Coffee Cup Stirling Engine
Of the sciences, maths is considered the one that's most simplified, derived from axioms, but Gödel showed that there are true statements which can't be proved. This finding has affected the grand designs of mathematicians. No longer do they look for a list of axioms and rules that can be used to generate all the theorems. It's made no difference to their other work because there's no way of knowing beforehand whether a proof that's being sought exists or not. Certainly no other fields of science have been affected.
Science seeks generalisations but ones which don't compromise. It seeks all-encompassing rules rather than rules which, though covering more cases, leave behind a slew of exceptions. 'Reality is complex whereas truth is simple' says John Light . But isn't truth an accurate statement of reality and thus at least as complex as its subject matter? The generalisations of science are simpler only because they're formulae, not lists of particulars.
Science is based on maths, and maths developed on Aristotlean logic where statements are either true or false. This excluded middle needn't permeate up through to science, and besides, new multivalue logics have been developed. Nevertheless it's true that science dislikes ambiguity, (which Empson considered a defining characteristic of poetry). At least when science is reductionist it usually states its assumptions.
The methods of science are popularly thought to be appropriate for only certain kinds of problems - truths of the world, not of people. Though 'the part played by new observation and experiment in the process of discovery in science is usually over-estimated' [26, p.28] and most of the interesting results come from 'reconsideration of known phenomena in a new context' [26, p.28], science depends on repeatable observation and prediction. In the main, science recognises these limits, perhaps anticipating that its time will come when new instruments and disciplines appear that can probe the previously hidden. For instance in the 70's interest grew in the study of chaos and complex systems, bridging, if only slightly, the gap between the domains. The use of neural nets in computing may signal progress towards a more human, pattern-matching type of computer.
Even if a science theory has been shown to present a fundamentally wrong model of reality, it can still be useful. Einstein's view of the universe superceded Newton's, but Newton's laws got us to the moon and back. Einstein's gravitation theory can't cope with quantum effects, but theoreticians still depend on it. Ultimate truth is not the only factor determining a theory's lifetime. The theoretical physicist Dirac said that when he had to choose between beauty and truth, he always chose beauty , expecting later experiments to prove him right.
Even in maths there is less certainty than is generally thought. In axiomatic systems (Euclidian Geometry, for example) there are sometimes disputes about which statements to use as axioms. In Number Theory the Axiom of Choice is contentious (thought by some to be not self-evident) but without it many other results couldn't be proved. And perhaps worse still, we know there'll always be true statements which can't be proved.
In science (and especially the arts) not all the past is discarded. 'Both tend to stability by precedents from the past' . An expression like e is an allusion to earlier work, just as much a shorthand as Eliot's burnished throne is.
Perhaps it's time to look in more depth at those who've excelled in both fields to see if they get similar satisfactions out of their dual endevears. William Empson and Valery were once maths students. Holub, Primo Levi, Goethe, da Vinci, Danny Abse and William Carlos Williams all pursued dual careers. It seems to me that the incidence of scientific and artistic talent in the one person is no more than one would expect were the talents independent. There seems little cross-influence except that poet-scientists use their science experiences as subject matter and some of them (Edward Lowbury, for instance) attribute their dislike of obscurity in art to their scientific upbringing.
Natural language has room for various language games. Maths hasn't - games aren't maths any more. Maths is one of the games that can be played in language - and you've got to have a net. Its range of expression is limited and consequently maths requires a longer apprenticeship before creative work can be done. Children can write poems but even undergraduate maths students can't express themselves.
Appreciation of the arts also requires a long apprenticeship. The difference between the cultures is that people looking at a Constable can say 'that's nice' because it's expected of them or because they like the countryside. Science rarely lets people off so lightly - scientists have to learn how appreciate as well as express, they have to learn the language. Poetasters can fake it.
I suspect that people with an impoverished appreciation of the arts (most of us) can only extract from art `truths' that they already know from lived experience. Playing Mozart to bushmen or reading Ashbery to almost anyone won't impress the audience. Art isn't clearly more natural or even richer than science.
Intuition and imagination are highly valued in the sciences, though one can plod along quite merrily without them (but then, one can be a very technically accomplished and 'successful' concert pianist, I'm told, yet have little feel for music).
I.A. Richards thought that 'the imaginative life is its own justification' [28, p.66]. More recently Holden in  says 'Poetry is like Pure Maths, an end in itself' but the Arts can contribute to the Zeitgeist, which in turn affects scientists. On the whole it appears that science has been more useful to the arts than vice versa - it has provided materials (oil paints), artforms (cinema) and subject matter (science fiction). It has changed the world that some think is art's duty to describe.
Attempts have been made to find analogies between arts and sciences. Buchanan thought that 'The symbolic elements of poetry are words, and the corresponding elements of mathematics are ratios' [2, p.18] and goes on to say that 'The mathematician sees and deals with relations, the poet sees and deals with qualities. Functions and adjectives respectively are the symbols through which they see and with which they operate' [2, p.135]. This promising start isn't built on. Elsewhere, the analogies are neither surprising nor interesting:
Methodology - A case has been made (in ) that the processes of art and science correspond. I think the likeness rather tenuous. They split the process into 3 contentious stages
Quantum Theory - In Quantum Theory, probabilities can be calculated but only when an observation is made can any certainty be established. Observation is said to 'collapse the probability function.' This has been used for an analogy to the way that a text is interpreted (dis-ambiguated) by the act of reading . But texts can be re-read!
Relativity - Connections are made between Einstein's Special Relativity and analytic cubism. Awareness of the equal importance of world viewpoints, the impossibility of absolute motion and time perhaps permeated via the Zeitgeist to artists; the link came from no deep mutual understanding.
Gödel - Gödel's findings (see above) have helped soften artists' views on science and has removed an aim of classical science. They have only made maths more obviously like the other sciences. The gap between science and the arts hasn't thereby been reduced.
Geometry - Mondrian is heavily geometric and minimalist. This doesn't make him more appealing to mathematicians. Equally, the 4-colour problem in maths isn't appealing to artists.
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