Fifty years ago today, fifty years ago this very hour in fact, at 9:32 AM EDT on July 16, 2019, Apollo 11 was launched.

Moonbound Apollo 11 clears the launch tower. NASA photo

And thus began a journey that, arguably, remains the greatest adventure in human history to date.

I was six years old in 1969, hooked on the novels of Jules Verne. With Apollo 11, Verne’s bold imagination became the reality of the day.

Galileo is the world’s third global satellite navigation system, built by the European Union, operating in parallel with the American GPS system and Russia’s GLONASS. It has been partially operational since 2016, with a full constellation if satellites expected to enter service this year.

But as of early Monday, July 15, Galileo has been down for nearly four days, completely inoperative in fact:

As of the time of this writing, no explanation is being offered, other than one article mentioning an unspecified issue with Galileo’s ground-based infrastructure.

It really is difficult to comprehend how such a failure can occur.

It is even more difficult to comprehend the silence, the lack of updates, explanations, or any information about the expected recovery.

And before I forget: Last week, wearing my release manager hat I successfully created a new version of Maxima, the open-source computer algebra system. As a result, Maxima is again named one of SourceForge’s projects of the week, for the week of June 10.

The release turned out to be more of an uphill battle than I anticipated, but in the end, I think everything went glitch-free.

Others have since created installers for different platforms, including Windows.

And I keep promising myself that when I grow up, I will one day understand exactly what git does and how it works, instead of just blindly following arcane scripts…

Here is a thought that has been bothering me for some time.

We live in a universe that is subject to accelerating expansion. Galaxies that are not bound gravitationally to our Local Group will ultimately vanish from sight, accelerating away until the combination of distance and increasing redshift will make their light undetectable by any imaginable instrument.

Similarly, accelerating expansion means that there will be a time in the very distant future when the cosmic microwave background radiation itself will become completely undetectable by any conceivable technological means.

In this very distant future, the Local Group of galaxies will have merged already into a giant elliptical galaxy. Much of this future galaxy will be dark, as most stars would have run out of fuel already.

But there will still be light. Stars would still occasionally form. Some dwarf stars will continue to shine for trillions of years, using their available fuel at a very slow rate.

Which means that civilizations might still emerge, even in this unimaginably distant future.

And when they do, what will they see?

They will see themselves as living in an “island universe” in an otherwise empty, static cosmos. In short, precisely the kind of cosmos envisioned by many astronomers in the early 1920s, when it was still popular to think of the Milky Way as just such an island universe, not yet recognizing that many of the “spiral nebulae” seen through telescopes are in fact distant galaxies just as large, if not larger, than the Milky Way.

But these future civilizations will see no such nebulae. There will be no galaxies beyond their “island universe”. No microwave background either. In fact, no sign whatsoever that their universe is evolving, changing with time.

So what would a scientifically advanced future civilization conclude? Surely they would still discover general relativity. But would they believe its predictions of an expanding cosmos, despite the complete lack of evidence? Or would they see that prediction as a failure of the theory, which must be remedied?

In short, how would they ever come into possession of the knowledge that their universe was once young, dense, and full of galaxies, not to mention background radiation?

My guess is that they won’t. They will have no observational evidence, and their theories will reflect what they actually do see (a static, unchanging island universe floating in infinite, empty space).

Which raises the rather unnerving, unpleasant question: To what extent exist already features in our universe that are similarly unknowable, as they can no longer be detected by any conceivable instrumentation? Is it, in fact, possible to fully understand the physics of the universe, or are we already doomed to never being able to develop a full picture?

I find this question surprisingly unnerving and depressing.

The Globe and Mail managed to publish today one of the saddest editorial cartoons I ever saw:

There really is nothing that I can add. The cartoon speaks for itself.

My research is unsupported. That is to say, with the exception of a few conference invitations when my travel costs were covered, I never received a penny for my research on the Pioneer Anomaly and my other research efforts.

