In many fields, especially ones related to the physical sciences, small angle approximations are often used. For example, when x is very small, the sine of x is said to equal x and to equal the tangent of x. These conjectures are then used as theorems in a proof. In my opinion, this is bad calculus. Such approximations generally serve their purpose well, but are merely a way to make a very long proof look very short. In calculations, it’s fine to be a little bit off, but not in proofs. We often come upon a difficult matter to prove and a professor takes the shortcut. While these shortcuts do work, they bring us to the right answer, they are not the real reason why that answer is right. In reality, the sine of x is never equal to x or the tangent of x unless x is absolutely zero. Consider the following proof:
1. Because the recursion can approach infinity at any rate (even an infinite rate), and in application, x is assumed to be a finite constant just greater than one, this indeterminate quantity (0^inf) evaluates to infinity. Thus, as we should expect, the anti-proof does not apply to an infinitely small x value or a zero x value.
2. At this point, one may evaluate the expression on the left by flipping and multiplying by the bottom, thus getting sin(0) = 0, which is true, but there is nothing stopping us from moving the cos(x) to the right side of the equation before the self-referencing recursion and thus ending with it on the other side, so the move of multiplying both sides by 1/0 is legal in this case, this way of doing it just makes it more clear. Of course, if one did start with the cosine on the other side, the proof might also evaluate to 0 = inf. Admittedly, this proof is somewhat ambiguous over all for these reasons, but the end result (ignoring all the paradoxes) is that a limit is a limit, and any finitely small angle will not satisfy this equation just as much as zero does not equal one, or infinity does not equal zero. If you plugin an infinitely small value for x, all the problems immediately dissolve and the limit is proven.
3. because, as was said in the previous note, the infinite quantity can approach infinity at any rate (an indeterminate rate), here it could be said to be approaching it at the same rate that x is “approaching” zero (even though x is a finite constant), and thus the limit holds (if we want it too…ha, ha, ha). Also note, it doesn’t present a problem for us that the power of the cosine had to approach infinity at a rate that would make the power of the cosine evaluate to an infinite quantity, because by continuing to manipulate that original rate of approach, we can also manipulate the rate at which the power of the cosine approaches infinity (which is what we need in this last step).
My Cartesian Point?
My point ultimately with all this silliness that no one actually knows (though we do, I think, know more about infinity and zero and the imagination of numbers– ok, i need to get better jokes–than we say we do) is that if you zoom in far enough (as we did here using the infinite recursion), the tangent of x, the sin of x, and x are just as far away from each other when x doesn’t equal zero as zero is from infinity. Thus, while the approximations are useful when dealing with actual quantities, they are just bad math when used in proofs (even though everyone does it… it’s pretty much like jumping off a cliff of infinite height), because in proofs, we depend on expressions being absolutely equal, and finitely small number is infinitely larger than an infinitely small number.
Answer: Most everything, but here is just a place to start….
Does any one have a reasonable answer to the most difficult questions reason may ask? What is zero over zero?
Lets start by considering the following true or false test.
Actually, this is more like a true AND/OR/XOR/NOR false test:
1. T || & !| !! F This statement is true.
2. T || & !| !! F This statement is false.
Both of those seem simple enough at first, but are not. The first temptation may be to mark the first one true and the second one false and be done with the matter, but that’s bad calculus. It is fine to mark the first one true; if you do so, you are saying that it is a true statement to say that it is a true statement to say that it is a true statement … on to infinity. But also, consider marking it false. If you do so, then you are saying that it is a false statement that that is a true statement, and that is also fine. Notice, marking it false creates, in a sense, a finite chain of logic, or so it appears to at first glance. In reality marking it either true or false both create an infinite, self-referencing, recursive chain of reasoning because the statement in itself is self-referencing and infinite regardless of what you mark it. However, it is sound reason to say either that a statement that claims itself to be true is a true statement (again, on to infinity) or that such a statement is false, but not both. It doesn’t make sense for a statement that says it is true to be simultaneously true and false. Therefore the statement is either true or false, to be marked ether with an OR or an XOR if you like.
