Saturday, December 31, 2011

The mathematical mind

The following is an excellent anonymous response (here) to the question: What is it like to have an understanding of very advanced mathematics?

I would characterize the mathematical mind as rigorous (precise, exact), flexible, and  subtle.

  • You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a few powerful general purpose "machines" (e.g., continuity arguments, the correspondences between geometric and algebraic objects, linear algebra, ways to reduce the infinite to the finite through various forms of compactness) combined with specific facts you have learned about your area. The number of fundamental ideas and techniques that people use to solve problems is, perhaps surprisingly, pretty small -- see for a partial list, maintained by Timothy Gowers.
  • You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It's not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.
  • You are comfortable with feeling like you have no deep understanding of the problem you are studying. Indeed, when you do have a deep understanding, you have solved the problem and it is time to do something else. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion. More on this in the next few bullets.
  • Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being aimlessly puzzled. For example, when answering a question about a high-dimensional space (e.g., whether a certain kind of rotation of a five-dimensional object has a "fixed point" which does not move during the rotation), you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don't know that they shouldn't be straining to visualize things for which they don't seem to have the visualizing machinery.) Instead...
  • When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about the examples into more impressive insights. For example, you might imagine two- and three- dimensional rotations that are analogous to the one you really care about, and think about whether they clearly do or don't have the desired property. Then you think about what was important to the examples and try to distill those ideas into symbols. Often, you see that the key idea in the symbolic manipulations doesn't depend on anything about two or three dimensions, and you know how to answer your hard question.

    As you get more mathematically advanced, the examples you consider easy are actually complex insights built up from many easier examples; the "simple case" you think about now took you two years to become comfortable with. But at any given stage, you do not strain to obtain a magical illumination about something intractable; you work to reduce it to the things that feel friendly.
  • To me, the biggest misconception that non-mathematicians have about how mathematicians think is that there is some mysterious mental faculty that is used to crack a problem all at once. In reality, one can ever think only a few moves ahead, trying out possible attacks from one's arsenal on simple examples relating to the problem, or using analogies with other ideas one understands. This is the same way that one solves problems in one's first real maths courses in university and in competitions. What happens as you get more advanced is simply that the arsenal grows larger, the thinking gets somewhat faster due to practice, and you have more examples to try, perhaps making better guesses about what is likely to yield progress.

    Indeed, most of the bullet points here summarize feelings familiar to many serious students of mathematics who are in the middle of their undergraduate careers; as you learn more mathematics, these experiences apply to "bigger" things but have the same fundamental flavor.
  • You go up in abstraction, "higher and higher". The main object of study yesterday becomes just an example or a tiny part of what you are considering today. For example, in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose points are functions or curves -- that is, you "zoom out" so that every function is just a point in a space, surrounded by many other "nearby" functions. Using this kind of zooming out technique, you can say very complex things in short sentences -- things that, if unpacked and said at the zoomed-in level, would take up pages. Abstracting and compressing in this way allows you to consider extremely complicated issues while using your limited memory and processing power.
  • The particularly "abstract" or "technical" parts of many other subjects seem quite accessible because they boil down to maths you already know. You generally feel confident about your ability to learn most quantitative ideas and techniques. A theoretical physicist friend likes to say, only partly in jest, that there should be books titled "______ for Mathematicians", where _____ is something generally believed to be difficult (quantum chemistry, general relativity, securities pricing, formal epistemology). Those books would be short and pithy, because many key concepts in those subjects are ones that mathematicians are well equipped to understand. Often, those parts can be explained more briefly and elegantly than they usually are if the explanation can assume a knowledge of maths and a facility with abstraction.

    Learning the domain-specific elements of a different field can still be hard -- for instance, physical intuition and economic intuition seem to rely on tricks of the brain that are not learned through mathematical training alone. But the quantitative and logical techniques you sharpen as a mathematician allow you to take many shortcuts that make learning other fields easier, as long as you are willing to be humble and modify those mathematical habits that are not useful in the new field.
  • You move easily between multiple seemingly very different ways of representing a problem. For example, most problems and concepts have more algebraic representations (closer in spirit to an algorithm) and more geometric ones (closer in spirit to a picture). You go back and forth between them naturally, using whichever one is more helpful at the moment.

