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Stories from SlateThe Perils of Hacking Math
http://www.slate.com/articles/health_and_science/science/2013/09/nsa_misuse_of_mathematics_secret_formulas_and_backdoor_cryptography.html
<p>Recently, I co-authored and published a <a href="http://arxiv.org/abs/1308.3444">math paper</a> that solved a 15-year-old mystery. But, unlike a book or a gadget, the work cannot be copyrighted or bought and sold. In fact, my co-author and I have made our paper available for free, for the whole world to see, on <a href="http://arxiv.org/">arXiv</a>, an online depository of scientific articles. This inherent democracy has always been the mark of mathematics: It belongs to us all, even if people are not aware of it. Mathematicians don't expect to be paid for their discoveries; we do math because we want to understand how the world works.</p>
<p>This principle has deep roots in history as well as in legal systems. No one can own mathematical knowledge; no one can claim ownership of a mathematical formula or idea as a personal possession. Though he discovered it, Albert Einstein couldn't patent his famous formula <em>E=mc<sup>2</sup></em>. In the landmark <a href="http://caselaw.lp.findlaw.com/scripts/getcase.pl?court=us&vol=409&invol=63"><em>Gottschalk v. Benson</em> decision</a>, the U.S. Supreme Court concluded:</p>
<blockquote>
A scientific truth, or the mathematical expression of it, is not a patentable invention. ... A principle, in the abstract, is a fundamental truth; an original cause; a motive; these cannot be patented, as no one can claim in either of them an exclusive right. ... He who discovers a hitherto unknown phenomenon of nature has no claim to a monopoly of it which the law recognizes.
</blockquote>
<p>Unfortunately, this time-honored and essential principle of freedom of mathematical information is now being compromised. According to <a href="http://www.theguardian.com/world/2013/sep/05/nsa-gchq-encryption-codes-security">published</a> <a href="http://www.nytimes.com/2013/09/06/us/nsa-foils-much-internet-encryption.html">reports</a>, the National Security Agency has attempted to undermine mathematical formulas used in widely used encryption systems. They did it both by using advances made in secret by mathematicians on their payroll and by intentionally subverting commonly used security protocols by installing "backdoors" that make these protocols easier to break.</p>
<p>The legality and broad implications of the NSA large-sale surveillance have already been discussed at great length. My point here, however, is that tampering with mathematics is by itself a dangerous precedent that raises a host of legal and ethical issues.</p>
<p>We should be especially alarmed by the reported attempts by the NSA to intentionally undermine cryptosystems. In a nutshell, to ensure that a third party can't read your email message, credit-card number, or password, communications sent over the Internet are encrypted. Many cryptosystems are based on sophisticated mathematical objects called "elliptic curves" (these are discussed in <a href="http://loveandmathbook.com/?p=5">my new book</a> <a href="http://www.amazon.com/gp/product/0465050743/ref=as_li_ss_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0465050743&linkCode=as2&tag=slatmaga-20"><em>Love and Math: The Heart of Hidden Reality</em></a>). There are plenty of elliptic curves to choose from. A cryptosystem based on a random one is virtually impregnable and hence protects our privacy. But it turns out that there are some elliptic curves that look random but actually allow for easy decryption; that's an example of a backdoor. It's a nontrivial mathematical problem to generate such curves (equipped with some extra data), but <a href="http://en.wikipedia.org/wiki/Dual_EC_DRBG">it can be done</a>. And according to the reports, the NSA has been pushing the National Institute of Standards and Technology, the body that sets encryption standards in the United States, and various vendors to adopt such special elliptic curves since <a href="http://www.wired.com/politics/security/commentary/securitymatters/2007/11/securitymatters_1115">as early as 2006</a>, knowing full well that they were prone to attacks. After these allegations came to light, encryption company RSA Security <a href="http://blog.cryptographyengineering.com/2013/09/rsa-warns-developers-against-its-own.html">issued an unprecedented advisory</a> noting that one of its widely used toolkits is based on the compromised algorithm and advising clients to stop using it.</p>
