Richard Stallman's personal site. Mirror 1

Richard Stallman's personal site.

https://stallman.org

For current political commentary, see the daily political notes.

RMS's Bio | The GNU Project

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How I do my computing

My computer

I use a Thinkpad T400s computer, which has a free initialization program (libreboot) and a free 

operating system (Trisquel GNU/Linux). It was not sold that way by Lenovo, however; small 

businesses buy them used, recondition them, and install the free software. 

This is one of the computers endorsed by the FSF.

Before using such Thinkpads, I used the Lemote Yeeloong for several years. At the time, it was 

the only laptop one could buy that could run a free initialization program and a free operating system. 

But it was never sold with a free operating system.

Before that, I used an OLPC for some weeks. The OLPC uses a nonfree firmware blob for the WiFi, 

so I could not use the internal WiFi device. No big problem, I used an external WiFi adaptor.
I stopped using it because the OLPC project decided to make their machine support Windows, 

so I did not want to appear to endorse it by visibly carrying it around. I could have continued 

using it privately with its free software installation, but I had no need for another computer to 

use only privately.

The results that seemed likely, millions of children running Windows on the OLPC, have not occurred. 

Instead we see millions of children running Windows on the Intel Classmate, or nowadays a 

Chromebook that sends the child's personal data to Google.

Before that I used machines that ran completely free GNU/Linux systems but had nonfree BIOSes. 

I tried for about 8 years to find a way to avoid the nonfree BIOS in some commercial machine.

GNU/Linux distro

I do not have a preferred GNU/Linux distro. I recommend all the ethical distros — namely, those 

that are 100% free software.

I've chosen not to have any preferences among those ethical distros. In fact, I am not in a position 

to judge them on other criteria: even to try them all would be a lot work of that I have no need to do.

What I do on my computer

Mostly I use a text console, for convenience's sake. Most of my work is editing text and that is more

 efficient on a text console. On the text console, the touchpad can't cause me any trouble if I touch it 

by accident.

I do use X11 for tasks that need a graphical interface. I have no preferred graphical environment or 

window manager. Since my interest in using graphical environments is small, I don't want to spend 

time comparing them.

This is not an ethical issue, just my own personal preference. On the ethical level, I think it is 

important for free software to provide convenient free graphical user interface software, which is 

why the GNU Project launched three projects to develop that. The third, GNOME, was successful, 

so we never needed a fourth one.

I spend most of my time editing in Emacs. I read and send mail with Emacs using M-x rmail 

and C-x m. I have no experience with any other email client programs. In principle I would be glad 

to know about other free email clients, but learning about them is not a priority for me and 

I don't have time.

I edit the pages on this site with Emacs also, although volunteer helpers install the political notes 

and urgent notes. I have no experience with other ways of maintaining web sites. In principle 

I would be glad to know about other ways, but learning about them is low priority for me and 

I have other things to do.

How this site is maintained

This site is maintained in a very simple way. I edit the pages such as this one manually as HTML. 

I only know simple HTML; others who know more wrote the parts at the top and bottom of pages, 

and the more complex formatting on the home page. Volunteer helpers install the political notes

 every day after receiving the text from me by email. A cron job "rolls over" the political notes page 

every two months. The photo galleries are generated with this perl script. The search feature on 

the site is done with this code.

An explanation of the concept of designing a "user experience" which also shows why I find it 

loathsome. This is why I want stallman.org to remain simple: not a "user experience" but rather 

a place where I present certain information, views and action opportunities to you.

Would you like to help do this? Write to rms at the site gnu.org.

How I use the Internet

• I have used the Internet since it first existed. I never used UUCP, though occasionally 

• I sent emails to addresses that involved transmission via UUCP.

• I am careful in how I connect to the internet.

Specifically, I refuse to connect through portals that would require me to identify myself, or 

I don't mind giving an identity that isn't really me, in order to connect, if that works.

I often connect in a person's home. The person of course knows who I am, but that does not 

• bother me. What I would object to is putting my identity in a database that can be searched. 

• I prevent that by changing my mac address at each location.

• to run any nontrivial nonfree Javascript code. I use LibreJS to prevent nonfree Javascript 

• code from running..

• I am careful in how I use the Internet.

I generally do not connect to web sites from my own machine, aside from a few sites 

I occasionally also browse unrelated sites using IceCat via Tor. Except for rare cases, 

I never pay for anything on the Web, because I cannot pay anonymously. Anything on the net that requires payment that would identify me, 

I don't do. (I made an exception for the fees for the stallman.org domain, since that is connected with me anyway.)

I avoid paying with credit cards generally. For freedom's sake, insist on paying cash.

• When a business pressures you to pay in an identified way, it's your chance to defend freedom by saying, 

"If you won't take my cash, no sale!"

• I do not identify myself to them. I think that is enough to prevent my browsing from being connected with me. 

IceCat blocks tracking tags and most fingerprinting methods.

• I have some special relationship with. I usually fetch web pages from other sites by sending mail to a program 

(see https://git.savannah.gnu.org/git/womb/hacks.git) that fetches them, much like wget, and then mails them back to me. 

Then I look at them using a web browser, unless it is easy to see the text in the HTML page directly. I usually try lynx first, 

then a graphical browser if the page needs it (using konqueror, which won't fetch from other sites in such a situation).

• I would not mind paying for a copy of an e-book or music recording on the Internet if I could do so anonymously, 

and it treated me justly in other ways (no DRM or EULA). But that option almost never exists. I keep looking for ways to make it exist.

• For searching, I have mostly used DuckDuckGo for the past few years. It does work with JS disabled, but you have 

to follow a link before you search.

I also sometimes use ixquick.com. My usual precautions should stop them from knowing it is me.

I no longer use Google search, not even occasionally, because it sends me a broken CAPTCHA. 

I suspect the reason it tries to send me a CAPTCHA is that I am coming through Tor. I would answer the CAPTCHA if that worked, but it does not.

I suspect that the reason the CAPTCHA is broken is that it depends on nonfree Javascript. 

I won't run that in my computer. I am not willing to let Google see where I am, so I won't bypass Tor. Therefore, I can't use Google search any more.

Social media

• I do not use any social networking sites because that way of working is inconvenient for me. 

That doesn't mean I think they are all unethical. Some are, some are not. Social networking sites raise their own set of ethical issues, 

completely different from the ethical issues of distributing software ( free vs proprietary), and there are big differences between them.

I have a Twitter account called rmspostcomments, which I use to log in on other sites to post comments on articles. I never post on Twitter.

 Someone made an account stallman_feed which I'm told posts something about my political notes.

 Any other Twitter account that claims to be mine is an impostor.

The rms account on gnusocial.no repeats the political notes from this site, but I do not post on it directly. That site runs GNU Social.

Aside from those two, any account on a social networking site that says it is mine is an impostor.

I do not post on 4chan. I have nothing against it in principle, but I am told a lot of the posts nowadays 

are right-wing bigotry which I condemn totally. I have occasionally answered questions for interviews for 4chan, 

but I have never posted anything there. Any posting there that says it is by me is by an impostor.

• I have never had a Facebook account, or a Google+ account. Some impostor created a Facebook account using my name. 

The page is not mine. The Google+ account using my name is also not mine.

I reject Facebook because it requires each used (i.e., person used by Facebook) to have just one account, which means 

that all the person's activities are grouped together. It also insists on knowing the person's usual name, and it is 

starting to demand a series of different photos.

I am proud to identify myself when stating my views; I can afford to do that because I am in a fairly safe position. 

There are people who rationally fear reprisals (from employers, gangsters, right-wing extremists, or the state) 

if they sign their name to their views. For their sake, let's reject any social networking site which insists on connecting an account to a person's real identity.

