What Makes Prime Numbers So Mysterious?

[mathwonk]

why do these flaky threads get so many hits? or for that matter, why does the national enquirer sell more copies than the ny times?

 

[HallsofIvy]

But I am confused as to why all the peano axioms start out with "if b is a natural number"... Am I missing something? I read Peano and feel as if he assumes all the natural numbers are set into position on the number line even before he finishes all the axioms. I figured his successor function was just a means of getting around. It gets confusing like the chicken and egg dilemma.

Primes, squares, addition, fractions, etc. all have to do with permitted "operations." But I still believe the natural numbers are still implicitly defined and do sit in place on the number line whether the operational axioms are defined yet or not.

?? The Peano axioms do not start out with "if b is a natural number" nor does Peano "assume all the natural numbers are set into position on the number line" (I have never seen any mention of "number line" in anything to do with Peano axioms). Peano's axioms DEFINE the natural numbers: the natural numbers are any set of things, together with a "successor function" that satisfies the Peano axioms.

 

[philiprdutton]

kind of my point

?? The Peano axioms do not start out with "if b is a natural number" nor does Peano "assume all the natural numbers are set into position on the number line" (I have never seen any mention of "number line" in anything to do with Peano axioms). Peano's axioms DEFINE the natural numbers: the natural numbers are any set of things, together with a "successor function" that satisfies the Peano axioms.

Sure Peano does not mention the number line. The problem that I have is that the "structure"... or shall I say "Scaffolding" of the traditional number line and the peano system are so so so so so so so close. I wanted to explore the link- try to build the common baseline structure between the two systems and see what you have. Then explore what it means to add in the additional "structure" which give you the two unique systems. I just can't help but want to explore the "intersecting scaffolding." I tried to explore this idea in earlier posts but it was a little bit of a challenge in terms of communication.

 

[HallsofIvy]

So basically what you are saying is that you do not understand what Peano's axioms are and how they give the natural numbers.

 

[phoenixthoth]

It seems odd that a definition of natural numbers would start with 1 is a natural number. (Or 0 if you like.) It seems simultaneously circular and a bit unclear (what is 1?).

The word structure is a bit vague to me, but if we're talking about orderings, then they aren't a part of the Peano axioms. After defining the natural numbers with the Peano axioms, one could then define various orders on the set, including the usual order which would say that n < m iff m is in the orbit of n under the successor function (i.e., there is a kth iterate of the successor function that when applied to n yields m). If one defines 1 to be the singleton containing the empty set, then the usual order does not have to be defined in this way: n < m iff n is an element of m. (That would work if the successor function is defined so that the successor of z is the union of z and the singleton containing z.) Anyway, order (which is what I think is meant by structure) is not a part of the definition but can be developed using the same tools used in the definition. One advantage of not inserting an order in the definition is to allow one to use unusual orders, if one so desires.

 

[CRGreathouse]

why do these flaky threads get so many hits? or for that matter, why does the national enquirer sell more copies than the ny times?

It's amusing to find order in chaos.

 

[CRGreathouse]

It seems odd that a definition of natural numbers would start with 1 is a natural number. (Or 0 if you like.) It seems simultaneously circular and a bit unclear (what is 1?).

1 is anything you like, as long as it performs as the axioms require. That's the strength of the axiomatic system.

 

[mathwonk]

found any lately?

 

[dodo]

I wanted to explore the link- try to build the common baseline structure between the two systems ...

I believe the idea of the Peano axioms is to BE that common baseline - for the integer numbers, as well as for strings of xxxx, or for other artifices that satisfy the axioms.

 

[philiprdutton]

orderings are there

It seems odd that a definition of natural numbers would start with 1 is a natural number. (Or 0 if you like.) It seems simultaneously circular and a bit unclear (what is 1?).

