Celebratio Mathematica — Mac Lane

[1]I. Langmuir, S. MacLane, and K. B. Blodget: “The effect of end losses on the characteristics of filaments of tungsten and other materials,” Phys. Rev. 35 : 5 (1930), pp. 478–503. article

The leads of a tungsten filament in vacuum cool the ends of the filament and so affect the voltage, candle power, electron emission and other properties of the filament. For long filaments, where there is a central portion at a uniform temperature \( T_m \), the temperature distribution near the lead is derived. A method for determining \( T_0 \), the temperature of the lead-filament junction, is given. Tables and formulas are presented which allow ready calculation of the effect of the leads on the properties of any long tungsten filament for which the current and diameter are known. From the more general results it has been found that the decrease in voltage due to the cooling of one lead may be represented by \[ \Delta V=0.154\frac{T_m}{1000}-0.081\frac{T_0}{1000}-2.1\cdot 10^{-8}T_0T_m-0.056 .\] There is an extension of the theory to cover the cases of filaments in gases, filaments of other materials, etc.

Part II of the paper gives figures from which may be found the properties of filaments so short that the first theory does not apply. Some experimental checks of the theory are given.

In general the results and the methods of application have been placed first, and the mathematical derivations have been placed at the end of each part.

For a short filament with leads cooled in liquid air a negative slope of the voltampere characteristic when the central temperature is much smaller than \( T_m \) is observed.

Katharine B. Blodget	Related
Irving Langmuir	Related

[2]S. MacLane: “General properties of algebraic systems,” Bull. Am. Math. Soc. 40 : 1 (1934), pp. 53–54. Unpublished address. JFM 60.0113.11 article

[3]S. MacLane: “A classification of finite distribution lattices,” Bull. Am. Math. Soc. 40 : 9 (1934), pp. 659–660. Unpublished address. JFM 60.0113.20 article

[4]S. MacLane: “Abbreviated proofs in logic calculus,” Bull. Am. Math. Soc. 40 : 1 (1934), pp. 37–38. Abstract for the author’s (German language) 1934 PhD thesis. JFM 60.0026.10 article

[5]S. MacLane: Abgekürzte Beweise im Logikkalkül [Abbreviated proofs in the logical calculus]. Ph.D. thesis, Georg-August-Universität Göttingen (Göttingen), 1934. Advised by P. Bernays and H. Weyl. An abstract was published (in English) in Bull. Am. Math. Soc. 40:1 (1934). phdthesis

Isaak Paul Bernays	Related
Hermann Weyl	Related

[6]S. MacLane: “Some unique separation theorems for graphs,” Bull. Am. Math. Soc. 40 : 9 (1934), pp. 659. Abstract for an article published in Am. J. Math. 57:4 (1935). JFM 60.0535.03 article

[7]S. MacLane: “A systematic theory of mathematical symbolism,” Bull. Am. Math. Soc. 40 : 3 (1934), pp. 214–215. Unpublished address. JFM 60.0026.08 article

[8]S. MacLane: “Problems of structure-theoretic type in mathematical logic,” Bull. Am. Math. Soc. 40 : 1 (1934), pp. 53. Unpublished address. JFM 60.0026.09 article

[9]S. MacLane: “On the representation of finite dual groups,” Bull. Am. Math. Soc. 40 : 3 (1934), pp. 216. Unpublished address. JFM 60.0113.06 article

[10]S. MacLane: “Abstract absolute values and polygonal irreducibility criteria,” Bull. Am. Math. Soc. 41 : 5 (1935), pp. 337. Unpublished address. JFM 61.0127.04 article

[11]S. MacLane: “A logical analysis of mathematical structure,” Monist 45 : 1 (January 1935), pp. 118–130. JFM 61.0050.01 Zbl 0011.00102 article

[12]S. MacLane: “The Schönemann–Eisenstein irreducibility criteria in terms of prime ideals,” Bull. Am. Math. Soc. 41 : 11 (1935), pp. 789. Abstract of article published in Trans. Am. Math. Soc. 43:2 (1938). JFM 61.0180.01 article

[13]S. MacLane: “A construction of prime ideals in terms of absolute values of an algebraic field,” Bull. Am. Math. Soc. 41 : 11 (1935), pp. 789–790. JFM 61.0180.02 article

[14]S. Mac Lane: “The ideal-decomposition of rational primes in terms of absolute values,” Proc. Natl. Acad. Sci. U.S.A. 21 : 12 (December 1935), pp. 663–667. JFM 61.1077.01 Zbl 0013.10301 article

[15]S. MacLane: “Some geometric aspects of abstract linear dependence,” Bull. Am. Math. Soc. 41 : 1 (1935), pp. 21–22. Unpublished address. JFM 61.0997.09 article

[16]S. MacLane: “Book review: Grundlagen der Mathematik, vol. I” [Book review: Foundations of mathematics, vol. I], Bull. Am. Math. Soc. 41 : 3 (1935), pp. 162–165. Book by D. Hilbert and P. Bernays (Springer, 1934). MR 1563043 JFM 65.0091.03 article

Isaak Paul Bernays	Related
David Hilbert	Related

[17]S. MacLane: “Book review: Signification de l’histoire de la pensée scientifique” [Book review: The meaning of the history of scientific thought], Bull. Am. Math. Soc. 41 : 11 (1935), pp. 776. Book by F. Enriques (Hermann, 1934). MR 1563194 article

[18]S. Mac Lane: “Abstract absolute values which give new irreducibility criteria,” Proc. Natl. Acad. Sci. U.S.A. 21 : 7 (July 1935), pp. 472–474. JFM 61.0117.02 Zbl 0012.09903 article

[19]S. MacLane: “Some unique separation theorems for graphs,” Am. J. Math. 57 : 4 (October 1935), pp. 805–820. An abstract for this article was published in Bull. Am. Math. Soc. 40:9 (1934). MR 1507114 JFM 61.1346.01 Zbl 0013.08102 article

[20]S. MacLane: “Book review: La causalité des théories mathématiques” [Book review: Causality of mathematical theories], Bull. Am. Math. Soc. 41 : 11 (1935), pp. 775. Book by G. Bouligand (Hermann, 1934). MR 1563191 article

[21]S. MacLane: “A construction for absolute values in polynomial rings,” Trans. Am. Math. Soc. 40 : 3 (1936), pp. 363–395. MR 1501879 JFM 62.1106.02 Zbl 0015.29202 article

[22]S. MacLane: “A construction for prime ideals as absolute values of an algebraic field,” Duke Math. J. 2 : 3 (1936), pp. 492–510. MR 1545943 JFM 62.0096.02 Zbl 0015.05801 article

[23]S. MacLane: “Note on some equations without affect,” Bull. Am. Math. Soc. 42 : 10 (1936), pp. 731–736. MR 1563414 JFM 62.0072.08 Zbl 0015.15001 article

[24]S. MacLane: “Some interpretations of abstract linear dependence in terms of projective geometry,” Am. J. Math. 58 : 1 (January 1936), pp. 236–240. MR 1507146 Zbl 0013.19503 article

[25]S. Mac Lane: “A structural characterization of planar combinatorial graphs,” Duke Math. J. 3 : 3 (1937), pp. 460–472. MR 1546002 JFM 63.0549.01 Zbl 0017.42702 article

[26]V. W. Adkisson and S. MacLane: “Planar graphs whose homeomorphisms can all be extended for any mapping on the sphere,” Am. J. Math. 59 : 4 (October 1937), pp. 823–832. MR 1507285 JFM 63.1158.05 Zbl 0017.42703 article

[27]S. Mac Lane: “A combinatorial condition for planar graphs,” Fundam. Math. 28 : 1 (1937), pp. 22–32. A Russian translation of this was published in Kibern. Sb., Nov. Ser. 7 (1970). JFM 63.0548.01 Zbl 0015.37501 article

[28]S. MacLane: “The Schönemann–Eisenstein irreducibility criteria in terms of prime ideals,” Trans. Am. Math. Soc. 43 : 2 (March 1938), pp. 226–239. An abstract for this article was published in Bull. Am. Math. Soc. 41:11 (1935). MR 1501940 JFM 64.0951.04 Zbl 0018.28901 article

[29]S. Mac Lane: “The uniqueness of the power series representation of certain fields with valuations,” Ann. Math. (2) 39 : 2 (April 1938), pp. 370–382. MR 1503414 JFM 64.0970.05 Zbl 0019.04901 article

[30]S. Mac Lane: “A lattice formulation for transcendence degrees and \( p \)-bases,” Duke Math. J. 4 : 3 (1938), pp. 455–468. MR 1546067 JFM 64.0072.02 Zbl 0019.39201 article

[31]S. MacLane and V. W. Adkisson: “Fixed points and the extension of the homeomorphisms of a planar graph,” Am. J. Math. 60 : 3 (July 1938), pp. 611–639. MR 1507342 JFM 64.0596.03 Zbl 0019.14201 article

Gehman [1926; 1929; 1936], Adkisson [1930; 1934], and Moore [1926] have found conditions under which a given homeomorphism of a continuous curve on the sphere can be extended to the sphere. In this paper we consider the problem of characterizing those graphs of the sphere for which every homeomorphism of the graph can be extended to the sphere. We restrict ourselves to cyclicly connected graphs and obtain a necessary and sufficient condition that such a graph has some one map on the sphere in which all of its homeomorphisms are extendable. This necessary and sufficient condition depends essentially on the behavior of certain fixed points, for it asserts merely that the fixed points of all homeomorphisms of order 2 must be distributed on the graph as are the fixed points of periodic homeomorphisms of the sphere.