Of course, Facebook is bad for many other reasons as well.

Google+ formerly required knowing the user's real name, but no longer. However, it does require identification 

in the form of a phone number. Meanwhile, Google+ has another unacceptable injustice: it requires running nonfree Javascript code to post a message.

E-mail service

People sometimes ask me to recommend an email service. The two ethical issues for an email service are (1) whether 

you can use it without running any nonfree software (including nonfree Javascript code from the site), and (2) whether it respects your privacy.

For issue 1, see the FSF's page. On issue 2, I have no way to verify that any email service is satisfactory. Therefore,

 I have no recommendation to offer.

However, I can suggest that it may be wise to use an email service that is not connected with your search engine. 

That way you can be almost sure that your email contents don't influence your search results. You shouldn't identify yourself to your search engine in any case.

Programming languages

• The most powerful programming language is Lisp. If you don't know Lisp (or its variant, Scheme), you don't know what 

it means for a programming language to be powerful and elegant. Once you learn Lisp, you will see what is lacking in most other languages.

Unlike most languages today, which are focused on defining specialized data types, Lisp provides a few data types which 

are general. Instead of defining specific types, you build structures from these types. Thus, rather than offering a way to define a list-of-this type and a list-of-that type, Lisp has one type of lists which can hold any sort of data.

Where other languages allow you to define a function to search a list-of-this, and sometimes a way to define a generic 

list-search function that you can instantiate for list-of-this, Lisp makes it easy to write a function that will search 

any list — and provides a range of such functions.

In addition, functions and expressions in Lisp are represented as data in a way that makes it easy to operate on them.

When you start a Lisp system, it enters a read-eval-print loop. Most other languages have nothing comparable to `read', 

nothing comparable to `eval', and nothing comparable to `print'. What gaping deficiencies!

While I love the power of Lisp, I am not a devotee of functional programming. I see nothing bad about side effects and 

I do not make efforts to avoid them unless there is a practical reason. There is code that is natural to write in a functional way, 

and code that is more natural with side effects, and I do not campaign about the question. I limit my campaigning to 

issues of freedom and justice, such as to eliminate nonfree software from the world.

Lisp is no harder to understand than other languages. So if you have never learned to program, and you want to start, 

start with Lisp. If you learn to edit with Emacs, you can learn Lisp by writing editing commands for Emacs. You can use 

the Introduction to Programming in Emacs Lisp to learn with: it is free as in freedom, and you can order printed copies from the FSF.

You can learn Scheme (and a lot of deep ideas about programming) from Structure and Interpretation of Computer Programs 

by Abelson and Sussman. That book is now free/libre although the printed copies do not say so.

Please don't buy books (or anything) from Amazon!

• My favorite programming languages are Lisp and C. However, since around 1992 I have worked mainly on free software activism, 

which means I am too busy to do much programming. Around 2008 I stopped doing programming projects. As a result, 

I have not had time or occasion to learn newer languages such as Perl, Python, PHP or Ruby.

I read a book about Java, and found it an elegant further development from C. But I have never used it. I did write some code 

in Java once, but the code was in C and Lisp (I simply happened to be in Java at the time ;-).

• By contrast, I find C++ quite ugly.

The flaws of C++, as I recall from when I studied the matter around 1990, include syntax and semantics. As for syntax, 

its grammar is ambiguous, and it is gratuitously incompatible with C, which blocks the smooth upgrade path from C to C++.

As for semantics, the abstract object facility of C++ is designed around the case where the real type of an object 

is known at compile time. However, in that case, abstract objects are equivalent to a naming convention for functions to call. 

The case where abstract objects add real power to a language is when the type is not known until run time. C++ does handle that,

 but it seems to be an afterthought, a poor relation.

I suspect that I would find plenty of ugliness in the template library, but I don't know. That was added to C++ after I studied it.

• I skimmed documentation of Python after people told me it was fundamentally similar to Lisp. 

My conclusion is that that is not so. `read', `eval', and `print' are all missing in Python.

How to learn programming

First, read a textbook about programming in some language, then manuals for several programming languages including Lisp. 

If this makes natural intuitive sense to you, that indicates your mind is well-adapted towards programming.

If they don't make intuitive sense to you, I suggest you do something other than programming. You might be able to do

 programming to some degree with a struggle, but if you find it a struggle you won't be very good at it. 

 What's the point of programming if it is a struggle instead of a fascination?

After that, you need to read the source code of real programs (or parts of them) and figure out what they do. 

Then start writing changes in them, to add features, or fix bugs if you can find out about specific bugs to fix. 

Ask some good programmers who are familiar with the code of those programs to read and critique your changes.

If you fix a bug in a free program that people are developing, the developers are likely to be glad to get fixes 

from you and will tell you the way to write them to make them good to install. Look at their TODO list for features you would like to implement. 

You will find it is a great satisfaction when the developers incorporate your changes.

Do this over and over and you will become good at developing software.

Please use your programming capability only for good, not for evil. Don't develop nonfree software, or service as a software substitute.

 Design systems not to collect personal information, and to allow anonymous use.

Non-free software issues

I firmly refuse to install non-free software or tolerate its installed presence on my computer or on computers set up for me.

However, if I am visiting somewhere and the machines available nearby happen to contain non-free software, through no doing of mine, 

I don't refuse to touch them. I will use them briefly for tasks such as browsing. This limited usage doesn't give my assent 

to the software's license, or make me responsible its being present in the computer, or make me the possessor of a copy of it, 

so I don't see an ethical obligation to refrain from this. Of course, I explain to the local people why they should migrate the 

machines to free software, but I don't push them hard, because annoying them is not the way to convince them.

Likewise, I don't need to worry about what software is in a kiosk, pay phone, or ATM that I am using. I hope their owners migrate

 them to free software, for their sake, but there's no need for me to refuse to touch them until then. (I do consider what 

 those machines and their owners might do with my personal data, but that's a different issue, which would arise just the same 

 even if they did use free software. My response to that issue is to minimize those activities which give them any data about me.)

That's my policy about using a machine once in a while. If I were to use it for an hour every day, that would no longer be 

"once in a while" — it would be regular use. At that point, I would start to feel the heavy hand of any nonfree software 

in that computer, and feel the duty to arrange to use a liberated computer instead.

Likewise, if I were to ask or lead someone to set up a computer for me to use, that would make me ethically responsible for

 its software load. In such a case I insist on free software, just as if the machine were my own property.

As for microwave ovens and other appliances, if updating software is not a normal part of use of the device, then it is not a computer.

 In that case, I think the user need not take cognizance of whether the device contains a processor and software, or is built some other way. 

 However, if it has an "update firmware" button, that means installing different software is a normal part of use, so it is a computer.

Skype (or any nonfree noninteroperable communication program) is a special case because of its network effect. Using Skype to

 talk with someone else who is using Skype is encouraging the other to use nonfree software. Doing so regularly is pressuring 

 the other to use nonfree software. So I refuse to use Skype under any circumstances (See more information.)

Streaming medias and DRM issues

• Streaming media dis-services such as Netflix and Spotify require nonfree client programs that impose digital restrictions mechanisms (DRM)

 intended to stop the user from saving a copy of the data being streamed through her own computer. You should never use DRM that you can't break,

 so you should not use these dis-services unless you can break their DRM.

An additional injustice of these and other streaming client programs is that they impose unjust contracts (EULAs) which restrict the 

user more strictly than copyright law itself. I do not agree to EULAs, period, and I urge you to join me in rejecting them.

These streaming dis-services are malicious technology designed to make people antisocial. (If you don't have a copy, you can't share copies.)