The word structure is a bit vague to me, but if we're talking about orderings, then they aren't a part of the Peano axioms. After defining the natural numbers with the Peano axioms, one could then define various orders on the set, including the usual order which would say that n < m iff m is in the orbit of n under the successor function (i.e., there is a kth iterate of the successor function that when applied to n yields m). If one defines 1 to be the singleton containing the empty set, then the usual order does not have to be defined in this way: n < m iff n is an element of m. (That would work if the successor function is defined so that the successor of z is the union of z and the singleton containing z.) Anyway, order (which is what I think is meant by structure) is not a part of the definition but can be developed using the same tools used in the definition. One advantage of not inserting an order in the definition is to allow one to use unusual orders, if one so desires.

Actually, I disagree with you when you say that "oderings are not part of the Peano axioms." The problem is that recursion uses an ordering whether you like it or not. It does not "look" like our standard notion of "ordering" but I believe it is there.

 

[philiprdutton]

oh nice...

I believe the idea of the Peano axioms is to BE that common baseline - for the integer numbers, as well as for strings of xxxx, or for other artifices that satisfy the axioms.

Very interesting thought! This is kind of where I am trying to go in terms of discussion (and maybe latter someone can throw in some "rigor").

 

[philiprdutton]

hmmm

Do you mean numeration system? You need the Peano Axioms to have NUMBERS- regardless of what base or Roman numerals or other numeration system you use for them. Numeration systems are just symbols you use for the numbers.

Perhaps on a technical note I should like to mention that you have just implied that numbers can not be defined separately from the operations on them such as addition, multiplication, etc. A long time ago, many posts back, I tried to explore this notion of defining the numbers without the darn boring operations... and guess what? You get all the crap for free instantly! In one fell swoop you get the numbers, the reference point, the primes, the multiplication, the yada-yada...

 

[philiprdutton]

temporal aspects

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.

I have taken some time off from this topic of prime numbers. Having recently re-read some of the earlier posts, I found the above quote particularly clean and clear in terms of communicating ideas. In earlier posts I was talking about using a bare Peano axiomatic system as a "metronome" system. Something that would return the symbols (whatever they might be) even though there was not anything that could provide the inter-number connectivity "glue" described above. (The metronome axiomatic system I was thinking of didnt have definitions for operations on the number nor did it have that mechanism which could give special connectivity between the numbers.)

However, if you use the system then you are automatically stuck in time (yes I know there is not time in the world of axiomatic systems but I am talking about human users ... not abstract users). Now, if you are stuck in time (because you are human) then when you use the metronome axiomatic system you "automatically" get the "interconnectivity" between the "things" returned by the metronome system. The temporal aspect of the "use case" provides it. (use case = a human is using it)

Now, combine the temporal aspect of the "use case" with the metronome and you *SHOULD* have the exact same thing as the Peano axiomatic system in all it's glory.This temporally hacked metronome axiomatic system *is* what i wanted to equate (or map) to the the Peano axiomatic system.I believe it is very interesting thought experiment. I am barely able to grasp formal methods or techniques to do this mapping. Nonetheless, I will continue to ponder this idea and hope that others find it interesting.

Philip

 

[CRGreathouse]

In earlier posts I was talking about using a bare Peano axiomatic system as a "metronome" system. Something that would return the symbols (whatever they might be) even though there was not anything that could provide the inter-number connectivity "glue" described above. (The metronome axiomatic system I was thinking of didnt have definitions for operations on the number nor did it have that mechanism which could give special connectivity between the numbers.)

In case you discuss this elsewhere and want people to understand, we did (as I recall) give the following logical interpretation to this: an axiomatic system with the axiom schemata "x is a number" for x = 1, 2, 3, ...

However, if you use the system then you are automatically stuck in time (yes I know there is not time in the world of axiomatic systems but I am talking about human users ... not abstract users). Now, if you are stuck in time (because you are human) then when you use the metronome axiomatic system you "automatically" get the "interconnectivity" between the "things" returned by the metronome system. The temporal aspect of the "use case" provides it. (use case = a human is using it)

It won't surprise you that I have no understanding of what you mean by "stuck in time".

I believe it is very interesting thought experiment. I am barely able to grasp formal methods or techniques to do this mapping.