    Indeed, some of the most powerful ideas in mathematics (e.g., duality, Galois theory [, algebraic geometry [ provide "dictionaries" for moving between "worlds" in ways that, ex ante, are very surprising. For example, Galois theory allows us to use our understanding of symmetries of shapes (e.g., rigid motions of an octagon) to understand why you can solve any fourth-degree polynomial equation in closed form, but not any fifth-degree polynomial equation. Once you know these threads between different parts of the universe, you can use them like wormholes to extricate yourself from a place where you would otherwise be stuck. The next two bullets expand on this.
  • Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable. Mathematicians develop a powerful attachment to elegance and depth, which are in tension with, if not directly opposed to, mechanical calculation. Mathematicians will often spend days thinking of a clean argument that completely avoids numbers and strings of elementary deductions in favor of seeing why what they want to show follows easily from some very deep and general pattern that is already well-understood. Indeed, you tend to choose problems motivated by how likely it is that there will be some "clean" insight in them, as opposed to a detailed but ultimately unenlightening proof by exhaustively enumerating a bunch of possibilities. In A Mathematician's Apology [, the most poetic book I know on what it is "like" to be a mathematician], G.H. Hardy wrote:

    "In both [these example] theorems (and in the theorems, of course, I include the proofs) there is a very high degree of unexpectedness, combined with inevitability and economy. The arguments take so odd and surprising a form; the weapons used seem so childishly simple when compared with the far-reaching results; but there is no escape from the conclusions. There are no complications of detail—one line of attack is enough in each case; and this is true too of the proofs of many much more difficult theorems, the full appreciation of which demands quite a high degree of technical proficiency. We do not want many ‘variations’ in the proof of a mathematical theorem: ‘enumeration of cases’, indeed, is one of the duller forms of mathematical argument. A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way."


    "[A solution to a difficult chess problem] is quite genuine mathematics, and has its merits; but it is just that ‘proof by enumeration of cases’ (and of cases which do not, at bottom, differ at all profoundly) which a real mathematician tends to despise."
  • You develop a strong aesthetic preference for powerful and general ideas that connect hundreds of difficult questions, as opposed to resolutions of particular puzzles. Mathematicians don't really care about "the answer" to any particular question; even the most sought-after theorems, like Fermat's Last Theorem [, are only tantalizing because their difficulty tells us that we have to develop very good tools and understand very new things to have a shot at proving them. It is what we get in the process, and not the answer per se, that is the valuable thing. The accomplishment a mathematician seeks is finding a new dictionary or wormhole between different parts of the conceptual universe. As a result, many mathematicians do not focus on deriving the practical or computational implications of their studies (which can be a drawback of the hyper-abstract approach!); instead, they simply want to find the most powerful and general connections. Timothy Gowers has some interesting comments on this issue, and disagreements within the mathematical community about it [ ].
  • Understanding something abstract or proving that something is true becomes a task a lot like building something. You think: "First I will lay this foundation, then I will build this framework using these familiar pieces, but leave the walls to fill in later, then I will test the beams..." All these steps have mathematical analogues, and structuring things in a modular way allows you to spend several days thinking about something you do not understand without feeling lost or frustrated.

    Andrew Wiles, who proved Fermat's Last Theorem, used an "exploring" metaphor:
    "Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it's completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it's all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they're momentary, sometimes over a period of a day or two, they are the culmination of—and couldn't exist without—the many months of stumbling around in the dark that precede them." [ ]
  • In listening to a seminar or while reading a paper, you don't get stuck as much as you used to in youth because you are good at modularizing a conceptual space and taking certain calculations or arguments you don't understand as "black boxes" and considering their implications anyway. You can sometimes make statements you know are true and have good intuition for, without understanding all the details. You can often detect where the delicate or interesting part of something is based on only a very high-level explanation. (I first saw these phenomena highlighted by Ravi Vakil, who offers insightful advice on being a mathematics student:
  • You are good at generating your own questions and your own clues in thinking about some new kind of abstraction. One of the things I've reliably heard from people who know parts of mathematics well but never went on to be professional mathematicians (i.e., write articles about new mathematics for a living) is that they were good at proving difficult propositions that were stated in a textbook exercise, but would be lost if presented with a mathematical structure and asked to find and prove some "interesting" facts about it. This kind of challenge is like being given a world and asked to find events in it that come together to form a good detective story. Unlike a more standard detective, you have to figure out what the "crime" (interesting question) might be, and you have to generate your own "clues" by building up deductively from the basic axioms. To do these things, you use analogies with other detective stories (mathematical theories) that you know and a taste for what is "surprising" or "deep". This is the hardest thing in this answer to articulate precisely but also the thing that I would guess is the "strongest" thing that mathematicians have in common.