Which is fine, I do it for fun after all. Still, in this day and age of crowdfunding, I couldn’t say no to the possibility that others, who find my efforts valuable, might choose to contribute.

Hence my launching of a Patreon page. I hope it is well-received. I have zero experience with crowdfunding, so this really is a first for me. Wish me luck.

Yes, it can get cold in Toronto. Usually not as cold as Ottawa, but winters can still be pretty brutal.

But this brutal, this late in the season?

Yes, according to The Weather Network earlier this morning, the temperature overnight will plummet to -59 degrees Centigrade next weekend.

Yikes. Where is global warming when we need it?

I run across this often. Well-meaning folks who read introductory-level texts or saw a few educational videos about physical cosmology, suddenly discovering something seemingly profound.

And then, instead of asking themselves why, if it is so easy to stumble upon these results, they haven’t been published already by others, they go ahead and make outlandish claims. (Claims that sometimes land in my Inbox, unsolicited.)

Let me explain what I am talking about.

As it is well known, the rate of expansion of the cosmos is governed by the famous Hubble parameter: $$H\sim 70~{\rm km}/{\rm s}/{\rm Mpc}$$. That is to say, two galaxies that are 1 megaparsec (Mpc, about 3 million light years) apart will be flying away from each other at a rate of 70 kilometers a second.

It is possible to convert megaparsecs (a unit of length) into kilometers (another unit of length), so that the lengths cancel out in the definition of $$H$$, and we are left with $$H\sim 2.2\times 10^{-18}~{\rm s}^{-1}$$, which is one divided by about 14 billion years. In other words, the Hubble parameter is just the inverse of the age of the universe. (It would be exactly the inverse of the age of the universe if the rate of cosmic expansion was constant. It isn’t, but the fact that the expansion was slowing down for the first 9 billion years or so and has been accelerating since kind of averages things out.)

And this, then, leads to the following naive arithmetic. First, given the age of the universe and the speed of light, we can find out the “radius” of the observable universe:

$$a=\dfrac{c}{H},$$

or about 14 billion light years. Inverting this equation, we also get $$H=c/a$$.

But the expansion of the cosmos is governed by another equation, the first so-called Friedmann equation, which says that

$$H^2=\dfrac{8\pi G\rho}{3}.$$

Here, \rho is the density of the universe. The mass within the visible universe, then, is calculated as usual, just using the volume of a sphere of radius $$a$$:

$$M=\dfrac{4\pi a^3}{3}\rho.$$

Putting this expression and the expression for $$H$$ back into the Friedmann equation, we get the following:

$$a=\dfrac{2GM}{c^2}.$$

But this is just the Schwarzschild radius associated with the mass of the visible universe! Surely, we just discovered something profound here! Perhaps the universe is a black hole!

Well… not exactly. The fact that we got the Schwarzschild radius is no coincidence. The Friedmann equations are, after all, just Einstein’s field equations in disguise, i.e., the exact same equations that yield the formula for the Schwarzschild radius.

Still, the two solutions are qualitatively different. The universe cannot be the interior of a black hole’s event horizon. A black hole is characterized by an unavoidable future singularity, whereas our expanding universe is characterized by a past singularity. At best, the universe may be a time-reversed black hole, i.e., a “white hole”, but even that is dubious. The Schwarzschild solution, after all, is a vacuum solution of Einstein’s field equations, wereas the Friedmann equations describe a matter-filled universe. Nor is there a physical event horizon: the “visible universe” is an observer-dependent concept, and two observers in relative motion or even two observers some distance apart, will not see the same visible universe.

