We are living in an age of, amongst other things, excellent graphic novels. One shining example, which I have just finished reading, is LOGICOMIX, a graphic novel biography of mathematician and philosopher Bertrand Russell. (Side note: can a biography still be called a graphic novel? Our terminology may need an update.)
Seeking an escape from his authoritarian religious upbringing, young Bertrand turned to mathematics as the one source of absolute certainty in his life. But the more he studied mathematics, the more he realized that underlying all the sophisticated theories of the time were arguments based more on intuition than full rigor. Driven by his quest for absolute truth, Russell embarked on a project to rebuild mathematics from the foundations up, and thereby establish its status as absolute truth.
Unfortunately, his project ran into major difficulties of the mathematical/philosophical variety (to say nothing of his equally great personal difficulties) including the famous paradox of Russell's own invention, the arguments of his student Wittigstein that logic was merely a tool for generating tautologies, and finally, Godel's proof that even in the self-consistent world of mathematics, there must always be true statements that cannot be proven.
In the end, though Russell and his contemporaries eventually succeeded in placing mathematics on a rigorous footing, the dream of a logically grounded "universal truth" had to be abandoned. Mathematics is only as true as the assumptions it rests on, and cannot even prove all that is true in its domain.
While the mathematical and philosophical ideas are well-illustrated for a lay audience, the heart of LOGICOMIX is Russell's personal struggle, first to find the universal truths in mathematics and then to accept their nonexistence. Like others engaged in this project, Russell's struggle with logic occasionally veered into a struggle with sanity. Through a meta-narrative of the book's creation, the authors debate the "logic and madness" theme, and ask whether some amount of detachment from reality a prerequisite for one who spends his or her life searching for its foundations.
This narrative of Russell's quest had personal resonance for me: I went through my own late-high-school/early-college phase of viewing mathematics as a bastion of truth in an illogical world. I wonder if many of my mathematical colleagues' careers had their genesis in the same yearning for certainty. I imagine we all eventually come to the same realization as Russell: that mathematics is a powerful tool for clear thinking, but the only "truth" it contains is ultimately tautological.
Disillusioned by his self-described "failure" but ultimately freed from his need for unblemished truth, Russell turns to more worldly concerns, including pacifist activism and the founding of a school with no rules (spoiler: it doesn't go well). The book ends on a bittersweet note as Russell encourages students to accept their lives in an uncertain world.
I had great pleasure following Russell's journey, and the many ideas and people encountered along the way. If anyone is interested in what really drives mathematicians, this book is heartily recommended.
Field of Science
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Change of address1 year ago in Variety of Life
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Change of address1 year ago in Catalogue of Organisms
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Earth Day: Pogo and our responsibility1 year ago in Doc Madhattan
Unsustainable
The following question was given as a homework problem in a course I'm TAing:
(Anyone who needs help with the mathematics of this problem can meet me in the comments.)
CNBC had an interesting program on the current financial crisis. They located one investor who noticed that since the late 1990's housing prices have been growing 10 percent every year (that is, each year, the average home price is 1.1 times the average price in the previous year) while income was only increasing by 5 percent each year (that is, each year, the average income was only 1.05 times the average of the previous year).This simple question goes right to the heart of the financial collapse. I would only add that, not only did this particular investor make billions off this observation, but our whole economy lost trillions, because the vast majority of financial decision makers were either unable or unwilling to make this same observation.
Explain why it is "absolutely clear that this situation could not go on forever", in the words of the investor (who made over a billion dollars because of this observation).
(Anyone who needs help with the mathematics of this problem can meet me in the comments.)
Human Cultural Transformation Triggered by Dense Populations
Biologically,modern humans first appeared 160,000 to 200,000 years ago. But the transition to complex human societies, with art, music, advanced tools, occurred a good deal more recently, and moreover, occured at different times in different parts of the world. An article in June's Science magazine (see a less technical write-up here) argues, based on historical evidence and computer simulations, that in each case the transition was triggered once the population density had reached a critical threshold. At this threshold, there is sufficient interaction to allow for complex ideas to be passed down through generations, enabling rapid cultural evolution.
This highlights an interesting evolutionary tension: as I've written before, evolutionary theory tells us that cooperative behaviors are more likely to evolve (biologically speaking) in populations that are dispersed over space rather than densely packed. But I'm beginning to think that cultural evolution may be different enough from biological evolution to require its own body of theory.
This highlights an interesting evolutionary tension: as I've written before, evolutionary theory tells us that cooperative behaviors are more likely to evolve (biologically speaking) in populations that are dispersed over space rather than densely packed. But I'm beginning to think that cultural evolution may be different enough from biological evolution to require its own body of theory.
