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From: John Jones on 24 Apr 2010 19:36
The Peano Axioms define the way that numbers, and hence arithmetic, is
"presented" to us.

But my question is this: In such a presentation do the Peano Axioms
"define" arithmetic or do they "limit" arithmetic?

The question is important.

1) If the Peano Axioms "define" arithmetic then they are necessary to
arithmetic and, hence, arithmetic is a branch of logic.
2) If the Peano Axioms "limit" arithmetic then the source of arithmetic
isn't logical or necessary. That is, it is contingent. And the sort of
contingency that that might be isn't immediately fathomable.

But I think that there is an alternative to these two options. Any ideas
what that alternative might be? That is, in what relationship does the
Peano Axioms stand to arithmetic?
From: Immortalist on 24 Apr 2010 20:02
On Apr 24, 4:36 pm, John Jones <jonescard...(a)btinternet.com> wrote:
> The Peano Axioms define the way that numbers, and hence arithmetic, is
> "presented" to us.
>
> But my question is this: In such a presentation do the Peano Axioms
> "define" arithmetic or do they "limit" arithmetic?
>
> The question is important.
>
> 1) If the Peano Axioms "define" arithmetic then they are necessary to
> arithmetic and, hence, arithmetic is a branch of logic.
> 2) If the Peano Axioms "limit" arithmetic then the source of arithmetic
> isn't logical or necessary. That is, it is contingent. And the sort of
> contingency that that might be isn't immediately fathomable.
>
> But I think that there is an alternative to these two options. Any ideas
> what that alternative might be? That is, in what relationship does the
> Peano Axioms stand to arithmetic?

If Peano's set of most basic functions of ALL mathematical operations
is true then your asking the wrong question. The question should not
be whether Peanos Axioms are necessary or contingent for math
operations but instead the proper question would be; what relation
math has to the theory of reality. Of course math is an attempt to
represent our abstract notions of what reality is with a set of
symbols and a group of rules.

"Axiom", in classical terminology, referred to a self-evident
assumption common to many branches of science. A good example would be
the assertion that

When an equal amount is
taken from equals, an
equal amount results.

At the foundation of the various sciences lay certain basic hypotheses
that had to be accepted without proof. Such a hypothesis was termed a
postulate. The postulates of each science were different. Their
validity had to be established by means of real-world experience.
Indeed, Aristotle warns that the content of a science cannot be
successfully communicated, if the learner is in doubt about the truth
of the postulates.

The classical approach is well illustrated by Euclid's elements, where
we see a list of axioms (very basic, self-evident assertions) and
postulates (common-sensical geometric facts drawn from our
experience).

A1 Things which are equal to the same thing are also equal to one
another.

A2 If equals be added to equals, the wholes are equal.

A3 If equals be subtracted from equals, the remainders are equal.

A4 Things which coincide with one another are equal to one another.

A5 The whole is greater than the part.

P1 It is possible to draw a straight line from any point to any other
point.

P2 It is possible to produce a finite straight line continuously in a
straight line.

P3 It is possible to describe a circle with any centre and distance.

P4 It is true that all right angles are equal to one another.

P5 It is true that, if a straight line falling on two straight lines
make the interior angles on the same side less than two right angles,
the two straight lines, if produced indefinitely, meet on that side on
which are the angles less than the two right angles.

http://planetmath.org/encyclopedia/Axiom.html
http://www.mathgym.com.au/history/pythagoras/pythgeom.htm

----------------------------

Peano's Axioms

Sure, every school child knew the rules of counting, addition,
subtraction, multiplication, and division; but whence came these
"rules"? What reason do we have to believe that these rules are valid.
Euclid (c. 300 BC) had placed a firm foundation under plane geometry
with his set of five axioms, from which the whole of classical
geometry could be derived deductively. This was Peano's goal for
arithmetic. To reach this goal, he began, as Euclid did, with five
axioms:

Axiom 1. 0 is a number.

Axiom 2. The successor of any number is a number.

Axiom 3. If a and b are numbers and if their successors are equal,
then a and b are equal.

Axiom 4. 0 is not the successor of any number.

Axiom 5. If S is a set of numbers containing 0 and if the successor of
any number in S is also in S, then S contains all the numbers.

