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SCIENCE 



[N. S. Vol. XLIV. No. 1146 



be deduced from the former with certainty. That 

 is to say, he believes that there are systems of co- 

 herent or consistent propositions, and he regards 

 it as his business to discover such systems. Any 

 such system is a branch of mathematics. 



A word might be said about pure and ap- 

 plied mathematics. "We may have a 

 branch of mathematics with its postulates 

 or axioms consistent and have no physical 

 interpretation of them. On the other hand, 

 we may make our postulates consistent with 

 what we believe to be the proper interpre- 

 tation of certain phenomena, and this would 

 be applied mathematics. It is to be ob- 

 served that after we have our conditions 

 once fixed by our interpretation of these 

 phenomena, we proceed to our conclusions 

 in a way which is wholly independent of 

 whether we have the right interpretation 

 or not, and are thus back in the domain of 

 pure mathematics. 



The popular conception of mathematics 

 has been that it devoted itself to problem 

 solving. You will see, however, that the 

 mathematician concerns himself not with 

 the solution of particular problems, but 

 with the principles which underlie the so- 

 lution of classes of problems. There is and 

 has been a lively interest in problem solv- 

 ing as is evidenced by the problem depart- 

 ments of various journals. To some the so- 

 lution of these problems has offered simply 

 the diversion which comes from the solu- 

 tion of a puzzle, to others they have offered 

 a real mathematical stimulus. 



There are two general methods of work- 

 ing — I mean of research — in mathematics, 

 the intuitional and postulational. In the 

 case of the first the worker jumps to his 

 conclusions, as it were, guided by some 

 analogy or by his sense of what the facts 

 should be or by his experience; and then 

 follows this drawing of conclusions by fill- 

 ing in his proof by rigorous deduction. In 

 the second method the postulates are kept 

 definitely in view and results are reached 



by deduction. Most discoveries are, I 

 think, made by the intuitional method. 

 Most progress can be made by leaping 

 across barriers and viewing the country be- 

 yond and then returning to build roads 

 and tunnels. It is true that when we at- 

 tempt to build the road it may not lead us 

 where we leaped, it may not lead us any- 

 where, and we must return to our starting 

 point. But we build with so much more 

 enthusiasm, with so much more skill, if we 

 think we know where the road leads. The 

 postulational method of work is more for- 

 mal and is a better tool for the road build- 

 ing than for spying out the land. 



We learned our arithmetic by the intui- 

 tional method. We said 1 + 1 = 2, not be- 

 cause of some set of postulates, but because 

 in our experience one and one gave some- 

 thing to which we attached the name two. 

 Now to set down our postulates and prove 

 that 1 + 1 = 2 is possible and profitable at 

 the proper time, but altogether out of place 

 in an elementary arithmetic. In plane 

 geometry we had our introduction to the 

 postulational method. In this subject we 

 started with a set of postulates explicitly 

 stated and deduced from them certain re- 

 sults. In discovering the facts of Eucli- 

 dean geometry, intuition is largely called 

 upon, while in setting those facts down in 

 a text-book we use the postulational 

 method. Euclidean geometry is so largely 

 intuitional in discovery because its postu- 

 lates agree with our notions of space. In 

 the non-Euclidean geometries we can not 

 trust our intuition and must depend di- 

 rectly on our postulates. 



If instead of saying that the whole is 

 equal to the sum of its parts, we say that a 

 part may equal the whole, our intuition is 

 no safe guide. Other examples might be 

 given. 



Research work in mathematics attracts 

 two classes of workers, those interested for 



