> Date: Tue, 9 Jun 2009 15:22:10 -0700
> From: [email protected]
> To: [email protected]
> Subject: Re: The seven step-Mathematical preliminaries
>
>
> Jesse Mazer wrote:
>>
>>
>>> Date: Tue, 9 Jun 2009 12:54:16 -0700
>>> From: [email protected]
>>> To: [email protected]
>>> Subject: Re: The seven step-Mathematical preliminaries
>>>
>>
>>> You don't justify definitions. How would you justify Peano's axioms
>> as being
>>> the "right" ones? You are just confirming my point that you are
>> begging the
>>> question by assuming there is a set called "the natural numbers"
>> that exists
>>> independently of it's definition and it satisfies Peano's axioms.
>>
>> What do you mean by "exists" in this context? What would it mean to
>> have a well-defined, non-contradictory definition of some mathematical
>> objects, and yet for those mathematical objects not to "exist"?
>
> A good question. But if one talks about some mathematical object, like
> the natural numbers, having properties that are unprovable from their
> defining set of axioms then it seems that one has assumed some kind of
> existence apart from the particular definition.
Isn't this based on the idea that there should be an objective truth about
every well-formed proposition about the natural numbers even if the Peano
axioms cannot decide the truth about all propositions? I think that the
statements that cannot be proved are disproved would all be ones of the type
"for all numbers with property X, Y is true" or "there exists a number (or some
finite group of numbers) with property X" (i.e. propositions using either the
'for all' or 'there exists' universal quantifiers in logic, with variables
representing specific numbers or groups of numbers). So to believe these
statements are objectively true basically means there would be a unique way to
"extend" our judgment of the truth-values of propositions from the judgments
already given by the Peano axioms, in such a way that if we could flip through
all the infinite propositions judged true by the Peano axioms, we would *not*
find an example of a proposition like "for this specific number N with property
X, Y is false" (which would disprove the 'for all' proposition above), and
likewise we would not find that for every possible number (or group of numbers)
N, the Peano axioms proved a proposition like "number N does not have property
X" (which would disprove the 'there exists' proposition above).
We can't actual flip through an infinite number of propositions in a finite
time of course, but if we had a "hypercomputer" that could do so (which is
equivalent to the notion of a hypercomputer that can decide in finite time if
any given Turing program halts or not), then I think we'd have a well-defined
notion of how to program it to decide the truth of every "for all" or "there
exists" proposition in a way that's compatible with the propositions already
proved by the Peano axioms. If I'm right about that, it would lead naturally to
the idea of something like a "unique consistent extension" of the Peano axioms
(not a real technical term, I just made up this phrase, but unless there's an
error in my reasoning I imagine mathematicians have some analogous
notion...maybe Bruno knows?) which assigns truth values to all the well-formed
propositions that are undecidable by the Peano axioms themselves. So this would
be a natural way of understanding the idea of truths "about the natural
numbers" that are not decidable by the Peano axioms.
Of course even if the notion of a "unique consistent extension" of certain
types of axiomatic systems is well-defined, it would only make sense for
axiomatic systems that are consistent in the first place. I guess in judging
the question of the consistency of the Peano axioms, we must rely on some sort
of ill-defined notion of our "understanding" of how the axioms should represent
true statements about things like counting discrete objects. For example, we
understand that the order you count a group of discrete objects doesn't affect
the total number, which is a convincing argument for believing that A + B = B +
A regardless of what numbers you choose for A and B. Likewise, we understand
that multiplying A * B can be thought of in terms of a square array of discrete
objects with the horizontal side having A objects and the vertical side having
B objects, and we can see that just by rotating this you get a square array
with B on the horizontal side and A on the vertical side, so if we believe that
just mentally rotating an array of discrete objects won't change the number in
the array that's a good argument for believing A * B = B * A. So thinking along
these lines, as long as we don't believe that true statements about counting
collections of discrete objects could ever lead to logical contradictions, we
should believe the same for the Peano axioms.
Jesse
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