Fundamental Problems of Mathematics, and Why It Isn't the Apocalypse Right Now

Let us start with a stunner to people's prejudices. All over the world today, we use Arabic numerals in mathematics because they are very simple and clear. If we go back to maybe 800 in the Common Era, the Arabs had a stunning mathematics knowledge, and even a very crude form of calculus. They also used the base-10 system very early, and by the way, this was never common in the world.

Base-10 is very advantageous for certain reasons, although really, you can use any base-system without any real difference in measuring quantities. The problem with other base-systems isn't that you have, "funny numbers," it is that base-10 is simply - more simple and more clear. Simplicity and clarity of notation can revolutionize the world - and in the case of Arabic numerals and base-10 - this is a big reason for our engineering advances - a simple massive shift in the clarity of mathematics notation.

Other base systems are quite interesting - binary for computers - senary in terms of primes - octal-base has some strange obscurities - we use hexadecimal in programming languages based in binary for certain reasons. All of these obscurities in base-systems are really a matter of notation - except in binary where an efficient processor switches on - or off.

A fundamental problem that goes back to Arabic mathematics is the problem with indeterminate sets. An indeterminate set is an infinitely large set, and a determinate set is a set of finite size. The first problem is that indeterminate sets are loosey-goosey by definition to start with. The second problem is that when you integrate an infinite set with a finite one, you get nonsense.

So we use logarithms to approximate the solutions for our engineering and for other calculations, as we have to integrate indeterminate sets and determinate sets all the time in mathematics. One reason we have to do this all of the time, is that our fundamental set in our set theory is the set of natural numbers, which is an indeterminate-infinite-number series.

The problem has been known since 800 of the Common Era by the Arabs. The problem goes back at least that far. So let us do some history. A philosopher named Bertrand Rusell and a mathematician named Alfred North Whitead came up with the idea that we might solve this problem by building our sets in hierarches. The umbrella of the set would widen as it went up the chain.

The set theory had some large advantages. What was the problem? It was cumbersome and unwieldy. It was too complex to ever be effectively used! So, mathematicians at the time said, "Very interesting Mr. Russell and Mr. Whitehead. However, I'm doing my research with logarithm tables and approximations, and my research is going just fine. Go to hell, Mr. Russell and Mr. Whitehead!"

So our next Whitehead was a man named Kurt Godel. He did an amazing and iron-clad deductive proof, and it went like this: For every set of axioms, there is a deductive error in those axioms which cannot be solved. The proof is a monster, and "needs more study," but again, "I see your amazing proof Mr. Godel. However, I am working very hard on my research with logarithm tables and approximations, and I don't have the time for your silly proof. Go to hell Mr. Godel!"

A further answer might be, "I hope you are aware that other than deductive logic, we also have a rather difficult branch of logic called induction." Inductive logic is the most important part of our data-based sciences today. You collect approximate data and interpret it by induction. That is how we do almost all of our amazing engineering!

In mathematics today, one of the things maniacs obsess over is called, "finistic reductionism." The idea would be to prove that all mathematics can be reduced to a finite set of finite sets. First, this is probably just as maniacal as these maniacs are. However, it is attractive to wonder if we could reduce our problems to both discrete and finite terms. It is likely utterly impossible, but it is attractive, and, "needs more study." Most people are too busy for such obscure and abstruse work, but I've put my mind to it some, and I am not - that type of maniac.

So, let us look at our fundamental natural numbers set. Remember, not many mathematicians delve into set theory much, but an advanced mathematician is aware of this problem. I am not inventing a new wheel here. I can't plug true math notation in, but less us look at a simple-negative-deductive proof involving the set of natural numbers.

1. The set of natural numbers is defined as: Set N = Set N + 1. This is the most common definition you'll see in math classes.
2. Select any element x from Set N.
3. Let x = x + 1
4. x does not equal x + 1, as x must equal x by the identity theorem, a = a.
5. Null proof. Set N can contain no elements, and is union with a null-set as defined.

This is our entire problem with indeterminate sets, particularly if they are of the form of an infinite series, and the set of natural numbers if the fundamental set of all of our mathematics. I've imagined a new definition, and it may be already suggested by mathematicians with a better education and more resources than my own. I can't be sure as I have no contact with any peers in mathematics.

It comes from my study of discrete mathematics, and uses a form of notation that started in inductive logic. It also uses the concept of an infinitely-limited series. I can't plug the exact notation in, but it goes like so:

Definition of Set N, the Set of Natural Numbers: For any element x that is discrete, Set N is union with {E+ = 1 --> |x+1|}.

That short statement is the result of three years of effort. Now, there are other big set theory problems. How do we define what the number one is? There is a branch of mathematics that is developing called number theory to try to understand that. No one really has the time for number theory. I have some ideas on the platter, but I'm not so sure at this point.

Also, how do you define an operation? Just take addition. We have two quantities and we arrive at a sum. Fine. What is that? That is a problem of what is called Modern Algebra. Few people even understand modern algebra well enough to do any work on it, and very few people have any time for it. I have some ideas on the platter, but I'm not so sure at this point.

The definition of a limit. That is a complex SOB. A finite-limit definition is way beyond most people, and I can't remember well-enough the beta-delta forms to list it here without going to look it up. An infinite-limit definition is worse, because of those same problems we see with infinite series and indeterminate sets.

However, my simple reduction to a discrete function of Set N, whether previously in mathematical minds or not, produces some simple and clear results. For the integers, we can simply do a positive and negative summation of the exact same equation. The zero-number is a different problem, and I have thought of something hilarious - comedy math - over zero numbers. However, I want to probe at that one some more before I release it into the world at large. That is farther along than the definition for the number one, or the definitions for operations.

This is one of the silver-linings of my life, is that while other people need to probe some other tough questions, I've been able to trawl through set theory, number theory discrete mathematics and modern algebra, because I simply have the time and the freedom to do so. I remember remarking once that, "What if all of this development has a flaw at the bottom of the pile?" - and that is the kind of analysis I'm working on.

There are flaws at the bottom of the pile. We've known about set theory problems for at least 1200 years or so, and Godel is right, no axiomatic system is perfect. However what can I do with my time to look at the cracks at the bottom of the pile? I'll show you some other way out things I've done with matrix geometry in another article, but the next one up is my attempt to construct a fundamental philosophical system of category and context. It isn't done or perfect, but it's important to me, and I hope someone might find some value in it. Off we go, after a short break for me and a bit of food.