Pointer

“ Our idea of a point, I apprehend to be simply our idea of the minimum visibile, the smallest portion of surface which we can see. A line, as defined by geometers, is wholly inconceivable. We can reason about a line as if it had no breadth; because we have a power, which is the foundation of all the control we can exercise over the operations of our minds; the power, when a perception is present to our senses, or a conception to our intellects, of attending to a part only of that perception or conception, instead of the whole. ”

“ A topic is taken, and without reference to the order of time or the words of the text, is unfolded as a proposition in Geometry—each thought being preliminary to that which follows, and the whole ending in the demonstration of some great truth, and the deduction of its legitimate corollaries. ”

“ How do we decide what the sense of a proposition is, or what we mean by a sentence which is spoken, written, or printed? We try to present to ourselves the significance of the different words that we have learned to use, and then endeavor to find sense in the proposition. ”

“ And it is remarkable that, among all the definitions of the proposition, for example, as the oratio indicativa, as the subsumption of an object under a concept, as the expression of the relation of two concepts, and as the indication of the mutable ground of appearance, there is, perhaps, not one in which the conception of reference to an object or correlate is not the important one. In the same way, the conception of reference to an interpretant or third, is always prominent in the definitions of argument. ”

“ The parts of the rational explanation or notion, into which that notion is divided, are prior to the notion, at least in some instances. But the notion of a right angle is prior to that of an acute angle or is one of the elements into which the notion of an acute angle is divided; for you cannot define an acute angle without introducing the right angle into your definition, nor can you define the semicircle without introducing the circle, nor the finger without introducing the man — the finger being such and such a part of the man. ”

“ WE have given the appellation of Ideas to certain comprehensive forms of thought,—as space, number, cause, composition, resemblance,—which we apply to the phenomena which we contemplate. But the special modifications of these ideas which are 31 exemplified in particular facts, we have termed Conceptions; as a circle, a square number, an accelerating force, a neutral combination of elements, a genus. ”

“ General Principles.—There is perhaps no other form of composition which is so generally in use as exposition or explanation. If you observe your own conversation and that of the people about you, you will find that a great deal of it is explanation. Every time you say in answer to some question about a remark you have just made, "Why, I mean that—," you are explaining the first remark. In almost all the recitations you make in school you are explaining something—a principle in arithmetic, or in physics, the construction of something in manual training, the meaning of a word, etc. ”

“ There are other kinds of singular points, the properties of which we shall now investigate, but, before doing so, we must define some expressions which we shall find useful in emancipating our ideas from the thraldom of systems of coordinates. ”

“ The fact that I have never studied the subject does not at all disqualify me from writing about it, if I were to follow the standard set up by some who have gone before me. Long essays have appeared from the pens of men who could hardly tell the difference between a pediment, and a cornice, or explain why a segment is not an angle or an angle a segment. ”

“ But the next step in composing, which has been mentioned throughout the book, is more necessary in argumentation than in any other form of expression. You must not only have an outline in mind for what you are about to say, but that outline should be written, and almost as much time and thought should be given to it as to the composition itself; for clear thought is the great essential in argumentation, and a carefully prepared outline is the greatest help to clear thought.
88. The Introduction.—There are three parts to every outline for a discussion or argument. ”

“ We can therefore easily understand why the mathematician and the symbolic logician are driven to discard the word and to build up their thought with the help of symbols which have, each of them, a rigidly unitary value. ”

“ Verbs of choosing, calling, naming, making, and thinking may take two objects referring to the same person or thing.
The first of these is the direct object, and the second, which completes the sense of the predicate, is called a predicate objective.
We chose Oscar president. [Oscar is the direct object of chose; president is the predicate objective.] I call John my friend. They thought the man a coward. Make my house your home. The predicate objective is often called the complementary object or the objective attribute. It is classed as a complement. ”

“ An illustration will make this plain: let us suppose the centre of a circle to be endowed with consciousness, and suppose we affirm that this centre can have no cognizance of the circumference, without being cognizant of itself (the centre) as well. ”

“ The Criterion is, that the person shall see the necessity of the Axioms belonging to each Idea;—shall accept them in such a manner as to perceive the cogency of the reasonings founded upon them. Thus, a person has a clear Idea of Space who follows the reasonings of geometry and fully apprehends their conclusiveness. The Explication of Conceptions, which we are speaking of as an essential part of real knowledge, is the process by which we bring the Clearness of our Ideas to bear upon the Formation of our knowledge. ”

“ Now let us make another agreement--that a red counter in a compartment shall mean that it is 'OCCUPIED', that is, that there are SOME Cakes in it. (The word 'some,' in Logic, means 'one or more' so that a single Cake in a compartment would be quite enough reason for saying "there are SOME Cakes here") . ”

“ In perception they can be distinguished as right and left, up and down, and so forth. But since what is right to one percipient is left to another, in conceptual space, where complete abstraction is made from the presence of an individual percipient, there is neither right nor left, up nor down, nor any other qualitative difference between one direction and another, all such differences being relative to the individual percipient. ”

