Exploring the Mystery of Prime Numbers

[philiprdutton]

creating a number system

If you want to play "da da da da" for a while, stress one "da" of every N, as in "da da DA da da DA ..."; if you put them all together,

2 da DA da DA da DA da DA da DA da DA da ...
3 da da DA da da DA da da DA da da DA da da DA ...
4 da da da DA da da da DA da da da DA da da da DA ...
5 da da da da DA da da da da DA da da da da DA da da da da DA ...
6 da da da da da DA da da da da da DA da da da da da DA ...
7 da da da da da da DA da da da da da da DA da da da da da da DA ...

a prime number is one where the first stressed DA's won't coincide with any DA for all smaller numbers.

(Which of course is a re-edition of the [/PLAIN]
Sieve of Eratosthenes.)

Okay, so basically you just created a numbering system. Given the counting system you just added some machinery to give you the ability to talk about where you stopped counting. Once you do this you can start looking at the patterns produced and start theorizing and start writing conjectures. But all that you discover is not related to the place on the number line. It is related to the nature of the extra machinery.

 

[philiprdutton]

a race

Yes, but that would be beside the point. They have no properties, so there's nothing making the counting "2" more or less like the Peano "2" than the Peano "7". I can put them in any order I want -- and in fact I could associate them with only the Peano primes, or only the Peano composites, or only the Peano powers of 2 that are squares.



Terminology. Remember that both Peano arithmetic and your counting system have an infinite number of axioms -- you have one axiom schema, which actually has omega members (one for each natural number). So yes, each of your axioms has only one statement it can make, but you can make an infinite number of statements.

That aside, I'm still not sure I quite follow. What is the motivation behind the branch terminology?

Basically, I meant to say that you "start" the counting system. You also "start" to count USING the Peano system. Now, for each step, there will be a result for each system. Let's say that there is a set of results for the counting system and a set of results for the Peano system WHEN USED as a counting system. Now, just map the two systems formally with these results in mind. If this could be done, then I guess you can say the two systems are equivalent in the sense of those sets of results. However, you can not impose the notion of prime from the Peano set of results back over to the counting system. That is all I am wanting to do. And I want to know what it means for the notion of primality. Just trying to open up discussion about all this in layman's terms.

 

[CRGreathouse]

But all that you discover is not related to the place on the number line. It is related to the nature of the extra machinery.

Of course some could argue -- and I think I would -- that this extra machinery is the number line, not the counting axiom schema. So far that's not even strong enough to tell us that "2" comes after "1".

 

[dodo]

I don't think there was too much extra machinery. I just replaced his notation (da da da <end> for the number 3) for another more easy to type (da da DA for 3), and defined addition as the concatenation of sequences, which is only natural when counting "da da da".

The prime definition, on the old notation, would not change the concept. It would say, "a prime is a number where the first 'da<end>' do not coincide with any 'da<end>' for smaller numbers". I merely replaced 'da<end>' by 'DA', and defined addition. The fact that you don't explicitly mention the <end> at the end does not make the concept of 'end' disappear.

 

[philiprdutton]

strong enough

Of course some could argue -- and I think I would -- that this extra machinery is the number line, not the counting axiom schema. So far that's not even strong enough to tell us that "2" comes after "1".

So, if the extra machinery is the number line, then Peano might possibly might not have been biased by an intuitive notion of a number line? Given Peano axioms (we just happen to be using Peano axioms for sake of discussion) what do they do (in context of discussion)? Do they:

A) create the number line?
-- or --
B) create the facility to "talk" about the number line?

 

[philiprdutton]

primes: what is it?

I don't think there was too much extra machinery. I just replaced his notation (da da da <end> for the number 3) for another more easy to type (da da DA for 3), and defined addition as the concatenation of sequences, which is only natural when counting "da da da".

The prime definition, on the old notation, would not change the concept. It would say, "a prime is a number where the first 'da<end>' do not coincide with any 'da<end>' for smaller numbers". I merely replaced 'da<end>' by 'DA', and defined addition. The fact that you don't explicitly mention the <end> at the end does not make the concept of 'end' disappear.

Are you saying that you are in agreement that the notion of prime is not due to a strict position on the "number line" (whatever the hell a number line is) and instead it is due to structure of the 'meta-data" you are adding when you change the appearance from "da" to "DA"?

 

[CRGreathouse]

Are you saying that you are in agreement that the notion of prime is not due to a strict position on the "number line" (whatever the hell a number line is) and instead it is due to structure of the 'meta-data" you are adding when you change the appearance from "da" to "DA"?

