Mathematical principles can be applied to other sciences. All causes operate according to mathematical laws; an effect being ever dependent on the quantity or a function of the agent, and generally on its position too. Mathematical principles cannot, indeed, as M. Comte has well explained, be usefully applied to physical questions, whenever the causes are either too inaccessible for their numerical laws to be ascertained, or are too complex for _us_ to compute the effect, or are ever fluctuating. And, in proportion as physical questions cease to be abstract and hypothetical, mathematical solutions of them become imperfect. But the great value of mathematical training is, that we learn to use its _method_ (which is the most perfect type of the Deductive Method), that is, we learn to employ the laws of simpler phenomena to explain and predict those of the more complex. CHAPTER XXV. THE GROUNDS OF DISBELIEF. The result of examining evidence is not always belief, or even suspension of judgment, but is sometimes positive disbelief. This can ensue only when the affirmative evidence does not amount to full proof, but is based on some approximate generalisation. In such cases, if the negative evidence consist of a stronger, though still only an approximate, generalisation, we think the fact improbable, and disbelieve it provisionally; but if of a complete generalisation based on a rigorous induction, it is disbelieved by us totally, and thought impossible. Hence, Hume declared miracles incredible, as being, he considered, contrary to a complete induction. Now, it is true that _in the absence of any adequate counteracting cause_, a fact contrary to a complete induction is incredible, whatever evidence it may be grounded on; unless, indeed, the evidence go to prove the supposed law inconsistent with some better established one. But when a miracle is asserted, the presence of an adequate counteracting cause _is_ asserted, viz. a direct interposition of an act of the will of a Being having power over nature. Therefore, all that Hume proved is, that we cannot believe in a miracle unless we believe in the power, and _the will_, of the Deity to interfere with existing causes by introducing new ones; and that, in default of such belief, not the most satisfactory evidence of our senses or of testimony can hinder us from holding a seeming miracle to be merely the result of some unknown natural cause. The argument of Dr. Campbell and others against Hume, however, is untenable, viz. that, as we do not disbelieve an alleged fact (which may be something conforming to the uniform course of experience) merely because the chances are against it, therefore we need never disbelieve any fact supported by credible testimony (even if contrary to the uniform course of experience). But this is to confound _improbability before the fact_, which is _not_ always a ground for disbelief, with _improbability after the fact_, which always is.