SUFFICIENT REASON. (Mathematics and Physics.) The prin ciple which is connected with these words might be, and frequently is, called the want of sufficient reason ; and even this term may appear inabeurats, for It should be the want of any possible amount of reason. Since, however, all that takes place must have a sufficient reason (whether we know It or not) for its happening, and everything which is ;asserted must be capable, If true, of being shown to have a sufficient reason, there is no objection to our using the words " want of sufficient reason " in the sense of absolute want of reason, in all matters connected with the exact sciences. If A be equal to s, there must not only be reason, but reason enough for it : anything short of reason enough is no reason at all, and anything short of proof enough is no proof at all.
The use of the word reason in the statement of this principle may itself be fairly objected to We are in the habit of speaking of mathe matical consequences in the same manner as of those to which the notion of cause and effect applies. When one proposition is made to subserve the proof of another, we call the first, one of the reasons of the second, just as we should say that the reason of a flood was the preceding heavy rain. But this mode of speaking must be objection able if the word reason be used in the same sense in both places. For, first, we are at liberty to deny the effect on denying that cause ; if the rain had not fallen, the flood would not have taken place. But when wo say that one mathematical proposition is the reason of another, in which position do we stand if we make an hypothetical denial of the first I Simply in that of persons who assert a contradiction of terms, and try to make rational consequences. Thus, the equality of the angles at the base of an isosceles triangle is one of the reasons (so called) why the tangent of a circle is at right angles to the radius ; rationally, the first is one of the simpler propositions, the necessity of which, when seen, helps us to see the necessity of the second and more complicated one. But the necessity of the first is not previous to that of the second, except in the order of our perceptions, when we follow Euclid. Suppose we were to ask, if the angles at the base of the isosceles triangle had not been equal, what effect would that circum stance have had upon the position of the tangent of a circle ? We might as well inquire what would our geometry have been if two straight lines had been capable of inclosing a space I We remember a book of arithmetic in which it was gravely asked, by way of exercise for the student, " If 6 had been the third part of 12, what would the quarter of 18 have been a question which can only be paralleled by " If a thing were both to exist and not to exist at one and the same moment, how many other non-existences would therefore become existences ?" Secondly, the term reason, in the sense of is wrong as applied to mathematical propositions, because when any one is made to prove the second, it generally happens that the second, when granted, may be made to prove the first. Thus [RIGHT ANGLE] of the
two propositions, " all right angles are equal," and "two Hues which coincide between two points, coincide beyond them," one must be assumed, and the other will then follow : but either may be the one assumed; the other will follow. Now it is absurd to say that of two things each is the previous cause of the other. The whole of this con fusion may be remedied by any one who will remember that one pro position is not the cause of another, but it is our perception, of the one which is made the instrument of bringing about our perception of the other. The constitution of our faculties is the previous cause of the necessity of mathematical propositions, but not of one before another, though in arriving at the perception of this necessity our cognisance of the necessity of one is made the previous cause of that of the necessity of another: To say that n is the consequence of a, is only to say that our knowledge of the truth of n is the consequence of our knowledge of that of A. • Taking care to use the word reason in the sense just alluded to, we assume that whatever is necessary has a possibility of being shown to be necessary, and that whatever is true has a possibility of being shown to be true. If this be a legitimate assumption, it then follows that whatever it is impossible to show to be true, must be false. But can there be such a thing as a proposition of which there shall be seen, not its falsehood, but the impossibility of demonstrating its truth I Can there arise a case in which we shall be so completely cognisant of all that may possibly be said for or against an assertion, as to affirm a necessary incapability of demonstration of one side or the other ? Such cases are universally admitted by mathematicians to exist ; and the final assertion which is made on the known impossibility of proving a contradiction, is said to be made on the principle of the want of sufficient reason. But this very dangerous weapon is never put into the bands of a beginner, in mathematics at least. And when we call it a dangerous weapon, we do not deny its utility, but we only state what is well known to every mathematical teacher, that a student who is allowed to proceed one step by this principle will S0011 ask per inission to make it the universal solvent of difficulties, and will be quite ready to urge that a proposition cannot be shown to be false, in preference to seeking for or following the demonstration that it is true. A beginner can easily admit a sound use of this pririciple, but can hardly distinguish it from the thousand inaccurate applications which his ignorance will make, if it be left in his own hands.