Inferring Social Security Numbers from Birth Data
An article in July's PNAS investigates the possibility of predicting a person's Social Security number from their birth date and place. Exploiting patterns in how SSN's are assigned, authors Alessandro Acquisti and Ralph Gross developed an algorithm which could correctly predict the first 5 digits of a social security number 44% of the time, for people born after 1988 (older SSNs are significantly harder to predict). The accuracy varied from state to state; for smaller states and recent birthdays, the algorithm could sometimes predict an entire SSN on the first try.
Think you're safe?
Think you're safe?
The Quandaries of Quantifying Complexity
My good friend and computer scientest Kyle Burke has recently started a highly interesting blog on his research field: combinatorial game theory. The idea of this field is to use games as a tool for studying issues of complexity. Though his blog is only a month old, some important foundational ideas have begun to rear their heads, one of which I'll explore in this post.
Understanding complexity is important for almost any human endeavor, but defining it in rigorous terms is notoriously difficult. For example, which is the more complicated game, chess or tic-tac-toe? Almost anyone would say chess, but suppose you had a computer that was designed only to play chess. In fact, this computer has no capacity for calculation; it simply has the best move for any given chess position hardwired into its architecture. To get this computer to play tic-tac-toe, you would have to program it to translate each tic-tac-toe position into an analagous chess position, so it could then find the best chess move and translate this move back into tic-tac-toe. This computer would certainly find chess an easier game to play.
Computer scientists have a way around this paradox: instead of looking at individual games or problems, they look at classes of problems. Each problem in the class has a certain size, and they look at how complexity increases in relation to size.
For example, you could easily imagine playing tic-tac-toe on boards of various sizes. Computer scientists can analyze how the complexity of tic-tac-toe varies with the size of the board. (Chess, on the other hand, doesn't generalize as easily to larger sizes, which makes it difficult to talk about its complexity.)
Unfortunately, if we are faced with a real-world issue (such as how to provide for the needs of a large population), we will want to know the complexity of the specific problem at hand, not how the complexity might theoretically scale with problem size. Part of the reason that complexity issues are so often ignored (to the detriment of many well-meaning policies and programs) is that defining and quantifying complexity is so unavoidably slippery.
Further reading
Understanding complexity is important for almost any human endeavor, but defining it in rigorous terms is notoriously difficult. For example, which is the more complicated game, chess or tic-tac-toe? Almost anyone would say chess, but suppose you had a computer that was designed only to play chess. In fact, this computer has no capacity for calculation; it simply has the best move for any given chess position hardwired into its architecture. To get this computer to play tic-tac-toe, you would have to program it to translate each tic-tac-toe position into an analagous chess position, so it could then find the best chess move and translate this move back into tic-tac-toe. This computer would certainly find chess an easier game to play.
Computer scientists have a way around this paradox: instead of looking at individual games or problems, they look at classes of problems. Each problem in the class has a certain size, and they look at how complexity increases in relation to size.
For example, you could easily imagine playing tic-tac-toe on boards of various sizes. Computer scientists can analyze how the complexity of tic-tac-toe varies with the size of the board. (Chess, on the other hand, doesn't generalize as easily to larger sizes, which makes it difficult to talk about its complexity.)
Unfortunately, if we are faced with a real-world issue (such as how to provide for the needs of a large population), we will want to know the complexity of the specific problem at hand, not how the complexity might theoretically scale with problem size. Part of the reason that complexity issues are so often ignored (to the detriment of many well-meaning policies and programs) is that defining and quantifying complexity is so unavoidably slippery.
Further reading
The Criminalization of Poverty
Barbara Ehrenreich had an excellent article in yesterday's New York Times on the many ways that being poor can land you in trouble with the law. One striking example:
The column was based on a report by the National Law Center on Homelessness and Poverty, which finds that the number of ordinances passed and tickets issued for crimes related to poverty has grown since 2006.
Hey, no one likes poverty, right? Let's pass a law!
In just the past few months, a growing number of cities have taken to ticketing and sometimes handcuffing teenagers found on the streets during school hours.
In Los Angeles, the fine for truancy is $250; in Dallas, it can be as much as $500 — crushing amounts for people living near the poverty level. According to the Los Angeles Bus Riders Union, an advocacy group, 12,000 students were ticketed for truancy in 2008.