Next, lets consider the second statement. This one is not as simple. It seems okay at first to just mark it true. But this is because when one does so, one does not see the infinite, self-referencing, recursive property it posses as clearly (just like how it was hard to see the infinite recursion in the first one when it was marked false, but it still exists nonetheless). So lets begin by marking it false to make it easier to see. If it is false that that statement is false, than that statement must be true, but if that statement is true, than by what the statement claims, it must be false. We quickly find that we are going in circles. Therefore, the statement can be neither true nor false because if it is true, then by the statement, it must be false and thus it is not true, and if it is false, than by the statement, it must be true and thus cannot be false. True and false are mutually exclusive qualities, therefore this statement is not true AND false, but rather neither true NOR false.
Bad calculus makes these complex problems seem so much easier but it is simply wrong. For example, it is easy to miss the complexity of the second statement by making a simple error. One might suppose the statement is false, and therefore consider it rewritten as “This statement is true.” Upon doing that, one might note that it is sound logic to, as was done in the first statement, mark such a statement false. And with that he or she would have concluded that the statement is false. This is bad calculus. The error lie in the way the statement was rewritten, it should not be “This statement is true,” but rather “That statement is true.” If this method of rewriting the statement is used, one must then ensure the rewritten statement is true in order for the marking (true or false) to be considered accurate. Thus, the statement that “That statement is true” which says “This statement is false” recreates the self-referencing logic discussed earlier, only be this method, it instead causes an infinite loop of rewriting the statement.
Now lets consider this same logic in terms of calculus.
First, however, I’d like to present the following model of the categories of logic and math if I may:
Algebra is arithmetic with logic, calculus is algebra with simplified algorithms for computing infinite and non-exiting quantities, and logic is calculus without arithmetic.
In calculus, I’d say these problems are most related to problems that involve an indeterminate. If you are not familiar with the concept of an indeterminate, take the following examples:
1^∞ 0*∞ 0/0
There are more, but this is a good start. The first an last of these examples are much like the logic discussed in the previous section because they seem to have an obvious default answer. For example, zero over zero is most likely “by default” equal to one, and the power of one to the infinity is like wise, one “by default.” However all three of these examples are like the logic discussed in the previous section in that, they could be said to equal most anything. I would likewise say that if the logic from the previous section were actually applied in a full rhetorical situation, the outcome could be anything depending on the situation. Lets examine how this works with a few proofs:
This definition is an example of 0/0 equaling the “default” (one). Now consider a second definition, again provable with L’hopital:
This is an example of 0/0 not equaling one, but equaling five. If you don’t believe me on this, use L’hopital: take the derivative of both the top and the bottom at the point x = 0 and divide them. The derivative of the top at x equals zero is five (5 * cos(5 * 0) = 5) and the derivative of the bottom is always one. Therefore, the limit of the function as x approaches zero is five.
This is the nature of an indeterminate; they equals different things in different cases. It is all a matter of how the incomputable quantities are related to each other. The most classic example is an integral. An integral, as you probably know, is any infinite sum of infinitely small parts. It is, in essence, 0*∞ or ∞/∞, both of which are the same thing. An integral may also equal different things in different cases, but by definition it must always be finite (assuming it is integer-dimensional and not fractal). This is simple logic, because if it were infinitely large, it would have to include ether infinite finite parts or at least one infinite part, and if it were infinitely small, either all the parts would have to have zero magnitude (I’m ignoring negatives for the sake of my point), or it could not be an infinite sum.
Of course your mind will immediately jump to something like this:
But notice, while this is an area under a curve, and thus can be found using an anti-derivative, it is not technically, by definition, an integral.