[32]S. MacLane: “Book review: The logical syntax of language,” Bull. Am. Math. Soc. 44 : 3 (1938), pp. 171–176. Book by R. Carnap (Harcourt, Brace and Co., 1937). MR 1563703 article

[33]S. Mac Lane and O. F. G. Schilling: “Infinite number fields with Noether ideal theories,” Am. J. Math. 61 : 3 (July 1939), pp. 771–782. MR 0000019 JFM 65.0093.01 Zbl 0021.20104 article

The abstract development of many different types of ideal theories in general rings, at the hands of W. Krull [1935] and other students of Emmy Noether suggests a natural converse problem: Can one characterize constructively the set of all rings in which a given type of ideal theory holds? In particular, in what rings does the complete Noether (or classical) prime ideal decomposition theory hold? For this second problem we obtain here a partial answer: if the elements of such a ring are integral in every valuation of the ring, then the ring is necessarily the set of algebraic integers from a classical (finite) algebraic number field or from a certain type of infinite algebraic number field. The other classical case of algebraic function fields is also considered, as well as the closely related problem of finding those fields for which every valuation is discrete.

[34]S. Mac Lane: “Steinitz field towers for modular fields,” Trans. Am. Math. Soc. 46 : 1 (July 1939), pp. 23–45. MR 0000017 Zbl 0021.38903 article

[35]S. MacLane and O. F. G. Schilling: “Zero-dimensional branches of rank one on algebraic varieties,” Ann. Math. (2) 40 : 3 (July 1939), pp. 507–520. MR 0000158 JFM 65.1140.02 Zbl 0023.29402 article

[36]S. MacLane: “Subfields and automorphism groups of \( p \)-adic fields,” Ann. Math. (2) 40 : 2 (April 1939), pp. 423–442. MR 1503470 JFM 65.0092.02 Zbl 0021.00501 article

[37]S. MacLane: “The universality of formal power series fields,” Bull. Am. Math. Soc. 45 : 2 (April 1939), pp. 888–890. MR 0000610 JFM 65.0093.02 Zbl 0022.30401 article

[38]S. Mac Lane: “Modular fields, I: Separating transcendence bases,” Duke Math. J. 5 : 2 (1939), pp. 372–393. MR 1546131 JFM 65.0092.01 Zbl 0021.10102 article

[39]S. MacLane: “Some recent advances in algebra,” Am. Math. Mon. 46 : 1 (January 1939), pp. 3–19. See also articles in Revista Mat. Hisp.-Am. (4) 6 (1946) and Studies in modern algebra (1963). MR 1524466 JFM 65.0094.01 Zbl 0021.10003 article

[40]S. Mac Lane: “Symbolic logic,” Am. Math. Mon. 46 : 5 (1939), pp. 289–296. Zbl 0021.09703 article

[41] S. Mac Lane: “Review: Katudi Ono, ‘Logische untersuchungen über die grundlagen der mathematik’ Journal of the Faculty of Science, Imperial University of Tokyo, I, 3:7 (1938) 329–389,” The Journal of Symbolic Logic 4 : 2 (1939), pp. 89–90. article

[42] S. Mac Lane: “Review: Alfred Tarski, ‘Der aussagenkalkul und die topologie’, Fundamenta Mathematicae 31 (1938), 103–134,” The Journal of Symbolic Logic 4 (1939), pp. 26–27. article

[43] S. Mac Lane: “Review of M. H. Stone, ‘Applications of the theory of Boolean rings to general topology’ Trans. Amer. Math. Soc. 41:3 (1937), 375–481,” Journal of Symbolic Logic 4 (1939), pp. 88–89. article

[44]V. W. Adkisson and S. Mac Lane: “Extending maps of plane Peano continua,” Duke Math. J. 6 : 1 (1940), pp. 216–228. MR 0001345 JFM 66.0970.03 Zbl 0026.36302 article

When can a homeomorphism \( T \) of a Peano continuum \( M \) on a sphere \( S \) to a set \( M^{\prime} \) on a sphere \( S^{\prime} \) be extended to a homeomorphism \( T^{\prime} \) which carries the whole sphere \( S \) into \( S^{\prime} \)? Gehman has solved this problem for the extension of \( T \) to a map of a plane containing such a space \( M \) [1926]. His condition can easily be modified to include the case of \( M \) and \( M^{\prime} \) on spheres [Adkisson 1934]. But this condition, which requires that certain “sides” of \( M \) be preserved by \( T \), is extrinsic to the given \( M \), and requires in fact that one establish the existence of certain arcs of \( S \) and \( S^{\prime} \) which may cut through \( M \) and its complement \( S - M \) in a complicated fashion. We establish here another necessary and sufficient condition for the extendibility of \( T \). This condition is intrinsic, and applies only to the triods of \( M \), where by a triod in \( M \) we mean a set \( \tau = [\alpha,\beta,\gamma] \) of three arcs of \( M \) with a common end point, the vertex of the triod, such that any two of these arcs intersect only in this vertex.

[45]S. MacLane and O. F. G. Schilling: “Normal algebraic number fields,” Proc. Natl. Acad. Sci. U.S.A. 26 : 2 (February 1940), pp. 122–126. Brief outline of results presented in an article in Trans. Am. Math. Soc. 50:2 (1941). MR 0001232 JFM 66.0122.01 Zbl 0024.01201 article

In this note we present a brief outline of investigations on the generalization of class field theory to arbitrary normal fields. A complete treatment of the results will be submitted for publication to the Transactions of the American Mathematical Society. In the sequel \( k \) denotes an algebraic number field of finite degree over the field of all rational numbers. Class field theory [Chevalley 1936], [Weil 1936; 1938], deals with the description of abelian extensions over \( k \) in terms of classes of ideals in \( k \). In other words, it is the analog of the theory of multiplicative functions on algebraic Riemann surfaces. It is possible to describe the class groups of the classical theory by means of the totality of all \( p \)-adic extensions \( k_p \) of \( k \); \( p \) denoting the prime divisors of \( k \).

[46]S. Mac Lane: “Modular fields,” Am. Math. Mon. 47 : 5 (May 1940), pp. 259–274. MR 0001969 JFM 66.0115.02 Zbl 0023.19804 article

[47]M. F. Becker and S. MacLane: “The minimum number of generators for inseparable algebraic extensions,” Bull. Am. Math. Soc. 46 : 2 (1940), pp. 182–186. MR 0001218 JFM 66.0115.01 Zbl 0022.30402 article

[48]S. Mac Lane: “Note on the relative structure of \( p \)-adic fields,” Ann. Math. (2) 41 : 4 (October 1940), pp. 751–753. MR 0002860 JFM 66.0115.03 Zbl 0025.10404 article

[49] S. Mac Lane: “Review: J.-L. Destouches, ‘Essai sur la forme générale des théories physiques’, Institutul de Arte Grafice ‘Ardealul’, Cluj, Romania, 1938,” The Journal of Symbolic Logic 5 : 1 (1940), pp. 23. article

[50] S. Mac Lane: “Review: F. Gonseth, ‘Philosophie mathématique,’ Actualités Scientifiques et Industrielles 837, Hermann, Paris, 1939,” The Journal of Symbolic Logic 5 : 2 (1940), pp. 77–78. article

[51]A. H. Clifford and S. MacLane: “Factor-sets of a group in its abstract unit group,” Trans. Am. Math. Soc. 50 (3 1941), pp. 385–406. MR 0005722 JFM 67.0072.04 Zbl 0025.38904 article

[52]G. Birkhoff and S. Mac Lane: A survey of modern algebra. Macmillan (New York), 1941. The fiftieth anniversary of this book was commemorated in an article in Math. Intell. 14:1 (1992). MR 0005093 Zbl 0061.04802 book

[53]S. Mac Lane and V. W. Adkisson: “Extensions of homeomorphisms on the sphere,” pp. 223–236 in Lectures in topology (University of Michigan, Ann Arbor, MI, 24 June–6 July 1940). Edited by R. L. Wilder and W. L. Ayres. University of Michigan Press (Ann Arbor, MI), 1941. MR 0005304 Zbl 0063.03694 incollection

Virgil William Adkisson	Related
Raymond Louis Wilder	Related
William Leake Ayres	Related

[54]S. Eilenberg and S. MacLane: “Infinite cycles and homologies,” Proc. Natl. Acad. Sci. U.S.A. 27 : 11 (November 1941), pp. 535–539. MR 0005351 Zbl 0061.40601 article

[55]S. Mac Lane and O. F. G. Schilling: “Normal algebraic number fields,” Trans. Am. Math. Soc. 50 : 2 (September 1941), pp. 295–384. A brief outline of these results was published in Proc. Natl. Acad. Sci. U.S.A. 26:2 (1940). MR 0005108 JFM 67.0981.03 Zbl 0061.06102 article

[56]F. K. Schmidt and S. MacLane: “The generation of inseparable fields,” Proc. Natl. Acad. Sci. U.S.A. 27 : 12 (December 1941), pp. 583–587. MR 0006153 Zbl 0063.06800 article

[57]S. Eilenberg and S. MacLane: “Group extensions and homology,” Ann. Math. (2) 43 : 4 (October 1942), pp. 757–831. MR 0007108 Zbl 0061.40602 article

[58]S. Eilenberg and S. MacLane: “Natural isomorphisms in group theory,” Proc. Natl. Acad. Sci. U.S.A. 28 : 12 (December 1942), pp. 537–543. See also article in Gaz. Mat. 29:109–112 (1968). MR 0007421 Zbl 0061.09203 article

[59]S. MacLane and O. F. G. Schilling: “A formula for the direct product of crossed product algebras,” Bull. Am. Math. Soc. 48 : 2 (1942), pp. 108–114. MR 0006152 Zbl 0061.06103 article

[60]S. Mac Lane and O. F. G. Schilling: “A general Kummer theory for function fields,” Duke Math. J. 9 : 1 (1942), pp. 125–167. MR 0006169 Zbl 0061.06101 article

