0/0- The Definitive Answer

After a quite heated, albeit illuminating class discussion on what 0/0 should be, we can finally put this question (to the best of our ability) to rest. To this aim, we should first discuss each position and see the evidence for each individually before coming to a definite conclusion (and oh boy is the conclusion a doozy.) So the choices supported by class discussion are:

0/0= 0, 1, ∞, or undefined/indeterminate

1. 0/0=0

Examine the function y(x)=0/x. For all x ≠ 0, y(x)=0. Graphically this looks like


So a logical conclusion would to allow the discontinuity to simply equal the limit as x approaches 0

So we would have when x=0, y(0)=0/0=0.

2. 0/0=1

Let us now look the function g(x)=x/x. For all x ≠ 0, g(x)=1. We can see this visually as


Again we should be tempted to fill this discontinuity by the limit as x approaches 0

When x=0, we would have g(0)=0/0=1


3. 0/0= ∞

For any a > 0, 

When a=1, this looks like

We know this is true for any a > 0, so we could think to simply extend this property to when a=0 as well (there is nothing particularly wrong with this assumption.) Therefore we would have 0/0= ∞

There is similar argument when a < 0, except there the limit goes to -∞, 
another logical answer.

4. 0/0 is undefined/indeterminate

This is nothing inherently wrong with refusing to answer the question. In fact why should we answer a question that seems to have multiple correct answers? All three approaches above seem somewhat logical, so we should be worrying about now about how to pin down a single answer.  After all, when we solve something like 2b=5 we don't get several logical non-equivalent answers. So where is the inconsistency? Where was our mistake? The reality is that all of the above answers are correct. 

Consider the equation bx=a, where we can see that x is solved for by x=a/b. Now, consider when a=0, or bx=0. From the properties of the 0, we see that either b or x must be zero. Since we don't want to let x be trivially zero, let us say b=0 instead. Then we have

0x=0

Here is the key part: because ANY number times 0 is 0, x can be any number. Again we can see that

x=0/0

follows trivially from this result. Therefore we concluded that 0/0 can be any number x that has the property 0x=0.

Another way to see this is with the following question: how many times can I add up zeros to get zero? Any number of times!

5. Conclusion

If 0/0 can be any number, what is the point in defining it? The concept of having a "solution" to 0/0 seems rather meaningless, since any number we could possible wish to be the answer is.  So, we shouldn't feel the need to provide a single answer. Our conclusion is then this: we in fact will refuse to answer the question. In other words, since any 0/0 can be any number, 0/0 is undefined/indeterminate.

Doing Math da Greek Way- Volume and Surface Area of a Sphere

As modern mathematicians, we are very aware of all the results of classical geometry, especially regarding areas and volumes. Heck, these problems are usually designated to poor freshmen in order to acquaint them with rudimentary calculus. However, my question is a bit different: how in tarnation did mathematicians figure these "simple" results using basic geometry? For the time being let us abandon calculus so that we can stick to our intuition and wits. If you wish to see the "official" calculus way, here is a link to my attempt to explain it. SO without further ado, let us do math the "Greek" way.


Let r be radius. Let us assume we already know the results of basic circles (I can't derive everything, you know!) such as Circumference =2π*r and Area=π*r^2 as well as squares such as Area=(side)^2. We can find the surface area of a sphere first. Since the surface area of a square box is 6(side)^2 (we are just adding 6 square areas up), we should expect our answer for surface area of a sphere to have the form of constant*(side)^2. Our first guess, a dumb guess if you will, would be (Circumference)^2=4π^2*r^2 since like the case of the square we have our length squared. Geometrically we can think of the circumference "sweeping" out area as it rotates. A link to a Mathematica file I made of this phenomena is right here. However, we can see that during the first animation that there is an overlap where area is counted more than one time. Upon measurement we would see that experiment matches this observation. Adjusting our thinking slightly, we can see that we don't need one of the circumferences to sweep out entirely. In fact one of these terms can simply be the diameter (2*r) while we let the other sweep. Physically this means that we can let the sweeping occur around a diameter. The second animation shows this accurately. So now we have

Surface Area=(2π*r)*(2*r)=4π*(r)^2

Upon measurement, we can agree that this is the correct expression.

Now onto volume. I can postulate on the volume of sphere by the fact that I know the volume of a cylinder with the length 2r (Volume=Area*2r=2pi*r^3). We know that the sphere could not have a greater volume since it would sit inside the cylinder. A VERY VERY keen observer would notice while looking at the cross section of a cylinder, a sphere, and cone all with the same radius, the cross sectional areas of each three objects would have the relation.

Cone+Sphere=Cylinder  .

So a dumb guess would be that same relation holds for the volume (not unfounded since the volumes and cross areas differ only from a "sweeping" mechanism), so we would have

Sphere=Cylinder-Cone=4π*r^2-2/3π*r^3

Sphere=4/3π*r^3

And by golly we have it.

On the Human Origins of Mathematics

The perceived beauty and interconnectedness of mathematics often leads many to conclude that math is wholly apart from man, and that advances in mathematics are in fact discoveries rather than creations. However, mathematics is simply one of many logic structures used to explain the world around us (i.e. science), meaning its creation is the result of human cooperation and world building (refer to Peter Beger's Sacred Canopy). Let us take a quick thought experiment to evaluate this statement: what is mathematics without humans studying it? Does math still exist then? A quick parallel to help us understand this concept is the difference between science and nature. Barring philosophical questions, we know that nature would exist without humans, but science would not. This is because science is our understanding of reality, it is not reality itself. Thus goes the saying, "Newton's Laws never moved a billiard ball."

Examining a brief history of mathematics we can see that mathematics has always come about as an explanation of the world we see. Imagine the earliest forms of algebra: I have 3 apples, and after picking 2 more, I have 5 now. The question would be: why do I have 5 now? Why not 4 or 6 or something else? The answer we know now is that this a physical manifestation of addition, by adding 2 I must increase my original number by 2. Thus we have addition as an explanation of physical events. Next we can examine geometry: how come the distance across this round stone seems to be proportional to the distance around it? I measure this stone and I can find this number that relates the two, which I call pi. Again I can see this physical phenomena in nature and create mathematics to try to understand it. Astronomy and the rest of the sciences all logically fall in suit of this idea, with people wanting explanations to why certain "stars" were at certain locations during the year, etc. In fact, in the case of sciences, the mathematics and the explanation of the phenomena are heavily interwoven to the point where they are interchangeable and complementary.

Since our conclusion is that mathematics does not exist without those studying it and therefore man made, we must lastly address the reason we do not "discover" mathematics. Let us bring up another question: does the mathematics exist before it is known? Since we've concluded that mathematics is an explanation, like science, let us draw another comparison between the two. Quantum mechanics, the funky physics of the small, was obviously not always known in the past and was formulated by scientists in the early 20th century. However, the reality of how these particles behaved was always true in nature, regardless of our understanding of it. Essentially what this means is that nature is right regardless if science is. With the case of mathematics, there is no equivalent world where mathematics exists regardless of humans. Any math we "see" is superimposed on the world; it is our metric to quantify reality. We yearn to explain that which is around us, the goal of academia. Meanwhile, reality simply is. Mathematics is mental; it is a framework we use to understand, and like science, it is not reality itself.



Edit:
I have another discussion on this topic that responds to a few of the criticisms of this argument, many of which were my own (I am not personally convicted either way but I want to facilitate conversation on this topic) I take a slightly modified position, of which I posted here.