Boutroux is chiefly known for his work in the history and philosophy of mathematics.

He was born in Paris on 6 December 1880 into a well connected family of the French intelligentsia.

A cousin of Aline, Raymond Poincaré was to be President of France.

He occupied the mathematics chair at Princeton University from 1913 until 1914.

He was an Invited Speaker of the ICM in 1904 at Heidelberg, in 1908 at Rome, and in 1920 at Strasbourg.