I am not good at math. The farthest I ventured into this field was basic trigonometry and advanced algebra–and I don’t think I retained very much of the latter. But as much as this subject is alien to me–a field in which I have not a scrap of skill–I can still recognize that it is important. It is when I read about a genius in math, and how they see and feel math, that I begin to dimly see how much is beyond my vision.
Benjamin Peirce (4 April 1809 – 6 October 1880) was an American mathematician who taught at Harvard University for forty years. He made contributions to celestial mechanics, number theory, algebra, and the philosophy of mathematics. He was a rare genius of math. Edward Waldo Emerson relates a story of George Flagg about Peirce: “His talk was informal, often far above their heads. ‘Do you follow me?’ asked the Professor one day. No one could say Yes. ‘I’m not surprised,’ said he; ‘I know of only three persons who could.’ At Paris, the year after, at the great Exposition, Flagg stood before a mural tablet whereon were inscribed the names of the great mathematicians of the earth for more than two thousand years. Archimedes headed, Peirce closed the list; the only American.”
Peirce could rhapsodize about math like a poet might write about nature. He said,
Geometry, to which I have devoted my life, is honoured with the title of the Key of Sciences; but it is the Key of an ever open door which refuses to be shut, and through which the whole world is crowding, to make free, in unrestrained license, with the precious treasures within, thoughtless both of lock and key, of the door itself, and even of Science, to which it owes such boundless possessions, the New World included. The door is wide open and all may enter, but all do not enter with equal thoughtlessness. There are a few who wonder, as they approach, at the exhaustless wealth, as the sacred shepherd wondered at the burning bush of Horeb, which was ever burning and never consumed. Casting their shoes from off their feet and the world’s iron-shod doubts from their understanding, these children of the faithful take their first step upon the holy ground with reverential awe, and advance almost with timidity, fearful, as the signs of Deity break upon them, lest they be brought face to face with the Almighty.
And in talking about symbolism in math he begins to sound like a poet discussing the use of symbolism in poetry:
Some definite interpretation of a linear algebra would, at first sight, appear indispensable to its successful application. But on the contrary, it is a singular fact, and one quite consonant with the principles of sound logic, that its first and general use is mostly to be expected from its want of significance. The interpretation is a trammel to the use. Symbols are essential to comprehensive argument. [...] In Algebra, likewise, the letters are symbols which, passed through a machinery of argument in accordance with given laws, are developed into symbolic results under the name of formulas. When the formulas admit of intelligible interpretation, they are accessions to knowledge; but independently of their interpretation they are invaluable as symbolical expressions of thought. But the most noted instance is the symbol called the impossible or imaginary, known also as the square root of minus one, and which, from a shadow of meaning attached to it, may be more definitely distinguished as the symbol of semi-inversion. This symbol is restricted to a precise signification as the representative of perpendicularity in quaternions, and this wonderful algebra of space is intimately dependent upon the special use of the symbol for its symmetry, elegance, and power.
I don’t get the meaning of the square root of minus one, but I do get the importance of symbols.
Peirce was, to those who studied math, both a character and a delight. As a pupil of Peirce, Thomas Wentworth Higginson recalled,
He gave us his “Curves and Functions”, in the form of lectures; and sometimes, even while stating his propositions, he would be seized with some mathematical inspiration, would forget pupils, notes, everything, and would rapidly dash off equation after equation, following them out with smaller and smaller chalk-marks into the remote corners of the blackboard, forsaking his delightful task only when there was literally no more space to be covered, and coming back with a sigh to his actual students. There was a great fascination about these interruptions; we were present, as it seemed, at mathematics in the making; it was like peeping into a necromancer’s cell, and seeing him at work; or as if our teacher were one of the old Arabian algebraists recalled to life.
Speaking of Peirce, Abbott Lawrence Lowell said,
Looking back over the space of fifty years since I entered Harvard College, Benjamin Peirce still impresses me as having the most massive intellect with which I have ever come in contact, and as being the most profoundly inspiring teacher I ever had. … As soon as he had finished the problem or filled the blackboard he would rub everything out and begin again. He was impatient of detail, and sometimes the result would not come out right; but instead of going over his work to find the error, he would rub it out, saying that he had made a mistake in a sign somewhere, and that we should find it when we went over our notes. Described in this way it may seem strange that such a method of teaching should be inspiring; yet to us it was so to the highest degree. We were carried along by the rush of his thought, by the ease and grasp of his intellectual movement. The inspiration came, I think, partly from his treating us as highly competent pupils, capable of following his line of thought even through errors in transformations; partly from his rapid and graceful methods of proof, which reached a result with the least number of steps in the process, attaining thereby an artistic or literary character; and partly from the quality of his mind which tended to regard any mathematical theorem as a particular case of some more comprehensive one, so that we were led onward to constantly enlarging truths.
I will never travel in such circles, or understand such things, but somehow I can still feel a bit of sympathetic delight with those who find such fascination in learning and understanding things.
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(1) http://en.wikiquote.org/wiki/Benjamin_Peirce
(2) http://en.wikipedia.org/wiki/Benjamin_Peirce
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