Nonetheless, these ideas, memes perhaps, show up regularly, in manuscripts submitted to journals of dubious quality, appearing in self-published books, or on the alternative manuscript archive viXra. And there are further variations on the theme. For instance, the so-called Planck power, divided by the Hubble parameter, yields $$2Mc^2$$, i.e., twice the mass-energy in the observable universe. This coincidence is especially puzzling to those who work it out numerically, and thus remain oblivious to the fact that the Planck power is one of those Planck units that does not actually contain the Planck constant in its definition, only $$c$$ and $$G$$. People have also been fooling around with various factors of $$2$$, $$\tfrac{1}{2}$$ or $$\ln 2$$, often based on dodgy information content arguments, coming up with numerical ratios that supposedly replicate the matter, dark matter, and dark energy content.

I almost forgot: a couple of months ago, I was interviewed over the telephone by a journalist who wanted to know my thoughts about one of my favorite moments in manned space exploration: The Apollo 8 “Genesis” moment, the reading of the opening verses of the Old Testament, on Christmas Day, 1968, by the astronauts of Apollo 8 as their spacecraft emerged from behind the Moon.

Today, something reminded me of this interview and I did a quick search. Lo and behold, there it is: My words, printed in The Boston Globe on December 23, 2018:

“It was a beautiful moment, and Genesis is part of our Western cultural heritage,” said Viktor Toth, an atheist and a senior research fellow at Carleton University in Ottawa, Canada, who played the lead role in the investigation of the Pioneer Anomaly, the mysterious acceleration of the Pioneer 10 and 11 spacecrafts in deep space. “This was an awe-inspiring thing: Human beings for the first time cut off from the Earth, and then they reemerged and saw the Earth again. The message was entirely appropriate.”

Though shortened, this pretty accurately reflects what I actually said during that roughly 10-minute conversation with the journalist.

Even as China was celebrating the first successful landing of a spacecraft on the far side of the Moon, NASA’s New Horizons continued to radio back data from its New Year’s Day encounter with Ultima Thule: a strange, “contact binary” asteroid in the Kuiper belt, far beyond Pluto.

Ultima Thule will remain, for the foreseeable future, the most distant celestial object visited by spacecraft. While there is the odd chance that New Horizons may find another target within range (as determined by the on-board fuel available, which limits trajectory corrections, and the aging of its nuclear power source that provides electricity on board), chances are it won’t happen, and it won’t be until another deep space probe is launched, quite possibly decades from now, before we get a chance to see a world as distant as Ultima Thule.

Another piece of news from the New Horizons project is that so far, the probe found no moon orbiting Ultima Thule. No Moon At All.

OK, the far side of the Moon is not really dark, but it is kind of hard to see. But now, from the department of unqualified good news: China successfully landed its spacecraft, Chang’e 4 (named after the Chinese Moon goddess), on the dark side of the Moon, and it has already sent us back some pictures.

This is big. Really big. To make it happen, China first had to launch a lunar orbiter, Queqiao (“Magpie bridge”), in order to maintain communication with the lander. And being on the far side of the Moon, the lander is completely shielded from radio signals from the Earth, which means an unprecedented opportunity to study radio signals of extrasolar origin.

Chang’e 4 also carried a rover, Yutu-2, which has since been deployed.

By any reasonable measure, this is a huge success for China’s space program, and for humanity overall. Hopefully, both lander and rover will remain operational and able to fulfill their scientific objectives.

Today, I answered a question on Quora about the nature of $$c$$, the speed of light, as it appears in the one equation everyone knows, $$E=mc^2.$$

I explained that it is best viewed as a conversion factor between our units of length and time. These units are accidents of history. There is nothing fundamental in Nature about one ten millionth the distance from the poles to the equator of the Earth (the original definition of the meter) or about one 86,400th the length of the Earth’s mean solar day. These units are what they are, in part, because we learned to measure length and time long before we learned that they are aspects of the same thing, spacetime.

And nothing stops us from using units such as light-seconds and seconds to measure space and time; in such units, the value of the speed of light would be just 1, and consequently, it could be dropped from equations altogether. This is precisely what theoretical physicists often do.