    Concretely, this amounts to being good at making definitions and formulating precise conjectures using the newly defined concepts that other mathematicians find interesting. One of the things one learns fairly late in a typical mathematics education (often only at the stage of starting to do research) is how to make good, useful definitions.
  • You are easily annoyed by imprecision in talking about the quantitative or logical. This is mostly because you are trained to quickly think about counterexamples that make an imprecise claim seem obviously false.
  • On the other hand, you are very comfortable with intentional imprecision or "hand waving" in areas you know, because you know how to fill in the details. Terence Tao is very eloquent about this here [ ]:

    "[After learning to think rigorously, comes the] 'post-rigorous' stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the 'big picture'. This stage usually occupies the late graduate years and beyond."

    In particular, an idea that took hours to understand correctly the first time ("for any arbitrarily small epsilon I can find a small delta so that this statement is true") becomes such a basic element of your later thinking that you don't give it conscious thought.
  • Before wrapping up, it is worth mentioning that mathematicians are not immune to the limitations faced by most others. They are not typically intellectual superheroes. For instance, they often become resistant to new ideas and uncomfortable with ways of thinking (even about mathematics) that are not their own. They can be defensive about intellectual turf, dismissive of others, or petty in their disputes. Above, I have tried to summarize how the mathematical way of thinking feels and works at its best, without focusing on personality flaws of mathematicians or on the politics of various mathematical fields. These issues are worthy of their own long answers!
  • You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems. There are only very few mathematical questions to which we have reasonably insightful answers. There are even fewer questions, obviously, to which any given mathematician can give a good answer. After two or three years of a standard university curriculum, a good maths undergraduate can effortlessly write down hundreds of mathematical questions to which the very best mathematicians could not venture even a tentative answer. (The theoretical computer scientist Richard Lipton lists some examples of potentially "deep" ignorance here: This makes it more comfortable to be stumped by most problems; a sense that you know roughly what questions are tractable and which are currently far beyond our abilities is humbling, but also frees you from being very intimidated, because you do know you are familiar with the most powerful apparatus we have for dealing with these kinds of problems.


A two part question to determine if you "think like a mathematician," from Prof. Eugene Luks, Bucknell University, circa 1979.

Part I: You're in a room that is empty except for a functioning stove and a tea kettle with tepid water in it sitting on the floor. How do you make hot water for tea?

Answer to Part I: Put tea kettle on stove, turn on burner, heat until water boils.

Part II: Next, you're in another room that is empty except for a functioning stove and a tea kettle with tepid water in it sitting on a table. How do you make hot water for tea?

Non-mathematician's answer to Part II: Put tea kettle on stove, turn on burner, heat until water boils.
Mathematician's answer to Part II: Put the tea kettle on the floor.

Why? Because a solution to any new problem is elegantly complete when it can be reduced to a previously demonstrated case.

Thursday, December 29, 2011

Algebra, algorithm and Uzbekistan

A page from al-Khwārizmī's Algebra (source)

The words "algebra" and "algorithm" were introduced to Latin Christendom through the 12th century Latin translation of the works of a Persian polymath, born in Khwarezm (in present day Uzbekistan), and working in the House of Wisdom in Baghdad during the Abbasid Caliphate.

He is al-Khwarizmi ("from Khwarezm", c.780-850) (Algoritmi in Latin, hence "algorithm"), a mathematician, geographer, and astronomer.

Algebra and Algorithm happen to be two subjects of which I have a modest degree of mastery. Their history illustrates the role of the Islamic civilization in bridging the Classical Greco-Roman world and the High Middle Ages and beyond.

