Views from Outside the Art Box

Feinman:

You will have to brace yourselves for this – not because it is difficult to understand, but because it is absolutely ridiculous: All we do is draw little arrows on a piece of paper – that's all!

Richard Feynman

QED: The Strange Theory of Light and Matter

Weyl:

All musicians agree that underlying the emotional element of music is a strong formal element.  It may be that it is capable of some such mathematical treatment as has proved successful for the art of ornaments.  If so, we have probably not yet discovered the appropriate mathematical tools. .... Andreas Speiser ... has taken a special interest in the group-theoretic aspects of ornaments [and] tried to apply combinatorial principles of a mathematical nature ... to the formal problems of music [in his Theorie der Gruppen von endlicher Ordnung [1924], chapter entitled "Die mathematische Denkweise"]

Hermann Weyl

Symmetry (p.52)

If nature were all lawfulness then every phenomenon would share the full symmetry of the universal laws of nature .... The mere fact that this is not so proves that contingency is an essential feature of the world.

Hermann Weyl

Symmetry (p.26)

Hardy:

[T]here is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.  Exposition, criticism, appreciation, is work for second-rate minds. (p.61)

What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men. (p.76)

We must guard against the fallacy common among apologists of science, the fallacy of supposing that the men whose work most benefits humanity are thinking much of that while they do it, that physiologists, for example, have particularly noble souls.  A physiologist may indeed be glad to remember that his work will benefit mankind, but the motives which provide the force and the inspiration for it are indistinguishable from those of a classical scholar or a mathematician. (p.?)

 [I]f a mathematician, or a chemist, or even a physiologist, were to tell me that the driving force in his work had been the desire to benefit humanity, then I should not believe him (nor should I think the better of him if I did). (p.79)

A mathematician, like a painter or a poet, is a maker of patterns.  If his patterns are more permanent than theirs, it is because they are made with ideas. (p.84)

[I have to quibble with him here & in the following quote.  The patterns of the poet and painter are made up of ideas no less than those of the mathematician.]

The mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way.  Beauty is the first test: there is no place in the world for ugly mathematics. (p.85)

The best mathematics is serious as well as beautiful – 'important' if you like, but the word is very ambiguous, and 'serious' expresses what I mean much better. (p.89)

[V]ery little of mathematics is useful practically, and ... that little is comparatively dull.  The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects.  We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas.  Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences. (p.89) 

There are two things at any rate which seem essential [in a 'significant' idea], a certain generality and a certain depth .... (p.103)

G.H. Hardy

A Mathematician's Apology

*

Doing vs. using.  It is not necessary to "do" mathematics in order to "use" it (this is different than the usual distinction of abstract vs. applied).  And the same in the sciences – physics, medicine, anthropology, genetics, chemistry.  What's not often evident is how widely this distinction can apply.  In music too often the performer is only "using" music, not "doing" it.  That's why too much musical practice is like acrobatics: there's no doubt the performance is masterful, but someone else inventd the tricks.

"Doing" need not involve creation, but at the very least it includes a drive to understanding the creation.  In musical performance this involves the need to impart this drive to someone called "the audience."

*

Whitehead:

It is the large generalization, limited by a happy particularity, which is the fruitful conception.

Alfred North Whithead

Science and the Modern World

(p.109) 

Mitchell:

If the earth had waited for a precedent, it never would have turned on its axis.

Maria Mitchell

(astronomer, 1818-89) 

[source?]

Einstein:

Feeling and longing are the motive force behind all human endeavor and human creation, in however exalted a guise the latter may present themselves to us. [source?]

... [searching for] the secrets of the Old One. [source?]

Goodwin:

There is no truth beyond magic .... One, when you've discovered the truth ... it does have the most extraordinary magical quality about it.  It's the payoff, to recognize the deep order ..., you feel you are in touch with something fundamental.  But there's also a poetic sense in it: reality is strange. Many people think reality is prosaic. I don't. We don't explain things away .... We get closer to the mystery.

Brian Goodwin (theoretical biologist)

Quoted in

Roger Lewin

Complexity: life at the edge of chaos

(Macmillan, 1992)

Morse:

Mathematics are the result of mysterious powers which no one understands, and in which the unconscious recognition of beauty must play an important part.  Out of an infinity of designs a mathematician chooses one pattern for beauty's sake and pulls it down to earth.

Marston Morse (mathematician)

Quoted by Stravinsky

[source: Conversations?]

Poincaré

Mathematicians do not deal in objects, but it relations between objects; thus, they are free to replace some objects by others so long as the relations remain unchanged. [source?]

Spengler

The sense of form of the sculptor, the painter, the composer, is essentially mathematical in its nature. [source?]

Hanson:

A theory is a cluster of conclusions in search of a premiss. (p.90)

What is it to supply a theory? It is to offer an intelligible, systematic, conceptual pattern for the observed data.  The value of this pattern lies in its capacity to unite phenomena which, without the theory, are either surprising, anomalous, or wholly unnoticed. (p.121)

The basic concept of microphysics is interaction. (p.122)

Gold is rarely discovered by one who has not got the lay of the land. (p.19)

Norwood Russell Hanson

Patterns of Discovery

Cantor:

To ask the right question is harder than to answer it. [George Cantor, source?] 

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Plato

I do not, however, think the attempt to tell mankind of these matters a good thing, except in the case of some few who are capable of discovering the truth for themselves with a little guidance.  In the case of the rest to do so would excite in some an unjustified contempt in a thoroughly offensive fashion, in others certain lofty and vain hopes, as if they had acquired some awesome lore. (Epistle VII, 341.e)