But then… I commented that something very similar takes place in aviation, where different units are used to measure horizontal distance (nautical miles, nmi) and altitude (feet, ft). So if you were to calculate the kinetic energy of an airplane (measuring its speed in nmi/s) and its potential energy (measuring the altitude, as well as the gravitational acceleration, in ft) you would need the ft/nmi conversion factor of 6076.12, squared, to convert between the two resulting units of energy.

As I was writing this answer, though, I stumbled upon a blog entry that discussed the crazy, mixed up units of measure still in use worldwide in aviation. Furlongs per fortnight may pretty much be the only unit that is not used, as just about every other unit of measure pops up, confusing poor pilots everywhere: Meters, feet, kilometers, nautical miles, statute miles, kilograms, pounds, millibars, hectopascals, inches of mercury… you name it, it’s there.

Part of the reason, of course, is the fact that America, alone among industrialized nations, managed to stick to its archaic system of measurements. Which is another historical accident, really. A lot had to do with the timing: metric transition was supposed to take place in the 1970s, governed by a presidential executive order signed by Gerald Ford. But the American economy was in a downturn, many Americans felt the nation under siege, the customary units worked well, and there was a conservative-populist pushback against the metric system… so by 1982, Ronald Reagan disbanded the Metric Board and the transition to metric was officially over. (Or not. The metric system continues to gain ground, whether it is used to measure bullets or Aspirin, soft drinks or street drugs.)

Yet another example similar to the metric system is the historical accident that created the employer-funded healthcare system in the United States that American continue to cling to, even as most (all?) other advanced industrial nations transitioned to something more modern, some variant of a single-payer universal healthcare system. It happened in the 1920s, when a Texas hospital managed to strike a deal with public school teachers in Dallas: For 50 cents a month, the hospital picked up the tab of their hospital visits. This arrangement became very popular during the Great Depression when hospitals lost patients who could not afford their hospital care anymore. The idea came to be known as Blue Cross. And that’s how the modern American healthcare system was born.

As I was reading this chain of Web articles, taking me on a tour from Einstein’s $$E=mc^2$$ to employer-funded healthcare in America, I was reminded of a 40-year old British TV series, Connections, created by science historian James Burke. Burke found similar, often uncanny connections between seemingly unrelated topics in history, particularly the history of science and technology.

A quote from 50 years ago is the most appropriate one tonight, considering that our world is just as troubled as the world of 1968:

And from the crew of Apollo 8, we close with good night, good luck, a Merry Christmas – and God bless all of you, all of you on the good Earth.

I know, I know, it’s not for the weather that we love Canada.

Still… plunging into winter in the middle of November?

The other day, a conservative friend of mine sent me a link. It was to a paper purportedly demonstrating that conservatives were more ideologically diverse than their liberal counterparts.

I found this result surprising, rather striking to be honest, and contrary to my own experience.

The study, as it appears, has not been published yet but it is available in an online manuscript archive. It certainly appears thorough. So how could it come to its striking conclusions?

I think I figured out the answer when I looked at the appendices, which provide details on how the research was conducted. Here is a set of questions that were used in two of the four studies discussed in the paper:

1. It is the responsibility of political leaders to promote programs that will help close the income gap between the rich and the poor.
2. There is no “right way” to live life; instead, everyone must create a way to live which works best for them.
3. Spending tax dollars on “abstinence education” rather than “sex education” is more effective in curbing teen pregnancy.
4. The more money a person makes in America, the more taxes he/she should pay.
5. The use of our military strength makes the United States a safer place to live.
6. America would be a better place if people had stronger religious beliefs.
7. The traditional (male/female) two-parent family provides the best environment of stability, discipline, responsibility and character.
8. America’s domestic policy should do more to ensure that living and working conditions are equal for all groups of people.
9. Flag burning should be illegal.
10. Our society is set up so that people usually get what they deserve.
11. Taxation should be used to fund social programs.
12. Gay marriage threatens the sanctity of marriage.

