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Its enduring value was simply a deeper understanding of the central concepts of mathematics and their basic laws and interrelationships. Their total translatability into just elementary logic and a simple familiar two-place predicate, membership, is of itself a philosophical sensation.
--
W.V. Quine (on the Principia Mathematica) in From Stimulus to Science (1995).

 

Bertrand Russell

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So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: starting with premisses which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary.

 

Bertrand Russell

 

The principles of logic and mathematics are true simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying. In other words, the truths of logic and mathematics are analytic propositions or tautologies.

 

Alfred Jules Ayer

 

The sphere of mathematics is here extended, in accordance with the derivation of its name, to all demonstrative research, so as to include all knowledge strictly capable of dogmatic teaching. Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims ; and neither law can rule nor theory explain without the sanction of mathematics. It deduces from a law all its consequences, and develops them into the suitable form for comparison with observation, and thereby measures the strength of the argument from observation in favor of a proposed law or of a proposed form of application of a law.
Mathematics, under this definition, belongs to every enquiry, moral as well as physical. Even the rules of logic, by which it is rigidly bound, could not be deduced without its aid. The laws of argument admit of simple statement, but they must be curiously transposed before they can be applied to the living speech and verified by,observation.

 

Benjamin Peirce

 

Well, we’re getting a little philosophical and serious, ok? Let’s go back to what we’re doing. One day we look at a map and this capital is K-Y-Z-Y-L and we decided it would be fun to go there because it’s so obscure and peculiar. It’s a game. It’s not serious; it does not involve some deep philosophical point of view about authorities or anything. It’s just fun of having an adventure to try to go to a land that we’ve never heard of, that we knew was an independent country once, no longer an independent country, find out what it’s like and discover as we went along that nobody went there for a long time, and it’s isolated. Made it more interesting. But, you know, many explorers liked to go to places that are unusual. And, it’s only for the fun of it. I don’t go for this philosophical interpretation of our deeper understanding of what we’re doing. We haven’t any deep understanding of what we’re doing. If we tried to understand what we’re doing, we’d go nutty.

 

Richard Feynman

 

Liberate yourself from concepts and see the truth with your own eyes. — It exists HERE and NOW; it requires only one thing to see it: openness, freedom — the freedom to be open and not tethered by any ideas, concepts, etc. ... When our mind is tranquil, there will be an occasional pause to its feverish activities, there will be a let-go, and it is only then in the interval between two thoughts that a flash of UNDERSTANDING — understanding, which is not thought — can take place.

 

Bruce Lee