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.

Sunday, July 29, 2007

number line vrs. counting line

Everyone has some understanding of the number line. I do not know if people just simply remember what they have been taught in grade school or if they intuitively have this uncanny understanding of the number line. Somewhere in between we humans know how to count using the number line. My question is about counting. Can you count without knowing numbers? If I ask you to count to 100 you can easily do this.

What if I tell you to do the same thing again but do not use the base 10 decimal system. In fact don't use any number based system other than unary. Can you count now? Sure you can. But you will soon loose track of where you are if you try to use your brain's short term memory or if you eat to much MSG. You will know not if you are getting close to the original number that I requested you to count to. You will not know if you have passed this number.

In this context, we have a new phenomenon. The number line is basically still there but we do not have any more reason to call it a number line. Let us call it a "counting line."

fooledbyprimes details....

I have been involved in a forum discussion on the unfortunate teaching about what is "prime." I will try to collect all the thoughts and post them here soon. For now, here is the link to the fooledbyprimes discussion on the www.physicsforums.com site.

(special thanks to user known as CRGreathouse for his patient dialog with me during that discussion)

Friday, July 27, 2007

The source of "prime" is not the number

I agree that "prime" is important. But I am trying to explain that a number is just a number. Try to define a prime number without using the word "product" nor the word "multiplication." Now, if the math professors can't do this then there is a serious problem with all the hype about primes. What will need to happen is that educators change their language-- change the way they talk about the phenomenon. The phenomenon is due to short cut addition. So, it is not the numbers that are cool but the short cuts we come up with for faster addition (in other words: multiplication).

In the pure mathematical universe time is not a factor so addition is just as fast as multiplication. However, when humans try to add in their head, they impose a time constraint. This is why the earliest mathematicians came up with multiplication. It is just a short cut since we humans are stuck in time. So, beg your professor to explain a prime number in terms other than multiplication and "product". I bet they can't do it or they will come up with an excuse about how they need to leave the room.

mystery of prime

What I am saying (in the first post entitled "Silly Primes") is that the prime numbers are not mystical. What is mystical is the relationship between the algorithmic process of counting and the notion of short-cuts (multiplication) and how the two "inter-twine". Are the two different entities? Yes. Counting is very pure. However, short-cuts require some sort of memory. The memory is in the form of additional "wiring"... like defining new kinds of number systems. Think about counting in a pure sense: the Egyptians, Babylonians, Greeks, Hebrews, Hindus, they all counted the same at the pure core level because they were all humans. But their short cut methods are what were different. Counting is simple, just repeat after me: "da, da, da, da, da, da, da....." Short-cutting and communicating about where the counting stops is a completely different ballgame and it is what produces the "mysterious" properties that we perceive in the primes.

I am very hopeful that new dialog will open up in the mathematics community. My challenge is still open: rewrite the fundamental theorem of arithmetic without using the words "product" or "multiplication."

Friday, July 13, 2007

Silly Primes

Not until recently has the whole prime number "culture" become a distraction to me. While a child the primes never really caught my attention. Even in college there was not much drawing me to the subject beyond the occasional newspaper headline proclaiming the exuberance of the mathematics community as some rather skinny, unkempt math geek held a new largest prime in high esteem.

One of the things that bothered me about primes is how messy they are. From the perspective of where they are on the number line one can't help but get the feeling that any equation related to their distribution is going to be ugly. Maybe I am a sucker for simplicity- just call it an eye for elegance!

Taking a look at the math culture's definition of a prime we find something like: "..a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself." Oh how boring! Of course the mathematicians tell us that primes build all the other numbers. Digging around one will find this formal statement called the fundamental theorem of arithmetic. It says, "every natural number greater than 1 can be written as a unique product of prime numbers." It appears to be very, very important to mathematics- afterall, it is the fundamental theorem of arithmetic!

I must admit I didn't investigate the prime number sequence at all other than taking a quick peek at the first 100 primes. Instead, I became intensely focused on the two related definitions given above. Take a look at the words in the definition and convince yourself which words convey the most "action"- the meat of the definitions so to speak. I came up with "natural number divisors" and "unique product." Now, I must say right away that I failed calculus II so I do not profess to be a brilliant mathematician (don't worry, I took the class again with a different professor and got an passing grade). There is one thing that I do know about math and it is this: multiplication is just repeated addition.

So, I wondered what would happen if the math culture rewrote the fundamental theorem of arithmetic without using the word "product." Wouldn't that be cool- a simplified version of the definition! Maybe... just maybe... we might find some new way to think about prime numbers and make some progress on the stubborn topic.