Re: Elements of Mathematics (MS 165)

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NEM 2, pp. 4–6
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PREFACE (2) (94a) (cont.)

In the fifth place, Georg Cantor and others have succeeded in analyzing the conceptions of infinity and of continuity, so as to render our reasonings concerning them far more exact than they had previously been;  and the fundamental researches that have largely occupied mathematicians of late years, into the theory of functions, do much to render geometrical reasonings more exhaustive and precise.

All these intellectual movements ought, in the opinion of very many mathematicians, to have their effects upon the system of teaching the elements of the subject.

But this is not all.  Pedagogy is an art which has come in the last sixty years to be based more and more upon modern scientific psychology, and upon modern views of logic.  If diligent and intelligent youths find difficulty in understanding mathematics, teachers no longer deem it becoming to flog them or to objurgate them, as they used to do, in the days when the second theorem of Euclid received the name of the "Asses' Bridge."  On the contrary, they consider the fact that a considerable percentage of the best minds imagine themselves to be utterly unable to comprehend mathematical reasoning to constitute an emphatic condemnation of that old system of teaching which had such a result.

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On Jul 18, 2012, at 11:04 AM, Jon Awbrey wrote:

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> NEM 2, pp. 4–6
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> 
> PREFACE (2) (94a) (cont.)
> 
> In the fourth place, the whole conception of metrical geometry has been revolutionized.  In 1837, the "Doctrine of Parallels" formed an urgent but unsolved problem.  The earnest and persistent efforts of Legendre and of many other eminent mathematicians had been powerless to clear up its difficulties.  Benjamin Peirce, in his treatise, thought to conquer them by the introduction of the idea of a _difference of direction_.  There can be no intelligent question that this idea brings strong help to mathematical inquiry.  Soon after its introduction by Peirce, it was taken up by Hamilton and by Grassmann with such effect as might have been anticipated.  But Professor Peirce himself subsequently admitted, with the rest of the mathematical world, that there was no solution for the question of parallels except from the idea which the pupils of Gauss, Bolyai, Lobatchewsky, Riemann, derived through their master, from Lambert, and ultimately from the Italian Jesuit, Saccheri.  The idea was that it is simply a question for observation of nature whether the sum of the angles of a triangle is less than, or more than, or possibly equal to two right angles.  Subsequently, Cayley (in 1854) and Klein (more fully, in 1873) showed that metrical geometry is simply the geometry of the _firmament_, or _absolute_, or infinitely distant part of space, which constitutes a surface which is one or another quadric surface, according to the system of measurement adopted, that is, according to the way in which rigid bodies move.  In the course of this inquiry, a fallacy in Euclid's 16th proposition was brought to light that had remained undetected for two thousand years.
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