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Brouwer's contribution to the study of foundations of mathematics is generally cepted, yet the greater part of his work has remained inaccessible because of language and for lack of a bibliography. In this study of the fundamental concepts of Brouwer's intuitionism and philosophy use has been made of all Brouwer1's published papers. Chapter I: Some relevant biographical details are given, as well as a survey of Brouwer's foundational works. A largely forgotten work, Leven. Kunst en Mystiek. has been included as a useful source of information on Brouwer's character, his general views, and his mystical tendencies. A bibliography of all Brouwer's work has been compiled, and is included. Chapter II: An analysis is made of Brouwer's philosophy, which determined his intuitionism as early as 1905.The central place in this philosophy is taken by Brouwer's theory of intuition and mathematics: intuition as the human mind actingindependently of all data of experience, and mathematics as nothing else but this intuitive mental activity. It is shown that this philosophy is the foundation of Brouwer's intuitionism; all his intuitionist theories and practices ultimately stem from his conception of mathematics. Chapter IIII: in particular, Brouwer's criticism of classical mathematics, logicism, formalism, and Poincare's neo-intuitionism is based on his absolute distinction between mathematics and language, i.e. expression of this mental activity in sounds or symbols. Neither language norlogic, the post-factum analysis of this language can contribute anything to mathematics; any device, such as the Principle of the Excluded Middle which claims to produce mathematical results from a purely verbal structure, is suspect. Chapter IV: Brouwer's conception of mathematics places greater emphasis on the active human role; this leads to an entirely new concept of the infinite sequence, of sets, and of the continuum,A survey is given of these fundamental notions of Brouwer's analysis, especially in as far they diverge from classical mathematics. Chapter V summarizes the main conclusions drawn in this work.

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The first International Congress of Mathematicians (ICM) was held in Zurich in 1897, setting the standards for all future ICMs. Whilst giving an overview of the congress itself, this thesis focuses on the Swiss organisers, who were predominantly university professors and secondary school teachers. As this thesis aims to offer some insight into their lives, it includes their biographies, highlighting their individual contributions to the congress. Furthermore, it explains why Zurich was chosen as the first host city and how the committee proceeded with the congress organisation. Two of the main organisers were the Swiss geometers Carl Friedrich Geiser (1843-1934) and Ferdinand Rudio (1856-1929). In addition to the congress, they also made valuable contributions to mathematical education, and in Rudio's case, the history of mathematics. Therefore, this thesis focuses primarily on these two mathematicians. As for Geiser, the relationship to his great-uncle Jakob Steiner is explained in more detail. Furthermore, his contributions to the administration of the Swiss Federal Institute of Technology are summarised. Due to the overarching theme of mathematical education and collaborations in this thesis, Geiser's schoolbook "Einleitung in die synthetische Geometrie" is considered in more detail and Geiser's methods are highlighted. A selection of Rudio's contributions to the history of mathematics is studied as well. His book "Archimedes, Huygens, Lambert, Legendre" is analysed and compared to E W Hobson's treatise "Squaring the Circle". Furthermore, Rudio's papers relating to the commentary of Simplicius on quadratures by Antiphon and Hippocrates are considered, focusing on Rudio's translation of the commentary and on "Die Möndchen des Hippokrates". The thesis concludes with an analysis of Rudio's popular lectures "Leonhard Euler" and "Über den Antheil der mathematischen Wissenschaften an der Kultur der Renaissance", which are prime examples of his approach to the history of mathematics.

In this thesis I present new insights into aspects of Peter Guthrie Tait's life and work, derived principally from largely-unexplored primary source material: Tait's scrapbook, the Tait–Maxwell school-book and Tait's pocket notebook. By way of associated historical insights, I also come to discuss the innovative and far-reaching mathematics of the elusive Frenchman, C.-V. Mourey. P. G. Tait (1831–1901) F.R.S.E., Professor of Mathematics at the Queen's College, Belfast (1854–1860) and of Natural Philosophy at the University of Edinburgh (1860–1901), was one of the leading physicists and mathematicians in Europe in the nineteenth century. His expertise encompassed the breadth of physical science and mathematics. However, since the nineteenth century he has been unfortunately overlooked—overshadowed, perhaps, by the brilliance of his personal friends, James Clerk Maxwell (1831–1879), Sir William Rowan Hamilton (1805–1865) and William Thomson (1824–1907), later Lord Kelvin. Here I present the results of extensive research into the Tait family history. I explore the spiritual aspect of Tait's life in connection with The Unseen Universe (1875) which Tait co-authored with Balfour Stewart (1828–1887). I also reveal Tait's surprising involvement in statistics and give an account of his introduction to complex numbers, as a schoolboy at the Edinburgh Academy. A highlight of the thesis is a re-evaluation of C.-V. Mourey's 1828 work, La Vraie Théorie des quantités négatives et des quantités prétendues imaginaires, which I consider from the perspective of algebraic reform. The thesis also contains: (i) a transcription of an unpublished paper by Hamilton on the fundamental theorem of algebra which was inspired by Mourey and (ii) new biographical information on Mourey.