Formula

“ In the common or Syllogistic Logic, a certain Formula of language is used in stating the reasoning, and is useful in enabling us more readily to apply the Criterion of Form to alleged demonstrations. This formula is the usual Syllogism; with its members, Major Premiss, Minor Premiss, and Conclusion. It may naturally be asked whether in Inductive Logic there is any such Formula? whether there is any standard form of words in which we may most properly express the inference of a general truth from particular facts? ”

“ In constructing the formulæ by which laws of nature are expressed, although the first object is to assign the Law of the Phenomena, philosophers have, in almost all cases, not proceeded in a purely empirical manner, to connect the observed numbers by some expression of calculation, but have been guided, in the selection of their formula, by some Hypothesis respecting the mode of connexion of the facts. ”

“ The first process therefore in the effectual study of the science must be one of simplification and reduction of the results of previous investigations to a form in which the mind can grasp them. The results of this simplification may take the form of a purely mathematical formula or of a physical hypothesis. ”

“ The real inference is always from particulars to particulars, from the observed instances to an unobserved one: but in drawing this inference, we conform to a formula which we have adopted for our guidance in such operations, and which is a record of the criteria by which we thought we had ascertained that we might distinguish when the inference could, and when it could not, be drawn. ”

“ The mathematical conception of the natural sciences would imply perfect mechanism, the reduction of all phenomena to quantity without quality, the representation of each phenomenon by means of a mathematical formula, which should be its adequate definition. ”

“ Formulæ for solving Problems in Syllogisms.
When once we have found, by Diagrams, the Conclusion to a given Pair of Premisses, and have represented the Syllogism in subscript form, we have a Formula, by which we can at once find, without having to use Diagrams again, the Conclusion to any other Pair of Premisses having the same subscript forms. ”

“ If, therefore, we could find a mathematical formula which was, as a matter of fact, verifiable in economical problems about prices, and so forth, we should rightly apply to mathematicians to help us with their methods. ”

“ Beings of the first class have a distinguishing formula, and only use names for plants and the lower animals. The second-class beings, the workers, may possibly use names among themselves, but of that I have no knowledge. My own distinguishing formula is MZ04, and as no two people have the same formula, much confusion is prevented. ”

“ In a progressive series of numbers, unless the formula which we adopt be really that which expresses the law of nature, the deviations of the formula from the facts will generally become enormous, when the experiments are extended into new parts of the scale. True formulæ for a progressive series of results can hardly ever be obtained from a very limited range of experiments: just as the attempt to guess the general course of a road or a river, by knowing two or three points of it in the neighbourhood of one another, would generally fail. ”

“ Thus, it would be certainly possible to describe every individual's physiognomy by means of a very complicated mathematical formula, and yet there is no one who would not prefer the usual mode of taking pictures. ”

“ In the first place, it must require much time to try different formulae. In the next place it may happen that, in the progress of human knowledge, more correct formula: may be discovered, and constants may be determined with far greater precision. ”

“ Yet the time will come when the professors of science will feel proud to expound, by formulas, the very invention which they have shown, by formulas, to be an absolute contradiction of all the laws of Nature. ”

“ That any formula should be capable of expressing a common character in the shapes of things so unlike as a tree and a cow, a flower and a centipede, is a remarkable fact; and is a fact which affords strong primâ facie evidence of truth. ”

“ When the more obvious of the inductions which can be made in any science from direct observations, have been made, and general formulas have been framed, determining the limits within which these inductions are applicable; as often as a new case can be at once seen to come within one of the formulas, the induction is applied to the new case, and the business is ended. But new cases are continually arising, which do not obviously come within any formula whereby the question we want solved in respect of them could be answered. ”

“ When the two can not be brought into juxtaposition, or not so that the observer is able to compare their several parts in detail, he must employ the indirect means of reasoning and general propositions. When we can not bring two straight lines together, to determine whether they are equal, we do it by the physical aid of a foot-rule applied first to one and then to the other, and the logical aid of the general proposition or formula, “Things which are equal to the same thing are equal to one another.” ”

“ There was a formula—a sort of list of things to say and do—which I recognized as something black and forbidden; something which I had read of before in furtive paragraphs of mixed abhorrence and fascination penned by those strange ancient delvers into the universe’s guarded secrets whose decaying texts I loved to absorb. ”

“ Now how are we to adjudicate among rival ontologies? Certainly the answer is not provided by the semantical formula “To be is to be the value of a variable”; this formula serves rather, conversely, in testing the conformity of a given remark or doctrine to a prior ontological standard. ”

“ When a problem in chemistry is given, or when certain formulas are to be calculated, a knowledge of mathematics is indispensable. [Or] When we face a problem in chemistry, or wish to calculate certain formulas, we find that a knowledge of mathematics is indispensable. ”

“ The mathematical questions which have given birth to the Calculus of Variations consist generally in the investigation of the maxima and minima of certain indeterminate integral formulas, which express the analytical law of such or such a phenomenon of geometry or mechanics, considered independently of any particular subject. ”

