Abstract
In the nineteenth century, the discovery of non-Euclidean geometries by German, Hungarian, and Russian
mathematicians, together with Georg Cantor’s discovery of a non-Euclidean arithmetic of infinite sums, raised
the question: what are the foundations of mathematics? There were three main answers—form, logic, and
intuition—each of which inspired early modern art. David Hilbert conceived of mathematics as a formal
axiomatic system which is an internally consistent, self-contained arrangement of abstract, meaning-free,
replaceable signs. Russian Constructivist artists adopted a formalist aesthetic and made paintings and sculptures
composed of meaning-free colors and forms arranged within an autonomous realm. German logician Gottlob
Frege and his follower, British mathematician Bertrand Russel, declared that mathematics is based in logic.
Logicism developed into British analytic philosophy, which was expressed by the sculptor Henry Moore. The
leading intuitionist mathematician, the Dutchman L.E.J. Brouwer, declared that abstract objects exist only in the
human mind and are known by intuition, an idea that inspired De Stijl painter Piet Mondrian. Of the three
answers, form has had the most lasting impact on mathematics and art, such as the Swiss Concrete artist Karl
Gerstner.
