David Hilbert was one of the acknowledged leaders of mathematics in the early 20th century. He’s often associated with the movement to formalise all mathematics on strictly logical principles, so it’s interesting to find him taking a broader perspective:

In mathematics… we find two tendencies present. On the one hand, the tendency towards abstraction seeks to crystallise the logical relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency towards intuitive understanding fosters a more immediate grasp of the objects one studies, a live rapport with them, so to speak, which stresses the concrete meaning of their relations.

*Preface to* Geometry and the Imagination *(1952)*

Learning to reconcile these formal and intuitive ways of dealing with mathematical ideas is perhaps the single most important — and least teachable — element of a mathematical education. Another, arguably, is modesty, as a second statement attributed to Hilbert doesn’t quite illustrate:

Physics is much too hard for physicists.

(DP)

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