That this may be better seen, let me briefly indicate the indispensable discipline which each class of sciences gives to the intellect, and also the wrong intellectual habits produced if that class of sciences is studied exclusively.
Entire absence of training in the Abstract Sciences leaves the mind without due sense of necessity of relation. Watch the mental movements of the wholly ignorant, before whom not even the exact and certain results of Arithmetic have been frequently brought, and it will be seen that there exists nothing like irresistible conviction that from given data there is an inevitable inference. That which to you has the aspect of a necessity, seems to them not free from doubt. Even men whose educations have made numerical processes and results tolerably familiar, will show, in a case where the implication is logical only, that they have not absolute confidence in the dependence of conclusion on premisses.
Unshakable beliefs in necessities of relation, are to be gained only by studying the Abstract Sciences, Logic and Mathematics. Dealing with necessities of relation of the simplest class, Logic is of some service to this end; though often of less service than it might be, for the reason that the symbols it uses are not translated into thought, and the connections stated not really represented. Only when, for a logical implication expressed in the abstract, there is substituted an example so far concrete that the interdependencies can be contemplated, is there an exercise of the mental power by which logical necessity is grasped. Of the discipline given by Mathematics, also, it is to be remarked that the habit of dealing with the necessities of numerical relation, though in a degree useful for cultivating the consciousness of necessity, is not in a high degree useful; because, in the immense majority of cases, the mind, occupied with the symbols used, and not passing beyond them to the groups of units they stand for, does not really figure to itself the relations expressed—does not really discern their necessities, and has not therefore the conception of necessity perpetually repeated. It is the more special division of Mathematics, dealing with Space-relations, which, above all other studies, yields necessary ideas, and so makes strong and definite the consciousness of necessity in general. A geometrical demonstration time after time presents premisses and conclusion in such wise that the relation alleged is seen in thought—cannot be passed over by mere symbolization. Each step exhibits some connection of positions or quantities as one that could not be otherwise; and hence the habit of taking such steps makes the consciousness of such connections familiar and vivid.
But, while mathematical discipline, and especially discipline in Geometry, is extremely useful, if not indispensable, as a means of preparing the mind to recognize throughout Nature the absoluteness
