Happy Quaternions Day

"Here as he walked by [in Dublin] on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i² = j² = k² = ijk = -1 & cut it on a stone of this bridge."

It's the 175th anniversary.

Moose Fight

For Moose Friday, from CBC New Brunswick, a video of two male moose battling it out.

Numberphile Mentions Our Paper

You have to look really quickly, but the youtube channel numberphile briefly mentions our recent paper on sums of palindromes:

Our paper appears onscreen at 3:30.

Columnists Go Ga-Ga Over Reagan Letter that Demonstrates What a Tool He Was

Karen Tumulty discovered a previously unpublished 1982 letter written by Ronald Reagan to his father-in-law, Loyal Davis, shortly before Davis's death. She, like many other columnists, think this illustrates what a wonderful guy Reagan was. Michael Gerson gushed, "This letter is remarkable and revealing. I am so grateful that Karen found it." Peter Wehner called it a "rather remarkable/moving historical document". Ron Fournier sighed, "What a beautiful letter". Glenn Kessler said, "Such a remarkable find. Pause the Twitter feuds for a moment and glimpse the personal faith of a president."

I think it's an interesting find, but not for the reasons that Tumulty et al. do. I think it illustrates at least three significant deficiencies in Reagan's character that many in the public don't know about (but anyone who followed his career closely knows all too well).

First, Reagan was just not that bright, and showed signs of senility in his second term. As Jonathan Chait wrote,

Lou Cannon’s biography describes President Reagan frequently misidentifying members of his own Cabinet, describing movie scenes as though they were real, changing his schedule in order to follow the advice of an astrologer, and bringing up a science-fiction movie, in which aliens cause the Soviets and Americans to come together, with such frequency that Colin Powell would joke to his staffers, “Here come the little green men again.” As Cannon concluded, “The sad, shared secret of the Reagan White House was that no one in the presidential entourage had confidence in the judgment or the capacities of the president.”

The letter confirms it. Reagan didn't know the difference between "prophesy" (the verb) and "prophecy" (the noun), and thought the correct plural was "prophesys".

Second, Reagan never let actual facts get in the way of a good story. Truth was unimportant to him. Again, anyone who's actually followed his career already knows this, but the general public doesn't -- they saw him as a genial, reliable grandfather figure. But as Stephen Greenspan wrote in Annals of Gullibility:

Many of these stories [of Reagan] were embellished or, quite typically, completely made up. One example is a story Reagan told about a football game between his high school from Dixon, Illinois, and a rival team from Mendota. In this story, the Mendota players yelled for a penalty at a crucial point in the game. The official had missed the play and asked Reagan what had happened. Reagan's sense of sports ethics required him to tell the truth, Dixon was penalized, and went on to lose the game by one touchdown. Wonderful story, except that it never happened.

This aspect of Reagan's character is also illustrated in the letter. He refers to "one hundred and twenty three specific prophesys [sic] about his [Jesus'] life all of which came true."

The claim that aspects of Jesus' life were correctly and miraculously foretold is a common one among Christian evangelicals. Oddly enough, however, the specific number of fulfilled prophecies varies widely from author to author. A google search gives "more than 300", "over 400", "hundreds", "191", "68", and many similar claims. However, most of these so-called prophecies can be dismissed right away because (a) they were not prophecies or (b) they actually referred to something other than Jesus or (c) they were extremely obscure or vague or (d) their correctness is seriously disputed.

The few that remain that might well be true because Jesus (assuming he existed) deliberately chose to take actions based on what the Old Testament said. In this case, the prophecy is correct, but not for any miraculous reason.

And of course, the value of true prophecies is negated by the prophecies that were falsified. One of the most important of Jesus' predictions -- (in Matthew 24) "Verily I say unto you, This generation shall not pass, till all these things be fulfilled." -- was falsified. None of the things Jesus claimed would happen occurred in the generation after his lifetime. The amount of ink Christians have expended trying to excuse this failed prophecy could probably fill a dozen swimming pools.

I doubt very much that Reagan investigated his 123 claims. He was not a scholar or expert in the Bible. Almost certainly he was just repeating some claim he had once heard -- this would be in line with other stories about Reagan, who had a large number of half-remembered quips and anecdotes he liked to relate, without concern for whether they were true.

Third --- and this is the most damning for me --- what the letter illustrates is the willingness of Reagan to take advantage of someone's pain and suffering to ram his religious beliefs down the throat of a dying man. Civilized people do not expect others to share their religious beliefs, and do not evangelize to vulnerable people. It is rude and it is grotesque and it is contemptible.

If, dear reader, you are a Christian and you have trouble understanding my point of view, let us try a thought experiment. Suppose you were on your deathbed, and you were very worried because, in your religion, the sins you know that you committed would likely condemn you to an afterlife of eternal damnation. Suppose I, your atheist relative, tried to console you by saying, "Look, your beliefs about Hell are all nonsense. You are not going to experience eternal damnation because THERE IS NO HELL. No heaven, either, by the way." Would you be grateful? My guess is no, but rest assured -- I would not do such a thing.

