These are the ideas of the twentieth century mathematician Abraham Robinson
Thesis: The results of the centuries arouse deep admiration but are built on
shifting sands. of foundations of mathematics now as well as in the
past.
Commonly accepted: math as a deductive science begin with Greeks
4 -5 century B.C.
What about the Babylonian? the Egyptians?
Was their mathematics not deductive?
What would the Babylonian mathematician say of Euclid?
Major discoveries were not the invention of new math but the codification
of elements of mathematical thought - explicit arguments notions
assumptions rules etc. which had been intuitive for a long time.
Thesis: The differentiation between Philosophy and Mathematics was less
distinct then than now.
Hence - Plato or Aristotle or some other philosophical school could have
influenced the development of the Axiomatic method even though none that we
know of (except Democritus (atomism and math treatises) contributed to
mathematics.
Aristotle established standards of rigor and completeness which went far
beyond the level actually reached at that time.
Euclid and Archimedes singled out those axioms which could not be taken for
granted and then proceeded to produce them with assumptions whose truth
seemed obvious - by means of rules of deduction seemingly equally obvious.
Yet Euclid did make the Archimedian Axiom as a definition because he
didn't wish to exclude the other. He needed it for the theory of proportion
and method of exhaustion.
Thesis: Axioms have always (until early 19 century) been
regarded as statements of .
However in Euclid there is an element of ``constructivism'' which
should hark back to pre-hellinic times and should strike a chord in the
hearts of those believing Mathematics has been pushed too far in a
formal-deductive direction and needs a more constructive approach in the
foundations.
Ironic fate: only after Geometry lost its standing as the basis
of all Mathematics was its axiomatic foundations finally reached the degree
of perfection which the public estimation had always given them.
Soon thereafter the first form axiomatic theories were proposed.
20 century: Set theory achieves the position
once occupied by Geometry
Set Theory also went through a similar evolution.
the initial assumptions of Set Theory were held to be
intuitively clear being based on natural laws of thought
Then Set Theory was put on a postulated basis including the
axiom of choice (Zorn's lemma) the least intuitive
At this point the axioms were still suppose to describe
reality, although Platonic.
Then came the realization that it is equally consistent
to deny or affirm some major assertions of Set Theory e.q. continuum
hypothesis mid 1960s this led the Set Theory as describing objective
reality to be dropped.
Set Theory and Mathematical Logic - advances have been made thru the
codification of notion (e.g. truth). There is every reason to believe the
codification will continue, bringing new advances.eject
Abraham Robinson
Born: 6 Oct 1918 in Waldenburg, Germany
Died: 11 April 1974 in New Haven, Connecticut
Robinson's family emigrated to Palestine in 1933, forced out of Germany
since they were Jewish. There Robinson studied mathematics under Fraenkel
and Levitzki. He went to the Sorbonne in 1939 but was forced to flee when
the Germans invaded. After reaching England on one of the last small boats
to evacuate refugees, he changed his name from Robinsohn and worked on
aerodynamics during World War II.
After the War he attended London University and received a Ph.D. in 1949
for pioneering work in model theory, the metamathematics of algebraic
systems. He went to Toronto in 1951 to take up a chair of applied
mathematics but left for Jerusalem in 1957 to fill Fraenkel's chair.
During a year visiting Princeton he made the discovery for which he is best
remembered, non-standard analysis. In 1967 he was appointed to Yale where
he died of cancer at the age of 55.
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