 Rejecting them is of the highest ethical priority.

A friend once asked me to watch a video with her that she was going to display on her computer using Netflix. I declined, saying that

 Netflix was such a threat to freedom that I could not treat it as anything but an enemy.

Out, out, damned Spotify! Flick off, Netflix!

• Every product with Digital Restrictions Management (DRM) is an attack on your freedom.

Therefore, one should not buy or tolerate any product with DRM handcuffs unless one personally possesses the means to break the handcuffs. 

For instance, don't use encrypted DVDs unless you have DeCSS or another comparable free program. And never use a Bluray disk unless you 

find a way to break its handcuffs. Don't use the Amazon Swindle or other e-book readers that trample readers' freedoms. Don't use music 

or video streaming "services" that impose DRM. (If they require a nonfree client program, it is probably for DRM or some sort of surveillance of users.)

Miscellaneous

• I never used Unix (not even for a minute) until after I decided to develop a free replacement for it (the GNU system). 

I chose that design to follow because it was portable and seemed fairly clean. I was never a fan of Unix; I had some criticisms

 of it too. But it was ok overall as a model.

• In the mid 90s I had bad hand pain, so bad that most of the day I could only type with one finger. The FSF hired typists 

for me part of the day, and part of the day I tolerated the pain. After a few years I found out that this was due to the 

hard keys of my keyboard. I switched to a keyboard with lighter key pressure and the problem mostly went away.

My problem was not carpal tunnel syndrome: I avoid that by keeping my wrists pretty straight as I type. There are several 

kinds of hand injuries that can be caused by repetitive stress; don't assume you have the one you heard of.

• I find it bizarre that people use the term "coding" to mean programming. For decades, we used the word "coding" for 

the work of low-level staff in a business programming team. The designer would write a detailed flow chart, then the 

"coders" would write code to implement the flow chart. This is quite different from what we did and do in the hacker 

community -- with us, one person designs the program and writes its code as a single activity. When I developed GNU programs, 

that was programming, but it was definitely not coding.

Since I don't think the recent fad for "coding" is an improvement, I have decided not to adopt it. I don't use the term "coding",

 except if I were talking about a business programming team which has coders.

• Why I coined the name POSIX.

________________________________________

Return to Richard Stallman's home page.

Please send comments on these web pages to rms@gnu.org.

Copyright (C) 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2015, 2018, 2019 Richard Stallman

Verbatim copying and distribution of this entire article is permitted in any medium, provided this notice is preserved.

Richard Stallman responde a Linus Torvalds

Aquí tenéis la réplica a la entrevista realizada el 30 de mayo por Página|12,  a Linus Torlvalds.

–Torvalds: “Yo nunca usé el nombre GNU”.
–Stallman: Es verdad, él tiene la costumbre de no reconocer nuestro trabajo.

–Torvalds: “Linux nunca fue un proyecto de la Free Software Foundation, y la FSF jamás tuvo nada que ver con él.
–Stallman: ” Es verdad, Linux el kernel es su trabajo y no participamos. Pero el sistema para el usuario es la combinación de GNU y Linux.

–Torvalds: “La mayoría de las herramientas no son GNU, tampoco, aunque el compilador GNU C fue y es un gran invento.”
–Stallman: El sistema inicial fue el sistema GNU con el kernel Linux. En ese sistema, los programas específicamente del proyecto GNU son una gran parte.

–Torvalds: “Hoy en día la mayoría del sistema no es GNU ni Linux.
–Stallman: GNU actualmente es 10 por ciento del total y Linux es uno por ciento. Esos porcentajes señalan nuestro logro de atraer a muchos programadores a extender el sistema con más programas libres. Pero, ¿por qué existe sistemas operativos libres? Porque nosotros del Proyecto GNU teníamos la meta de hacer uno, y desde 1984 hemos luchado para hacerlo. Pienso que por eso, más que por haber escrito la mayor contribución de cualquier proyecto, el nombre del sistema debería incluir “GNU”. Este es el único sistema operativo que existe con el motivo de liberar a los usuarios, y necesitan recordarlo, porque olvidar la lucha por la libertad es el primer paso para perder la libertad.

Michael atiyah. sir Michael atiyah

Recuerdo la conferencia de Michael Atiyah en la faculta de ciencias de la UNAM.

Se vei que manejaba totalmente el tema. Un tipo de mediana o corta estatura y muy severo en su trato.

Me autógrafo el cartel

Sir Michael Atiyah obituary

One of the greatest British mathematicians since Isaac Newton

Michael Atiyah at his home in Edinburgh in 2004. Photograph: Murdo MacLeod/The Guardian

Ian Stewart

Tue 15 Jan 2019 15.41 GMT

8

The last time I met Michael Atiyah, who has died aged 89, was at Tate Modern in London; not the most likely place to run into probably Britain’s greatest mathematician since Isaac Newton, but entirely consistent with his wide-ranging enthusiasm for his subject. It was June 2012, and I joined him and the flamboyant French mathematician Cédric Villani in a panel discussion: Mathematics, a Beautiful Elsewhere. The title says it all.

We have sulphuric acid to thank for Atiyah’s decision to become a mathematician. Early in 1940, as Britain and France fought over his homeland of Lebanon, his parents sent him to Victoria college in Cairo. In a 1984 interview he said that while there he got very interested in chemistry, but eventually decided that making “sulphuric acid and all that sort of stuff” was not for him: “Lists of facts, just facts ...” From that time on, mathematics became his passion. “I never seriously considered doing anything else.” Atiyah’s work was to have a profound influence on today’s mathematics.

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Atiyah was a geometer, in the sense of visual thinking allied to abstract symbolism, a new attitude that swept through mathematics in the middle of the 20th century. You thought about it like geometry but wrote about it like algebra, and very esoteric algebra at that. His research divides into four main periods, to some extent overlapping – in the 1950s, algebraic geometry; in the 60s and early 70s, K-theory; the 60s to 80s, index theory; and the late 70s to mid-80s, gauge theory, where his ideas became extremely influential in quantum physics.

Algebraic geometry originally developed from a deep link between geometry and algebra promoted in the 1600s by René Descartes. Start with Euclid’s plane and introduce coordinates – pairs of numbers describing the location of a point, much as latitude and longitude determine a point on the Earth’s surface. Geometric properties of curves can then be described by algebraic equations, so questions in geometry can be tackled using algebra, and vice versa.

In the late 1800s and early 1900s, a new kid appeared on the mathematical block: topology, in which geometrical shapes can be deformed as if made of elastic. Classical features such as lengths and angles lose their meaning, and are replaced by concepts such as being connected, knotted, or having a hole like a doughnut.

Topology turned out to be fundamental to many areas of mathematics. Techniques were devised to associate with a topological space various “invariants”, which reveal when spaces can or cannot be deformed into each other.

One of the most powerful invariants, homology, was established by Emmy Noether, the greatest female mathematician of the late 1800s and early 1900s. She reinterpreted, in terms of abstract algebra, rudimentary methods for counting features such as the number of holes in a surface.

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In effect, Noether explained that as well as counting holes and associated structures, we can ask how they combine, and extract topological information from the answer.

Atiyah began his research career in algebraic geometry, but under the influence of his supervisor, William Hodge, at Cambridge, he quickly moved into an adjacent field, differential geometry, which studies concepts such as curvature — how a space deviates from the flat plane of Euclid. There he made big advances in the interactions between algebraic geometry, differential geometry and topology.

Euclid’s investigations of a circle includes its tangents: straight lines that touch it at one point, like a road supporting a bicycle wheel. Similarly, a sphere has a family of tangent planes, one for each point on its surface. A general family of this kind is called a vector bundle: “bundle” because the sphere ties all the planes together, and “vector” because higher-dimensional analogues of lines and planes are called vector spaces.

