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doubt of the great law in physics, that heavy bodies descend to the earth in a line directed towards its centre. But we can conceive of the opposite of this without involving any contradiction or absurdity. In other words, they might have been subjected, if the Creator had so determined, to the influence of a law requiring them to move in a different direction. But, on a thorough examination of a demonstrative process, we shall find ourselves unable to admit even the possibility of the opposite.
$284. Demonstrations do not admit of different degrees of belief.
When our thoughts are employed upon subjects which come within the province of moral reasoning, we yield different degrees of assent; we form opinions more or less probable. Sometimes our belief is of the lowest kind; nothing more than mere presumption. New evidence gives it new strength; and it may go on, from one degree of strength to another, till all doubt is excluded, and all possibility of mistake shut out. It is different in demonstrations; the assent which we yield is at all times of the highest kind, and is never susceptible of being regarded as more or less. This results, as must be obvious on the slightest examination, from the nature of demonstrative reasoning.
In demonstrative reasonings we always begin with certain first principles or truths, either known or taken for granted; and these hold the first place, or are the foundation of that series of propositions over which the mind successively passes, until it rests in the conclusion. In mathematics, the first principles, of which we here speak, are the definitions.
We begin, therefore, with what is acknowledged by all to be true or certain. At every step there is an intuitive perception of the agreement or disagreement of the propositions which are compared together. Consequently, however far we may advance in the comparison of them, there is no possibility of falling short of that degree of assent with which it is acknowledged that the series commenced.-So that demonstrative certainty may be judged to amount to this. Whenever we arrive at the last step or the conclusion of a series of propositions, the
mind, in effect, intuitively perceives the relation, whether it be the agreement or disagreement, coincidence or want of coincidence, between the last step or the conclusion, and the conditions involved in the propositions at the commencement of the series; and, therefore, demonstrative certainty is virtually the same as the certainty of intuition. Although it arises on a different occasion, and is, therefore, entitled to a separate consideration, there is no difference in the degree of belief.
285. Of the use of diagrams in demonstrations.
In conducting a demonstrative process, it is frequently the case that we make use of various kinds of figures or diagrams. The proper use of diagrams, of a square, circle, triangle, or other figure, which we delineate before us, is to assist the mind in keeping its ideas distinct, and to help in comparing them together with readiness and correctness. They are a sort of auxiliaries, brought in to the help of our intellectual infirmities, but are not absolutely necessary, since demonstrative reasoning, wherever it may be found, resembles any other kind of reasoning in this most important respect, viz., in being a comparison of our ideas.
In proof that artificial diagrams are only auxiliaries, and are not essentially necessary in demonstrations, it may be remarked, that they are necessarily all of them imperfect. It is not within the capability of the wit and power of man to frame a perfect circle, or a perfect triangle, or any other figure which is perfect. We might argue this from our general knowledge of the imperfection of the senses; and we may almost regard it as a matter determined by experiments of the senses themselves, aided by optical instruments. "There never was," says Cudworth, "a straight line, triangle, or circle, that we saw in all our lives, that was mathematically exact, but even sense itself, at least by the help of microscopes, might plainly discover much unevenness, ruggedness, flexuosity, angulosity, irregularity, and deformity in them."**
Our reasonings, therefore, and our conclusions will not * Treatise concerning Immutable Morality book iv., ch. iii.
apply to the figures before us, but merely to an imagined perfect figure. The mind can not only originate a figure internally and subjectively, but can ascribe to it the attribute of perfection. And a verbal statement of the properties of this imagined perfect figure is what we understand by a DEFINITION, the use of which, in this kind of reasoning in particular, has already been mentioned.
§ 286. Of signs in general as connected with reasoning. The statements in the last section will appear the less exceptionable when it is recollected that in all cases reasoning is purely a mental process. From beginning to end, it is a succession of perceptions. Neither mathematical signs nor words constitute the process, but are only its attendants and auxiliaries. We can reason without diagrams or other signs employed in mathematics, the same as an infant reasons before it has learned artificial language.