[61] S. Mac Lane: “Review: F. Gonseth, ‘Les entretiens de Zurich sur les fondements et la méthode des sciences mathématiques, 6–9 décembre 1938’ S. A. Leemann frères & Cie., Zurich, 1941,” The Journal of Symbolic Logic 7 : 1 (1942), pp. 35–37. article

[62]S. Eilenberg and S. MacLane: “Relations between homology and homotopy groups,” Proc. Natl. Acad. Sci. U.S.A. 29 : 5 (May 1943), pp. 155–158. MR 0007982 Zbl 0061.40701 article

[63]S. MacLane: “A conjecture of Ore on chains in partially ordered sets,” Bull. Am. Math. Soc. 49 : 8 (1943), pp. 567–568. MR 0009018 Zbl 0060.06105 article

[64]S. MacLane and O. F. G. Schilling: “Groups of algebras over an algebraic number field,” Am. J. Math. 65 : 2 (April 1943), pp. 299–308. MR 0007752 Zbl 0060.07905 article

In this note we shall discuss some new aspects of the theory of normal simple algebras over an algebraic number field. The algebras considered are split by a finite normal extension which is the join of two normal subfields. Moreover, the ramifications of the algebras are to be prime to the discriminant of the given normal field. We propose to find conditions insuring that the group of algebras split by the top field be the join of the groups of algebras split by the given component fields. This query leads to certain curious facts concerning the associated Galois groups. One of our results states that the group of algebras for any abelian field is always the join of the group of algebras split by any set of subfields whose join is the top field. In the general case we have succeeded in finding a partial description of the Galois groups in terms of the condition on the associated algebras. The method of proof of our theorems consists in a reduction of problems on algebras to problems on groups by means of the theory of invariants of algebras.

[65]S. Eilenberg and S. MacLane: “General theory of natural equivalences,” Trans. Am. Math. Soc. 58 : 2 (September 1945), pp. 231–294. MR 0013131 Zbl 0061.09204 article

[66]S. Eilenberg and S. MacLane: “Relations between homology and homotopy groups of spaces,” Ann. Math. (2) 46 : 3 (July 1945), pp. 480–509. MR 0013312 Zbl 0061.40702 article

[67]S. Mac Lane: “Some recent advances in algebra,” Revista Mat. Hisp.-Am. (4) 6 (1946), pp. 191–216. See also articles published in Am. Math. Mon. 46:1 (1939) and Studies in modern algebra (1963). MR 0018163 Zbl 0060.06806 article

[68]S. Eilenberg and S. MacLane: “Determination of the second homology and cohomology groups of a space by means of homotopy invariants,” Proc. Natl. Acad. Sci. U.S.A. 32 : 11 (November 1946), pp. 277–280. MR 0019307 Zbl 0061.40707 article

[69]S. MacLane: “Book review: The development of mathematics,” Am. Math. Mon 53 : 7 (August–September 1946), pp. 389–390. Book by E. T. Bell (2nd ed., McGraw-Hill, 1945). MR 1526489 article

[70]S. Eilenberg and S. MacLane: “Cohomology theory in abstract groups, II: Group extensions with a non-Abelian kernel,” Ann. Math. (2) 48 : 2 (April 1947), pp. 326–341. MR 0020996 Zbl 0029.34101 article

The problem of group extensions resides in the construction of the manifold of all groups with a given normal subroup \( K \) and a corresponding given factor group \( Q \). When the given subgroup \( K \) is abelian, this manifold leads to the various two-dimensional cohomology groups of \( Q \) with coefficients in \( K \). When \( K \) is not abelian, and when the automorphisms which the cosets of \( Q \) induce on \( K \) are specified, it is possible [Baer 1934] that no extension of the specified character can exist. In this paper we obtain a necessary and sufficient condition for the existence of such an extension, formulated in terms of a certain three-dimensional cocycle. In the case when such an extension can exist, we show that the manifold of such extensions is equivalent to the manifold of extensions of a certain abelian subgroup. This analysis also leads to an interpretation of the three-dimensional cohomology group \( Q \) over an abelian coefficient group \( G \).

[71]S. Eilenberg and S. MacLane: “Algebraic cohomology groups and loops,” Duke Math. J. 14 : 2 (1947), pp. 435–463. MR 0021939 Zbl 0029.34102 article

[72]S. Eilenberg and S. MacLane: “Cohomology theory in abstract groups, I,” Ann. Math. (2) 48 : 1 (January 1947), pp. 51–78. MR 0019092 Zbl 0029.34001 article

[73]S. MacLane: “A nonassociative method for associative algebras,” Bull. Am. Math. Soc. 54 : 10 (1948), pp. 897–902. MR 0026993 Zbl 0031.34302 article

[74]S. MacLane: “Groups, categories and duality,” Proc. Natl. Acad. Sci. U.S.A. 34 : 6 (June 1948), pp. 263–267. MR 0025464 Zbl 0031.24702 article

[75]S. MacLane: “Symmetry of algebras over a number field,” Bull. Am. Math. Soc. 54 (1948), pp. 328–333. MR 0025442 Zbl 0032.10801 article

[76]S. Eilenberg and S. MacLane: “Cohomology and Galois theory, I: Normality of algebras and Teichmüller’s cocycle,” Trans. Am. Math. Soc. 64 : 1 (July 1948), pp. 1–20. MR 0025443 Zbl 0031.34301 article

[77]S. MacLane: “Cohomology theory in abstract groups, III: Operator homomorphisms of kernels,” Ann. Math. (2) 50 : 3 (July 1949), pp. 736–761. MR 0033287 Zbl 0039.25703 article

[78]S. Eilenberg and S. MacLane: “Homology of spaces with operators, II,” Trans. Am. Math. Soc. 65 : 1 (1949), pp. 49–99. Part I was by Eilenberg alone, Trans. Amer. Math. Soc. 61 (1947), pp. 378–417. MR 0033001 Zbl 0034.11101 article

[79]S. Eilenberg and S. MacLane: “Cohomology theory of Abelian groups and homotopy theory, II,” Proc. Natl. Acad. Sci. U.S.A. 36 : 11 (November 1950), pp. 657–663. MR 0039252 Zbl 0039.19002 article

[80]S. Eilenberg and S. MacLane: “Relations between homology and homotopy groups of spaces, II,” Ann. Math. (2) 51 : 3 (May 1950), pp. 514–533. MR 0035435 Zbl 0036.12602 article

[81]S. Eilenberg and S. MacLane: “Cohomology theory of Abelian groups and homotopy theory, I,” Proc. Nat. Acad. Sci. U.S.A. 36 : 8 (August 1950), pp. 443–447. MR 0038072 Zbl 0037.39504 article

[82]S. MacLane: “Duality for groups,” Bull. Am. Math. Soc. 56 : 6 (1950), pp. 485–516. MR 0049192 Zbl 0045.29905 article

[83]S. MacLane and J. H. C. Whitehead: “On the 3-type of a complex,” Proc. Natl. Acad. Sci. U.S.A. 36 : 1 (January 1950), pp. 41–48. MR 0033519 Zbl 0035.39001 article

[84]S. Eilenberg and S. MacLane: “Cohomology theory of Abelian groups and homotopy theory, III,” Proc. Natl. Acad. Sci. U.S.A. 37 : 5 (May 1951), pp. 307–310. MR 0042706 Zbl 0042.41406 article

For each abelian group \( \Pi \) and for each integer \( n \) we consider the complexes \( K(\Pi,n) \) described in [Eilenberg and Mac Lane 1950a] and the suspension maps \[ S:K(\Pi,n)\to K(\Pi,n+1) \] raising the dimensions by 1. We recall that \( K(\Pi,n) \) gives the singular homology and cohomology theory for any arcwise connected space \( X \) in which only the \( n \)th homotopy group \( \pi_n(X) = \Pi \) is non-trivial (\( n > 1 \)). We show now that the augmented complex \( K(\Pi,n) \) is, after a change in dimension, chain equivalent to the complex \( A^{n-1}(\Pi) \) which gives the “abelian” homology theory of \( \Pi \) in level \( n-1 \). This proves the main conjecture (26) of [Eilenberg and Mac Lane 1950b].

[85]S. Eilenberg and S. MacLane: “Homology theories for multiplicative systems,” Trans. Am. Math. Soc. 71 : 2 (1951), pp. 294–330. MR 0043774 Zbl 0043.25403 article

[86]S. MacLane: “Book review: The theory of algebraic numbers,” Bull. Am. Math. Soc. 57 : 2 (1951), pp. 162–163. Book by by H. Pollard (MAA, 1950). MR 1565291 article

[87]S. MacLane: “Cohomology theory of Abelian groups,” pp. 8–14 in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 3 August–6 September 1950), vol. 2. American Mathematical Society (Providence, RI), 1952. MR 0045115 Zbl 0049.01401 incollection

[88]S. Eilenberg and S. MacLane: “Cohomology groups of Abelian groups and homotopy theory, IV,” Proc. Natl. Acad. Sci. U.S.A. 38 : 4 (April 1952), pp. 325–329. Parts I–III were titled “Cohomology theory of…”. MR 0048025 Zbl 0046.16701 article

[89]S. Eilenberg and S. Mac Lane: “On the groups of \( H(\Pi,n) \), I,” Ann. Math. (2) 58 : 1 (July 1953), pp. 55–106. MR 0056295 Zbl 0050.39304 article

[90]G. Birkhoff and S. Mac Lane: A survey of modern algebra, revised edition. Macmillan (New York), 1953. Revision of the 1941 original. MR 0054551 Zbl 0052.25402 book

