You know that you are a mathematician when:

.. when you see an advertisement that reads "... in partnership with H&O", and the first thought that comes to your mind is: I know what it means H - stands for quaternions, O - stands for octonions and ... What?!?.

I hope it doesn't sound like math is the only thing I think about...

Anyway, it is a good thing that when mathematicians think too much about math they are at least not very dangerous. You just need to remember that if the person you are talking to doesn't know math well enough - it is not a good idea to talk with him about cantor sets, cardinalities, vector spaces, etc... Even if you feel that it is simple and you are sure that you can explain it, hold you self.

Invented or discovered?

Is math invented or discovered? It is a very ancient question. It dates back to the ancient Greek mathematicians. They believed that everything is a number - and they took this philosophy very serious. In fact, according to a legend the man who discovered that is irrational payed for this with his life. He proved that while sailing in the sea with his friends, and when he told them, they threw him out of the boat. For them such a proof was totally unacceptable.

In our days this sound weird. Killing for money and power is more or less common. But killing because of a mathematical question? This is not something you hear all the time.
But anyway, back to the question. If math is invented what does it mean? Does it mean that it is a mere game of thought? But if so why there are wrong answers? And if it is only a game of thought, then why it actually describes something real? Take E8 for example. Some people believe that it describes the shape of our universe and also all the particles are described in this symmetry.
The math is build on a logical basis. In fact, it is an old joke that math professors are the only one who can say that what they teach is always true and correct and it will stay so. But it does not explain why so often pure mathematics is found to either be useful in natural sciences or to describe something that exists in the real world.

If math is discovered, what is this strange "universe" that we are discovering then? The term Mathiverse is sometimes used for this. Also in some mathematical jokes we are told about things like "function city", "fractal forest" and so on. But are they real? Well the functions are real, but I really doubt that they have a city to live in. I cannot prove this of course, so all I can do is to say what I think about it, and it is fine if you think differently.

Personally, I believe that math is discovered. But I don't believe that we can talk about mathematical "entities" as if they were humans - they don't have cities, and they don't drink tea. So for me the term Mathiverse is just a reference to a very real set of objects. They are real because we can use them, and they are important because a lot of what we have depends on how well we understand them. Even the fact that you are reading it now is only possible because someone understood math well enough to build this monitor.

By the way the in the first paragraph is brought to you by a script that integrates latex to blogger (unfortunately there is no official way to type formulas here). You can get it here, and you can find its author here.

Strike in the University

You probably don't know about this but all the universities in Israel (there are seven) are partially close for about two month now. The professors are on strike so they just don't show up.
They have a serious disagreement with the treasury. In Israel the professors salary is payed by the universities, but all the agreements about it are done with the government. Or more precisely are not done at all. The last contract they had ended a few years ago and the government simply refused to do a new contract. It lasted in this way for a few years and all went well - except for the inflation. Because of inflation the professors lost about 35% of their salary. At the end of last year they tried to begin talks with government officials about this but without any success.
This year they decided not to show up to classes until their demands are met. It didn't help. As far as I know the government refused to acknowledge the problem and they even said they don't care if the semester will be canceled because of this.
The students are the main victims in all of this, but for different reasons they prefer to wait and see what happens. Officially they support the professors, but many of them a very angry.
Personally I support the professors. I didn't go to any of the demonstrations and I don't plan to go. Partially it is because half of my courses are taught despite the strike but mainly because I simply don't see any point in such demonstrations. The government already showed clearly what they want. Their plan is to modify the universities so that they will bring immediate profit. It was already stated in a public enough way. As a side note there is even a poster near the entrance to the Hebrew University - "We want education not education ltd." So what is needed is a change of policy. And for this to happen ... only God knows what can cause this.
I hope the strike will end soon and that the professors will get what they want.

Readability level

I recently did a check of this blog readability level on website grader. The site checks different specs including readability, google rank and so on. Strangely I got a Phd level. Since I tried to write it as simple as I could I was very surprised and amused by this result.

Anyway, I recently stumbled upon a site about octomatic number system. It is a system of base 8.
As of itself it is not very interesting - as I already wrote in a previous post, Evolution of numbers, all number systems that can be used for geometry are isomorphic to each other. However on this site they also offer to use different signs for the numbers:

What is interesting is the second row from below. The signs here are directly connected to binary. You can see it for your self if we will line to be 1 and empty space zero we will get:
000 001 010 011 100 101 110 111
And from the image you can tell immediately that 111=7. The same rule works for large numbers: 100111001 is for example 4*8*8+7*8+1=313.

The bottom row is a handwritten form. Personally I doubt that it is a good form, but it is their choice...

Should you go shopping?

