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.

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 seperate 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 :)

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

Code (Text):

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.

How do you make parts of the two systems synonymous? Lets 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?

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?

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.