<p>Secrecy in cryptography is nothing new. We remember <a href="http://en.wikipedia.org/wiki/Bletchley_Park">Bletchley Park</a>, where mathematicians such as Alan Turing, working in secret, were able to decode German communications during World War II. But what’s different now is the ubiquity of the security protocols that are being compromised. Encryption is now woven in the very fabric of our daily lives. That’s why creation of secret means for breaking commonly used cryptosystems by the government is so troubling.</p>
<p>Furthermore, by secretly installing backdoors into these systems, the NSA makes all of us more vulnerable to outside attacks. If these backdoors allow the NSA to easily break these systems, what's to stop other players from maliciously doing the same? They may steal this information from the NSA or a rogue on the inside may disclose or sell it. Besides, others may discover these backdoors on their own.</p>
<p>Mathematics is a great equalizer. A young man from India named Srinivasa <a href="http://en.wikipedia.org/wiki/Srinivasa_Ramanujan">Ramanujan</a> received no formal training but in the early 20<sup>th</sup> century was able to make dazzling mathematical discoveries that stumped professional mathematicians. For a more recent example, consider this: In 1973, three mathematicians working for the U.K.’s Government Communications Headquarters discovered a new method of encryption. Their discovery was kept secret by GCHQ, but shortly afterward two other mathematicians rediscovered the same thing and published their result (now called <a href="http://en.wikipedia.org/wiki/Diffie-Hellman">Diffie-Hellman key exchange</a> in their honor).</p>
<p>Who's to say that the sophisticated math the NSA has been keeping secret from the rest of the world will not be discovered by someone else?</p>
<p>You can hide a formula, but you can't prevent others from finding it. One might only need a pencil and a piece of paper to do that. And once the secret is out in the open, it’s not just Big Brother that will be watching us—other “brothers” will be spying on us, intercepting our messages, and hacking our bank accounts.</p>
<p>We live in a new era in which mathematics has become a powerful weapon. It can be used for good—we all benefit from technological advances based on math—but also for ill. When the nuclear bomb was built, theoretical physicists who had inadvertently contributed to creating something monstrous were forced to confront deep ethical questions. What is happening now with mathematics may have similarly grave implications. Members of my community must initiate a serious discussion about our role in this brave new world. We need to find mechanisms to protect the freedom of mathematical knowledge that we love and cherish. And we have to help the public understand both the awesome power of math and the serious consequences that await all of us if that power is misused.</p>Mon, 30 Sep 2013 16:42:13 GMThttp://www.slate.com/articles/health_and_science/science/2013/09/nsa_misuse_of_mathematics_secret_formulas_and_backdoor_cryptography.htmlEdward Frenkel2013-09-30T16:42:13ZThe National Security Agency is undermining fundamental principles of mathematical knowledge.Health and ScienceThe NSA Is Misusing Mathematics for Dangerous Ends100130930008mathmathematicsgovernment surveillancensanational security agencyEdward FrenkelSciencehttp://www.slate.com/articles/health_and_science/science/2013/09/nsa_misuse_of_mathematics_secret_formulas_and_backdoor_cryptography.htmlfalsefalsefalseThe NSA is misusing math for alarming ends. @edfrenkel on the dangers.The NSA Is Misusing Mathematics for Dangerous EndsPhoto by Kerem Yucel/iStock/ThinkstockThe time-honored principle of freedom of mathematical information is now being compromised.Prime Numbers Hide Your Secrets
http://www.slate.com/articles/health_and_science/science/2013/06/online_credit_card_security_the_rsa_algorithm_prime_numbers_and_pierre_fermat.html