When I look at this list, it is clear to me that questions 1, 2, 4, 8 and 11 are standard “liberal” pushbutton items, whereas the rest are “conservative” in nature.

But look more closely. Items 1, 2, 4, 8 and 11 are, insofar as liberal views are concerned, very mild and mainstream. Closing the income gap? Taxing income? (Not even a mention of progressive taxation.) Social programs? I know of no liberal who would disagree with these broad concepts. In fact, I can think of many conservatives who would readily subscribe at least to some of these ideas.

Now look at the conservative questionnaire items. Flag burning? A great many conservatives in the US believe firmly that this right is strongly protected by the First Amendment. Gay marriage? Sure, it’s an issue for some, but for many conservatives, it’s either something that they are neutral about (or may even support it) or often, it’s an issue that comes up more as a matter of states’ rights vs. the federal government, without a priori opposing the idea.

In short, when I looked closely I realized that whereas the “liberal” questions accurately reflect the liberal mainstream, the “conservative” questions are more representative of a liberal caricature of what conservatives are thought to be. By way of example, the “liberal” analog of some of these “conservative” questions would be something like, “Research that demonstrates differences on the basis of gender or race should be banned”, or some similar conservative caricature of liberal “identity politics” or “social justice warriors”.

The results, therefore, are not surprising after all. Since most liberals agree on mainstream liberal ideas, the liberal side comes across as ideologically monolithic; and since many conservatives take issue with narrowly defined, often religiously motivated line items, they come across as more diverse, more heterogeneous.

Ironically, then, the liberal bias of the researchers resulted in a paper that, contrary to their expectations, appeared to show that the conservative side is more ideologically tolerant than their liberal counterparts. In reality, though, I think the paper merely demonstrates the garbage-in-garbage-out principle that is so well known in computer science: when your research is flawed, your results will be just as flawed.

Allow me to preface this post with the following: I despise Donald J. Trump, the infantile, narcissistic, racist, misogynist “leader of the free world” who is quite possibly a traitor and may never have become president without help from his Russian buddy Putin. Also, when it comes to matters that I consider important, I am a small-l liberal; I support, for instance, LGBTQ rights, the right to have an abortion, or the legalization of cannabis, to name a few examples. I celebrate the courage of #MeToo victims. I reject racism and misogyny in all forms, open or covert.

Yet I am appalled by some of the things that happened lately in academic circles, sadly justifying the use of the pejorative term “SJW” (social justice warrior) that is so popular on the political right. A few specific cases:

1. Last month, the European nuclear research institution CERN held a workshop with the title, High Energy Physics and Gender. One of the speakers was the Italian physicist Alessandro Strumia. Strumia offered a semi-coherent presentation, whimsically titled Experimental test of a new global discrete symmetry. In it, Strumia argued that men are over-represented in physics because they perform better. In the presentation, he offered some genuine data, but he also offered what may be construed as a personal attack, in the form of a short list of three names: those of two women who were hired by Italy’s nuclear research institute INFN, along with Strumia’s, who was rejected despite his much higher citation count. Strumia’s research is questionable. His conclusions may be motivated by his bitterness over his personal failures. His approach may be indefensible. All of which would be justification to laugh at him during his presentation, to not accept his work for publication in the workshop proceedings, and perhaps to avoid inviting him in the future.

But CERN went a lot further. They retroactively removed Strumia’s presentation altogether from the conference archive, and have since administratively sanctioned him, putting his future career as a physicist in question. When this response was questioned by some, there came the retroactive justification that his one slide containing the three names constitutes a “personal attack”, violating CERN policy.

I don’t agree with Strumia. I don’t like him or respect his research. But I have to ask: If he is not allowed to offer his views at a conference dedicated to “high-energy physics and gender” without fear of severe repercussions, where can he?

Now you might ask why he should be given a platform at all. Because this is (supposedly) science. And science thrives on criticism and controversial views. If we only permit views that preach to the choir, so to speak, science dies. I’d much rather risk getting offended by clowns like Strumia from time to time.

