Category: mathematics

Mathematicians warn of ideology polluting their discipline

May 25, 2023 • 11:30 am

It looks as if today will be about ideology infecting science—in this case, mathematics. One would think that math would be relatively impervious to the ideological tides inundating other sciences, but one would be wrong. This article from the Torygraph (click on screenshot, or on the archived version here), discusses nonbinding but injurious ideological guidelines given to college teachers of math in the UK. These guidelines have nothing to do with improving math education, of course, but everything to do with propagandizing students with certain approved political views.

Excerpts are indented:

More than 50 of Britain’s leading mathematicians have accused standards bosses of politicising the curriculum with new diversity guidance.

Academics at top UK universities have signed an open letter criticising guidance on academic standards that states that values of Equality, Diversity and Inclusion (EDI) “should permeate the curriculum and every aspect of the learning experience”.

The guidance was published in March by the Quality Assurance Agency (QAA), an independent body that receives membership fees from more than 300 UK higher education providers and distributes advice on courses.

In an open letter, the mathematicians write: “We reject the QAA’s insistence on politicising the mathematical curriculum.

“We believe the only thing that should permeate the mathematics curriculum is mathematics. Academics should teach from a perspective informed by their academic experience, not from a political perspective determined by the QAA.

“Students should be able to study mathematics without also being required to pay for their own political indoctrination.”

I believe the letter of protest to the QAA guidelines is here, though it may be an earlier version. The link to the guidelines themselves (given in the letter) seems to be gone, but the letter’s signers paraphrase some guidelines:

A particular concern is that the new edition states: “the curriculum should present a multicultural and decolonised view of MSOR, informed by the student voice.”

We abhor racism, but one can abhor racism without subscribing to the theory of decoloniality.

The theory of decoloniality is a postmodernist critique of the “European paradigm of rational knowledge”. We believe that history suggests that mathematics is not a particularly European paradigm. On the contrary there are many examples where the same mathematical ideas have been developed independently across cultures. As just one example, the Japanese mathematician Seki and the Swiss mathematician Bernoulli both studied what are now called Bernoulli numbers. We agree that where practical the mathematical community should use terminology that gives nonWestern mathematicians proper credit, but this is not the meaning of decoloniality.

The QAA suggests promoting a decolonialist perspective as follows:

Students should be made aware of problematic issues in the development of the MSOR content they are being taught, for example some pioneers of statistics supported eugenics, or some mathematicians had connections to the slave trade, racism or Nazism.

The mathematicians are correct; math curricula should be about math alone.  But what is the QAA recommending? This is hard to believe, but seems to be true:

The QAA guidance suggests that professors should note that “some early ideas in statistics were motivated by their proposers’ support for eugenics, some astronomical data were collected on plantations by enslaved people, and, historically, some mathematicians have recorded racist or fascist views or connections to groups such as the Nazis”.

Maths professors said that the agency wanted to teach “a skewed view of the history of mathematics”. They noted that the QAA did not recommend teaching “the universality of mathematical truth, the use of statistics to disprove historical racial theories or about the Jewish mathematicians persecuted by Nazis”.

If you take this tactic, then every single academic subject must devote its time to showing how famous achievers in its area were politically impure. If you want to discuss things like how slaves collected astronomical data, do it in a history or sociology of science class.

But the scariest thing in these guidelines—and I can’t verify this because I can’t find the guidelines themselves—is that the QAA did NOT recommend teaching “the universality of mathematical truth, the use of statistics to disprove historical racial theories or about the Jewish mathematicians persecuted by Nazis”.  Is mathematical truth not universal?  Yes, I know that Euclidean geometry differs from non-Euclidian geometry, but that itself is a universal truth. And they recommending teaching how mathematicians promoted slavery, racism, and Nazism, but, curiously, don’t recommend teaching how slaves enriched astronomy or how Jewish mathematicians were persecuted by Nazis? And, as a secular Jew, I want to know why Jewish persecution get a pass here.

In truth, none of this should be in math class, but I find it deeply weird that of all the philosophies held by some mathematicians, including the morality of slavery and of Nazism, they leave out Jews, who of course were the very victims of Nazi persecution, just as slaves were the victims of racism.

But there’s pushback beyond the letter:

Dr John Armstrong, a reader in financial mathematics at King’s College London, and a signatory of the letter, said: “Education for sustainable development may sound like a positive thing, but when you look into what that is, what they are promoting is encouraging all students to become activists on issues of social justice.

“It’s really quite a remarkable thing to change education from goals such as understanding, learning and appreciating art and shift everything towards consideration of social justice.”

It is simply bizarre that we all sit back and accept this explicit injection of ideology into science, a practice that not only takes time away from science (and, in this case, math), but tries to turn young mathematicians into ideologues. Were I a parent, I’d want my children to decide their views themselves, not have propaganda stuffed down their throats by math teachers.  These are bizarre times we live in, but we can’t let those who are most vocal foist their politics onto children who want to learn math or science (or anything else, for that matter).

