Minima

“ MAXIMA AND MINIMA. One of the principal uses of the process of differentiating is to find out under what conditions the value of the thing differentiated becomes a maximum, or a minimum. This is often exceedingly important in engineering questions, where it is most desirable to know what conditions will make the cost of working a minimum, or will make the efficiency a maximum. ”

“ Such expansions are just as important in analysis, geometry and mechanics as expansions of functions of one variable. Among the problems which are solved by means of such expansions are the problem of maxima and minima for functions of more than one variable (see Maxima and Minima) . ”

“ His leisure time was mostly devoted to mathematics, which he studied with irresistible passion. Unlike Descartes and Pascal, he led a quiet and unaggressive life. Fermat has left the impress of his genius upon all branches of mathematics then known. A great contribution to geometry was his De maximis et minimis. About twenty years earlier, Kepler had first observed that the increment of a variable, as, for instance, the ordinate of a curve, is evanescent for values very near a maximum or a minimum value of the variable. Developing this idea, Fermat obtained his rule for maxima and minima. ”

“ But when from the illusion created by this common property we think we shall distinguish something in the minimum, one part on one side and another part on the other side, it must be another minimum equal to the first which catches our eye. ”

“ In other words, the minimum in each sphere of being contains implicitly in itself the whole reality of that sphere. The minimum is its substance, not merely the ultimate of analysis, but the actual source, the [Pg 232] dynamic origin of reality, as God is implicitly the whole universe and also the source of the universe as it actually exists. It is because the minimum is all reality, is the maximum, that the knowledge of it gives us that of the whole. ”

“ Give me the statistics of pyrotechnic powder burnt by a people, and I shall tell you the standard measure of their souls and bodies. If the figure be a maximum, then the physical and moral measure will be the minimum, for the ratio is inverse. ”

“ Now that quantity which is apprehended only as unity, and in which plurality can be represented only by approximation to negation = 0, I call an intensive quantity. Every reality, therefore, in a phenomenon has intensive quantity, that is, a degree. ”

“ In proportion, therefore, to our ignorance of the causes on which the empirical law depends, we can be less assured that it will continue to hold good; and the further we look into futurity, the less improbable is it that some one of the causes, whose co-existence gives rise to the derivative uniformity, may be destroyed or counteracted. ”

“ The maximum and minimum, however, may in one sense coincide, so that we know the maximum to be everywhere, since from [Pg 238] what has been said it is evident that the maximum consists in the minimum and the minimum in the maximum, as in the many is the one, in the one the many. Yet reason and nature may more readily separate the minimum from the maximum than the maximum from the minimum. Therefore the immeasurable universe is nothing but centre everywhere; eternity nothing but a moment always ”

“ For instance, no experience of an individual, however near he may stand to us, can be perfect, so that we could have a logical right to make a complete estimate of him ; all estimates are rash, and must be so. Finally, the standard by which we measure, our nature, is not of unalterable dimensions,—we have moods and vacillations, and yet we should have to recognise ourselves as a fixed standard in order to estimate correctly the relation of any thing whatever to ourselves. ”

“ As the comparison of numbers with one another, their determination as greater or less, is only possible on the assumption of a unit, a common measure to which each may be referred, so the comparison of bodies with one another, as to quantity and quality alike, demands a corporeal minimum, to which their differences must be reduced. This relation to knowledge carries with it the relativity of the minimum according to the subject-matter with which the knower is for the time being concerned. ”

“ You have no standard to estimate by, no measure to make comparisons with. Given a certain small quantity of higher pleasure in collision with a certain large quantity of lower, how can you decide between them? To work the sum you must reduce the data to the same denomination. You must go to quantity or nothing; you decline to go to quantity, and hence you can not get any result. But if you refuse to work the sum, you abandon the greatest amount of pleasure principle. ”

“ It is said in the Bible that God has ordered all things according to weight, number and measure. Omnia in pondere et numero et mensura disposuit Deus. Weight is equilibrium, number is quantity, measure is proportion—these three, and these are the eternal or divine basis of the science of Nature. Here now is the formula of equilibrium: Harmony results from the analogy of contraries. Number is the scale of analogies, the proportion of which is measure. The entire occult philosophy of the Zohar might be termed the science of equilibrium. ”

“ That for which it is the way to roughly measure, Yet it is measured in small amounts, It is of secondary importance to the rough amounts, And that for which it is the way to measure small, but was measured roughly, The rough amount is of secondary importance to the small. ”

“ The paradoxes of what is called chance, or hazard, might themselves make a small volume. All the world understands that there is a long run, a general average; but great part of the world is surprised that this general average should be computed and predicted. There are many remarkable cases of verification; and one of them relates to the quadrature of the circle. I give some account of this and another. ”

“ Units proper doubtless do not exist apart from the experience of quantity, and I do not mean to say that apart from quantity no distinction is possible, or again that quantity could be developed rationally from anything more simple than itself. And I have emphasized the words ‘as a matter of fact’ in order to leave these questions on one side, since they can be neglected provisionally. Numerical sameness, in the same way, is the persistence of any such bare distinction through diverse contexts, no matter what these contexts are. ”

“ Now here again, of all the paths that lead from one position to another, there is one for which the average action is a minimum, and there is only one. The principle of least action suffices, then, to determine the differential equations which define the variations of the parameters q. ”

“ Knowledge being scientific, as it can apply number and measure, an attempt is in order to see if a scientific order of values cannot be built "on a number and measure scale of force," ascent in the scale signifying increase of value, descent diminution of value—all other estimations being prejudices, naïvetés, misunderstandings. ”

