Proceedings of the American Academy Workshop on the 

      Evolution of Modern Mathematics held at the American Academy of 

      Arts and Sciences in Boston, Massachusetts, August 7--9, 1974. 

      Edited by Garrett Birkhoff with the assistance of Sue Ann 

      Garwood ......................................................... 425--615



Forward 

      John Voss ....................................................... 425--427



Preface 

      Garrett Birkhoff and Sue Ann Garwood ............................ 429--430



Participants in the Workshop .......................................... 431--432



Opening Remarks for the Conference 

      Garrett Birkhoff and I. B. Cohen ................................ 433--436



Part A: Historiography            



      Introduction 

           Garrett Birkhoff ........................................... 437--438



      The mathematician, the historian and the history of mathematics 

           J. V. Grabiner ............................................. 439--447



      What is good history and who should do it? 

           Kenneth O. May ............................................. 449--455



      Progress in mathematics 

           Elaine Koppelman ........................................... 457--463



      The role of oral communication 

           Henry Tropp ................................................ 465--468



      Ten ``laws'' concerning conceptual change in mathematics 

           Michael J. Crowe ........................................... 469--470



      A role for the history of mathematics 

           Phillip S. Jones ........................................... 471--472       



Part B: History of Foundations            



      Introduction 

           Garrett Birkhoff ........................................... 473--474



      Preliminary notes on the historical significance of quantification 

           and of the axioms of choice in the development of 

           mathematical analysis 

           I. Grattan-Guinness ........................................ 475--488



       Russell's logical progress: some new light from manuscript sources 

           I. Grattan-Guinness ........................................ 489--495



       The cradle of modern topology, according to Brouwer's inedita 

           Hans Freudenthall .......................................... 495--502



Part C: Foundations of Mathematics            



       Introduction 

           Garrett Birkhoff ........................................... 503--505



       The crisis in contemporary mathematics 

           Errett Bishop .............................................. 507--517



       Commentary on Bishop's talk 

           Nancy Kopell and Gabriel Stolzenberg ....................... 519--521



       Remarks against foundational activity 

           Gerald Sacks ............................................... 523--528



       What is mathematical truth? 

           Hilary Putnam .............................................. 529--533



Part D: Algebra            



       Introduction 

           Garrett Birkhoff ........................................... 535--536



       Introductory remarks on algebra, topology and analysis 

           Jean Dieudonné ............................................. 537--548



       Some historical remarks on number theory 

           Alan Baker ................................................. 549--553



       Some remarks on harmonic analysis 

           George Mackey .............................................. 555--562



       Mathematical progress without fusion 

           Thomas Hawkins ............................................. 563--566



       High-school algebra in algebraic geometry 

           Shreeram S. Abhyankar ...................................... 567--572



Part E: Analysis            



       Introduction 

           Garrett Birkhoff ........................................... 573--574



       Opening remarks 

           Morris Kline .................................................... 575



       The relation of functional analysis to concrete analysis in 20th 

       century mathematics 

           Felix E. Browder ........................................... 577--590



       The role of Fourier series in the development of analysis 

           Antoni Zygmund ............................................. 591--594



       A central object in mathematics: the Brownian motion 

           J. P. Kahane ............................................... 595--600



       How mathematics progresses 

           Joaquin B. Diaz ............................................ 601--602



Appendices            



        Additional discussion questions 

            Sue Ann Garwood ........................................... 603--604



        Bibliographies 

            Garrett Birkhoff and Sue Ann Garwood ...................... 605--615



INDEX OF AUTHORS OF ARTICLES AND REVIEWS 

        IN VOLUME II........................................................ 619



INDEX OF AUTHORS OF PUBLICATIONS REVIEWED 

        OR ABSTRACTED IN VOLUME II..................................... 620--624                              

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