William Paul Thurston (October 30, 1946 – August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.

Thurston was a professor of mathematics at Princeton University, University of California at Davis and Cornell University. He was also a director of the Mathematical Sciences Research Institute.

MathOverflow makes all answers posted to the website available under a Creative Commons licence. In particular, all answers posted before 08 Apr 2011 (UTC) are available under the terms of the Creative Commons Attribution-ShareAlike 2.5 Generic (CC BY-SA 2.5) licence. Thurston wrote the answer I am about to share on 30 Oct 2010. Due to the licence terms, this post too is available under the terms of the same licence.

Thurston posted his answer while replying to a MathOverflow question: What's a mathematician to do?. The question enquires about how an ordinary mathematician can contribute to mathematics. Thurston's answer from mathoverflow.net/a/44213 is reproduced below:

It's not mathematics that you need to contribute to. It's deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyond our understanding. We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way. That is why you do well to follow your heart and your passion. Bare reason is likely to lead you astray. None of us are smart and wise enough to figure it out intellectually.

The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat's Last Theorem, or the Poincaré conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.

The world does not suffer from an oversupply of clarity and understanding (to put it mildly). How and whether specific mathematics might lead to improving the world (whatever that means) is usually impossible to tease out, but mathematics collectively is extremely important.

I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. There are many ideas in mathematics that may be hard to get, but are easy once you get them. Because of this, mathematical understanding does not expand in a monotone direction. Our understanding frequently deteriorates as well. There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand.

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining --- they depend very heavily on the community of mathematicians.

In the comments to the answer, one of the commenters was Suresh Venkatasubramanian who was a professor in the School of Computing at the University of Utah back then. He is now a professor of Computer Science and Data Science at Brown University. In his comment, Suresh proposed that this answer be called Thurston's Paean. Here is his complete comment:

This seems like an ideal counterpoint to Hardy's Lament. I'm calling it Thurston's Paean :). Seems poignant now that he has passed.

Thurston's answer does appear to be a perfect complement to Hardy's lament in the 1940 essay A Mathematician's Apology. While Hardy's lament is remarkably beautiful and introspective, it may also feel a little depressing at places. Thurston's post on the other hand is full of hope and purpose that goes beyond the actual work of doing mathematics. Indeed Thurston's Paean is a befitting title for his answer.

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One striking characteristic of Grothendieck's mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called "Grothendieck prime". In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. "You mean an actual number?" Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, "All right, take 57."

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I recently found this interesting paragraph from an article titled Ten Lessons I Wish I Had Been Taught that is based on a talk presented by Gian-Carlo Rota in Apr 1996:

Running overtime is the one unforgivable error a lecturer can make. After fifty minutes (one microcentury as von Neumann used to say) everybody's attention will turn elsewhere even if we are trying to prove the Riemann hypothesis. One minute overtime can destroy the best of lectures.

That's fine advice. In fact, the whole article is full of good advice like this. Although it was written primarily for mathematicians, a lot of what is said in the article applies quite well to professionals in other fields too.

The excerpt I have quoted above got me thinking about exactly how long a microcentury is. It couldn't be exactly 50 minutes, could it?

Wiktionary on Microcentury

The English Wiktionary entry for microcentury (as of revision 59316064 on 7 May 2020) mentions:

A time period of a millionth of a century, equal to 52 minutes and 34 seconds.

Not a standard unit of measurement and used mostly humorously to denote the maximum length of a lecture.

This looks incorrect to me. This is based on the oversimplified assumption that a century contains 36500 days, that is, it assumes that a century is a span of 100 years where each year has exactly 365 days. If a century were to have exactly 36500 days, then indeed it would have 3 153 600 000 seconds and one millionth of it would be 3153.6 seconds which is equivalent to 52 minutes 33.6 seconds. This looks consistent with the Wiktionary entry. However, an actual century on the calendar does not have exactly 36500 days. Some years are leap years, so the actual number of days in a century is more than that.

Assumptions

Let us find out how long a microcentury is as accurately as possible. We will count the leap years. We will ignore leap seconds because they are irregularly spaced and unpredictable. We will also ignore the following gap between 2 Sep 1752 and 14 Sep 1752 when the British Empire switched from the Julian calendar to the Gregorian calendar:

$ cal 9 1752
   September 1752
Su Mo Tu We Th Fr Sa
       1  2 14 15 16
17 18 19 20 21 22 23
24 25 26 27 28 29 30

The above output can be obtained by running the cal command as shown above on a Unix or Linux system. Ignoring this gap is equivalent to assuming that we are working with the Gregorian calender since the year 1 AD.

