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Jacob Ridder (1894-1977)

Jacob Ridder (Harderwijk 3 january 1894 -- Zeist 11 august 1977) was a lector/professor in Groningen from 1931 till 1962.

[Naar L.R.J. Westermann, Rijksuniversiteit Groningen, Nieuw Archief voor Wiskunde 3 XXVI, 1-12, 1979 ]

Jacob Ridder was born in the city of Harderwijk on 3 january 1894 and died in Zeist on 11 august 1977. His life of hard work was almost exclusively devoted to mathematics. In 1915 when he started to study mathematics and physics at the University of Leiden, he had already held a two years place as a primary schoolteacher and he had passed for several certificates (; wiskunde l.o., staatspraktijk diploma boekhouden, staatsexamen B). At the end of his studies in Leiden in 1918 he was advised to attend the lectures of the French Mathematician A. Denjoy, who was appointed at the University of Utrecht in order to make the latest advancements in analysis better known in the Netherlands. It was there that Ridder promoted in 1921 with Prof. Denjoy as a promotor. Ridder was a secondary school teacher in mathematics and physics at Tiel, Gorinchem, Breda from 1920-1922 and from 1922 until 1931 at the Baarnsch Lyceum. His intensive efforts of mathematical study and research during these years were rewarded with an appointment as a lecturer at the University of Groningen in 1931. In 1950 his position was changed into an extra-ordinary professorship and in 1951 in an ordinary professorship in analysis, particularly measure theory and its applications and the foundations of mathematics. Professor Ridder retired in 1962 and almost unnoticed left Groningen on his own request. He settled in Zeist, continuing his mathematical research unabated. Ridder was elected as a member of the Koninklijke Nederlandsche Akademie van Wetenschappen in 1954. His death on 11 august 1977 followed a short hospitalization necessitated by an unlucky fall from a staircase at home.

Ridder wrote a 160 scientific papers, brought together under 114 titles at the end of this survey. His research started with a thesis on rather general limit-interchanging theorems and gradually turned to measure and integration theory during the 20s. It was quite natural that the general concern for unification to match the many results and concepts within real analysis, especially measure and integration theory, was also felt by Ridder. Indeed, among his ppers covering the years 1930-1946 those dealing with rather general integration procedures, abstract structures and their integration theories attract attention. This research and that on the elements of real analysis will have wakened his interest in mathematical logic and the foundations of math-ematics. His articles on these topics cover the period 1946-1958. The later papers mostly deal with integration theory. Although several of Ridder's publications could be classified within the theory of complex functions or potential theory they belong essentially by method and inspiration to real analysis.

Ridder's research is characterized by an abundance of special results and attention for detailed elaborations. In this survey I can only try to trace some general lines of his scientific work. He was mathematically educated at a time and under conditions impacted by the turbulent decades for real analysis and the foundations of mathematics at the end of the former and the begin of this century.

His work, [2]-[30], of the years following his thesis of 1921 makes clear that his attention gradually moved away from traditional analysis towards recent developments of real analysis, measure- and integration theory. He took his share in the elaboration of ideas of Lebesgue, Denjoy, Perron, Saks, Banach and Caratheodory, to mention a few.

Themes that return again and again in his papers are the specific distinction between finite-additivity and $\sigma$-additivity of measures, the nature of exceptional sets, modifications of integration concepts and procedures in order to enlarge the classes of suinmable functions and particularly the comparison and unification of the integral concepts of Riemann, Stieltjes, Lebesgue, Young, Denjoy, Perron, a.o.

Although from a present days point of view one looks for general and lasting lines, I think that Ridder, with his efforts to investigate and elaborate now ideas in so much detail helped to pave the way to a more definite state of affairs in measure and integration theory. Both aspects of the integral as a summation convention ( the geometric approach as Ridder calls it) and the integral as an inverse of the derivation ( especially Denjoy and Perron integrals ) were extensively discussed by Ridder. Only relatively few of his papers deal with the integral as essentially a functional, [67], [89], [108].

