"An incontestable claim of mathematics to importance in our civilization is that it is indispensable in a scientific explanation of what we observe in nature, i.e., the phenomena of nature. Of the several fields of elementary mathematics, the calculus may be called the motion-picture machine of mathematics which catches natural phenomena in the act of changing, or, as Newton called it, in a state of flux. Other fields of mathematics are to be likened to the camera which shows a still picture (of nature) as it appears at a given instant without regard to the possible appearance the following instant."
"In mathematics, as in the world about us, when one quantity depends on a second quantity, or when the value of one symbol depends on the value of another symbol, the first is said to be a function of the second. If the second quantity, or the second symbol, is thought of as taking on a number of arbitrary values (e.g., the angle A when it increases or decreases), it is called an independent variable and the function which depends on it is called a dependent variable. It may happen that a function depends on more than one independent variable."
"Mathematical theories have been of great service in many experimental sciences in correlating the results of observations and in predicting new data afterward verified by observation. This has happened particularly in geometry, physics, and astronomy. But the relationship between a mathematical theory and the data which it is designed to relate is often misunderstood. When such a theory has been successful as a correlating agent, the conviction is likely to become established that the theory has a unique relationship to nature as interpreted for us by the observations. Furthermore, it is sometimes inferred that nature behaves in precisely the way which the mathematics indicates. As a matter of fact, nature never does behave in this way, and there are always more mathematical theories than one whose results depart from a given set of data by less than the errors of observation."
"Neither the principle of cause and effect nor the principle of uncertainty can be precisely characteristic of the behavior of nature. They are merely most interesting theorems in two different theories by means of which we endeavor to correlate and interpret observed data. The ultimate choice between the two theories must be determined by convenience or by their relative accuracies of fit with observation, and not because of any supposedly precise correspondence with nature on the part of either one of them." (Mayme I Logsdon, "A Mathematician Explains", 1935)
"[…] the purposes of an applied mathematical science are twofold: first to correlate and systematize data which may otherwise appear heterogeneous and unrelated in character, and second to predict by logical processes new results which might be difficult or impossible to discover by experimental methods alone."
"The purpose of a coordinate system is twofold. It enables
one person to describe the position of points or objects in such a manner that
others listening or reading may know exactly what points or objects are meant.
And it is a device for linking algebra and geometry so that an algebraic
equation corresponds to a geometric locus and, conversely, a geometric locus
corresponds to one or more algebraic equations."
"The underlying notion of the integral calculus is also that
of finding a limiting value, but this time it is the limiting value of a sum of
terms when the number of terms increases without bound at the same time that
the numerical value of each term approaches Zero. The area bounded by one or
more curves is found as the limiting value of a sum of small rectangles; the
length of an arc of a curve is found as the limiting value of a sum of lengths
of straight lines (chords of the arc); the volume of a solid bounded by one or
more curved surfaces is found as the limiting value of a sum of volumes of
small solids bounded by planes; etc."
"The words 'maximum' and 'minimum' are used here in a technical sense. Maximum value of the function, for example, does not mean (as one might well suppose) the greatest value which the function attains for any value of x but, merely, the greatest value which it attains when, having been increasing, it ceases increasing and begins to decrease. In other words, the ordinate of a maximum point on a curve is greater than the ordinates of other nearby points. In a similar manner the ordinate of a minimum point is less than the ordinates of other nearby points."
"There is probably no one word which is more closely
associated in everyone's mind with the mathematician than the word equation.
The reason for this is easy to find. In the language of mathematics the word 'equation' plays the same role as that played by the word 'sentence' in a spoken and written language. Now the sentence is the
unit for the expression of thought; the equation is the unit for the expression
of a mathematical idea."
"To square a circle means to find a square whose area is
equal to the area of a given circle. In its first form this problem asked for a
rectangle whose dimensions have the same ratio as that of the circumference of
a circle to its radius. The proof of the impossibility of solving this by use
of ruler and compasses alone followed immediately from the proof, in very recent
times, that π cannot be the root of a polynomial equation with rational
coefficients."
"When an induction, based on observations, is made, it is not intended that it shall be accepted as a universal truth, but it is advanced as a hypothesis for further study. Additional observations are then made and the results compared with the results expected from the hypothesis. If there is more deviation between the experimental results and the computed results than can be expected from the inaccuracies of observation and measurement, the scientist discards the' hypothesis and tries to formulate another."
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