Re: Epistemology 201: The Science of Science

From: Lester Zick (lesterDELzick_at_worldnet.att.net)
Date: 02/10/05 

Date: Thu, 10 Feb 2005 21:59:55 GMT

On Thu, 10 Feb 2005 12:45:12 -0500, Chairman of the David Hilbert
Appreciation Society <mathgeekXXXXII@hotmail.com> in
comp.ai.philosophy wrote:

>Lester Zick wrote:
>> On Wed, 09 Feb 2005 16:15:35 -0500, Chairman of the David Hilbert
>> Appreciation Society <mathgeekXXXXII@hotmail.com> in
>> comp.ai.philosophy wrote:
>>
>>
>>>Lester Zick wrote:
>>>
>>>>On Wed, 09 Feb 2005 10:54:55 -0500, Chairman of the David Hilbert
>>>>Appreciation Society <mathgeekXXXXII@hotmail.com> in
>>>>comp.ai.philosophy wrote:
>>>>
>>>>
>>>>
>>>>>Tony Orlow (aeo6) wrote:
>>>>>
>>>>>[...]
>>>>>
>>>>>
>>>>>
>>>>>>I have not been exposed to the Peano axioms until now, but a preliminary
>>>>>>look at them leaves me with the question of how one goes from saying
>>>>>>that every natural number has a successor that is a natural number, to
>>>>>>defining addition of two natural numbers. I must be missing something if
>>>>>>it's so widely accepted, or maybe not. I don't mind being at odds with
>>>>>>the mathematical community any more than I mind being corrected. Can
>>>>>>someone explain this?
>>>>>
>>>>>The first step would be to prove the existence of a function
>>>>>f: N x N -> N which satisfies a rough notion of what addition
>>>>>is.
>>>>
>>>>
>>>>This makes it look like the successor function requires addition.
>>>
>>>Why?
>>
>>
>> Because you say "The first step would be to prove the existence of a
>> function f: N x N -> N which satisfies a rough notion of what addition
>> is" which it appears requires a successor function for proof but
>> doesn't explain how the successor function differs from addition.
>
>The successor operation is unary, it doesn't operate on pairs of
>arguments, like addition, so it's not the same.

But there are two arguments, the increment 1 and that which is
incremented? I will grant that the increment function alone requires
a smallest increment 1 but I think the second operand is implied.

>> In other words, it looks to me like the successor function is just a
>> reduction of the addition function to its simplest form and not an
>> explanation for addition in different form that would prove addition
>> without assuming and requiring addition in some form.
>
>Right. The successor operation is a primitive operation. It's
>equivalent to addition by 1, but according to this axiomatization,
>we must define addition in terms of the successor operation because
>it's all that we've got.

Okay. I can see this. The problem I have is that a great many assume
this makes the definition of addition recursive rather than circular.
If we admit the definition is actually circular I don't mind admitting
the definition is elementary as far as addition is concerned. But I
also think we can do better in terms of a definition for addition. As
it stands it looks like we need two axiomatic definitions, one for
addition and one for subtraction, presumably through some kind of
analogous regressor function.

>Alternatively, as part of another axiomatization, one could
>simply postulate the existence of a function that satisfies
>all of the desired properties of addition from the start;
>then you could prove the existence of a successor function
>from that.

I'm more inclined to a definition for addition in terms of subtraction
that would be finitely regressable. In other words I see + = - -. This
would satisfy the requirement for a definition of addition in non
circular terms. The definition of subtraction on the other hand would
be finitely regressable to itself in the sense that anything different
from differences would be self contradictory but that there could be
differences between differences which result in addition mechanically.

In any event I certainly appreciate your sensible and responsive
answers.

>It turns out that these two approaches aren't equivalent.
>
>> Regards - Lester
>
>
>

Regards - Lester

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... the increment 1 and that which is ... The successor operation is a primitive operation. ... If we admit the definition is actually circular I don't mind admitting ... I'm more inclined to a definition for addition in terms of subtraction ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... the increment 1 and that which is ... The successor operation is a primitive operation. ... If we admit the definition is actually circular I don't mind admitting ... I'm more inclined to a definition for addition in terms of subtraction ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... >>of integer addition and subtraction come later, ... >>terms of counting (i.e. in terms of the successor operation). ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... >>of integer addition and subtraction come later, ... >>terms of counting (i.e. in terms of the successor operation). ...
    (sci.physics)
  • Re: Epistemology 201: The Science of Science
    ... >>of integer addition and subtraction come later, ... >>terms of counting (i.e. in terms of the successor operation). ...
    (sci.math)