1) Philosophy (i.e. Language). A clear thinking genius - Ludwig Wittgenstein of Austria - finally resolved the most intractable problems of philosophy by proving that they couldn't be solved, or even expressed.
Philosophers had tied themselves in knots trying to put into words the things they felt to be important, but, no matter which languages they used, they couldn't do it.
Ludwig realized that, as human languages are constructs, they cannot have an `essence' of their own. Therefore all `essential' truths are inexpressible in any human language.
Well, if any of us had thought about it, that now seems fairly obvious - here's an everyday example:
An author's `great thought' has to be cramped and mangled merely to get it into words. Then, when those words are heard or read by another - with a uniquely different brain, and a different personal background, maybe even a different cultural history - the received impression, the end result, must differ from the original thought - which never quite made it into words anyway.
[Peter Watts has nicely doubly-extremed the analogy in "Blindsight" - quote - "Like trying to describe dreams with smoke signals" ]
Also - the same applies to "logic" which is only manipulation of a system of human-constructed symbols; the system suffers from same lack of `essential' truths. It's therefore incomplete - incapable of final proofs.
2) Science (i.e. Mathematics). Folk like Turing and Russell (an early mentor of Wittgenstein's) had tried to analyse the failure of attempts to systematize Mathematics - once thought to be total `logic' and therefore `universal'.
1931, the Czech-born mathematician Kurt Gödel eventually proved that Mathematics is `incomplete' and must always remain so.
That is, Mathematics is a human construct attempting to logically describe the Universe but its terms are merely imposed on parts of reality, they cannot describe - in a provable way - reality itself.
As he said, in Gödel's Paradox, there will be mathematical statements (about physical reality) which may or may not be true, but which are unprovable by any mathematical means.
Well, that was staring us in the face, but `science', and mathematicians in particular, wanted to believe that they had a handle on `reality'.
Looking at science more truthfully, we can shake a charm bracelet and watch one charm swing on its chain - that's a pendulum, which science can accurately describe mathematically; but when you hold the bracelet by one charm and swing the whole thing - that's at least two pendulums, one hanging from another, which science hasn't got a hope of describing accurately.
Or, look at the sun and the moon. The Moon orbits the Earth, but science can describe that only by ignoring the Sun; similarly science can only describe Earth's orbit around the Sun by ignoring the Moon.
So a mere `three body problem' has no formulaic solution; it has to be done by successive approximations.
Because scientists are getting `pretend' descriptions of just a simple part of the Universe, when a human scientist (or anybody else) says anything about the real, physical universe, it will be only partly true; wide, complex systems are `indescribable', i.e. not able to be truthfully expressed at all.
Like Einstein said:
"When mathematics tries to talk about reality - it's inaccurate, when mathematics is accurate - it's not talking about reality"
"Not everything that can be counted counts, and not everything that counts can be counted."
Another example is the armchair experts' dictum on "rogue waves". For years they'd calculated and decreed that a giant wave would arise only once per thousand years or so.
Now however, the reports of actual sailors are being confirmed by latest satellite recordings of wave-heights. Several giant waves occur, around the world, every week or so! See Nat'l Geographic report - [text version]
What went wrong? First - the experts hadn't bothered to gather _all_ available facts. Second - they'd chosen the wrong mathematics.
3) Seems no-ones's addressed the same problem in Music.
Received opinions about music are mostly from posers and music snobs - no `provable' statements.
As before, let's take a look at reality:
An emotional concept of a composer - if s/he has one - is changed merely by attempting to write it down in musical notation: it's limited by the `language'.
Then the artistes & their conductor / producer put their gloss on it. Finally the piece will be interpreted by each listener, almost wholly dependent on their mood, culture and background, and so uniquely for each individual.
I.e. personally quite like and am even moved by a wide range, from blue-grass to sentimental folk-song to some of Mahler or even Schubert - both of which are also `sentimental'. However, have come to realize that there can be no essential `truth' in music, and even its `beauty' is internal and unique to each listener.
Note: those `music snobs' are actually quite sad people; spending time learning the technicalities & `buzz-words' of music means losing its emotional impact - the composer's prime intention. So those music snobs have lost the real pleasure of music - a clear case of retributive kharma.
In fact much music seems to become associated with the happy or sad circumstances in which it was first heard, as Noel Coward probably meant when he referred to "the power of cheap music". Then the composer's intention (if not technical ability) is irrelevant.
E.g. - a young Chinese friend from Hong Kong told me, while listening to some music, that she has a strong dislike for the Radetzky March (a cavalry march set to the pace of trotting horses) because her gambling-mad father would drag the whole family to the racecourse each weekend - and the Radetzky March was always played at the opening.
Update Oct. 2013 - Latest research shows - as seems to be personally true - that musical tastes tend to `mellow' as you grow into maturity - although some rebels might disagree.
He wrote it in mathematical symbols but it can be said as a simple English sentence:-
"This statement cannot be proved to be true by an all-embracing system of logic."
- if the statement is true, the logical system cannot be all-embracing;
- if the statement is false, an all-embracing logical system could prove it to be true;
which are both paradoxical.
Reason why Kurt Gödel investigated mathematics as a logical system
It's said that in 1950, working at Los Alamos National Laboratory, the physicist Enrico Fermi was in casual conversation with colleagues Emil Konopinski, Edward Teller and Herbert York. The men were discussing UFO reports and ET possibilities.
Then, during lunch Fermi suddenly exclaimed, "Where are they?" or "Where is everybody?". He then calculated that Earth should have been visited long ago and many times over. [wiki]
Since then that so-called paradox has been used by `skepti-bunkers' as an argument against the existence of ETs or of any intelligent life elsewhere in the Universe.
That argument rests on some unassailable logic (Fermi's), followed by shaky assumptions and an even shakier proclamation:
If intelligent life has arisen elsewhere, then logically -
1) it is most likely to have arisen in many places;
2) it is very improbable that humans were the first;
3) it is extremely likely that some groups are tens or hundreds of millions of years more advanced than us.
1) that such advanced groups would have explored our galaxy;
2) that such exploration would mean Earth had been visited many times;
3) that the `explorers' would have made themselves known to us, either directly or by leaving message(s) to be found by us (because each `visit' might be at intervals of more than a million years);
"We say there is no evidence of extraterrestrial visitors ... therefore intelligent life cannot exist elsewhere in the Universe"
The falsity of that claim and of the whole `paradox' can be seen if you ask two questions:
"Can gnats, ticks and bacteria `be aware of' or `understand' the presence of humans who might occasionally walk around / over them?"
"Do humans bother to `communicate with' or to `leave messages for' any gnats, ticks or bacteria observed during a country walk?"
I.e. - a senior US astronomer / astrophysicist recently said that if intelligent life exists anywhere else in the Universe, statistical logic says they would probably see us as "lower than we think of bacteria".