[91]S. Eilenberg and S. MacLane: “Acyclic models,” Am. J. Math. 75 : 1 (January 1953), pp. 189–199. MR 0052766 Zbl 0050.17205 article

[92]S. Eilenberg and S. MacLane: “On the groups \( H(\Pi,n) \), III: Operations and obstructions,” Ann. Math. (2) 60 : 3 (November 1954), pp. 513–557. MR 0065163 Zbl 0057.15302 article

[93]S. Mac Lane: Curso de topologia geral [Course in general topology]. Notas de Matemática 11. Instituto de Matemática Pura e Aplicada do Conselho Nacional de Pesquisas (Rio de Janeiro), 1954. In Portuguese. MR 0068819 book

[94]S. MacLane: “The homology products in \( K(\Pi,n) \),” Proc. Am. Math. Soc. 5 : 4 (August 1954), pp. 642–651. MR 0063674 Zbl 0059.16405 article

[95]S. Eilenberg and S. MacLane: “On the groups \( H(\Pi,n) \), II: Methods of computation,” Ann. Math. (2) 60 : 1 (July 1954), pp. 49–139. MR 0065162 Zbl 0055.41704 article

[96]S. MacLane: “Of course and courses,” Am. Math. Mon. 61 : 3 (March 1954), pp. 151–157. MR 1528667 article

[97]S. Mac Lane: Lehrgang der allgemeinen Topologie [Course on general topology]. Notas de Matemática 11. 1954. In German. Zbl 0058.16501 book

[98]S. Eilenberg and S. MacLane: “On the homology theory of Abelian groups,” Can. J. Math. 7 (1955), pp. 43–53. MR 0066375 Zbl 0064.02701 article

[99]S. Mac Lane: “Slide and torsion products for modules,” Rend. Semin. Mat., Univ. Politec. Torino 15 (1955–1956), pp. 281–309. MR 0082488 Zbl 0072.02702 article

[100]S. MacLane: “Book review: Homological algebra,” Bull. Am. Math. Soc. 62 : 6 (1956), pp. 615–624. Book by H. Cartan and S. Eilenberg (Princeton University Press, 1956). MR 1565811 article

Samuel Eilenberg	Related
Henri Cartan	Related

[101]S. Mac Lane: “Homologie des anneaux et des modules” [Homology of rings and modules], pp. 55–80 in Colloque de topologie algébrique [Colloquium on algebraic topology] (Centre Belge de Recherches Mathématiques, Louvain, Belgium, 11–13 June 1956). Publications du Centre Belge de Recherches Mathématiques 11. Georges Thone (Liège), 1957. MR 0094374 Zbl 0084.26703 incollection

[102]S. MacLane: “A proof of the subgroup theorem for free products,” Mathematika 5 : 1 (June 1958), pp. 13–19. MR 0097442 Zbl 0085.01403 article

[103]S. Mac Lane: “Extensions and obstructions for rings,” Ill. J. Math. 2 : 3 (1958), pp. 316–345. MR 0098773 Zbl 0081.03303 article

[104]S. MacLane: “Metric postulates for plane geometry,” Am. Math. Mon. 66 : 7 (August–September 1959), pp. 543–555. MR 0107833 Zbl 0087.35102 article

[105]S. MacLane: “Triple torsion products and multiple Künneth formulas,” Math. Ann. 140 : 1 (1960), pp. 51–64. MR 0114843 Zbl 0093.02101 article

[106]S. MacLane: “Group extensions by primary abelian groups,” Trans. Am. Math. Soc. 95 : 1 (April 1960), pp. 1–16. MR 0131449 Zbl 0091.02203 article

[107]S. Mac Lane: “Locally small categories and the foundations of set theory,” pp. 25–43 in Infinitistic methods (Warsaw, 2–9 September 1959). Pergamon (Oxford), 1961. MR 0191941 Zbl 0116.00902 incollection

[108]S. Mac Lane: “An algebra of additive relations,” Proc. Natl. Acad. Sci. U.S.A. 47 : 7 (July 1961), pp. 1043–1051. MR 0126477 Zbl 0123.01102 article

[109]F. Haimo and S. Mac Lane: “The cohomology theory of a pair of groups,” Ill. J. Math. 5 : 1 (1961), pp. 45–60. MR 0122857 Zbl 0099.25204 article

[110]S. Mac Lane: “The bar construction for differential Hopf algebras,” pp. 58–64 in Colloquium on algebraic topology (Aarhus, Denmark, 1–10 August 1962). Aarhus University, 1962. Zbl 0168.21002 incollection

[111]S. Mac Lane: Homology. Die Grundlehren der mathematischen Wissenschaften 114. Academic Press (New York), 1963. In German. A Russian translation was published in 1966. MR 0156879 Zbl 0133.26502 book

[112]S. Mac Lane: “Natural associativity and commutativity,” Rice Univ. Stud. 49 : 4 (1963), pp. 28–46. MR 0170925 Zbl 0244.18008 article

[113]S. Mac Lane: “Some recent advances in algebra,” pp. 9–34 in Studies in modern algebra. Edited by A. A. Albert. MAA Studies in Mathematics 2. Prentice-Hall (Englewood Cliffs, NJ), 1963. See also articles in Revista Mat. Hisp.-Am. (4) 6 (1946) and Am. Math. Mon. 46:1 (1939). Zbl 0199.04802 incollection

[114]S. Mac Lane: “Some additional advances in algebra,” pp. 35–58 in Studies in modern algebra. Edited by A. A. Albert. MAA Studies in Mathematics 2. Prentice-Hall (Englewood Cliffs, NJ), 1963. Zbl 0199.04803 incollection

[115]G. Birkhoff and S. Mac Lane: A survey of modern algebra, 3rd edition. Macmillan (New York), 1965. Republication of the 1941 book. MR 0177992 book

[116]S. MacLane: “Categorical algebra,” Bull. Am. Math. Soc. 71 : 1 (1965), pp. 40–106. MR 0171826 Zbl 0161.01601 article

[117]G. Birkhoff and S. Mac Lane: A brief survey of modern algebra, 2nd edition. Macmillan (New York), 1965. Condensed version of A survey of modern algebra (1941). MR 0180595 book

[118]S. MacLane: “Book review: Abelian categories: An introduction to the theory of functors,” Am. Math. Mon 72 : 9 (November 1965), pp. 1043–1044. Book by P. Freyd (Harper & Row, 1964). MR 1533506 article

[119]Proceedings of the conference on categorical algebra (La Jolla, CA, 1965). Edited by S. Eilenberg, D. K. Harrison, S. Mac Lane, and H. Röhrl. Springer (Berlin), 1966. MR 0202786 Zbl 0178.29005 book

Helmut Röhrl	Related
Samuel Eilenberg	Related
David Kent Harrison	Related

[120]S. Mac Lane: Homology. Mir (Moscow), 1966. In Russian. Russian translation of German 1963 original. Zbl 0149.26203 book

[121]S. Mac Lane and G. Birkhoff: Algebra. Macmillan (New York), 1967. MR 0214415 Zbl 0153.32401 book

[122]S. Mac Lane: Homology, reprint of 1st edition. Die Grundlehren der Mathematischen Wissenschaften 114. Springer (Berlin), 1967. In German. Republication of original 1963 edition. MR 0349792 book

[123]A. Dold, S. Mac Lane, and U. Oberst: “Projective classes and acyclic models,” pp. 78–91 in Reports of the Midwest category seminar. Lecture Notes in Mathematics 47. Springer (Berlin), 1967. MR 0236245 incollection

Albrecht Dold	Related
Ulrich Oberst	Related

[124]J. Bénabou, R. Davis, A. Dold, J. R. Isbell, S. MacLane, U. Oberst, and J.-E. Roos: Reports of the Midwest category seminar. Lecture Notes in Mathematics 47. Springer (New York), 1967. MR 0220788 Zbl 0165.33001 book

Robert Davis	Related
Albrecht Dold	Related
Ulrich Oberst	Related
John Rolfe Isbell	Related
Jan-Erik Roos	Related
Jean Bénabou	Related

[125]S. MacLane: “The future role of the federal government in mathematics,” Am. Math. Mon. 74 : 1, Part II (January 1967), pp. 92–100. Fiftieth anniversary issue. MR 1534184 article

[126]S. Eilenberg and S. Mac Lane: “Natural isomorphisms in group theory,” Gaz. Mat. (Lisboa) 29 : 109–112 (1968), pp. 77–82. See also article in Proc. Nat. Acad. Sci. U.S.A. 28:12 (1942). MR 0254150 Zbl 0271.18001 article

[127]Reports of the Midwest category seminar, II. Edited by S. MacLane. Lecture Notes in Mathematics 61. Springer (Berlin), 1968. MR 0230781 Zbl 0165.00201 book

[128]S. Mac Lane: “Foundations for categories and sets,” pp. 146–164 in Category theory, homology theory and their applications (Battelle Memorial Institute, Seattle, WA, 24 June–19 July 1968), vol. II. Edited by P. J. Hilton. Lecture Notes in Mathematics 92. Springer (Berlin), 1969. MR 0242919 Zbl 0218.18003 incollection

[129]S. MacLane: “Possible programs for categorists,” pp. 123–131 in Category theory, homology theory and their applications (Battelle Memorial Institute, Seattle, WA, 24 June–19 July 1968), vol. I. Edited by P. J. Hilton. Lecture Notes in Mathematics 86. Springer (Berlin), 1969. MR 0245635 Zbl 0215.38901 incollection

[130]S. Mac Lane: “One universe as a foundation for category theory,” pp. 192–200 in Reports of the Midwest category seminar, III. Edited by S. Mac Lane. Lecture Notes in Mathematics 106. Springer (Berlin), 1969. MR 0249486 Zbl 0211.32202 incollection