<p>Prime numbers are all the rage these days. I can tell something’s up when random people start asking me about the <a href="http://www.slate.com/articles/health_and_science/do_the_math/2013/05/yitang_zhang_twin_primes_conjecture_a_huge_discovery_about_prime_numbers.single.html">randomness of primes</a>—without even knowing that I’m a mathematician! In the past couple of weeks we’ve heard about a beautiful result on the <a href="http://www.wired.com/wiredscience/2013/05/twin-primes/">gaps between primes</a> and about <a href="http://www.newyorker.com/online/blogs/elements/2013/05/why-cicadas-love-prime-numbers.html">cicadas’ prime-numbered life cycles</a>. Our current love affair with primes notwithstanding, many people have wondered whether this is all just abstract theoretical stuff or whether prime numbers have real-world applications.</p>
<p>In fact, they have applications to something as ubiquitous and mundane as making a purchase online. Every time you enter your credit card number on the Internet, prime numbers spring into action. Before your card number is sent over the wires, it must be encrypted for security, and once it’s received by the merchant, it must be decrypted. One of the most common encryption schemes, the RSA algorithm, is based on prime numbers. It uses a “public key,” information that is publicly available, and a “private key,” something that only the decoding party (merchant) has. Roughly speaking, the public key consists of a large number that is the product of two primes, and the private key consists of those two primes themselves. It’s very difficult to factor a given large number into primes. For example, it took researchers two years recently to <a href="http://eprint.iacr.org/2010/006.pdf">factor a 232-digit number</a>, even with hundreds of parallel computers. That’s why the RSA algorithm is so effective.</p>
<p>Prime numbers are whole numbers greater than 1 that are not divisible by any whole number other than 1 and itself. The first few are 2, 3, 5, 7, 11, 13 … To explain how the RSA algorithm works, I need to tell you first about something called Fermat’s little theorem. It was discovered by Pierre Fermat, the same French mathematician who came up with the famous <a href="http://en.wikipedia.org/wiki/Fermat's_Last_Theorem">Fermat’s last theorem</a>. Fermat had a penchant for being cryptic; in the case of his last theorem, he left a note on the margin of a book stating his theorem and adding: “I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.” Call it the 17<sup>th</sup>-century version of a Twitter proof. Many professional mathematicians and amateurs tried to reproduce Fermat’s purported proof, and it took more than 350 years to come up with a <a href="http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html">real one</a>.</p>
<p>To understand what Fermat’s little theorem means, we need to learn how to do arithmetic “modulo N.” This is something, in fact, we do all the time when adding angles. If you rotate an object by 180 degrees, and then by another 270 degrees, the net result will be rotation by 90 degrees. That’s because 180 + 270 = 450, and then we subtract 360 from it, because rotation by 360 degrees means no net rotation at all. This is what mathematicians call addition “modulo 360.” Likewise, we can do addition modulo any whole number N instead of 360. (<a href="http://blogs.scientificamerican.com/guest-blog/2013/05/21/an-unheralded-breakthrough-the-rosetta-stone-of-mathematics/">Another familiar example is adding hours, where N = 12.</a>) And we can also do multiplication modulo N.</p>
<p>Now suppose that N is a prime number. Then we have the following remarkable fact: Raising <em>any</em> number to the N<sup>th</sup> power modulo N, we get back that same number. For example, if N = 5, then the 5<sup>th</sup> power of 1 is 1 and the 5<sup>th</sup> power of 2 is 32, which is equal to 2 modulo 5 because after you subtract the closest multiple of 5 (in this case, you subtract 30, or 5 × 6), you are left with 2 (because 32 = 5 × 6 + 2). Likewise, the 5<sup>th</sup> power of 3 is equal to 3 modulo 5, and so on. This is Fermat's little theorem. Fermat first divulged it in a letter to a friend. “I would send you a proof of it,” he wrote, “but I am afraid it’s too long.” He was such a tease, this Fermat. Unlike the proof of his last theorem, however, the proof of the little one is surprisingly simple—it could even fit in the margin of a book. I would write it here, but my editor tells me that my article is already too long. No worries though, you can read the proof <a href="http://math.berkeley.edu/~frenkel/RSA.pdf">in this excerpt</a> from <a href="http://www.amazon.com/gp/product/0465050743/ref=as_li_ss_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0465050743&linkCode=as2&tag=slatmaga-20">my book <em>Love and Math</em></a>.</p>