I strongly recommend that you read up on some basic logic/axiomatic mathematics, as it will help you (1) understand and (2) communicate better with mathematically-minded people. Here's what looks to be a decent, understandable introduction:

The Essence of Logic by John J. Kelly

 

[philiprdutton]

I am just simply saying that when a human counts out loud or repeatedly says "da, da, da, da,...,da" then, the human is essentially in the process of building the number system. If I stop counting then I have not finished building the system but that's beside the point right now. Because I am a being in time, every time I say "da" I have essentially bound the "da" to the previous "da" because they happen sequentially. Because of the fact that one "da" occurs after the other "da" in time, the Peano style Successor Function "role" is played by time itself.

This goes back to another question I have had: How does one map a time-based system (like human metronome counting) to an abstract Peano style algorithmic process? The reason I have not read much material on axiomatic systems is precisely because I have *not* expected them to consider such a mapping (which is what I am interested in). I simply expect the material to teach axiomatic writing and not *mapping* from time based systems to abstract systems. Please don't say you can't "bridge" the two worlds because that would really confuse me considering I am using the number system every day of my life and thus, according to my perception, the two worlds have been bridged.

 

[CRGreathouse]

I am just simply saying that when a human counts out loud or repeatedly says "da, da, da, da,...,da" then, the human is essentially in the process of building the number system. If I stop counting then I have not finished building the system but that's beside the point right now. Because I am a being in time, every time I say "da" I have essentially bound the "da" to the previous "da" because they happen sequentially. Because of the fact that one "da" occurs after the other "da" in time, the Peano style Successor Function "role" is played by time itself.

This goes back to another question I have had: How does one map a time-based system (like human metronome counting) to an abstract Peano style algorithmic process?

As a string of symbols. x, x', x'', x''', x'''', x''''', ... would be the standard way, but plenty of others, all essentially unary numbers, suggest themselves. But when you allow yourself this kind of structure you're opening the door to all the things I thought you were trying to avoid.

 

[philiprdutton]

exploring

As a string of symbols. x, x', x'', x''', x'''', x''''', ... would be the standard way, but plenty of others, all essentially unary numbers, suggest themselves. But when you allow yourself this kind of structure you're opening the door to all the things I thought you were trying to avoid.

While being lost in exploration, I've almost forgot what I was avoiding. :) If you use the time based metronome then what is the reference point? There is no clear way to encode it like there is with the Peano axioms.

 

[philiprdutton]

number systems

While being lost in exploration, I've almost forgot what I was avoiding. :) If you use the time based metronome then what is the reference point? There is no clear way to encode it like there is with the Peano axioms.

Given the above, I now have a revised quote to offer the world:
"Counting is all too easy. Figuring out how to talk about where you started or where you stopped is the hard part." - Philip Ronald Dutton


I think the Peano axiomatic system fails in regards to tricking the amatuer into thinking that the facility to record where you stop and where you start is provided by the Peano axioms themselves. In reality, the axiomatic systems DO NOT provide such facility. What exactly is this called anyway? Is there a formal definition of this particular "chasm" that I am referring to?

 

[F-Meson]

Frankly, I'm surprise you guys have gotten this far without clearly explaining to Mr. Dutton that "1" is undefined.

The statement "1 is a natural number." is used to define "natural number." It is not used to define "1." "1" is not defined. "1" is never defined. At no point is "1" defined. We do not define "1." Got that?

"1" is undefined.

 

[Hurkyl]

philiprdutton: I think at this point, it would be worthwhile for you to recollect your thoughts, and restate just exactly what it is that you are thinking and seeking (and hopefully to do it precisely!). This is often a good to do.

And there is the fringe benefit that you have a much better chance of getting new responses if people don't have to review 15 pages of comments.

 

[dodo]

"1" is undefined.

I'm curious: when you define addition after the Peano axioms, the first defining axiom typically goes something like: x + 1 = Sx. Doesn't this say something about '1', at least in relation to addition?

Meaning, once you (a) define subtraction (not as the addition of inverses, since there are no inverses in N; just as the solution for x of the equation y = x + a, whenever there is one), and (b) define a total order relation >=, ... aren't you tempted to define some kind of 'integer metric' d(), and say that 1 = d(x,Sx), thus giving some meaning to '1'?