Oh, and in light of the letter, the QAA has added this:

A spokesman for the QAA said: “Subject benchmark statements are written by groups of academics from the relevant discipline. Institutional autonomy and academic freedom are crucial principles, and therefore the statements do not mandate academics to teach specific content – they are a reflective tool to support course design and are not compulsory. We agree with the letter’s assertion that course content should be taught by academics in line with their own expertise and academic judgment.”

Indeed. Why, then, did they insist on producing a benchmark statement? And, as one of my friends asked, “How did it all go off the rails?”  It’s almost as if we’re being subject to “extraordinary popular delusions and the madness of crowds,” as the famous book was called.  This foisting of ideology on education is our version of Tulip Mania.

A trigonometric proof of the Pythagorean theorem at last? The news said it was done by two high school students!

March 28, 2023 • 10:30 am

The latest mathematical news involves two high school students from New Orleans who have found a new way to prove the Pythagorean Theorem, which of course states that for a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.  Now of course there are many proofs of this theorem (the link above gives some), but, according to the Guardian article referenced below (click on screenshot), and the POCIT article below that (also click on screenshot), one form of proof is missing: a proof based on trigonometry. The Guardian article reports that this proof was supplied by Calcea Johnson and Ne-Kiya Jackson from St Mary’s Academy.

Here’s a tweet from the American Mathematical Society:

Click to read the Guardian article:

 

Now this is somewhat above my pay grade, but perhaps a mathematician will weigh in. The Guardian notes this:

The 2,000-year-old theorem established that the sum of the squares of a right triangle’s two shorter sides equals the square of the hypotenuse – the third, longest side opposite the shape’s right angle. Legions of schoolchildren have learned the notation summarizing the theorem in their geometry classes: a2+b2=c2.

As mentioned in the abstract of Johnson and Jackson’s 18 March mathematical society presentation, trigonometry – the study of triangles – depends on the theorem. And since that particular field of study was discovered, mathematicians have maintained that any alleged proof of the Pythagorean theorem which uses trigonometry constitutes a logical fallacy known as circular reasoning, a term used when someone tries to validate an idea with the idea itself.

Johnson and Jackson’s abstract adds that the book with the largest known collection of proofs for the theorem – Elisha Loomis’s The Pythagorean Proposition – “flatly states that ‘there are no trigonometric proofs because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem’.”

What is the circularity? To me (and again I may be wrong) it’s instantiated in the section below from the Wikipedia article.  The supposed trigonometric “proofs” of the Theorem depends on an identity:

The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions.

The identity is

. . . and that is proven simply from the observation of right triangles and the definition of sines and cosines (see here).  Thus, since this equation is derived from definitions and the use of a right triangle, it cannot itself be used to prove the theorem. Solid proofs have depended on arguments from geometry, rearranging similar triangles, algebra, and differentials. What is said to be new here is the proof of the Theorem using trigonometry, but not in a circular way. Because I haven’t seen the students’ proof, I can’t comment on it, and probably don’t have the chops, either. I’ve put one mathematician’s video interpretation below.

A lot of the press has gotten this mixed up, like this article below (click to read):

It says this:

The Pythagorean theorem is a fundamental theorem in trigonometry that describes the relationship between the three sides of a right-angled triangle. It is expressed with the formula a² + b² = c².

The theorem holds true in every plausible example and has been around since the days of the Ancient Greeks.

American mathematician Elisha Loomis argued that no mathematician has been able to establish its truth without using circular logic i.e. without using Pythagorean Theorem. However, others have argued it can be proved using the notion of similar triangles.

Well, that’s a bit misleading, as you can establish the truth of the Pythagorean theorem without using circular logic—so long as the proof is not a trigonometric one. And that’s what the teens are said to have overcome. However, I noticed this morning that an addendum had been added to the article.

This article was corrected on March 28, 2023. The article previously stated that the Pythagorean theorem had yet to be proven without using circular logic. The article has been amended to acknowledge that this argument was posited by American mathematician Elisha Loomis and is not accepted by all mathematicians.

Even so, the correction (which doesn’t itself invalidate what the students did) still leaves out the fact that even if Loomis was wrong, his claim was not that it couldn’t be proven without using circular logic, for it can be. What he said is that it can’t be proven trigonometrically without using circular logic.

Here’s a video which is one person’s interpretation of Johnson and Jackson’s proof based on the slides he saw of their presentation at the AMS meeting:

 

It’s quite clever, and doesn’t depend on the use of the Pythagorean Identity, merely the definition of “sine”.  Note that the person who made this video isn’t 100% sure that he’s presenting Johnson and Jackson’s proof.

So, were they the first people to prove the Pythagorean theorem without trigonometry? I did some Googling, and found this at the site cut-the-knot.org (click to read), which apparently came out before 2019.

It merely reprises another trigonometric proof of the Pythagorean Theorem, one given in the paper below. This is from the article above:

Now, Jason Zimba has showed that the theorem can be derived from the subtraction formulas for sine and cosine without a recourse to sin²α + cos²α = 1. [JAC: the latter is  of course the Pythagorean Trigonometric Identity.]

