Harristotelian Logic

A while back I talked a bit about why I don't believe fractions are objects. Not surprisingly, that was met with a great deal of derision on the newsgroup. I don't know how many of you can appreciate what it feels to be an object of ridicule for presenting such thoughts, but I do.

Now, there has been a lot of discussion about a ring I use in my proof of Fermat's Last Theorem.

That ring is formed by adjoining the square roots of the integers to the integers and also adjoining the numbers b1,…,bp to the resulting ring.

An assertion was made that the ring might have f as a unit, where f is an important variable in my proof that is a prime number (like 3, 5, or 7).

While dealing with this objection, I realized something interesting, since people kept arguing with me after I felt I'd clearly shown that f was not a unit in my ring.

It looked like an impasse until I realized I could produce a contradiction with their argument using a simple example:

Consider the ring formed by adjoining the number x to the ring of integers.

Now, prove that 1/3 is not in the ring.

I realized, as I also realized that none of these people had the skills to answer the challenge, that a property of being a fraction or being fractional, as I've called it, is what I'm abstracting, but current mathematics doesn't seem to have it.

I've tried to generalize the definition of fraction myself, and consistently failed.

However, that comes from thinking fractions are objects. As, I've talked about fractions like 1/2 not being objects but being operators, I thought it'd be worth it to explain what I mean.

It basically has to do with the label of object.

That is, a fundamental postulate is that something is either an object or an object operator.

I don't like that "something", but I seem to be stuck with it.

Then 1/2 is not in and of itself and object because you can't have 1/2 of an object unless the object is a set with two objects.

Then 1/2 means take 1 of the 2 objects within the set the operator is being applied to, which is the way it's always used in the real world.

For instance, you can have a pie, but that pie can be cut into two pieces. So you now have a set of two pieces and to take 1/2 the pie, you take one of the two pieces.

There are no exceptions to the use of 1/2 as an operator in the real world. That is, 1/2 is never in and of itself an object but is used on a set with two objects, where the objects can themselves be sets.

(Consider, what's 1/2 of an electron?)

So, I've used the term set without defining it, when I think it's clear that a set is a group of one or more objects.

So what is a group by my definition, why it's a set, of course!

Circularity among definitions is not a surprise when you're dealing with things at the fundamental level. If you look at established mathematics at the level of traditional set theory, you will find axioms, and if you push the axioms, you will see they are circular.

Or, you can just get a dictionary, look up a word, and then look up the words that make up the definition, and continue with this, and you will find yourself going in a circle.

To get all of mathematics I need only a few operations for my objects. I need addition, multiplication, and extraction.

I've used extraction by considering 1 piece of pie out of two that make up the entire pie.

I don't think I need to explain the addition and subtraction operators.

Now, what's neat about this is that gives me two types of numbers as objects. There are the integers, and the numbers of the continuous field. For instance, sqrt(2) (note we define it with an operator) is an object that multiplies times sqrt(2) to give the object 2 and is a member of the continuous field.

Now here's where things get fascinating (at least to me).

While i is an object, e and pi are not, as they are both operators.

That's fun to explain, and it goes back to the real world.

Consider that when you see a circle, for instance, drawn on your computer screen, what you're seeing are a lot of little dots. The number of dots can be given by multiplying pi times the diameter of the drawn circle, which defines the pi operator. Note: The result is an integer.

It turns out that if you work things out this way, you can build all of mathematics without the interesting little ambiguities that plague modern mathematics, like the one I pointed out at the top of this post, and no, I'm not thinking of any more off-hand, I just feel they have to be there.

Oh well, time for a lot more ridicule. Since I expect nothing but ridicule, I'm not likely to read any replies to this post.