Infinitesimals

In the previous post, I wrote about division by zero. In this post I want to talk about one particular case when such division, and its definition are important. As you probably guessed from the title, this post is about calculus. (In this post I am talking only about one variable calculus).

One of the most basic questions in calculus is finding slopes of functions. The simplest example of such a problem is to find the slope of a linear function, f(x)=mx+b.
In this case we get a straight line, so the slope is uniform. To find is we need to calculate the difference in y divided by the difference in x:

(f(x+h)-f(x))/h=(m(x+h)+b-mx-b)/h=m(h)/h=m

For a less friendlier functions, we cannot talk about globe slope, but only about slope of a certain part of the function, or even only in one single point. And this is were the problem appeared. We want to know the slope in every point, but how can we calculate it? Firstly we need to define what such a slope is. For example lets look on the function f(x)=x^2 and on the point x=3. Lets look on the points f(3.5) and f(3). If we connect those two points by a straight line, we can calculate the slope of that line. If we draw this on a paper, you will see that the function and the line are really close to each other on a small area around 3. Therefore we can thing about the slope of this line as an approximation to the slope of the function. But, obviously if instead of 3.5 we will take 3.1, we will get a better approximation. In the end we can think about the slope as:

slope=(f(3+h)-f(3))/h , h=0

And here we have it - devision by zero. It is obvious that there is no way around it. If h is not zero we get only an approximation and one which can be improved easily enough. The solution to this problem was found by Newton and Leibniz. Their idea was to define a non negative "number" that is smaller that any positive number. Such a number is called infinitesimal. The only real infinitesimal is zero, but if we agree to imagine that there is another such number, they we get many such numbers. This is because if dx is an infinitisimal that 0.5dx is also an infinitisimal. Since dx is not zero, we can divide by it. And becasue it is smaller than any positive number we can disregard it as if it was zero. It is simple to find the slope of x^2 usinig dx:

slope=(f(3+dx)-f(3))/dx=(9+6dx+dx^2-9)/dx=(6dx+dx^2)/dx=6+dx=6

Although the result is correct, we no longer use infinitesimals, but use limits instead. The reason for this is that infinitesimals are problematic. The problem lies in the very definition - it is not clear what do we mean by a number that is smaller than any positive number, but not negative or zero. We also treat it as both zero and as not zero. However, they are still in use in physics. The reasons for this is that while they are not rigorous enough for mathematicians to use, they give good intuition and they appear rather naturally in physical problems.

Division by Zero

We all know that division by zero is undefined. But what does it mean? Firstly, it is not "completely" undefined. For example, it is possible to say that:

1/0=lim1/x, x->0

This particular definition is not very good, because the limit doesn't exist. But we can always suppose that we look only on positive x and the the limit is defined to be infinity. Alternatively, it is possible to use geometric series. The formula for the sum of the geometric series says that:

1+q+q^2=q^3+...=1/1-q,

If we will take q to be 1 we will get: 1/0=1+1+1+1+....=infinity

As you can see I just wrote two different definitions to division be zero. However, this doesn't solve anything. The problem is that infinity is not a number. We could just as well say that 1/0=watermelon. Mathematically it is basically the same. Therefore when we divide by zero we no longer deal with numbers, and this what makes such division undefined.
Now, what about other definitions? A very long time ago, when "zero" appeared in mathematics there was an attempt to define division be zero by simply stating that division by zero gives a fraction whose denominator is zero. But this is again not a number.
We also cannot define 1/0=a where a is a number. The reason for this is that in this case we get:

1=0*1/0=0*a=0

And if this happens we get that the only number that we have is 0, because all the "other" numbers are equal to it. Obviously this is not an interesting situation. Because of this it is necessary that the devision by zero is not defined.

Lets look on some other example of undefined identities. The first one is 0^0. The reason this one is undefined is very simple:

0^0=0^(1-1)=0^1*0^-1=0/0=0*1/0

And we are back to division by zero. It is important to note that it is sometimes assumed that 0^0=1, but this is mostly done as an alternative to saying that in a polynomial x^n in which n can be zero x is not zero.
A more complex example is (-3)^x where is a real number (that is, x is not rational). Obviously, there is nothing special in number (-3), this expression is undefined for all negative numbers. For positive numbers the definition uses limits. For example:

3^x=lim(3^q), limq=x, q- rational

This definition works well for positive numbers, but for negarive numbers we have the problem that the square root is not real. Because of this if we don't allow complex numbers we get many undefined points in the series, and if we allow such numbers the series doesn't converge.