The topology of a vector bundle provides information about the underlying space. The tangents to a circle, for example, form a cylinder. As proof: rotate each tangent line through a right angle, out of the plane of the circle, and you get a cylinder. There is another vector bundle associated with a circle, in which the lines are twisted to form the famous Möbius band, a surface that differs topologically from a cylinder since it has only one side. Atiyah applied these ideas to “elliptic curves”, actually doughnut-shaped surfaces with interesting number-theoretic properties.

His next topic, K-theory, is a far-reaching extension of Noether’s homology invariant. A cylinder and a Möbius band are topologically distinct because their associated bundles have different twists. K-theory exploits vector bundles to capture higher dimensional analogues of such twists.

The topic underwent a period of rapid development in the 60s, stimulated by remarkable links to other major areas of mathematics, and it provided topologists with a powerful toolkit of invariants.

Atiyah, often jointly with other leading mathematicians, was a driving force behind these developments. Important themes were the cobordism theory of René Thom (how one circle splits into two as you move down a pair of trousers from the waist to the leg holes, only done for multidimensional spaces) and the periodicity theorem, first proved by Raoul Bott, showing that higher K-groups repeat in a cycle of length eight.

Index theory has its origins in the observation that topological features of a landscape, such as the numbers of mountain peaks, valleys and passes, are related to each other. To get rid of a peak by flattening it out you must also get rid of a pass, for example. The index organises such phenomena, and can be used, in suitable circumstances, to prove that a peak must exist in some region.

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A landscape is a metaphor for the graph of a mathematical function, and a sweeping generalisation relates the number of solutions of a differential equation to a more esoteric topological index.

Differential equations relate rates of change of various quantities to each other, and are ubiquitous in mathematical physics; the Atiyah-Singer Index Theorem, proved jointly with the American mathematician Isadore Singer in 1963, reveals a highly significant link between a topological index and the solutions of a differential equation.

In an appropriate mathematical setting this can lead to a proof that a solution must exist, so the Atiyah-Singer index has widespread applications to physics. Forty years after their discovery, the pair were jointly awarded the Abel prize of the Norwegian Academy of Science and Letters, in 2004.

Gauge theory arose in physics, formalising certain symmetries of quantum fields and particles. The first example arose from James Clerk Maxwell’s equations for the electromagnetic field (1861), where certain mathematical transformations can be applied without changing the physics.

In 1954 Chen Ning Yang and Robert Mills extended this idea to the strong interaction, which holds together each quantum particle in the atomic nucleus. Symmetry turned out to be vital to quantum mechanics – for example, the recently discovered Higgs boson, which endows particles with mass, acts by breaking certain symmetries – and gauge symmetries have huge importance.

Atiyah contributed key ideas to their mathematics, using his index theory to study instantons (particles that wink into existence and immediately wink out again) and magnetic monopoles (particles like a north magnetic pole without any corresponding south pole).

In 1983 his PhD student Simon Donaldson used these ideas to prove a remarkable theorem: contrary to what almost all topologists expected, four-dimensional space has infinitely many distinct differentiable structures – utterly different in this respect from any other dimension. The broader context for all this work is superstring theory, a conjectured unification of quantum theory and Albert Einstein’s relativity.

Atiyah was born in London, one of four children of Edward, a Lebanese civil servant, and his wife, Jean (nee Levens), who was born in Yorkshire of Scottish descent. The family moved to Khartoum, Sudan, where Michael went to school before boarding at Victoria College in Cairo and then moving to Manchester Grammar school at 16 to prepare for Cambridge. He was always keen on mathematics. An inspiring teacher introduced him to projective geometry and William Rowan Hamilton’s algebra of quaternions, and he read about number theory and group theory – all of which clearly influenced his later mathematical interests.

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In 1949, after two years of national service, he studied at Trinity College, Cambridge, remaining there for his PhD. He held positions at the Institute for Advanced Study in Princeton (including a professorship 1969-72), and at Cambridge and at Oxford, where he was Savilian professor of geometry 1963-69 and Royal Society research professor 1973-90. He became a fellow of the Royal Society in 1962, and was the society’s president from 1990 to 1995. In 1966 he won a Fields medal, the highest honour for any mathematician.

In 1990 he became master of Trinity College, Cambridge, and director of the Isaac Newton Institute for Mathematical Sciences, Cambridge. He was knighted in 1983 and made a member of the Order of Merit in 1992. After retiring from Trinity in 1997 he moved with his wife, Lily (nee Brown), whom he had married in 1955, to Edinburgh.

Atiyah was always a keen advocate of public engagement, giving popular talks on the beauty of mathematics and his lifelong passion for the subject. Small and compact, with a quiet, precise delivery, he could nevertheless hold an audience spellbound. That is how I remember him, on that day in Tate Modern, telling non-mathematicians why we do it, what it is for, and what it feels like.

He and Lily had three sons: John, David and Robin. John died in a climbing accident in 2002; Lily died last year. Michael is survived by David and Robin.

 Michael Francis Atiyah, mathematician, born 22 April 1929; died 11 January 2019

Interview with Interview with

Michael Michael Atiyah and Isadore Singer Atiyah and Isadore Singer

Interviewers: Martin Raussen and Christian Skau

from left to right: I. Singer, M. Atiyah, M. Raussen, C. Skau

see what happens”.

ATIYAH No doubt: Singer had a strong

expertise and background in analysis and

differential geometry. And he knew certainly more physics than I did; it turned out

to be very useful later on. My background

was in algebraic geometry and topology, so

it all came together. But of course there are

a lot of people who contributed in the background to the build-up of the Index

Theorem – going back to Abel, Riemann,

much more recently Serre, who got the

Abel prize last year, Hirzebruch,

Grothendieck and Bott. There was lots of

work from the algebraic geometry side and

from topology that prepared the ground.

And of course there are also a lot of people

who did fundamental work in analysis and

the study of differential equations:

Hörmander, Nirenberg… In my lecture I

will give a long list of names2

; even that

one will be partial. It is an example of

international collaboration; you do not

work in isolation, neither in terms of time

nor in terms of space – especially in these

days. Mathematicians are linked so much,

people travel around much more. We two

met at the Institute at Princeton. It was nice

to go to the Arbeitstagung in Bonn every

year, which Hirzebruch organised and

where many of these other people came. I

did not realize that at the time, but looking

back, I am very surprised how quickly

these ideas moved…

Collaboration seems to play a bigger role

in mathematics than earlier. There are a

lot of conferences, we see more papers

that are written by two, three or even more

authors – is that a necessary and commendable development or has it drawbacks as well?

ATIYAH It is not like in physics or chemistry where you have 15 authors because

they need an enormous big machine. It is

not absolutely necessary or fundamental.

But particularly if you are dealing with

areas which have rather mixed and interdisciplinary backgrounds, with people who

have different expertise, it is much easier

and faster. It is also much more interesting

for the participants. To be a mathematician

on your own in your office can be a little bit

dull, so interaction is stimulating, both psychologically and mathematically. It has to

be admitted that there are times when you

go solitary in your office, but not all the

time! It can also be a social activity with

lots of interaction. You need a good mix of

both, you can’t be talking all the time. But

talking some of the time is very stimulating. Summing up, I think that it is a good

development – I do not see any drawbacks.

SINGER Certainly computers have made

collaboration much easier. Many mathematicians collaborate by computer instantly;

 it’s as if they were talking to each other.

I am unable to do that. A sobering counterexample to this whole trend is

Perelman’s results on the Poincaré conjecture: He worked alone for ten to twelve

years, I think, before putting his preprints

on the net.