As expected, there is an impressive Wikipedia article on al-Khwarizmi (here), where we learn the following:

In the twelfth century, Latin translations of al-Khwarizmi's work on the Indian numerals, introduced the decimal positional number system to the Western world.  His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations in Arabic. In Renaissance Europe, he was considered the original inventor of algebra, although we now know that his work is based on older Indian or Greek sources.  He revised Ptolemy's Geography and wrote on astronomy and astrology.

Some words reflect the importance of al-Khwarizmi's contributions to mathematics. "Algebra" is derived from al-jabr, one of the two operations he used to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the Latin form of his name. His name is also the origin of (Spanish) guarismo and of (Portuguese) algarismo, both meaning digit.

His name may indicate that he came from Khwarezm (Khiva), then in Greater Khorasan, which occupied the eastern part of the Greater Iran, now Xorazm Province in Uzbekistan. Abu Rayhan Biruni calls the people of Khwarizm "a branch of the Persian tree"


Further interesting facts: Khwarezm, a region to the south of the Aral Sea, saw the rise of the Khwarazmian Empire (1077-1231, culturally Persianate, ethnically Turkic),  first as vassals of the Seljuqs, Kara-Khitan, and later as independent rulers,  until the Mongol invasions of the 13th century. Xorazm (or Khorezm, a variant spelling of Khwarezm) is the name of a present day province of Uzbekistan.

Interestingly, the Khwarezmians were briefly in control of Jerusalem, during 1244-1247. (See this for details)

Frederick II of the Holy Roman Empire gained control of Jerusalem by treaty with the Ayyubid (the dynasty founded by Saladin) sultan al-Kamil in 1229. Control of the city changed hands several times between the Ayyubids and the Christians  from 1239 to 1244. In 1244 Jerusalem was in Christian hands.

The Khawarezmi Turks, displaced by the advancing Mongols and allied with the Ayyubids, invaded Jerusalem (The siege of Jerusalem in 1244) on July 11, 1244, and the city's citadel, the Tower of David, surrendered on August 23. The Khwarezmians then ruthlessly decimated the city's population, leaving only 2,000 people, Christians and Muslims, still living in the city. This attack triggered the Europeans to respond with the Seventh Crusade, although the forces of King Louis IX of France never even achieved success in Egypt, let alone advancing as far as Palestine.

Egyptian Ayyubid Sultan al-Malik al-Salih then decided to use his new Mamluk army to eliminate the Khwarezmians, and Jerusalem soon returned to Egyptian Ayyubid rule in 1247.

The Khanate of Khiva existed in the historical region of Khwarezm from 1511 to 1920, except for a period of Persian occupation by Nadir Shah between 1740–1746.

Panoramic view of Khiva, Khorezm Province, Uzbekistan, a UNESCO World Heritage Site (for high resolution view: here)

Saturday, December 10, 2011

First Anglo-Japanese trade agreement (1613): a Bodleian treasure


In 1613 Shogun Tokugawa Ieyasu made a personal agreement with representatives of the East India Company, granting them trade privileges in Japan.

The East India company had been drawn to Japan by what it imagined would be abundant selling opportunities.

The exhibit at the Bodleian Library, dated 12 October 1613, is thought to be one of two copies given to John Saris, commander of the Clove, which reached Hirado on 11 June 1613, almost two years and two months after leaving England.

Ieyasu's seal is visible at the top of the document, which grants Saris privileges for trade in Japan

Continue here for an image and a translation of the agreement, and a video.

Thursday, December 1, 2011

Suharto, Marcos, Mobutu and Sani Abacha: in their own league

Suharto, Marcos, Mobutu and Sani Abacha are in their own league where corruption is concerned. The appalling greed of some past ASEAN (Association of Southeast Asian Nations) leaders (Suharto, Marcos, Joseph Estrada) is world class. This is old news (from 2004), but worth a reminder.

Suharto, Marcos and Mobutu head corruption table with $50bn scams

The Guardian,  26 March 2004 (source)

Mohammed Suharto, Ferdinand Marcos and Mobutu Sese Seko ripped off up to $50bn (£28bn) from the impoverished people of Indonesia, the Philippines and Zaire, a sum equivalent to the entire annual aid budget of the west, anti-bribery campaigners said yesterday.

