of the geometry physical space or physical space-time then physical space-time must have a structure corresponding to the fundamental concept in the mathematics. And a different mathematical language, built on a different primitive concept, requires that physical space-time have a different structure if it is to be accurately described. I will argue that standard geometry has been built on the wrong conceptual foundation to apply optimally to space-time. I will sketch an alternative geometrical language, and explain how it could directly reflect the structure of the physical world. ... Physicists seeking such a mesh between mathematics and physics can only alter one side of the equation. The physical world is as it is, and will not change at our command. But we can change the mathematical language used to formulate physics, and we can even seek to construct new mathematical languages that are better suited to represent the physical structure of the world. The Theory of Linear Structures, whatever else its virtues, provides and example of how this can be done. If it is correct, then we might see how the time itself creates the geometry of space-time, and also makes space-time exactly the sort of thing that is well described using this mathematical language. @philipthrift -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/c34b6c62-19d4-4216-85a0-d8b1f076edd7%40googlegroups.com.
