this is a cool video. dr. richard feynman, one of the most important physicists of the 20th century, was a great speaker. as a professor, he communicated a passion for physics that it's difficult to emulate. here he improvises (to an audience of freshmen at cornell university) the difference between mathematics and physics.
i take issue with feynman's presentation of mathematics:
... the mathematicians only are dealing with the structure of the reasoning and they don't care what they're talking of (0.23) ... they don't even need to know what they're talking about...what?
if a mathematician works within a given a system S (already proven), one would generally accept that S is true. mathematicians don't believe that mathematical hypotheses are beyond falsification. i have no idea of what dr. feynman means when he explains:
... if you state the axioms ... if you say such and such are so, and such and such are so, what then? (0.38) then the logic can be carried out without knowing what the words such and such mean... (0:45)what words is he referring to?
first, axioms can be, 1) logical and 2) non-logical.
1) x→y, x, therefore y, (logical, modus ponens).
2) (xy)z=x(yz) = xyz (for any x, y, z), (non-logical, say in arithmetic).
second, mathematicians deal with symbols, not words (unless feynman means words=symbols). now, that doesn't mean they don't have an "idea" of what they're doing. what's an "idea" in mathematics? i don't know exactly (i think that's a deep discussion), but for sure, an idea doesn't have to have a string of words in it. here are some examples: 1- a music phrase, 2- an image (not an idea?), 3- a potential ingredient for my soup (i intend it as flavor), 4- a strategy of reductio for a particular logical problem.
then dr. feynman adds: "... if the statements about the axiom a carefully formulated and complete enough... it's not necessary to known the meaning of these words."
is dr. feynman protesting the truth-preserving qualities of deduction?
all men are immortal
socrates is a man
therefore, socrates is immortal.
granted, the internal structure of deduction is what makes the unsound argument above valid. it isn't a problem of mathematics that deduction is isolated from physical laws. feynman should applaud that deduction is protected from the ebb & flow of reality!
i take issue with this characterization of mathematicians:
... mathematicians prepare abstract reasoning that's ready to be used if you only had a set of axioms about the real world... (1:26)without axiomatics you would not have mathematics. are we not in agreement that math is deductive? is so, where do you expect math to mine from?
it is as if feynman resents mathematics's deductive exceptionality, a sort of independence from the "real world" (it's not true that all math is strictly insulated from reality. math begins as a practical science):
... you have to have a sense of the connection of the words (he definitely means symbols) with the real world (1:51) ... into english??what about quantum mechanics? i don't believe one needs to translate schröedinger's equation into into --never mind english-- any language. math's symbolic language is universal.
at some point feynman realizes that he's bending the discussion too much towards physics (by 3:37):
... and later on always turns out that the poor physicist has to come by, excuse me, when you wanted to tell me about the four dimensions ... (audience laughs).i love feynman.