The Curry paradox, „If this statement is true, then Y.“, can be used to ‚prove‘ any statement Y. It exhibits the antinomies that arise with self-referential statements. Compared to Russell’s paradox, this paradox does not need set theory.
The Banach-Tarski paradox – Part 2
The Banach-Tarski paradox
Kurt Gödel (1906-1978)
English:
But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.
Kurt Gödel
Bertrand Russell (1872-1970)
English:
Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
Bertrand Russell (Mysticism and Logic, 1919)
Aristotle (384-322 BC)
English:
Those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or definitions, it is not true that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.
Aristotle (Metaphysics, Book XIII, 1078a.33)
Faces Of Abstraction 