This is a simple proof that you shouldn't. You cannot find anything useful in shops. And I am going to prove you this right now. The proof is by induction. I am going to prove that in any group of shops no matter how large, there is nothing interesting or useful.
Lets start with n=0 this is with the empty set. There are no shops in it so there is nothing interesting in them. Now lets suppose that we proved this for n=k. We just need to prove it now for n=k+1.
Lets therefore look at a random group of k+1 shops - {a1, a2, a3, a4 ....a(K+1)}. It can be viewed as a union of two groups - {a1,...,ak} and {a2,...,a(K+1)}. Since they both have the some number of shops in them - k, we know from our assumption that there is nothing interesting or useful in any shop in both of them. And since we don't get anything new when we unite the groups there is nothing interesting in the union also. And this completes the proof.

Now lets see another example. Lets suppose you have a random group of people. Is it true that all of them male or all of them female? In other words is it possible by choosing people randomly to get a group where there are both men and women? By now you probably think that I am crazy since I actually asked such a question, right?
Well lets see what math says about this. For n=1 we have a group with one human in it - either a a man or a woman obviously. Now lets suppose that we proved for n=k that any random group of size k has only men or only women. As in the previous example we can now look on a K+1 group as a union of two groups. And since we know that any group of the size k contains only men or only women they both contain only men or only women. Because a2 belongs to both groups and naturally has only one gender :) all the rest have to be the same gender - that is either male of female. And this completes the proof.

So now, after you saw this what do you think about mathematics?

Update: I guess I should have written this part in the first place but anyway: These proofs are wrong. It can be seen easily - just by simple example. In fact all the idea behind those proofs is that induction must be used with great care - or you might proof something that is plain wrong. A good way to check yourself is to try and prove your statement for n=3 and n=4. In the examples here there was no proof whatsoever. Induction is a very valuable and powerful tool but it is meant to be used for proving that some statement is correct for all n, when you know that it is correct for at least some n, and not for proving something you know that is not correct.

Evolution of numbers

We all know how numbers look. And sometimes it is even hard to imagine that they can have a different form that what is familiar to us. But while math is a rather ancient science our number system and our numbers are not very old. This photo shows their development in time:

As you can see even numbers in the middle ages were different from modern numbers. And I am sure that you would not even think that there is any connection between our numbers and Hindu.

It is a surprising fact however that there is just one number system. You will probably say it is not correct. But it is. Yes, there were a lot of different number system in history. Even in the present we have more then one. Take the binary system for example.
But if we want a number system that is good enough for both algebra and geometry, that is a system that lets us measure distance we discovery that there is only one such system - they are all isomorphic to each other. It means that you can give different names to numbers, you can have it in base 17 (for example) and not in base 10 but it will still be the same thing. In mathematical language - we have only one mathematical structure that is a complete ordered field.

A bit of humor

Theorem: Every positive integer is interesting.
Proof: By contradiction, assume that there exists an uninteresting positive integer. Then there must be a smallest uninteresting positive integer. But that's pretty interesting! Therefore a contradiction!

(I really doubt that you will manage to proof this one in school..)


Top ten excuses for not doing homework:
1. I accidentally divided by zero and my paper burst into flames.
2. It was Isaac Newton's birthday.
3. I could only get arbitrarily close to my textbook. I couldn't actually reach it.
4. I have the proof, but there isn't room to write it in this margin.
5. I was watching the World Series and got tied up trying to prove that it converged.
6. I have a solar powered calculator and it was cloudy.
7. I locked the paper in my trunk but a four-dimensional dog got in and ate it.
8. I couldn't figure out whether I am the square root of negative one or i is the square root of negative one.
9. I took time out to snack on a doughnut and a cup of coffee. I spent the rest of the night trying to figure which one to drink.
10. I could have sworn I put the homework inside a Klein bottle, but this morning I couldn't find it



Q: What is the world's longest song?
A: "Aleph-nought Bottles of Beer on the Wall."

(Aleght-nought is infinity. It is from Group theory)

A story:
Some famous mathematician was to give a keynote speech at a conference. When he was asked for an advance summary, he said he would present a proof of Fermat's Last Theorem -- but they shouldn't announce it. Although when he arrived, he spoke on a much more prosaic topic. Afterwards the conference organizers asked why he said he'd talk about the theorem and then didn't. He replied that this was his standard practice, just in case he was killed on the way to the conference.

Another story:
I don't know if the previous story was real but this one is:
WHEN G. H. Hardy faced a stormy sea passage from Scandinavia to England, he took out an unusual insurance policy. Hardy scribbled a postcard to a friend with the words: "Have proved the Riemann hypothesis". God, Hardy reasoned, would not let him die in a shipwreck, because he would then be feted for solving the most famous problem in mathematics. He survived the trip.

Lottery:
A mathematician organizes a lottery in which the prize is an infinite amount of money. When the winning ticket is drawn, and the jubilant winner comes to claim his prize, the mathematician explains the mode of payment:
"1 dollar now, 1/2 dollar next week, 1/3 dollar the week after that..."

(This is a reason for not playing with mathematician in math games. Also as a interesting fact, you chance of winning in a real world lottery is less then the possibility that you will die this week.)