....If there is some property P which we believe to be true of all the
whole numbers, mathematicians agree that we cannot simply show a few
examples where P is true and then conclude that it is, therefore, true
for all numbers. Since the set {0, 1, 2, 3,...} is infinite, we cannot
conclude from a finite number of examples, no matter how large, that
property P holds for all the numbers in {0, 1, 2, 3,...}. However,
Peano's fifth axiom, the principle of mathematical induction gives us
the means for determining whether property P is true for all numbers
in {0, 1, 2, 3...}. The induction axiom says that we need carry out
only two steps to determine the truth of P for the infinite set of
whole numbers.

First, we must show that 0 has property P.

Second, we must show that, for any number n, if n has property P, then
this implies that the succesor of n, namely n+1, also has property P.

Imagine a line of an infinite number of dominoes standing upright. Two
things need to happen for all the dominoes to fall with a single push:

(1) the first domino must fall, and

(2) each domino in the line must be close enough to the next domino so
that when it falls it causes the next domino to fall.

Note than both conditions must hold in order for all the dominoes to
fall. If the first domino does not fall, then the chain reaction never
begins. On the other hand, if the first domino does fall but somewhere
along the line the separation between a domino and its successor is so
great that they never make contact, then the chain reaction stops
there. This is the essence of Peano's fifth axiom.

One of the great beauties of the Peano axioms is that they make
possible the generation of an infinite set of numbers from a finite
number of symbols. Essentially, Peano hands us two items, the number 0
and the concept "successor", and an "instruction manual," i.e, his
five axioms, and promises us that with just these we can build an
entire system of arithmetic.

As a first step in building this arithmetic system, we may generate
all the natural numbers(positive integers) as follows. Let us agree
that if n is any number, rather than writing out "the succesor of n,"
we will write "s(n)." Now Peano has given us 0. Axiom 1 says that 0 is
a number and Axiom 2 says that the successor of any number is a
number. Therefore, s(0) is a number, s(s(0)) is a number, s(s(s(0)))
is a number, and so on. Clearly this process, if continued forever, is
sufficient to generate all the natural numbers. As each new natural
number is "born" by our process, we may want to give it a name. For
instance we may wish to call s(0) "Sam" or "Samantha" or, preferably,
"1." To our newly generated s(s(0)), which clearly is s(1), we will
give the name "2." We shall call s(s(s(0)))=s(s(1))=s(2) by the name
"3" and so forth. Thus the set {1, 2, 3,...} is generated.

Having generated the natural numbers, we can next begin to define
operations on those numbers. For example, Peano defines addition as
follows: For any natural numbers n and k: i. n+0=n and ii. n+s(k)=s(n
+k), where s(k) still means "the successor of k." So the addition "2 +
1" is interpreted as 2 + 1 = 2 + s(0) = s(2+0) = s(2) = 3.

Multiplication can be defined in a similar way and then subtraction is
defined in terms of the addition of inverse elements and division is
defined in terms of the multiplication of inverse elements. In order
to do this, the negative integers are defined as the additive inverses
(or opposites) of the natural numbers and the rational numbers of the
form 1/n, where n is not 0, are defined as multiplicative inverses of
the natural numbers and their opposites. In this way, we expand the
number system to include all the integers and all the rational numbers
of the form 1/k, where k is not 0. Then we must account for division
of the form p/q where p/q turns out to be neither an integer nor the
multiplicative inverse of an integer. Thus we bring in the rest of the
rational numbers to allow for this. In this way, we can continue to
build up a richer number system. We will need irrational numbers to
account for the solutions of equations such as x2 = 2 and complex
numbers to account for solutions of equations such as x2 + 1 = 0. The
important thing is that all these expansions of the number system can
be accomplished by definitions, without adding any more axioms or
primitive terms to Peano's original system. That is not to say that
this expansion is trivial; for example, it took some true
inventiveness on the part of mathematicians such as Dedekind
(1831-1916), Cantor (1845-1918) to come up with an acceptable
definition of a real number. However, once this was accomplished,
mathematicians could feel more comfortable about the foundations of
the real number system, which has been the setting for the development
of arithmetic, algebra, geometry, and modern mathematical analysis
including the calculus. Thus, starting with 0, the idea of a successor
for each number, and his five axioms, Peano provided a simple but
solid foundation upon which to construct the edifice of modern
mathematics.

