Infinities in QFT (and physics in general)

[PeroK]

Taken from wiki on “Limit of a function”.

“Formal definitions, first devised in the early 19th century, are given below. Informally, a function fassigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist”

only in the wiki the words “sufficiently” and “arbitrarily” are italicized. Rightly so?

That's explicitly an "informal" explanation. The formal definition makes things rigorous, explicit and dependent only on the properties of (finite) real numbers.

 

[Jimster41]

I get that you can keep the notion of “tangent point” on a curve without shrinking the length of the deltas all the way to 1/infinity by invoking “embedded affine sub spaces”… so the tool is well defined, and wildly useful. I am not suggesting it isn’t, or that people shouldn’t learn it fapp. But don’t those notions (foliations of the continuum etc.) have their own problems?

So, real question: what about two tangent builders converging across the surface of a sphere getting way more and more tiny step-wise heading toward 1/infinity but always avoiding it by setting up affine sub spaces as they go. Where do they meet? What non-infinitesimal real valued “point” What guarantees the resulting tangent point is a point? Is it a single affine sub space? If so how does it go from two to one? How come they don’t just end up in a fight to the death down to 1/infinity trying to own the tangent point? Or do they form a stable harmonic oscillator? Or does the sphere provide the discrete foliation at the end of the day, where they somehow meet for non-infinite coffee?

 

[PeroK]

A tangent to a curve is a line like any other line. Two lines intersecting in a single point is not mathematically problematic.

There's no issue with never having learned rigorous mathematics, but it is an issue if you try to tell those of us who have that we're crazy and imagining "flying elephants".

 

[Jimster41]

So are you asserting reality is a smooth, infinitely differentiable a continuum? Has that been proven?
My only problem is that lines and point have these “negligible widths and sizes” which sounds a lot like “infinity”.

 

[PeroK]

So are you asserting reality is a smooth, infinitely differentiable a continuum? Has that been proven?

Mathematics is not dependent on the fundamental nature of spacetime.

 

[Jimster41]

Interesting, and our brains are or are not?
I would argue the fact mathematics had to be invented to support detailed agreement between individuals is totally dependent on the fundamental nature of spacetime.

 

[PeroK]

You mean that in order for our brains to conceive of a spacetime continuum, spacetime must be continuous?

The problem is that we can conceive fundamentally incompatible systems. So, however nature is configured we have the capacity to conceive of it otherwise.

 

[Jimster41]

fair enough. Perhaps a dialectic oscillator.
For my part I think I wonder if the opposite of your first statement, if true, could help explain the second.

IOW if spacetime was continuous there would not be brains, just brain (or some barely conceivable “one”). I.e. Discrete spacetime could help explain discrete experience/phenomena in general. I find it hard to understand how a continuum got itself into such a big various mess.

 

[Interested_observer]

This is only part of the goal. The real goal is to understand Nature in a way that allows us to make the best use of it. This needs much more than just making physics computations the predictions of which agree with observations.

I would say that the real goal is to understand Nature, period! Then we would KNOW how to make the best use of our knowledge.