I am the first to champion the scientific method, as a means to explore and explain our natural world, and by extension, our physical Universe. [...] The Universe contains space and time, but does not exist in space and time.
If you don't mind I would love to pick your brains on this. To my thinking, if space and time and everything therein can be accurately explained by mathematical methods then surely the only reliable way we can imagine a landscape outside of the universe is by mathematical reasoning? Obviously the laws of physics go to shit at the boundaries of the universe, but could a robust mathematical theory of what goes on beyond that boundary be meaningful at all?
Hi Chris, sorry for the late response...
Your question underlies a very deep, philosophical debate that has been going on for centuries. The question is whether Mathematics is purely a construction of the human mind, or exists as an objective reality. I happen to come down on the side with those who believe it exists on its own, regardless of human consciousness (mathematical realism). In other words, mathematical objects/entities/constructs exists independent of the human mind. We don't invent mathematics, but rather "discover" it. Realists view Mathematical enties as existing a priori or human experience.
Conversely, there is a school of thought that asserts all abstract entities, including numbers, mathematical functions, etc., are "fictions" that do not have an objective existence. Fictionalists reject the truth of the statement that "2 + 2 = 4", because it presumes the existence of abstract mathematical entities (numbers), which they deny. It is kind of a crazy and subtle argument, but it is consistent. Fictionalist view mathematical statements as false in reality, but true in the "fiction" of the story (in this case, the story of human mathematics). An analogy would be the statement "Luke Skywalker is a jedi knight". This statement is true within the fiction of George Lucas' Star Wars story, but is strictly speaking false in reality, because jedi knight's don't exist. Fictionalists argue that abstract entities do not exist in space and time, and have no causal powers. As such, they are utterly unlike any entities that we know about. Of course, the main challenge to mathematical fictionalism is the general applicability of mathematics to our world of experience, and the specifically to it's utility in scientific inquiry and describing our natural world.
Ultimately, the question boils down to:
A) Is mathematics invented, or discovered?
B) Can abstract objects exist outside of space and time?
One other point, Chris. You may have been using the word "boundary" as a metaphor...but in case you meant it in a physical sense, I just wanted to clarify that our Universe has no physical boundary. Current cosmological observation/evidence indicates our observable (local) Universe's geometry is flat (k=0, Omega =1). The global topology of our Universe is another issue, and without getting into specifics, is much more complicated. One possible topology could be infinite and boundaryless (e.g. Euclidian flat space) or a finite, Compact space (e.g, the surface of a sphere). Adding to the subtlety, an boundaryless flat space can still be topologically finite and Compact. A torus is a case of a topologically flat, compact manifold.
However, in all cases, there exists no physical boundary or "edge".