What Makes Prime Numbers So Mysterious?

[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.

 

[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.