“ The method in which we simultaneously contemplate two figures, and recognize a correspondence between certain points in the one figure and certain points in the other, is one of the most powerful and fertile methods hitherto known in science. ”

“ Perhaps the 1st point could include the 2d and the 3d. Be careful not to split your general subject up into very many parts. See, too, that no point is repeated, that no point foreign to the subject is introduced, and that all the points together exhaust the subject as nearly as may be. Look to the arrangement of the points. There is a natural order; (6) could not precede (5) ; nor (5) , (4) ; nor (4) , (1) .
TO THE TEACHER.—Question the pupils carefully upon every point taken up in this Lesson. ”

“ If you examine a separate piece of the watch, and proceed to draw your conclusions according to the well-known laws of mechanics as to the construction of the other necessary parts, in order to give to that one part, which you actually perceive, the whole determinateness which you perceive belongs to it, do you, in this your function of drawing conclusions, actually see and feel, or perceive with your external senses, those other parts? ”

“ It is only for some artificial purpose that we ever look at the proposition in the aspect in which the predicate also is thought of as a class-name, either including the subject only, or the subject and something more. ”

“ I've seen the professor work lots of sums on the blackboard, and he claimed anything could be done with x's and y's and a's, and such things, by mixing them up with plenty of plusses and minuses and equals, and so forth. ”

“ When functions are mentioned or implied in the notation, the reader will always understand single-valued functions, unless the contrary is distinctly intimated, or the case is one in which the distinction is obviously immaterial. ”

“ This approximation grows more and more nearly exact as the difference grows less; in technical language, the laws of perspective account for the differences to the first order of small quantities, and other laws are only required to account for second-order differences. ”

“ 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. ”

“ We found that whatever be the form of the proposition, and whatever its nominal subject or predicate, the real subject of every proposition is some one or more facts or phenomena of consciousness, or some one or more of the hidden causes or powers to which we ascribe those facts ”

“ Position and proportion are the means of emphasis in a sentence. By placing the important words near the major marks of punctuation, by arranging the parts in a climax or a period, by forcing words out of the natural order, and by subduing unimportant details, a sentence is massed to give the important elements their relative emphasis. Last, the parts of a sentence should be arranged so that their relations shall be clear and unmistakable. Proximity of related parts, parallel construction for parallel ideas, and connectives are the surest means of securing Coherence in a sentence. ”

“ It will be hereafter seen that when any important consequences seem to follow, as in mathematics, from an essential proposition, or, in other words, from a proposition involved in the meaning of a name, what they really flow from is the tacit assumption of the real existence of the objects so named. ”

“ In fact, however, mankind have always given general names to objects as well as attributes, and indeed before attributes: but the general names given to objects imply attributes, derive their whole meaning from attributes; and are chiefly useful as the language by means of which we predicate the attributes which they connote. ”

“ Position.—The sight has this peculiar prerogative, that it apprehends the place of its objects directly and primarily. We see where an object is at the same instant that we see what it is. If we see two objects, we see their relative position. We cannot help 303 perceiving that one is above or below, to the right or to the left of the other, if we perceive them at all. ”

“ Until one has gained almost complete mastery of the meanings of marks and of the subtle sense relations between the groups of words constituting language, he cannot interpret such relations when indicated by marks or by their omission. ”

“ The peculiar simplicity of an instantaneous point has a twofold origin, one connected with position, that is to say with its character as a punct, and the other connected with its character as an event-particle. ”

“ The second kind of imperfect definition, therefore, in which the name of a class is defined by any of its accidents,—that is, by attributes which are not included in its connotation,—has been rejected from the rank of genuine Definition by all logicians, and has been termed Description. ”

“ Thus in studying a given kind of geometry the pure mathematician is studying a certain class of relations defined by means of certain abstract logical properties which take the place of what used to be called axioms. ”

“ Whether, therefore, we conform to the practice of those who have made the subject their particular study, or to that of popular writers and common discourse, the province of logic will include several operations of the intellect not usually considered to fall within the meaning of the terms Reasoning and Argumentation. ”

“ In the same way we derive from several general ideas of the same degree another general idea, and so on successively, step by step, always proceeding according to the natural order of things, by constant analysis, using expressive signs, as with mathematicians in passing from calculation by the fingers to calculation by numerals, and from this to calculation by letters, and who, calling upon the eyes to aid Reason, depict the inward analogy of quantities by the outward analogy of symbols. ”

“ With its aid objects permit themselves to be described as multiples of one another, and as occupying positions in relation to one another. When all objects are described strictly in terms of one another, they are expressed in terms of arbitrary units, and located in terms of arbitrary spacial or temporal axes of reference. ”

“ The truths of geometry are summed up and embodied in its definitions and axioms.... Let us turn to the axioms, and what do we find? A string of propositions concerning magnitude in the abstract, which are equally true of space, time, force, number, and every other magnitude susceptible of aggregation and subdivision. Such propositions, where they are not mere definitions, as some of them are, carry their inductive origin on the face of their enunciation.... ”