ATIYAH Fortunately, there are many different kinds of mathematicians, they work

on different subjects, they have different

approaches and different personalities –

and that is a good thing. We do not want

all mathematicians to be isomorphic, we

want variety: different mountains need different kinds of techniques to climb.

SINGER I support that. Flexibility is

absolutely essential in our society of mathematicians.

Perelman’s work on the Poincaré conjecture seems to be another instance where

analysis and geometry apparently get

linked very much together. It seems that

geometry is profiting a lot from analytic

perspectives. Is this linkage between different disciplines a general trend – is it

true, that important results rely on this

interrelation between different disciplines? 

And a much more specific question: What do you know about the status

of the proof of the Poincaré conjecture?

SINGER To date, everything is working

out as Perelman says. So I learn from

Lott’s seminar at the University of

Michigan and Tian’s seminar at Princeton.

Although no one vouches for the final

details, it appears that Perelman’s proof

will be validated.

As to your first question: When any two

subjects use each other’s techniques in a

new way, frequently, something special

happens. In geometry, analysis is very

important; for existence theorems, the

more the better. It is not surprising that

some new [at least to me] analysis implies

something interesting about the Poincaré

conjecture.

ATIYAH I prefer to go even further – I

really do not believe in the division of

mathematics into specialities; already if

you go back into the past, to Newton and

Gauss… Although there have been times,

particularly post-Hilbert, with the axiomatic approach to mathematics in the first half

of the twentieth century, when people

began to specialize, to divide up. The

Bourbaki trend had its use for a particular

time. But this is not part of the general attitude to mathematics: Abel would not have

distinguished between algebra and analysis. And I think the same goes for geometry and analysis for people like Newton.

It is artificial to divide mathematics into

separate chunks, and then to say that you

bring them together as though this is a surprise. On the contrary, they are all part of

the puzzle of mathematics. Sometimes you

would develop some things for their own

sake for a while e.g. if you develop group

theory by itself. But that is just a sort of

temporary convenient division of labour.

Fundamentally, mathematics should be

used as a unity. I think the more examples

we have of people showing that you can

usefully apply analysis to geometry, the

better. And not just analysis, I think that

some physics came into it as well: Many of

the ideas in geometry use physical insight

as well – take the example of Riemann!

This is all part of the broad mathematical

tradition, which sometimes is in danger of

being overlooked by modern, younger people who say “we have separate divisions”.

We do not want to have any of that kind,

really.

SINGER The Index Theorem was in fact

instrumental in breaking barriers between

fields. When it first appeared, many oldtimers in special fields were upset that new

techniques were entering their fields and

achieving things they could not do in the

field by old methods. A younger generation immediately felt freed from the barriers that we both view as artificial.

ATIYAH Let me tell you a little story

about Henry Whitehead, the topologist. I

remember that he told me that he enjoyed

very much being a topologist: He had so

many friends within topology, and it was

such a great community. “It would be a

tragedy if one day I would have a brilliant

idea within functional analysis and would

have to leave all my topology friends and

to go out and work with a different group

of people.” He regarded it to be his duty to

do so, but he would be very reluctant.

Somehow, we have been very fortunate.

Things have moved in such a way that we

got involved with functional analysts without losing our old friends; we could bring

them all with us. Alain Connes was in

functional analysis, and now we interact

closely. So we have been fortunate to

maintain our old links and move into new

ones – it has been great fun.

We would like to have your comments on

the interplay between physics and mathematics. There is Galilei’s famous dictum

from the beginning of the scientific revoMathematics and physics

INTERVIEW

EMS September 2004 25

lution, which says that the Laws of Nature

are written in the language of mathematics. Why is it that the objects of mathematical creation, satisfying the criteria of

beauty and simplicity, are precisely the

ones that time and time again are found to

be essential for a correct description of the

external world? Examples abound, let me

just mention group theory and, yes, your

Index Theorem!

SINGER There are several approaches in

answer to your questions; I will discuss

two. First, some parts of mathematics were

created in order to describe the world

around us. Calculus began by explaining

the motion of planets and other moving

objects. Calculus, differential equations,

and integral equations are a natural part of

physics because they were developed for

physics. Other parts of mathematics are

also natural for physics. I remember lecturing in Feynman’s seminar, trying to

explain anomalies. His postdocs kept

wanting to pick coordinates in order to

compute; he stopped them saying: “The

Laws of Physics are independent of a coordinate system. Listen to what Singer has to

say, because he is describing the situation

without coordinates.” Coordinate-free

means geometry. It is natural that geometry appears in physics, whose laws are

independent of a coordinate system.

Symmetries are useful in physics for

much the same reason they’re useful in

mathematics. Beauty aside, symmetries

simplify equations, in physics and in mathematics. So physics and math have in common geometry and group theory, creating a

close connection between parts of both

subjects.

Secondly, there is a deeper reason if your

question is interpreted as in the title of

Eugene Wigner’s essay “The

Unreasonable Effectiveness of

Mathematics in the Natural Sciences”3

.

Mathematics studies coherent systems

which I will not try to define. But it studies coherent systems, the connections

between such systems and the structure of

such systems. We should not be too surprised that mathematics has coherent systems applicable to physics. It remains to be

seen whether there is an already developed

coherent system in mathematics that will

describe the structure of string theory. [At

present, we do not even know what the

symmetry group of string field theory is.]

Witten has said that 21st century mathematics has to develop new mathematics,

perhaps in conjunction with physics intuition, to describe the structure of string theory.

ATIYAH I agree with Singer’s description

of mathematics having evolved out of the

physical world; it therefore is not a big surprise that it has a feedback into it.

More fundamentally: to understand the

outside world as a human being is an

attempt to reduce complexity to simplicity.

What is a theory? A lot of things are happening in the outside world, and the aim of

scientific inquiry is to reduce this to as simple a number of principles as possible.

That is the way the human mind works, the

way the human mind wants to see the

answer.

If we were computers, which could tabulate vast amounts of all sorts of information, we would never develop theory – we

would say, just press the button to get the

answer. We want to reduce this complexity to a form that the human mind can

understand, to a few simple principles.

That’s the nature of scientific inquiry, and

mathematics is a part of that. Mathematics

is an evolution from the human brain,

which is responding to outside influences,

creating the machinery with which it then

attacks the outside world. It is our way of

trying to reduce complexity into simplicity,

beauty and elegance. It is really very fundamental, simplicity is in the nature of scientific inquiry – we do not look for complicated things.

I tend to think that science and mathematics are ways the human mind looks and

experiences – you cannot divorce the

human mind from it. Mathematics is part

of the human mind. The question whether

there is a reality independent of the human

mind, has no meaning – at least, we cannot

answer it.

Is it too strong to say that the mathematical problems solved and the techniques

that arose from physics have been the

lifeblood of mathematics in the past; or at

least for the last 25 years?

ATIYAH I think you could turn that into

an even stronger statement. Almost all

mathematics originally arose from external

reality, even numbers and counting. At

some point, mathematics then turned to ask

internal questions, e.g. the theory of prime

numbers, which is not directly related to

experience but evolved out of it.

There are parts of mathematics where the

human mind asks internal questions just

out of curiosity. Originally it may be physical, but eventually it becomes something

independent. There are other parts that

relate much closer to the outside world

with much more interaction backwards and

forward. In that part of it, physics has for a

long time been the lifeblood of mathematics and inspiration for mathematical work.

There are times when this goes out of fashion or when parts of mathematics evolve

purely internally. Lots of abstract mathematics does not directly relate to the outside world.

It is one of the strengths of mathematics

that it has these two and not a single

lifeblood: one external and one internal,

one arising as response to external events,

the other to internal reflection on what we

are doing.

