Addition

“ Write the tens to be carried each time in a smaller figure underneath the units, so that afterwards any column can be added over again without repeating the entire operation. By the practice of addition the eye and mind soon become accustomed to act rapidly, and this is the art of addition. ”

“ If we are engaged in teaching addition combinations, we should be absolutely certain that we have taught every possible combination. If we want to be sure that children know how to write numbers up to one million, we must have given them drill on all of the possible difficulties. If children are always to respond correctly when problems involving two steps in reasoning are presented, we must have been careful to provide for the purpose of drill all of the combinations of situations involving addition, multiplication, subtraction, and division which can occur. ”

“ Gradgrind, who was the person overhearing, stepped forth into the light and said, ‘Louisa, never wonder!’
Herein lay the spring of the mechanical art and mystery of educating the reason without stooping to the cultivation of the sentiments and affections. Never wonder. By means of addition, subtraction, multiplication, and division, settle everything somehow, and never wonder. Bring to me, says M’Choakumchild, yonder baby just able to walk, and I will engage that it shall never wonder. ”

“ Yet we find that by carefully graduating our lines we can make it intuitive and accessible to a child who has learned that the common operations of addition and subtraction can be represented with material objects. The generation of negative and positive quantities follows quite naturally. ”

“ These various operations might be brought within the compass of the science, and the additional advantage be obtained of a very simple definition, if, by an extension of the term, sanctioned by high authorities, we were to define logic as the science which treats of the operations of the human understanding in the pursuit of truth. ”

“ All great masters are chiefly distinguished by the power of adding a second, a third, and perhaps a fourth step in a continuous line. Many a man had taken their first step. With every additional step you enhance immensely the value of your first. ”

“ Induction, then, is that operation of the mind, by which we infer that what we know to be true in a particular case or cases, will be true in all cases which resemble the former in certain assignable respects. In other words, Induction is the process by which we conclude that what is true of certain individuals of a class is true of the whole class, or that what is true at certain times will be true in similar circumstances at all times. ”

“ For again we divide that which walks into rational and irrational; and then selecting from the species, apprehended by division, those next to man, and combining them into one formula, we state the definition of a man, who is an animal, mortal, terrestrial, walking, rational.
Whence Division furnishes the class of matter, seeking for the definition the simplicity of the name; and the definition of the artisan and maker, by composition and construction, presents the knowledge of the thing as it is; not of those things of which we have general notions. ”

“ Induction infers a rule. Now, the belief of a rule is a habit. That a habit is a rule active in us, is evident. That every belief is of the nature of a habit, in so far as it is of a general character, has been shown in the earlier papers of this series. Induction, therefore, is the logical formula which expresses the physiological process of formation of a habit. Hypothesis substitutes, for a complicated tangle of predicates attached to one subject, a single conception. ”

“ As has been already pointed out, this is all that art can do. To test a generalization, by showing that it either follows from, or conflicts with, some stronger induction, some generalization resting on a broader foundation of experience, is the beginning and end of the logic of induction. ”

“ It is necessary to proceed by observation, by experiment, and then by induction, but with prodigious mistrust of induction. Induction consists in drawing conclusions from the particular to the general, from a certain number of facts to a law. ”

“ In induction, because present knowledge falls short, the major step of analysis necessary to abstraction of the essential is impossible, and therefore the search for new facts must come first, whereas in deduction, the analysis of the particular situation results in a search for generals and a classification of the situation in question. (3) In induction many particular facts may be necessary before one concept or principle is made adequate, while in deduction many concepts or principles may be examined before one particular is classified. ”

“ To begin with the conditions of direct Induction.—An Induction is an universal real proposition, based on observation, in reliance on the uniformity of Nature: when well ascertained, it is called a Law. ”

“ Extension is quantitative, while intension is qualitative. An extensional use means to point out or number objects, while an intensional use means to describe by naming qualities. To name is to use a term in extension—to describe is to use a term in intension.
To divide a term into its kinds we must regard it in an extensional sense; e. g., the term man may be divided into Caucasian, Mongolian, Malay, Ethiopian, American Indian.
To define a term we must regard it in an intensional sense; e. g., man is a rational animal. ”

“ Induction always signifies that observations are instituted, experiments made, experience regarded, and from this the universal determination is derived.
We have already called to mind how important it is to lead on to the content as the content of actuality, of the present; for the rational must have objective truth. The reconciliation of spirit with the world, the glorification of nature and of all actuality, must not be a Beyond, a Futurity, but must be accomplished now and here. ”

“ The indispensable foundation, therefore, of a scientific formula of induction, must be a survey of the inductions to which mankind have been conducted [pg 331] in unscientific practice; with the special purpose of ascertaining what kinds of uniformities have been found perfectly invariable, pervading all nature, and what are those which have been found to vary with difference of time, place, or other changeable circumstances. ”

“ Of Induction, therefore, we shall say no more at present, than that it at least is, without doubt, a process of real inference. The conclusion in an induction embraces more than is contained in the premisses. The principle or law collected from particular instances, the general proposition in which we embody the result of our experience, covers a much larger extent of ground than the individual experiments which are said to form its basis. ”

“ When, from the observation of a number of individual instances, we ascend to a general proposition, or when, by combining a number of general propositions, we conclude from them another proposition still more general, the process, which is substantially the same in both instances, is called Induction. ”