When the infant has once put his finger in the fire, he avoids the repetition of the experiment, reasoning in this way, that there is a resemblance between one flame and another, and that what has once caused him pain, will be likely, under the same circumstances, to cause the same sensation. When the infant sees before him some glittering toy, he reaches his hand towards it, and is evidently induced to do so by a thought of this kind, that the acquisition of the object will soon follow the effort of the hand, as it has a similar effort previously made. Here is reasoning without words; it is purely internal; nevertheless, no one will presume to say that words are not great helps in reasoning. And thus in demonstrative reasoning, although diagrams, and numerical and algebraic signs are assistances, they do not constitute the process; nor can it be even said that they are indispensably essential to it.
"Some geometricians," says Buffier (First Truths, pt. i., ch. 6), “ are led into a palpable error in imagining that things demonstrated by Geometry exist, out of their thought, exactly similar to the demonstration formed of them in their mind. They must be quickly sensible of their mistake, if they will but reflect a moment on the VOL. I-H H
perfect globe, the imaginary properties of which are demonstrated in Geometry, though the thing itself has no real existence in nature. Geometry shows nothing of the existence of things, but only what they are, suppo sing them to exist really such as they are conceived by the mind. And, indeed, were all created things existing annihilated, geometry would not lose a single point of its demonstrations; the circle would still remain a round figure, of which all the points of the circumference would be equally distant from the centre."
◊ 287. Of the influence of demonstrative reasoning on the mental character.
A considerable skill in demonstrative reasoning is on a number of accounts desirable, although it cannot be denied that very frequent practice and great readiness in it are not always favourable; so that it seems proper briefly to mention some of the effects, both propitious and unpropitious, on the mental character.
(1.) A frequency of practice in demonstrative reasoning greatly aids in giving one a ready command of his attention. And this is said for two reasons. First, because the subjects of such reasoning are not objects of the senses, but immaterial; are conceptions rather than existences; the abstractions of things rather than things themselves; and, consequently, are not distinctly comprehensible without considerable effort. And, second, because, in this species of reasoning, the propositions follow each other in such regular order and so closely, and so great is the importance of perceiving the agreement or disagreement of each succeeding one with that which goes before, that a careless, unfixed, and dissipated state of the mind seems to be utterly inconsistent with carrying on such a process with any sort of success to the conclusion. As, therefore, the strictest attention is here so highly necessary, the more a person subjects himself to this discipline, the more ready and efficient will be the particular application of the mind to which we give that And we often find distinguished individuals in political life and in the practice of the law who are desirous of holding their mental powers in the most prompt
and systematic obedience, imposing on themselves exercises in geometry and algebra for this purpose.
(II.) This mode of reasoning accustoms one to care and discrimination in the examination of subjects.--In all discussions where the object is to find out the truth, it is necessary to take asunder all the parts having relation to the general subject, and bestow upon them a share of our consideration. And, in general, we find no people more disposed to do this than mathematicians; they are not fond of reasoning, as Mr. Locke expresses it. in the lump, but are for going into particulars, for allowing everything its due weight and nothing more, and for resolutely throwing out of the estimate all propositions which are not directly and fully to the point.-It must further be said, as a general remark closely connected with what has just been observed, that those departments of science which require demonstrative reasoning are promotive of a characteristic of great value-a love of the truth.
(III.) Demonstrative reasoning, although this beneficial result is not exclusively appropriate to this mode of reasoning, gives to the mind an increased grasp or comprehension. This result, it is true, will not be experienced in the case of those who have merely exercised themselves in the study of a few select demonstrations; it implies a familiarity of the mind with long and complicated trains of deductions. A thorough mathematician, who has made it a business to exercise himself in this method of reasoning, can hardly have been otherwise than sensible of that intellectual comprehension, or length and breadth of survey, which we have in view; since one demonstration is often connected with another, much in the same way as the subordinate parts of separate demonstrations are connected with each other; and he therefore finds it necessary, if he would go on with satisfaction and pleasure, to gather up and retain, in the grasp of his mind, all the general and subordinate propositions of a long treatise.
188. Further considerations on the influence of demonstrative rea
But, on the other hand, there are some results of a