Something about Apologetics.

[Categories+Sheaves]

The reason for that is quite involved, but that too is based on a concept of self-reference. Cantor's set theory is indeed a self-referenced system. The change I propose would produce a system that it not self-referenced.

Thus Godel's incompleteness theorem would not apply to my set theory.

But that's a whole other story.

Indulge me.

[Abracadabra]

The reason for that is quite involved, but that too is based on a concept of self-reference. Cantor's set theory is indeed a self-referenced system. The change I propose would produce a system that it not self-referenced.

Thus Godel's incompleteness theorem would not apply to my set theory.

But that's a whole other story.

Indulge me.

I would love to share my views on this. But it would truly require a book, or at least an in-depth series of lectures to properly convey.

And if I were going to put that much energy into it, it sure would be nice to have an audience that is seriously interested in hearing it. Trying to argue against a hostile audience is a waste of everyone's time.

However, if you are sincerely interested I'll try to explain a few things very briefly.

First off, "What is the definition of number?"

That's worth a chapter of discussion right there.

But to keep things brief, intuitively humans recognize the concept of 'number' as a collection of 'individual things'.

The individuality of the things is of paramount importance. Yet this is taken for granted. However it would be absolutely impossible to 'count' something if you can't speak about individual 'units' to count.

In short, you must realize that, fundamentally, the idea of quantity is an idea of many or few individual things. We address this in terms of putting individual things in 'collections', and then recognizing the resulting 'quantitative property' of the collection as being a concept we call "number".

So our cardinal definition and understanding of number is fundamentally based on the concept of recognizing the individuality of things in collections and from that observing the quantitative property of the collection as a whole.

That is what we mean by number and that is how we should formally define the concept in a cardinal sense.

That was actually proposed by Giuseppe Peano at the same time Georg Cantor was proposing his idea of an "empty set" as being the foundation of set theory.

Please realize that set theory is truly nothing other than our formalized understanding of what we mean by "number".

Also, please realize that this very formalization of the definition of number was taking place at the beginning of the 20th century precisely because the sciences had exploded due to the success of Newtonian physics as well as due to the introduction of the calculus.

Mathematicians were being pressured to come up with a more formalized definition for the very concept of number. In fact, prior to this time the idea of number was taken for granted as being merely intuitive!

So Peano introduces the idea of a system starting with the number ONE.

Peano said, "Let ONE be defined as the quantity of unity".

The mathematical community questioned this. "What do you mean by unity?" they asked. Peano tried to explain how we use our intuitive understanding of the individuality of things to define what we mean by 'ONE' unit of thing.

This idea was hard to pin down in a very rigorous and precise way. The mathematical community also objected on grounds that this would tie mathematics to something 'concrete' in the physical world (i.e. this vague notion of individuality or [/i]unity[/i]). So they saw this as being cumbersome, and unnecessary 'clutter', to something that they believed should be defined in a more elegant simple way.

Well, Georg Cantor came to their rescue. He proposed that we start with an idea of an "empty set". A collection that does not contain any elements at all. Let's call that "zero". Then we can use that concept to define the number one by saying that the number one is the set containing the empty set. This is clearly a different idea and thus ONE is distinct from ZERO, and so on.

This impressed a lot of mathematicians and the community began to adopt this notion. They now had a totally self-referenced axiomatic system that did not require any contaminating notions of "individuality" or "unity". This idea of a collection that is not a collection of anything seemed to fit that role.

And thus modern Set Theory was born as a self-referenced system.

It did not come without the need to band-aids however. Many band-aids were placed on this set theory as it evolved.

The question then really comes down to the question of whether this approach truly addresses the notion of 'unity' or the property of 'individuality' directly? Or it is just a way of sweeping the concept under the carpet and avoiding have to address it at all?

I personally believe it's the the latter. Moreover, it has caused mathematicians to completely ignore whether the things they are quantifying even have recognizable or legitimate properties of individuality or unity before they even start to quantify them.

The very notion of an empty set has swept all of that right under the carpet.

~~~~~

So what's my point?

Well, let's go back and look more closely at Peano's original idea. Let's start a set theory based on the idea of ONE, and face the dragon of addressing what we mean by ONE.

What do we mean when we say that something has the quantitative property of "oneness"?

That is most certainly not something that we should be sweeping under the carpet. This property of individual is paramount to our very understanding and meaning of number as a quantitative property of a collection of individual things.

This prerequisite recognition of a quantitative property of individuality is paramount to our comprehension and meaning of the concept of number.

Question: Can a set theory be constructed based on this foundation?

Answer: Yes it most certainly can be.

Would it be fundamentally different from Cantors empty set theory?
Yes, it most certainly would be.

Would it be closer to what we mean by 'number' in a phenomenological sense of our physical world. Absolutely!

It would be a set theory that is "more correct" in terms of what we actually mean by the concept of "number".

What would be the results of such a formalism?

Well, for one thing, sets couldn't contain elements that have no quantitative property of individuality (or mathematical unity).

That would eliminate using an 'empty set' as an valid element or definition of the number one right there. It would also eliminate the paradox of multiple sized infinities (but that's a whole other story)

Furthermore, for precisely the same reason that the mathematical community objected to Peano's suggestion in the first place, Kurt Godels incompleteness theorem would not longer apply to this new Set Theory.

Well, why not?

Because this abstract concept of 'individuality' is necessarily a property of the things that are being quantified. In other words the system is no longer self-referenced (which is what Godel's theorem requires), instead it has now become dependent upon a property that is external to the logical system (i.e. the property of individuality of the things that are being quantified).

Now you may think, "But that would mean that mathematics can only address physical or tangible things!"

No it doesn't mean that at all. You can imagine the most intangible abstract things you like. The only thing that is required is that when you imagine such objects, in order to apply mathematics to them, you must first have a meaningful description of what you mean by an "individual thing". You must recognize what it takes for your units to be considered to be an 'individual thing'. You can't just take that for granted like mathematicians currently do.

Fortunately for mathematicians most things that they attempt to quantify do coincidentally have well-behaved properties of individuality. But no thanks to mathematical formalism!

In fact, physicist have been addressing the notion of the importance of "units" of measure all the time. They discovered very quickly the importance of recognizing what it takes for something to be considered to be an 'individual unit' in a physical sense.

And that is precisely what 'saved' engineering and physics from an otherwise sloppy mathematical formalism. It's the engineers and physicists that we have to thank for that. Not the abstract mathematicians, or Georg Cantor's empty set theory.