[131]Reports of the Midwest category seminar, III. Edited by S. MacLane. Lecture Notes in Mathematics 106. Springer (Berlin), 1969. MR 0249477 Zbl 0184.03504 book

[132]S. Mac Lane: “Coherence and canonical maps,” pp. 231–242 in Convegni del Dicembre del 1968 e del Marzo del 1969 [Conferences of December 1968 and March 1969] (INDAM, Rome, 9–12 December 1968 and 24–28 March 1969). Symposia mathematica 4. Academic Press (London), 1970. MR 0302736 Zbl 0225.18007 incollection

[133]S. Mac Lane and G. Birkhoff: Algèbre [Algebra], tome I: Structures fondamentales. Cahiers Scientifiques 35. Gauthier-Villars (Paris), 1970. With a preface by J. Dieudonné. French translation of part of the 1965 original. MR 0266716 Zbl 0201.01801 book

Garrett Birkhoff	Related
Jean Alexandre Eugène Dieudonné	Related

[134]S. Mac Lane: “Hamiltonian mechanics and geometry,” Am. Math. Mon. 77 : 6 (June–July 1970), pp. 570–586. MR 0266080 Zbl 0212.56703 article

[135]S. Mac Lane: “The Milgram bar construction as a tensor product of functors,” pp. 135–152 in The Steenrod algebra and its applications (Battelle Memorial Institute, Columbus, OH, 30 March–4 April 1970). Edited by F. P. Peterson. Lecture Notes in Mathematics 168. Springer (Berlin), 1970. Conference to celebrate N. E. Steenrod’s sixtieth birthday. MR 0273618 Zbl 0216.45405 incollection

Franklin P. Peterson	Related
N. E. Steenrod	Related

[136]S. Mac Lane: “The influence of M. H. Stone on the origins of category theory,” pp. 228–241 in Functional analysis and related fields (University of Chicago, 20–24 May 1968). Edited by F. E. Browder. 1970. Zbl 0261.18001 incollection

Felix E. Browder	Related
Marshall Harvey Stone	Related	Volume

[137]S. Mac Lane: “A combinatorial condition for planar graphs,” Kibern. Sb., Nov. Ser. 7 (1970), pp. 133–144. Russian translation of article published in Fundam. Math. 28:1 (1937). Zbl 0213.50601 article

[138]Reports of the Midwest category seminar, IV. Edited by S. MacLane. Lecture Notes in Mathematics 137. Springer (Berlin), 1970. MR 0263888 Zbl 0198.00201 book

[139]S. Mac Lane: Categories for the working mathematician. Graduate Texts in Mathematics 5. Springer (New York), 1971. A German translation was published as Kategorien: Begriffssprache und mathematische Theorie (1972). MR 0354798 Zbl 0232.18001 book

[140]G. M. Kelly and S. MacLane: “Erratum: ‘Coherence in closed categories’,” J. Pure Appl. Algebra 1 : 2 (July 1971), pp. 219. Erratum for an article published in J. Pure Appl. Algebra 1:1 (1971). MR 0292906 Zbl 0215.09703 article

[141]S. Mac Lane: “Categorical algebra and set-theoretic foundations,” pp. 231–240 in Axiomatic set theory (UCLA, 10 July–5 August 1967), part I. Edited by D. S. Scott. Proceedings of Symposia in Pure Mathematics 13. American Mathematical Society (Providence, RI), 1971. MR 0282791 Zbl 0219.02044 incollection

[142]S. Mac Lane and G. Birkhoff: Algèbre [Algebra], tome II: Les grands théorèmes. Cahiers Scientifiques 36. Gauthier-Villars (Paris), 1971. In French. French translation of part of the 1965 original. MR 0277320 Zbl 0213.28901 book

[143]G. M. Kelly and S. MacLane: “Coherence in closed categories,” J. Pure Appl. Algebra 1 : 1 (January 1971), pp. 97–140. An erratum for this article was published in J. Pure Appl. Algebra 1:2 (1971). MR 0283045 Zbl 0212.35001 article

[144]Reports of the Midwest category seminar, V. Edited by J. W. Gray and S. MacLane. Lecture Notes in Mathematics 195. Springer (Berlin), 1971. MR 0279147 Zbl 0222.18001 book

[145]G. M. Kelly and S. Mac Lane: “Closed coherence for a natural transformation,” pp. 1–28 in Coherence in categories. Edited by S. Mac Lane. Lecture Notes in Mathematics 281. Springer (Berlin), 1972. MR 0374237 Zbl 0245.18003 incollection

[146]S. Mac Lane: Kategorien: Begriffssprache und mathematische Theorie [Categories: Conceptual language and mathematical theory]. Hochschultext. Springer (Berlin), 1972. In German. German translation of Categories for the working mathematician (1971). MR 0354799 Zbl 0243.18001 book

[147]G. M. Kelly, M. Lalplaza, G. Lewis, and S. Mac Lane: Coherence in categories. Edited by S. Mac Lane. Lecture Notes in Mathematics 281. Springer (Berlin), 1972. MR 0330253 Zbl 0234.00006 book

Gregory Maxwell Kelly	Related
M. Lalplaza	Related
Gene L. Lewis	Related

[148]S. MacLane: “Sets, topoi, and internal logic in categories,” pp. 119–134 in Logic colloquium ’73 (Bristol, July 1973). Edited by H. E. Rose and J. C. Shepherdson. Studies in Logic and the Foundations of Mathematics 80. North-Holland (Amsterdam), 1975. MR 0384545 Zbl 0313.18001 incollection

J. C. Shepherdson	Related
Harvey Ernest Rose	Related

[149]S. Mac Lane: Homology, 3rd corrected edition. Die Grundlehren der Mathematischen Wissenschaften 114. Springer (Berlin), 1975. Corrected edition of 1963 original. Zbl 0328.18009 book

[150]S. Mac Lane: “Topology and logic as a source of algebra,” Bull. Am. Math. Soc. 82 : 1 (January 1976), pp. 1–40. Retiring Presidential address. MR 0414648 Zbl 0324.55001 article

[151]S. Mac Lane: “The work of Samuel Eilenberg in topology,” pp. 133–144 in Algebra, topology, and category theory: A collection of papers in honor of Samuel Eilenberg. Edited by A. Heller and M. Tierney. Academic Press (New York), 1976. MR 0414284 Zbl 0338.55001 incollection

Samuel Eilenberg	Related
Myles Tierney	Related
Alex Heller	Related

[152]G. Birkhoff and S. MacLane: A survey of modern algebra, 4th edition. Macmillan (New York), 1977. Republication of the 1941 book. Zbl 0365.00006 book

[153]S. Mac Lane: “Origins of the cohomology of groups,” Enseign. Math. (2) 24 : 1–2 (1978), pp. 1–29. MR 497280 Zbl 0379.18012 article

This paper is a small essay on the history of contemporary mathematics. It will examine the questions: What were the origins of the theory of the cohomology of groups? What were the essential steps in the development of this theory? What were the effects of this development in related fields of mathematics? These questions seem appropriate to a conference in Zurich, because major steps in the development of this subject took place here at the Eidgenössische Technische Hochschule. These questions may also be typical of questions that can be put about the development of other parts of mathematics in the twentieth century. Here are some of these questions: How does the interchange of ideas between different fields of mathematics come about? Which ideas (or, which research papers) are of essential novelty or originality and which are derivative? Do some ideas arrive before their time, and so are neglected? What are the differences between mathematical developments seen beforehand, or seen after the fact — and is there not a third perspective, that of mathematical ideas as they are in the process of development?

[154]S. Mac Lane: Selected papers. Edited by I. Kaplansky. Springer (New York), 1979. With contributions by Alfred Putnam, Roger Lyndon, Kaplansky, Samuel Eilenberg and Max Kelly. MR 544841 Zbl 0459.01024 book

Max Kelly	Related
Samuel Eilenberg	Related
Irving Kaplansky	Related	Volume
Alfred L. Putnam	Related
Roger C. Lyndon	Related

[155]S. Mac Lane and G. Birkhoff: Algebra, 2nd edition. Macmillan (New York), 1979. Republication of the {1967 original|MR:0214415}. MR 524398 Zbl 0428.00002 book

[156]I. Kaplansky: “The early work of Saunders Mac Lane on valuations and fields,” pp. 519–524 in Saunders Mac Lane: Selected papers. Edited by I. Kaplansky. Springer (New York), 1979. incollection

[157]M. Kelly: “Saunders Mac Lane and category theory,” pp. 527–543 in Saunders Mac Lane: Selected papers. Edited by I. Kaplansky. Springer (New York), 1979. incollection

Irving Kaplansky	Related	Volume
Max Kelly	Related

[158]A. Putnam: “A biographical note,” pp. ix–xii in Saunders Mac Lane: Selected papers. Edited by I. Kaplansky. Springer (New York), 1979. incollection

Alfred L. Putnam	Related
Irving Kaplansky	Related	Volume

[159]R. Lydon: “Saunders Mac Lane as a shaper of mathematics and mathematicians,” pp. 515–518 in Saunders Mac Lane: Selected papers. Edited by I. Kaplansky. Springer (New York), 1979. incollection

R. Lydon	Related
Irving Kaplansky	Related	Volume

[160]S. Mac Lane: “A late return to a thesis in logic,” pp. 63–66 in S. Mac Lane: Selected papers. Edited by I. Kaplansky. Springer (New York), 1979. incollection

[161]S. Mac Lane: “The genesis of mathematical structures, as exemplified in the work of Charles Ehresmann,” pp. 353–365 in Third colloquium on categories, part I (Amiens, France, 8–July 1980), published as Cahiers Topologie Géom. Différentielle 21 : 4 (1980). MR 606381 Zbl 0441.18001 incollection