“ Each category, or determination, is a general principle by which we seek to make experience a consistent whole and render it a concrete universal; and each has its own way of relating the two root factors, viz., unity and difference. ”

“ Throughout which series of instances we see, that as the number of objects grouped together in thought increases, the concept, formed of a few typical samples joined with the notion of multiplicity, becomes more and more a mere symbol ”

“ We have, therefore, the warrant of a rigid induction for considering it probable, in a degree undistinguishable from certainty, that the known conditions requisite for the sun’s rising will exist to-morrow. ”

“ The collection of the parts, the parts themselves united. The idea of the parts then enters into the idea of the whole. What is the meaning of greater? One thing is said to be greater than another, when, besides containing an equal quantity, it also contains something else. ”

“ All cases, then, tend to show that deduction involves generalization from an observed case on a single axiom; the axioms, therefore, of induction are the axioms of reasoning, and the generalization of particulars is the work of reason. ”

“ Adaptation of means to an end may be produced in two ways that we at present know of; by processes of natural selection, and by the agency of an intelligence in which an image or idea of the end preceded the use of the means. In both cases the existence of the adaptation is accounted for by the necessity or utility of the end. ”

“ If objects are compared in reference to the succession {107} of our conceptions or perceptions, we gain the ideas of such relations as are expressed by the terms firstly, secondly, and thirdly. If objects are compared in reference to the degree of any quality, we gain an idea of such relations as are expressed by the terms brighter, sweeter, harder, louder. If objects are compared in reference to proportion, we gain ideas of such relations as are expressed by the terms an eighth, a half. ”

“ Perfect Induction necessitates a knowledge of all the particulars forming a class; that is, all the individual objects, persons, things or facts comprising a class must be known and enumerated in this form of Induction. ”

“ Thus a thing stands all the higher in the vital scale in proportion as, from the beginning to the end of its vital manifestations, it exhibits a larger number of successive and simultaneous changes, and as these changes are more heterogeneous and more closely linked together, and in more definite relations to one another. ”

“ Acts that produce external changes imply adjustment of means to ends. The creation of external things is such an act. Unless an actor is identical with the product itself, he must therefore be subject to the external conditions of adjustment, i.e., he must be finite. ”

“ This is why there is real utility in the process of deduction, which goes from the general to the general, or from the general to the particular, as well as in the process of induction, which goes from the particular to the particular, or from the particular to the general. It is an old debate among philosophers whether deduction ever gives new knowledge. ”

“ The Methods of Induction which are treated of in this and the preceding chapter, and which are specially applicable to causes governed by relations of Quantity or of Resemblance, commonly lead us to Laws of Phenomena only. Inductions founded upon other ideas, those of Substance and Cause for example, appear to conduct us somewhat further into a knowledge of the essential nature and real connexions of things. ”

“ For how, in fact, can we conceive combinations, otherwise than as definite in kind and quantity? If we were to suppose each element ready to combine with any other indifferently, and indifferently in any quantity, we should have a world in 26 which all would be confusion and indefiniteness. ”

“ Distinguish between induction as a mode of inference and induction as a method.
(7) State and explain the law of universal causation. Illustrate fully.
(8) Make evident that a cause may involve many antecedents.
(9) State and explain by illustration the law of uniformity of nature.
(10) Verify by illustration the notion that the “fact of causation” conditions all induction.
(11) Which of the two laws is empirical, “causation” or “uniformity”? Why?
(12) Show that induction is a form of thinking.
(13) Why should the law of uniformity of nature convince man that nature is honest? ”

“ Now, Induction is mainly a process for finding the causes of effects: and in attempting to fathom the mode of tracing causes and effects in physical science, I soon saw that in the more perfect of the sciences, we ascend, by generalization from particulars, to the tendencies of causes considered singly, and then reason downward from those separate tendencies, to the effect of the same causes when combined. I then asked myself, what is the ultimate analysis of this deductive process; the common theory of the syllogism evidently throwing no light upon it. ”

“ I say these things after all with the sense, so founded on past experience, that, in closer quarters and the intimacy of composition, pre-noted arrangements, proportions and relations, do most uncommonly insist on making themselves different by shifts and variations, always improving, which impose themselves as one goes and keep the door open always to something more right and more related. ”

“ In proportion as in what she offers us we see life without rearrangement do we feel that we are touching the truth; in proportion as we see it with rearrangement do we feel that we are being put off with a substitute, a compromise and convention. ”

“ But there remain a large number of the recognised problems of philosophy in regard to which the method advocated gives all those advantages of division into distinct questions, of tentative, partial, and progressive advance, and of appeal to principles with which, independently of temperament, all competent students must agree. ”

“ Every considerable change historically known to us in the condition of any portion of mankind, when not brought about by external force, has been preceded by a change, of proportional extent, in the state of their knowledge, or in their prevalent beliefs. ”

“ In the other case, the answers were given in words conveying ideas which were supposed to be those most acceptable to the teacher.
It is often helpful to state the opposite of the common expression of a generalization and to suggest that you are willing to maintain this point of view. The best lesson that the writer ever conducted on induction and deduction was begun with the statement: “Induction always begins with a generalization and moves to the consideration of particulars. Deduction always begins with a particular and moves to a generalization.” ”

“ It is of the utmost importance for understanding the logic of induction, that we should form a distinct conception of what is meant by chance, and how the phenomena which common language ascribes to that abstraction are really produced. ”