The paper it refers to is this:

J. Zimba, On the Possibility of Trigonometric Proofs of the Pythagorean Theorem, Forum Geometricorum, Volume 9 (2009) 275-278

And I found it online. You can read Zimba’s paper by clicking on the screenshot:

Zimba says he can derive the Pythagorean Trigonometric Identity independently of the Pythagorean Theorem, and thus you can use the former to prove the latter without circularity. He defines sine and cosine without using right angles, and then employs a subtraction procedure to come up with the Pythagorean Identity. That identity then establishes the truth of the Pythagorean Theorem. Here’s a bit of the ending, but it’s a short paper.

The author of the cut-the-knot.org piece says that Zimba’s proof is correct. But what do I know? If it is, then Johnson and Jackson were not the first people to produce a non-circular trigonometric proof of the Pythagorean theorem. But even if that’s the case, if the video gives an accurate take of the two students’ proof, what they did is still remarkably clever.

I suppose mathematicians will weigh in on the Internet in the next few days, so stay tuned.

Nature on “decolonizing” mathematics

February 2, 2023 • 12:30 pm

The latest issue of Nature, one of the world’s most prestigious scientific journals, has a long (4-page) feature about the “decolonization” of mathematics. As we’ve learned to expect from this kind of article, it points out gender and ethnic inequities among mathematicians, ascribes them to structural racism existing today, and seen as ubiquitous in math, and and then proposes untested ways to achieve equity in math (proportional representation of groups) by infusing the teaching of math with aspects of local culture.

The problem with this paper, like similar “decolonization” screeds, is that while it certainly means well (I agree that everyone should have the chance to learn math), and is sensitive to differences among cultures, it gives no evidence that “decolonizing” mathematics (that is, removing its “whiteness” and “Westernness”, and using as math subjects features of the local culture) actually works. It’s a gift package of suggestions and assertions wrapped around, well, nothing.  This doesn’t meant that the suggestions are not worthwhile, but there’s nothing to be gained by blaming inequities, which could be due to a number of factors, to existing bigotry and racism in math, for which it offers no evidence.  More important, the “course” they chart has to be shown to actually lead to more understanding than alternatives.

Here’s where blame is affixed. You don’t have to be a rocket scientist to know that it adheres to white men.

Maths is built on a modern history of elevating the achievements of one group of people: white men. “Theorems or techniques have names associated to them and most of the time, those names are of nineteenth-century French or German men,” such as Georg Cantor, Henri Poincaré and Carl Friedrich Gauss, all of whom were white, says John Parker, head of the mathematical sciences department at Durham University, UK. This means that the accomplishments of people of other genders and races have often been pushed aside, preventing maths from being a level playing field. It has also squelched wider access to rich mathematical ideas developed by people of different backgrounds — such as Chike Obi, James Ezeilo and Adegoke Olubummo, a trio credited by the website Mathematicians of the African Diaspora with having pioneered modern maths research in Nigeria. Another example is Mary Golda Ross, a Cherokee mathematician and engineer who was a founding member of ‘Skunk Works’, a secretive division of the US aerospace manufacturer Lockheed. There, she developed early designs for space travel and satellites, among other things.

Where is the evidence that high quality and non-white mathematicians, of which until recently there were very few, are now being pushed aside by racism? I don’t doubt that there was discrimination in the past against women and minorities, but even then I keep thinking of the Indian Srinivasa Ramanujan, an immensely talented autodidact from Tamil Nadu who in 1913 sent a bunch of his theorems and proofs to G. H. Hardy at Cambridge, who instantly recognized the man’s talent and arranged for him to study at Cambridge. I can’t imagine anyone more “minoritized” in the UK than Ramanujan, dark of skin, poor, and humble of origin. And yet people helped him, and he’s still regarded as a giant in the field. Would people push him aside today—or anyone like him? I doubt it, just as I doubt that mathematics is presently rife with structural racism—that the playing field is still “far from level”. If “level” means “equal opportunity”, then I’d say we’re pretty close. If it means “equal outcoms”, I’d say, yes, it’s not level. But that’s not what a tilted playing field means: it means that right now there is not equal opportunity. Yes, the pipeline needs to fill up after a past of sexism and bigotry, but the article gives evidence for “structural bias” or “system bias” at the pipeline’s distal end.

Here’s what the advocates of decolonization advocate to replace the kind of math teaching we have today:

Edward Doolittle, a mathematician at First Nations University of Canada in Regina, contrasts Indigenous mathematics with the mainstream, global way of teaching maths, in which instructors essentially present the same content regardless of where they’re teaching.

Doolittle, who’s also a Mohawk person from Six Nations in southern Ontario, says that calculus courses are structured so similarly that he could teach the subject “anywhere the students speak English”, and even take over teaching a course midstream.

By contrast, he says that Indigenous mathematics involves getting inside a culture and examining the mathematical thinking in it. He draws a further distinction between Indigenous mathematics and the practice of what he calls “indigenizing mathematics”, which, he says, involves searching for cultural examples to use in courses taught in the global version of mathematics.

Indigenizing mathematics tweaks the curriculum when it isn’t feasible to fully immerse students in ideas from an Indigenous culture, Doolittle says. “It’s very hard, if not impossible, to break out of” the global mathematics system, he notes. By indigenizing mathematics, instructors can stay within the parameters of what they’re required to cover while broadening the cultural scope of their curriculum.

