Exploring the Mystery of Prime Numbers

[philiprdutton]

counting: my definition

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

YES! That is what I mean when I say "count." I am sorry I had to keep perverting the standard meaning of "count" but I felt it necessary to push the thinking as far as possible down this course of study using that term.

 

[CRGreathouse]

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?

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? 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.

All of those are number systems. You're saying that decimal is just as fast as binary, but unary is slower. I would say that unary is slower than binary for numbers greater than 1, binary is slower than decimal for numbers greater than 1, ternary is slower than decimal for numbers greater than 2, hextal is slower than decimal for numbers greater than 1295, decimal is slower than hexadecimal for numbers greater than 99999, and so on.

 

[philiprdutton]

web

...
...
...
This doesn't make a "number line" so much as a web.

Considering your web system: Sure I can count (my def) with it:

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

is simply as follows:

step da { }
step da,da {{}}
step da,da,da {{}, {{}}}
step da,da,da,da {{}, {{}}, {{{}}}}
step da,da,da,da,da {{}, {{{}}}}
etc... {{{}, {{{}}}}, {{}}}

 

[CRGreathouse]

YES! That is what I mean when I say "count." I am sorry I had to keep perverting the standard meaning of "count" but I felt it necessary to push the thinking as far as possible down this course of study using that term.

So consider this system.

Axiom 1. A point exists.
Axiom 2. From any point, you may draw a 1-unit arrow down and to the left. The end of the arrow is a point.
Axiom 3. From any point, you may draw a 1-unit arrow down and to the right. The end of the arrow is a point.

The metalogic of the system is that two diagrams are equal iff they have the same arrow structure, and one diagram is larger than another iff the first contains all the arrows of the second but the two are not equal.

So "/\" > "/" > "" and "/\" > "\" > "", but not ("/" > "\") and not ("\" > "/"). The system can make many different theorems ("diagrams" in its own terminology) but they don't work like the natural numbers, or any sensible number line at all.

 

[CRGreathouse]

Considering your web system: Sure I can count (my def) with it because each statement is numbered:

But someone else could use those axioms and come up with theorems in a different order. You don't want different things to be equal to each other, do you?

 

[philiprdutton]

speed

All of those are number systems. You're saying that decimal is just as fast as binary, but unary is slower. I would say that unary is slower than binary for numbers greater than 1, binary is slower than decimal for numbers greater than 1, ternary is slower than decimal for numbers greater than 2, hextal is slower than decimal for numbers greater than 1295, decimal is slower than hexadecimal for numbers greater than 99999, and so on.

No. I am saying unary is the slowest of them all because unary is essentially a system where you have to count (my def) "out loud" in order to express the point where you are in the counting line.

visually:

the number of decimal 10 is viewed as:

10
but in unary it is:
...

You have some correct points about relative speeds which I missed but still, nothing is slower than unary. Unary = counting (my def)

 

[CRGreathouse]

No. I am saying unary is the slowest of them all because unary is essential a system where you have to count "out loud" in order to express the number.

visually:

the number of decimal 10 is viewed as:

10
but in unary it is:
...

Well the standard form for decimal 10 in unary would be 1111111111, but that's beside the point. Of course both could be written with different symbols, but that's just a simple replacement issue.

You have some correct points about relative speeds which I missed but still, nothing is slower than unary. Unary = counting (my def)

I suppose one could construct systems which are slower than unary...

I presume the equal sign above means "is a kind of"?

Where were you going with this?

 

[philiprdutton]

going somewhere

Yes my equal sign is "a kind of."

Now, we can see that both the counting system and the Peano system are unary speed systems (for practical human purposes). Essentially, the Peano system at it's CORE has a counting system (my def).

So, before you can build a Peano system you must have the counting system.

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 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.

 

[CRGreathouse]

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 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.

That's a claim, but what you want is a proof or an example. What axioms can you remove from Peano arithmetic so it can still count but not talk about primes? I may have actually given an example of this earlier on the thread...

 

[philiprdutton]

adding prime to a system

That's a claim, but what you want is a proof or an example. What axioms can you remove from Peano arithmetic so it can still count but not talk about primes? I may have actually given an example of this earlier on the thread...

Actually, I did not originally care about removing pieces from the Peano system. I was originally thinking about this in terms of how to build up from scratch a basic system that did not support primes but had some commonality with a system like Peano. But now that you mention it, it would be a great exercise to see how much must be removed from Peano in order that notion of "prime" can not be supported.