“ And so the balance sways and the story runs—here a gain, there a loss, and Nature’s great average struck across the two, whereof the sum total forms one of the factors in that mighty equation in which the result will equal the unknown quantity of her purpose. ”

“ As to equality or any exact proportion, we can only guess at it from a single consideration; except in very short numbers, or very limited portions of extension; which are comprehended in an instant, and where we perceive an impossibility of falling into any considerable error. In all other cases we must settle the proportions with some liberty, or proceed in a more artificial manner. ”

“ But if it comes to be discovered that variations of quality in any class of phenomena, correspond regularly to variations of quantity either in those same or in some other phenomena; [Pg 249] every formula of mathematics applicable to quantities which vary in that particular manner, becomes a mark of a corresponding general truth respecting the variations in quality which accompany them: and the science of quantity being (as far as any science can be) altogether deductive, the theory of that particular kind of qualities becomes, to this extent, deductive likewise. ”

“ If we except those cases in which the problem has no definite meaning because the data are such that the integral must be infinite, it is evident that a minimum must always exist, and (on account of the quadratic form of the integral) that it is unique. ”

“ Of natural powers, some are unlimited, others limited in quantity. By an unlimited quantity is of course not meant literally, but practically unlimited: a quantity beyond the use which can in any, or at least in present circumstances, be made of it. ”

“ Socrates: Yes, that is usually what people mean by ratios. The learned people call numbers made from the ratios, rational. Does the name rational number suit you to call a number which can be expressed as the ratio of two whole numbers, whether they be odd or even whole numbers?
Boy: You want me to call the numbers made from ratios of whole numbers something called rational? A ratio makes a rational number?
Socrates: Yes boy, can you do that?
Boy: Certainly, Socrates.
Socrates: Do you agree with the way I told him this, Meno? ”

“ If we formed our groups, not by equal numbers, but by equal amount of variation, we should probably find, in accordance with the law of averages, that the central group in which the mean characteristics prevailed was much more numerous than the extremes, perhaps twice, or even three times, as great as either of them, and forming such a series as the following—10 maximum, 30 mean, 10 minimum development. ”

“ As an empirical rule, the principle of the minimum is, no doubt, essential to the scientific method. To erect it into a metaphysical axiom, however, is preposterous; for simple explanations are not necessarily true explanations. ”

“ Size, on the other hand, and its alteration regarded merely as size, is for the qualitative determination, when it is unable to assert itself in terms of measure, an indifferent determination. That is to say, measure is quantity, precisely in so far as it can give to itself an aspect which qualifies it, and thereby a qualitative determination is united to the purely quantitative one. ”

“ Now when we are referring to averages it must be remembered that the minimum number of different occurrences necessary in order to justify the average is that which enables each of them to present itself once. ”

“ The process of measure, instead of being only the wrong infinite of an endless progression, in the shape of an ever-recurrent recoil from quality to quantity, and from quantity to quality, is also the true infinity of coincidence with self in another. In measure, quality and quantity originally confront each other, like some and other. But quality is implicitly quantity, and conversely quantity is implicitly quality. ”

“ And I for my part cannot but consider the talk of the contemporary sociological school about averages and general laws and predetermined tendencies, with its obligatory undervaluing of the importance of individual {262} differences, as the most pernicious and immoral of fatalisms. Suppose there is a social equilibrium fated to be, whose is it to be,—that of your preference, or mine? There lies the question of questions, and it is one which no study of averages can decide. ”

“ The illusion of numerical inequality is marked for many subjects. (2) The judgment is quite possibly a function of the two factors—object-size and group-vacancies. If we recall the fact that the small-object group is more scattered than the other, we shall note that the leading class here is like the leading class in Table I, and we may fairly reckon this factor as of importance in the issue. Of the incomplete introspective notes on this question, those of only one observer speak clearly for the size. ”

“ The first point, therefore, to be noted in regard to what is called the uniformity of the course of nature, is, that it is itself a complex fact, compounded of all the separate uniformities which exist in respect to single phenomena. These various uniformities, when ascertained by what is regarded as a sufficient induction, we call, in common parlance, Laws of Nature. ”

“ That, however, is apart from the question I have been debating in this work, and my purpose will have been amply served if I have been successful in throwing some light upon a little known subject, and, above all, in leading my readers to the conclusion I have long since arrived at—that our system is, fundamentally, the best in existence, the most convenient for the needs of man, and that it would be an act of stupendous and costly folly, to change to the metric system. ”

“ And when we deliberately set ourselves to obtain uniformities by taking an average result, derived from comparison of a multitude of subjects, our results are, of course, always mechanical in appearance, because the element of individual purpose and initiative has been excluded by ourselves from our data in the very process of taking the average. ”

“ At first, no doubt, looking at the nature of quantity as an indifferent and external character of Being, we are disposed to answer these questions in the negative. And yet, as we must admit, this indifferent increase and diminution has its limit: a point is finally reached, where a single additional grain makes a heap of wheat; and the bald-tail is produced, if we continue plucking out single hairs. ”

“ The larger the number we take out of any mixed crowd, the clearer become the symptoms of order, the more nearly will the average length of each selected class be the same. These few cases will serve as simple examples of a property of things which can be traced almost everywhere, to a greater or less extent, throughout the whole field of our experience. ”

“ Among the foundations of the Higher Mathematics is also the Idea of a Limit. The Idea of a Limit cannot be superseded by any other definitions or Hypotheses, The Axiom which we employ in introducing this Idea into our reasoning is this:—What is true up to the Limit is true at the Limit. This Idea and Axiom are the bases of all Methods of Limits, Fluxions, Differentials, Variations, and the like. ”