Veblen attended school in Iowa City before entering the University of Iowa in 1894, receiving his A.B. in 1898. After a year spent as a laboratory assistant at the University of Iowa, Veblen spent a year at Harvard University where he was awarded a second B.A. in 1900, before going to the University of Chicago to undertake research. Archibald writes in :-
He received the major part of his mathematical training at the University of Chicago from that inspiring trio Bolza, Maschke, and Eliakim Moore. Under their direction he laid the basis for the important work he was later to achieve in the fields of foundations of geometry, projective geometry, topology, differential invariants and spinors. His often quoted dissertation under Moore, on a system of axioms of Euclidean geometry, followed the trend of development of Pasch (1882) and Peano (1889, 1894) rather than that of Hilbert (1899) and Pieri (1899).In fact Veblen first impressed Moore when he attended his seminar in 1901. When Moore published On the projective axioms of geometry in the following year he acknowledged Veblen's contribution, writing:-
... queries and remarks of members of this course, in particular Mr O Veblen, have been a source of much stimulus.Veblen's doctoral dissertation, supervised by Moore, was entitled A System of Axioms for Geometry and he was awarded his doctorate from the University of Chicago in 1903. In it Veblen gave an axiom system based on point and order rather than on the traditional notions of point, line and plane. He presented twelve axioms for Euclidean geometry which he proved to be an complete system of axioms and he also proved the independence of the axioms. He spent another two years at Chicago after the award of his doctorate as an associate in mathematics. During this time he effectively supervised the doctoral studies of Robert Moore, officially a student of Eliakim Moore's, who was awarded a doctorate in 1905 for a dissertation entitled Sets of Metrical Hypotheses for Geometry. During this period Veblen worked on putting his own thesis into a form for publication and A system of axioms for geometry appeared as a 41 page paper in the Transactions of the American Mathematical Society in 1904. Already at this time he had begun to undertake research in topology (or analysis situs as it was then called) and he published Theory on plane curves in non-metrical analysis situs in 1905.
Leaving Chicago, Veblen taught mathematics at Princeton University from 1905 to 1932. His initial appointment to Princeton in 1905 was as a preceptor, a new position which Woodrow Wilson, the president of Princeton, introduced in an attempt to raise the level of scholarship at the university. Veblen married Elizabeth Mary Dixon Richardson in 1908. She was the sister of Owen Richardson who was the professor of physics at Princeton, and later went on to receive the Nobel prize for physics in 1928. Veblen and his wife Elizabeth had no children. Two years after his marriage, in 1910, Veblen was promoted to professor of mathematics at Princeton.
During the First World War he served first as a captain, later as a major in the army. He sought a commission in the Ordnance Department in the spring of 1917 and was appointed to the Office of Ballistics Research at the new Aberdeen Proving Ground in Maryland. After basic training during the summer of 1917, he arrived at the Proving Ground on 4 January 1918. Veblen headed the Division of Experimental Ballistics which was the mathematical research facility at the Proving Ground. He began collecting data on the range of a 2.95 inch gun. He then worked on the problem of transferring raw data into tables of results and for this he had a staff of men to undertake the lengthy hand calculations. He employed Joseph Ritt to take charge of these computers. His work led to new ballistics theory. See  for details of the work he undertook during this period.
Veblen was honoured by Princeton in 1926 when they named him Henry B Fine Professor of Mathematics. In the academic year 1928-29 he taught at Oxford as part of an exchange with G H Hardy. In 1929 funds were provided for Fine Hall at Princeton and Veblen provided most of the ideas that went into its design. He wanted the mathematicians in Fine Hall to be able to:-
... group themselves for mutual encouragement and support. [It had to be a place where] the young recruit and the old campaigner [could have] those informal and easy contacts that are so important to each of them.However he also wanted a room reserved for professors since:-
... not always understood by those who try to bring about closer relations between faculty and students [is] that in all forms of social intercourse the provisions for privacy are as important as those for proximity.In 1932 he spent time in Germany and lectured at Göttingen, Berlin and Hamburg. Zund writes that this experience :-
... gave him a first hand glimpse of the approaching turbulence in Germany, and he subsequently worked tirelessly to help place refugees who came to the United States.Leon Warren Cohen, in an interview in 1984, recalled Veblen's attitude towards placing refugee mathematicians:-
It was the Depression. Young American mathematicians were finding it hard to get appointments, and the question of whether to bring in foreign mathematicians to occupy positions which would then not be available to American mathematicians was debated. Veblen took what I would call the broader view. I hesitate to attribute views to Veblen, but the considerations that seem to have actuated him were two: a concern for the welfare of mathematics itself, and a humane concern for certain individuals who had talent. Veblen was a grand man, and the people for whom he made it possible to come to the United States made a great contribution to mathematics. G D Birkhoff opposed him on this. ... [Birkhoff said] "If these distinguished people come and take the positions, the young American mathematicians will become hewers of wood and drawers of water."Veblen helped organise the Institute for Advanced Study in Princeton in 1932, resigning the Henry B Fine Professorship to become the first professor at the Institute in the same year. He had Alexander, Einstein, von Neumann and Weyl, all mathematicians he had chosen, as original Institute members.