We essentially are talking about two directions (top down or bottom up approaches) with the same goal: to explore "when" the notion of "prime" is added to a system (which in our discussion has been Peano.

 

[dodo]

While you create new systems, try to add to them a very desirable property than Peano's have: the ability to express very big (or infinite) objects in a finite, even very compact, space of symbols. This is what makes better a system like "1 is a number, Sa is a number if a is" than "foo is a number, foo foo is a number, foo foo foo is a number, ...", or than "1 is a number, S1 is a number, SS1 is a number, SSS1 is a number, ...".

 

[CRGreathouse]

We essentially are talking about two directions (top down or bottom up approaches) with the same goal: to explore "when" the notion of "prime" is added to a system (which in our discussion has been Peano.

My basic answer would be when you make a system strong enough to support a discrete chain/total order, you essentially have a way to talk about divisibility and thus primality.

 

[dodo]

I have some confusion here. The relation < defines a total order on R, yet that doesn't make R isomorphic to N. Without that isomorphism, you get 7 is not a prime because it is divisible by 7/2.

 

[philiprdutton]

no features please

While you create new systems, try to add to them a very desirable property than Peano's have: the ability to express very big (or infinite) objects in a finite, even very compact, space of symbols. This is what makes better a system like "1 is a number, Sa is a number if a is" than "foo is a number, foo foo is a number, foo foo foo is a number, ...", or than "1 is a number, S1 is a number, SS1 is a number, SSS1 is a number, ...".

I am not interested in building a system with "very desirable property than Peano's have." I have a specific reason why I am limiting the functionality of the counting system.

 

[dodo]

No problem; as long as you don't count over 100, you won't spend too much paper.

But CR has a point up there. Divisibility is actually a very basic concept, that can come from addition, not necessarily multiplication. For instance, define numbers with dots, x, xx, xxx, xxxx..., and addition as concatenation. Then define divisibility by these two axioms:
1) Every number divides itself, f.i. xxx divides xxx.
2) If b divides a, then b divides a+b. That is, if xx divides xx, then xx divides xxxx, and also xx divides xxxxxx ...

 

[philiprdutton]

talking about prime

My basic answer would be when you make a system strong enough to support a discrete chain/total order, you essentially have a way to talk about divisibility and thus primality.

My basic thought is that "talking about" does not mean it is a "hard asset" of the system. For example, the Peano axioms lend themselves quiet well to "algorithmic" discussions due to the successor function- it lends it self quite well to a mechanical "stepping" system. However, that doesn't mean there is any real ability for the system to "step around" on the number line. It could very well just as easily magically make the numbers "poof" into existence (since there is no notion of time constraint ).

Anyway, my question about your response is: what system is weaker than a discrete chain/total order? (also, I am not totally sure what you mean by discrete chain/total order but I have a good guess that it is something that just "ticks").

 

[philiprdutton]

divisibility

No problem; as long as you don't count over 100, you won't spend too much paper.

But CR has a point up there. Divisibility is actually a very basic concept, that can come from addition, not necessarily multiplication. For instance, define numbers with dots, x, xx, xxx, xxxx..., and addition as concatenation. Then define divisibility by these two axioms:
1) Every number divides itself, f.i. xxx divides xxx.
2) If b divides a, then b divides a+b. That is, if xx divides xx, then xx divides xxxx, and also xx divides xxxxxx ...

Okay I see the problem. I want a system that just "ticks." Your system is nested ticking. In my system I don't want to do anything except be able to produce the next "tick". There is no notion of nested ticking in my system. How can this be formalized with an axiomatic system? I will just call it a ticking system. Is such a ticking system the most basic kind of axiomatic system with the least amount of feature? In such a system I think divisibility is not definable. Basically, you push off the job and definition of divisibility to the observer or user.

 

[philiprdutton]

msg

No problem; as long as you don't count over 100, you won't spend too much paper.

Exactly the point I made earlier. It also helps if you do not eat very much MSG (monosodium glutamate) since it is an excitotoxin and directly attacks cells in the short term memory area of the brain making it hard to count in unary...

