Front Page Titles (by Subject) CHAPTER II.: COMPOUND QUANTITATIVE REASONING (CONTINUED). - The Principles of Psychology
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CHAPTER II.: COMPOUND QUANTITATIVE REASONING (CONTINUED). - Herbert Spencer, The Principles of Psychology 
The Principles of Psychology (London: Longman, Brown, Green and Longmans, 1855).
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COMPOUND QUANTITATIVE REASONING (CONTINUED).
§ 21. The results reached in the last chapter do not, apparently, help us very far on the way to a theory of Quantitative Reasoning. Such an intuition as that expressed in the axiom educed, can form but one amongst the many intuitions which, joined together, constitute a mathematical argument. A moment's reflection will show that however many times quoted, or applied in thought, the axiom—Relations which are equal to the same relation are equal to each other, can never do anything else than establish the equality of some two relations by the intermediation of a series of relations severally equal to both: and there are few if any cases, save those furnished by algebraic and allied processes, in which the equality of two relations is the fact to be arrived at; or could be thus arrived at if it were. The proposition—“If two circles touch each other externally, the straight line which joins their centres shall pass through the point of contact,” is one with which such an axiom can have no concern: and the same is manifestly the case with the great majority of geometrical truths. Some more general cognition, then, has to be found.
Guidance in the search for such a cognition, may be drawn from the consideration that if a truly fundamental one, it must be involved not only in all other kinds of quantitative reasoning, but also in the kind exemplified in the preceding chapter. It must underlie both. This being an à priori necessity, it follows that as, in the case of algebraic reasoning, the foregoing axiom expresses in general language the sole cognition by which the successive steps are rationally co-ordinated, the required fundamental cognition must be somehow involved in it. I seems therefore, that our best course will be to continue the line of analysis already commenced.
If then, ceasing to consider in its totality the complex axiom—Relations which are equal to the same relation are equal to each other, we go on to inquire what are the simpler elements of thought into which it is proximately decomposable; we at once see that it twice over involves a recognition of the equality of some two relations. Before it is possible to predicate that the relations A : B and E : F being severally equal to the relation C : D, are equal to each other; it must first be predicated that the relation A : B is equal to the relation C : D; and that the relation C : D is equal to the relation E : F. Hence the intellectual act which we have now to consider, is the establishment of a relation of equality between two relations. And this is the intellectual act of which we are in search. An intuition of the equality of two relations is implied in every step, alike of that quantitative reasoning which deals with homogeneous magnitudes, and that which deals with magnitudes that are not homogeneous—is the ultimate ratiocinative act into which every complete mathematical argument is resolvable. Let us take as our first field for the exemplification of this fact, the demonstration of geometrical theorems.
§ 22. Before analysing the steps by which a proposition is proved, we may with advantage contemplate the substance of a proposition; and consider by what process the mind advances from that particular case of it which the demonstration establishes, to the recognition of its general truth. Let us take as an example, the proposition—“The angles at the base of an isosceles triangle are equal to each other.”
To establish this, the abstract terms are forthwith abandoned, and the proposition is re-stated in a concrete form. Let A B C be an isosceles triangle of which the side A B is equal to the side A C; then the angle A B C shall be equal to the angle A C B. By a series of steps which need not be here specified, the way is found from these premisses to this conclusion. It is definitely demonstrated that the angle A B C is equal to the angle A C B. But now mark what takes place. As soon as this particular fact has been proved, the general fact is immediately re-enunciated and held to be proved. We pass directly from the concrete inference—the angle A B C is equal to the angle A C B, to the abstract inference—therefore the angles at the base of an isosceles triangle are equal to each other. Q. E. D. Be the cogency of every step in the demonstration what it may, the truth of the proposition at large hinges entirely upon the cognition that what holds in this case holds in all cases. What now is the nature of this cognition? It is a consciousness of the equality of two relations: on the one hand, the relation subsisting between the sides and angles of the triangle A B C; and on the other hand, the relation subsisting between the sides and angles of another isosceles triangle, of any isosceles triangle, of all isosceles triangles. Whatever theory be espoused respecting the mode in which we figure to ourselves a class—whether in the present case the abstract fact be recognized only after it has been seen to hold in this isosceles triangle, and in this, and in this; or whether after it has been seen to hold in some ideal type of an isosceles triangle; does not in the least affect the position that the thing discerned is the equality of the relations presented in successive concepts. If we use the letter A to symbolize the premised fact (viz. that in the triangle A B C the sides A B and A C are equal), and the letter B to symbolize the fact asserted (viz. that the angle A B C is equal to the angle A C B); then, after establishing a certain relation (of coexistence) between A and B in this one case, we go on to affirm that the same relation holds between some other A and B, or all As and Bs: or strictly speaking, not the same relation, but an equal relation. And as, for this affirmation, we can assign no reason, it manifestly represents a simple intuition.
But not only do we pass from the special truth to the general truth by an intuition of the equality of two relations: a like intuition is implied in each of the steps by which the special truth is reached. In the demonstration of such special truth, the truths previously established are explicitly or implicitly referred to; and the relations that subsist in the case in hand are recognized as equal to relations which those previously established truths express. This will be at once seen on subjecting a demonstration to analysis. The one belonging to the foregoing theorem is inconveniently long: we shall find a fitter one in Proposition xxxii.
“If the side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.”
“Let A B C be a triangle, and let one of its sides B C be produced to D; then the exterior angle A C D is equal to the two interior and opposite angles C A B, A B C; and the three interior angles of the triangle, namely A B C, B C A, C A B, are together equal to two right angles.”
Thus, alike in each step by which the special conclusion is reached, and in the step taken from that special conclusion to the general one, the essential operation gone through is the establishment in consciousness of the equality of two relations. This is the bare abstract statement of the thing effected. If this is not done, nothing is done. And as, in each such cognition, the mental act is undecomposable—as for the assertion that any two such relations are equal, no reason can be assigned save that they are perceived to be so; it is manifest that the whole process of thought is thus expressed.
§ 23. Perhaps it will be deemed scarcely needful specifically to prove that each step in an algebraic argument is of the same nature. But though, by showing that the axiom—Relations which are equal to the same relation are equal to each other, twice involves an intuition of the above described kind, it may have been implied that the reasoning which proceeds upon that axiom, is built up of such intuitions; yet it will be well definitely to point out that only in virtue of such intuitions do the successive transformations formations of an equation become allowable. Unless it is perceived that a certain modification made in the form of the equation, leaves the relation between its two sides the same as before—unless it is seen that each new relation established is equal to the foregoing one, the reasoning is vicious and the result erroneous. A convenient mode of showing that the mental act continually repeated in one of these analytical processes is of the kind described, is suggested by an ordinary algebraic artifice. When a desired simplification may be thereby achieved, it is usual to throw any two forms of an equation into a proportion: a procedure in which the equality of the relations is specifically asserted. Here is an illustration: not such an one as the algebraist would choose; but one which will serve present purposes.
or, as it is otherwise written,
and if proof be needed that this mode of presenting the facts is legitimate, we may at once obtain it by multiplying extremes and means; whence results the truism—
This clearly shows that the mental act determining each algebraic transformation, is one in which the relation expressed by the new form of the equation is recognized as equal to the relation which the previous form expresses. Only in virtue of this equality is the step valid: and hence the intuition of this equality must be the essence of the step.