The concern for unification must have brought him to a series of monographs on integration in abstract spaces and lattices ("Strukturen") following work of Caratheodory, [43], [57], [71]. These studies and his articles on topology and Boolean algebra, [65], [75], containing an elaboration and comparison of separation concepts of general topology on the one hand and duality of lattice theory on the other hand, are intermediate between his work on real analysis as such and his research on mathematical logic of the period 1946-1958.

In the papers, [71], [77], Ridder extends the Russell-Whitehead-Bernays axioms for propositional calculus (and restricted predicate calculus) and this enables him to conceive it as a countable Boolean algebra; it admits then a $2^n$-valued interpretation. In the following much broader articles, [80], Ridder studies, with ideas developed so far as a starting point, general axiomatic systems for propositional calculi of which e.g. Johansson's minimal calculus, the intuitionistic calculus and the classical calculus are special cases. Here also he gives a thorough elaboration emphasizing particularly duality, the decision-problem and itself's solution. Next he turns to modal logic and to applications to topological lattices, [84], [85], [88].

His work since 1960 demonstrates the same interest in measure and integration theory as before. New ideas on integration procedures as well on abstract settings for integration theory are digested. Ridder stays strikingly well acquainted with literature and adapts results quickly.

Ridder's interest was attracted by the concepts and the basic methods, while his point of view was somewhat formalistic. He seemed to favour syntaxis above semantics. When he speaks of applications it is often to denote a specialization to a more restrictive theoretical case. Some nice subtle "applied" results he achieved on sufficient conditions for monogeneous functions via general versions of Cauchy-like integral theorems, [19], [56], [68].

Ridder was a very conscientious mathematician, overlooking no detail. He was not in the first place a source of unexpected new ideas, but his strength was mainly in pressing all mathematical truth out of ideas and themes, which were (recently) brought to light and here he was indefatigable. Besides he was very well up in a much larger part of mathematics as on which he published or lectured.

His lectures were always thoroughly prepared and well presented. Although he was exacting for his students, one did not apply for his assistance or support in vain. In personal contact he was rather reticent and he showed sometimes unexpected reactions that were not easily understood, even if they could be explained from his past. I suppose that he was personally vulnerable and the means he applied to protect himself in his rather isolated position were sometimes difficult to follow. But his main hold and devotion was mathematics, which he practiced with great enthousiam. He had, in my opinion, the conviction that mathematics demanded inexorable devotion and this he applied in the first place to himself. It was a guiding principle for his teaching too. He conceived it as his task to bring a student to skill and exactness of thinking and to inform him on the developments of analysis from the $19^{th}$ century along the bridge of real analysis to a recent state. His students will respectfully agree that he succeeded.

Publications by J. Ridder:

