108 Irving Fisher —Mathematical investigations

“ mechanics ” proper a few years later the same boy studies “ fallingbodies ” he finds it helpful to use the formula v —gt which contrastswith the preceding formula only in that space (s) is replaced byspace per unit of time ( v) and velocity (w) by velocity acquired perunit of time (g). The increased complexity of the magnitudesmakes a formula relatively desirable. Yet for some minds the latterformula is of no use. Experience in teaching this very subject hasconvinced me that there are a few who understand it better withoutthe aid of the formula, but they are just those individuals whosecomprehension of the relations involved is the vaguest and theweakest.

The formulae-, diagrams and models are the instruments of higherstudy. The trained mathematician uses them to clarify and extendhis previous unsymbolic knowledge. When he reviews the mathe-matics of his childhood, the elementary mechanics is to himillumined by the conceptions and notation of the calculus and qua-ternions. To think of velocity, acceleration, force, as fluxions is notto abandon but to supplement the old notions and to think ofmomentum, work, energy, as integrals is greatly to extend them.Yet he is well aware or ought to be that to load all this on thebeginner is to impede his progress and produce disgust. So also thebeginner in economics might be mystified, while the advancedstudent is enlightened by the mathematical method.

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The utility of a mathematical treatment varies then according tothe characteristics of the user, according to the degree of his mathe-matical development and according to the intricacy of the subjecthandled. There is a higher economics just as there is a higherphysics, to both of which a mathematical treatment is appropriate.It is said that mathematics has given no new theorems to economics.This is true and untrue according to the elasticity of our terms-The challenge of Cairnes might be answered by a counter challengeto show the contents of Cournot, Walras , or Auspitz und Lieben inany non-mathematical writer.

If I may venture a speculation, those who frown on the mathe-matical economist because he “ wraps up his mysterious conclusionsin symbols ” seem to me in some cases to point their finger at those“ conclusions ” which when “ unwrapt ” of symbols they recognize asold friends and lustily complain that they are not new; at the sametime they seem to ignore completely those “ mysterious ” conelu-