Using that approach, “we have respected the knowledge of Indigenous people and are furthering our ties with Indigenous people” while still teaching students core topics, he says. For example, when teaching statistics courses, Doolittle has discussed a simplified version of the Peach Stone Game, which is based on making wagers and is played in his community. “You can analyse this in terms of a binomial probability distribution,” or the chances of two outcomes over time, he says.

“I would like to encourage many of my colleagues to engage in indigenization efforts, and hopefully to turn up interesting examples from their local area,” Doolittle says.

As for how to “indigenize” math, the article gives a couple of examples beyond the Peach Stone Game: teaching about Polynesian navigation to Hawaiians  in Hawaii and using aspects of local culture to teach math in five African countries (“the next Einstein will be African” is the motto of this five-nation consortium, the African Institute for Mathematical Sciences, or AIMS). And that’s about it.  There is a lot of noise, but, as of yet, little to show that this kind of training produces results better than “non-indigenous” training. If it does work, more power to them. So far, most of the “indigenizing” appears to be mainly trying to increase the diversity of people going into math. That’s great, too, but it’s not a revolution in teaching math.

And even some of these endeavors involve bringing in mathematicians who aren’t indigenous. Here’s what AIMS does:

Faculty members at the centres are hired from African countries, often through partnerships with local universities. AIMS also hosts visiting lecturers from outside Africa who teach courses that range from a few weeks to two months in length. Bringing in outside researchers exposes students to top talent while they continue to expand their roots in Africa’s mathematical communities.

But isn’t it counterproductive to bring in “top talent”, probably white people, who undoubtedly teach math in decidedly non-Indigenous ways?

It’s clear that while I have no strong beef against using local culture or examples to teach math—or any form of science—this will go only so far (what happens when you get to really high-level math?), and if you’re going to do something like this, it’s better to start by showing in pilot projects that it really works.  Blaming whiteness or the West on holding down math education in places like Africa (where whites are actually a minority), is no longer tenable, and even counterproductive.

But here’s the part I most object to. Durham University in the UK is itself mounting a decolonization effort that involves Ric Crossman, a statistician, and John Parker, head of Durham’s maths department. Here’s their philosophy of education:

Durham’s senior mathematicians felt that their curriculum-reform process had to be led by the students, because otherwise “we’re in the awful situation of deciding for ourselves what’s best for them”, Crossman says. That, Parker adds, would be at odds with the concept of decolonization, because colonization “was some group of people thinking they knew best for some other group of people”.

What an AWFUL situation!  It’s certainly feasible for some students to tell you the best ways they can absorb mathematics, but this will certainly differ among students, and not every student knows. But to put the curriculum and all the teaching methods in the hands of the students, ignoring the experience of teachers who have spent years finding out which forms of pedagogy work in general, is a recipe for disaster. It’s simply invidious to denigrate the expertise of teachers by comparing it to “colonizers.” But such are the rhetorical tactics that progressives have learned to use.

h/t: Carl

Video: Alternative math takes over

January 18, 2023 • 12:45 pm

This video, “Alternative Math,” has been around for six years, and has won 15 awards for short features and funny videos. The sad thing is that while it’s funny, it’s also true: truer now than it was when it was made. It documents the “2 + 2 = 5” alternative-truth mentality that is represented by “other ways of knowing.” But it also has a funny ending, so be sure to watch the whole thing (it’s nine minutes long).

The IMDb summary (which also has info about the film and the cast) is this: “A well meaning math teacher finds herself trumped by a post-fact America.”

Enjoy!

Four awarded the Fields Medal for mathematics, including only the second woman to get it (and she’s Ukrainian)

July 5, 2022 • 9:00 am

Every four years since 1936, the prestigious Fields Medal is awarded to a maximum of four young mathematicians (under 40) for outstanding accomplishments. The awarding organization is the International Congress of the International Mathematical Union.

It’s seen as the “Nobel Prize in Mathematics,” even though it isn’t formally a Nobel. But in one way it’s better: it’s awarded every four years instead of yearly. Since the Nobel Prize in any area can be given to up to three people, the maximum number of Nobelists in four years is twelve—compared to four for the Fields.

The down side, if there is one given the immense prestige the Fields confers, is that it doesn’t come with a lot of dosh—about $15,000 Canadian. In contrast, a Nobel Prize comes with a sum of 10 million Swedish kroner—almost exactly one million U.S. dollars. (If there are two winners it’s split evenly, if three the division is decided by the Swedes.) $15,000 won’t enable you to buy a beach house, as Feynman did with his Nobel money. But money seems of much smaller consequence than the fact that winners are topped for the life with the halo “Fields Medal Winner.” (See the movie “Good Will Hunting.”)

The Fields was just awarded to four people, including only the second woman ever to win. And she’s from Ukraine!

Click to read:

The details and accomplishments of the four are in the article, but here are their names and institutions:

Hugo Duminil-Copin; Institut des Hautes Études Scientifiques, France andUniversity of Geneva, Switzerland

June Huh;  Princeton University, US

James Maynard;  Oxford University, UK

Maryna Viazovska;  École Polytechnique Fédérale de Lausanne, Switzerland

Viazovska is the only woman to win the prize besides Maryam Mirzakhani of Stanford, who won in 2014 and is of Iranian descent.

