There have been some note able polymaths: Newton, Leibniz and arguably Groethendieck.
The word polymath derives from the Greek Polk and mathesis. Mathesis derives from the root Mathematikos from which Mathematikos derives. The essential idea is thoughtfully, meditative manipulation of the objects and experiences in and of the " world". The concept of world being scalable.
It may have been a rare word even in Pythagoras time, but it is clearly associated with the Pythagorean " school" where it was the highest qualification, a doctor of philosophy.
The word mathesis, rare and perhaps used only by classical scholars, means a doctrine, and thus it is appropriate to associate it with a doctorate, now called a Phd.
A polymath therefore would be skilled in a broad range of subjects and have a broad range of qualifications if they were enamoured by this kind of social grooming. Those polymaths outside this system would be no less skilled, just " unqualified".
The qualification of a polymath is a kind of joke on the system. The polymath is really unconstrained by a system designed to socialise and protect vested interests. It is because of the wide range of interests that polymaths are so difficult to handle.
The system of which I speak is set up and established by he few to control the many. On the other hand, the many, biologically support this kind of system in the sense of role differentiation. Thus sinister as it may be portrayed, there is biological evidence of the naturalness of these systems, and the conformal contributory benefit , a kind of symbiosis.
We may observe that the rich get richer and the poor poorer, but not understand the biological reasons for this, placing purely social interpretations on his phenomenon. Nevertheless, the qualification root is manifestly a system of control, giving those in the controlling seat the option of selecting from the best within their social model. The reward of status is just one of a number of blandishments on offer, which are used to effectively control the system, somewhat.
The polymath represents a threat to current interests of those in control, because they may discover something that upsets heir applecart! On the other hand, when the controlling group is in crisis, which it frequently is, the polymath may assist in solutions that move the situation on. The clever polymath will use this to their advantage, and may te control of the controlling group.
In the meantime, this highlights he extremes of subject boundary divisions, and how they arise fom personal interest considerations rather than natural divisions.
He division ino subjects can be traced back to Aristotle, a platonists. Aristoe appears to have bern autistic to the level of. aspbergers syndrome. He spent all of his life in a ritual of categorisation.
The categories he developed spread across the whole range of human enquiry, and fom the basis of the Trivium and the Quadrivium, which were the main subject categorisations in his school the Lyceum. It is a natural consequence that his students, called peripatetic because they had to meet in a place which was a public avenue in Athens, much like the stoics met in a public market place with a roof, so probably better represented by a mall, called a Stoa; that they should categorise still further.
When Aristotle fell out of favour, his scholars scattered throughout the Greek empire and earned a living by teaching the categories. As they became more widespread, subject territories became a feature of contention. In order to attract the highest patronage, a teacher would keep secret their personal methods, especially if they were proved effective. There tufts would either support their tutor or vote with there feet nd fid someone they could support.
This often broke out into Acadmic warfare, and the casualty was often the scope of the subject. Smaller nd smaller divisions of the main categories were introduced by this process, with high levels of emotional and livelihood interest.
The concept behind Ausdehnungslehre is Ausdehnungs Groesse. This refers directly to sn extension of Atithmetic through extending the quantities or magnitudes it could apply to. These magnitudes turn out to be in general Clifford nodes, but the full,concept is a mesh of these nodes? We find that Euler was considering these things throughout his mathematical career, but the modern story begins with Hmilton nd Grassmann and the liberation of Algebra.
This shows the boundary wars in action. The theory of Monads, both the Leibnizin and the Euclidesn, leads to the theory of rithmd. These Arithmoi were the main modelling tools and metrical tools for comparing space quantitively. It was generally regarded as the highest intellectual achievement. Because of this, it took control of the goal of education, pedagoguery, and certain aesthetic factors contributed to this.
Algebra was a area where new ground coould be broken, and the bodies in arithmetic could be explored