Something about Apologetics.

[Abracadabra]

Unless, of course, your definition of 'continuum' is not related to the mathematical one. In which case, what are you taking 'continuum' to mean?

There's only one concept of a continuum as far as I'm aware. It's the concept that things are continues rather than discrete. In fact, this was the very concepts that were being argued in ancient Greece.

Is reality a continuum or does it break down at some level and is actually made of discreet discontinuous quanta?

That was the question.

Zeno argued for the quantized reality as opposed to a continuum.

The people who favored the continuum ultimately won in terms of shaping mathematical thinking and this is why we currently have a mathematics that based on the idea of a continuum.

However, science and physics have shown us that reality does not think it's continuous. On the contrary, reality demands quantization, and thus the science and physics communities have acknowledged that we live in a quantum universe.

Yet, mathematicians still believe they are living in a continuum.

Quantum number line? Do elaborate.

Rational numbers are quantized.

We invented the 'reals' and forced them to become a part of the number. In doing so we had no choice but to demand that the number line itself must be a continuum.

This is totally unnecessary. The 'real numbers' have no physical existence in reality. That's an illusion created by incorrect abstract thinking.

In other words, there does not exist in physical reality a physical 'quantity' that can be described by a 'real number'. No such absolute quantization is possible in our physical universe.

I realize that you are going to give the standard response and point to the irrationality of the diagonals of squares, and the relationship of Pi, etc.

But those are not absolute quantities. On the contrary those are relative relationships between quantities that are actually dependent upon a very non-ridge and non-static fabric of spacetime.

In truth, if you could physically measure the diagonal of a square or the diameter or circumference of a physical circle down to the quantum level you'd soon discover that it's either here or there, but can't be in between.

In other words, by the time you got down to the quantum level your theoretical continuum would break down and the quanta of reality would become apparent.

The imagined 'irrationality' that we are abstractly placing on these relative measurements is not "Real". It's not based on an infinitely divisible continuum. That was an incorrect assumption to begin with.

We have no need for "Real Numbers" in an absolute sense.

And that is the important key concept to understand. The only time we require "Real Numbers" is in the case of irrationality, which only arises in self-referenced relative situations. It's not an absolute quantitative property of the universe like our current mathematical formalism is attempting to treat it.

We have no need to support these notions of "real numbers" as though they must be an integral part of our idea of absolute quantity.

In fact, by doing that, we are actually loosing site of important information.

For example, mathematicians are never going to recognize that all irrational relationship arise from self-referenced relative situations.

The reason they are never going to recognize this is because they have already accepted irrational 'numbers' as completely independent absolute ideas of quantity.

That's a mistake. No such absolute quantities exist in our physical universe.

Real number (basically irrational numbers truly), are always (and may I repeat always) due to relative self-referenced situations.

They have nothing to do with any idea of an 'absolute' quantity and our attempt to force them to become such concepts is misplaced.

We don't need anything on the number line but rational numbers, and they are discrete (i.e. quantized).

So until the mathematical community recognizes the difference between absolute quantities and relative self-referenced quantities, our mathematics will remain flawed (i.e. incorrect with respect to the true quantitative nature of reality)

We live in a quantum universe. Not a continuous one like had been thought by the ancient Greeks.

Yet, our mathematics is still based on these ancient notions that it must support a continuum.

That's simply the wrong picture of reality. And we even know this to be true from science. Yet we still cling to these ancient incorrect ideas that we live in a continuum and that it makes sense to build a mathematical formalism based on those ancient and incorrect ideals.

We need to backtrack and rethink some of this stuff.

In fact, it's not nearly as bad as it sounds. We really only need to backtrack a few hundred years actually. Just to the dawn of the formalization of Set Theory. That's were these erroneous ideas became formalized. There we some bad decisions made at the beginning of the 20th century and we need to go back and readdress those.

Calculus itself wouldn't change much at all actually. There's nothing wrong with calculus. It's various axioms of set theory that's really at the base of the problem.

But I'm sure humans will sort it all out eventually.

It's just a matter of time.

Unfortunately though most mathematicians are quite happy just accepting things the way they are. They don't see where there's a problem.

In fact many of them take the stance that it's not even the job of mathematics to correctly reflect the real world. They view it as a totally abstract product of pure human thought.

And of course they are perfectly correct! That is indeed precisely what mathematics has become. Mathematicians are so lost in pure thought they they don't even care whether mathematics matches up with the true quantitative nature of reality. That's become a totally unimportant to them.

And thus with that frame of mind they are naturally going to go off in la-la land making up mathematics on their own whim with total disregard to whether or not is means anything. As long as it seems to be "self-consistent" with respect to its axioms that's all they care about.

But even a whim that is 'self-consistent' is still just a whim.

If the axioms don't match reality, then mathematics has nothing to do with reality. And that's the bottom line right there.

If we want a mathematics that can help take us to the stars, it had damn well better be correct and true with respect to describing the true nature of the quantitative essence of reality.

Whims won't make it to the stars no matter how logically consistent they are with respect to arbitrary man-made axioms.