The reader will also remember that one of the most frequent operations in mathematical reasoning is to substitute for a quantity its equal, as known either by assumed, natural, or self-evident conditions. Whenever a quantity appears twice over in a problem, we may apply what we learn of its relations in one place to its relations in the other. All reasoning in mathematics, as in other branches of science, thus involves the principle of treating equals equally, or similars similarly. In whatever way we employ quantitative reasoning in the remaining parts of this work, we never can desert the simple principle on which we first set out.
Reasoning by Inequalities.
I have stated that all the processes of mathematical reasoning may be deduced from the principle of substitution. Exceptions to this assertion may seem to exist in the use of inequalities. The greater of a greater is undoubtedly a greater, and what is less than a less is certainly less. Snowdon is higher than the Wrekin, and Ben Nevis than Snowdon; therefore Ben Nevis is higher than the Wrekin. But a little consideration discloses sufficient reason for believing that even in such cases, where equality does not apparently enter, the force of the reasoning entirely depends upon underlying and implied equalities.
In the first place, two statements of mere difference do not give any ground of inference. We learn nothing concerning the comparative heights of St. Paul’s and Westminster Abbey from the assertions that they both differ in height from St. Peter’s at Rome. We need something more than inequality; we require one identity in addition, namely the identity in direction of the two differences. Thus we cannot employ inequalities in the simple way in which we do equalities, and, when we try to express what other conditions are requisite, we find ourselves lapsing into the use of equalities or identities.
In the second place, every argument by inequalities may be represented in the form of equalities. We express that a is greater than b by the equation
where p is an intrinsically positive quantity, denoting the difference of a and b. Similarly we express that b is greater than c by the equation
and substituting for b in (1) its value in (2) we have
Now as p and q are both positive, it follows that a is greater than c, and we have the exact amount of excess specified. It will be easily seen that the reasoning concerning that which is less than a less will result in an equation of the form
Every argument by inequalities may then be thrown into the form of an equality; but the converse is not true. We cannot possibly prove that two quantities are equal by merely asserting that they are both greater or both less than another quantity. From e > f and g > f, or e < f and g < f, we can infer no relation between e and g. And if the reader take the equations x = y = 3 and attempt to prove that therefore x = 3, by throwing them into inequalities, he will find it impossible to do so.
From these considerations I gather that reasoning in arithmetic or algebra by so-called inequalities, is only an imperfectly expressed reasoning by equalities, and when we want to exhibit exactly and clearly the conditions