THE SCIENCES OF THE IDEAL 157 



ideal truth with the precision of which only mathematical research 

 is capable. As a fact, then, the mathematician and the philosopher 

 deal with ideal truth in ways which are not only contrasted, but 

 profoundly interconnected. The mathematician, in so far as he con- 

 sciously distinguishes significant from trivial problems, and ideal 

 systems, is a philosopher. The philosopher, in so far as he seeks 

 exactness of logical method, in his reflection, must meanwhile aim 

 to be, within his own limits, a mathematician. He, indeed, will not 

 in future, like Spinoza, seek to reduce philosophy to the mere develop- 

 ment, in mathematical form, of the consequences of certain arbitrary 

 hypotheses. He will distinguish between a reflection upon the unity 

 of the system of truth and an abstract development of this or that 

 selected aspect of the system. But he will see more and more that, 

 in so far as he undertakes to be exact, he must aim to become, in 

 his own way, and with due regard to his own purposes, mathemat- 

 ical; and thus the union of mathematical and philosophical inquiries, 

 in the future, will tend to become closer and closer. 



II 



So far, then, I have dwelt upon extremely general considerations 

 relating to the unity and the contrast of mathematical and philo- 

 sophical inquiries. I can well conceive, however, that the individual 

 worker in any one of the numerous branches of investigation which 

 are represented by the body of students whom I am privileged to 

 address, may at this point mentally interpose the objection that all 

 these considerations are, indeed, far too general to be of practical 

 interest to any of us. Of course, all we who study these so-called 

 normative sciences are, indeed, interested in ideas, for their own 

 sakes in ideas so distinct from, although of course also somehow 

 related to, phenomena. Of course, some of us are rather devoted to 

 the development of the consequences of exactly stated ideal hypo- 

 theses, and others to reflecting as we can upon what certain ideas and 

 ideals are good for, and upon what the unity is of all ideas and ideals. 

 Of course, if we are wdse enough to do so, we have much to learn 

 from one another. But, you will say, the assertion of all these things 

 is a commonplace. The expression of the desire for further mutual 

 cooperation is a pious wish. You will insist upon asking further: 

 " Is there just now any concrete instance in a modern type of research 

 which furnishes results such as are of interest to all of us? Are 

 we actually doing any productive work in common? Are the philo- 

 sophers contributing anything to human knowledge which has a 

 genuine bearing upon the interests of mathematical science? Are 

 the mathematicians contributing anything to philosophy?" 



These questions are perfectly fair. Moreover, as it happens, they 