I’ll highlight Maryna Viazovska to applaud not only the advance of women in math, but as a boon to the much-beleaguered Ukraine. Here’s what the NYT says about her in a summary by Kenneth Chang:

Maryna Viazovska, a Ukrainian who is now a professor at the Swiss Federal Institute of Technology in Lausanne, is known for proofs for higher-dimensional equivalents of the stacking of equal- sized spheres. She is also only the second woman ever to win the Fields Medal.

Of the 60 mathematicians who won Fields Medals before this year, 59 were men. In 2014, a Stanford mathematician, Maryam Mirzakhani, was the first and, until now, the only woman to receive one.

“I feel sad that I’m only the second woman,” Dr. Viazovska said. “But why is that? I don’t know. I hope it will change in the future.”

Dr. Viazovska’s work is a variation of a conjecture by Johannes Kepler more than 400 years ago. Kepler is best known for realizing that the planets move around the sun in elliptical orbits, but he also considered the stacking of cannonballs, asserting that the usual pyramid stacking was the densest way that they could arranged, filling up just over 75 percent of the available space.

Kepler could not prove that statement, however. Neither could anyone else until Thomas Hales, then at the University of Michigan, succeeded in 1998 with a 250-page proof and, controversially, the help of a computer program.

Proving something similar for the packing of equal-size spheres in dimensions higher than three has been impossible so far — with a couple of exceptions.

In 2016, Dr. Viazovska found the answer in eight dimensions, showing that a particularly symmetric packing structure known as E8 was the best possible, filling about one-quarter of the volume. Within a week, she and four other mathematicians showed that a different arrangement known as the Leech lattice was the best possible packing in 24 dimensions. In high dimensions, the filled volume is not very full, with the Leech lattice of 24-dimensional spheres occupying about 0.2 percent of the volume.

What’s so special about eight and 24 dimensions?

“I think that’s a mystery,” Dr. Viazovska said. “It’s just in these dimensions, certain things happen which don’t happen in other dimensions.”

She said that a method that generally gives an upper bound on the packing density turns out to be the exact solution in these cases.

High-dimensional sphere packings are related to the error-correcting techniques used to fix garbles in the transmission of information.

She said that the Russian invasion of Ukraine had taken its toll on her family. “It’s very difficult,” she said.

Her parents still live near Kyiv, Dr. Viazovska said, while her sisters, nephew and niece left and joined her in Switzerland.

Here’s the Fields Medal (caption from Wikipedia, the Latin translation is “Rise above oneself and grasp the world”), and a photo of Viazovska:

Photo of the obverse of a Fields Medal made by Stefan Zachow for the International Mathematical Union (IMU), showing a bas relief of Archimedes (as identified by the Greek text). The Latin phrase states: Transire suum pectus mundoque potiri

Maryna Viazovska, from the Guardian:

h/t: Tom

The intellectual vacuity of mathematical arguments against evolution

June 2, 2022 • 12:00 pm

UPDATE: Somehow I missed that Jason has a new book that expands on this problem (I didn’t see it on the Amazon site). Here’s the cover, and click on it to go to the site:

*****************

Jason Rosenhouse is a professor of math at James Madison University in Virginia and also a friend. Besides teaching and researching in his field, he’s also written a lot about applying math to popular culture, including books on Sudoku and the perplexing Monty Hall Problem. But to me his biggest contribution has been his series of books and writings about creationism. Jason has not only immersed himself in creationist culture, attending lots of meetings to suss out the psyche of anti-evolutionists, but also written about it in both books and articles (see his 2012 book Among the Creationists: Dispatches from the Anti-Evolutionist Front Line).

He’s just come out with an article in the Skeptical Inquirer (see below) in which he summarizes how Intelligent Design creationists use mathematical arguments to show that evolution is impossible, and then Rosenhouse debunks the tactics they use. Jason writes very well and very clearly, so this article is accessible to the layperson. It’ll give you a strarter background on the creationists’ arguments (yes, IDers are creationists), and why those arguments re misguided.

Click to read (it’s free).

Jason explicates and then demolishes two ID arguments against evolution. Quotes from Jason are indented, my own prose is flush left.

1.) The probability of evolution producing complex features, like bacterial flagella, is almost nil. 

The ID argument rests on the idea that if the probability of an amino acid in a protein, say tyrosine, being in a specific position is small, then the probability of getting a protein of 100 amino acids with tyrosine in the right position and the other 19 amino acids in the other right positions is effectively zero. (They simply multiply probabilities for each site together.) But, as Jason shows, that’s not the way that evolution works. Proteins are built up step by step, with each step adopted only if it incrementally improves fitness. The probability-multiplying argument is so transparently false that I’m surprised people believe it, but of course most people don’t have a decent understanding of probability.

Jason:

However, this argument is premised on the notion that genes and proteins evolve through a process analogous to tossing a coin multiple times. This is untrue because there is nothing analogous to natural selection when you are tossing coins. Natural selection is a non-random process, and this fundamentally affects the probability of evolving a particular gene.

