[counting axioms] - A New Kind of Science: The NKS Forum

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counting axioms

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Posted by: Philip Ronald Dutton

Given the Peano axioms we can produce the natural numbers. If operations are defined in terms of the successor function then how would one go about defining a prime number? You can not use the word "product" or "multiplication."


Moving along, let us define a very basic system and call it "the counting system." This system is axiomatic just like the Peano system. With the counting system we just want to be able to move "forward" along the so-called "number line." We do not have any operational functions other than what allows us to take a so-called "algorithmic step."

Essentially, this system can mimic a unary counting system;however, it does not give the user any facility for "talking" about what would actually be the "number" at any given point of the "count." I am perverting the word "count" but I shall continue to do so. Normally the word "count" invokes the emotional assumption that the system lets you talk about "numbers."

Quickly consider the following post:

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 too much MSG. You will not know 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."


Now, the counting system (in the form of a very limited axiomatic system) stands alone and can not be talked about in terms of "numbers", "number systems", nor "prime numbers".

Can the Peano system simulate the counting system? (Don't worry...I have a specific place where I am trying to go with this thought! I will post the remainder shortly.)


Posted by: Philip Ronald Dutton

Say that we create the counting system as explained above. It does not let you talk about numbers or operations. I assume that the Peano system can simulate the counting system OR the Peano system actually has (at its core) a counting system.

Actually let us use the assumption that the Peano system actully has a counting system "inside" of it somewhere.


The complement (as in set theory) of the counting system within the Peano system is what causes the notion of "prime"... NOT the counting system.

That is where I am trying to go.

Visually take two concentric circles. The inner circle is the counting system which is a sub feature or building block of the Peano system. The outer circle is the whole Peano system. The complement of the inner circle is what creates the ability to talk about primes.

So, from an NKS perspective: My question is related to counting systems as defined above (albeit somewhat informally). I really feel that they are a building block which is very fundamental to various branches of mathematics and also to NKS.

QUICKLY: let me change the name of "counting system" to "ticker system" ... basically I am talking about the axiomatic version of a metronome. It should be very easy to write an axiomatic metronome.

Given the metronome system, I would not be surprised if most of the axiomatic mathematic systems (or brances of study) have an metronome system built in. Maybe it is just harder to spot.

Now, concerning NKS, another question is how much of the behavior of simple programs is actually due to the metronome? First guess is "not much." So, having said that, perhaps the behavior we see is usually due to the allowed operations and not the underlying "number system" or "grid" or whatever "form" is being explored.

Finally, I have a small QUIZ which I hope illuminates some of the reasons why I am talking about all this.

QUIZ: Given an axiomatic system that produces the natural numbers, define "Prime Number" in strict terms of addition instead of in terms of division/multiplication.


Here are the first few paragraphs of the definition of "prime number" from Mathworld:


A prime number (or prime integer, often simply called a "prime" for short) is a positive integer p>1 that has no positive integer divisors other than 1 and p itself. (More concisely, a prime number p is a positive integer having exactly one positive divisor other than 1.) For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization 24==2^3.3), making 24 not a prime number. Positive integers other than 1 which are not prime are called composite numbers.

Prime numbers are therefore numbers that cannot be factored or, more precisely, are numbers n whose divisors are trivial and given by exactly 1 and n.


In an abstract way, I will be trying my best to map the metronome system to the natural numbers that come out of something like Peano's axioms. As I tried to explain above in the first post, they both seem to be using the same "FORM"... one looks like a number line in the standard sense and the other is the same number line but stripped of the ability to talk about it in terms of "numbers". One is a number line the other is the "metronome line" but they appear to be the same thing in terms of their "form" and their "construction" via the axioms.

I went through hoops trying to figure out if Peano's system came before or after the number line... but that is a topic for another day. I am assuming here that the Peano system creates the number line.

PS: I just happen to be talking of Peano but surely other natural number generating axiomatic systems could have been used in my discussions.

PS: Often, the layman (me included) think about prime numbers in terms of where they lay on the number line ONLY. I think this is a disastrous result of math education across the globe. But the fact is you just look at the definition of "prime number" and you can clearly see that "primality" is due to the operations like "multiplication/division". Otherwise, the definition of prime number would be an infinite list EXPLICITLY stating where each prime number is on the number line or it would be a function (which nobody has found).


Posted by: Philip Ronald Dutton

Consider the following NKS question: What is the simplest program possible for all program types (mobile automaton, ca, tag systems, etc.).

Let us assume there is one and that it is called "SIMPLE 1.0." Now, will "SIMPLE 1.0" be something that can be written down using the axiomatic system?

Obviously it is not going to be useful but that is besides the point right now.


Posted by: Philip Ronald Dutton

Since the beginning of this post, I have learned a lot about axiomatic systems, the guy Peano, and "counting systems". I admit I am not an expert but I continue to press forward. Here is my lastest attempt to communicate more clearly on the subject of axiomatic counting (it will definitely be of interest in terms of NKS.. which is where I hope to arrive soon).

Continuing...

Let us think in terms of a metronome for a few minutes. No, it isn't the creepy guy in 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.

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 am viewing the wikipedia list of axioms). 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 (according to my error-prone interpretation) 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 unless your system has a reference point (it is a little obvious I admit).

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 (or shall I say "dogma") we desperately tried to remove. Also, sometimes, attempts are often made which 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 which might be less fruitful if you are trying to solve the Reimann Hypothesis. Right now I am just trying to solve the puzzle of spelling that guy's name!


Posted by: Philip Ronald Dutton

Are the Peano axioms simply an interface to recursion? The Peano axioms, seem to have encoded a reference point such that the recursion "black box" becomes useful enough that we can talk about numbers and operations on them. However, if that reference point is left undefined then in some sense the recursion is still available for use. Of course it will be hard to find many uses. I can think of only one abstract use: a metronome? It is a little mind boggling.

From an NKS perspective what simple program frameworks (CA, mobile automata) can be used to create the simplest "metronome" ?



Posted by: Philip Ronald Dutton

Think about ANY algorithm and pretend that we will let it run but we will first "delete" all the internal steps. We only want the "shell". Now, isn't this the simplest program? For whatever simple program framework you choose, this simple program must implicity already exist before you can do anything. It is like a Turing machine that is turned on and just keeps moving the tape. Let us call all of these types of programs by the name "metronome" for sake of discussion.

Questions:
*Is the metronome usually "installed" in most axiomatic systems?

*Does the formalized mathematical notion of recursion/induction have a metronome?

*If we never allow "metronomes" then, can there be any mathematics which are "created" using the axiomatic method?




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