John, (01)
I would rather not follow your daily digest method of keeping volume down. (02)
To me the most important thing is not the absolute number of messages,
but that the relevance of each message is easily identified. Mixing
responses greatly reduces the relevance of each message. (03)
It takes me much more time to read through all your responses to
others, before stumbling upon one to myself. (04)
Of course it totally messes up Gmail threads. (05)
I may turn out to be wrong though, because it would force people to
consolidate their arguments, which might improve the coherence of
arguments themselves. (06)
Perhaps a compromise of separate threads and consolidation is best. (07)
On Wed, Jan 27, 2010 at 4:35 PM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:
>
> ...I made the point that pure mathematics differs
> from the other sciences because it does not make any statements
> or claims about the physical world. The basic methods of
> mathematics have not changed since our stone age ancestors:
>
> 1. Adopt some rules or axioms that define abstract patterns
> (e.g., the numbers, geometrical forms, etc.).
>
> 2. Use those rules to generate examples of those patterns.
> The stone age rules are to count on your fingers or make
> notches on a stick. These are observations, but they
> are not observations of physical phenomena.
>
> 3. Analyze them to determine the possible range of patterns.
> This kind of analysis is usually called a proof. Formal
> proofs were codified by Euclid, but the stone age people
> who built Stonehenge showed a high degree of sophistication.
> They probably used some informal methods of analysis.
>
> The use of rules to generate examples could be called 'empirical',
> but Chaitin added the prefix 'quasi-' because those patterns were
> generated by mental processes, not by physical mechanisms. That
> is much more than a quibble. (08)
I didn't disagree with your basic premise, so much as the vehemence
with which you asserted it. (09)
To me it is a difference of emphasis. Mathematics is elaborated from
axioms, I agree. But mathematicians will change their axioms to fit
one or other real-world problem. I think non-Euclidean geometry is in
that class. Attitudes to infinity are perhaps more modern examples. (010)
Whether you call that "empirical" is to me a question of language. (011)
But unless anyone else is learning anything from the exchange, I'm
quite happy to admit I see the distinction you wish to make, and
acknowledge mathematics looks at problems differently. (012)
-Rob (013)
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