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**Some thoughts on the History of Mathematics**

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.

Mon Apr 21 11:34:22 CDT 1997