To see why, suppose we toss 100 coins in the hopes of obtaining 100 heads. One approach is to throw all 100 coins at once, repeatedly, until all 100 happen to land heads at the same time. Of course, this is exceedingly unlikely to occur. An alternative approach is to flip all 100 coins, leave the ones that landed heads as they are, and then toss again only those that landed tails. We continue in this manner until all 100 coins show heads, which, under this procedure, will happen before too long. The creationist argument assumes that evolution must proceed in a manner comparable to the first approach, when really it has far more in common with the second.

That’s a very good explanation.

IDers, however, have made the argument a bit more sophisticated:

Let us return to coin-tossing. Suppose we toss a coin 100 times, thereby producing a chaotic jumble of heads and tails. It was very unlikely that just that sequence would appear, but we do not suspect trickery. After all, something had to happen. But now suppose we obtained 100 Hs or a perfect alternation of Hs and Ts. Now we probably would suspect trickery of some kind. Such sequences are not only improbable but also match a recognizable pattern. ID proponents argue that it is the combination of improbability and matching a pattern that makes them suspect that something other than chance or purely natural processes are at work. They use the phrase “complex, specified information” to capture this idea. In this context, “complex” just means “improbable,” and “specified” means “matches a pattern.”

As applied to biology, the argument goes like this: Consider a complex, biological adaptation such as the flagellum used by some bacteria to propel themselves through liquid. The flagellum is a machine constructed from numerous individual proteins working in concert. Finding this exact functional arrangement of proteins is extremely unlikely to happen by chance. Moreover, they continue, the structure of the flagellum is strongly analogous to the sort of outboard motor we might use to propel a boat. Therefore, the flagellum exhibits both complexity and specificity, and it therefore must be the product of intelligent design.

That is, natural selection, say critics like William Dembski, can’t create “complex specified design”. But we have no idea what organismal features would imply intelligent design (“specificity”) rather than selection. Further, as for “complexity”, Jason says this:

The argument likewise founders on the question of complexity. According to ID proponents, establishing complexity requires carrying out a probability calculation, but we have no means for carrying out such a computation in this context. The evolutionary process is affected by so many variables that there is no hope of quantifying them for the purposes of evaluating such a probability.

In summary, any anti-evolutionist argument based on probability theory can simply be dismissed out of hand. There is no way to carry out a meaningful calculation, and adding “specificity” to the mix does nothing to improve the argument.

2.) Because mutations are degrading processes, much more likely to make DNA coding for a protein less adapted to the environment than more adaptive, there is no way that new genetic information can be created. Ergo, complexity, much less adaptation, can’t increase. rgo God—the Creator of Complexity. In some ways this resembles the old Second Law of Thermodynamics argument against evolution: entropy must increase, and evolution appears to violate entropy by making matter less random.  Thus we need God to get the entropy down.

The problem with that is, of course, the Second Law holds only in a closed system, but evolution occurs in an open system: the Earth in its surrounding universe. Evolution is fueled by radiation from the Sun, which involves an increase in entropy, and any decrease in entropy produced by evolution is more than compensated for by the increased entropy produced by generating evolution’s fuel: solar energy.  Ergo, in the whole system, the Second Law is obeyed.

There’s already one way known whereby new genetic “information” can increase: gene duplication.  Sometimes due to errors in replication, a gene is duplicated, and we have two copies instead of one (there are always two copies in a diploid genome, but I’m talking about what happens when a gene on one chromosome duplicates in addition. When this happens, there is an opportunity for that new copy of the gene to diverge in function from the old one, for the old one’s still around doing its thing. The new copy can do a new thing. Ergo, new information.  This in fact has happened a gazillion times in evolution: all of our globins, for instance (alpha, beta, fetal hemoglobin, and myoglobin) were produced by gene duplication and subsequent divergence. In Antarctic fish, an enzyme used to digest food has, after duplication, evolved into a blood antifreeze protein to allow them to inhabit waters below the freezing point.

Jason mentions gene duplication (I’m just giving examples), and then goes into the “No Free Lunch” ideas of Dembski and others, showing that these ideas irrelevant to the possibility of evolution.  I’ll let you read that part for yourself (read the whole thing!), and will just give two more quotes from Jason:

Even if we accept everything Dembski and his coauthors are saying about these theorems, this whole line of attack simply amounts to nothing. Most of us did not need difficult mathematical theorems to understand that Darwinian evolution can work only if nature has certain properties. The search problem confronted by evolution arises ultimately from the laws of physics, but it is well outside biology’s domain to wonder why those laws are as they are. Dembski and his cohorts argue that the fundamental constants of the universe encode information of a sort that can arise only from an intelligent source, but they have no more basis for this claim than they did for the comparable claim about genetic sequences.