We will call a year that is a multiple of 100 to be a centurial year. Further, we will not debate whether a centurial year begins a new century or ends one, that is, we don't care whether the current century runs from 2001 to 2100 or if it runs from 2000 to 2099. The computation presented in the next section works equally well for any span of 100 years.

Computation

Any span of 100 years contains exactly one centurial year, that is, a year that is a multiple of 100. A centurial year is a leap year if and only if it is also a multiple of 400. Apart from the centurial year, a century contains 24 occurrences of years that are multiples of 4 and these are all leap years. From these facts, we can conclude that a span of 100 years contains:

• Exactly 25 leap years if the centurial year within the span is a multiple of 400.
• Exactly 24 leap years, otherwise.

Therefore a century has either 36524 days or 36525 days. In other words, a century has either 3 155 673 600 seconds or 3 155 760 000 seconds. Here is a quick demonstration of this with a simple Python program:

#!/usr/bin/env python3

import datetime

for year in range(1, 2400, 100):
    delta = datetime.date(year + 100, 1, 1) - datetime.date(year, 1, 1)
    print('{:04}-{:04}: {} d = {} s'
          .format(year, year + 99, delta.days, delta.total_seconds()))

Here is the output of this program:

0001-0100: 36524 d = 3155673600.0 s
0101-0200: 36524 d = 3155673600.0 s
0201-0300: 36524 d = 3155673600.0 s
0301-0400: 36525 d = 3155760000.0 s
0401-0500: 36524 d = 3155673600.0 s
0501-0600: 36524 d = 3155673600.0 s
0601-0700: 36524 d = 3155673600.0 s
0701-0800: 36525 d = 3155760000.0 s
0801-0900: 36524 d = 3155673600.0 s
0901-1000: 36524 d = 3155673600.0 s
1001-1100: 36524 d = 3155673600.0 s
1101-1200: 36525 d = 3155760000.0 s
1201-1300: 36524 d = 3155673600.0 s
1301-1400: 36524 d = 3155673600.0 s
1401-1500: 36524 d = 3155673600.0 s
1501-1600: 36525 d = 3155760000.0 s
1601-1700: 36524 d = 3155673600.0 s
1701-1800: 36524 d = 3155673600.0 s
1801-1900: 36524 d = 3155673600.0 s
1901-2000: 36525 d = 3155760000.0 s
2001-2100: 36524 d = 3155673600.0 s
2101-2200: 36524 d = 3155673600.0 s
2201-2300: 36524 d = 3155673600.0 s
2301-2400: 36525 d = 3155760000.0 s

Thus one millionth of a century has 3155.6736 or 3155.7600 seconds, that is 52 minutes 35.6736 seconds or 52 minutes 35.7600 seconds.

Conclusion

If we round off the number of seconds in a microcentury to one decimal place, we can say that a microcentury has 52 minutes 35.7 seconds.

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There was no Burt in that office as far as I knew, so I became a little confused. I was led into a conference room where I could meet him. Mildly disoriented, I opened the conference room door. Then I saw Burt, the person I was about to meet. As soon as I saw him, I immediately recognised who he was. He was Dr Burt Kaliski, the Chief Scientist of RSA Laboratories. He had flown in that morning to meet certain teams in our office. During his visit, he had carved out some time from his schedule to meet me.

I had never met or talked to Dr Kaliski before. But I was very well aware of his work and his accomplishments many years before joining RSA. At a previous job, while implementing digital signatures for a banking product, I had spent a lot of time reading the RFCs for PKCS #1, PKCS #5 and PKCS #7, which were all authored by Dr Kaliski. It was an honour to meet the person whose work had taught me a lot and helped me begin my career as a software developer in the area of information security.

He began the meeting by explaining that he happened to learn about my wish to work on more challenging projects. He enquired about the type of projects I desired to work on. He listened to my thoughts patiently. He offered some advice and then wished me the best for my future. Little did I know back then that the short 30 minute interaction I had with him would end up changing the trajectory of my career.

Soon after the meeting, I followed his advice, talked to several teams, found out what they were working on and finally decided to join a team that worked on a network security product. It turned out to be the best decision of my career! While working on that product, I learnt to design and implement efficient algorithms for petabytes of data. Some parts of the project relied heavily on probability theory, so it was a joy to be able to take a subject that was a hobby until then and apply it to solve actual real-world problems. Working on that product offered the intellectual challenges that my younger self yearned for back then. I think the most interesting experience there was my work on parser generators. It involved writing algorithms in C++ to parse a formally specified language which in turn was used to generate in-memory FSMs to parse and analyse network packets and events. Parsing one language and generating in-memory parsers for another language proved to be intellectually demanding as well as satisfying. I have fond memories of my time there working on parser generators alongside remarkably skilled engineers from whom I learnt a lot.

