Wisdom

And yet, I am doubtful Post Date: Thu, 7 Aug 2008 17:45:10 +0000
Had Stewart read Vico, we are inclined to believe that he would not have complained of the confusion with which various ancient and modern authors have explained this doctrine. " The peculiarity of that species of evidence which is called de monstrative, and which so remarkably distinguishes our mathematical conclusions from those to which we are led in other branches of science, is a fact which must have arrested the attention of every person who possesses the slightest acquaintance with the elements of geometry. And yet, I am doubtful if a satisfactory account has hitherto been given of the circumstance in which it arises.

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1 The inaccuracy Post Date: Thu, 7 Aug 2008 18:03:21 +0000
Mr Locke tells us, that what constitutes a demonstration is intuitive evi dence of every step ; and I readily grant, that if in a single step such evidence should fail, the other parts of the demonstration would be of no value. It does not, however, seem to me that it is on this consi deration that the demonstrative evidence of the conclusion depends, not even when we add to it another which is much insisted on by Dr Reid, that in demonstrative evidence our first principles must be intuitively certain. 1 The inaccuracy of this remark I formerly pointed out when treating of the evidence of axioms ; on which occa sion I also observed, that the first principles of our reasonings in mathematics are not axioms, but definitions.

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"It Post Date: Thu, 7 Aug 2008 18:22:03 +0000
It is in this last circum stance (I mean the peculiarity of reasoning from definitions) that the true theory of mathematical demonstration is to be found ; and I shall accordingly endeavor to explain it at considerable length, and to state some of the more important consequences to which it leads. " That I may not, however, have the appearance of claiming, in be half of the following discussion, an undue share of originality, it is necessary for me to remark that the leading idea which it contains has been repeatedly started, and even to a certain length prosecuted, by different writers, ancient as well as modern ; but that, in all of them, has been so blended with collateral considerations, although foreign to the point in question, as to divert the attention both of writer and reader, from that single principle on which the solution of the problem hinges. "It was already remarked, in the first chapter of this part, that whereas, m all other sciences, the propositions which we attempt to establish, express facts real or supposed,in mathematics, the propo sitions which we demonstrate only assert a connection between cer tain suppositions and certain consequences.

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In the other sciences admitting Post Date: Thu, 7 Aug 2008 18:41:33 +0000
Our reasonino s , there- lore, m mathematics, are directed to an object essentially different rom what we have in view, in any other employment of our intellec tual faculties ; not to ascertain truths with respect to actual exist ences, hut to trace the logical filiation of consequences which follow from an assumed hypothesis. If, from this hypothesis, we reason with correctness, nothing, it is manifest, can be wanting to complete the evidence of the result ; as this result only asserts a necessary connec- 1011 between the supposition and the conclusion. In the other sciences admitting that every ambiguity of language were removed, and that every step of our deductions were rigorously accurate, our conclu sions would still be attended with more or less of uncertainty, bein- ultimately founded on principles which may, or may not, correspond exactly with the fact.

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We have said that, considering Post Date: Thu, 7 Aug 2008 19:01:13 +0000
" This is exactly Vice s doctrine of the cause of the difference in the degrees of evidence and certainty; although he makes a general sys tem, m order to explain the problem of intelligence, whaUhe Scotch man only assigns as a fact to show the reason of mathematical evi dence. Pere Buffier explains the same thing with great clearness. We have said that, considering the indefatigable laboriousness which distinguishes the Germans, it would not be strange if they had read the scholastics.

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By observing the traces Post Date: Thu, 7 Aug 2008 19:11:20 +0000
In confirmation of this, we notice that Leibnitz urgently recommends the reading of them, and the more modern Ger mans are not likely to forget the advice of so able an author. Among various passages of Leibnitz, we select the following ex tract, because it seems to us rather curious : k Truth is more spread than one would believe ; but it is often colored, also often covered over, and even weakened, mutilated, and corrupted by additions which spoil ,t, or render it less useful. By observing the traces of truth in the ancients, or, to speak more generally, in all who have preceded us, we dig gold from dirt, and draw the diamond from its mine, liaht from darkness, and this would really be perennis quadam Philosophia.

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ve great and beautiful ideas Post Date: Thu, 7 Aug 2008 19:27:21 +0000
" It may even be said that some progress would be observable in knowledge. The Orientals ha.ve great and beautiful ideas of the Divinity; the Greeks added reasoning and a form of science ; the Fathers of the Church rejected whatever was bad in Greek philoso- phy ; but the scholastics labored to usefully employ whatever waa acceptable in Pagan philosophy.

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" (Lettre 3 leme a M Post Date: Thu, 7 Aug 2008 19:43:01 +0000
I have frequently said : aurum latere in stercore illo schohtatico barbariei, and I wish we could find some one versed in the Irish and Spanish philosophy, to cull from it what is good ; I am sure he would find his labor repaid by many and beautiful truths. There was once a Swiss who mathematized schol- astically ; his works are but little known ; although, from what I have seen of them, I should judge them to be profound and worthy of con sideration." (Lettre 3 leme a M Remond de Montmort.

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This study, setting aside its Post Date: Thu, 7 Aug 2008 19:53:03 +0000
) Thus speaks Leibnitz, one of the nm c t eminent men of modern times, and of whom Fontenelle said : " He led the van in all the sciences. 1 See, then, if he was wrong in recommending the study of those authors to all desirous of acquiring a profound knowledge of philosophy. This study, setting aside its intrinsic utility, is of great advantage in judg ing, with knowledge of the cause, a school, which, whatever its worth, occupies a page in the history of the human mind.

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(29) THE immateriality Post Date: Thu, 7 Aug 2008 20:07:00 +0000
(28) THE author to whom I allude (317) is Fenelon, who, under the name of common sense, includes also the criterion of evidence, as may be seen by this extract: " What is common sense 1 ? Does it not consist in the first notions which all men have of the same things? This common sense, which always and everywhere is the same, which precedes all examination, and even holds it in ridicule on certain questions, in which one laughs instead of examining; which renders man unable to doubt, no matter how great his efforts may be; this sense which belongs to all men, which only waits to be consulted in order to discover itself and show us the evidence or the absurdity of the question, is not this what I call my ideas? Here, then, are these, general ideas or notions which I cannot contradict or examine, but ac cording to which I examine and judge every thing, so that, instead of replying, I laugh when any thing is proposed clearly in opposition tc : what ihese immutable ideas represent." There is no doubt that Fenelon speaks of evidence in this extract, since, besides using this very term, he refers to immutable ideas ; by common sense he understands the general ideas by which we judge of all things, or in other words, the ideas from which evidence proceeds. (29) THE immateriality of the souls of brutes is not a discovery of modern philosophy, the scholastics maintained it, and carried their ideas on this subject so far as to assert that no vital principle can be a body.

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