Metamathematical problems
- 1 September 1973
- journal article
- Published by Cambridge University Press (CUP) in The Journal of Symbolic Logic
- Vol. 38 (3) , 500-516
- https://doi.org/10.2307/2273049
Abstract
When a logician approaches the world of mathematics, he may have in mind one or more of several purposes. He may try to find in mathematics a framework for formalizing commonly accepted laws of thought or perhaps laws of thought that are not commonly accepted. He may want to assist the mathematician by providing him with firm foundations for his theories. But it may also be the case that the logician wishes to use his own characteristic tools—formalized languages, explicit relations between symbols and objects, rigidly expressed and controlled rules of deduction—in order to gain a better understanding of the various and variegated kinds of structures, methods, theories and theorems that are to be found in mathematics. We may then expect him to adopt the attitude of the physicist or psychologist who (whatever his professed philosophy) feels that he deals with phenomena of the external world, whose rules cannot be imposed by him arbitrarily. He, or those that come after him, may indeed use the understanding thus gained in order to modify these phenomena, but as a scientist he would not regard this possibility as his only justification. For many years now, I have concentrated on the third of the lines of approach sketched above, and it seemed natural that I should discuss it again on the present occasion. However, today I do not wish to emphasize past developments but, using some of them as a background, I propose to enumerate a number of open problems. These problems seemed to me of some interest not only for their own sake but also because their solution might well require weapons whose introduction would close definite gaps in our armory.Keywords
This publication has 10 references indexed in Scilit:
- Analytic function theory (2nd edition), by Einar Hille. Vol. I. Pp xi, 308. $9·95. Vol. II Pp xii, 496. $9·95. 1973. SBN 0 8284 0269 8/0270 1 (Chelsea)The Mathematical Gazette, 1975
- Non-standard analysis and homologyFundamenta Mathematicae, 1972
- Zum Symposium über die Grundlagen der Mathematik1Dialectica, 1971
- Applications of Model Theory to Algebra, Analysis, and Probability.The American Mathematical Monthly, 1971
- Diophantine Problems Over Local Fields IAmerican Journal of Mathematics, 1965
- ber PolynomabbildungenMathematische Annalen, 1953
- A note on systems of homogeneous algebraic equationsBulletin of the American Mathematical Society, 1945
- Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme IMonatshefte für Mathematik, 1931
- Die Frage der endlich vielen Schritte in der Theorie der PolynomidealeMathematische Annalen, 1926
- Theorie der algebraischen Functionen einer VeränderlichenPublished by Walter de Gruyter GmbH ,1882