<p>This is neat, but what does it have to with Internet security? We need to devise a <em>two-step</em> procedure: First encrypt a credit card number and then decrypt it, so that if we follow both steps we get back the original number. The good news from Fermat’s little theorem is that raising a card number to a prime power modulo that prime is a procedure that gives us back the original number. The bad news is that because a prime is not divisible by anything, there is no way to break this procedure into two steps. However, Ron Rivest, Adi Shamir, and Leonard Adleman, after whom the RSA algorithm is named, were not discouraged. They took Fermat’s idea one step further. They asked: What if we take N which is the product of <em>two</em> primes—call them <em>p</em> and <em>q</em>. Then we have the following analogue of Fermat’s little theorem: Raising any number to the power (<em>p </em>– 1)(<em>q </em>– 1) + 1 will give us back the same number modulo N. For example, N = 15 is the product of <em>p </em>= 3 and <em>q </em>= 5. Then (<em>p </em>– 1)(<em>q </em>– 1) + 1 = (3 – 1)(5 – 1) + 1 = 9. If you raise any number to the 9<sup>th</sup> power, you get back the same number modulo 15. It looks like a miracle, but in fact <a href="http://math.berkeley.edu/~frenkel/RSA.pdf">the proof is no more complicated</a> than that of Fermat’s little theorem.</p>
<p>And now we can use it for encryption: For the given prime numbers <em>p</em> and <em>q</em>, the combination (<em>p </em>– 1)(<em>q </em>– 1) + 1 will typically be a number that is not a prime. Hence it can be represented as the product of two whole numbers, call them <em>r</em> and <em>s</em>. (In our example, (<em>p </em>– 1)(<em>q </em>– 1) + 1 = 9, so we can take <em>r </em>= 3 and <em>s </em>= 3.) Raising a number to the power (<em>p </em>– 1)(<em>q </em>– 1) + 1 can now be broken into two steps: raising it to the <em>r</em><sup>th</sup> power and then raising it to the <em>s</em><sup>th</sup> power.</p>
<p>Here’s how it works in practice: The merchant picks two large prime numbers <em>p</em> and <em>q</em> (there are various algorithms for <a href="http://www.numberempire.com/primenumbers.php">generating primes</a>), takes their product N, and chooses <em>r</em> and <em>s</em>. He or she then makes <em>r</em> and N known to everyone; this is the public key. The encryption consists of raising a credit card number to the <em>r</em><sup>th</sup> power modulo N. Anyone can do it (on a computer, because the numbers involved are quite large). The decryption, on the other hand, consists of raising the resulting number to the <em>s</em><sup>th</sup> power modulo N. This gives back the original credit card number (<a href="http://math.berkeley.edu/~frenkel/RSA.pdf">see here for more details</a>). The merchant keeps the number <em>s</em> secret. Therefore the transmission of the credit card numbers is secure. The only way to find <em>s</em>, and hence to be able to decrypt the card numbers, is to determine the prime factors <em>p</em> and <em>q</em> of the number N. For sufficiently large N, however, using known methods of prime factorization, <a href="https://sites.google.com/site/danzcosmos/why-rsa-encryption-is-secure">it may take many months</a> to find <em>p</em> and <em>q</em>, even on a network of powerful computers. But if one could come up with a more efficient way to factor numbers into primes (for example, by using a <a href="http://www.sciencedaily.com/releases/2012/08/120819153743.htm">quantum computer</a>), then one would have a tool for breaking the RSA algorithm. That’s why a lot of research is directed toward factoring numbers into primes. Scores of legitimate mathematicians are working on this, and perhaps not so legitimate ones as well.</p>
