×

Set theoretic naturalism. (English) Zbl 0882.03004

The author tries to develop a sound philosophical view on the foundations of set theory. She cites Einstein who once made a remark to the effect that sound philosophical views sometimes are beneficial to the development of science itself. In earlier publications she took a so-called realist position, siding with Gödel, who, as is well known, maintained that the question if the continuum hypothesis is true is meaningful, although we now know that the answer is not given by the axioms of set theory. She also sympathized with some views of Quine’s, who argued that it would be “dishonest” to deny the reality of mathematical objects as they figure so prominently in physical theories that we are prepared to accept. In this paper, the author wants to qualify her earlier views and seems anxious not to interfere with set theory as it is developed by the mathematicians. She confesses herself to “naturalism” rather than “realism”. The naturalist, as defined by Quine, rejects any restriction of the methods of science from a not strictly scientific point of view. The author accordingly circumscribes her task as “to account for set theory as it is practiced, not as some philosophy would have it to be”. She argues that a careful reading of Gödel’s texts reveals that her present position is still not very far from Gödel’s, as also Gödel seems anxious to defend existing mathematics against philosophically motivated attacks, and his sometimes offensive realism perhaps may be considered, partly, as issuing forth from this basic attitude. The author shows in some detail what makes the difference between her present naturalist attitude and her former realist attitude in the discussion on a particular question on how to extend the axioms of set theory.
The cautious and wise “naturalist” position chosen by the author is, in a sense, a very safe one.
Hopefully, it does not prevent her from taking seriously the objections sometimes brought in against set-theoretical practice.

MSC:

03A05 Philosophical and critical aspects of logic and foundations
03Exx Set theory
00A30 Philosophy of mathematics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Descriptive Set Theory (1980) · Zbl 0433.03025
[2] Problems in the Philosophy of Mathematics pp 82– (1967)
[3] Steps toward a constructive nominalism 12 pp 105– (1947)
[4] Collected Works II (1990)
[5] From Frege to Gödel pp 200– (1908)
[6] Realism, Mathematics and Modality (1989) · Zbl 1098.00500
[7] From Mathematics to Philosophy (1974) · Zbl 0554.03002
[8] The Evolution of Physics (1938)
[9] The works of the mind pp 180– (1947)
[10] DOI: 10.2307/2025002 · doi:10.2307/2025002
[11] DOI: 10.1093/philmat/2.2.148 · Zbl 0795.00007 · doi:10.1093/philmat/2.2.148
[12] DOI: 10.2307/2184627 · doi:10.2307/2184627
[13] Albert Einstein: Philosopher-Scientist (1949) · Zbl 0038.14804
[14] DOI: 10.1007/BF01049179 · Zbl 0780.03001 · doi:10.1007/BF01049179
[15] Philosophy of Mathematics Today · Zbl 0182.00303
[16] DOI: 10.1093/philmat/3.3.248 · Zbl 0962.00514 · doi:10.1093/philmat/3.3.248
[17] Does V equal L? 58 pp 15– (1993)
[18] DOI: 10.2307/2026712 · doi:10.2307/2026712
[19] Realism in Mathematics (1990)
[20] DOI: 10.2307/2026715 · doi:10.2307/2026715
[21] Mathematical Thought from Ancient to Modern Times (1972) · Zbl 0277.01001
[22] Mathematical logic and foundations of set theory pp 84– (1970)
[23] Cantorian Set Theory and Limitation of Size (1984)
[24] Ontology and the Vicious-Circle Principle (1973)
[25] Pursuit of Truth (1990) · Zbl 0733.03002
[26] Proof, logic, and formalization pp 8– (1992)
[27] DOI: 10.2307/2026033 · doi:10.2307/2026033
[28] DOI: 10.1007/978-94-009-1902-0_1 · doi:10.1007/978-94-009-1902-0_1
[29] Philosophy of Mathematics Today · Zbl 0182.00303
[30] Ontological relativity and other essays pp 69– (1969)
[31] Mathematics, Matter and Method pp 323– (1979)
[32] Philosophy of Mathematics (1983)
[33] Subtle is the Lord (1982)
[34] Non-well-founded Sets (1988)
[35] DOI: 10.1007/BF02394569 · Zbl 0186.29603 · doi:10.1007/BF02394569
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.