We often think that our intuition, and so called gut reactions, the businessman’s “seat of the pants” reflections leading to a course of action are invariably and intrinsically true and authentic; infallible sources connected to some primal verity embedded within our genetic makeup that must, by necessity be infallible. True or a necessary illusion. Or clinging to wreckage after a shipwreck on the high seas?
But what of the process? Is it from a power of analysis or from native capacity, something innate that is part of our genetic code. It has been suggested that the truth suddenly emerges from a chance combination of the individual’s arrangement of the data. Obviously, we can put it that way if so desired, but the idea of chance, of random luck seems to be significantly beyond the facile attribution to an idea of mere randomness. If an individual’s own native capacity is restrained and lmited, will a sliver of light appear? And, can quality teaching and training compensate for the absence of native shortcomings?
In a considerable measure, yes; but within the broader context, the answer probably cannot be affirmative. After a certain level, a barrier is reached, something like the Heinlein quote:“Never try to teach a pig to sing; it wastes your time and it annoys the pig.” So, its likely that remarkable, sparkling, mathematical skill can ever be acquired by someone with little natural mathematical endowment. A considerable measure of success may be met when say mathematics is taught to the “average” , the cluster in the middle of the bell curve, but no one can succeed in making them mathematicians. Something akin to the Nim Chimpsky experiment that held the thesis that chimps could learn sign language. That’s not entirely appropriate, but suffices to show there is a chasm to proportions requiring something magical within the faint hope clause to succeed in crossing.
Westaway said such individuals never succeed in making such boys mathematicians. He claimed that a student whose average hovered around sixty per cent – my personal comfort zone- is in all probability indebted to the instructor. Contrast this with the student who is an A or A+ type; they probably owe their relative brilliance much more to nature than nurture.
F.W. Westaway:The term a priori is ambiguous. Literally it signifies that the knowledge to which it applies is derived from something prior to it, i.e. is derivative, inferred, mediate. The metaphor involved in ” prior ” suggests an infinite series of premisses. But the term a priori is also often used to indicate that certain general truths come to the mind, to begin with, as heaven-born conceptions of universal validity, and are thus ” prior “ to all experience. Strictly, however, all a priori truths are derived truths. But derived from what?…Read More:http://www.archive.org/stream/craftsmanshipint032576mbp/craftsmanshipint032576mbp_djvu.txt
The mind seems to have a natural capacity for dictating the forms in which its particular experimental data may be combined. We may therefore correctly speak of the mind’s creative powers, though not of its innate ideas.
The mind’s undoubted power of detecting identity and difference, co-existence and succession, seems to be original and inborn. Still, the power is exercised only on a contemplation of actual things, from without or from within, and all such primitive judgments are individual. The mind compares two things and proclaims them to agree or disagree. The judgment is immediate, and it is felt to be necessary; it is irresistible and does not admit of doubt; it seems to be independent and to hang upon nothing else, and seems therefore to be primitive. But although the power is innate, this does not mean that the judgments themselves are innate….
As primitive judgments are immediate, they are sometimes described as intuitive:
Westaway:An intuition seems to be a general judgment immediately pronounced concerning facts perceived. But an intuitive judgment is as liable to error as is a reasoned judgment.
There is a natural tendency to ascribe to intuition a peculiar authority, for it seems to confront us with an irresistible force foreign to the products of voluntary and reflective experience. But knowledge derived from intuition is as much experiential knowledge as directly conscious knowledge, and it is just as fallible.
If we put on one side our purely primitive judgments, it seems very probable that, fundamentally, intuition and reasoning are identical, the former being instantaneous, the latter involving the notion of succession or progress. The difference then would be merely difference of time, every judgment of the mind being preceded by a process of reasoning, whether the individual is able to recollect it or not….
…There are times when a great new truth suddenly comes to the mind of a mathematician. The combination of factors contributing to it seem to be a garnered knowledge derived from accumulated experience, a complete analysis of the
given, a conscious connected reasoning, a systematic method of working, a natural capacity, and, finally, a flash of intuition. At some particular moment, the new truth flashes upon the vision as if light from all the other contributing factors was suddenly focused on the same point.
Does not something of the same kind happen on a small scale when an intelligent schoolboy is solving a difficult problem? All ordinary methods of systematic attack may have failed him, yet light suddenly comes. Whence? Who shall say? Something from the rules of logistic, doubtless; something from the boy’s store of mathematical knowledge;…Read More:http://www.archive.org/stream/craftsmanshipint032576mbp/craftsmanshipint032576mbp_djvu.txt