 

[philiprdutton]

axiomatic nesting

Okay I see the problem. I want a system that just "ticks." Your system is nested ticking. In my system I don't want to do anything except be able to produce the next "tick". There is no notion of nested ticking in my system. How can this be formalized with an axiomatic system? I will just call it a ticking system. Is such a ticking system the most basic kind of axiomatic system with the least amount of feature? In such a system I think divisibility is not definable. Basically, you push off the job and definition of divisibility to the observer or user.

Not to get off topic but I must ask because I am so curious: Is the Peano system inherently nested? Does it have built-in nesting? Do all axiomatic systems have built-in nesting? Is nesting just coming about because of the way the system is being "used" by the "user"?

 

[dodo]

I'm not sure of what you mean by "nested". I think you mean,

x, xx, xxx, xxxx... are numbers​

is "not nested", while

x is a number; also, if A is a number then Ax is a number​

is "nested". I think most people here would say both are one and the same.

 

[philiprdutton]

ticking

I'm not sure of what you mean by "nested". I think you mean,

x, xx, xxx, xxxx... are numbers​

is "not nested", while

x is a number; also, if A is a number then Ax is a number​

is "nested". I think most people here would say both are one and the same.

But if the system is viewed as an algorithmic process, then how do you distinguish? Especially if we are talking about a system that only can only tick. How can we limit the expressiveness of an axiomatic system so that all you can do is "poke" it so that it "ticks". Can we have a one-to-one input/output system. Axiom systems like Peano have many ways to "input" your "statements" to make them "produce" an output. I do not know how the formal mathemticians talk about the "usage" of the axiomatic systems at this level of abstraction, but I see it with the input/output metaphor.

Nested counting is where at each step of the count, the process starts again from "one."

x 1
xx 1,2
xxx 1,2,3
xxxx 1,2,3,4
xxxxx 1,2,3,4,5
etc.I am saying why waste so much effort? Just do this:
x 1
x 2
x 3
x 4
x 5
etc.

I had to put the numbers in there for visualization but I am saying that I just want a ticking system.

Anyway, Here is my focus:
I want to define a ticking system using the axiomatic method. But, I do not want the system to do anything except tick! No nesting. Can this be done with the axiomatic system or is it too flexible at it's core such that it can not make such limitations? This is a short side study on the nature of axiomatic systems.

 

[NeoDevin]

I am saying why waste so much effort? Just do this:
x 1
x 2
x 3
x 4
x 5

In this system, what differentiates 5 from 2? Other than the fact that they were created by different axioms? This system has no notion of order, unless you explicitly put it in, in which case you get nesting as you put it.

 

[dodo]

In other words, this ticking machine seems to have no internal state. When it ticks and says 'x', there is no way of telling if it is the first 'x', the 5th or the 625th. On the other hand, if it *does* have a state, then you should consider how the state is represented, and call this representation a 'number'.

 

[philiprdutton]

differentiating

In this system, what differentiates 5 from 2? Other than the fact that they were created by different axioms? This system has no notion of order, unless you explicitly put it in, in which case you get nesting as you put it.

Yes. It is my point. You cannot differentiate those numbers. I just put them in the post for sake of moving forth in the discussion. If you view the system as an algorithmic process, then it kind of does have order. Okay. So are you saying that using a formal axiomatic theory, I can not create a ticking system with order AND which does not provide notions of divisibility, prime, and anything else related to numbers or number bases? I am interested in the answer which is why all this time I am not concerned about cool stuff like addition or division.

 

[philiprdutton]

yes

In other words, this ticking machine seems to have no internal state. When it ticks and says 'x', there is no way of telling if it is the first 'x', the 5th or the 625th. On the other hand, if it *does* have a state, then you should consider how the state is represented, and call this representation a 'number'.

You are correct in saying you can not figure out if it is the first 'x' or the 5th 'x' or the 625th 'x'. I said already that I want the user of the system to worry about that. I don't want the user of the system to expect that the system to tell them via some particular feature of the system. They just use the ticking system like a metronome (a metaphor obviously).

Let me put it this way- can there be an axiomatic version of a metronome?

 

[philiprdutton]

internal state

In other words, this ticking machine seems to have no internal state. When it ticks and says 'x', there is no way of telling if it is the first 'x', the 5th or the 625th. On the other hand, if it *does* have a state, then you should consider how the state is represented, and call this representation a 'number'.

Do axiomatic systems like Peano have internal state? Can some axiomatic systems have an internal state and others not have one?