I think that once you put connections in between the numbers (so that "3" is right between "2" and "4", where in your axiom schema right now there's nothing special connecting the three) you can form the concept of primes.

 

[philiprdutton]

positions

I think that once you put connections in between the numbers (so that "3" is right between "2" and "4", where in your axiom schema right now there's nothing special connecting the three) you can form the concept of primes.

Actually I do not want to be able to have the concept of primes. That is why I left it as a counting system. Leaving it that way, I want to map the bare counting system to the Peano system's version of the counting system. After all, I am assuming that the Peano system can indeed "simulate" the bare counting system. Is this possible? Yes I think it is. Why do it? For the sake of understanding the notion of "prime" separate from the notion of the position of the "item" on the "number line."

 

[CRGreathouse]

Actually I do not want to be able to have the concept of primes. That is why I left it as a counting system. Leaving it that way, I want to map the bare counting system to the Peano system's version of the counting system. After all, I am assuming that the Peano system can indeed "simulate" the bare counting system. Is this possible? Yes I think it is. Why do it? For the sake of understanding the notion of "prime" separate from the notion of the position of the "item" on the "number line."

But your line doesn't have position right now. "7" is just as close to "1048576000000000" as it is to "6".

 

[philiprdutton]

Algorithmically speaking... I can just as easily interpret the Peano axioms algorithmically since Peano successor function is very "algorithmic." Yes my counting system does not have position in terms that you speak of. It only has algorithmic position. Can we map this notion of algorithmic position (or step of execution - comp. sci. terms) to the "positional" stuff that we get out of the Peano axioms?

 

[CRGreathouse]

What do you mean by algorithmic position?

 

[philiprdutton]

mapping retake

But your line doesn't have position right now. "7" is just as close to "1048576000000000" as it is to "6".

I was referring to the positional "stuff" that you get out of the Peano system.

Let try this: Just ignore positional stuff in the Peano system and try to get it to produce what we are calling a "counting system." Let us say the Peano system can do many things. One of the things it can do (we hope) is just simulate the basic counting system. we have been talking about. Now, equate these two systems Once you map them, then allow the positional stuff to come back into view on the Peano side. With it comes the notion of prime but you can not impose that notion of prime back onto the basic counting system that was mapped to the peano counting system.

 

[philiprdutton]

steps

What do you mean by algorithmic position?

Steps. How does the peano successor function produce the successor? In zero time? Does the framework allow one to talk about the successor function in terms of "steps." How does the successor function "compute" the successor of x? Is it a magical filter that pumps out numbers but does not let you look into it ?

Obviously, time is not a factor in the "ether of mathematics and abstractness" but what is preventing me from saying there are 2 steps from S(4) to S(6) ?

 

[philiprdutton]

successor

Steps. How does the peano successor function produce the successor? In zero time? Does the framework allow one to talk about the successor function in terms of "steps." How does the successor function "compute" the successor of x? Is it a magical filter that pumps out numbers but does not let you look into it ?

Obviously, time is not a factor in the "ether of mathematics and abstractness" but what is preventing me from saying there are 2 steps from S(4) to S(6) ?

Actually, I think about it more and I am convinced that the Peano system definitely allows you to look at what is happening in terms of algorithm. Algorithm implies steps. Perhaps Peano wanted an algorithmic viewpoint, I don't know.

 

[philiprdutton]

Time for some side question:

Is there a way to map one axiomatic system to another? Has it been done for systems that are similar? Can it be done from one simple system to a system that is more feature rich but which still simulates the simpler system?

What exactly does it mean to map a formal axiomatic system to SOMETHING ELSE. Is this the technique employed by Godel in his most excellent work?

 

[CRGreathouse]

I was referring to the positional "stuff" that you get out of the Peano system.

Let try this: Just ignore positional stuff in the Peano system and try to get it to produce what we are calling a "counting system." Let us say the Peano system can do many things. One of the things it can do (we hope) is just simulate the basic counting system. we have been talking about. Now, equate these two systems Once you map them, then allow the positional stuff to come back into view on the Peano side. With it comes the notion of prime but you can not impose that notion of prime back onto the basic counting system that was mapped to the peano counting system.

Still not following. What do you mean when you say you "equate the two systems"?