He finishes like this:

Everyone agrees that complex adaptations require a special sort of explanation. Scientists argue that actual biological systems show copious evidence of having resulted through evolution by natural selection. Anti-evolutionists reject this claim, but the ensuing debate, such as it is, has nothing to do with mathematics. This makes you wonder why anti-evolutionists insist on padding their work with so much irrelevant and erroneous mathematical formalism. The answer is that their literature has far more to do with propaganda than it does with serious argument. Mathematics is unique in its ability to bamboozle a lay audience, making it well suited to their purposes. But for all its superficial sophistication, anti-evolutionary mathematics is not even successful at raising interesting questions about evolution.

Jason knows whereof he speaks, as he knows both math and evolution.

More bias in Scientific American, this time in a “news” article

January 15, 2022 • 1:00 pm

Scientific American has tendered a news piece in their “Mathematics” section, reporting on a schism in the math community. I’ve followed this schism for a while but haven’t written about it. As I understand it, what happened is that last October the Association for Mathematical Research (AMR) was formed, breaking away from the two older associations, the Mathematical Association of America (MAA) and the American Mathematical Society (AMS), primarily because the latter two societies were becoming too woke, trying to dilute the mathematical goals of their organization with social-justice considerations, considerations favoring the performative and “progressive” ideology we know too well.

While the article starts off okay, giving the facts above, it quickly devolves into somewhat of a hit piece on the new AMR for being racist and sexist. This is in line with the total lack of objectivity of Scientific American, which, as we all know, has diverted much of its mission to teach science so that it can further social justice, though in a misguided and ineffective way. In this piece, the bias of Sci. Am. is reflected in both the imbalance of quotations from pro- and anti-AMR people (much more from the latter) and in its own commentary and slant.

Now I tend to be opposed to the new direction Sci Am is taking, so I may be biased, but I don’t think I am: I think this article is what’s slanted, not me. But read it for yourself by clicking on the screenshot.  The dissing of the AMR starts with the subheading, where critics get their say without any mention of why the AMR was formed.

This bit is pretty accurate, as far as I know, though you can see a bit of pro-woke bias nosing in:

A new organization called the Association for Mathematical Research (AMR) has ignited fierce debates in the math research and education communities since it was launched last October. Its stated mission is “to support mathematical research and scholarship”—a goal similar to that proclaimed by two long-standing groups: the American Mathematical Society (AMS) and the Mathematical Association of America (MAA). In recent years the latter two have initiated projects to address racial, gender and other inequities within the field. The AMR claims to have no position on social justice issues, and critics see its silence on those topics as part of a backlash against inclusivity efforts. Some of the new group’s leaders have also spoken out in the past against certain endeavors to diversify mathematics. The controversy reflects a growing division between researchers who want to keep scientific and mathematical pursuits separate from social issues that they see as irrelevant to research and those who say even pure mathematics cannot be considered separately from the racism and sexism in its culture.

Then, throwing off the mantle of objectivity, the author goes full steam ahead. All quotes from the piece are indented:

Criticism of the AMR (selected bits)

With bias, harassment and exclusion widely acknowledged to exist within the mathematics community, many find it dubious that a professional organization could take no stance on inequity while purporting to serve the needs of mathematicians from all backgrounds. “It’s a hard time to be a mathematician,” says Piper H, a mathematician at the University of Toronto. In 2019 less than 1 percent of doctorates were awarded to Black mathematicians, and just 29 percent were awarded to women.

. . .Louigi Addario-Berry, a mathematician at McGill University in Montreal, wrote about the AMR on his blog. He told Scientific American he is speaking up because “I think this is an organization whose existence, development and flourishing will hurt a lot of members of the mathematical community who I respect. It is being founded by people who have publicly stated views I find harmful—both hurtful to me as an individual and detrimental to the creation of an inclusive and welcoming mathematical community.”

Hass responded in a statement to Scientific American: “The focus of the AMR is on supporting mathematical research and this goal benefits all members of the mathematics community.” But Addario-Berry questions how the AMR can be neutral on social justice issues when some of its leaders have previously taken strong public stances on some of these topics.

This is very strange. It’s like saying that the University of Chicago cannot be organizationally neutral on social-justice issues when many of its faculty have taken strong stands one way or the other. Can the author not conceive of an organization being officially politically neutral even though its members may have strong views? This isn’t rocket science. It’s just the University of Chicago.

There’s some discussion both ways about UC Davis math professor Abigail Thompson’s criticism of requiring diversity statements for faculty jobs (see my post here), and a note that Thompson is secretary of the new AMR. But that’s seems like an attempt to tarnish the AMR by picking out members who themselves opposed wokeness. It says nothing about the organization’s own stance, which is indeed neutrality. Thompson is also listed as one of the “current vice presidents of the American Mathematical Society” in her Wikipedia bio. But none of this really has to do with the issue at hand, except to try to cast aspersions on Thompson and, by extension, the AMR. But wait! There’s more!

Another AMR founding member and a member of its board of directors, Robion “Rob” Kirby, is a mathematician at the University of California, Berkeley. In a post entitled “Sexism in Mathematics???” on his Web site, he wrote, “People who say that women can’t do math as well as men are often called sexist, but it is worth remembering that some evidence exists and the topic is a legimate [sic] one, although Miss Manners might not endorse it.”

