Numinus, Dr Who,
et al,
The Question here, may in itself, be an anceint "paradox" in a newer form.
(Merely an opinion.)
(Non-Original THOUGHT)
How do we count? What do we mean when something is "uncountable." What is a "set of numbers?"
There are, as I understand it, two lines of thought:
- Mathematical objects are real objects.
- Mathematical objects are concepts.
Thumbnail Source: http://platosheaven.blogspot.com/2005/12/do-numbers-exist.html
I'll be honest, I am one of those few who oppose the Platonic view that numbers are "real" and they exist independently of human thought. The same would go for the set of spectrum detection (colors) and similar formal constructs; or "points" along a continuum (time or dimension).
What is 'real' is something that can be described immutably -- independent of subjective perceptions.
Remember the platonic argument that the common-sense, physical world has, at best, only 'fluid' existence? Well, it is confirmed by quantum mechanics -- the properties of the physical world can only be described by a wave function (probability) and is dependent on the observer's subjective perception. Even the fundamental units of length, mass and time are relative and entirely dependent on the motion of an observer's reference frame.
Only ideas fit the definition of real.
(COMMENT)
Each question we have, in which we use mathematics is used as a solution tool, is not always going to have an derivable answer through mathematics that fits which other commutations. As we all know, scientifically, we see that individual evaluations of "relativity" and "quantum mechancis" are testable and verifiable. But again, yes --- the unforgiving truth is that, mathematically, they do not match-up
(at least not yet).
What exactly do you mean by 'solution tool'?
One becomes aware of the physical world through one's senses. One becomes aware of the rules that govern the physical world through mathematics.
The reality of a thing as described by our sense is mutable. The reality of a thing as described by ideas is immutable.
The conclusion is inescapable -- the laws of nature exists independent of the material world it governs.
The concept and implication of a "
Null Set"
(prove something does not exist) assumes that, in reality, there is such a thing
(very real) known as a "
Set"
(of something).
Actually, the existence of the null set can only be described in the world of ideas and there is absolutely nothing in the physical world that even comes close to it.
In set theory, a null set is a set without any members. Or it is euclidean or cartesian space you populate with infinite number of points.
But what exactly is it in the physical world? You imagine an empty container, do you not? But there is still the container and the description 'empty' can only be made relative to the container, no?
But intuitively, the essence of being 'empty' is very real.
First we must prove that the concept of a "Set" is even real; and not just a human construct. If it is, merely a human construct, then what does that mean in its application to the concept of a Supreme Being?
While it may seem to be a lower order question, it is actually a question (while very simple) of the highest order.
Most Respectfully,
R
The concept of set is real. Otherwise, nothing is real.
The set of all x such that x is described as some operator. All scientific laws are couched this way. Your very existence is dependent on a conception of sets -- you are a person residing in a particular part of the globe enjoying certain civil rights and bound by certain obligations.
Person, part of the globe, civil rights, obligations -- they are real simply because they can be described as sets. Without the idea of sets, nothing can be said about you. And something that cannot be described in any way doesn't exist.