[1] Enkele algemeene limietstellingen met toepassingen op reëele functies, Thesis, Utrecht (1921).
[2] Limieten van reeele en analytische functies, Handel, $18^e$ Nat. en Gen. Congres, p. 90-92 (1921).
[3] Over de gelijkheid van herhaalde en meervoudige limieten, Nw. Archief (2) 14 p. 102-121 (1923).
[4] Over reeksen van analytische functies, I, II, Nw. Archief (2) 14 p. 312-314 (1925) ; (2) 15 p. 1-8 (1925).
[5] Over het differentieeren van een reeks, Nw. Archief (2) 15 p. 9-13 (1925).
[6] Over bovenste en onderste limieten, Nw. Archief (2) 15 p. 34-48 (1925).
[7] Over reeksen van analytische functies, Chr. Huygens 4 p. 1-8 (1925/26).
[8] Over bovenste en onderste limieten, Nw. Archief (2) 15 p. 103-105 (1926).
[9] Reeksen van differentieerbare en analytische functies, Nw. Archief (2) 15 p. 106-110 (1926).
[10] Over de integraaldefinities van Riemann en Lebesgue, Chr. Huygens 4 p. 346-350 (1925/26); 5 p. 205 (1926/27).
[11] Over de stelling van Vtali, Chr. Huygens 5 p. 159-162 (1926/27).
[12] Ueber das Riemannsche Integral. Nw. Archief (2) 15 p.-321-329 (1928).
[13] Ueber stetige additieve Intervallfunktionen in der Ebene und ihre Derivierten I, II, Nw. Archief (2) 16 p. 55-69 (1929); (2) 16 p. 50-59 (1930).
[14] Ueber approximativ stetige Funktionen van zwei (und mehrere) Veränderlichen, Fund. Math. 13 p. 201-209 (1929).
[15] Die Uebertragung eines Satzes von Fubini auf Mengen, meszbar (J), Nw. Archief (2) 16 p. 12-16 (1929).
[16] Ueber Analytizität komplexer Funktionen, Nw. Archief (2) 16 p. 17-19 (1929)
[17] Ueber Grenzfunktionen von Funktionen, meszbar (J) oder integrierbar (R), Nw. Archief (2) 16 p. 20-21 (1929).
[18] Over Riemann integratie bij functies van meer veranderlijken, Handel, $22^e$ Nat. en Gen. Congres, p. 98-99 (1929).
[19] Ueber den Cauchyschen Integralsatz für reelle und komplexe Funktionen, Math. Annalen 102 p. 132-156 (1929).
[20] Ein Satz über iterierte Integrale und seine Anwendung zur Untersuchung der Analytizität von komplexen Funktionen, Math. Zeitschrift 31 p. 141-148 (1929).
[21] Das Riernannsche Integral und das Masz (J), C.R. Soc. Sc. et Lt. Varsovie 22 p. 118-142 (1929).
[22] Ueber additive Intervallfunktionen nebst einer Anwendung bei nicht meszbaren Mengen, Nw. Archief (2) 16 p. 60-75 (1930).
[23] Sur un théorème dans la theorie de la totalisation, Nw. Archief (2) 16 p. 76-79 (1930).
[24] Einige charakteristische Eigenschaften von meszbaren Mengen und Funktionen, Math. Annalen 103 p. 697-709 (1930).
[25] Ueber Derivierte und Ableitungen, C.R. Soc. Sc. et Lt. Varsovie 23 p. 1-11 (1930).
[26] Ueber approximative Ableitungen hei Punkt- und Intervallfunktionen, Fund. Math. 15 p. 324-328 (1930).
[27] Ein Satz über iterierte Integrale (R), Nw. Archief (2) 16 p.-43-45 (1930).
[28] Eine Eigenschaft der fast überall nicht differenzierbaren, stetigen Funktionen, Nw. Archief (2) 17 p. 1-2 (1931).
[29] Einige notwendige und hinreichende Bedingungen für die Summierbarkeit von Funktionen, Nw. Archief (2) 17 p. 28-32 (1931).
[30] Sur deux classes de fonctions introduites par M. Denjoy, Nw. Archief (2) 17 p. 33-50 (1931).
[31] De ontwikkeling van het integraalbegrip, Openbare les. Groningen (1931).
[32] Over primitieve functies. Handel, $23^e$ Nat. en Gen. Congres p. 120-121 (1931).
