The historical proof of the calculus bids us imagine some infinite series of shrinking numbers. It lets this series approach a limit. This limit is usually conceived of as a point. As an example let us imagine a sphere. Let the radius of the sphere be our given number. Now, let our sphere begin shrinking. The given number will get smaller, of course. The calculus supposes that as the given number gets smaller it gets closer to zero. It approaches zero. This implies that our shrinking sphere will physically approach a limit or a zero—that it will approach being a point. But it won’t. A shrinking sphere will not approach a point, not physically, metaphysically, mathematically, conceptually, really, or abstractly. This is why:
Size is a relative term. It is relative to other things and other times. You may be smaller than another thing, or smaller than you were yesterday, but other than that “small” has no meaning. A shrinking balloon has a limit. You can only let so much air out. It can’t get smaller than a deflated balloon. But if you take a sphere in physical or mathematical space and treat it only as structure, then there is no upper or lower limit on size. You can make it infinitely large or infinitely small. Large and small are opposite directions in extension, but they are the same conceptually. Just as a thing can go on expanding forever it can go on shrinking forever. Zero is precisely as far away as infinity. An infinite regression “toward” zero is exactly the same mathematically as an infinite progression toward infinity.
Most people have a somewhat easier time imagining a large infinity than a small infinity. Especially since I am not talking about a negative infinity. I am not talking about negative numbers here at all, you must realize. I am talking about a regress toward zero. Smaller and smaller fractions, or the like. Just as a large number does not really ever approach infinity, a small positive number does not really approach zero. Any infinite progression or regression does not approach ending. It does not end, therefore it cannot logically approach ending.
Everything I have said here applies to “size” in general, not just to material or physical size. Let us subtract out all the physical content from the discussion above. Let the numbers be numbers alone, and not refer to any physical parameter like length. We still have an inherent concept of size that we cannot subtract out. Numbers have size no matter how abstract we make them. Two is bigger than one in pure math as well as in applied math. If so, then let us ask, “If we move from 2 to 1, have we approached zero?” An exact analogy is the question, “If we move from 2 to 3, have we approached infinity?”
Of course, if we are talking about integers then we have no exact analogy: the answer to the first question is yes and the second no, since obviously the next smallest integer after 1 is 0. But in the second question, we are infinitely far away from infinity at 2, and we are infinitely far away from infinity at 3. We have not approached infinity. At the highest number we can imagine, we are still infinitely far away from the end of the series of integers, by definition. In fact, if we have approached the end of the series, then the series is not infinite.
Next, let us leave integers, since some will invoke Cantor to start inserting doubts into my reasoning so far. Let us move to real numbers, which have a higher order of infinity, for those who believe in such things. Let us ask the two questions again. “If we start at 1 and move down, have we approached zero?” And, “If we start at 1 and move up, have we approached infinity?” I think that it is clear that both questions are basically equivalent. We are dealing with an infinite series in either case. Neither series can possibly end, by definition. In fact, the proof of the calculus depends on using an infinite series. If a Cantorian or anyone else proved that a series actually had an end, then it would not be an infinite series, and it would not be the series the calculus is talking about. The calculus applies, axiomatically, to infinite series.
If this is so, then an infinite series of progressively smaller numbers does not in fact approach zero. The smallest number you can think of is still infinitely far away from zero. Therefore it is no closer to zero than 1 is, or a million billion.
All this is hard to imagine for some, since zero is not just like infinity in other ways. Zero has a slot on the number line. We reach it all the time in normal calculations. But we never reach infinity in normal calculations, and it has no slot on the number line. Zero is a limit we can point to on a ruler; infinity is not a limit we can point to on a ruler. For this reason, most people, or perhaps all people, have not yet seen that an infinite regression does not approach zero. Zero is not logically approachable by an infinite series of diminishing numbers. A diminishing series either approaches zero, or it is infinite. It cannot be both.
Therefore, the first postulate of the calculus is a contradiction. Not a paradox, a contradiction. Meaning that it is false. The calculus begins, “Given an infinite series that approaches zero….” But you cannot be given an infinite series that approaches zero.