Steps. How does the peano successor function produce the successor? In zero time? Does the framework allow one to talk about the successor function in terms of "steps." How does the successor function "compute" the successor of x? Is it a magical filter that pumps out numbers but does not let you look into it ?

Obviously, time is not a factor in the "ether of mathematics and abstractness" but what is preventing me from saying there are 2 steps from S(4) to S(6) ?

When you're working with the pure Peano axioms, there's noting you can say about time, space, or other complexity. If you choose a particular model of the Peano axioms, then you can talk about it.

 

[CRGreathouse]

Is there a way to map one axiomatic system to another? Has it been done for systems that are similar? Can it be done from one simple system to a system that is more feature rich but which still simulates the simpler system?

It's done a lot, sure. A system A can be shown to be consistent relative to a (stronger) system B by constructing a model of A inside B. Kelley-Morse set theory can model ZFC (and prove its consistency too, showing that KM is actually stronger).

What exactly does it mean to map a formal axiomatic system to SOMETHING ELSE. Is this the technique employed by Godel in his most excellent work?

Which work? Godel made several major contributions to mathematics... do you mean his Incompleteness Theorem, or perhaps his earlier Completeness Theorem?

 

[philiprdutton]

Godel's proof

Which work? Godel made several major contributions to mathematics... do you mean his Incompleteness Theorem, or perhaps his earlier Completeness Theorem?

I was referring to the one which states that "for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms." (copied from wikipedia)

 

[CRGreathouse]

I was referring to the one which states that "for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms." (copied from wikipedia)

That's the Incompleteness Theorem. I don't think his proof used the mapping technique you mention.

 

[philiprdutton]

question about mapping

It's done a lot, sure. A system A can be shown to be consistent relative to a (stronger) system B by constructing a model of A inside B. Kelley-Morse set theory can model ZFC (and prove its consistency too, showing that KM is actually stronger).

You said "consistent"... does that mean "equivalent?"

 

[philiprdutton]

godel mapping

That's the Incompleteness Theorem. I don't think his proof used the mapping technique you mention.

Well he used some kind of mathematical mapping.

 

[philiprdutton]

which proof?

That's the Incompleteness Theorem. I don't think his proof used the mapping technique you mention.

Sorry I am talking about the more famous of the two.

I think maybe his Godel number function is the "mapping" I am referring to. I thought it was a generic mathematics technique.

 

[CRGreathouse]

You said "consistent"... does that mean "equivalent?"

No. In fact those two are inequivalent -- and you should know the reason, since you just posted it: the Incompleteness Theorem. No sufficiently strong theory* can prove its own consistency, so since KM proves ZFC to be consistent the two can't be equal (unless one is inconsistent, in which case they're both equal to the theory "for all p, p" in which everything is true).

* Any theory containing Peano arithmetic is strong enough.

 

[CRGreathouse]

Sorry I am talking about the more famous of the two.

The Incompleteness Theorem is the more famous of the two, and it's the one you quoted. The Completeness Theorem is the one that essentially says that in first-order logic, provability <--> truth.

 

[philiprdutton]

Godel encoding

Godel encoding was used by Godel as follows: "Gödel used a system of Gödel numbering based on prime factorization. He first assigned a unique natural number to each basic symbol in the formal language of arithmetic he was dealing with."

Can I assign each natural number from the Peano system to the counting systems' statements which we have been talking about here? I want each step of the counting algorithm output to be assigned the associated natural number.

(sorry human language is making it difficult to be formal and to keep all my terms in proper context. Obviously, there is no existing association when I said "associated natural number" but you know what I mean...)

 

[philiprdutton]

enlightening...

No. In fact those two are inequivalent -- and you should know the reason, since you just posted it: the Incompleteness Theorem. No sufficiently strong theory* can prove its own consistency, so since KM proves ZFC to be consistent the two can't be equal (unless one is inconsistent, in which case they're both equal to the theory "for all p, p" in which everything is true).

* Any theory containing Peano arithmetic is strong enough.

Okay, sorry for my confusion thus far. I really wanted to get close to the idea of showing that the counting system and the peano system are both using the "number line" in a "synchronized" fashion.

If each system "creates" a "number line"... and, each "number line" has the same form, then I want to equate the two systems on that basis.

 

[CRGreathouse]

Can I assign each natural number from the Peano system to the counting systems' statements which we have been talking about here? I want each step of the counting algorithm output to be assigned the associated natural number.