Why does the Bus Riders Union care? Because it estimates that 80 percent of the “truants,” especially those who are black or Latino, are merely late for school, thanks to the way that over-filled buses whiz by them without stopping. I met people in Los Angeles who told me they keep their children home if there’s the slightest chance of their being late. It’s an ingenious anti-truancy policy that discourages parents from sending their youngsters to school.
The column was based on a report by the National Law Center on Homelessness and Poverty, which finds that the number of ordinances passed and tickets issued for crimes related to poverty has grown since 2006.
Hey, no one likes poverty, right? Let's pass a law!
The Evolution of Bad Ideas
It is by now common wisdom that our current financial crisis is due in large part to misplaced incentives in our financial system. Analysts and fund managers were rewarded for short-term thinking and risk-taking. If we can rework our financial system to reward long-term, careful planning, it is often argued, we can avoid collapses like this in the future.
While I agree that misplaced incentives were a fundamental problem, the question of how to change this is rather more deep and complex than I think many people realize.
Our economy is, of course, an evolutionary system. Successful businesses grow in size and their practices are imitated by others; unsuccessful businesses vanish. This process has led to many good business practices, even in the financial sector.
However, evolution does not always yield the best outcomes, in biology or in economics. Our recent crisis illustrates two key limitations of evolutionary systems, limitations which allow bad ideas to evolve over good ones.
The first problem has to do with time lags. Suppose Financial Company A comes up with an idea that will yield huge sums of money for five years and then drive the company to bankruptcy. They implement the idea, obfuscating the downside, and soon the company is rolling in cash. Investors line up to give them money, magazines laud them, and other companies begin imitating them.
Not so Company B. Company B believes in long-term thinking, and can see this idea for the sham it is. They persue a quiet, sound strategy, even when their investors begin pulling money out to invest in A.
We would like to think that in the end, Company B will be left standing and reap them benefits of their foresight. But there is a fundamental problem of time-scales here: by the time A folds, B may already be out of business, due to lack of interest from investors. In theoretical terms, there is a fundamental problem when the evolutionary process proceeds faster than the unfolding of negative consequences. In these situations, good ideas never have a chance to be rewarded, evolutionarily speaking.
One might argue that investors, not to mention government regulators and ratings agencies, should have forseen the flaw in A's plan. But this highlights a second limitation of the evolutionary process: it favors complexity. Simple bad ideas can be detected by intelligent agents, but complex ones have a chance to really stick. If Company A's idea was so complicated that no one aside from a few physicists could figure it out, investors and regulators could easily be fooled.
It's not clear to me how to patch these flaws in the evolutionary system. Increased transparency and oversight will help, but unless we can somehow cap the complexity of financial instruments (difficult) or slow down the evolutionary process (impossible), I'm not sure how we'll avoid similar crashes in the future.
While I agree that misplaced incentives were a fundamental problem, the question of how to change this is rather more deep and complex than I think many people realize.
Our economy is, of course, an evolutionary system. Successful businesses grow in size and their practices are imitated by others; unsuccessful businesses vanish. This process has led to many good business practices, even in the financial sector.
However, evolution does not always yield the best outcomes, in biology or in economics. Our recent crisis illustrates two key limitations of evolutionary systems, limitations which allow bad ideas to evolve over good ones.
The first problem has to do with time lags. Suppose Financial Company A comes up with an idea that will yield huge sums of money for five years and then drive the company to bankruptcy. They implement the idea, obfuscating the downside, and soon the company is rolling in cash. Investors line up to give them money, magazines laud them, and other companies begin imitating them.
Not so Company B. Company B believes in long-term thinking, and can see this idea for the sham it is. They persue a quiet, sound strategy, even when their investors begin pulling money out to invest in A.
We would like to think that in the end, Company B will be left standing and reap them benefits of their foresight. But there is a fundamental problem of time-scales here: by the time A folds, B may already be out of business, due to lack of interest from investors. In theoretical terms, there is a fundamental problem when the evolutionary process proceeds faster than the unfolding of negative consequences. In these situations, good ideas never have a chance to be rewarded, evolutionarily speaking.
One might argue that investors, not to mention government regulators and ratings agencies, should have forseen the flaw in A's plan. But this highlights a second limitation of the evolutionary process: it favors complexity. Simple bad ideas can be detected by intelligent agents, but complex ones have a chance to really stick. If Company A's idea was so complicated that no one aside from a few physicists could figure it out, investors and regulators could easily be fooled.
It's not clear to me how to patch these flaws in the evolutionary system. Increased transparency and oversight will help, but unless we can somehow cap the complexity of financial instruments (difficult) or slow down the evolutionary process (impossible), I'm not sure how we'll avoid similar crashes in the future.
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