Wednesday, April 30, 2008

binary numbers represented the standard way

Binary numbers, when written using standard westernized character sets, often resemble a string of characters limited to either a "1" or a "0" in arbitrary order and position. Take the normal, everyday decimal number 5. This, when translated to modern binary is "101." I have often pondered these binary character strings. When we humans "read" such binary numbers we usually read from left to right. Of course this is all quite obvious. However, consider a machine that produces such outputs in some way.

Perhaps we can visualize, for sake of an example, a digital lie-detector or a CAD engineer's plotter. Assume that there are mechanical plot heads for each print pixel instead of one plot head which moves back and forth. So we have an array of plot heads each happy to produce or to not produce for the particular line being "printed." Now, visualize this print machine to be as wide as the length of your regional Google campus. As a human operator looking down at the machine we could easily push the pause button stopping all operations. We look at the the last printed line (perhaps under microscope) and we see- essentially- a string of on and off pixels.

If we dare imagine this output to be a binary number then we are perhaps humbly amused. Perhaps my wild imagination would be too distracted from managing the plot machine causing the company to loose many dollars. Assume the operator would fancy the idea that the machine was plotting a sequence of binary numbers. Perhaps the operator would amuse himself by thinking that the machine was printing all the prime numbers- one right after another but in binary.

Since it is late on friday afternoon, the operator decides to quickly translate the first 200 binary numbers into decimal. He then plans to check the sequence of numbers with the sequences he is familiar with.

His first and foremost task is to figure out where the far right digit resides. Remember he is looking at quite a wide plot. For sake of discussion, assume that the spot wherein he is standing happens to have been the same spot in the plot where the machine first started to print binary numbers. Assume for example that the first number printed was "1" and the operate simply had to look down at the machine while standing in the middle of the printing facility. He did not have to walk 3 minutes to the end of the building to see the far right of the plot machine.

Now, we know the first number is a one because there was only one ink pixel. But what if the machine was trying to print the binary equivalent of the number 3?? The machine would still only print 1 pixel because the true binary representation is "10" but the zero represents a "do not print pixel" mark. So, the final point of this story is that the operator can not distinguish between the binary representations of the decimal number 1 and the decimal number 3!!

This example- I hope- provides a vivid delineation of the quite demanding requirement for all such talk of numbers to have a predefined reference point for the zero position.

Until next time,
"Happy Numerics!"

Note: The plot machine above is essentially a 1D cellular automaton. It is impossible to view the output of such an automaton as representing binary numbers. The best one can do is to say that they will read from left to right and that the far right "1" is the next to last position. Since all odd binary numbers have "1" in that position, one would essentially be interpreting only odd numbers. Also, while performing such explorations, one must realize that in order to be complete, it would be best to interpret from right to left as well.

Thursday, August 9, 2007

in Peano's direction

Fact: The Successor function can only move away from the reference point.
Question: Does any aspect of the Peano system utilize the direction toward the reference point?

Now, if the successor function is not used for "going toward the reference point", then what is the mechanism that allows this directional procession? What allows you to go back like that? There is no other function defined and it does not appear to be coming from some feature "underneath" the formal framework of axiomatic systems. So, my guess is that the expressive capabilities of the axioms is what is being used to move backwards in such a case???

(If "moving backward" is not a notion you want to entertain, then perhaps an alternate view is that moving from "2" to S(1) is a symbol decoding function- something that decodes a symbol into it's appropriate parameterized successor-function "call". If so, then are the axioms creating this decoding function?)

Wednesday, August 1, 2007

Peano recursion

Given that the Peano axiom set is stripped of the axiom that says "0 is a natural number," I still believe the underlying "form" of what was once called a "number line" would still remain completely the same despite loosing the ability to define what a number is. Peano, by stating that "1 is a natural number" has basically "encoded" a reference point into the system. However, without the axiom, a user could just define their own reference point outside of the system and just use what is left in the Peano axiom set as a "metronome." The combination of "reference" point and "metronome system" is basically enough to completely build all the numbers. In other words, from an algorithmic perspective if you have memory (for the reference point) and metronome, you can get all the numbers, addition, multiplication, "prime", etc. all in one complete magical "poof!". Honestly, I am not totally comfortable with the idea that numbers and the operations are not completely separable; however, I can intuitively understand the phenomenon.

Peano's axioms just give the user a way to input things into a recursive blackbox which then turns around and spits out a number. It is essentially an interface to recursion. It is a system which, once the recursion is kicked into gear, there is nothing you can do except wait until the "answer" comes back. You can't peer into the recursive "machinery" to glean or use "internal" information. If you take the "0 is a natural number" axiom out then the black box remains but is basically "disoriented".

I am very curious about the Peano axiomatic system minus the said axiom. How would one find practical application for this in a mathematical sense? It is a valid axiomatic system and deserves a share of study.

Furthermore, it is now my understanding that the following statements hold (more or less):

*Recursion without a reference point is basically a metronome.
*Recursion without a reference point is just unary "counting/ticking."
*Recursion can only be used to define numbers when given a seed.
*Recursion is a powerful thing (mystery) which requires an interface to be used; hence, Peano defined his axioms.
*Pure recursion does not have a reference point.
*Every time you do anything via the Peano axioms, essentially what happens is that the system starts from "0"! This is amazing!

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 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.