That being said, I’d like to bring up a well-known bit of theory that is actually horrible calculus: the Coastline Paradox. This is very a well-known theory, if you are unfamiliar, click the link. It is, however, simply wrong. The only reason the Coastline Paradox is so popular is because it is aesthetically pleasing on a plain level. It is “merely poetry,” (take a look at C. S. Lewis’ essay: Is Theology Poetry? if you want to know what I’m talking about). Convincing others that the length around the coast of Great Britain is infinite makes people feel smart, but there is nothing intellectually honest about it.
Before I go on, I must note that I by no means reject fractal theory; it is fascinating, but I do reject the Coastline Paradox as a miserable anti-example of the highly intriguing theory.
Lets consider the Paradox as it is made to appear. It seems at first, that the proposal is that the coast Great Britain, modeled by some sort of continuous function, has an infinite length. This means that we are, from the start, ignoring any rise and fall of the tide. The length of a differentiable and integrable function is beautifully defined by:
And for any cusps or discontinuities, the length may be taken from part to part and then summed. However, in the case of the coast of Great Britain, there can be no discontinuities for obvious reasons, and no cusps because even if it were physically possible to have an actual geometric cusp, having factored out the rise and fall of the waves, we do not have enough precision to account for one. But lets ignore precision for a minute (and we must be careful when we do that). Lets suppose Great Britain is frozen in time and the waves are, therefore, neither rising nor falling. The idea is that if we keep zooming in we will have more and more tinny measurements, and thus, on an infinitely small-scale, we have infinite measurements and the coast is infinitely long. This, in itself, is nonsense. Every time it is explained, it is done so in a matter designed to pull the wool over your eyes rather than to introduce you to fractals. Yes, the smaller the measuring device you use to measure the coast, the more measurements you’ll take, but also the smaller each measurement will be. The coastline paradox is portrayed to the public using bad calculus. It is entirely focused on the fact that the length is an infinite sum, and tries to evade the fact that that sum is one of infinitely small parts. It is, by definition, and integral, and integrals always evaluate to a finite quantity so long as they are integer-dimensional rather than fractal.
If the Coastline Paradox holds any weight at all, it is not the weight that it is appreciated for. What it is appreciated for is the idea that integrals can have no finite limit (in integer-dimensions), which is absolute rubbish. It is like a fallacy in the plot of a tragedy implemented inconspicuously in order to achieve a particular aesthetic (see Aristotle’s Poetics). This is absolutely false. It’s bad calculus! All the Coastline Paradox would be suggesting if it held any weight would be that all of the physical universe is fractal dimensional, but mind you, this is entirely unrelated to the coast of Great Britain, and such a detail would only evade such a theory. Nonetheless, lets examine the theory:
If you are unfamiliar with fractal geometry, consider the Koch Flake, a fascinating piece of geometric calculus. Here’s a diagram:
The idea is that after infinite iterations, the flake will have an infinite length between any two points. This is true. As you may find for yourself with a simple geometric proof, each iteration makes the flake 4/3 longer in perimeter than it was in the last one, and thus when n is the number of iterations and l is the original length, the flakes perimeter can be expressed by the following:
Therefore, because the limit of this expression as n goes to infinity is infinity, after infinite iterations, the flake will have an infinite perimeter between any two points. It is also interesting to note that because the rate at which the area of the flake is increasing is decreasing with each iteration, the area will diverge at a finite quantity. The result: a flake with an infinite perimeter and finite area, a member of a fractal dimension. This is the logically sound grounds on which the Coastline Paradox was likely originally based, but the coastline model is irrelevant. The problem is that the implied model for the coastline of Great Britain is one of an integer-dimension, namely, two-dimensional, but the whole idea of the Paradox is based on the assumption that it is not two-dimensional, but has a dimension of infinite detail, a fractal dimension. All the Paradox is really saying is that the atom is fractal dimensional. There is, of course, no evidence to prove this one way or the other. It is like saying the universe is infinite. One can assume the universe is infinite or not, and base ones reasoning on such an assumption, but not prove it. Therefore, the Coastline Paradox is merely a fancy way of saying “I believe in string theory.”