Mathematics is a process of staring hard enough with enough perseverence at at the fog of muddle and confusion to eventually break through to improve… - William Rowan Hamilton

As soon as a thought or word becomes a tool, one can dispense with actually ‘thinking’ it, that is, with going through the logical acts involved in verbal formulation of it. As has been pointed out, often and correctly, the advantage of mathematics—the model of all neo-positivistic thinking—lies in just this ‘intellectual economy.’ Complicated logical operations are carried out without actual performance of the intellectual acts upon which the mathematical and logical symbols are based. … Reason … becomes a fetish, a magic entity that is accepted rather than intellectually experienced.

Halakhic man, well furnished with rules, judgments, and fundamental principles, draws near the world with an a priori relation. His approach begins with an ideal creation and concludes with a real one. To whom may he be compared? To a mathematician who fashions an ideal world and then uses it for the purpose of establishing a relationship between it and the real world. ... The essence of the Halakhah, which was received from God, consists in creating an ideal world and cognizing the relationship between that ideal world and our concrete environment.

There is no way to guarantee in advance what pure mathematics will later find application. We can only let the process of curiosity and abstraction take place, let mathematicians obsessively take results to their logical extremes, leaving relevance far behind, and wait to see which topics turn out to be extremely useful. If not, when the challenges of the future arrive, we won’t have the right piece of seemingly pointless mathematics to hand.