Selasa, 28 Februari 2017

On fundamental problems with calculus



Hello Victor and Prof. Florentin,

I fully agree that "calculus" is a deeply troubled theory of mathematics,
which lead to the more troubled theory of mathematics called topology
which ended up with a largest play ground for "problem solvers" to
enumerate problems and publish papers endlessly.

Going back to the calculus, historically it is false that this was
developed by Newton and Leibniz. About three hundred years before it was
developed in India.  They did not use it for building Physics. They just
wanted to build a mathematical model of celestial system to be used for
agriculture. I do not know if these two developments are independent.

The issue in Calculus a la Europe is that it was impossible to define the
concept of "limit" articulately. It took about two centuries for European
(mostly German) mathematicians to gain this concept properly using the
theory of complete ordered field and epsilon-delta argument. This approach
developed into general topology from which the so called algebraic
topology came out.

It was unfortunate that this method though worked fine had little to do
with the original calculus used in early physics in which naive concept of
infinitesimals were used. In fact, the topological calculus was developed
as an anti-thesis to the infinitesimal based calculus. This is what most
student learns in the university. Rarely infinitesimal caluclus is taught
thought we have modern and complete infinitesimal calculus as developed by
Abraham Robinson in 1960. The reason for all of this is because the naive
concept of infinitesimals is apparently paradoxical as even George Cantor
complained. Certainly it is trouble some to figure out what kind of
numbers are "positive numbers each of which is smaller than all positive
real numbers".

Robinson was a most important mathematical logician (model theorist) in
the history of mathematics though due to the advanced and difficult nature
of his work, it went over the head of popular scientists and did not get
appreciation which so badly deserved. It is my view that this had a lot to
do with the anti-logic culture of theoretical physics. As you know well,
logic was developed to very high level by the Scholastic Philosophers in
Vatican who wanted to prove that God exists.  Unlike Physicists, they were
quite open and honest. They realized that in all of their proofs for the
existence of God, they assumed the existence of God in one way or the
other. So, they said predicate logic will not do. This lead them to the
development of modal logic in which they tried to prove the necessity of
God instead of existence. As the bloody history of theoretical physics
proves, basically admitting the error and defeat is the last thing for the
King of science would do. This ended up with the situation where even
secondary school students become highly critical of this entire activity.
The final outcome is that the King of Science left this ugly expensive
CERN and highly questionable authority to dominate at any cost.
Notwithstanding, as a professional logician, I can tell you that we
logicians are still learning from the Vatican cosmology, though we learned
very little from the so called theoretical physics. The problem is that
all we see in theoretical physics as we know of now looks nothing but a
mountain of errors and deceptions created by earthly expectations.  After
all, to be fair, we logicians learned from QM as the development of
quantum logic. This strangest logic became a fashion for a decade or so,
long time ago and nobody even remembers.

Anyhow, regarding QM, it is a very good example of how logically
inconsistent theory whose inconsistency is well concealed by the politics
of intimidation, black mailing and name calling can create  total illusion
and thrive as "science fiction" as exactly happening now.


So, going back to infinitesimal calculus of Robinson. To be honest only
very strong experts in mathematical logic can follow what he did. It
starts with a strange theory of the first order real numbers. This is
needed as we define infinitesimal calculus as the nonstandard model of the
first order calculus, taking advantage of the weakness of the first order
theory. Using fully developed model theory (to its development Robinson
was a major contributor), using the ultra power construction and
collapsing it to quotient structure using the provable equality of the
first order real number theory, he obtained full infinitesimal calculus.
In easier language, as we define real  numbers as infinite sequences of
rational numbers separating rationals and irrationals represented as
infinite sequences of rationals, Robinson considered  infinite sequences
of real numbers as nonstandard real numbers and separated those which
converges and which do not as real numbers and infinity. As the reciprocal
of infinity he defined infinitesimals.

After all of this basically without using limit, we can develop full
calculus. Limitless calculus meant topology-less calculus. Indeed the
topology of infinitesimal calculus is trivial T0 topology.

I think one of the reason why Robinson was never appreciated in mainstream
mathematics community has a lot to do with the pressure from the community
of mathematical physicists who felt threatened  by his work which will
make their work unnecessary complication. After fleeing from Nazi
persecution, he fled to Israel and then came to Canada. He became a member
of mathematics department at U of Toronto. Despite his great work, he was
not appreciated and so he moved to UC Irvin to chair the math department
and shortly after that he died young by cancer. The sad fact is that he
never worked for places like Stanford or Harvard.

However, we do have some problem with Robinson's infinitesimal calculus.
His infinitesimals have little to do with what Leibniz and Newton meant.
Physical interpretation of Robinson's infinitesimals is not as obvious as
it should be. I learned that there are some other infinitesimal calculus
developed after Robinson. But, I do not think any of them are as powerful
as Robinson's and have any better fit with infinitesimals physics needs.
For example I do not see how we can associate a fixed charge for a
mathematical infinitesimal. This is a common way to use infinitesimals in
Physics.

What is tragic is that due to the specialization and stupid ego trips, the
communication between physics and pure mathematicians ceased to exist.
Despite this concern, I am already a senior citizen and have not much time
left. So, I hope young researchers will take up this kind of very
fundamental problems for the advancement of science.

AK







Dear Prof. Akira and Prof. Florentin


Greetings,

Just thought you will like this paper:

http://vixra.org/pdf/1305.0033v1.pdf

Humbly yours,


Victor Christianto
*Founder and Technical Director, www.ketindo.com
E-learning and consulting services in renewable energy
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Http://www.facebook.com/vchristianto
Twitter: @Christianto2013
Phone: (62) 812-30663059
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http://www.unesco.chair.network.uevora.pt/media/kunena/attachments/731/ChristologyReloaded_Aug2016.pdf
http://fs.gallup.unm.edu/APS-Abstracts/APS-Abstracts-list.htm
http://independent.academia.edu/VChristianto
Http://researchgate.net/profile/Victor_Christianto/
Http://id.linkedin.com/pub/victor-christianto/b/115/167
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