[33] Ueber den Perronschen Integralbegriff und seine Beziehung zu den R-, L- und D-Integralen, Math. Zeitschrift 34 p. 234-269 (1931).
[34] Quelques théorèmes sur les fonctions primitives, C.R. Ac. Sc. Paris p. 1115-1118 (1930); Nw. Archief (2) 17 p. 169-172 (1932)
[35] Ueber die Bedingung (N) von Lusin und das allgemeine Denjoysche Integral, Math. Zeitschrift 35 p. 51-57 (1932).
[36] Zwei Sätze bei approximativ steti gen Funktionen, Nw. Archief (2) 17 p. 276-280 (1932).
[37] Der Perronsche Integralbegriff, Math. Zeitschrift 37 p. 161-169 (1933).
[38] Ueber approximativ stetige Denjoy-Integrale, Fund. Math. 21 p. 1-10 (1933)
[39] Ueber das allgemeine Denjoysche Integral, Fund. Math. 21 p. 11-19 (1933).
[40] Das Riemann-Stieltjessche Integral, Prac. Mat.-fiz. Varsovie 41 p. 65-95 (1933)
[41] Ueber die gegenseitigen Beziehungen verschiedener approximativ stetiger Denjoy-Perron Integrale, Fund. Math. 22 p. 136-162 (1934).
[42] Ueber die T- und N-Bedingungen und die approximativ steti gen Denjoy-Perron Integrale, Fund. Math. 22 p. 163-179 (1934).
[43] Integration in abstrakten Räumen, Fund. Math. 24 p. 72-117 (1935).
[44] Ueber Perron-Stieltjessche und Denjoy-Stieltjessche Integrationen, Math. Zeitschrift 40 p. 127-160 (1935).
[45] Ueber Denjoy-Perron Integration von Funktionen zweier Variablen, C.R. Soc. Sc. et Lt. Varsovie 28 p.-5-16 (1935).
[46] On the Verblunsky's Generalization of the Denjoy Integrals, Sc. Rep. Tôhoku Imp. Un. (1) 24 p. 344-351 (1935).
[47] Denjoysche und Perronsche Integration, Math. Zeitschrift 41 p. 184-199 (1936).
[48] Reduktion des Doppelintegrals in abstrakten Räumen, Nw. Archief (2) 19 p. 31-39 (1936).
[49] Ueber die gegenseitigen Beziehungen einiger "trigonometrischer" Integrationen, Math. Zeitschrift 42 p. 322-336 (1937).
[50] Cesàro-Perron Integration, C.R. Soc. Sc. et Lt. Varsovie 29 p. 126-152 (1937).
[51] Das spezielle Perron-Stieltjessche Integral, Math. Zeitschrift 43 p. 637-681 (1938).
[52] Das aligemeine Perron-Stieltjessahe Integral, Math. Annalen 116 p. 76-103 (1938).
[53] Totalisation par rapport â une fonction a variation bornée, C.R. Ac. Sc. Paris 209 p. 623-625 (1939).
[54] Nouvelles propriétés de la totalisation par rapport a und fonction a variation bornée generale, C.R. Ac. Sc. Paris 209 p. 670-672 (1939).
[55] Ueber approximative Differentiationen von Reihen, Nw. Archief (2). 20 p. 301-306 (1940).
[56] Harmonische, subharmonische und analytische Funktionen, Ann. Sc. Norm. Pisa (2) 9 p. 277-287 (1940).
[57] Masz- und Integrationstheorie in Strukturen, Acta Math. 73 p. 131-173 (1941).
[58] Ueber k-fache approximative Differentiationen von Reihen, Nw. Archief (2) 21 p. 25-27 (1941).
[59] Ueber den Greenschen Staz in der Ebene, Nw. Archief (2) 21 p. 28-32 (1941)
[60] Ueber das Flächenmasz im dreidimensionalen Raum, Nw. Archief (2) 21 p. 33-56 (1941). Errata in (2) 21 p. 268-269 (1943).
[61] Der Bairesche Satz bei Intervallfunktionen, Nw. Archief (2), p. 57-58 (1941).
[62] Over de additieve functionaalvergelijking en een additieve functionaalcongruentie, Euclides 18 p. 84-92 (1941/42).
[63] Ueber Haibtangenten an Punktmengen, Nw. Archief (2) 21 p. 168-193 (1943).
[64] Denjoy-Stieltjessche und Perron-Stieltjessche Integration in k-dim. Euklidischen Raum, Nw Archief (2) 21 p. 212-241 (1943).
[65] Ueber topologische Eigenschaften von Struckturen, Verhanci. Ak. A'dam Sect. 1, 18, nr. 4 p. 1-43 (1944).
