Tuesday, July 31, 2007

metro gnome

Let us think in terms of a metronome for a few minutes. No, it isn't the creepy guy at the subway station bathroom! Everyone knows what a metronome is. You have a heart and it beats. When I count numbers I am doing a few types of things but at the core of what happens is essentially a "tick, tick, tick,..." effect. Sure I have to extrapolate and manipulate some meta-language in order to actually tell someone what number I am at in my counting process. I already talked about such notions of counting in the previous posts.

The Peano Axioms clearly make use of a successor function which essentially helps the system navigate and get around on the number line. Now, of course the Peano axioms give the user the ability to perform addition and multiplication. I wanted to use the Peano system as a metronome (call it a "counting system" if you want). Once I realized it was possible to do this I decided to go ahead and strip off the extra features like addition and multiplication from the Peano system making the newer light-weight axiomatic system which simply gives me my kicks... I mean "ticks."

Removing those extra features from the Peano system was actually quite a puzzling ambition. I struggled with my understanding about how to get rid of the addition and multiplication yet keep the "counting." If you look at the Peano Axioms you can not "see" anything resembling an explicit definition of addition or multiplication. So how was I going to remove it? Again, the Peano Axioms make exclusive use of a successor function. It is the "thing" which lets a user move from one number to the next on the so-called "number line. " At first, it is quite odd that you can not have a succession capability without addition or multiplication. After all, in my mind, I have always thought of the numbers on the number line as objects which simply exist. I always thought you could at least get from one to the other somehow without resorting to the likes of addition or multiplication.

After some tough mental thought experimentation, online forum dialog, and further study I realized that there really was only one way to sever Peano addition and multiplication and still have a metronome/counting system. It is a rather simple surgery. One must only cut out the axiom that says: "1 is a natural number."

What this means is that things like addition and multiplication are not possible without a reference point. There are some other profound insights available when we complete the surgery. Consider, finally, that the notion of "prime" is destroyed without addition and multiplication. However, the form of the number line has not been changed (a consequence which is of extreme importance in my opinion). We just have no reason to call it a "number" line.

What this tells me for the numbers is that the positional aspect of numbers on the number line have nothing to do with whether or not they are prime numbers.

Interestingly, if you think about the metronome/counting system too long, your mind will quickly try to rebuild the additional wiring such that you can view that number/counting line in terms of the features we desperately tried to remove. Also, sometimes, a few mathematicians try to understand the primes in terms of the positional aspects of the numbers only and not in terms of the operational features of the axiomatic system.

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