This is an extremely special case, not realised in most of the series in which limits exist. As regards the second proposition, it does not constitute a definition of “limit,” but a property which belongs to limits of a certain particular sort, namely limits of magnitudes in compact series of magnitudes in which, given a magnitude a, and a difference of magnitude c, there is always a magnitude b, other than a, such that the difference of a and b is less than c. “Rate of change” is a phrase which can only be interpreted by means of a differential coefficient, and therefore we must first define the differential coefficient before we can speak of the rate of change. The idea of negligible difference is present in it.”Īs to the first of these propositions, it should be observed that, if the modern “arithmetization” of mathematics is not all a mistake, it puts the cart before the horse. He proceeds: “Now, this second proposition bears the mark of the cloven hoof. (2) “If there be a fixed magnitude to which a variable magnitude can be made as nearly equal as we please, and if it be impossible that the variable magnitude can ever be exactly equal to the fixed magnitude, the fixed magnitude is called the limit of the variable magnitude.” (1) “A differential Coefficient expresses the rate of change of a function with respect to its independent variable.” Haldane sums up “the broad working conceptions of the Differential Calculus” in two propositions (p. I will endeavour to explain these points, though it should be said that only careful study makes it possible to grasp them thoroughly or to see their bearings. He would there have learnt that the whole subject proceeds without ever introducing the infinitesimal, that the fundamental conception is that of a limit, and that a limit is something quite different from what non-mathematicians suppose it to be. 5) are fully justified, no such criticisms are applicable to the present theory of the Calculus. Haldane had read any such authority, he would have found that, while his criticisms of Leibniz (p. Haldane has not (apparently) had the good fortune to come across any of the recognised modern authorities on the Calculus, such (for example) as the Encyklopädie der mathematischen Wissenschaften. They do not say that the teacher they have in mind is ignorant, but if he is not, the things they say are no longer true. In these quotations they describe, very accurately, the state of mind of an intelligent pupil who is being taught the Calculus by an ignorant teacher. These authors are, unfortunately, unknown to me except through Mr. Buckingham, in support of his contentions. For this reason, what he says about the Infinitesimal Calculus is in the main not relevant to the present state of that science. Haldane speaks of “modern logic,” he must be understood as meaning the logic of Hegel (and his disciples) and when he speaks of “the conception of infinity,” he must be understood as meaning the conception of infinity as it existed in Hegel’s day. I may remark, however, that if I had the bestowal of the adjectives “true” and “false,” I should interchange them, since I think the “false” infinite logically faultless, and the “true” infinite a mere chimæra. I shall, therefore, say no more about the “true” infinite, confining myself to what Mr. To one who holds, as I do, that there are no contradictions about an infinite series, this connexion of course fails. 1): “An infinite series suggests, or ought to suggest, nothing analogous to an infinite God.” Apparently he conceives, however, that there is a dialectical connexion between them – that the infinite series, when its contradictions are brought to light and synthesised, is found to be merely an inadequate expression of an infinite God. Haldane himself seems to recognise this absence of connexion, for he says (p. These two parts have, so far as I can see, no connexion except that the word “infinite” happens to be used both for a mathematical concept and for a (quite different) philosophical concept. THIS address falls into two parts, of which the first may be described as an endeavour to restate shortly and intelligibly the parts of Hegel’s greater Logic which deal with mathematics, while the second, having dismissed the “false” infinite, proceeds to some meditations on the “true” infinite, i.e., the Absolute.
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