After all, when Peano axioms are extended to include (or start with) 0, the nature of 0 as an identity element (with respect to addition) comes naturally out of the axioms and the definition of addition. I would think that some relation between Sx and 1 would come out as well.

 

[philiprdutton]

counting... what is it?

philiprdutton: I think at this point, it would be worthwhile for you to recollect your thoughts, and restate just exactly what it is that you are thinking and seeking (and hopefully to do it precisely!). This is often a good to do.

And there is the fringe benefit that you have a much better chance of getting new responses if people don't have to review 15 pages of comments.

Clearly stated: when humans or machines count, is there anything in the process that can be mapped over to the Peano axiomatic system?

Or:


Is any "slice" (or part) of the Peano axiomatic system mappable to the human/machine counting process?



Axiomatic systems like Peano's don't have a time component. Human/machine counting DOES have a time component. My simple question is where do the two overlap or is there any way that the two separate systems intersect OR CAN BE MADE TO INTERSECT?


Clear as mud?


Why am I interested in this? Because I am about to solve quantum psuedo nuclear reactive lukewarm fission! Just kidding. I am purely and simply in love with the exploration stated above about finding a common ground between the two systems am quite surprised that the pros around here can not point me to one single piece of literature that attempts to explore the differences. Perhaps I assume too much: that a modern discussion about "counting" would attempt to explain how counting relates to the natural numbers. We are always taught counting first as children. Perhaps it also comes relatively naturally. Then, in later years of mathematic dogma we are introduced to the Peano axiomatic system or other equivalent axiomatic systems.

Imagine that you were taught the peano axiomatic system first before you ever figured out what counting was all about. And then you were taught about counting. Would you not want to know how this new counting related to the Peano system? Would you not be curious to understand what makes the two different or if there was any overlap?

Hades, maybe I can rephrase my question one more time for kicks:
Can the Peano system be used to simply count? yes or no?

Here is another:
Is the peano system (or any other beloved axiomatic number system) required for counting?

other questions:
How does the human/machine counting system provide for encoding the stopping and starting point of the count? Aren't these facilities also similarly provided the Peano system even though the Peano system is Axiomatic thus requiring no time?


finally:
does counting imply an existing axiomatic system which defines the natural numbers?



PS: just answer the last question... anyone! (please :)

 

[Hurkyl]

Maybe explicitly writing a counting algorithm would help clear things up.

Input:  a collection of objects
Output: the cardinality of the collection

. Let count = 0
. Mark each object as "uncounted"
. While there exists an uncounted object:
   . Let X be any uncounted object
   . Let count = (successor of count)
   . Mark X as "counted"
. Report count as the answer

(There are lots of ways to "mark" an object: you could just remember; you could make two piles, one for counted and one for uncounted; you could use a marker and make an actual mark on the object when it's "counted"; et cetera)

And if you really wanted to analyze the time evolution of the counting algorithm, you could tabulate the state of the algorithm at each time.


This is one method we humans actually use to count. This algorithm uses Peano arithmetic and computes a natural number -- obviously we cannot use this particular algorithm if we have not yet learned Peano arithmetic.

 

[philiprdutton]

half way there

Maybe explicitly writing a counting algorithm would help clear things up.

Input:  a collection of objects
Output: the cardinality of the collection

. Let count = 0
. Mark each object as "uncounted"
. While there exists an uncounted object:
   . Let X be any uncounted object
   . Let count = (successor of count)
   . Mark X as "counted"
. Report count as the answer

(There are lots of ways to "mark" an object: you could just remember; you could make two piles, one for counted and one for uncounted; you could use a marker and make an actual mark on the object when it's "counted"; et cetera)

And if you really wanted to analyze the time evolution of the counting algorithm, you could tabulate the state of the algorithm at each time.This is one method we humans actually use to count. This algorithm uses Peano arithmetic and computes a natural number -- obviously we cannot use this particular algorithm if we have not yet learned Peano arithmetic.

