Sandy Lindsay was the son of the Rev. Thomas Martin Lindsay (1845–1914) by his marriage to Anna Dunlop (1845–1903).
In 1903 he won the Shaw fellowship in moral philosophy at Edinburgh University, as had his father, the first recipient of this award. He was assistant lecturer in philosophy at the Victoria University of Manchester from 1904–1906, when he was elected a fellow and tutor in philosophy at Balliol College, Oxford. During the First World War he served in France, was mentioned twice in dispatches, and was a Lieutenant-colonel. He was Professor of Moral Philosophy at the University of Glasgow (1922–24) and was Master 1924-49. He was president of the Aristotelian Society from 1924 to 1925. In 1924 he became master of Balliol College and became Vice-Chancellor of the University of Oxford from 1935-38. He worked with Lord Nuffield who donated £1m to fund a new physical chemistry laboratory and a postgraduate college for social studies, Nuffield College, Oxford in 1937.
At Oxford, Lindsay was a leading figure in the adult education movement. On his retirement from Balliol, Lindsay was appointed the first Principal of the University College of North Staffordshire which opened in 1949 and is now Keele University. In 1938, Lindsay stood for Parliament in the Oxford by-election as an 'Independent Progressive' on the single issue of opposition to the Munich Agreement, with support from the Labour and Liberal parties as well as from many Conservatives including the future Prime Ministers Winston Churchill, Harold Macmillan and Edward Heath, but lost to the official Conservative candidate, Quintin Hogg.
Lindsay married Erica Violet Storr (1877 - 28 May 1962), daughter of Francis Storr, in 1907 and they had three sons.