SINGER Your statement is too strong. I

agree with Michael that mathematics is

blessed with both an external and internal

source of inspiration. In the past several

decades, high energy theoretical physics

has had a marked influence on mathematics. Many mathematicians have been

shocked at this unexpected development:

new ideas from outside mathematics so

effective in mathematics. We are delighted

with these new inputs, but the “shock”

exaggerates their overall effect on mathematics.

Can we move to newer developments with

impact from the Atiyah-Singer Index

Theorem? I.e., String Theory and

Edward Witten on the one hand and on

the other hand Non-commutative

Geometry represented by Alain Connes.

Could you describe the approaches to

mathematical physics epitomized by these

two protagonists?

ATIYAH I tried once in a talk to describe

the different approaches to progress in

physics like different religions. You have

prophets, you have followers – each

prophet and his followers think that they

have the sole possession of the truth. If you

take the strict point of view that there are

several different religions, and that the

intersection of all these theories is empty,

then they are all talking nonsense. Or you

can take the view of the mystic, who thinks

that they are all talking of different aspects

of reality, and so all of them are correct. I

tend to take the second point of view. The

main “orthodox” view among physicists is

certainly represented by a very large group

of people working with string theory like

Edward Witten. There are a small number

of people who have different philosophies,

one of them is Alain Connes, and the other

is Roger Penrose. Each of them has a very

specific point of view; each of them has

very interesting ideas. Within the last few

years, there has been non-trivial interaction

between all of these.

They may all represent different aspects

of reality and eventually, when we under

Newer developments

INTERVIEW

26 EMS September 2004

stand it all, we may say “Ah, yes, they are

all part of the truth”. I think that that will

happen. It is difficult to say which will be

dominant, when we finally understand the

picture – we don’t know. But I tend to be

open-minded. The problem with a lot of

physicists is that they have a tendency to

“follow the leader”: as soon as a new idea

comes up, ten people write ten or more

papers on it and the effect is that everything

can move very fast in a technical direction.

But big progress may come from a different direction; you do need people who are

exploring different avenues. And it is very

good that we have people like Connes and

Penrose with their own independent line

from different origins. I am in favour of

diversity. I prefer not to close the door or

to say “they are just talking nonsense”.

SINGER String Theory is in a very special

situation at the present time. Physicists

have found new solutions on their landscape - so many that you cannot expect to

make predictions from String Theory. Its

original promise has not been fulfilled.

Nevertheless, I am an enthusiastic supporter of Super String Theory, not just because

of what it has done in mathematics, but also

because as a coherent whole, it is a marvellous subject. Every few years new developments in the theory give additional

insight. When that happens, you realize

how little one understood about String

Theory previously. The theory of D-branes

is a recent example. Often there is mathematics closely associated with these new

insights. Through D-branes, K-theory

entered String Theory naturally and

reshaped it. We just have to wait and see

what will happen. I am quite confident that

physics will come up with some new ideas

in String Theory that will give us greater

insight into the structure of the subject, and

along with that will come new uses of

mathematics.

Alain Connes’ program is very natural –

if you want to combine geometry with

quantum mechanics, then you really want

to quantize geometry, and that is what noncommutative geometry means. Non-commutative Geometry has been used effectively in various parts of String Theory

explaining what happens at certain singularities, for example. I think it may be an

interesting way of trying to describe black

holes and to explain the Big Bang. I would

encourage young physicists to understand

non-commutative geometry more deeply

than they presently do. Physicists use only

parts of non-commutative geometry; the

theory has much more to offer. I do not

know whether it is going to lead anywhere

or not. But one of my projects is to try and

redo some known results using non-commutative geometry more fully.

If you should venture a guess, which

mathematical areas do you think are

going to witness the most important developments in the coming years?

ATIYAH One quick answer is that the

most exciting developments are the ones

which you cannot predict. If you can predict them, they are not so exciting. So, by

definition, your question has no answer.

Ideas from physics, e.g. Quantum

Theory, have had an enormous impact so

far, in geometry, some parts of algebra, and

in topology. The impact on number theory

has still been quite small, but there are

some examples. I would like to make a

rash prediction that it will have a big

impact on number theory as the ideas flow

across mathematics – on one extreme number theory, on the other physics, and in the

middle geometry: the wind is blowing, and

it will eventually reach to the farthest

extremities of number theory and give us a

new point of view. Many problems that are

worked upon today with old-fashioned

ideas will be done with new ideas. I would

like to see this happen: it could be the

Riemann hypothesis, it could be the

Langlands program or a lot of other related

things. I had an argument with Andrew

Wiles where I claimed that physics will

have an impact on his kind of number theory; he thinks this is nonsense but we had a

good argument.

I would also like to make another prediction, namely that fundamental progress on

the physics/mathematics front, String

Theory questions etc., will emerge from a

much more thorough understanding of

classical four-dimensional geometry, of

Einstein’s Equations etc. The hard part of

physics in some sense is the non-linearity

of Einstein’s Equations. Everything that

has been done at the moment is circumventing this problem in lots of ways. They

haven’t really got to grips with the hardest

part. Big progress will come when people

by some new techniques or new ideas really settle that. Whether you call that geometry, differential equations or physics

depends on what is going to happen, but it

could be one of the big breakthroughs.

These are of course just my speculations.

SINGER I will be speculative in a slightly

different way, though I do agree with the

number theory comments that Sir Michael

mentioned, particularly theta functions

entering from physics in new ways. I think

other fields of physics will affect mathematics - like statistical mechanics and condensed matter physics. For example, I predict a new subject of statistical topology.

Rather than count the number of holes,

Betti-numbers, etc., one will be more interested in the distribution of such objects on

noncompact manifolds as one goes out to

infinity. We already have precursors in the

number of zeros and poles for holomorphic

functions. The theory that we have for

holomorphic functions will be generalized,

and insights will come from condensed

matter physics as to what, statistically, the

topology might look like as one approaches infinity.

Mathematics has become so specialized, it

seems, that one may fear that the subject

will break up into separate areas. Is there

Continuity of mathematics

INTERVIEW

EMS September 2004 27

Isadore Singer and Sir Michael Atiyah receive the

Abelprize from King Harald. (Photo: Scanpix)

a core holding things together?

ATIYAH I like to think there is a core

holding things together, and that the core is

rather what I look at myself; but we tend to

be rather egocentric. The traditional parts

of mathematics, which evolved - geometry,

calculus and algebra - all centre on certain

notions. As mathematics develops, there

are new ideas, which appear to be far from

the centre going off in different directions,

which I perhaps do not know much about.

Sometimes they become rather important

for the whole nature of the mathematical

enterprise. It is a bit dangerous to restrict

the definition to just whatever you happen

to understand yourself or think about. For

example, there are parts of mathematics

that are very combinatorial. Sometimes

they are very closely related to the continuous setting, and that is very good: we have

interesting links between combinatorics

and algebraic geometry and so on. They

may also be related to e.g. statistics. I think

that mathematics is very difficult to constrain; there are also all sorts of new applications in different directions.

It is nice to think of mathematics having

a unity; however, you do not want it to be

a straitjacket. The centre of gravity may

change with time. It is not necessarily a

fixed rigid object in that sense, I think it

should develop and grow. I like to think of

mathematics having a core, but I do not

want it to be rigidly defined so that it

excludes things which might be interesting.

You do not want to exclude somebody who

has made a discovery saying: “You are outside, you are not doing mathematics, you

are playing around”. You never know!

That particular discovery might be the

mathematics of the next century; you have

got to be careful. Very often, when new

ideas come in, they are regarded as being a

bit odd, not really central, because they

look too abstract.