All of that happened seven years ago. A couple of months ago, while recounting that 30 minute meeting with Dr Kaliski, I could not recall if I had ever thanked him properly for his kindness and generosity. Seven years ago, there was no compelling reason for him to meet me. I was then just a regular software engineer who was hired into the company a year ago. We had never talked before. But he took the time to offer me help, suggestions and advice on how I could make a good career. Unsure if I had expressed my gratitude to him for how he helped me sculpt my career, I wrote to him recently to thank him and explain how that one meeting had a large and positive impact on my life.

Dr Kaliski replied a few days later and he expressed his happiness to know that the steps I took in the subsequent years worked out well for me. Before ending his message, he wrote a little heart-warming note that I am going to remember forever. Quoting it verbatim here:

“One of my goals is to be able to provide encouragement to others who are developing their careers, just as others have invested in mine, passing good blessings from one generation to another.”

Thank you, Burt! I will do likewise.

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I think that it's extraordinarily important that we in computer science keep fun in computing. When it started out, it was an awful lot of fun. Of course, the paying customers got shafted every now and then, and after a while we began to take their complaints seriously. We began to feel as if we really were responsible for the successful, error-free perfect use of these machines. I don't think we are. I think we're responsible for stretching them, setting them off in new directions, and keeping fun in the house. I hope the field of computer science never loses its sense of fun.

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It's easier to type $pi than \( \pi, \) especially if you don't have Unicode. And it will be easy to maintain the program in case the value of \( \pi \) ever changes.³⁷⁹

The corresponding footnote mentions:

³⁷⁹ It nearly did change by a legislative act in the state of Indiana. http://www.urbanlegends.com/legal/pi_indiana.htm

I searched the web and found that the original urbanlegends.com website is no longer there. However while searching for it, I came across this brilliant piece of humour archived in the article titled Alabama's Slice of Pi on Snopes. It is a fictitious report on a state legislature redefining the value of \( \pi \) to 3.

The Snopes article mentions that this piece of humour was first posted in a newsgroup. Then people started circulating it as hoax from there. Here are some of the intriguing and funny bits from it:

Lawson called into question the usefulness of any number that cannot be calculated exactly, and suggested that never knowing the exact answer could harm students' self-esteem.

Scientists have arbitrarily assumed that space is Euclidean, he says. He points out that a circle drawn on a spherical surface has a different value for the ratio of circumfence to diameter.

In fact, with a little geometry we can see that if a flatlander living on a globe with diameter \( D \) draws a circle of diameter \( d \) assuming that he is on a flat surface, then the ratio of the circumference \( c \) to the diameter \( d \) is \[ \frac{c}{d} = \frac{\pi D}{d} \sin{\frac{d}{D}}. \] Here are a few more excerpts from the Snopes article:

"These nabobs waltzed into the capital with an arrogance that was breathtaking," Learned said. "Their prefatorial deficit resulted in a polemical stance at absolute contraposition to the legislature's puissance."

One member of the state school board, Lily Ponja, is anxious to get the new value of pi into the state's math textbooks, but thinks that the old value should be retained as an alternative. She said, "As far as I am concerned, the value of pi is only a theory, and we should be open to all interpretations." She looks forward to students having the freedom to decide for themselves what value pi should have.

By the way, the real event that the footnote in the Learning Perl book mentioned is the Indiana Pi Bill.

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An even more impressive instance of remote debugging occurred on NASA's 1998 Deep Space 1 mission. A half year after the space craft launched, a bit of Lisp code was going to control the spacecraft for two days while conducting a sequence of experiments. Unfortunately, a subtle race condition in the code had escaped detection during ground testing and was already in space. When the bug manifested in the wild--100 million miles away from Earth--the team was able to diagnose and fix the running code, allowing the experiments to complete. One of the programmers described it as follows:

Debugging a program running on a $100M piece of hardware that is 100 million miles away is an interesting experience. Having a read-eval-print loop running on the spacecraft proved invaluable in finding and fixing the problem.

The original source of this story is an article called Lisping at JPL written by Ron Garret in 2002. This story occurs in the section called 1994-1999 - Remote Agent of the article.

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