I am a polymath. I refer you to my answers on Quora, and my works as evidence. Now it is for you to decide if I am actually one.
For instance, Socrates claims to not be a Sophist. But in certain circumstances Plato makes him look precisely like what he is against. Thus, what are we to take from that when we become disillusioned with Socrates and his difference from the Sophists as Nietzsche did for instance. Socrates was an anti-polymath, and the sophists were seen as the polymaths, and that is one of the things that Socrates held against them, he thought of them as Hydra, having many heads, and said that they escape into holes with many exits in the course of arguments. But Plato can be seen as a Polymath, if you read all his dialogues. So we have a Polymath with a character who is not a polymath, and who is against Sophists who are normally polymaths, from whom Socrates is hard to distinguish except by external factors like the fact that Socrates is from Athens and not a foreigner. and that Socrates does not ask for money to engage in conversation. Rather than knowing anything himself, Socrates is seen as one who is good at asking, hard to answer questions bout the real meaning of abstract words. So eventually you are completely caught up in Plato’s irony and you don’t know what to think. Basically to your point Plato and Socrates have the ideal of not being Sophists but in the end it seems that they themselves may be sophists. And so their claims must be pointing to something else than the nature of sophistry, something beyond sophistry like the Nondual for instance. Something not recognized in the tradition that followed, and something truly lost in oblivion in our tradition due to active suppression.
Being a Polymath in our society is highly discouraged. Everyone is a specialist. And specialism is in fact nihilism. Thus if you are to overcome nihilism oneself one is forced to try to become a polymath, but in that search for knowledge one puts oneself beyond the pale of academia, because the whole purpose of academia is to control knowledge and who might claim to have it. And everyone says that it is impossible for anyone to know everything in our age, but no one claims that it is impossible to know everything that is significant. That is still open as a goal. But then how do you decide what is significant? My measure is whether or not it relates to the structure of the Western Worldview as it is rooted in the Indo-European worldview. And significance is gained by comparing that to the various nondual traditions like Taoism, Buddhism, DzogChen, Sufism, etc. Significance comes from ones problematic. My problematic the nature of Western Science in relation to the structure of the Western worldivew, and we do this by studying anomalous cases like Acupuncture that has no scientific explanation, but is recognized to work by the establishment even though no one knows why. These anomalies suggest we might have blindspots in our own scientific approach to the world, which come from the structure of our worldview, which is now world dominant. Significance comes from the spread of the Kurgans due to horsepower, Colonialization, and now Globalisation by the Indo-Europeans whose worldview has become world dominant. This coincides with the ultimate nihilistic act which is destroying the planet, i.e. the ultimate terrorist act of destroying the planet and taking everyone else including all other species with you, which this world dominant Worldview seems to be in the process of attempting to realize. The fact that it cannot control itself to stop the emissions that is causing global warming and leading to a greenhouse planet suggests that there is something fundamentally self-destructive in this worldview, which is terrorizing the rest of the planet. In some respects Terrorism is a reflection of ourselves in the mirror of the world. The first terrorists were European anarchists. We developed the weapons of mass destruction which are being used to kill masses of people. If we had not spread them all over the globe we would not find them being used by others. So it seems fairly clear that the Western worldview is its own worst enemy, and unfortunately the enemy of all, including the other species on the planet.
So it is from this global crisis that we take the significance of our problematic. And it is from this crisis that we take the energy to pursue the quest for self-knowledge whereever it may lead. And it is the fact that it leads to many disparate fields that produces the polymatic qualities, which are a side effect of the intellectual journey being taken over a lifetime. And in fact I would guess that all polymaths have a similar motivation, they find something which is fascinating and they pursue it whereever it may lead in the pursuit of understanding, and knowledge of many subjects picked up as tools along the way is the result. They are not seeking to be polymaths, but they are seeking an elusive query, that continually hides in various fields of inquiry or endeavor and the only way to continue the pursuit is to master to some extent those various fields.
So the sign of a true polymath in my opinion is one who has a deep enough problematic that it cannot be bound by specialization, and who thus becomes a renegade from the Academic control structures built to reign in and control knowledge.
A Crank on the other hand is someone who is obsessed with something which is not related to the cutting edge of the tradition and does not recognize the tradition and its judgment on what is valuable and what is not valuable. Every polymath is somewhat of a crank, because they are willing to develop ideas that totally break the mold of the tradition. But the crank really does not understand the tradition, and thus pursues a vision completely out of kilter with it. The Polymath on the other hand is so involved with learning pursuing his goal that he just happens to learn a lot along the way, without regard to whether the knowledge is useful or fits into normal categories manufactured by Academia to control knowledge, The Crank is the person who is filtered out by the academic control system. The polymath does not care about the boundaries for learning established by the Academic knowledge control system because he is pursuing a problematic that is a crosscutting concern and too busy doing that to bother with specialization and the peer pressure of peer reviewed publications. The the true polymath has no peer. Because all the peers implicitly recognize the boundaries of specialization and are loath to transgress those boundaries.
This brings us to the trees in the Garden (http://en.wikipedia.org/wiki/Five_Trees). But it appears there were actually five trees trees of life, of immortality, Knowledge, comprehension, and knowledge of good and evil (http://www.bardic-press.com/thomas/saying19.htm). Aristotlec says “There are five virtues of thought: technê, epistêmê, phronêsis, sophia, and nous (1139b15). ” (http://plato.stanford.edu/entries/episteme-techne/). Throwing Blake into the equation we have several lists of terms we might try to reconcile.
5 trees nonduals Aristotle
Immortality Source sophia
Life Fate phronesis
Good & Evil Good (metis)
Comprehension Right techne
Knowledge Order episteme
(Information) InfoEntropy (senses)
5 trees Blake
(rivers origin) Albion
Good & Evil Luvah
Comprehension Urizen (reason)
Knowledge Beulah Land
(Information) (created world)
Only a Polymath can come up with a table like this. Whether it is meaningful or not you have to judge for yourself. If it is not meaningful then you would have to judge me a crank. If it is meaningful then it means that there is a lot more to life than just knowing a lot of things, and being a polymath is merely the most superficial of characteristics that we would desire as human beings if we could have all the depth we might be able to attain.
Nous also called intellect or intelligence, is a philosophical term for the faculty of the human mind which is described in classical philosophy as necessary for understanding what is true or real, very close in meaning to intuition. It is also often described as a form of perception which works within the mind (“the mind’s eye”), rather than only through the physical senses. The three commonly used philosophical terms are from Greek, νοῦς or νόος, and Latin intellectus and intelligentia respectively. http://en.wikipedia.org/wiki/Nous
Note: “Phronesis is the historically implicated, communally nurtured ability to make good sense of relatively singular contexts in ways appropriate to their relative singularity.” (https://sites.google.com/site/praxisandtechne/Home/architecture/knowledge/episteme/phronesis)
The Polymath merely collects knowledge though his fascination on his intellectual quest after what is sought from his problematic. This is indeed only the surface. What we need is something deeper that takes from all the trees in paradise rather than only one.
19. Jesus said, “Blessed is he who exists from the beginning before he comes to be. If you are my students and listen to my words, these stones will become your servants. For you have five trees in Paradise, which do not move in summer or in winter, and their leaves do not fall down. Whoever knows them will not taste death.”