Ten lessons for the math

1. Lecturing

Four requirements for a lesson or is not obvious to everyone if I think about the lessons I've been listening to 40 years.

a. Each lesson should have a theme.

German philosopher Hegel once said that a class or use the word "and" is not a good philosopher. I think he was right, at least for the lecture. Each lecture should raise a topic and repeat it constantly like a song many words. The listener is like a herd slowly moving towards being led away. If we just set a theme, we have the chance to be heard towards the right direction. If we lead in many directions, the herd will stampede over copper. The listener will lose interest and go back to everyone that they had stopped listening to lectures continue to monitor.

b. Never too-hour lecture.

Teaching is an error too now can not be forgiven. After 50 minutes (one of von Neumann century as is often said), people will not be interested in lectures even when we are proving the Riemann hypothesis. One minute the lecture will ruin the best sermon.

c. Contact us to listeners.

When we entered the room to see if anyone noticed the number of listeners that his work-related lectures. Take immediate redeployment of the lectures that he will be mentioned. In this way, we have at least one person watching lectures and a friend. Everyone came to hear my lectures are a secret hope that their work will be mentioned.

d. Give the audience something they can take home.

This is a Struik advice. Not easy to implement tips. We can easily set up a lecture on what will be the ones to hear and remember this is not what people expect lectures. I often meet MIT alumni have heard my lectures. Most admitted that they had forgotten the lecture content and all the mathematical knowledge that I think has been communicated to them. However, they would be happy to repeat the joke funny, joke stories, or comments on the sidelines of a certain bug me.

2. Engineering the blackboard

a. Let's wipe out the old chalk marks on the table.

It is very important to remove all traces of residual powder after wiping tables. By starting with a blackboard chalk marks had not secretly give the listener the impression that lectures are not flawless.

b. Start writing from the upper left corner of the table.

What should we write on the board corresponds to what a person wants to listen intently writing in their notebooks. So slow with letters written to and not abbreviated. Those who have heard the recording with us in good faith and we should help them take notes. When using lights, additional time should we interpret the reference pages by giving comments or unimportant to remind the listener has time to copy the reference page. We all fall into the illusion that the audience will have time to read a copy of the page they were given lectures after lectures. It just wished only.

3. Announcing the results multiple times

After my thesis research in a number of functional analysis. I bought a collection of works of F. Riesz soon as books, thick and heavy it was published. But when I started browsing can not find the book very thick, almost cardboard. Strangely, the article is reprinted with Riesz large letters. I liked the article because they are Riesz beautifully written and causing the reader a sense definitive.

When I read the book collection is a work of Riesz feelings emerge. The editors took advantage of every little thing in which Riesz announced. Clearly the work of Riesz not much. Surprisingly this is the work published many times. Riesz often also published a rough draft of an idea in a magazine of Hungary unknown. A few years later he sent a series of messages posted in the form of Comptes Rendus Académie française with ideas that are more detailed. A few more years passed and he will post the final report in French or English.

Koranyi, who has studied Riesz, told me that Riesz often teach the same subject year after year while reflecting on the article written last. Not surprisingly, this article is perfect.

Examples of Riesz worthy follow. About this math is divided into many small groups, each group has a habit, the symbols and its own concepts. So need to present a result in many different forms, each form can be used for a particular group. If not, the cost will be the someone will reveal to us with a result of language and other symbols and they will rightly claim that it is the result of them.

4. He will surely be remembered by his overview article

 Let us consider two examples, starting with Hilbert. When it comes to Hilbert, we think some of his famous theorem as the Hilbert basis theorem. But Hilbert's name is usually remembered for arithmetic works Overview (Zahlbericht) or Base geometry book or his textbooks on integral equations.

The name "Hilbert space" given by Stone and von Neumann in recognition of Hilbert textbook on integral equations in which the word "universal" is defined for the first time, at least 20 years before the subjects Quantum mechanics was born. This syllabus is almost a review is based on the work of Hellinger and many other mathematicians whose names were forgotten today.

Similarly, establishments geometry book is the book that made a name familiar to everyone Hilbert math does not contain any of his original works and reap the results of many works such as Kohn geometry, Schur , Wiener (Wiener is not Schur and we've heard of), Pasch, Pieri and many Italian mathematician.

Also, the book details the school, a building facility has revolutionized the theory subjects, carried out a review article that the newspaper's Bulletin Board German mathematician Hilbert set for writing.