The only statements you have in the counting system are the natural numbers and whatever your underlying logic allows with that:

"1 is in N"
"1 is in N and 7 is in N"
"(1 is in N and 7 is in N) or 6 is not in N"

You can certainly give a Godel numbering to your counting system's statements, but I don't understand to what end you are doing that. Also, do you mean statements or just theorems? Are you including false ones like "1 is not in N"?

Also, what algorithm do you mean?

 

[CRGreathouse]

Okay, sorry for my confusion thus far. I really wanted to get close to the idea of showing that the counting system and the peano system are both using the "number line" in a "synchronized" fashion.

If each system "creates" a "number line"... and, each "number line" has the same form, then I want to equate the two systems on that basis.

Still not getting it. Let me break this down and you can help me through what I don't get.

1. Each system creates a number line.
1a. Your counting scheme creates a number line.
1b. Peano arithmetic creates a number line.
2. The Peano number line has the same form as your counting scheme's number line.
3. If two number lines have the same form, they are equivalent in some sense.

What's a number line? That is, what properties does something need before you'll call it that? Surely any sensible definition will make 1b true, but some could make 1a false.

What do you mean when you say "form"? I would think this means the two share certain properties, but which?

In what sense do you want to equate the systems? Usually this would mean that systems which fit certain properties can prove a certain collection of facts about their members, but which?

 

[philiprdutton]

form

Still not getting it. Let me break this down and you can help me through what I don't get.

1. Each system creates a number line.
1a. Your counting scheme creates a number line.
1b. Peano arithmetic creates a number line.
2. The Peano number line has the same form as your counting scheme's number line.
3. If two number lines have the same form, they are equivalent in some sense.

What's a number line? That is, what properties does something need before you'll call it that? Surely any sensible definition will make 1b true, but some could make 1a false.

I do not know what a number line is nor "WHEN" it gets created in relation to either system. That is why I asked about what "comes first" in Peano: the number line that we all were taught as kids or the axioms. Also, I casually referred to Peano in terms of whether or not he was biased by the notion of "number line." Maybe for fun we could talk about a "counting line" since each system can at least produce or use one. Whenever I talk of number line I am referring to that "form" which has become so damn intuitive that I can't prevent it from affecting my thinking about math in general.

What do you mean when you say "form"? I would think this means the two share certain properties, but which?

In what sense do you want to equate the systems? Usually this would mean that systems which fit certain properties can prove a certain collection of facts about their members, but which?

Yes. I meant that the two systems "store results" in the same "form." A linear form with "points." I want to equate the two systems in terms of how their counting features use the form. Then I would allow the peano system to fully express itself and reveal the notion of prime... but then I would be able to say that you can have your prime but not in terms of the counting features only. If you can not have the prime in terms of the counting features only then that invariably says something about not having "prime" in terms of the FORM that each systems "use" (or "create").

My hope is that I can find a simple way to prevent the millions of people who know of the "prime" numbers from attributing the notion of "prime" to the "place in the form in which that number happens to reside."

(note: in my interpretation, a number can not reside anywhere until you have defined a way to talk about that number in terms of where you stopped on the counting line in order to "arrive" there.

 

[CRGreathouse]

Maybe for fun we could talk about a "counting line" since each system can at least produce or use one.

I would argue that your counting system can't actually count, and thus isn't a "counting line" as such. That's why knowing what you mean by form is so important to me.

Yes. I meant that the two systems "store results" in the same "form."

But Peano arithmetic has many more true results than your counting system, and I still don't know what "form" is.

A linear form with "points."

But the counting system of yours isn't linear, is it?

I want to equate the two systems in terms of how their counting features use the form.

Truly, I understand almost none of the key words in this sentence: "equate", "counting features", and "form".

(note: in my interpretation, a number can not reside anywhere until you have defined a way to talk about that number in terms of where you stopped on the counting line in order to "arrive" there.

As I understand it, your counting system does not have a way to "talk about that number in terms of where you stopped on the counting line in order to 'arrive' there".

 

[philiprdutton]

number line vrs. counting line

Here is something that I wrote which might give you insight into the madness going on in my head. :)

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. Can you count now? Sure you can. But you will soon loose track of where you are. 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."

the above is from this post

With this above line of thinking, I arrived at the point where I had to use "da,da,da,da,...,da" as a way to describe what happens after you abandon all the number BASED systems.

 

[philiprdutton]

iterate

I would argue that your counting system can't actually count, and thus isn't a "counting line" as such. That's why knowing what you mean by form is so important to me.