Some pre-calculus problems get around this problem by summing the series. The ancient Greeks solved problems with infinite series, such as the paradoxes of Zeno (e.g. Achilles and the Tortoise), by summing the series. This has been seen as a sort of pre-calculus, and rightly so, since it deals with both infinite series and limits. But when a series is summed, it no longer matters whether or not the series “approaches” the limit or not. It is beside the point. It simply does not matter whether the series actually reaches or approaches the limit, not in a physical sense nor a mathematical sense. All that is necessary is to show that the sum cannot exceed the limit. Since this is so, it may logically be assumed that the sum does indeed approach the limit; and what is more, that the sum reaches it. However, the terms in the series do not. The terms in the series do not approach or reach the limit.
In post-Newtonian math, it has been the custom to give a foundation to the derivative, and thereby to the differential calculus, by first assuming an infinite series and then letting it approach a limit. The wording is normally something like that with which I started off this paper. But this proof is not a proof of any integral or of the integral calculus. That is to say, we are not dealing with any summations at this point in the historical proof. Rather, the proof is a proof that determines the derivative. Only later do we use the derivative to define the integral and give a foundation to the integral calculus.
Therefore, when the proof of the derivative lets the series approach a limit, it is quite simply wrong to do so. The terms in the series do not approach the limit; only the sum of the series approaches the limit. In differential calculus, we are not dealing with sums. We are dealing with differentials, which are simply numbers gotten from differences (gotten by subtraction).
To be even more specific, the proof of the derivative, and of the differential calculus—as taught in contemporary courses—starts with a given differential. We are usually given a curve. We take a differential from the curve, x2 – x1 for instance. We then let that differential diminish by choosing further x’s that are closer and closer to x2. We then mathematically monitor what is happening to y differentials as the x differentials diminish. We want to know what happens when the x differential hits the limit at the point x2. So it is clear that summations or sums have absolutely nothing to do with differential calculus. We are not summing any series of x’s. We are following diminishing x’s, which are individual terms in the series. I have shown above that these terms do not in fact approach the limit. Therefore the proof fails.
In my long paper on the calculus, I show many other ways the proof fails, and some of these are more critical than the one I have related here. However, this one is also worthy of notice, since it leads us into other interesting arguments that I will take up in subsequent papers.
A member of the status quo will argue that I am just caviling—inventing problems. He will say, “It is clear that a diminishing series, and the terms in that series, do approach the limit, or zero in the case you have given. To show this, all I have to do is point at the curve. If we are taking smaller and smaller differentials, then of course those differentials are getting closer to x2. Look at the line itself. The distance is getting shorter, so x1 must be getting closer to x2. That is all the proof is claiming.”
But notice that my antagonist is now using a physical definition of distance. When I attack it on physical grounds, the status quo claims that calculus is pure math, unsullied by physics. When I attack it on logical grounds, the status quo hides behind physical statements. It points to the line, showing me that the line is shorter. But it is showing me a length, and a length is a parameter. A length is not pure math.
My answer is this. Yes, the segment of the curve gets shorter as the differential diminishes. But what is this segment of the curve? Over any interval, the drawn curve or mathematical curve is a summation. The complete curve is an overlay of all possible variables in the problem. A segment of this curve is an overlay of all possible variables over the given interval. The curve, and its length, has nothing to do with the individual terms in the series of differentials. What we are concerned with in the differential calculus, and in the proof in question, are the individual terms in the infinite series, not the summation of these terms. So that showing that the length of the curve gets shorter is not to the point. It is a misdirection in argument.
The important point—the one that really matters—is the one I have set forth above. The terms in the infinite series do not approach zero or the limit or the point. The terms in the series in the question at hand are differentials, and they do not approach the limit. They are always infinitely far away from it, as long as “far” is understood in mathematical terms. Therefore it is meaningless to let a differential approach a limit. Differentials do not approach limits, by definition and all the rules of logic.
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