[66] Ueber haimonische Funktionen, Math. (Timoésoara) 21 p. 5-9 (1945); extended version in Nw. Archief (2) 22 p. 162-170 (1946).
[67] Ueber Stieltjessche Integration und ihre Anwendung zur Darstellung linearer Funktionale I, II, Nw. Archief (2) 22 p. 171-188 (1946); (2) 22 p. 220-240 (1948).
[68] Ueber areolär-harmonische Funktionen, Acta Math. 78 p. 205-289 (1946).
[69] Ueber areolär-monogene Funktionen, Nw. Archief (2) 22 p. 200-206 (1946).
[70] Zur Masz- und Integrationstheorie in Strukturen I, II, Proc. Ak. A'dam 49 p. 167-184 (1946)1).
[71] Ueber den Aussagen- und den engeren Prädikatenkalkül I, II, Proc. Ak. A'dam 49 p. 1153-1164 (1946); 50 p. 24-30 (1947).
[72] Einige einfache Anwendungen der areolären Ableitungen und Derivierten, Proc. Ak. A'dam 50 p. 151-156 (1947).
[73] Eine Bemerkung uber das Masz in Strukturen, Proc. Ak. A'dam 50 p. 151-156 (1947).
[74] Einige Anwendungen des Dualitätsprinzips in topologischen Strukturen, Proc. Ak. A'dam 50 p. 341-350 (1947).
[75] Zur Reduktion der n-fache Integrale in abstrakten Räumen, Nw. Archief (2) 22 p. 312-323 (1948)
[76] Logic of propositions, Synthese 6 p. 496-502 (1947/48).
[77] Ueber mehrwertige Aussagenkalküle und mehrwertige engere Prädikatenkalküle I, II, III, Proc. Ak. A'dam 51 p. 670-680, 836-845, 991-995 (1948).
[78] Sur quelques logiques multivalentes, Proc. 10 th Int. Congress Phil. A'dam 1948. Vol. 1 p. 728-730 (1949).
[79] Stieltjessche Integrale, Proc. Ak. A'dam 52 1129-1134 (1949).
[80] Formalistische Betrachtungen über intuitionistische und verwandte logische Systeme I- IV, Proc. Ak. A'dam 53 p. 327-336, 446-455, 787-799, 1375-1389 (1950); V-VII, 54 p. 94-105, 169-177, 226-236 (1951).
[81] Aard en structuur der wiskunde, Oratie Groningen (1950).
[82] Bemerkungen zur vorangehenden Note von H. Schärf, Proc. Ak. A'dam 54 p. 223-225 (1951).
[83] De bepaalde integraal, Simon Stevin 29 p. 1-12 (1952).
[84] Ueber modale Aussagenlogiken und ihren Zusammenhang mit Strukturen I, II, Proc. Ak. A'dam 55 p. 213-223, 459-467 (1952); III -IV , 56 p. 1-11, 99-110, 378-388 (1953); 57 p. 2-8, 117-128, 389-396 (1954)
[85] Die Gentzenschen Schluszverfahren in modalen Aussagenlogiken I-III . Proc. Ak. A'dam A 58 p. 163-169, 170-177, 270-274, 275-276 (1955) .
[86] Die Einführung von beschrânkt- und total-additivem Masz I, II, Proc. Ak. A'dam A 59 p. 143-154, 155-165 (1956).
[87] Integration von Differentialkoeffizienten höhere Ordung, Proc. Ak. A'dam A 60 p. 364-368 (1957).
[88] ''Die Gentzenschen Schluszverfahren in modalen Aussagenlogiken I (co-authors: K. Matsumoto, M. Ohnishi), Proc. Ak. A'dam A 60 p. 481-491; II- III, A 61 p. 16-22, 23-27 (1958)
[89] Das abstrakte Integral I, II, III, Proc. Ak. A'dam A 62 p. 211-220, 221-238, 361-375 (1959).
[90] Grondslagen der wiskunde, didaktische aspecten. Faraday 28 p. 14-21 (1958/59).
[91] Bemerkungen zu der Theorie einen abstrakten Integrals, Proc. Ak. A'dam A 63 p. 124-131 (1960).
[92] Der Fubinische Satz I, II, Proc. Ak. A'dam A 64 p. 38-49, 50-57 (1961).
[93] Hinreichende Bedingungen für analytische, harmonische uz harmonische Funktionen I, II, III (co-author: L.R.J. Westermann), Proc. Ak. A'dam A 64 p. 188-196 261-271, 272-279 (1961)
[94] Integration von Ortsfunktionen II II, Proc. Ak. A 65 p. 1-16, 17-25, 169-180 (1962).
