“We need an analytic expression to specify the exact relationship between the values of x and y. This relationship between the values of the independent variable (in this case x) and the dependent variable (in this case y) is called a function.”
But what is a variable? In the same way that Euclid defines a point as only “that which has no part”, a suitable definition for a variable is only “that which varies” or maybe “a magnitude that varies” not because the definition is a negation (which is not the case for the definition of a variable), but because it is a naming of the single essential characteristic. A point and a variable are also similar because they are both their definition and no more. The name ‘variable’ is perfect and complete: it is entirely appropriate and expresses the whole nature of the being of a variable.
Analytic notation is the same in this way. It is wholly explicit and univocal. There can be no mistake about what an analytic expression means: it is all already there. It aims to communicate no more and no less. It is a language that signifies, only and perfectly, mathematical objects and the mathematical operations performed on them. When we say or write, for instance, the equation y=1-x2, we intend to express that the value of some variable named y is equal to the value of some other variable named x multiplied by itself once and then subtracted from 1. Analytic notation has departed, or has always been separate, from the world of geometrical objects, although it may retain some of the words such as ‘x squared’. We are no longer signifying the square constructed from two lines whose length is x. Instead, an exponent is a way to indicate the multiplication of a value by itself a certain amount, and to multiply something is to add it to itself to a particular extent. (Incidentally, this symmetry is pleasing: multiplication is repeated addition and exponents are repeated multiplication.) After performing these processes, what we are left with is only another undifferentiated, pure, unattached magnitude the same as all others. To say that it is undifferentiated is to say that it does not exclusively refer to a particular magnitude of, for example, inches or minutes, or area or length.
Modern math as a whole is a free and removed goings-on as well. As far as possible, all mathematical operations are generalized and liberated from what might be called their ‘real-world’ roots. For instance, although it may seem unadvisedly ambitious or without sufficient underpinning, we may multiply any magnitudes by any magnitudes whatsoever, as we may raise any magnitude to the power of any magnitude whatsoever. Originally, these concepts of multiplication and exponents may have interacted more harmoniously with real-world physical objects, but since they have been generalized in the above way, they are now less able and less required to refer back and indicated things about, say, how many bicycles I sold yesterday. The same is true of the trigonometric operations of sine, cosine, tangent and all the rest. Perhaps to take the sine of an angle used to be to consider the ratio between various sides of a right triangle that contains the angle in question. Well, no longer! We regularly take the sine of angles greater than 90° and less than 0°, among other affronts, although there can be no 90°-91°-?° or 90°-0°-90° triangle.
Math has the advantage of possessing a language that is perfect in a way, as well as the disadvantage of only really addressing and informing us of topics far removed from the human experience. There is much in math that can be applied to the physical world and used to give an account, as when a function happens to be able to model whatever phenomena, but this is secondary and not a property of math itself. A specific happenstance comes to mind. As it turns out, e2?i-1=0. Confronted by this dazzling and surprising confluence of intensely enigmatic constants, we are tempted to ask questions like What are the implications of this or What could this mean, but there are no answers which will satisfy us or are pertinent to our existence as humans because the subject matter is (literally) unimaginably abstracted.
It is important to place the notion of a function in this context. It differs in one respect from an equality like the one treated above y=1-x2, in that its variables are dependent and independent. The value of the independent variable is chosen by the mathematician and determines or causes the value of the dependent variable. Because of this, there can only be, at most, one value of the dependent variable for any given value of the independent variable. The manual offers this: that “it is intolerable in our calculations to have a choice of two or more values of the dependent variable”. The dependent variable would not truly be dependent or determined if this were the case. And that is pretty much all that can be said about functions.
About the integral and the derivative of a function, then. The derivative of a function provides the instantaneous rate of change of the function; the integral provides the value of the area between the curve, the x-axis, and the limits of integration. Because integration involves calculating areas, one might say that it is related to multiplication, and more specifically, that it is a generalization of multiplication. While multiplication supplies only the area of rectilinear areas, integration supplies the area of any figure whose sides can be modeled by a function. It is tempting to call integration a generalization of multiplication in a second way as well. As mentioned above, multiplication can be used to add the same magnitude to itself some extent (adding the same number to itself again and again). To integrate is to sum some vast number, maybe a limitless number, of evanescing magnitudes. Multiplication is limited because it can only process a single, particular magnitude. It may seem that integration is the adding up of any diverse magnitudes whatever, and thus is not limited as multiplication is, if ‘evanescing magnitudes’ can be understood to differ from one another.
Bizarrely, the derivative of the integral of a function is that very function, and the integral of the derivative of a function is that function as well. Why the instantaneous rate of change should have anything to do with the area under the curve is mysterious, and may very well be unyielding to inquiry. It is not merely that these two things, rate of change and area, are related, but that they are in some way the opposite or the reciprocal of each other. That a function has such a simple nature, that it ‘means’ very little, and that what little meaning it has is confined to some bare and distant world, all conspire to indicate that the integral and the derivative of a function have a very limited meaning as well, especially if they are somehow not primary or independent beings but derived from or dependent on the being of a function. George Berkeley goes much farther down this unhappy, skeptical path in tirade when he insinuates, among other things, that Newton does not ‘clearly conceive’ whatever objects he is conversant with and that he does not proceed upon sound principles.
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