Veblen's interest in the foundations of geometry led to his work on the axiom systems of projective geometry. He published Finite projective geometries with W H Bussey in 1906, Collineations in a finite projective geometry (1907), and Non-Desarguesian and non-Pascalian geometries (1908). Together with John Wesley Young he published Projective geometry (1910-18). The first of the two volumes in this book were jointly written by the two authors but the second volume was due the Veblen alone. The introduction to this work justifies the study of foundations:-
Even the limited space devoted in this volume to the foundations may seem a drawback from the pedagogical point of view to some mathematicians. To this we can only reply that, in our opinion, an adequate knowledge of geometry cannot be obtained without attention to its foundations. We believe, moreover, that the abstract treatment is particularly desirable in projective geometry, because it is through the latter that the other geometric disciplines are most readily coordinated. Since it is most natural to derive the geometrical disciplines associated with the names of Euclid, Descartes, Lobachevsky etc. from projective geometry than to derive projective geometry from one of them, it is natural to take the foundations of projective geometry as the foundations of all geometry.We mentioned above that Veblen's first work on topology appeared just before he arrived in Princeton. He went on to establish Princeton as one of the leading centres in the world for topology research. Analysis Situs (1922) provided the first systematic coverage of the basic ideas of topology and contributed to the development of modern topology.
Soon after Einstein's theory of general relativity appeared Veblen turned his attention to differential geometry. This work led to important applications in relativity theory, and much of his work also found application in atomic physics. His work The invariants of quadratic differential forms (1927) is a systematic treatment of Riemann geometry while his work, written jointly with his student Henry Whitehead, The foundations of differential geometry (1933) gives the first definition of a differentiable manifold. In Projective relativity theory (1933) he gave a new treatment of spinors, used to represent electron spin.
Veblen was an active member of the American Mathematical Society, serving the Society as vice-president in 1915 and president in 1923-24. He was the Colloquium Lecturer for the Society in 1916 when he gave a series of lectures on topology. As president of the American Mathematical Society he was the leader of American mathematics and his efforts to increase the funding for research is explained in  where Feffer writes:-
Between the world wars, all the scientific professions in the United States underwent tremendous growth. The wartime experiences of scientific leaders whetted their appetites for the continuation of some kind of concentrated, well-funded research programs. Turning not to government but instead to philanthropy, physicists and chemists worked to parlay their postwar prestige into greater support for unfettered research. Leaders of the American mathematical community also wanted to expand their base of support but found themselves facing unique obstacles. Attempts to raise money to support mathematical research had mixed results. [Most important were] the efforts led by Oswald Veblen in the 1920s to collect funds to support mathematics through Princeton University and the American Mathematical Society in the context of this climate of expansion for the physical sciences.Veblen was elected to the American Philosophical Society and the National Academy of Sciences (United States) and was honoured with memberships of many other societies around the world. For example he was a member of the London Mathematical Society, serving on the council in 1928 when he was replacing Hardy at Oxford. Oxford further honoured him with an honorary D.Sc. in 1929, while in the same year he was honoured by the University of Oslo on the occasion of the centenary celebrations for Abel. He was elected to the Danish Academy of Sciences, the Academy of Sciences (Paris), the Polish Academy of Sciences, the Royal Society of Edinburgh and a number of other national academies in, for example, Ireland, Italy, and Peru. In addition to an honorary degree from Oxford, he received similar honours from the universities of Edinburgh, Glasgow, Hamburg, and Oslo. He was made a Knight of Norway's Royal Order of St Olaf, an honour which had also been bestowed on his father.
As to his personal characteristics he was :-
... An engaging person of great modesty and personal charm.L W Cohen said:-
Veblen had many wonderful characteristics. He was a kind man.Veblen loved the outdoors and it was his initiative which led to Princeton purchasing a woodland site for the Institute for Advanced Study so that members could go for countryside walks. His own home was in eighty acres of woodland which he named Herrontown Wood. He and his wife left Herrontown Wood to Mercer County as a place where:-
... you can get away from cars and just walk and sit.In his last years Veblen became partially blind and he devised a number of aids for blind people, one of which was manufactured and distributed by the American Foundation for the Blind.
Article by: J J O'Connor and E F Robertson