 

[dodo]

I fail to see how a system with no numbers fits in a number theory forum, but for the sake of the discussion let me provide an example. The boolean operator 'not' behaves as you want. Or, if you prefer, a function f(x) = 1 - x. But unless we begin counting time, or sequence steps, the only thing we produce is the set {0,1}, with no additional structure, operations or functionality.

 

[CRGreathouse]

Yes. It is my point. You cannot differentiate those numbers. I just put them in the post for sake of moving forth in the discussion. If you view the system as an algorithmic process, then it kind of does have order. Okay. So are you saying that using a formal axiomatic theory, I can not create a ticking system with order AND which does not provide notions of divisibility, prime, and anything else related to numbers or number bases? I am interested in the answer which is why all this time I am not concerned about cool stuff like addition or division.

You want your system to have elements ('number' which function as atoms or ur-elements) and a (presumably transitive) order on those elements, and you want to know if all systems with those properties can talk about primality in some restricted way or not? Is that right?

 

[philiprdutton]

building blocks

I fail to see how a system with no numbers fits in a number theory forum, but for the sake of the discussion let me provide an example. The boolean operator 'not' behaves as you want. Or, if you prefer, a function f(x) = 1 - x. But unless we begin counting time, or sequence steps, the only thing we produce is the set {0,1}, with no additional structure, operations or functionality.

The reason is not sake of discussion. The reason is of great importance. One of the points all all this discussion is that you have built on top of something to get "numbers". The thing (stepping stone) you start with is the tick system. So, whether or not you agree with what I am doing, I really need help trying to formalize the tick system without using something that already has notions of "number". Unfortunately, that math education imposed upon us does not even begin to get into these concepts.

Look at the peano system as one holistic system or look at the Peano system in terms of modular building blocks. If you take the modular building block approach then I am saying that before you can construct numbers you have to build on top of the tick system. Therefore, it has lots to do with the topic of Number Theory.

 

[CRGreathouse]

Do axiomatic systems like Peano have internal state? Can some axiomatic systems have an internal state and others not have one?

Internal state?

 

[dodo]

Do axiomatic systems like Peano have internal state? Can some axiomatic systems have an internal state and others not have one?

I think we take the machine analogy too far. Axiomatic systems are meant to provide a set of initial assumptions from which to construct the rest of the building. They are not algorithms, but logical propositions accepted by convention as true (so that all derived statements can be proven true). By themselves, they do not travel in time; our reasonings and explanations do, but only because our talking does.

One of the points all all this discussion is that you have built on top of something to get "numbers". The thing (stepping stone) you start with is the tick system.

I'm getting lost. I thought you said you didn't want the ticks to represent numbers, since there is no way of telling the 5th from the 625th.

 

[philiprdutton]

order and elements

You want your system to have elements ('number' which function as atoms or ur-elements) and a (presumably transitive) order on those elements, and you want to know if all systems with those properties can talk about primality in some restricted way or not? Is that right?

Yes you are following me. I felt originally that if a system has the two things (elements and order) you could not guarantee the ability to talk of "primality."

But now I am side tracking to a system where order is out the window. I just want a ticking system (unforntunatley, the algorithmic interpretation of step, next step, next step, next step, etc. DOES indeed imply order.

 

[philiprdutton]

internal state

Internal state?

Sorry, that term was NOT brought into the discussion by me!

 

[CRGreathouse]

But now I am side tracking to a system where order is out the window. I just want a ticking system (unforntunatley, the algorithmic interpretation of step, next step, next step, next step, etc. DOES indeed imply order.

So you want a countably infinite number of elements, but no comparisons between them. Seems like the axiom schema "For each natural number n, there is an element distinct from at least n others" of your counting scheme is what you want. Of course I can't think of a way to use that at all -- not for finding/defining primes, not for counting sheep, not for anything. It's essentially identity calculus.

 

[philiprdutton]

stop using

So you want a countably infinite number of elements, but no comparisons between them. Seems like the axiom schema "For each natural number n, there is an element distinct from at least n others" of your counting scheme is what you want. Of course I can't think of a way to use that at all -- not for finding/defining primes, not for counting sheep, not for anything. It's essentially identity calculus.

Sure I did not expect that anyone would want to use this system. Given what you just said about the identity calculus my question is can the Peano axiom system be built from the identity calculus?

Lets just say I am interested in decomposing the Peano axioms into building blocks (or feature sets) much like you would decompose some number into a combination of primes.