SINGER Countries differ in their attitudes

about the degree of specialization in mathematics and how to treat the problem of too

much specialization. In the United States I

observe a trend towards early specialization driven by economic considerations.

You must show early promise to get good

letters of recommendations to get good first

jobs. You can’t afford to branch out until

you have established yourself and have a

secure position. The realities of life force a

narrowness in perspective that is not inherent to mathematics. We can counter too

much specialization with new resources

that would give young people more freedom than they presently have, freedom to

explore mathematics more broadly, or to

explore connections with other subjects,

like biology these days where there is lots

to be discovered.

When I was young the job market was

good. It was important to be at a major university but you could still prosper at a

smaller one. I am distressed by the coercive effect of today’s job market. Young

mathematicians should have the freedom

of choice we had when we were young.

The next question concerns the continuity

of mathematics. Rephrasing slightly a

question that you, Prof. Atiyah are the origin of, let us make the following

gedanken experiment: If, say, Newton or

Gauss or Abel were to reappear in our

midst, do you think they would understand the problems being tackled by the

present generation of mathematicians –

after they had been given a short refresher course? Or is present day mathematics

too far removed from traditional mathematics?

ATIYAH The point that I was trying to

make there was that really important

progress in mathematics is somewhat independent of technical jargon. Important

ideas can be explained to a really good

mathematician, like Newton or Gauss or

Abel, in conceptual terms. They are in fact

coordinate-free, more than that, technology-free and in a sense jargon-free. You

don’t have to talk of ideals, modules or

whatever – you can talk in the common

language of scientists and mathematicians.

The really important progress mathematics

has made within 200 years could easily be

understood by people like Gauss and

Newton and Abel. Only a small refresher

course where they were told a few terms –

and then they would immediately understand.

Actually, my pet aversion is that many

mathematicians use too many technical

terms when they write and talk. They were

trained in a way that if you do not say it 100

percent correctly, like lawyers, you will be

taken to court. Every statement has to be

fully precise and correct. When talking to

other people or scientists, I like to use

words that are common to the scientific

community, not necessarily just to mathematicians. And that is very often possible.

If you explain ideas without a vast amount

of technical jargon and formalism, I am

sure it would not take Newton, Gauss and

Abel long – they were bright guys, actually!

SINGER One of my teachers at Chicago

was André Weil, and I remember his saying: “If Riemann were here, I would put

him in the library for a week, and when he

came out he would tell us what to do next.”

Next topic: Communication of mathematics: Hilbert, in his famous speech at

the International Congress in 1900, in

order to make a point about mathematical communication, cited a French

mathematician who said: “A mathematical theory is not to be considered complete until you have made it so clear that

you can explain it to the first man whom

you meet on the street”. In order to pass

on to new generations of mathematicians the collective knowledge of the previous generation, how important is it

that the results have simple and elegant

proofs?

ATIYAH The passing of mathematics on

to subsequent generations is essential for

the future, and this is only possible if

every generation of mathematicians

understands what they are doing and distils it out in such a form that it is easily

understood by the next generation. Many

complicated things get simple when you

have the right point of view. The first

proof of something may be very complicated, but when you understand it well,

you readdress it, and eventually you can

present it in a way that makes it look

much more understandable – and that’s

the way you pass it on to the next generation! Without that, we could never make

progress - we would have all this messy

stuff. Mathematics does depend on a sufficiently good grasp, on understanding of

the fundamentals so that we can pass it on

in as simple a way as possible to our successors. That has been done remarkably

successfully for centuries. Otherwise,

how could we possibly be where we are?

In the 19th century, people said: “There is

so much mathematics, how could anyone

make any progress?” Well, we have - we

do it by various devices, we generalize,

we put all things together, we unify by

new ideas, we simplify lots of the constructions – we are very successful in

mathematics and have been so for several

hundred years. There is no evidence that

this has stopped: in every new generation,

there are mathematicians who make enormous progress. How do they learn it all?

It must be because we have been successful communicating it.

SINGER I find it disconcerting speaking

to some of my young colleagues, because

they have absorbed, reorganized, and simplified a great deal of known material into

a new language, much of which I don’t

understand. Often I’ll finally say, “Oh; is

that all you meant?” Their new conceptuCommunication of mathematics

INTERVIEW

28 EMS September 2004

al framework allows them to encompass

succinctly considerably more than I can

express with mine. Though impressed

with the progress, I must confess impatience because it takes me so long to

understand what is really being said.

Has the time passed when deep and

important theorems in mathematics can

be given short proofs? In the past, there

are many such examples, e.g., Abel’s

one-page proof of the addition theorem

of algebraic differentials or Goursat’s

proof of Cauchy’s integral theorem.

ATIYAH I do not think that at all! Of

course, that depends on what foundations

you are allowed to start from. If we have

to start from the axioms of mathematics,

then every proof will be very long. The

common framework at any given time is

constantly advancing; we are already at a

high platform. If we are allowed to start

within that framework, then at every stage

there are short proofs.

One example from my own life is this

famous problem about vector fields on

spheres solved by Frank Adams where the

proof took many hundreds of pages. One

day I discovered how to write a proof on

a postcard. I sent it over to Frank Adams

and we wrote a little paper which then

would fit on a bigger postcard. But of

course that used some K-theory; not that

complicated in itself. You are always

building on a higher platform; you have

always got more tools at your disposal

that are part of the lingua franca which

you can use. In the old days you had a

smaller base: If you make a simple proof

nowadays, then you are allowed to

assume that people know what group theory is, you are allowed to talk about

Hilbert space. Hilbert space took a long

time to develop, so we have got a much

bigger vocabulary, and with that we can

write more poetry.

SINGER Often enough one can distil the

ideas in a complicated proof and make

that part of a new language. The new

proof becomes simpler and more illuminating. For clarity and logic, parts of the

original proof have been set aside and discussed separately.

ATIYAH Take your example of Abel’s

Paris memoir: His contemporaries did not

find it at all easy. It laid the foundation of

the theory. Only later on, in the light of

that theory, we can all say: “Ah, what a

beautifully simple proof!” At the time, all

the ideas had to be developed, and they

were hidden, and most people could not

read that paper. It was very, very far from

appearing easy for his contemporaries.

I heard you, Prof. Atiyah, mention that one

reason for your choice of mathematics for

your career was that it is not necessary to

remember a lot of facts by heart.

Nevertheless, a lot of threads have to be

woven together when new ideas are developed. Could you tell us how you work best,

how do new ideas arrive?

ATIYAH My fundamental approach to

doing research is always to ask questions.

You ask “Why is this true?” when there is

something mysterious or if a proof seems

very complicated. I used to say – as a kind

of joke – that the best ideas come to you during a bad lecture. If somebody gives a terrible lecture, it may be a beautiful result but

with terrible proofs, you spend your time

trying to find better ones, you do not listen

to the lecture. It is all about asking questions – you simply have to have an inquisitive mind! Out of ten questions, nine will

lead nowhere, and one leads to something

productive. You constantly have to be

inquisitive and be prepared to go in any

direction. If you go in new directions, then

you have to learn new material.

Usually, if you ask a question or decide to

solve a problem, it has a background. If you

understand where a problem comes from

then it makes it easy for you to understand

the tools that have to be used on it. You

immediately interpret them in terms of your

own context. When I was a student, I

learned things by going to lectures and reading books – after that I read very few books.

I would talk with people; I would learn the

essence of analysis by talking to Hörmander

or other people. I would be asking questions because I was interested in a particular

problem. So you learn new things because

you connect them and relate them to old

ones, and in that way you can start to spread

around.

