Hermann Weyl's Intuitionistic Mathematics

1995 ◽  
Vol 1 (2) ◽  
pp. 145-169 ◽  
Author(s):  
Dirk van Dalen
Keyword(s):  
Common Knowledge ◽  
Foundations Of Mathematics ◽  
Extensive Discussion ◽  
Traditional Construction ◽  
Dedekind Cuts ◽  
Historical Issues ◽  
Intuitionistic Mathematics ◽  
Mathematical Construction ◽  
The Continuum ◽  
Short Episode

Dedicated to Dana Scott on his sixtieth birthday.It is common knowledge that for a short while Hermann Weyl joined Brouwer in his pursuit of a revision of mathematics according to intuitionistic principles. There is, however, little in the literature that sheds light on Weyl's role and in particular on Brouwer's reaction to Weyl's allegiance to the cause of intuitionism. This short episode certainly raises a number of questions: what made Weyl give up his own program, spelled out in “Das Kontinuum”, how did Weyl come to be so well-informed about Brouwer's new intuitionism, in what respect did Weyl's intuitionism differ from Brouwer's intuitionism, what did Brouwer think of Weyl's views,…? To some of these questions at least partial answers can be put forward on the basis of some of the available correspondence and notes. The present paper will concentrate mostly on the historical issues of the intuitionistic episode in Weyl's career.Weyl entered the foundational controversy with a bang in 1920 with his sensational paper “On the new foundational crisis in mathematics”. He had already made a name for himself in the foundations of mathematics in 1918 with his monograph “The Continuum” [18]; this contained in addition to a technical logical-mathematical construction of the continuum, a fairly extensive discussion of the shortcomings of the traditional construction of the continuum on the basis of arbitrary—and hence also impredicative—Dedekind cuts.

How connected is the intuitionistic continuum?

1997 ◽  
Vol 62 (4) ◽  
pp. 1147-1150 ◽  
Author(s):  
Dirk Van Dalen
Keyword(s):  
Real Function ◽  
Peculiar Feature ◽  
Constructive Mathematics ◽  
Fan Theorem ◽  
First Time ◽  
Uniformly Continuous ◽  
Intuitionistic Mathematics ◽  
Excluded Middle ◽  
Do So ◽  
The Continuum

In the twenties Brouwer established the well-known continuity theorem “every real function is locally uniformly continuous,” [3, 2, 5]. From this theorem one immediately concludes that the continuum is indecomposable (unzerlegbar), i.e., if ℝ = A ∪ B and A ∩ B = ∅ (denoted by ℝ = A + B), then ℝ = A or ℝ = B.Brouwer deduced the indecomposability directly from the fan theorem (cf. the 1927 Berline Lectures, [7, p. 49]).The theorem was published for the first time in [6], it was used to refute the principle of the excluded middle: ¬∀x ∈ ℝ (x ∈ ℚ ∨ ¬x ∈ ℚ).The indecomposability of ℝ is a peculiar feature of constructive universa, it shows that ℝ is much more closely knit in constructive mathematics, than in classically mathematics. The classically comparable fact is the topological connectedness of ℝ. In a way this characterizes the position of ℝ: the only (classically) connected subsets of ℝ are the various kinds of segments. In intuitionistic mathematics the situation is different; the continuum has, as it were, a syrupy nature, one cannot simply take away one point. In the classical continuum one can, thanks to the principle of the excluded third, do so. To put it picturesquely, the classical continuum is the frozen intuitionistic continuum. If one removes one point from the intuitionistic continuum, there still are all those points for which it is unknown whether or not they belong to the remaining part.

An intuitiomstic completeness theorem for intuitionistic predicate logic

1976 ◽  
Vol 41 (1) ◽  
pp. 159-166 ◽  
Author(s):  
Wim Veldman
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Logic ◽  
Completeness Theorem ◽  
Point Of View ◽  
Foundations Of Mathematics ◽  
Strong Completeness ◽  
Recursive Predicate ◽  
Image Position ◽  
Primitive Recursive ◽  
Intuitionistic Mathematics

