Do you believe in god or math?

[Abracadabra]

I'm sorry, "well-defined" in what sense? I'm familiar with functions or relations being "well-defined" in the sense that f(x) is unambiguous for all x in the domain of f.

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.

What do you mean "well-defined units of quantity"? What makes a unit of quantity 'well-defined'?

That 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.

And you do understand that the kind of ambiguity you're introducing is entirely accounted for in vector fields, right?

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,...

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.

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.

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.

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You're using "valid" in a strange sense; "1 + 0 = 1" and "No field contains zero divisors" are true regardless of physical reality. Their validity does not depend on whether there are physical objects that correspond isomorphically to them (i.e., the physical objects have the same relations between them as their corresponding mathematical objects).

I'll have to confess that I'm not sure what you are attempting to get at with the above paragraph. So I'll have to pass on commenting on this one for now save for commenting on the following:

(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.

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

Your argument seems to be simple, "Maybe our current mathematical models don't accurately describe reality." That doesn't prove that no mathematical model could accurately describe physical reality.

Well I haven't claimed to have "proven" anything. Nor have I stated that "no" mathematical model could accurately describe physical reality. On the contrary, I have my own ideas of how such a mathematical model could be constructed. However, the "mathematical model" that I'm talking about would require totally different axioms than the current mathematical formalism has.

So would that constitute "just another model in mathematics"?

Or would that be a whole new mathematics?

I mean, I would need to toss out the Peano axioms and replace them with something quite different. In fact, the whole notion of "Set Theory" would need to be revised. But it's not as bad as it sounds actually.

Any object (such as a system of rules or particles or whathaveyou) or collection of such objects that has a structure, relational properties, or properties in general can be described mathematically.

That's only true in the physical world if the objects in question meet the requirements of well-defined "oneness". Or if we can get away with pretending that they do.

It might not be with anything that looks like a "number", but mathematics isn't limited to numerics. Any line of reasoning is at its core mathematics; any structural property is mathematical. Any relational property is mathematical. Any system in which logic holds is mathematical.

Well your just using the term "mathematical" as an umbrella term to cover any logical system that can describe structure of any kind.

My point is that Our Current Mathematical Formalism isn't equipped to properly describe the true nature of the universe.

So if I introduce a new formalism that might have the capability of properly described the behavior of the physical world, then that too would be considered to be "mathematics'.

Fine.

But that doesn't change my point that Our Current Mathematical Formalism isn't equipped or designed properly to deal with this task.

So even if you're right that no model can accurately describe the microuniverse, that itself is a mathematical description of the microuniverse.

Perhaps so, but it would be a rather useless description would it not?

[quote]Proving that certain problems (like squaring the circle, solving the general quintic, or modeling the microuniverse) are impossible is within the domain of mathematics.[quote]

Sure, there's a lot of things that our current mathematical formalism can deal with and make truthful statements about. I have never questioned that at all.

But that's totally irrelevant to the points I'm addressing.

I'm talking about moving beyond that. I'm not talking about destroying what already exists.

Not to imply that I alone could do this, but what I'm talking about would be similar to how Relativity expanded upon Newtonian physic. It didn't exactly destroy it altogether. But it clearly did destroy some of the basic ideas associated with the older physics (such as the concepts of absolute time and space).

I'm talking about the same type of thing here. The new approach to mathematics would not destroy the old altogether, but it would require some adjustments to certain foundational concepts for sure.