[162]S. Mac Lane: “Mathematical models of space,” Am. Sci. 68 : 2 (March–April 1980), pp. 184–191. MR 563333 article

[163]S. Mac Lane: “Mathematics,” pp. 104–110 in “Science” Centennial, published as Science 209 : 4452. Issue edited by P. H. Abelson and R. Kulstad. July 1980. MR 576525 Zbl 1225.00002 incollection

Philip H. Abelson	Related
Ruth Kulstad	Related

[164]S. Mac Lane: “History of abstract algebra: Origin, rise, and decline of a movement,” pp. 3–35 in American mathematical heritage: Algebra and applied mathematics (El Paso, TX, November 1975 and Arlington, TX, October 1976). Edited by J. D. Tarwater, J. T. White, C. Hall, and M. E. Moore. Mathematics Series 13. Texas Tech Press (Lubbock, TX), 1981. MR 641700 incollection

John Thomas White	Related
Marion Edward Moore	Related
Carl E. Hall	Related
J. Dalton Tarwater	Related

[165]S. Mac Lane: “Mathematical models: A sketch for the philosophy of mathematics,” Am. Math. Mon. 88 : 7 (August–September 1981), pp. 462–472. MR 628015 Zbl 0468.00019 article

[166]S. Mac Lane: “Award for distinguished service to Professor Ralph P. Boas, Jr.,” Am. Math. Mon. 88 : 2 (February 1981), pp. 85–86. Zbl 0448.01012 article

[167] S. Mac Lane: “Mathematics at the University of Göttingen, 1931–1933,” pp. 65–78 in Emmy Noether: A tribute to her life and work. Marcel Dekker (New York), 1981. incollection

[168]S. Mac Lane: “Why commutative diagrams coincide with equivalent proofs,” pp. 387–401 in Algebraists’ homage: Papers in ring theory and related topics (New Haven, CT, 2–5 June 1981). Edited by D. Amitsur, J. Saltman, and G. B. Seligman. Contemporary Mathematics 13. American Mathematical Society (Providence, RI), 1982. Presented in honor of Nathan Jacobson. MR 685975 Zbl 0504.18004 incollection

David Amitsur	Related
Joseph Saltman	Related
George Benham Seligman	Related
Nathan Jacobson	Related

[169]S. Mac Lane: Proof, truth and confusion, 1982. The 1982 Ryerson Lecture, University of Chicago (1982). misc

[170]S. Mac Lane: “The health of mathematics,” Math. Intell. 5 : 4 (1983), pp. 53–55. A Czech translation was published in Pokroky Mat. Fyz. Astronom. 31:1 (1986). MR 746896 article

[171]S. Mac Lane: “Samuel Eilenberg and topology,” Wiadom. Mat. 25 : 2 (1984), pp. 229–242. In Polish. MR 786118 article

[172]S. Mac Lane: “Spectral complications in cohomology computations,” pp. 11–23 in Contributions to group theory. Edited by K. I. Appel, J. G. Ratcliffe, and P. E. Schupp. Contemporary Mathematics 33. American Mathematical Society (Providence, RI), 1984. MR 767094 Zbl 0553.20031 incollection

John G. Ratcliffe	Related
Kenneth Ira Appel	Related	Volume
Paul Eugene Schupp	Related

[173]S. Mac Lane: “Diagrams, equations and theories in categories,” pp. 143–149 in Universal algebra and its links with logic, algebra, combinatorics and computer science (Darmstadt, Germany, 4–6 February 1983). Edited by P. Burmeister, B. Ganter, C. Herrmann, K. Keimel, E. Poguntke, and R. Wille. Research and Exposition in Mathematics 4. Heldermann (Berlin), 1984. MR 792509 Zbl 0547.18002 incollection

Rudolf Wille	Related
Bernhard Ganter	Related
Christian Herrmann	Related
E. Poguntke	Related
Klaus Keimel	Related
Peter Burmeister	Related

[174]S. Mac Lane: “Topos theory, the confluence of geometry and logic,” pp. 31–42 in Mathematics in the Gulf area (Riyadh, Saudi Arabia, 17–21 October 1982). Edited by Y. Al-Khamees. Arab Bureau of Education for the Gulf States, Science Department (Riyadh), 1984. Zbl 0568.18003 incollection

[175]S. MacLane and R. Paré: “Coherence for bicategories and indexed categories,” J. Pure Appl. Algebra 37 : 1 (1985), pp. 59–80. MR 794793 Zbl 0567.18003 article

[176]S. Makleĭn: “Applications of categorical algebra,” pp. 69–75 in International conference on current problems in algebra and analysis (Moscow–Leningrad, 24–27 September 1984), published as Uspekhi Mat. Nauk 40 : 4(244) (1985). In Russian. English version was published in Russ. Math. Surv. 40:4 (1985). MR 807786 incollection

[177] L. Henkin, D. H. Lehmer, E. Lehmer, E. Scott, J. Kelley, L. Gaal, D. Gale, M. Davis, S. Mac Lane, I. Niven, E. Pitcher, L. Blum, and S. Feferman: “Julia Bowman Robinson: 1919–1985,” Notices Am. Math. Soc. 32 : 6 (November 1985), pp. 739–742. MR 811002 Zbl 0571.01026 article

Derrick Henry Lehmer	Related
Emma Markovna Trotskaia Lehmer	Related
Solomon Feferman	Related
Lisl Gaal	Related
David Gale	Related
Ivan Niven	Related
Elizabeth Leonard Scott	Related
Leon A. Henkin	Related
Arthur Everett Pitcher	Related
Lenore Carol Blum	Related
Martin David Davis	Related
John Leroy Kelley	Related

[178]S. Mac Lane: “Applications of categorical algebra,” Russ. Math. Surv. 40 : 4 (1985), pp. 73–80. Russian version was published in Uspekhi Mat. Nauk 40:4(244) (1985). Zbl 0588.18009 article

[179]S. Mac Lane: “Mathematical logic is neither foundation nor philosophy,” Philos. Math. (2) 1 : 1–2 (1986), pp. 3–14. MR 858787 Zbl 1156.03308 article

[180]S. Mac Lane: “Are we all just specialists?,” Math. Intell. 8 : 4 (1986), pp. 74–75. MR 858303 article

[181]S. Mac Lane: “Correction: ‘Nicholas Bourbaki, collective mathematician: An interview with Claude Chevalley’,” Math. Intell. 8 : 2 (1986), pp. 5. MR 832590 article

[182]S. Mac Lane: “O zdraví matematiky” [The health of mathematics], Pokroky Mat. Fyz. Astronom. 31 : 1 (1986), pp. 44–48. Czech translation of article published in Math. Intell. 5:4 (1983). MR 841415 article

[183]S. Mac Lane: “Criteria for excellence in mathematics,” Bull. Soc. Math. Belg. Sér. A 38 (1986), pp. 301–302. Czech translation published in Pokroky Mat. Fyz. Astronom. 34:5 (1980). MR 885540 Zbl 0628.01019 article

[184]S. Eilenberg and S. Mac Lane: Collected works. Academic Press (Orlando, FL), 1986. MR 856225 Zbl 0613.01018 book

[185]S. Mac Lane: “Topology becomes algebraic with Vietoris and Noether,” J. Pure Appl. Algebra 39 : 3 (1986), pp. 305–307. MR 821895 Zbl 0577.01030 article

Leopold Vietoris	Related
Max Noether	Related

[186]S. Mac Lane: Mathematics, form and function. Springer (New York), 1986. MR 816347 Zbl 0675.00001 book

[187]S. Mac Lane: “Concepts and categories in perspective,” pp. 323–365 in A century of mathematics in America, part I. Edited by P. Duren, R. A. Askey, and U. C. Merzbach. History of Mathematics 1. American Mathematical Society (Providence, RI), 1988. An addendum to this article was published in History of Mathematics, part III (1989). MR 1003181 Zbl 0665.01002 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Richard Allen Askey	Related	Volume

[188]S. Mac Lane: “Group extensions for 45 years,” Math. Intell. 10 : 2 (1988), pp. 29–35. MR 932159 Zbl 0657.01011 article

[189]S. Mac Lane and G. Birkhoff: Algebra, 3rd edition. Chelsea (New York), 1988. Republication of the {1967 original|MR:0214415}. MR 941522 Zbl 0641.12001 book

[190]S. Mac Lane: “Decker’s sharper Künneth formula,” pp. 242–256 in Categorical algebra and its applications (Louvain-La-Neuve, Belgium, 26 July–1 August 1987). Edited by F. Borceux. Lecture Notes in Mathematics 1348. Springer (Berlin), 1988. MR 975974 Zbl 0655.18010 incollection

[191]S. Mac Lane: “Kritéria významnosti v matematice” [Criteria for excellence in mathematics], Pokroky Mat. Fyz. Astronom. 34 : 5 (1989), pp. 281–282. Czech translation of an article published in Bull. Soc. Math. Belg. A 38 (1986). MR 1024522 article

[192]S. Mac Lane and J. Rayner: “Gloria Olive,” New Zealand Math. Soc. Newslett. 45 (April 1989), pp. 22–23. MR 1003939 article

John Rayner	Related
Gloria Olive	Related

[193]S. Mac Lane: “Topology and logic at Princeton,” pp. 217–221 in A century of mathematics in America, part II. Edited by P. Duren, R. A. Askey, and U. C. Merzbach. History of Mathematics 2. American Mathematical Society (Providence, RI), 1989. MR 1003130 Zbl 0682.01016 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Richard Allen Askey	Related	Volume