Sure, you can count using the counting system. I just never said you could interpret each position as a number in the sense of what can be done once you define a number base system.

Yes, indeed you can count with the counting system. My definition of count will have to become something like: "take another algorithmic step"

I am interested in a system that let's me move forward in the "line" and I don't care at this point about whether or not you can label each position. I know this system is going to be almost useless for most people. But if you remember that the Peano system is basically able to simulate this counting system then you can not deny that lots of stuff in mathematics is related to a counting system... .it might just be harder to recognize that fact since there are so many other things you can do with Peano like the fancy multiplication or addition.

 

[CRGreathouse]

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. Can you count now? Sure you can. But you will soon loose track of where you are. 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.

You're getting into linguistics now!*

First of all, counting is not a natural thing, and there are people who do not count (most famously the Piraha of South America). Babies and animals can spot the difference between 1, 2, 3, 4-5, 6-9, and so forth, but more particular nuances are generally the area of counting which is a human construct.

But even people who can't count can use tally sticks to record and compare numbers. Essentially every truly ancient civilization used them in some form or other: notches carved into pieces of wood or whatever was convenient. (The Inca used knots in ropes instead.)

But even people who can't count and don't use devices like tally sticks, abacuses, or the like can compare numbers by setting up bijections. Imagine you want to compare the number of sheep I have to the number you have. Just pass one of yours and one of mine through a gate until one of us has none left. If we both have none left we had the same number; otherwise the one with more left has more.

This works even if, like the Piraha, you have no abstract concept of "number".

* Fortunately I've picked that up as a hobby (having read a few textbooks on the subject recommended to me by my friend who has a degree in the field).

 

[CRGreathouse]

Sure, you can count using the counting system. I just never said you could interpret each position as a number in the sense of what can be done once you define a number base system.

Yes, indeed you can count with the counting system. My definition of count will have to become something like: "take another algorithmic step"

In the systems I give, you may suppose modus ponens is the only underlying logic.

Consider the system:
Axiom 1. A

You can take as many algorithmic steps as you like with this system:
1. A (1)
2. A (1)
3. A (1)
4. A (1)
. . .

Thus it let's you count in your terminology. Perhaps you mean taking steps that are essentially different from those before?

Consider this system:
Axiom 1. A
Axiom 2. A --> B
Axiom 3. B --> A

We can take as many algorithmic steps as you like:
1. A (1)
2. A --> B (2)
3. B (MP)
4. B --> A (3)
5. A (MP)
. . .

Alternatively:

Axiom 1. A
Axiom 2. For all x, x --> x.

1. A (1)
2. A --> A (2)
3. (A --> A) --> (A --> A) (2)
4. A --> (A --> A) (MP)
. . .

Plenty of algorithmic steps, but there's no real way to count with this one. For a more concrete system, consider forming sets:

{}
{{}}
{{}, {{}}}
{{}, {{}}, {{{}}}}
{{}, {{{}}}}
{{{}, {{{}}}}, {{}}}

Sets that are subsets of others can be said to be smaller, but some sets are incomparable -- neither is smaller. This doesn't make a "number line" so much as a web.

I am interested in a system that let's me move forward in the "line" and I don't care at this point about whether or not you can label each position. I know this system is going to be almost useless for most people. But if you remember that the Peano system is basically able to simulate this counting system then you can not deny that lots of stuff in mathematics is related to a counting system... .it might just be harder to recognize that fact since there are so many other things you can do with Peano like the fancy multiplication or addition.

I don't think the Peano axioms simulate arithmetic; I think they define how something has to act to be arithmetic.

I see set theory as the basis for mathematics more than counting, but I'm sure a counting system could be used as an alternate basis. My field (number theory) would find that particularly natural.

 

[philiprdutton]

Here is another thought:

What is faster? Counting in binary or counting in decimal? Neither. You get there at the same rate.

Who talks about numbers faster? A people who communicate about numbers only using the binary system or a people group who communicate about numbers only using base 10. They both use the same language. If you have to listen to one of these people speak out loud as they count then who takes the longest at each number when using their own number base to communicate?

Now, if you do not use any number based system when "counting out loud" you are just going to have to make a noise over and over... "buh,buh,buh,buh,buh...buh."

What is the slowest possible way for a human to count out loud? By not using a number based system to describe what point the count is currently at. They are still counting. Just not describing it with fancy short cuts. So, number systems are basically short cuts. They are an encoding which prevents people from having to "count" when exchanging numbers verbally, on paper, or whatever.