As a bonus, here is a little proof that 2=1. Can you see where is the error?

a=b.
a^2=ab ,
a^2+a^2=a^2+ab,
2a^2=a^2+ab,
2a^2-2ab=a^2-ab,
2(a^2-ab)=1(a^2-ab)
2(a^2-ab)/(a^2-ab)=1(a^2-ab)/(a^2-ab)
2=1

The reliability of wikipedia

As you all know there are many articles on wikipedia, and their quality varies greatly. However the main problem is that in many cases it is impossible to tell if a particular article is accurate or not. For example, I just found the following article on wikipedia:

Beard-second
The beard-second is a unit of length inspired by the light year, but used for extremely short distances such as those in nuclear physics. The beard-second is defined as the length an average physicist's beard grows in a second, or about 5 nanometers[1].

One beard-second equals 50 Ångströms (10^-10 m). 20000 Beard seconds equal 1 RCH. 2000 Beard seconds = 1 RBC.

Google search supports the beard-second for unit conversions.[2]

Unsurprisingly, this article is being considered for deletion in accordance with Wikipedia's deletion policy. Suprisingly, the part about Google supporting beard second in unit conversion is actually true. After a quick search on the internet, it seems that the beard second was invented as a teaching exercise a few years ago.

In this case, the article seems to be a joke, but in reality it is about a funny teaching exercise....

Changes

As you all probably noticed I didn't post anything for a long time. The main reason for this is that I was overloaded with homework. Also, the things I wanted to do started to pile up. The unread count in google reader went to about 800, sent pages on SU are now at over 100 etc. I also have unanswered comments that piled up..
In the following days I hope to clean this all up, and continue to post more regularly. Also in order to prevent this pile up from happening in the future, I decided to take some measures. Those that are relevant to this blog are written below:

1. Removing the skribit widget - while I got some votes and even one suggestion from it, it is not active enough to be really useful. Update: I will of course firstly write about all the topics that are currently in the skribit widget.
2. Closing the comments for antonymous users - I don't get too much comments, but anonymous comments are somewhat annoying when you have two or more such comments on one post.
3. Closing the comments on my kiba dock posts - there are lots of comments there wich ask for help, and I don't really have the time to help people with computer problems over the internet.

Now, if you have an opinion about this changes you are welcomed to voice it. As of now none of the changes are made, and if enough people will comment saying that they want things to remain as they are now, I will reconsider the changes.

The future

Found on the web - Pizza 2015:

Operator: "Thank you for calling Pizza Hut. May I have your..."

Customer: "Hi, I'd like to order."

Operator: "May I have your NIDN first, sir?"

Customer: "My National ID Number, yeah, hold on, eh, it's 6102049998-45-54610."

Operator: "Thank you, Mr. Sheehan. I see you live at 1752 Meadowland Drive, and the phone number's 494-2399. Your office number over at Lincoln Insurance is 745-2302 and your cell number's 266-2566. Which number are you calling from, sir?"

Customer: "Huh? I'm at home. Where d'ya get all this information?"

Operator: "We're wired into the system, sir."

Customer: (Sighs) "Oh, well, I'd like to order a couple of your All-Meat Special pizzas..."

Operator: "I don't think that's a good idea, sir."

Customer: "Whaddya mean?"

Operator: "Sir, your medical records indicate that you've got very high blood pressure and extremely high cholesterol. Your National Health Care provider won't allow such an unhealthy choice."

Customer: "Dang. What do you recommend, then?"

Operator: "You might try our low-fat Soybean Yoghurt Pizza. I'm sure you'll like it."

Customer: "What makes you think I'd like something like that?"

Operator: "Well, you checked out 'Gourmet Soybean Recipes' from your local library last week, sir. That's why I made the suggestion."

Customer: "All right, all right. Give me two family-sized ones, then. What's the damage?"

Operator: "That should be plenty for you, your wife and your four kids, sir. The 'damage,' as you put it, heh, heh, comes to $49.99."

Customer: "Lemme give you my credit card number."

Operator: "I'm sorry sir, but I'm afraid you'll have to pay in cash. Your credit card balance is over its limit."

Customer: "I'll run over to the ATM and get some cash before your driver gets here."

Operator: "That won't work either, sir. Your checking account's overdrawn."

Customer: "Never mind. Just send the pizzas. I'll have the cash ready. How long will it take?

Operator: "We're running a little behind, sir. It'll be about 45 minutes, sir. If you're in a hurry you might want to pick 'em up while you're out getting the cash, but carrying pizzas on a motorcycle can be a little awkward."

Customer: "How the heck do you know I'm riding a bike?"

Operator: "It says here you're in arrears on your car payments, so your car got repo'ed. But your Harley's paid up, so I just assumed that you'd be using it."

Customer: "@#%/$@&?#!"