[194]S. Mac Lane: “Addendum: ‘Concepts and categories in perspective’,” pp. 439–441 in A century of mathematics in America, part III. Edited by P. Duren, R. A. Askey, H. M. Edwards, and U. C. Merzbach. History of Mathematics 3. American Mathematical Society (Providence, RI), 1989. Addendum to an article in History of Mathematics, part I (1988). MR 1025356 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Harold Mortimer Edwards, Jr.	Related
Richard Allen Askey	Related	Volume

[195]S. Mac Lane: “The development of mathematical ideas by collision: The case of categories and topos theory,” pp. 1–9 in Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 22–27 August 1988). Edited by J. Adámek and S. Mac Lane. World Scientific (Teaneck, NJ), 1989. MR 1047887 incollection

[196]S. Mac Lane: “The education of Ph.D.s in mathematics,” pp. 517–523 in A century of mathematics in America, part III. Edited by P. Duren, R. A. Askey, H. M. Edwards, and U. C. Merzbach. History of Mathematics 3. American Mathematical Society (Providence, RI), 1989. MR 1025361 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Harold Mortimer Edwards, Jr.	Related
Richard Allen Askey	Related	Volume

[197]S. Mac Lane: “The applied mathematics group at Columbia in World War II,” pp. 495–515 in A century of mathematics in America, part III. Edited by P. Duren, R. A. Askey, H. M. Edwards, and U. C. Merzbach. History of Mathematics 3. American Mathematial Society (Providence, RI), 1989. MR 1025360 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Harold Mortimer Edwards, Jr.	Related
Richard Allen Askey	Related	Volume

[198]S. Mac Lane: Some major research departments of mathematics, 1989. AMS-MAA Joint Lecture Series (Providence, RI, August 1988). American Mathematical Society (Providence, RI). DVD. MR 1054898 Zbl 0925.01032 misc

[199]Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 22–27 August 1988). Edited by J. Adámek and S. Mac Lane. World Scientific (Teaneck, NJ), 1989. International conference CAT-TOP. MR 1047886 Zbl 0849.18001 book

[200] G. L. Alexanderson and S. Mac Lane: “A conversation with Saunders Mac Lane,” College Math. J. 20 : 1 (1989), pp. 2–25. MR 979840 Zbl 0995.01520 article

[201]S. Mac Lane: “Mathematics at the University of Chicago: A brief history,” pp. 127–154 in A century of mathematics in America, part II. Edited by P. Duren, R. A. Askey, and U. C. Merzbach. History of Mathematics 2. American Mathematical Society (Providence, RI), 1989. MR 1003124 Zbl 0667.01023 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Richard Allen Askey	Related	Volume

[202]S. Mac Lane: “The reasonable effectiveness of mathematical reasoning,” pp. 115–121 in Mathematics and science. Edited by R. E. Mickens. World Scientific (Teaneck, NJ), 1990. MR 1082757 incollection

[203]S. Mac Lane: Fifty years of “Mathematical Reviews”, 1990. AMS-MAA Joint Lecture Series (Louisville, KY, January 1990). American Mathematical Society (Providence, RI). Videocassette. MR 1104659 misc

[204]S. Mac Lane: Algebra as a means of understanding mathematics, 1991. AMS-MAA Joint Lecture Series (Columbus, OH, August 1990). American Mathematical Society (Providence, RI). Videocassette. MR 1153663 Zbl 0925.00031 misc

[205]G. Birkhoff and S. Mac Lane: “‘A survey of modern algebra’: The fiftieth anniversary of its publication,” Math. Intell. 14 : 1 (1992), pp. 26–31. A discussion of the 1941 book. MR 1147344 Zbl 0753.01012 article

[206]S. Mac Lane: “The protean character of mathematics,” pp. 3–13 in The space of mathematics: Philosophical, epistemological, and historical explorations (San Sebastiàn, Spain, September 1990). Edited by J. Echeverria, A. Ibarra, and T. Mormann. Grundlagen der Kommunikation und Kognition. de Gruyter (Berlin), 1992. MR 1214607 Zbl 0848.00006 incollection

Andoni Ibarra	Related
Javier Echeverria	Related
Thomas Mormann	Related

[207]S. Mac Lane and I. Moerdijk: “Reals and forcing with an elementary topos,” pp. 373–385 in Logic from computer science (Berkeley, CA, 13–17 November 1989). Edited by Y. N. Moschovakis. MSRI Publications 21. Springer (New York), 1992. MR 1236017 Zbl 0756.03025 incollection

Yiannis Nicolas Moschovakis	Related
Ieke Moerdijk	Related

[208]S. Mac Lane: “Coherence theorems and conformal field theory,” pp. 321–328 in Category theory 1991 (Montreal, 23–30 June 1991). Edited by R. A. G. Seely. CMS Conference Proceedings 13. American Mathematical Society (Providence, RI), 1992. MR 1192155 Zbl 0789.18006 incollection

[209]S. Mac Lane: “Is Mathias an ontologist?,” pp. 119–122 in Set theory of the continuum (Berkeley, CA, 16–20 October 1989). Edited by H. Judah, W. Just, and W. H. Woodin. MSRI Publications 26. Springer (New York), 1992. MR 1233816 Zbl 0786.03001 incollection

Adrian Richard David Mathias	Related
William Hugh Woodin	Related
Haim Judah	Related
Winfried Just	Related

[210]S. Mac Lane and I. Moerdijk: Sheaves in geometry and logic: A first introduction to topos theory. Universitext. Springer (New York), 1992. A corrected reprint was published in 1994. Zbl 0822.18001 book

[211] M. Atiyah, A. Borel, G. J. Chaitin, D. Friedan, J. Glimm, J. J. Gray, M. W. Hirsch, S. MacLane, B. B. Mandelbrot, D. Ruelle, A. Schwarz, K. Uhlenbeck, R. Thom, E. Witten, and C. Zeeman: “Responses to ‘Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics’, by A. Jaffe and F. Quinn,” Bull. Am. Math. Soc., New Ser. 30 : 2 (April 1994), pp. 178–207. Zbl 0803.01014 ArXiv math/9404229 article

M. F. Atiyah	Related	Volume
Benoit B. Mandelbrot	Related
Edward Witten	Related
David Ruelle	Related
René Thom	Related
Daniel Harry Friedan	Related
E. Christopher Zeeman	Related
Arthur Jaffe	Related
Karen Uhlenbeck	Related	Volume
Armand Borel	Related
James G. Glimm	Related
Jeremy J. Gray	Related
Albert S. Schwarz	Related	Volume
Frank Stringfellow Quinn, III	Related
Gregory John Chaitin	Related
Morris W. Hirsch	Related

@article {key0803.01014z,
	AUTHOR = {Atiyah, Michael and Borel, Armand and
		 Chaitin, G. J. and Friedan, Daniel and
		 Glimm, James and Gray, Jeremy J. and
		 Hirsch, Morris W. and MacLane, Saunders
		 and Mandelbrot, Benoit B. and Ruelle,
		 David and Schwarz, Albert and Uhlenbeck,
		 Karen and Thom, Ren\'e and Witten, Edward
		 and Zeeman, Christopher},
	TITLE = {Responses to ``Theoretical mathematics:
		 {T}oward a cultural synthesis of mathematics
		 and theoretical physics'', by {A}.~{J}affe
		 and {F}.~{Q}uinn},
	JOURNAL = {Bull. Am. Math. Soc., New Ser.},
	FJOURNAL = {Bulletin of the American Mathematical
		 Society. New Series},
	VOLUME = {30},
	NUMBER = {2},
	MONTH = {April},
	YEAR = {1994},
	PAGES = {178--207},
	DOI = {10.1090/S0273-0979-1994-00503-8},
	NOTE = {ArXiv:math/9404229. Zbl:0803.01014.},
	ISSN = {0273-0979},
}

[212]S. Mac Lane and I. Moerdijk: Sheaves in geometry and logic: A first introduction to topos theory, reprinted edition. Universitext. Springer (New York), 1994. MR 1300636 book

[213]S. Mac Lane: “Jobs in the 1930s and the views of George D. Birkhoff,” Math. Intell. 16 : 3 (1994), pp. 9–10. MR 1281748 Zbl 0818.01007 article

[214]S. Mac Lane: “Will Fermat last?,” Math. Intell. 16 : 3 (1994), pp. 65. The development of Fermat’s Last Theorem in verse form. Zbl 0825.00004 article

[215]S. MacLane: Categories for the working mathematician. Graduate Texts in Mathematics 5. Springer, 1994. Corrected republication of 1971 original. Zbl 0705.18001 book

[216]S. Mac Lane: Homology. Classics in Mathematics 114. Springer (Berlin), 1995. Reprint of the 1975 edition. MR 1344215 Zbl 0818.18001 book

[217]S. Mac Lane: “Categories in geometry, algebra and logic,” Math. Japon. 42 : 1 (1995), pp. 169–178. MR 1344646 Zbl 0831.18001 article

[218] S. Mac Lane: “A matter of temperament,” The New York Review of Books 42 : 15 (Oct. 5 1995), pp. 56. Letter, with reply by Freeman J. Dyson. article

[219]S. Mac Lane: “Mathematics at Göttingen under the Nazis,” Notices Am. Math. Soc. 42 : 10 (1995), pp. 1134–1138. MR 1350011 Zbl 1042.01536 article

[220]S. Mac Lane: “The development and prospects for category theory,” pp. 129–136 in The European colloquium of category theory (Tours, France, 25–31 July 1994), published as Appl. Categ. Struct. 4 : 2–3. Issue edited by P. Damphousse and R. Guitart. 1996. MR 1406093 Zbl 0854.18002 incollection

Samuel Eilenberg	Related
René Guitart	Related
Pierre Damphousse	Related

[221]S. Mac Lane: “Samuel Eilenberg’s work in category theory,” pp. 137–138 in The European colloquium of category theory (Tours, France, 25–31 July 1994), published as Appl. Categ. Struct. 4 : 2–3. Issue edited by P. Damphousse and R. Guitart. 1996. MR 1406094 Zbl 0853.18001 incollection

