Precise language serves the purposes of logic, science, and mathematics.
Tuesday, July 20, 2010
THE IDEAS OF VARIABLE AND FUNCTION
``By pairing simultaneous specimens of the distance traveled by a falling stone and of the duration of the motion, Galileo discovered not only that ![[;s;]](http://www.codecogs.com/gif.latex?s "s")
is a function of ![[;t;]](http://www.codecogs.com/gif.latex?t "t")
, but that ![[;s = 16t^2;]](http://www.codecogs.com/gif.latex?s%20=%2016t%5E2 "s = 16t^2")
. Similarly, ![[;s = \frac{1}{64}v^2;]](http://www.codecogs.com/gif.latex?s%20=%20%5Cfrac%7B1%7D%7B64%7Dv%5E2 "s = \frac{1}{64}v^2")
and ![[;v = 32 t;]](http://www.codecogs.com/gif.latex?v%20=%2032%20t "v = 32 t")
. Each such formula may (in analogy to the description of ![[;L = 7\frac{1}{2}ft.;]](http://www.codecogs.com/gif.latex?L%20=%207%5Cfrac%7B1%7D%7B2%7Dft. "L = 7\frac{1}{2}ft.")
as an equality of a quantity, ![[;L;]](http://www.codecogs.com/gif.latex?L "L")
, and a ``denominate number,'' ![[;7\frac{1}{2}ft.;]](http://www.codecogs.com/gif.latex?7%5Cfrac%7B1%7D%7B2%7Dft. "7\frac{1}{2}ft.")
) be described as an equality of a variable and a denominate function. We reformulate these equalities in terms of pure numbers and functions by writing ( in ![[;ft.;]](http://www.codecogs.com/gif.latex?ft. "ft.")
) = ![[;7\frac{1}{2};]](http://www.codecogs.com/gif.latex?7%5Cfrac%7B1%7D%7B2%7D "7\frac{1}{2}")
and ( sim. ) = ![[;16\cdot j^2;]](http://www.codecogs.com/gif.latex?16%5Ccdot%20j%5E2 "16\cdot j^2")
where ( sim. ) stands for the following function: the set of all ordered pairs (![[;\sigma, \tau;]](http://www.codecogs.com/gif.latex?%5Csigma,%20%5Ctau "\sigma, \tau")
) of values of and which are paired by virtue of the pairing of the specimens. More briefly, ![[;s_t = 16\cdot j^2;]](http://www.codecogs.com/gif.latex?s_t%20=%2016%5Ccdot%20j%5E2 "s_t = 16\cdot j^2")
, ![[;s_v =\frac{1}{64}\cdot j^2;]](http://www.codecogs.com/gif.latex?s_v%20=%5Cfrac%7B1%7D%7B64%7D%5Ccdot%20j%5E2 "s_v =\frac{1}{64}\cdot j^2")
, ![[;v_t = 32\cdot j;]](http://www.codecogs.com/gif.latex?v_t%20=%2032%5Ccdot%20j "v_t = 32\cdot j")
. The mass of a piece of lead is related to the historical time by a constant function.''
![[;\textsc{KARL MENGER};]](http://www.codecogs.com/gif.latex?%5Ctextsc%7BKARL%20MENGER%7D "\textsc{KARL MENGER}")
wrote an article in 1955 in the Proceedings of the National Academy of Sciences, introducing three kinds of variable; namely, logical, scientific, and mathematical.
Precise language serves the purposes of logic, science, and mathematics.
Precise language serves the purposes of logic, science, and mathematics.
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