William Feller is another example. Feller was remembered as the author of the best book on probability. Few people appear to confirm this can name all the work of Feller. Most people are not aware that funds research Feller convex geometry.

Let me digress with a personal recollection. Sometimes I have

published in a branch of philosophy called scientific phenomena (Phenomenology). After the publication of the first paper in this subject, I am very annoyed when people told me at a meeting of the scientific and philosophical phenomenon exists (existential philosophy) a veiled way that everything I write in the paper are already known, and I was forced to review its published standards in science phenomena.

One more thing that works is the basis of science is the phenomenon of language written in a heavy German philosophy. Traditionally, there is no illustration of what is discussed. One day I decided to publish a doubt with a bit of paper is actually a re-post a few paragraphs from a book by Husserl, plus a few examples. At the next meeting of the scientific and philosophical phenomenon exists, I am waiting for the worst thing that can happen is a phenomenon leading scientists rushed to me with a smile.

He praised my article all said and encouraged me to develop new ideas and originality of this article.

5. Each mathematician only a few tips

Long ago a well-known scholars have made some disparaging remarks of Erdos works. I admire Erdos contributions to mathematics and annoyed when old mathematician said it confirmed that all work can Erdos reduced to a few tips that Erdos had always rely on them in the proof. The thing that the arithmetic is not noticed by other mathematicians, even the best people, and based on a few tips that they use again and again. Take a look at Hilbert. Book two of the anthology contains the works of Hilbert of Hilbert's paper on invariant theory. I am determined to read some of this article. Sadly, some of the beautiful results of Hilbert was forgotten. But when reading the proof of Hilbert for some deep theorems of invariant theory, I was surprised to see that the proofs are some tips to use the same. Thus Hilbert only a few tips!

6. Do not worry about the error

Again I started with Hilbert. When the German collection of published works by Hilbert and gave him a birthday occasion one of his later they realize that they can not publish the article in its original form because they contain too much errors, including the very serious error. So they hired mathematicians (female) Todd unemployed OlgaTaussky-reviewed articles of Hilbert and repair all errors. Olga worked for three years and all errors have been corrected without much change theorems content. The only exception is an article written by Hilbert's age when he was not able to edit well. It is a proven hypothesis Continuum

papers published in Mathematische Annalen early thirties. Finally, the Hilbert was given a copy of the brand new collection on their birthdays. Hilbert was falling apart and thoroughly considered not detect anything. ...

There are two types of errors. Type fatal error will destroy the whole theory of error uncertainties will also be useful when checking the correctness of a theory.

7. Using the method of Feynman

Feynman liked following advice on how to become a genius. You need to constantly keep in mind a number of issues he likes, although most of the time they stayed there. Every once he heard or read of a trick or a new result, then I try to see what it can to help each of you does not matter. You can also get lucky and sometimes people will say, "How did he solve the problem? Maybe he is a genius!"

8. Generous citing

I'm always offended when I read an article that I feel that I do not like to be quoted. Can confirm that this is true of everyone. One day I did try an experiment. After writing a relatively long article, I started doing a whole draft citations. Then I suddenly decided to quote some articles not related at all to the content of my article to see

what will happen.

Unexpectedly I received a letter from the authors that their articles irrelevant to my article. Both letters were written with an emotional voice. Both authors are congratulated me be the first to have recognized their contributions in that field.

9. Writing the introduction of information

Few people today are reading an article from beginning to end. If we want others to read your paper, then we have to give them incentives to persuade. An introduction to long and rambling brief history of the issue, citing the full contribution of people and content described attractive article will help attract some readers.

As an editor of the journal Advances in Mathematics I usually send the article back to the authors asking them to write longer. Once I receive a reply from the author said that it was paper and magazine Annals of Mathematics denied for too long introduction.

10. Be prepared for old age

He Ulam you have my late often remarked that his life is divided into two distinct parts. in the beginning he was always the youngest in the group, while in the second part he is always the oldest person without a transition period.

Now I see Ulam remarked very true. Old age does not seem to gradually and we accept it the hard way. This depends on a basic awareness and we need time to get used to it. We must understand that to a certain age it is no longer seen as an individual anymore. I will become an institution (Institution), and will be treated as one treats an institution. They expect us to use the shares as an object or as an architectural wonder.

The longer we write or not does not matter anymore. If the articles are not good, they will say, "What were you hoping for? Hardened He's gone! '". And if one of our articles that become interesting, they will say "I expect nothing else? Whole life he has studied this before!" The only reaction is so delicate fun of his new role as a