Samuel Eilenberg	Related
René Guitart	Related
Pierre Damphousse	Related

[222]S. Mac Lane and G. Birkhoff: Algèbre. Et solutions développées des exercices: Structures fondamentales, les grands théorèmes, théorie de Galois [Algebra. With detailed solutions of the exercises: Fundamental sturctures, major theorems, Galois theory]. Les Grands Classiques Gauthier-Villars. Éditions Jacques Gabay (Paris), 1996. In French, with a preface by J. Dieudonné. Republication of 1970 French translation and 1971 French translation of parts of the1965 original, with additional material. MR 1452598 book

Garrett Birkhoff	Related
Jean Alexandre Eugène Dieudonné	Related

[223]S. Mac Lane: “Stephen Cole Kleene: A reminiscence,” Ann. Pure Appl. Logic 81 : 1–3 (1996), pp. 3–7. MR 1407050 Zbl 0855.01030 article

[224]S. MacLane and I. Moerdijk: “Topos theory,” pp. 501–528 in Handbook of algebra, vol. 1. Edited by M. Hazewinkel. Elsevier/North-Holland (Amsterdam), 1996. MR 1421810 Zbl 0858.18001 incollection

Michiel Hazewinkel	Related
Ieke Moerdijk	Related

[225]S. Mac Lane: “Structure in mathematics,” pp. 174–183 in Mathematical structuralism, published as Philos. Math. (3) 4 : 2. Issue edited by S. Shapiro. 1996. MR 1397752 Zbl 0905.18001 incollection

The article considers structuralism as a philosophy of mathematics, as based on the commonly accepted explicit mathematical concept of a structure. Such a structure consists of a set with specified functions and relations satisfying specified axioms, which describe the type of the structure. Examples of such structures such as groups and spaces, are described. The viewpoint is now dominant in organizing much of mathematics, but does not cover all mathematics, in particular most applications. It does not explain why certain structures are dominant, not why the same mathematical structure can have so many different and protean realizations. ‘Structure’ is just one part of the full situation, which must somehow connect the ideal structures with their varied examples.

[226]S. MacLane: “Abstract algebra uses homomorphisms,” Am. Math. Mon. 103 : 4 (April 1996), pp. 330–331. With response by U. Leron and Ed Dubinsky. MR 1542826 Zbl 0962.00517 article

Edward Leonard Dubinsky	Related
Uri Leron	Related

[227]S. Mac Lane: “Requiem for the skillful,” Notices Am. Math. Soc. 44 : 2 (1997), pp. 207–208. MR 1426108 Zbl 0934.01019 article

[228]S. Mac Lane: “Van der Waerden’s ‘Modern algebra’,” Notices Am. Math. Soc. 44 : 3 (1997), pp. 321–322. MR 1435205 Zbl 1036.01007 article

[229]S. Mac Lane: “Categorical foundations of the protean character of mathematics,” pp. 117–122 in Philosophy of mathematics today (Budapest, 20–22 May 1993). Edited by E. Agazzi and G. Darvas. Episteme 22. Kluwer Academic (Dordrecht), 1997. MR 1798375 incollection

Evandro Agazzi	Related
György Darvas	Related

[230]S. Mac Lane: “Garrett Birkhoff and the ‘Survey of modern algebra’,” Notices Am. Math. Soc. 44 : 11 (1997), pp. 1438–1439. MR 1488571 Zbl 0908.01019 article

[231]S. Mac Lane: “Despite physicists, proof is essential in mathematics,” pp. 147–154 in Proof and progress in mathematics (Boston, MA, 12 February 1996), published as Synthese 111 : 2. Issue edited by A. Kanamori. 1997. MR 1465267 Zbl 1052.00512 incollection

[232]G. Birkhoff and S. Mac Lane: A survey of modern algebra, 5th edition. AKP Classics. AK Peters (Wellesley, MA), 1997. Republication of the 1941 book. Zbl 0863.00001 book

[233]J. Green, J. LaDuke, S. Mac Lane, and U. C. Merzbach: “Mina Spiegel Rees (1902–1997),” Notices Am. Math. Soc. 45 : 7 (1998), pp. 866–873. MR 1633722 Zbl 0973.01070 article

Uta Caecilia Merzbach	Related
Judith Green	Related
Mina Rees	Related
Alice Jeanne LaDuke	Related

[234]S. Mac Lane: “The Yoneda lemma,” Math. Japon. 47 : 1 (1998), pp. 156. MR 1606360 Zbl 0918.18002 article

[235]H. Bass, H. Cartan, P. Freyd, A. Heller, and S. Mac Lane: “Samuel Eilenberg (1913–1998),” Notices Am. Math. Soc. 45 : 10 (1998), pp. 1344–1352. MR 1646820 Zbl 0908.01023 article

Samuel Eilenberg	Related
Alex Heller	Related
Hyman Bass	Related
Peter J. Freyd	Related
Henri Cartan	Related

[236]S. Mac Lane: Categories for the working mathematician, 2nd edition. Graduate Texts in Mathematics 5. Springer (New York), 1998. Republication of 1971 original. MR 1712872 Zbl 0906.18001 book

[237]S. Mac Lane: “The origins of mathematical abstraction,” pp. 309–311 in Special volume on the occasion of the 60th birthday of Professor Michael Barr (McGill University, Montreal, 29–31 May 1997), published as J. Pure Appl. Algebra 143 : 1–3. Issue edited by F. W. Lawvere and R. A. G. Seely. 1999. MR 1731048 Zbl 0934.18001 incollection

Francis William Lawvere	Related
Robert Andrew George Seely	Related
Michael Barr	Related

[238]S. Mac Lane: “Higher homotopies, pacts, and the bar construction,” pp. 5–7 in Homotopy invariant algebraic structures: A conference in honor of Mike Boardman (Baltimore, MD, 7–10 January 1998). Edited by J.-P. Meyer and J. Morava. Contemporary Mathematics 239. American Mathematical Society (Providence, RI), 1999. MR 1718070 Zbl 0938.55026 incollection

Jean-Pierre Meyer	Related
Jack Johnson Morava	Related
John Michael Boardman	Related

[239]S. Mac Lane: “Contrary statements about mathematics,” Bull. London Math. Soc. 32 : 5 (2000), pp. 527. Commentary on “Strong statements of analysis” by A. R. D. Mathias, which appeared in the same issue of the Bulletin. MR 1767704 Zbl 1025.03053 article

[240]H. Bass, H. Cartan, P. Freyd, A. Heller, and S. MacLane: “Samuel Eilenberg: 1913–1998,” Biographical Memoirs 79 (2000). article

Samuel Eilenberg	Related
Alex Heller	Related
Hyman Bass	Related
Peter J. Freyd	Related
Henri Cartan	Related

[241]S. Mac Lane: “The lambda calculus and adjoint functors,” pp. 181–184 in Logic, meaning and computation: Essays in memory of Alonzo Church. Edited by C. A. Anderson and M. Zelëny. Synthese Library 305. Kluwer Academic (Dordrecht), 2001. MR 2067971 Zbl 1023.03010 incollection

Alonzo Church	Related
Curtis Anthony Anderson	Related
Michael Zelëny	Related

[242]S. Mac Lane: “Samuel Eilenberg and categories,” pp. 127–131 in Category theory 1999 (Coimbra, Portugal, 13–17 July 1999), published as J. Pure Appl. Algebra 168 : 2–3. Issue edited by J. Adamek, P. T. Johnstone, and M. Sobral. 2002. MR 1887153 Zbl 0992.18001 incollection

Maria Manuela Sobral	Related
Jiří Adámek	Related
Samuel Eilenberg	Related
Peter Tennant Johnstone	Related

[243]S. Mac Lane: Saunders Mac Lane: A mathematical autobiography. A K Peters (Wellesley, MA), 2005. With a preface by David Eisenbud. MR 2141000 Zbl 1089.01010 book

[244]C. McLarty: “Saunders Mac Lane (1909–2005): His mathematical life and philosophical works,” Philos. Math. (3) 13 : 3 (2005), pp. 237–251. With selected bibliography emphasizing Mac Lane’s philosophical writings. MR 2192173 Zbl 1094.01010 article

[245]C. McLarty: “Saunders Mac Lane and the universal in mathematics,” Sci. Math. Jpn. 63 : 1 (2006), pp. 25–29. MR 2202158 Zbl 1099.01513 article

[246]C. McLarty: “The last mathematician from Hilbert’s Göttingen: Saunders Mac Lane as philosopher of mathematics,” Br. J. Philos. Sci. 58 : 1 (2007), pp. 77–112. MR 2301283 Zbl 1122.01017 article

While Saunders Mac Lane studied for his D.Phil in Göttingen, he heard David Hilbert’s weekly lectures on philosophy, talked philosophy with Hermann Weyl, and studied it with Moritz Geiger. Their philosophies and Emmy Noether’s algebra all influenced his conception of category theory, which has become the working structure theory of mathematics. His practice has constantly affirmed that a proper large-scale organization for mathematics is the most efficient path to valuable specific results — while he sees that the question of which results are valuable has an ineliminable philosophic aspect. His philosophy relies on the ideas of truth and existence he studied in Göttingen. His career is a case study relating naturalism in philosophy of mathematics to philosophy as it naturally arises in mathematics.

[247]W. Lawvere: “Saunders Mac Lane,” pp. 237–251 in New dictionary of scientific biography, 2nd edition, vol. 5. Edited by N. Koertge. Scribner (New York), 2007. incollection

Francis William Lawvere	Related
Noretta Koertge	Related

Complete Bibliography