Whatever may be the psychological hypothesis to which it ie referred, it is certain that there is a real distinction between a mathematical fact and one of any other kind. If we say that an un supported bit of lead will fall to the ground, we state a fact of which we are as certain, in the sense of reliance, as we are of the other pro-' position that two straight lines cannot inclose a space. But in the for mer preposition an exception, or even a permanent alteration of the law, is conceivable by the imagination : in the latter the mind would sensible of absurdity if it attempted to construct the idea of an inclesure bounded by only two straight lines. No distinctive phrases can be too strong to express the essential difference of these two asser tions; but it is a misfortune that all terms which create a sufficient distinction are linked to one or another theory of the human mind. If the mathematical student can receive these terms as indicative of the difference of species, without bending before an hypothesis about the conformation of his own reason, he will do well to adopt them; if, ou the other hand, ho feel compelled to agree with any one system of mental philosophy, he will neither impede nor advance his mathematical career.
The unavoidable certainty and definite character of mathematical conclusions have obtained for mathematics the name of exact science : but to this name it has not exclusive right. The laws under which wo must think are the foundation of a science which has an equal claim with mathematics to any epithet which indicates either necessity or precision. Accordingly logic and mathematics are separate branches of exact science. There are but three things of which we cannot divest ourselves so long as we imagine ourselves to retain both existence and consciousness of existence : they are thought, space, and time. With everything else there is a possibility of dispensing; that is, the imagination can conceive everything got rid of, and out of existence, except its own consciousness in some kind of activity, and the space and time without which it cannot conceive existence. The necessary laws of thought are the subject matter of logic : the necessary pro perties of space and time are the subject matter of mathematics. Number is an offspring of the notion of time; enumeration is a succes sion in time; in no other way can number be distinguished from multitude. And geometry is, without need of illustration, the offspring of the notion of space.
There has long been something of schism between the cultivators of logic and these of mathematics. To establish this fact, and tr..
speculate on the reasons of it, would require more space than we have to give. The effect has been unfortunate. The mathematician dis penses with the analysis of the laws of thought ; the logicians avoid that field in which the laws of thought are applied to necessary matter. This state of things is perhaps mending, but until the im provement is very decided, neither mathematics nor logic will be what they ought to be, as component parts of liberal education.
The sciences of which we speak may be considered either as dis ciplines of the mind, or as instruments in the investigation of nature and the advancement of the arts. In the former point of view their object is to strengthen the power of logical deduction by frequent examples; to give a view of the difference between reasoning on probable premises and on certain ones, by the construction of a body of results which in no case involve any of the uncertainty arising from the previous intro.
duction of that which may be false ; to form the habit of applying the attention closely to difficulties which can only be conquered by thought, and over which the victory is certain if the right means be used ; to establish confidence in abstract reasoning by the exhibition of pro. ceases whose results may be verified in many different ways ; to help in enabling' the student to acquire correct notions and habits of generalisation; to give caution in receiving that which at first sight appears good reasoning; to instil a correct estimate of the powers of the mind, by pointing out the enormous extent of the consequences which may be developed out of a few of its most fundamental notions, and its incapacity to imagine, much leas to attain, the boundary of knowledge; to methodise the invention of the means of expressing thought, and to make apparent the advantages of system and analogy in the formation of language and symbols; to sharpen the power of inveatigationa and the faculty of suggesting new combinations of the resources of thought; to enable the historical student to look at men of different races, opinions, and habits, in those parts of their minds where it might bo supposed d priori that all would most nearly agree; and to give the luxury of pursuing a study in which self-interest cannot lay down premises nor deduce conclusions.
As instruments in the investigation of nature and the advancement of the arts it is the object of the mathematical sciences to give correct habits of judgment and ready means of expression in matters involving degree and magnitude of all kinds; to teach the method of decomposing phenomena, and ascending from the complicated forms of manifestation to the simple law which regulates them ; to trace the necessary con sequences of any law, assumed on suspicion, in order to compare those consequences with phenomena ; to construct hypothetical repre sentations of laws, or approximations to laws, which shall sufficiently represent phenomena ; to convert processes of known accuracy, but complicated operation, into others which make up in simplicity for a certain amount of inaccuracy, and to devise means for judging of that amount of inaccuracy, and confining it within given limits ; to ascertain the most probable result of observations or experiments which are dis cordant with each other from errors of measurement or unknown causes of disturbance; to point out the species of experiments which should be made to obtain a particular sort of information, or to decide between two laws which existing phenomena both indicate as of nearly equal credibility ; to make all those investigations, which are necessary for the calculation of results to be used in practice, as in nautical astronomy, application of force and machinery, and conduct of money transactions; in a word, though that word by itself would have not presented a sufficiently precise idea, to find out truth in every matter in which nature is to be investigated, or her powers and those of the mind to be applied to the physical progress of the human race, or their advancement in the knowledge of the material creation.