<p>To an outsider, the RSA algorithm appears like a card trick: You pick a card from a stack, hide it (this is like encryption), and after some manipulations the magician produces your card—bazinga! Well, that's pretty much what the RSA algorithm does … except that the role of magic is now played by math.</p>Mon, 03 Jun 2013 09:55:00 GMThttp://www.slate.com/articles/health_and_science/science/2013/06/online_credit_card_security_the_rsa_algorithm_prime_numbers_and_pierre_fermat.htmlEdward Frenkel2013-06-03T09:55:00ZWhat happens when you enter your credit card number online?Health and SciencePrime Numbers and Pierre Fermat Keep Your Secrets Safe Online100130603006online securitymathonline securitymathEdward FrenkelSciencehttp://www.slate.com/articles/health_and_science/science/2013/06/online_credit_card_security_the_rsa_algorithm_prime_numbers_and_pierre_fermat.htmlfalsefalsefalsePrime Numbers and Pierre Fermat Keep Your Secrets Safe OnlinePrime Numbers and Pierre Fermat Keep Your Secrets Safe OnlinePhoto Illustration by Justin Sullivan/Getty ImagesHow exactly are credit card purchases kept secure?Don’t Listen to E.O. Wilson
http://www.slate.com/articles/health_and_science/science/2013/04/e_o_wilson_is_wrong_about_math_and_science.html
<p>E.O. Wilson is an eminent Harvard biologist and best-selling author. I salute him for his accomplishments. But he couldn’t be more wrong in his <a href="http://online.wsj.com/article/SB10001424127887323611604578398943650327184.html">recent piece</a> in the<em> Wall Street Journal</em> (adapted from his new book <a href="http://www.amazon.com/gp/product/0871403773/ref=as_li_ss_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0871403773&linkCode=as2&tag=slatmaga-20"><em>Letters to a Young Scientist</em></a>), in which he tells aspiring scientists that they don’t need mathematics to thrive. He starts out by saying: “Many of the most successful scientists in the world today are mathematically no more than semiliterate … I speak as an authority on this subject because I myself am an extreme case.” This would have been fine if he had followed with: “But you, young scientists, don’t have to be like me, so let’s see if I can help you overcome your fear of math.” Alas, the octogenarian authority on social insects takes the opposite tack. Turns out he actually believes not only that the fear is justified, but that most scientists don’t need math. “I got by, and so can you” is his attitude. Sadly, it’s clear from the article that the reason Wilson makes these errors is that, based on his own limited experience, he does not understand what mathematics is and how it is used in science.</p>
<p>If mathematics were fine art, then Wilson’s view of it would be that it’s all about painting a fence in your backyard. Why learn how to do it yourself when you can hire someone to do it for you? But fine art isn’t a painted fence, it’s the paintings of the great masters. And likewise, mathematics is not about “number-crunching,” as Wilson’s article suggests. It’s about concepts and ideas that empower us to describe reality and figure out how the world really works. Galileo famously said, “The laws of Nature are written in the language of mathematics.” Mathematics represents objective knowledge, which allows us to break free of dogmas and prejudices. It is through math that we learned Earth isn’t flat and that it revolves around the sun, that our universe is curved, expanding, full of dark energy, and quite possibly has more than three spatial dimensions. But since we can’t really imagine curved spaces of dimension greater than two, how can we even begin a conversation about the universe without using the language of math?</p>
<p>Charles Darwin rightfully spoke of math endowing us “with something like a new sense.” History teaches that mathematical ideas that looked abstract and esoteric yesterday led to spectacular scientific advances of today. Scientific progress would be diminished if young scientists were to heed Wilson’s advice.</p>
<p>It is interesting to note that Wilson’s <a href="http://www.nature.com/nature/journal/v466/n7310/abs/nature09205.html">recent article</a> in <em>Nature</em> and his book claiming to show support for so-called <a href="http://www.amazon.com/gp/product/0871404133/ref=as_li_ss_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0871404133&linkCode=as2&tag=slatmaga-20">group selection</a> have been sharply criticized, by <a href="http://www.prospectmagazine.co.uk/magazine/edward-wilson-social-conquest-earth-evolutionary-errors-origin-species/">Richard Dawkins</a> and <a href="http://whyevolutionistrue.wordpress.com/2010/08/30/a-misguided-attack-on-kin-selection/">many others</a>. Some of the critics pointed out that <a href="http://www.nature.com/nature/journal/v471/n7339/abs/nature09834.html">one source of error</a> was in Wilson’s math. Since I’m not an expert in evolutionary theory, I can’t offer an opinion, but I find this controversy interesting given Wilson’s thesis that “great scientists don’t need math.”</p>
