Buku Natural Philosophy by Wilhelm Ostwald

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Natural Philosophy by Wilhelm Ostwald

Author:Wilhelm Ostwald

Language: eng

Format: mobi, epub

Tags: Science — Philosophy

Published: 2013-09-21T16:00:00+00:00

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Natural Philosophy by Wilhelm Ostwald

37. The Function.

The concept of continuity makes possible the development of another concept of greater universality, which can be characterized as an extension of the concept of causation (p. 31). The latter is an expression of the experience, if A is, B is also, in which A is understood to be a definite thing at first conceived of as immutable. Now it may happen that A is not immutable, but represents a concept with continuously changing characteristics. Then, as a rule, B will also be of that nature, so that every special value or state of B corresponds to every special value or state of A.

Here, in place of the reciprocal relation of two definite things, we have the reciprocal relation of two more or less extended groups of similar things. If these things are continuous, as is assumed here (and which is extremely often the case), both groups or series, even though they are finite, contain an endless quantity of individual cases. Such a relation between two variable things is called a function. Although this concept is used chiefly for the reciprocal relation of continuous things, there is nothing to hinder its application to discrete things, and accordingly we distinguish between continuous and discontinuous functions.

 

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Natural Philosophy by Wilhelm Ostwald

Author:Wilhelm Ostwald , Date: July 5, 2019

,Views: 35

Author:Wilhelm Ostwald

Language: eng

Format: mobi, epub

Tags: Science — Philosophy

Published: 2013-09-21T16:00:00+00:00
37. The Function.

The concept of continuity makes possible the development of another concept of greater universality, which can be characterized as an extension of the concept of causation (p. 31). The latter is an expression of the experience, if A is, B is also, in which A is understood to be a definite thing at first conceived of as immutable. Now it may happen that A is not immutable, but represents a concept with continuously changing characteristics. Then, as a rule, B will also be of that nature, so that every special value or state of B corresponds to every special value or state of A.

Here, in place of the reciprocal relation of two definite things, we have the reciprocal relation of two more or less extended groups of similar things. If these things are continuous, as is assumed here (and which is extremely often the case), both groups or series, even though they are finite, contain an endless quantity of individual cases. Such a relation between two variable things is called a function. Although this concept is used chiefly for the reciprocal relation of continuous things, there is nothing to hinder its application to discrete things, and accordingly we distinguish between continuous and discontinuous functions.

The intellectual progress involved in the conception of the reciprocal relation of entire series or groups to one another, as distinguished from the conception of the relations between individual things, is of the utmost importance and in the most expressive manner characterizes the difference between modern scientific thought and ancient thought. Ancient geometry, for example, knew only the cases of the acute, right, and obtuse angled triangle, and treated them separately, while the modern geometrician represents the side of the triangle as starting from the angle zero and traversing the entire field of possible angles. Accordingly, unlike his colleague of old, he does not ask for the particular principles bearing upon these particular cases, but he asks in what continuous relation do the sides and angles stand to one another, and he lets the particular cases develop from out of one another. In this way he attains a much profounder and more effectual insight into the whole of the existing relations.

It is in mathematics in especial that the introduction of the concept of continuity and of the function concept arising from it has exercised an extraordinarily deep influence. The so-called Higher Analysis, or Infinitesimal Analysis, was the first result of this radical advance, and the Theory of Functions, in the most general sense, was the later result. This progress rests on the fact that the magnitudes appearing in the mathematical formulas were no longer regarded as certain definite values (or values to be arbitrarily determined), but as variable, that is, values which may range through all possible quantities. If we represent the relation between two things by the formula B = f(A), expressed in spoken language by B is a function of A, then in the old conception A and B are each individual things,

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