The problem of treating the semantics of intuitionistic logic within the framework of intuitionistic mathematics was first attacked by E. W. Beth [1]. However, the completeness theorem he thought to have obtained, was not true, as was shown in detail in a report by V. H. Dyson and G. Kreisel [2]. Some vague remarks of Beth's, for instance in his book, The foundations of mathematics, show that he sustained the hope of restoring his proof. But arguments by K. Gödel and G. Kreisel gave people the feeling that an intuitionistic completeness theorem would be impossible [3]. (A (strong) completeness theorem would implyfor any primitive recursive predicate A of natural numbers, and one has no reason to believe this for the usual intuitionistic interpretation.) Nevertheless, the following contains a correct intuitionistic completeness theorem for intuitionistic predicate logic. So the old arguments by Godel and Kreisel should not work for the proposed semantical construction of intuitionistic logic. They do not, indeed. The reason is, loosely speaking, that negation is treated positively.Although Beth's semantical construction for intuitionistic logic was not satisfying from an intuitionistic point of view, it proved to be useful for the development of classical semantics for intuitionistic logic. A related and essentially equivalent classical semantics for intuitionistic logic was found by S. Kripke [4].

N. A. Šanin. On the constructive interpretation of mathematical judgments. English translation of XXXI 255 by Elliott Mendelson. American Mathematical Society translations, ser. 2 vol. 23 (1963), pp. 109–189. - A. A. Markov. On constructive functions. English translation of XXXI 258(1) by Moshe Machover. American Mathematical Society translations, vol. 29 (1963), pp. 163–195. - S. C. Kleene. A formal system of intuitionistic analysis. The foundations of intuitionistlc mathematics especially in relation to recursive functions, by Stephen Cole Kleene and Richard Eugene Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 1–89. - S. C. Kleene. Various notions of realizability:The foundations of intuitionistlc mathematics especially in relation to recursive functions, by Stephen Cole Kleene and Richard Eugene Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 90–132. - Richard E. Vesley. The intuitionistic continuum. The foundations of intuitionistlc mathematics especially in relation to recursive functions, by Stephen Cole Kleene and Richard Eugene Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 133–173. - S. C. Kleene. On order in the continuum. The foundations of intuitionistlc mathematics especially in relation to recursive functions, by Stephen Cole Kleene and Richard Eugene Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 174–186. - S. C. Kleene. Bibliography.The foundations of intuitionistlc mathematics especially in relation to recursive functions, by Stephen Cole Kleene and Richard Eugene Vesley, Studies in logic and the foundations of mathematics, North-Holland Publishing Company, Amsterdam1965, pp. 187–199.

1966 ◽  
Vol 31 (2) ◽  
pp. 258-261 ◽  
Author(s):  
Georg Kreisel
Keyword(s):  
English Translation ◽  
American Mathematical Society ◽  
Formal System ◽  
Publishing Company ◽  
Mathematical Society ◽  
Foundations Of Mathematics ◽  
North Holland Publishing Company ◽  
Recursive Functions ◽  
North Holland Publishing ◽  
The Continuum

scholarly journals The changing role of continuity and discontinuity in the history of philosophy and mathematics

2017 ◽  
Vol 36 (1) ◽  
Author(s):  
Danie F.M. Strauss
Keyword(s):  
Infinite Divisibility ◽  
History Of Philosophy ◽  
Greek Philosophy ◽  
History Of ◽  
And Mathematics ◽  
Changing Role ◽  
Intuitionistic Mathematics ◽  
Continuity And Discontinuity ◽  
The Continuum

The aim of this article is to highlight the inevitability of employing discreteness and continuity as primitive (indefinable) modes of explanation in the history of philosophy and mathematics. It embodies the general challenge to account for the coherence of what is unique. Gödel emphasises the coherence of ‘primitive concepts’. Greek philosophy already discovered the spatial whole and/or parts relation with its infinite divisibility. During and after the medieval era philosophers toggled between an atomistic appreciation of the continuum and its opposite, for example found in the thought of Leibniz who postulated his law of continuity (lex continui). The discovery of incommensurability (irrational numbers) by the Greeks caused the first foundational crisis of mathematics, as well as its geometrisation. Leibniz and Newton did not resolve the problems surrounding the limit concept and soon it induced the third foundational crisis of mathematics. It caused Frege and the ‘continuum theoreticians’ to assign priority to the continuum – discreteness is a catastrophe. Recently Smooth Infinitesimal Analysis appreciated what is ‘continuous’ as constituting ‘an unbroken or uninterrupted whole’. Intuitionistic mathematics once more proceeded from an emphasis on the whole and/or parts relation. In spite of alternating attempts to understand continuity exclusively, either in arithmetical or in spatial terms, the history of philosophy and mathematics undeniably confirms that the co-conditioning role of these two modes of explanation remains a constant element in reflections on continuity and discontinuity. (The role of continuity and discontinuity within the disciplines of physics and biology will be discussed in a separate article.)