Operator: "I'd advise watching your language, sir. You've already got a July 2006 conviction for cussing out a cop."

Customer: (Speechless)

Operator: "Will there be anything else, sir?"

Customer: "No, nothing. Oh, yeah, don't forget the two free liters of Coke your ad says I get with the pizzas."

Operator: "I'm sorry sir, but our ad's exclusionary clause prevents us from offering free soda to diabetics."

Tautology and theories

What is a tautology? Simply put it is a statement that is always true. More specifically, it is a statement that is true because of its structure. Usually such a statement is not informative. The easiest way to explain this is by example, so lets look on the following statements:

1. All apples are round.
2. All apples are either round or not round.

I have never seen an apple that wasn't round, so the first statement is a correct one. However it is not a tautology. It is perfectly possible for a square shaped apple to exist, you just need to make it grow inside a box. Therefore this statement is not always true.
The second statement is always true, and therefore a tautology, but it doesn't say anything. That is, if all we know about apples is that an apple is either round or not round we don't know anything about apples.
It doesn't mean that tautologies are useless, they have both use and importance in certain cases.

Lets look on tautologies in a more formalistic way. In the example above I assumed we have an object "apple" and a property "round". However, this is not necessary. All we need to write this example is two symbols: P, Q. If we rewrite the example using this symbols we get:
1. P->Q
2. P->(Q(or)(notQ))

I don't want to explain the notation, if you don't know it you are welcome to use wikipedia.
This notation is far better that the previous one. With this notation we are free to chose the meaning of the symbols. For example we can suppose that the meaning of P is x=7, and the meaning of Q is x+4=0. In this case 1 is usually false (but not always) but 2 is true.
From this we see that the second statement is completely independent from reality. It doesn't matter if we use it to tell something about apples or about mathematics, it will always remain true. This is the true meaning of being a tautology.

Sometime ago I read a post that claimed that all theories are tautologies. From the above it should be obvious that this is wrong. But there is a certain moment that causes this confusion. A theory is basically a collection of statements (theorems) that can be logically concluded from a certain set of prepositions. Those prepositions are divided in two groups - tautologies and axioms. (We don't really need to include tautologies, but because they are always true they get included automatically). Axioms are not tautologies. They are just "rules" that we choose by ourselves. In physics, those rules are based on reality and experiment, but this doesn't have to be the case. For example - the parallel postulate of Euclid. It seems natural to us to think that it is correct. But it is possible to built a geometry without it. Naturally in such a geometry all the statement that were derived from the parallel postulate are no longer correct (they might be correct, but not necessarily).

To sum up this discussion, a theory is correct as long as the axioms from which it was built are correct. However, if we try to apply it to a "world" in which the axioms are not true, the theory will not work. By the way, this is a major problem for physics and economics. They can build wonderful theory only to find out that the world we live in is different from their initial assumptions. In mathematics this is not a problem because there is no desire to describe our world, but to built a mathematical structure.
This also makes easy to explain the statement that a theorem once proved is true forever - a theorem is dependent only on its own conditions, so as long as they are satisfied the theorem will always be true.

Limits of applicability

A question: In physics, the world is considered to be continues. We can see this for example in the fact that we use integrals instead of finite sums in mechanics. However, we know that the objects in our world are made of atoms, which are in turn made of more elementary particles so they cannot be considered continues in a mathematical meaning. So can we apply mathematical result that are based on the assumption that the world is continues, to solve physical problems?

Since as I already said we use integrals in mechanics, it should be obvious that we can do this. But this is not true in all cases. For example lets consider the following theorem:

For any compact and convex set K in R^n and any continues function F from K to K, there is a point x in K such that F(x)=x.

This theorem is known as the general version of the Brouwer Fixed Point Theorem.

It should be obvious that a glass of water can be viewed as a compact and convex set in R^3. Also mixing the water using a spoon can be considered a continues function. Therefore, according to the theorem there is a point that didn't move. But this is obviously wrong. This is because the theorem gives us a point, but that point doesn't have to correspond to a particle and therefore saying that it didn't move is meaningless.

Why can we consider the world to be continues in one case and not the other? The reason for this is size. When we want to solve a problem in our scale (that is, we are dealing with objects that we can see and with results that can be observed directly), we can safely assume that the world is continues and use the corresponding tools. But this is only because the atoms are so small that for us there is no practical difference if we assume that the objects are continues.
But when like in the example with the water, the result we want to calculate (in that case a point) is not something we can see with our own eyes (we cannot see if an atom moved or not) assuming that the world is continues is wrong.

The above classification is very basic and inaccurate but it gives a general understating of the situation. In the next post, I plan to continue with this topic by discussing applicability of theories to reality.