Do you believe in god or math?

[CliveStaples]

Yes, I'm familiar with the descriptions of behaviors of entire functions too. I'm not addressing that. In fact, that's a far loftier topic than even needs to be addressed for the issues that I'm addressing.

...erm, I wasn't talking about entire functions (how in the world did you jump to functions specific to the complex numbers?). I was talking about well-defined functions (typically the definition of a function requires that it be well-defined in the sense that if f(x) = y and f(x) = z then y=z).

WThat is a very good question, and is far closer to the issue that I'm attempting to get at. But I confess that it's not going to be easy to get at it on a public forum like this. And, yes, I am addressing physics more so than the so-called pure axiomatic mathematics.

In other words, I'm far more concerned with the ability of the so-called pure axiomatic mathematics to correctly describe the physical quantitative essence of the physical world.

That really is the core of my concern.

Then your problem isn't with the validity of math. Your problem is in arguments that certain physical systems are modeled by certain mathematical systems.

No it is not. I too understand vector fields and their notations and representations. You're still going to run into the same problems of how well those vector fields are actually quantitatively representing any physical reality that may be associated with them, or that they may be designed to describe.

Fortunately for us, they do work quite well for normal macro phenomena. I think our mathematical success describing things like the behavior of electromagnetic fields, etc, is more than sufficient evidence for that.

I have no problem with the success of mathematics in terms of basic classical physics, and even in terms of Relativistic physics. As long as things remain a macroscopic size and behave fairly classically (or Relativistically) mathematics will work pretty good. Maybe even "perfectly" if everything is taken into account properly.

That's not the concern. But what we are about to discuss next is where problems begin,...

...you do know that quantum physics involves quite a lot of highly developed mathematical theory, yes? And that empirical observations have borne them out (cf. quantum teleportation)?

Sure you do. You absolutely do have an idea of what I am referring to, you just aren't yet aware of it because you haven't yet understood what I'm trying to get at. I simply haven't yet explained enough details.

It's really quite simple and intuitive. Think about any physical object. What would you require of a physical object in order to claim that you have "One" and only "One" of that object?

Would you run off to read the book of mathematical Axioms to see if the physical object qualifies as being "One" object?

No, of course you wouldn't. There's nothing in those axioms that would help you decide.

Your recognition of having "One" of something is entirely up to you. It's up to you to decide what constitutes "Oneness". In some cases that can be a very simple task. In other cases, it can be quite difficult.

You're just talking about measure theory, now. What you decide to call "1" depends on what you're trying to measure. If you're trying to measure "how many", then you're doing something like counting; the corresponding measure is the Cantor measure. If you're trying to measure "how much", then you're interested in something like length, area, volume, or some higher-dimension analog; the corresponding measure is the Lebesgue measure.

If you're using the "how many do I have of these" measure, then you just have to see what sort of structure "how many do I have of these" has. For instance, is there an amount which when taken together with "how many do I have of these", yields the same amount? If so, that amount is "0". And so forth.

I like to call this the "boogieman syndrome". The reason being that I use a concept of boogeymen to illustrate this point:

Suppose I show you flash cards with pictures of physical objects on them and ask you to give me the mathematical number that quantifies what you see on the cards. This is actually how humans are taught the very concept of "number" in kindergarten and preschool.

So as long as I show you cards with well-defined objects on them you have no problem at all instantly giving me a "number" to quantify them.

However what if I showed you a card that had really weird abstract paintings of "boogeymen" on it. Some had two heads, some had three arms, some had no arms or legs at all. Some appeared to have bodies that leave the right side of the card and reappear on the left side of the card. In that case would that qualify as a single boogeyman, or maybe two different boogeymen each not being completely shown?

What about the boogeymen that appear to be connected like Siamese twins? Is that one bogeyman, or two?

In short, I can show you a flash card where you would find it quite difficult to place an firm quantitative value to properly describe the the number of objects that you are viewing.

It's not a "trick". It's a genuine demonstration that shows that unless you can firmly describe the quantitative nature of objects, you really can't even apply mathematics to that situation at all with any degree of certainty or success.

So where does that bring us?

Well, let's repeat your same question and address it again in light of the above information:

What is a "unfit of quantitative property"? I've studied mathematics and (some) physics (including quantum mechanics) and I have no idea what you're referring to.

In Quantum Mechanics anything beneath the Planck level loses the ability to be firmly described in terms of it's "oneness". In other words, it becomes lost in a superposition of states (kind of like what I was trying to get at with the Boogeyman Flash Card) and its quantitative property of "oneness" breaks down. It no longer behaves in a strict "quantitative" way that our macro mathematics requires.

And this is why our current macro mathematical formalism must necessarily break down at the quantum level.

Superposition of states has literally nothing to do with the Planck length. Superposition of states occurs with particles that are above the Planck length.

It seems like you think that if cardinal numbers aren't sufficient to describe quantum phenomena, then "math is insufficient." But that's kind of absurd; you just need something different than the cardinal numbers. You need something like probability measures (instead of Cantor measures). Which is precisely what they use in quantum mechanics.

(i.e., the physical objects have the same relations between them as their corresponding mathematical objects)

That situation may or may not hold depending on how well-defined the quantitative nature of the objects in question is. Like I say, if they are stable macro classical objects that are "Well-defined" in they property of "oneness" then sure. But if they are like quantum boogeymen spread out in states of superposition, then clearly that's not going to hold.

Okay, I'm going to stop you here. I literally have no idea what the phrase "stable macro classical objects that are 'Well-defined' in the[] property of 'oneness'" means. What does it mean to be a "stable" object? What does it mean to be a "macro" object? What does it mean to be a "classical" object (GR is non-classical but describes what I assume you mean by 'macro' objects)? What is "the property of oneness" and what does it mean for it to be "well-defined"?

And we already know that mathematics breaks down in this case and can only make probabilistic calculations at that point. No news there.

I had to come back for this one. Mathematics doesn't break down; determinism breaks down. There's an entire field of mathematics devoted to non-deterministic processes, starting with the development of probability theory. Probabilistic calculations aren't a breakdown of mathematics; they are rather an application of mathematics.