Maybe the supplied algorithm uses Peano arithmetic and computes a natural number. However, I am not convinced that the axiomatic Peano system indeed does any "marking." "Inside" the axiomatic systems there is no marking. How can there be? "Marking" requires a time component. I agree that we humans and machines can count. However I believe only we in the physical world are capable of "marking"... axiomatic system's can't do it. I just do not understand how an axiomatic system can do any "marking". It seems imagined. Perhaps the marking of the "zero" is the boundary of the empty set when analyzing the set theoretic version of the Peano system. But for the peano system itself, i am afraid I disagree that there is real way for the system to mark itself despite the "0/1 is a natural number" statement. I don't trust that statement's marking power.

 

[Hurkyl]

"Peano arithmetic" is not synonymous with "counting algorithm"...

 

[philiprdutton]

building

"Peano arithmetic" is not synonymous with "counting algorithm"...

How do you make parts of the two systems synonymous? Let's say you were going to build both systems from the ground up: A and B. Build it "modularly" as in: "build piece by piece". Start with the common pieces. How far can you get before you loose commonality?

 

[Hurkyl]

The counting algorithm is not a formal system; it's an algorithm! As stated, I can't really make sense of your inquiry.

I can guess at other things; maybe you are interested in comparing a formalization of the theory of computation with peano arithmetic? But that doesn't seem like a meaningful comparison.

Or maybe you are interested in the fact that if I had a theory of collections, then I could use the properties of Peano arithmetic to deduce properties of the counting algorithm. Then I could attempt to reverse the process -- if I assume certain properties of the counting algorithm, then I could try and derive Peano arithmetic.

 

[philiprdutton]

sortof ...

The counting algorithm is not a formal system; it's an algorithm! As stated, I can't really make sense of your inquiry.

I can guess at other things; maybe you are interested in comparing a formalization of the theory of computation with peano arithmetic? But that doesn't seem like a meaningful comparison.

Or maybe you are interested in the fact that if I had a theory of collections, then I could use the properties of Peano arithmetic to deduce properties of the counting algorithm. Then I could attempt to reverse the process -- if I assume certain properties of the counting algorithm, then I could try and derive Peano arithmetic.

Yes: I am interested in comparing a formalization of the theory of "counting" (not the theory of computation... or are they the same?? haha don't answer that! ) with the peano arithmetic. I think it is a valid comparison.

More precisely to rephrase my thoughts and your guess:
I want to formalize "counting".. then, see what needs to be added to it this formalization to end up with something equivalent to Peano system. I believe formalized "counting" is more "basic" than Peano arithmetic. I want to explore this curiousity. If there is a "bridge" between the two systems then maybe you can do things like your second "guess" above.

Thanks for the input thusfar, it is very helpful.



Side question:
Do you think there is a "theory of counting?" or a "axiomatic counting system"? If there is no time in "axiomatic systems" then how can Peano make the successor function "stop" just long enough to give a name to the next number?

 

[ramsey2879]

Yes: I am interested in comparing a formalization of the theory of "counting" (not the theory of computation... or are they the same?? haha don't answer that! ) with the peano arithmetic. I think it is a valid comparison.

More precisely to rephrase my thoughts and your guess:
I want to formalize "counting".. then, see what needs to be added to it this formalization to end up with something equivalent to Peano system. I believe formalized "counting" is more "basic" than Peano arithmetic. I want to explore this curiousity. If there is a "bridge" between the two systems then maybe you can do things like your second "guess" above.

Thanks for the input thusfar, it is very helpful.



Side question:
Do you think there is a "theory of counting?" or a "axiomatic counting system"? If there is no time in "axiomatic systems" then how can Peano make the successor function "stop" just long enough to give a name to the next number?

Axioms of counting based on the Peano Axioms
1. 0 is a natural number that corresponds to an empty set(peano axiom 5)
2.For every element of a non empty set A there is a corresponding natural number x that is unique, x = x. That is, equality is reflexive. (Peano axiom 1)
3. Every corresponding natural number x is an unique successor of 0 or of another corresponding natural number x. (peano axium 6)
4. For the set corresponding natural number x, there is an maximun x = n which has a successor that corresponds to no element in set A. (peano axiom 7)
5. n is the count of set A (peano axium 9)
Edit I posted an improvement on these axioms at my blog.