<p>One thing should be clear: While our perception of the physical world can always be distorted, our perception of the mathematical truths can't be. They are objective, persistent, necessary truths. A mathematical formula means the same thing to anyone anywhere—no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand year from now. And that’s why mathematics is going to play an increasingly important role in science and technology.</p>
<p>One of the key functions of mathematics is the ordering of information. With the <a href="http://www.amazon.com/gp/product/1451654960/ref=as_li_ss_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=1451654960&linkCode=as2&tag=slatmaga-20">advent of the 3-D printing and other new technology</a>, the reality we are used to is undergoing a radical transformation: Everything will migrate from the layer of physical reality to the layer of information and data. We will soon be able to convert information into matter on demand by using 3-D printers just as easily as we now convert a PDF file into a book or an MP3 file into a piece of music. In this brave new world, math will be king: It will be used to organize and order information and facilitate the conversion of information into matter.</p>
<p>It might still be possible to be “bad in math” (though I believe that anyone can be good at math if it is explained in the right way) and be a good scientist—in some areas and probably not for too long. But this is a handicap and nothing to be proud of. Granted, some areas of science currently use less math than others. But then practitioners in those fields stand to benefit even more from learning mathematics.</p>
<p>It would be fine if Wilson restricted the article to his personal experience, a career path that is obsolete for a modern student of biology. We could then discuss the real question, which is how to improve our math education and to eradicate the fear of mathematics that he is talking about. Instead, trading on that fear, Wilson gives a misinformed advice to the next generation, and in particular to future scientists, to eschew mathematics. This is not just misguided and counterproductive; coming from a leading scientist like him, it is a disgrace. Don’t follow this advice—it’s a self-extinguishing strategy.</p>Tue, 09 Apr 2013 17:46:49 GMThttp://www.slate.com/articles/health_and_science/science/2013/04/e_o_wilson_is_wrong_about_math_and_science.htmlEdward Frenkel2013-04-09T17:46:49ZMath can help you in almost any career. There’s no reason to fear it.Health and ScienceE.O. Wilson Is Wrong About Math and Science100130409013mathevolutionEdward FrenkelSciencehttp://www.slate.com/articles/health_and_science/science/2013/04/e_o_wilson_is_wrong_about_math_and_science.htmlfalsefalsefalseE.O. Wilson Is Wrong About Math and ScienceE.O. Wilson Is Wrong About Math and SciencePhoto by Cindy Ord/Getty Images for World Science FestivalBiologist Edward O. Wilson doesn't think scientists need to learn math. He's wrong.Don’t Let Economists and Politicians Hack Your Math
http://www.slate.com/articles/technology/technology/2013/02/should_algebra_be_in_curriculum_why_math_protects_us_from_the_unscrupulous.html
<p>Imagine a world in which it is possible for an elite group of hackers to install a “backdoor” not on a personal computer but on the entire U.S. economy. Imagine that they can use it to cryptically raise taxes and slash social benefits at will. Such a scenario may sound far-fetched, but replace “backdoor” with the Consumer Price Index (CPI), and you get a pretty accurate picture of how this arcane economics statistic has been used.</p>
<p>Tax brackets, Social Security, Medicare, and various indexed payments, together affecting tens of millions of Americans, are pegged to the CPI as a measure of inflation. The fiscal cliff deal that the White House and Congress reached a month ago was almost derailed by a proposal to change the formula for the CPI, which Matthew Yglesias <a href="http://www.slate.com/articles/business/moneybox/2012/12/chained_cpi_a_sneaky_plan_to_cut_social_security_and_raise_taxes_by_changing.html">characterized</a> as “a sneaky plan to cut Social Security and raise taxes by changing how inflation is calculated.” That plan was scrapped at the last minute. But what most people don’t realize is that something similar had already happened in the past. A new book, <a href="http://www.amazon.com/gp/product/0547317271/ref=as_li_ss_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0547317271&linkCode=as2&tag=slatmaga-20"><em>The Physics of Wall Street</em></a> by James Weatherall, tells that story: In 1996, five economists, known as the Boskin Commission, were tasked with saving the government $1 trillion. They observed that if the CPI were lowered by 1.1 percent, then a $1 trillion could indeed be saved over the coming decade. So what did they do? They proposed a way to alter the formula that would lower the CPI by <em>exactly</em> that amount!</p>
