Re: An uncountable countable set

In article <45193e6f@xxxxxxxxxxxxxxxxxxx>,
Tony Orlow <tony@xxxxxxxxxxxxx> wrote:

Han de Bruijn wrote:
Tony Orlow wrote:

Virgil wrote:

Mathematicians know better.

Define "better". Those that work in various areas of science share a
notion which defines science. Theories which have no means of
verification are not science, but philosophy. In mathematics,
verification really consists of corroboration by other means,
agreement between different approaches. In science, where you find a
contradiction with your theory, it needs revision. So, the scientific
approach to mathematics requires some criterion for universal
consistency, as measured by the predictions of the various theories
that comprise it. Where two theories collide, one or both is in error.
I think that's better.

Precisely ! In mathematics, there are contradictory approaches, such as
constructivism (Brouwer) against axiomatism (Hilbert). Its practicioners
are asked to be "nice" to each other and to "reconciliate" the different
points of view, which turns out to be a hopeless task. Such a situation
would be unthinkable if mathematics aimed to be a science.

Han de Bruijn


Well, Han, I'm not sure I agree with the statement that reconciliation
is hopeless. Is it hopeless to reconcile the wave nature of elementary
entities with their particle nature?

It is close to hopeless to expect those who reject the law of the
excluded middle (constructionists) and those who insist on it
(formalists) to agree.






There is confusion about my "definition" of infinitesimals, because I
can see the validity both in nilpotent infinitesimals and in those that
are further infinitely divisible.

Until TO can come up with an axiom system which simultaneously allows
his infinitesimals to be both nilpotent and not, he is in trouble.



Constructivism and Axiomatism are two sides of a coin. They can be
reconciled in larger framework, I think.

Better men that TO will ever be have tried.
.

Relevant Pages

  • =?UTF-8?B?UmU6INmB2LHZiNi0INmF2KzZhdmI2LnZhyDaqdin2YXZhCDaqdiq2KfYqNmH2KfbjCDZhQ==?= =?UTF-8
    ... SEPARATION SCIENCE AND TECHNOLOGY Volume I BIOSEPARATION OF PROTEINS ... Organometallic Chemistry v.27 by Royal ... Valtchev, Svetlana Mintova, Michael Tsapatsis],Copyright © ... Unified Integration (Pure and Applied Mathematics (Academic Pr)), ...
    (sci.math.num-analysis)
  • Re: Q!Re: Shall We All Leave At Once?
    ... EVERY ancient historian accepted Zeus as being a deified king of Crete and the ancestor of the Greeks that worshiped him. ... >>> Yove not studied real science or philosophy. ... All you do is parrot other peoples biased ignorant personal options who have never read the source material either. ... >> Mathematics doesn't predict anything. ...
    (rec.arts.drwho)
  • The uncorrected simplifications/oversimplifications of Hamiltons Rule (Re: Removing Lewontins Fallac
    ... >> proposition of mathematics it is not a valid proposition of science. ... There are no axiomatic truths within the sciences only within mathematics. ... Hamilton's tautology for the diploid case. ... of non random patterns in gene freq. ...
    (sci.bio.evolution)
  • Re: Q!Re: Shall We All Leave At Once?
    ... challenged fool. ... Learn to comprehend English! ... or science. ... "based on mathematics" in the same way journalism is based on word ...
    (rec.arts.drwho)
  • Re: An uncountable countable set
    ... notion which defines science. ... approach to mathematics requires some criterion for universal ... Is it hopeless to reconcile the wave nature of elementary ... There is confusion about my "definition" of infinitesimals, ...
    (sci.math)