Differential equations

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"Jim Hines"
Senior Member
Posts: 88
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by "Jim Hines" »

> In medicine, we often hear that cells respond to the rate
>of change of some sensory input, which would appear to contradict

Organisms, organizations (e.g. revenues), and mechanisms (e.g.
speedometer) sense the rate of change via an average change over some
**finite** time interval. That is they sense average changes. An
average change is related to, but not the same thing as a mathematical
derivatives.

Liebnitz conceptualized a derivative as "average change" over an
**infinitesimal** time interval; and Newton conceptualized the
derivative as the limit of the average change over a time interval as
that interval approached zero. And Karl Weierstrass in lectures cited
by Hein in 1872, created a (set theoretic) conceptualization of
derivatives that didnt depend on thinking in terms of change or time
intervals at all, but only depended on real numbers that can get as
small as you like. These are the ways people still think of the
derivative. Because these ideas all depend on approaching or arriving
at something **infinitely** small, its unlikely anyone could ever
actually observe a true derivative in nature, if one exists.

FYI: Lakoff and Nunezs __Where Mathematics Comes From__, published by
Basic Books in 2000 contains an up to date consideration of how humans
think of derivatives (and other mathematical ideas) as well as the
objective reality (or lack thereof) of derivatives (and other
mathematical "things").

Jim Hines
From: "Jim Hines" <jhines@MIT.EDU>
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