<p>This raises a question: Is economics being used as science or as after-the-fact justification, much like economic statistics were manipulated in the Soviet Union? More importantly, is anyone paying attention? Are we willing to give government agents a free hand to keep changing this all-important formula whenever it suits their political needs, simply because they think we won’t get the math?</p>
<p>Ironically, in a recent <a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?pagewanted=all&_r=2&">op-ed in the<em> New York Times</em></a>, social scientist Andrew Hacker suggested eliminating algebra from the school curriculum as an “onerous stumbling block,” and <em>instead</em> teaching students “how the Consumer Price Index is computed.” What seems to be completely lost on Hacker and authors of similar proposals is that the calculation of the CPI, as well as other evidence-based statistics, is in fact a difficult mathematical problem, which requires deep knowledge of all major branches of mathematics including … advanced algebra.</p>
<p>Whether we like it or not, calculating CPI necessarily involves some abstract, arcane body of math. If there were only one item being consumed, then we could easily measure inflation by dividing the unit price of this item today by the unit price a year ago. But if there are two or more items, then knowing their prices is not sufficient. We also need to know the levels of consumption today and a year ago; economists call these “baskets.” Of course, we can easily find a typical consumer’s expenditure today by multiplying today’s consumption levels by the current prices and adding them up. But to what number from a year ago should we compare it? If the consumption levels were static, we would compute last year’s expenditure by multiplying the same consumption levels by last year’s prices and adding them up. We would then be able to measure inflation by dividing this year’s expenditure by last year’s. But consumption tends to change—in part because our tastes change, but also in response to price variations. The inflation index must account for this, so we have to find a way to compare the baskets today and a year ago. This turns out to be a hard mathematical problem that has perplexed economists for more than a century and still hasn’t been completely solved. But even to begin talking about this problem, we need a language that would enable us to operate with symbolic quantities representing baskets and prices—and that’s the language of algebra!</p>
<p>In fact, we need much more than that. As Weatherall explains in his book, to implement a true cost-of-living index, one actually has to use the so-called “gauge theory.” This mathematics is at the foundation of a <a href="http://blogs.discovermagazine.com/cosmicvariance/2012/11/22/thanksgiving-7/">unified physical theory</a> of three forces of nature: electromagnetism, the strong nuclear force, and the weak nuclear force. (Many Nobel Prizes have been awarded for the development of this unified theory; it was also used to predict the Higgs boson, the elusive elementary particle <a href="http://www.slate.com/articles/technology/future_tense/2012/07/higgs_boson_announcement_from_cern_why_the_god_particle_is_so_important_.html">recently discovered at the Large Hadron Collider</a> under Geneva.) The fact that gauge theory also underlies economics was a <a href="http://pirsa.org/06050010/">groundbreaking discovery</a> made by the economist Pia Malaney and mathematical physicist Eric Weinstein around the time of the Boskin Commission. Malaney, who was at the time an economics <a href="http://lists.perimeterinstitute.ca/pipermail/piuwcomplex/attachments/20090205/7f2cccd3/attachment-0002.pdf">Ph.D. student</a> at Harvard, tried to convey the importance of this theory for the index problem to the Harvard professor Dale Jorgenson, one of the members of the Boskin Commission, but to no avail. In fact, Jorgenson responded by throwing her out of his office. Only recently, George Soros’ Institute for New Economic Thinking finally gave Malaney and Weinstein long overdue recognition and is <a href="http://ineteconomics.org/grants/geometric-marginalism">supporting their research</a>. But their work still remains largely ignored by economists.</p>