If you come with a problem, and you need

to move to a new area for its solution, then

you have an introduction – you have already

a point of view. Interacting with other people is of course essential: if you move into a

new field, you have to learn the language,

you talk with experts; they will distil the

essentials out of their experience. I did not

learn all the things from the bottom

upwards; I went to the top and got the

insight into how you think about analysis or

whatever.

SINGER I seem to have some built-in sense

of how things should be in mathematics. At

a lecture, or reading a paper, or during a discussion, I frequently think, “that’s not the

way it is supposed to be.” But when I try

Individual work style

out my ideas, I’m wrong 99% of the time. I

learn from that and from studying the ideas,

techniques, and procedures of successful

methods. My stubbornness wastes lots of

time and energy. But on the rare occasion

when my internal sense of mathematics is

right, I’ve done something different.

Both of you have passed ordinary retirement age several years ago. But you are

still very active mathematicians, and you

have even chosen retirement or visiting

positions remote from your original work

places. What are the driving forces for

keeping up your work? Is it wrong that

mathematics is a “young man’s game” as

Hardy put it?

ATIYAH It is no doubt true that mathematics is a young man’s game in the sense that

you peak in your twenties or thirties in terms

of intellectual concentration and in originality. But later you compensate that by experience and other factors. It is also true that

if you haven’t done anything significant by

the time you are forty, you will not do so

suddenly. But it is wrong that you have to

decline, you can carry on, and if you manage to diversify in different fields this gives

you a broad coverage. The kind of mathematician who has difficulty maintaining the

momentum all his life is a person who

decides to work in a very narrow field with

great depths, who e.g. spends all his life trying to solve the Poincaré conjecture –

whether you succeed or not, after 10-15

years in this field you exhaust your mind;

and then, it may be too late to diversify. If

you are the sort of person that chooses to

make restrictions to yourself, to specialize in

a field, you will find it harder and harder –

because the only things that are left are harder and harder technical problems in your

own area, and then the younger people are

better than you.

You need a broad base, from which you

can evolve. When this area dries out, then

you go to that area – or when the field as a

whole, internationally, changes gear, you

can change too. The length of the time you

can go on being active within mathematics

very much depends on the width of your

coverage. You might have contributions to

make in terms of perspective, breadth, interactions. A broad coverage is the secret of a

happy and successful long life in mathematical terms. I cannot think of any counter

example.

SINGER I became a graduate student at the

University of Chicago after three years in

the US army during World War II. I was

older and far behind in mathematics. So I

was shocked when my fellow graduate students said, “If you haven’t proved the

INTERVIEW

EMS September 2004 29

Riemann Hypothesis by age thirty, you

might as well commit suicide.” How infantile! Age means little to me. What keeps me

going is the excitement of what I’m doing

and its possibilities. I constantly check [and

collaborate!] with younger colleagues to be

sure that I’m not deluding myself – that

what we are doing is interesting. So I’m

happily active in mathematics. Another reason is, in a way, a joke. String Theory needs

us! String Theory needs new ideas. Where

will they come from, if not from Sir Michael

and me?

ATIYAH Well, we have some students…

SINGER Anyway, I am very excited about

the interface of geometry and physics, and

delighted to be able to work at that frontier.

You, Prof. Atiyah, have been very much

involved in the establishment of the

European Mathematical Society around

1990. Are you satisfied with its development since then?

ATIYAH Let me just comment a little on

my involvement. It started an awful long

time ago, probably about 30 years ago.

When I started trying to get people interested in forming a European Mathematical

Society in the same spirit as the European

Physical Society, I thought it would be easy.

I got mathematicians from different countries together and it was like a mini-UN: the

French and the Germans wouldn’t agree; we

spent years arguing about differences, and –

unlike in the real UN – where eventually at

the end of the day you are dealing with real

problems of the world and you have to come

to an agreement sometime; in mathematics,

it was not absolutely essential. We went on

for probably 15 years, before we founded

the EMS.

On the one hand, mathematicians have

much more in common than politicians, we

are international in our mathematical life, it

is easy to talk to colleagues from other

History of the EMS

countries; on the other hand, mathematicians are much more argumentative. When

it comes to the fine details of a constitution,

then they are terrible; they are worse than

lawyers. But eventually – in principle – the

good will was there for collaboration.

Fortunately, the timing was right. In the

meantime, Europe had solved some of its

other problems: the Berlin Wall had come

down – so suddenly there was a new Europe

to be involved in the EMS. This very fact

made it possible to get a lot more people

interested in it. It gave an opportunity for a

broader base of the EMS with more opportunities and also relations to the European

Commission and so on.

Having been involved with the set-up, I

withdrew and left it to others to carry on. I

have not followed in detail what has been

happening except that it seems to be active.

I get my Newsletter, and I see what is going

on.

Roughly at the same time as the collapse

of the Berlin Wall, mathematicians in general – both in Europe and in the USA –

began to be more aware of their need to be

socially involved and that mathematics had

an important role to play in society. Instead

of being shut up in their universities doing

just their mathematics, they felt that there

was some pressure to get out and get

involved in education, etc. The EMS took

on this role at a European level, and the

EMS congresses – I was involved in the one

in Barcelona – definitely made an attempt to

interact with the public. I think that these

are additional opportunities over and above

the old-fashioned role of learned societies.

There are a lot of opportunities both in terms

of the geography of Europe and in terms of

the broader reach.

Europe is getting ever larger: when we

started we had discussions about where

were the borders of Europe. We met people

from Georgia, who told us very clearly, that

the boundary of Europe is this river on the

other side of Georgia; they were very keen

to make sure that Georgia is part of Europe.

Now, the politicians have to decide where

the borders of Europe are.

It is good that the EMS exists; but you

should think rather broadly about how it is

evolving as Europe evolves, as the world

evolves, as mathematics evolves. What

should its function be? How should it relate

to national societies? How should it relate

to the AMS? How should it relate to the

governmental bodies? It is an opportunity!

It has a role to play!

Could you tell us in a few words about your

main interests besides mathematics?

SINGER I love to play tennis, and I try to

do so 2-3 times a week. That refreshes me

and I think that it has helped me work hard

in mathematics all these years.

ATIYAH Well, I do not have his energy! I

like to walk in the hills, the Scottish hills – I

have retired partly to Scotland. In

Cambridge, where I was before, the highest

hill was about this (gesture) big. Of course

you have got even bigger ones in Norway. I

spent a lot of my time outdoors and I like to

plant trees, I like nature. I believe that if you

do mathematics, you need a good relaxation

which is not intellectual – being outside in

the open air, climbing a mountain, working

in your garden. But you actually do mathematics meanwhile. While you go for a long

walk in the hills or you work in your garden

– the ideas can still carry on. My wife complains, because when I walk she knows I am

thinking of mathematics.

SINGER I can assure you, tennis does not

allow that!

Thank you very much on behalf of the

Norwegian, the Danish, and the European

Mathematical Societies!

The interviewers were Martin Raussen,

Aalborg University, Denmark, and

Christian Skau, Norwegian University of

Science and Technology, Trondheim,

Norway.

1 More details were given in the laureates’

lectures.

2 Among those: Newton, Gauss, Cauchy,

Laplace, Abel, Jacobi, Riemann,

Weierstrass, Lie, Picard, Poincaré,

Castelnuovo, Enriques, Severi, Hilbert,

Lefschetz, Hodge, Todd, Leray, Cartan,

Serre, Kodaira, Spencer, Dirac,

Pontrjagin, Chern, Weil, Borel,

Hirzebruch, Bott, Eilenberg,

Grothendieck, Hörmander, Nirenberg.

3 Comm. Pure App. Math. 13(1), 1960