In fact, I don’t think that Kirby is right; as far as I knew, men and women in secondary school perform equally well in math, but the women excel in reading. Women like reading more than math, and thus they tend to go on more often to the humanities. Whatever is responsible for inequity between men and women, it’s not skill.

Or course conservatives are going to leave an organization disproportionately if it becomes too woke, for wokeness is the purview of the Left, not the Right. You don’t have to be a conservative to try to keep your discipline pure, but if you’re a liberal like me who doesn’t like performative wokeness, you’re going to have to live being associated with some politically inconvenient bedfellows. At any rate, the statement above doesn’t represent someone supporting the new AMR, it’s Sci Am’s attempt to denigrate it.

Then there’s this:

The AMS and the MAA have publicly acknowledged the need to work toward a more inclusive mathematical community. Last year an AMS task force released a 68-page report that, in the organization’s words, details “the historical role of the AMS in racial discrimination; and recommends actions for the AMS to take to rectify systemic inequities in the mathematics community.” In 2020 an MAA committee stated that the mathematics community must “actively work to become anti-racist” and “hold ourselves and our academic institutions accountable for the continued oppression of Black students, staff, and faculty.” It also addressed Black mathematicians specifically, saying, “We are actively failing you at every turn as a society and as a mathematics community. We kneel together with you. #BlackLivesMatter.”

In contrast, the AMR has not released any official statements about injustice.

Okay, that’s pretty snarky, but is followed by something even snarkier:

“I am supposed to believe, in the year 2021, that this omission is not itself an act of racism?” asks Piper H, who spoke to Scientific American late last year. “How am I, as a 40-year-old Black American mathematician, parent, and person who has paid a bit of attention to American history and American present, supposed to believe that AMR’s refusal to address the actual obstacles that real mathematicians face to doing mathematical research and scholarship is anything other than an insult and a mockery?”

This is pure Kendian mishigass: if your organization doesn’t make an explicitly anti-racist statement, then your organization is racist. Note that they add that Hass denies that the AMR’s silence on diversity is a message (see below).

. . . It’s not just a coincidence that the AMR was founded on the heels of a greater push for diversity within the AMS,” wrote Lee Melvin Peralta, a mathematics education graduate student at Michigan State University, in the November 16, 2021, newsletter of the Global Math Department, an organization of math educators. The AMR, Peralta added, “seems more like a separatist organization for those people who are striving for some kind of ‘purity’ within mathematics away from ‘impure’ considerations of race, gender, class, ability, sexual orientation, and socioeconomic status (among others).”

And, at the end of the article, there’s this parting shot:

Some of the AMR’s founding members have left the organization amid the controversy. “To create an organization to do something positive requires the trust and goodwill of the community that it wants to affect. And this is something that the AMR does not have at this point,” wrote Daniel Krashen, a mathematician at the University of Pennsylvania, in a November 14, 2021, Twitter thread. “I have no desire to negatively impact the mathematical community by my actions and words. I see that some people feel less safe and less heard by my actions, and for this I apologize. I have decided to withdraw my membership.”

Less safe? How has Krashen made anybody less safe or less heard? For crying out loud, this whole article is a megaphone handed to the critics of the AMR! Nobody has been silenced and the only harm has come to people’s feelings. (That said, I of course oppose those social conditions that have denied women or minorities entry into the math “pipeline.”)

Defense of the AMR:

Joel Hass, a mathematician at the University of California, Davis, and current president of the AMR, describes the group as “definitely focused on being inclusive.” He adds that the AMR “welcomes all to join us in supporting mathematical research and scholarship. In early 2022 we plan to open membership to anyone in the world who wishes to join us. There will be no fees or dues. By removing financial barriers to entry, we will make it easier to have participation from anyone across the world. Mathematical research is a truly global endeavor that transcends nation, creed and culture.”

. . . Hass denies that the AMR’s founding had anything to do with the antiracism push at the AMS or the MAA. “The changes in the research environment caused by the COVID pandemic revealed new opportunities for the development and communication of mathematical research, allowing for incorporation of new technologies and international activities,” he says. “We felt there was room for a new organization that would explore these.” Hass adds that “the AMS and MAA are wonderful organizations that we hope to work with, along with other organizations such as SIAM [Society for Industrial and Applied Mathematics], ACM [Association for Computing Machinery] and many non-U.S.-based groups.”

I think Hass is being disingenuous here, for what I’ve heard is that the AMR is a reaction to the wokeness of the other two organizations. I don’t see that as a sign of racism; I see it as a sign of trying to keep an objective discipline from being diverted into political pursuits.*********

So there we have it:  four mathematicians criticizing the AMR for racism/sexism or “harm”, and one defending its mission. That’s not to mention the way that Scientific American has structured the article, providing a critical sub-header for the title and ending with a critical slam.

I’m not by any means a fan of the views of all AMR members: in fact I’ve just criticized two statements of their members. But with this article, Sci. Am. is casting its lot in with the woke, as it always does. There is no rationale, they’re saying, for a mathematics organization that is not explicitly devoted to achieving Social Justice.

This is my view, which of course might be conditioned by my extreme dislike for the direction that Sci. Am. is taking. So read for yourself and let me know if the piece seems objective to you.