<p>So that’s where we find ourselves today: Politicians are still eager to exploit backdoor mathematical formulas for their political needs, economists are still willing to play along, and no one seems to care about finding a scientifically sound solution to the inflation index problem using adequate mathematics. And the public—well, very few people are paying attention. And if we follow Hacker’s prescriptions and further dumb down our math education, there won’t be anyone left to understand what’s happening behind closed doors.</p>
<p>Irrespective of one’s political orientation, one thing should be clear: In this brave new world, in which formulas and equations play a much bigger role than ever before, our ignorance of mathematics is being abused by the powers that be, and this will continue until we start taking math seriously for what it is: a powerful weapon that can be used for good and for ill.</p>
<p>Alas, instead of recognizing this new reality, we keep giving forum to paragons of mathematical illiteracy.</p>
<p>In his book, Weatherall made an admirable effort to start a serious conversation about the need for a new mathematical theory of the CPI. But guess who <a href="http://www.nybooks.com/articles/archives/2013/jan/10/how-he-got-it-right/?pagination=false">reviewed this book</a> in the<em> New York Review of Books</em>? Andrew “we don’t need no algebra” Hacker! There is nothing wrong with healthy debate; it can only be encouraged. But something is wrong when an opinionated individual who has demonstrated total ignorance of a subject matter gets called on over and over again as an expert on that subject.</p>
<p>We have to break this vicious circle. As Richard Feynman eloquently said, “People who wish to analyze nature without using mathematics must settle for a reduced understanding.” Now is the time <em>not</em> to reduce math curriculum at schools, but to <em>expand</em> it, taking advantage of new tools in education: computers, iPads, the wider dissemination of knowledge through the Internet. Kids become computer literate much earlier these days, and they can now learn mathematical concepts faster and more efficiently than any previous generation. But they have to be pointed in the right direction by teachers who inspire them to think big. This can only be achieved if math is not treated as a chore and teachers are not forced to spend countless hours in preparation for standardized tests. Math professionals also have a role to play: Schools should invite them to help teachers unlock the infinite possibilities of mathematics to students, to show how a mathematical formula can be useful in the real world and also be elegant and beautiful, like a painting, a poem, or a piece of music.</p>
<p>Working together, we should implement the 21<sup>st</sup> century version of the Second Amendment: Everyone shall have the right to bear “mathematical arms”—to possess mathematical knowledge and tools needed to protect us from arbitrary decisions by the powerful few in the increasingly math-driven world. So that the next time someone wants to alter a formula that affects us all, we won’t be afraid to ask: “Wait a minute, what does this formula mean and why are you changing it?”</p>Fri, 08 Feb 2013 19:14:01 GMThttp://www.slate.com/articles/technology/technology/2013/02/should_algebra_be_in_curriculum_why_math_protects_us_from_the_unscrupulous.htmlEdward Frenkel2013-02-08T19:14:01ZOf course kids need to learn algebra.TechnologyEveryone Should Have the Right To Bear Mathematical Arms100130208008matheconomicseducationmatheconomicseducationEdward FrenkelTechnologyhttp://www.slate.com/articles/technology/technology/2013/02/should_algebra_be_in_curriculum_why_math_protects_us_from_the_unscrupulous.htmlfalsefalsefalseEveryone Should Have the Right To Bear Mathematical ArmsEveryone Should Have the Right To Bear Mathematical ArmsPhoto by Ralph Morse/Time & Life Pictures/Getty ImagesAlbert Einstein's office