2. Meanwhile, a few weeks ago, we learned of Helen Pluckrose, James Lindsay and Peter Boghossian, who prepared and submitted 20 completely bogus papers to reputably social science journals. Here are a few gems:
• The paper titled Human Reactions to Rape Culture and Queer Performativity in Urban Dog Parks in Portland, Oregon argues that dog parks are “rape-condoning” places of rampant “canine rape culture”. Accepted, published and recognized for excellence in the journal Gender, Place and Culture.
• The paper, Going in Through the Back Door: Challenging Straight Male Homohysteria and Transphobia through Receptive Penetrative Sex Toy Use argues that heterosexual men should practice anal self-penetration using sex toys in order to decrease transphobia and increase feminist values. Accepted and published in Sexuality & Culture.
• The paper, An Ethnography of Breastaurant Masculinity: Themes of Objectification, Sexual Conquest, Male Control, and Masculine Toughness in a Sexually Objectifying Restaurant demonstrates how papers, even when they rely on made-up bogus data, are accepted when they problematize the attraction of heterosexual males to women. Accepted and published in Sex Roles.
• The paper with the ominous title, Our Struggle is My Struggle: Solidarity Feminism as an Intersectional Reply to Neoliberal and Choice Feminism was accepted for publication in the journal Affilia, despite the fact that it is just a paraphrasing of Adolf Hitler’s opus Mein Kampf (My struggle), with feminist and grievance-related buzzwords replacing Nazi hate terms.
1. How could such nonsensical papers be accepted for publication? Perhaps because life, in this case, imitates art: because of papers like those written by Rochelle Gutiérrez, who apparently believes that mathematics education as currently practiced is just a vehicle to spread white supremacism. In her paper, When Mathematics Teacher Educators Come Under Attack (published by the journal Mathematics Teacher Educator of the National Council of Teachers of Mathematics) she argues (citing her earlier work) that there exists “a direct link between White supremacist capitalist patriarchy and mathematics”. In an earlier paper, she introduces her invention, “Mathematx” (supposedly an ethnically neutral, LGBTQ-friendly alternative to the white supremacist term “mathematics”), with the intent to “underscore with examples from biology the potential limitations of current forms of mathematics for understanding/interacting with our world and the potential benefits of considering other-than-human persons as having different knowledges to contribute.” The reader might be forgiven if they thought that these were just further hoax papers by Pluckrose, Lindsay and Boghossian, but nope; these papers are for real, penned by an author who plays an influential role in the shaping of mathematics education in the United States.
2. Meanwhile, another paper has been “disappeared” in a manner not unlike how persons were “disappeared” in communist or fascist dictatorships. Theodore P. Hill’s paper, An Evolutionary Theory for the Variability Hypothesis, discusses the mathematical background of what has been known as the “greater male variability hypothesis”: an observation, dating back to Charles Darwin’s times, that across a multitude of species, males often show greater variability in many traits than females. (Simply put, this may mean that a given group of males may contain more idiots and more idiot savants than an equal size group of females.)

Unlike Strumia, Hill does not appear to have a personal agenda. The stated goal of the paper was neither to promote nor to refute the idea but to see whether a simple mathematical basis might exist that explains it. After being rejected (even following initial acceptance) by other journals, it was finally published in the New York Journal of Mathematics, only to be taken down (its page number and identifier assigned to a completely different paper) three days later after the editors received a complaint and a threat of losing support.

One of the justifications for this paper’s removal (and for these types of actions in general) is that such material may discourage young women from STEM fields. Apart from the intellectual dishonesty of removing an already published paper due to political pressure, I think this is also the ultimate form of covert sexism. The message to young women who are aspiring engineers and scientists is, “You, womenfolk, are too weak, too innocent to be able to think critically and reject ideological bias masquerading as science. So let us come and defend you, by ensuring that you are not exposed to vile ideas that your fragile little minds cannot handle.”