The continuum and first-order intuitionistic logic

1992 ◽  
Vol 57 (4) ◽  
pp. 1417-1424 ◽  
Author(s):  
D. van Dalen
Keyword(s):  
Prima Facie ◽  
Order Logic ◽  
Point Of View ◽  
Ordered Sets ◽  
Practice Model ◽  
Original Proof ◽  
First Order ◽  
The One ◽  
Intuitionistic Mathematics ◽  
The Continuum

Ever since Cantor, we have known that the reals and the rationals are not isomorphic (as equality structures, i.e., sets). Logically speaking, however, they are not all that different; in first-order classical logic they are elementarily equivalent, since the theory of infinite sets is complete. The same holds for ℝ and ℚ as ordered sets; again the theory of dense linear order without end points is complete.From an intuitionistic point of view these matters are more complicated; e.g., the theory of equality of ℚ is decidable, whereas the one of ℝ patently is not. This, in a roundabout way, shows that ℚ and ℝ are not isomorphic; of course, there is no need for such a detour, as Cantor's original proof [2] is intuitionistically correct, and Brouwer's new proof [1] is another alternative intuitionistic argument.In view of the fact that ℚ and ℝ behave so strikingly differently with respect to first-order logic, one is easily tempted to look for elementary equivalences among the subsets of ℝ. Until quite recently most model theoretic investigations of intuitionistic theories made use of special (artificial) notions of “model”, e.g., Kripke models, sheaf models,…; but there is no prima facie reason why one should not practice model theory much the same way as traditional model theorists do. That is to say on the basis of a naive set theory, or, in our case, of naive intuitionistic mathematics.This paper uses the method of (k, p)-isomorphisms of Fraïssé, and it is briefly shown that one half of the Fraïssé theorem holds intuitionistically.

C. The Continuous Spectrum of Pulsating Variable Stars

1967 ◽  
Vol 28 ◽  
pp. 177-206
Author(s):  
J. B. Oke ◽  
C. A. Whitney
Keyword(s):  
Continuous Spectrum ◽  
Variable Stars ◽  
Velocity Fields ◽  
The Other ◽  
Line Spectrum ◽  
Direct Information ◽  
Thermodynamic Structure ◽  
Diagnostic Interpretation ◽  
Pulsating Variable Stars ◽  
The Continuum

Pecker:The topic to be considered today is the continuous spectrum of certain stars, whose variability we attribute to a pulsation of some part of their structure. Obviously, this continuous spectrum provides a test of the pulsation theory to the extent that the continuum is completely and accurately observed and that we can analyse it to infer the structure of the star producing it. The continuum is one of the two possible spectral observations; the other is the line spectrum. It is obvious that from studies of the continuum alone, we obtain no direct information on the velocity fields in the star. We obtain information only on the thermodynamic structure of the photospheric layers of these stars–the photospheric layers being defined as those from which the observed continuum directly arises. So the problems arising in a study of the continuum are of two general kinds: completeness of observation, and adequacy of diagnostic interpretation. I will make a few comments on these, then turn the meeting over to Oke and Whitney.

Photo-electric Hβ and uvby photometry

1966 ◽  
Vol 24 ◽  
pp. 170-180
Author(s):  
D. L. Crawford
Keyword(s):  
Narrow Band ◽  
Line Strength ◽  
Interference Filter ◽  
B Stars ◽  
Relative Line ◽  
The Mean ◽  
F Stars ◽  
Two Parameters ◽  
Filter Photometry ◽  
The Continuum

Early in the 1950's Strömgren (1, 2, 3, 4, 5) introduced medium to narrow-band interference filter photometry at the McDonald Observatory. He used six interference filters to obtain two parameters of astrophysical interest. These parameters he calledlandc, for line and continuum hydrogen absorption. The first measured empirically the absorption line strength of Hβby means of a filter of half width 35Å centered on Hβand compared to the mean of two filters situated in the continuum near Hβ. The second index measured empirically the Balmer discontinuity by means of a filter situated below the Balmer discontinuity and two above it. He showed that these two indices could accurately predict the spectral type and luminosity of both B stars and A and F stars. He later derived (6) an indexmfrom the same filters. This index was a measure of the relative line blanketing near 4100Å compared to two filters above 4500Å. These three indices confirmed earlier work by many people, including Lindblad and Becker. References to this earlier work and to the systems discussed today can be found in Strömgren's article inBasic Astronomical Data(7).

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