http://www.bookrags.com/sciences/mathematics/peano-axioms-wom.html

From: John Jones on 24 Apr 2010 22:25
Immortalist wrote:
> On Apr 24, 4:36 pm, John Jones <jonescard...(a)btinternet.com> wrote:
>> The Peano Axioms define the way that numbers, and hence arithmetic, is
>> "presented" to us.
>>
>> But my question is this: In such a presentation do the Peano Axioms
>> "define" arithmetic or do they "limit" arithmetic?
>>
>> The question is important.
>>
>> 1) If the Peano Axioms "define" arithmetic then they are necessary to
>> arithmetic and, hence, arithmetic is a branch of logic.
>> 2) If the Peano Axioms "limit" arithmetic then the source of arithmetic
>> isn't logical or necessary. That is, it is contingent. And the sort of
>> contingency that that might be isn't immediately fathomable.
>>
>> But I think that there is an alternative to these two options. Any ideas
>> what that alternative might be? That is, in what relationship does the
>> Peano Axioms stand to arithmetic?
>
> If Peano's set of most basic functions of ALL mathematical operations
> is true then your asking the wrong question.

If the Peano Axioms are necessarily "true" then arithmetic is either
necessary or contingent.
So I am asking the right question when I ask which is the case.

> The question should not
> be whether Peanos Axioms are necessary or contingent for math
> operations but instead the proper question would be; what relation
> math has to the theory of reality. Of course math is an attempt to
> represent our abstract notions of what reality is with a set of
> symbols and a group of rules.
>

Yes
From: Zerkon on 26 Apr 2010 08:05
On Sun, 25 Apr 2010 00:36:06 +0100, John Jones wrote:

> The Peano Axioms define the way that numbers, and hence arithmetic, is
> "presented" to us.
>
> But my question is this: In such a presentation do the Peano Axioms
> "define" arithmetic or do they "limit" arithmetic?
>
> The question is important.
>
> 1) If the Peano Axioms "define" arithmetic then they are necessary to
> arithmetic and, hence, arithmetic is a branch of logic. 2) If the Peano
> Axioms "limit" arithmetic then the source of arithmetic isn't logical or
> necessary. That is, it is contingent. And the sort of contingency that
> that might be isn't immediately fathomable.
>
> But I think that there is an alternative to these two options. Any ideas
> what that alternative might be? That is, in what relationship does the
> Peano Axioms stand to arithmetic?

Numbers as a representative of thought.

1. For every natural number x, x = x. That is, equality is reflexive.

Is absolute equality reflective or is it undefinable? Is it even
discernible?
From: bigfletch8 on 26 Apr 2010 11:47
On Apr 26, 8:05 pm, Zerkon <Z...(a)erkonx.net> wrote:
> On Sun, 25 Apr 2010 00:36:06 +0100, John Jones wrote:
> > The Peano Axioms define the way that numbers, and hence arithmetic, is
> > "presented" to us.
>
> > But my question is this: In such a presentation do the Peano Axioms
> > "define" arithmetic or do they "limit" arithmetic?
>
> > The question is important.
>
> > 1) If the Peano Axioms "define" arithmetic then they are necessary to
> > arithmetic and, hence, arithmetic is a branch of logic. 2) If the Peano
> > Axioms "limit" arithmetic then the source of arithmetic isn't logical or
> > necessary. That is, it is contingent. And the sort of contingency that
> > that might be isn't immediately fathomable.
>
> > But I think that there is an alternative to these two options. Any ideas
> > what that alternative might be? That is, in what relationship does the
> > Peano Axioms stand to arithmetic?
>
> Numbers as a representative of thought.
>
> 1. For every natural number x, x = x. That is, equality is reflexive.
>
> Is absolute equality reflective or is it undefinable? Is it even
> discernible?- Hide quoted text -
>
> - Show quoted text -

It is conceivable, but never concieved. Such is the nature of the
realtive mind.

BOfL
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