“ It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem. I should [Pg 5] say first of all, this: that it shall be possible to establish the correctness of the solution by means of a finite number of steps based upon a finite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated. This requirement of logical deduction by means of a finite number of processes is simply the requirement of rigor in reasoning. ”

“ Theorems and formulae are appropriated wholesale without acknowledgment, and a production which may be described as the organized result of a century of patient toil presents itself to the world as the offspring of a single brain. ”

“ Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascods, so pages of formulae will not get a definite result out of loose data. ”

“ The formulae at which we arrive must be such that a person of any nation, by substituting for the different symbols the numerical value of the quantities as measured by his own national units, would arrive at a true result. ”

“ The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas; ”

“ In abstract mathematics, of course operations alter those particular relations which are involved in the considerations of number and space, and the results of operations are those peculiar results which correspond to the nature of the subjects of operation. But the science of operations, as derived from mathematics more especially, is a science of itself, and has its own abstract truth and value; just as logic has its own peculiar truth and value, independently of the subjects to which we may apply its reasonings and processes. ”

“ We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. These rules of inference constitute the major part of the principles of formal logic. We then take any hypothesis that seems amusing, and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. ”

“ The idea of the analytic function on the other hand includes the whole wealth of functions most important to science, whether they have their origin in number theory, in the theory of differential equations or of algebraic functional equations, whether they arise in geometry or in mathematical physics; and, therefore, in the entire realm of functions, the analytic function justly holds undisputed supremacy. ”

“ We have now arrived at this step,—that we are in possession of a class of symbols , , , &c. satisfying certain laws, and applicable to the rigorous expression of any categorical proposition whatever. It will be our next business to exhibit a few of the general theorems of the calculus which rest upon the basis of those laws, and these theorems we shall afterwards apply to the discussion of particular examples.
Of the general theorems I shall only exhibit two sets: those which relate to the development of functions, and those which relate to the solution of equations. ”

“ To illustrate, — suppose that a mathematician wants to prove something about the value of π, or about the universal laws of Arithmetic, or about the properties of a continuous function, or about the sum of an infinite series, or about the mathematical relationships of two infinite collections of ideal objects. What he is concerned to demonstrate, lies in the realm of the infinite, and of the eternally valid. And our direct experience gives us only the passing data and the fragmentary ideas of the moment. ”

“ While mathematics, and the mathematical sciences, supply us with a typical example of the ascertainment of truth by reasoning; those physical sciences which are not mathematical, such as chemistry, and purely experimental physics, shew us in equal perfection the other mode of arriving at certain truth, by observation, in its most accurate form, that of experiment. ”

“ We have secondly the science of the processes of thought dealing with concepts, judgments and reasoning, i.e., logic, and we have finally the science of those objects which the thought creates freely for its own purposes and which are independent from the content of the world, i.e., mathematics, which leads to the qualitative aspect of general mathematics and the quantitative aspect of concrete mathematics. For our purposes it may be sufficient to separate externally algebra, arithmetic, analysis and geometry. ”

“ The development of this pure mathematics, which comes to be a logic of the mathematical sciences, has been made possible by such a generalization of number theory and theories of the elements of space and time that the rigor of mathematical reasoning is secured, while the physical scientist is left the widest freedom in the choice and construction of concepts and imagery for his hypotheses. The only compulsion is a logical compulsion. ”

“ By dropping the use of diagrams, and substituting, in the demonstrations, general phrases for the letters of the alphabet, we might prove the general theorem directly, that is, we might demonstrate all the cases at once; and to do this we must, of course, employ as our premises, the axioms and definitions in their general form. But this only means, that if we can prove an individual conclusion by assuming an individual fact, then in whatever case we are warranted in making an exactly similar assumption, we may draw an exactly similar conclusion. ”

“ Scientific thought in general, and especially in philosophy, consists of notions, intuitions, and judgments, if only the latter term be taken, in its usual logical sense, to signify the combining together notions or intuitions. Now, of the true mathematical mode of proceeding with notions according to the pure and simple acceptation of the Pythagorean mystery of numbers, and, secondly, of what in its inmost essence is a grammatical method—their combination in methodical thought—we have already spoken. ”

“ Logic, broadly speaking, is distinguished by the fact that its propositions can be put into a form in which they apply to anything whatever. All pure mathematics—Arithmetic, Analysis, and Geometry—is built up by combinations of the primitive ideas of logic, and its propositions are deduced from the general axioms of logic, such as the syllogism and the other rules of inference. ”

“ The true method of discovery in symbolic logic, and probably also the best method for introducing the study to a learner acquainted with other parts of mathematics, is the analysis of actual examples of deductive reasoning, with a view to the discovery of the principles employed. ”

“ The two former we infer from the testimony adduced, or from the traces of those past occurrences which still exist; the latter, from the premises laid down in books of geometry, under the title of definitions and axioms. Whatever we are capable of knowing must belong to the one class or to the other; must be in the number of the primitive data, or of the conclusions which can be drawn from these. ”