I call these incidents “irritants” when it comes to free speech.

On the one hand, publications dedicated to social science and education publish even the most outrageously bogus research so long as it kowtows to the prevailing sociopolitical agenda.

On the other hand, obscure research is thrust into the spotlight by intolerant “SJW”-s who seek to administratively suppress ideas that they find offensive. While this goal is technically accomplished (Strumia’s presentation and Hill’s paper were both successfully “unpublished”), in reality they achieve the exact opposite: they expose these authors to a much greater audience than they otherwise would have enjoyed. The message, meantime, to those they purportedly protect (e.g., women, minorities) is one of condescension: these groups apparently lack the ability to think critically and must be protected from harmful thoughts by their benevolent superiors.

Beyond all that, these actions also have negative consequences on academic life overall. In addition to suppressing controversial research, they may also lead to self-censorship. Indeed, I am left to wonder: Would I have the courage to write this blog entry if I myself had an academic career to worry about?

Last but not least, all this is oil on the fire. Those on the right, fans of Jordan Peterson and others, who are already convinced that the left is dominated by intolerant “SJW”-s, see their worst fears confirmed by these irritants, and thus their hostility increases towards the scientific establishment (including climate science, political economics, genuine social science research on refugees and migration, health, sexual education, etc.) with devastating consequences for all of us living on this planet.

If we truly believe in our small-l liberal values, it includes defending free speech even when it is vile speech. It also includes respecting others, including women and minorities, not misguidedly protecting them from hurtful ideas that they are supposedly too weak and fragile to handle. And it includes defending the freedom of scientific inquiry even if it is misused by self-absorbed losers. After all, if we can publish the nonsensical writings of Gutiérrez, surely the world won’t come to an end if Hill’s paper is published or if Strumia’s presentation remains available on the CERN workshop archive.

Just got back from The Perimeter Institute, where I spent three very short days.

I had good discussions with John Moffat. I again met Barak Shoshany, whom I first encountered on Quora. I attended two very interesting and informative seminar lectures by Emil Mottola on quantum anomalies and the conformal anomaly.

I also gave a brief talk about our research with Slava Turyshev on the Solar Gravitational Lens. I was asked to give an informal talk with no slides. It was a good challenge. I believe I was successful. My talk seemed well received. I was honored to have Neil Turok in the audience, who showed keen interest and asked several insightful questions.

I just watched a news conference held by the University of Waterloo, on account of Donna Strickland being awarded the Nobel prize in physics.

This is terrific news for Canada, for the U. of Waterloo, and last but most certainly not least, for women in physics.

Heartfelt congratulations!

Years ago, I accepted the role of release manager for the Maxima computer algebra system. It proved to be more laborious than I assumed (mostly for two things: assembling changelogs, and dealing with build glitches) but it still has its upside. Right now, it is my pleasure to announce that Maxima 5.42 has been released on the unsuspecting public. Enjoy!

Michael Atiyah, 89, is one of the greatest living mathematicians. Which is why the world pays attention when he claims to have solved what is perhaps the greatest outstanding problem in mathematics, the Riemann hypothesis.

Here is a simple sum: $$1+\frac{1}{2^2}+\frac{1}{3^2}+…$$. It is actually convergent: The result is $$\pi^2/6$$.

Other, similar sums also converge, so long as the exponent is greater than 1. In fact, we can define a function:

\begin{align*}\zeta(x)=\sum\limits_{i=1}^\infty\frac{1}{i^x}.\end{align*}

Where things get really interesting is when we extend the definition of this $$\zeta(x)$$ to the entire complex plane. As it turns out, its analytic continuation is defined almost everywhere. And, it has a few zeros, i.e., values of $$x$$ for which $$\zeta(x)=0$$.