Releasing a list of the top 10 most corrupt politicians of the past two decades, headed by the former Indonesian dictator, Transparency International warned that the scale of political corruption was undermining hopes for prosperity in the developing world and damaging the global economy.

No country is immune from corruption, the report says, citing the lax controls over political financing in Greece, the close connection between companies and the Bush administration and the unchallenged power of Italy's prime minister, Silvio Berlusconi, over his country's media.

But the most egregious examples of wholesale looting have occurred in the developing world, TI said, depriving the desperately poor of vital state resources.

"The abuse of political power for private gain deprives the most needy of vital public services, creating a level of despair that breeds conflict and violence," said Peter Eigen, TI's chairman.

Most of the names on the list were protected by western governments who turned a blind eye to their criminal activities in exchange for support during the cold war.

Suharto, regarded as a bulwark against communism in Asia, stole as much as $35bn from his impoverished country during his three decades in power, before being deposed in 1998 in a popular uprising that was triggered by the Asian economic crisis.

He was charged with looting up to $500m from the state through various charities controlled by his family, but the judges ruled that he was too ill to stand trial.

Marcos, whose wife's 3,000-piece shoe collection became a byword for the corrupt excesses of his regime, was backed by successive US administrations.

Efforts to track down the estimated $10bn he embezzled during his 20 years in power were frustrated by years of strict banking secrecy laws in Switzerland. In August last year, 14 years after his death, the Swiss courts finally authorised the release of $657m to the Philippines' authorities.

Mobutu cleverly used the threat of an invasion from the then Marxist government of Angola to quieten concerns in the west about his increasingly blatant looting from one of the most resource-rich countries in Africa. Washing ton leaned on the International Monetary Fund to continue lending, despite the doubts of IMF officials.

By the time he was overthrown in 1997, Mobutu had stolen almost half of the $12bn in aid money that Zaire - now the Democratic Republic of Congo - received from the IMF during his 32-year reign, leaving his country saddled with a crippling debt.

Western multinationals must take responsibility for allowing corruption to flourish, TI says. "The corporate sector has a vital role to play in ending the abuse of power," the report says.

Bribery of local officials by western business is still widespread, despite global initiatives to stamp it out, such as the UN convention against corruption. The culture of secrecy surrounding access to resources in the developing world allows corrupt governments to hide revenues from their populations.

Companies from Australia, Sweden, Switzerland, Austria and Canada topped TI's list of bribe-payers last year, despite the introduction of anti-corruption laws to comply with an Organisation for Economic Co-operation and Development convention banning bribery of foreign officials.

"We are very much aware that a lot of the responsibility for corruption in the developing world has been with northern companies and northern institutions," said Mr Eigen.

Britain was singled out for dragging its feet on the implementation of the OECD convention. It only outlawed the bribing of officials abroad two years ago, and no one has been prosecuted so far.

Oil, a curse more often than a blessing for poor countries, is a significant factor in corruption. "The flow of oil money is so vast it can distort decision-making in poor producer countries and the rich world alike," the report says.

On the take

Head of state Mohammed Suharto
Place, time Indonesia, 1967-98
Amount $15bn-$35bn

Head of state Ferdinand Marcos
Place, time Philippines, 1972-86
Amount $5bn-$10bn

Head of state Mobutu Sese Seko
Place, time Zaire, 1965-97
Amount $5bn

Head of state Sani Abacha
Place, time Nigeria, 1993-98
Amount $2bn-$5bn

Head of state Slobodan Milosevic
Place, time Serbia, 1972-86
Amount $1bn

Head of state Jean-Claude Duvalier
Place, time Haiti, 1971-86
Amount $300m-$800m

Head of state Alberto Fujimori
Place, time Peru, 1990-2000
Amount $600m

Head of state Pavlo Lazarenko
Place, time Ukraine 1996-97
Amount $114m-$200m

Head of state Arnoldo Alemán
Place, time Nicaragua, 1997-2002
Amount $100m

Head of state Joseph Estrada
Place, time Philippines, 1998-2001
Amount $78m-$80m