Saturday, February 5, 2011

Further addenda

William James offers an abbreviated version of our Constructivist Fable (the Woodchuck Mathematician / Babylonian mathematics):  William James, The Principles of Psychology (1890), vol. II, p. 340:

By many measurements of triangles, one might find their area always equal to their height multiplied by half their base, and one might formulate an empirical law to that effect.  But a reasoner saves himself all this trouble by seeing that it is the essence (pro hac vice) of a triangle to be the half of a parallelogram  whose area is the height into the entire base.  To see this  he must invent additional lines.

That final observation is key.  To progress, we must advance beyond what is given visibly, and develop our intuitions of Invisibilia.
     These invented additional lines may stand as metonym for many such inspired devices.  Thus, to plumb the propterties of polynomials with exclusively real coefficients, we must consider a much wider world, and treat these polynomials as functions of a complex variable.

~ ~ ~

Although we renounced the project of quoting theistic language from pre-twentieth-century figures, as being but shooting fish in a barrel, the matter does retain some interest when the scientist used theistic ideas in a substantive way, to help guide his science, and not as mere window-dressing.  Thus, Ivar Ekeland, in The Best of All Possible Worlds (2000), p. 72, quotes Euler (1744) arguing for the truth and central role of the Principle of Least Action:

Since the constitution of the universe is perfect, and completed by an all-wise creator, absolutely nothing happens in this world  which canot be explained by some argument of maximum or minimum.  This is why there is no doubt at all that all effects observed in the world  can be explained from final causes, with the method of maxima and minima, with the same success as from efficient causes.

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