[95] Die Gleichheit von iterierten Riemann-Stieltjesschen Integralen in abstrakten Râumen I, II, Proc. Ak. A'dam A 65 p. 369-384, 489-498 (1962).
[96] Mehrfache iterierte Riemann-Stieltjesschen Integrale in abstrakten Räumen, Proc. Ak. A'dam A 66 p. 154-166 (1963).
[97] Erweiterung eines von Montel und Tolstoy herrührenden Satzes, Proc. Ak. A'dam A 66 p. 488-495 (1963); II, A 69 p.,505-514 (1966).
[98] Hinreichende Bedingung-en für die Analytizität komplexwertiger Funktionen, Proc. Ak. A'dam A 67 p. 1-9 (1964).
[99] ''Ein Einheitliches Verfahren zur Definition von absolut- und bedingt-konvergenten Integralen I- Vc, Proc. Ak. A'dam A 68 p. 1-13, 14-30, 31-39, 165-177, 365-375, 376-387, 705-721, 722-735, 736-745 (1965); VI, A 69 p. 248-257 (1966); Vil-VIII, A 70 p. 1-7, 8-17, 305-316 (1967); IX, A 72 p. 10-17 (1969).
[100] Ein einfacher Eindeutigkeitssatz für analytische Funktionen, Proc. Ak. A'dam A 70 p. 373-374 (1967).
[101] Die ailgerneine Riemann-Integration in topologischen Räumen, A, B, C, D, Proc. Ak. A'dam A 71 p. 12-23, 137-148, 239-252, 363-377 (1968)
[102] Aequivalenzen von Integraldefinitionen im Sinne von Denjoy, von Perron und von Riemann, Proc. Ak. A'dam A 72 p. 201- 212 (1969)
[103] Anwendung von Riemann-Summen in Definitionen von Integralen, Proc. Ak. A'dam A 72 p. 309-326 (1969).
[104] Methoden zur Erweiterung von Maszen in $\sigma$-Somenringen, Proc. Ak. A'dam A 73 p. 361-375 (1970).
[105] Methoden zur Erweiterung von (beschränkt) additiven Maszen in Somenringen, Proc. Ak. A'dam A 73 p. 376-384 (1970).
[106] Methoden zur Erweiterung von beschränkt additiven Produktmaszen, und ihre Anwendung zur Einführung von "Riemann"- wie auch von "Lebesgue" - Integralen I, II, Proc. Ak. A'dam A 74 p. 201-215, 216-221 (1971).
[107] Dualität in den Methoden zur Erweiterung von beschränkt - wie von totaladditiven Maszen I, II, Proc. Ak. A'dam A 74 p. 389-398, 399-410 (1971); III, A 76 p. 393-396 (1973)
[108] Dualität bei Daniell-Stoneschen Methoden zur Einführung und Erweiterung von Integralen nicht-negativer Funktionen in abstrakten Râumen IA -II, Proc. Ak. A'dam A 75 p. 281-294, 295-301, 302-313 (1972)
[109] Duale Konstruktionen von Riemann - wie auch von Lebesgue-Integralen nicht-negativer Funktionen in abstrakten Räumen I, I, Proc. Ak. A'dam A 77 p. 16-28, 29-39 (1974).
[110] "Duale Verhältnisse bei regulären äuszeren und regulären inneren H-Kapazitäten'', Proc. Ak. A'dam A 77 p. 189-194 (1974).
[111] ''Beispiele von Choquet'schen Kapazitäten aus der Theorie der Caratheodory'schen Masze", Proc. Ak. A'dam A 79 p. 50-56 (1976)
[112] Bemerkungen über reguläre und nicht regulare äuszere und innere H-Kapazitäten, Proc. Ak. A'dam A 79 p. 1661-68 (1976).
[113] Spezielle Beispiele für äuszere und innere Kapazitäten (im Sinne von Choquet), Proc. Ak. A'dam A 79 p. 440-451 (1976).
[114] ,,Dualitât bei Suslinschen Mengen; Anwendung in Theorie von äuszeren und inneren Kapazitäten'', Proc. Ak. A'dam A 80 p. 425-431 (1977)


[1] LUIKENS, H., Riemann-Stieltjes integratie bij functies van twee of meer veranderlijken, (1937), (Prof. Dr. J.G. van der Corput formally promotor).
[2] WESTERMANN, L . R .J., Hinreichende Bedingungen für analytische, harmonische und subharmonische Funktionen, (1963).

Publications of J. Ridder: see MathSciNet

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