The so-called trivial zeros of $$\zeta(x)$$ are the negative even integers: $$x=-2,-4,-6,…$$. But the function also has infinitely many nontrivial zeros, where $$x$$ is complex. And here is the thing: The real part of all known nontrivial zeros happens to be $$\frac{1}{2}$$, the first one being at $$x=\frac{1}{2}+14.1347251417347i$$. This, then, is the Riemann hypothesis: Namely that if $$x$$ is a non-trivial zero of $$\zeta(x)$$, then $$\Re(x)=\frac{1}{2}$$. This hypothesis baffled mathematicians for the past 130 years, and now Atiyah claims to have solved it, accidentally (!), in a mere five pages. Unfortunately, verifying his proof is above my pay grade, as it references other concepts that I would have to learn first. But it is understandable why the mathematical community is skeptical (to say the least).

A slide from Atiyah’s talk on September 24, 2018.

What is not above my pay grade is analyzing Atiyah’s other claim: a purported mathematical definition of the fine structure constant $$\alpha$$. The modern definition of $$\alpha$$ relates this number to the electron charge $$e$$: $$\alpha=e^2/4\pi\epsilon_0\hbar c$$, where $$\epsilon_0$$ is the electric permeability of the vacuum, $$\hbar$$ is the reduced Planck constant and $$c$$ is the speed of light. Back in the days of Arthur Eddington, it seemed that $$\alpha\sim 1/136$$, which led Eddington himself onto a futile quest of numerology, trying to concoct a reason why $$136$$ is a special number. Today, we know the value of $$\alpha$$ a little better: $$\alpha^{-1}\simeq 137.0359992$$.

Atiyah produced a long and somewhat rambling paper that fundamentally boils down to two equations. First, he defines a new mathematical constant, denoted by the Cyrillic letter $$\unicode{x427}$$ (Che), which is related to the fine structure constant by the equation

\begin{align*}\alpha^{-1}=\frac{\pi\unicode{x427}}{\gamma},\tag{1.1*}\end{align*}

where $$\gamma=0.577…$$ is the Euler–Mascheroni constant. Second, he offers a definition for $$\unicode{x427}$$:

\begin{align*}\unicode{x427}=\frac{1}{2}\sum\limits_{j=1}^\infty 2^{-j}\left(1-\int_{1/j}^j\log_2 x~dx\right).\tag{7.1*}\end{align*}

(The equation numbers are Atiyah’s; I used a star to signify that I slightly simplified them.)

Atiyah claims that this sum is difficult to calculate and then goes into a long-winded and not very well explained derivation. But the sum is not difficult to calculate. In fact, I can calculate it with ease as the definite integral under the summation sign is trivial:

\begin{align*}\int_{1/j}^j\log_2 x~dx=\frac{(j^2+1)\log j-j^2+1}{j\log 2}.\end{align*}

After this, the sum rapidly converges, as this little bit of Maxima code demonstrates (NB: for $$j=1$$ the integral is trivial as the integration limits collapse):

(%i1) assume(j>1);
(%o1)                               [j > 1]
(%i2) S:1/2*2^(-j)*(1-integrate(log(x)/log(2),x,1/j,j));
log(j) + 1
---------- + j log(j) - j
(- j) - 1          j
(%o2)             2          (1 - -------------------------)
log(2)
(%i3) float(sum(S,j,1,50));
(%o3)                         0.02944508691740671
(%i4) float(sum(S,j,1,100));
(%o4)                         0.02944508691730876
(%i5) float(sum(S,j,1,150));
(%o5)                         0.02944508691730876
(%i6) float(sum(S,j,1,100)*%pi/%gamma);
(%o6)                         0.1602598029967022


Unfortunately, this does not look like $$\alpha^{-1}=137.0359992$$ at all. Not even remotely.

So we are all left to guess, sadly, what Atiyah was thinking when he offered this proposal.

We must also remember that $$\alpha$$ is a so-called “running” constant, as its value depends on the energy of the interaction, though presumably, the constant in question here is $$\alpha$$ in the infrared limit, i.e., at zero energy.