~~~~

However, in today world of physics pure abstract mathematics is becoming of paramount importance, precisely because physics is moving into a realm where their theories are becoming almost solely mathematical.

This is where the faulty set theory of pure mathematics is going to cause problems. We are now entering and age where it needs to be fixed.

And because of this it probably will be fixed in due time.

But right now, it's based on a totally erroneous idea that was basically introduced by one man at the dawn of the 20th century.

That's where the fundamental mistake has been made. And we need to go back to that point in time (mathematically speaking) and readdress those mistakes. Only then can we get back on track with a viable mathematical formalism.

And like I say, this isn't going to change much in calculus, or trig, or geometry, or algebra, or much of anything. What it's going to change is Set Theory and however set theory may have had an effect on these other disciplines.

I've already shown how all irrational numbers can be seen to be relative self-referenced effects, rather than absolute cardinal properties in terms of collection of individual things. That's one very powerful insight that comes out of this.

Why is that so powerful? Well, now anytime we run into an irrational number we'll know to look for what's causing the self-referenced situation. And we'll understand better what's going on.

~~~~~

Another thing that falls out of this is that we can then calculate Planck's constant using 'pure mathematics' much like we have calculated the value of Pi. But that's way down the road. The basic flaws in set theory need to be addressed first. Then we can move on to more interesting stuff like this.

[Categories+Sheaves]

Thank you for going into that.

Not much more to add here. I'm a fan of that empty set theory, you're not. But if you ever do write this stuff out/formalize it more, I would like to take a look at it.

[Abracadabra]

Thank you for going into that.

Not much more to add here. I'm a fan of that empty set theory, you're not. But if you ever do write this stuff out/formalize it more, I would like to take a look at it.

As much as a ramble on in forums, I'm actually a horrible author. I've started to write a book on this topic more times that I care to count. I still have those old manuscripts. I'm never happy with them and I keep starting the book over again. I'm currently not working on it at all. I'd really rather produce a lecture video series and present it that way instead. I don't have the equipment to do that at the moment.

I also have extreme ADHD which causes me to be easily distracted into other things. I'm one of those people who has so many pokers in the fire that I've put the fire out. (ha ha)

I've been thinking a lot about Planck's Constant and how it most likely is like Pi. As a mathematician, I'm sure you're fully aware that Pi is a constant that is entirely dependent upon flat space. In other words, the value of Pi changes if you change the geometry from Euclidean to spherical or hyperbolic. The value of Pi is actually a property of flat space.

It's not some magical 'number'. It's just a quantitative property of flat space.

I've become increasingly aware the the same thing is true of Planck's Constant. It's very much like Pi. It's value is dependent upon the curvature of space. I'm not sure if physicists are actually aware of this. In some ways they might be, but in other ways I don't think they have fully grasped the significance of this. Especially in terms of unifying General Relativity and Quantum Mechanics.

When General Relativity and Quantum Mechanics actually 'meet' in a significant way there is necessarily always an extremely warpage of spacetime involved. Thus, not only is the value of Pi going to change dramatically, but so will the value of Planck's Constant. And other constants as well.

The mystery of what's going on in a blackhole, for example, baffles the scientists because all known physics seems to break down. However, so does the fabric of spacetime.

In fact, since time itself slows down at an increasing rate as the curvature of spacetime becomes more intense, this means that time, inside a black hole also slows down to a near standstill.

In fact, here's something to think about.

If you fell into a black hole by the time you reached the supposed singularity the entire universe will have ended.

In other words during your seemingly short trip (from your perspective) falling into the black hole, the universe exterior to the black hole will have aged almost infinitely.

So by the time you 'hit bottom' (if there is such a thing) then the universe external to the black hole would be infinitely old.

Here's another interesting thought.

Realizing the truth of what I have just observed above, and adding to that Stephen Hawking's theory that black holes will eventually "evaporate away" given enough time (through Hawking radiation), then, if you fell into a black hole, but the time you got to the singularity the universe would have aged infinitely. And thus the very black hole that you had fallen into will have evaporated by the time you reach the singularity!

Wow!

I mean, seriously. Stop and think about that for a moment.

This basically means that everything that falls toward the singularity of a black hole can ultimately never reach that singularity because by the time they get there (due to time dilation) the black hole will have evaporated completely.

Thus in a sense, it's meaningless to even speak about there being a singularity at the center of black hole because by the time anything reaches it, the universe will be over and the black hole will have evaporated completely.

~~~~~

Ok, well I warned you that I have severe ADHD, so perhaps now you can see how I go off on tangents so easily.

But that's they way I am. I try to address a particular subject and end up going off in every imaginable direction except where I had planned as my destination.

So if I ever succeed in writing a completed book, that will be a miracle and I'll have to return to Christianity and start believing in Gods that speak from clouds again I guess. (ha ha)