There are other aspects of Reagan's character on exhibit in his letter -- a lack of judgment, a deficiency of skepticism, and an overwhelming gullibility. But I think I've said enough: the letter is an appalling document. The fact that people celebrate it as praiseworthy indicates a fundamental sickness at the heart of modern Christian America.

I Did Warn You

When I read the latest dreck from the "Walter Bradley Center for Natural and Artificial Intelligence", all I could think was: I did warn you.

Of course, it didn't really take that much cleverness. The "Center" is a project of the Discovery Institute, a think tank so committed to dissembling about evolution that it's often been called the "Dishonesty Institute". And, as I pointed out, the folks working at the "Center" aren't exactly luminaries in the area they purport to critique.

This latest column is by Michael Egnor, a surgeon whose arrogance (as we've seen many times before) is only exceeded by his ignorance. Despite knowing nothing about computer science, Egnor tries to explain what machine learning is. The results are laughable.

Egnor starts by making an analogy between a book and a computer. He says a book "is a tool we use to store and retrieve information, analogous in that respect to a computer". But this comparison misses the single most essential feature of a computer: it doesn't just store and retrieve information, it processes it. A book made of paper typically does not; the words are the same each time you look at it.

Egnor goes on to construct an analogy where the book's binding cracks preferentially where people use it. But to be a computer you need more kinds of processing capabilities than cracked bindings. Not just any processing; there's a reason why machines like the HP-35, despite their ability to do trig functions and exponentials, were called "calculators" and not "computers". To be genuinely considered a "computer", a machine should be able to carry out basic operations such as comparisons and conditional branching. And some would say that a computer isn't a real computer until it can simulate a Turing machine. A book with a cracked binding isn't even close.

Egnor goes on to elaborate on his confusion. "The paper, the glue, and the ink are the book's hardware. The information in the book is the software." Egnor clearly doesn't understand computers! Software specifies actions to be taken by the computer, as a list of commands. But a book doesn't typically specify any actions, and if it does, those actions are not carried out by the "paper" or "glue" or "ink". If anything carries out those actions, it is the reader of the book. So the book's hardware is actually the person reading the book. Egnor's analogy is all wrong.

Egnor claims that computers "don't have minds, and only things with minds can learn". But he doesn't define what he means by "mind" or "learn", so we can't evaluate whether this is true. Most people who actually work in machine learning would dispute his claim. And Egnor contradicts himself when he claims that machine learning programs "are such that repeated use reinforces certain outcomes and suppresses other outcomes", but that nevertheless this isn't "learning". Human learning proceeds precisely by this kind of process, as we know from neurobiology.

Finally, Egnor claims that "it is man, and only man, who learns". This will be news to the thousands of researchers who study learning in animals, and have done so for decades.

When a center is started by people with a religious axe to grind, and staffed by people who know little about the area they purport to study, you're guaranteed to get results like this. Computer scientists have a term for this already: GIGO.

Robert Marks: Four Years and Still No Answer -- and More Baylor Hijinks

Once upon a time, the illustrious Baylor professor Robert Marks II made the following claim: "we all agree that a picture of Mount Rushmore with the busts of four US Presidents contains more information than a picture of Mount Fuji".

I don't agree, so I asked the illustrious Marks for a calculation or other rationale supporting this claim.

After three months, no reply. So I asked again.

After six months, no reply. So I asked again.

After one year, no reply. So I asked again.

After two years, no reply. So I asked again.

After three years, no reply. So I asked again.

Now it's been four years. Still no reply.

The illustrious Marks also recently supervised a Ph. D. thesis of Eric Michael Holloway. In it, the author apparently makes some dubious claims. He claims that "meaningful information...cannot be made by anything deterministic or stochastic". But if you want to actually read this Ph. D. thesis and learn how this startling claim is proven, you're out of luck. And why is that? It's because Eric Holloway has imposed a 5-year embargo on his thesis, meaning that no one can read it for five years, unless Eric Holloway approves. And when I asked to see a copy, I was refused.

Now, if there were some shenanigans going on -- for example, if a Ph. D. thesis were of such low quality that you wouldn't want anyone else to know about it -- what better way to hide that fact than to impose a ridiculously lengthy embargo? Perhaps an embargo so long that the supervisor would be safely retired by then and not subject to any investigation or sanction?

Then again, perhaps Eric Holloway is just following the example of his illustrious supervisor, who is adept at ducking questions for years.

Creationist Physicist Doesn't Understand Mathematics, Either

If there's one consistent aspect of creationism, it's that people lacking understanding and training are put forth as experts. Here we have yet another example, from the creationist blog Uncommon Descent. There physicist Rob Sheldon is quoted as saying

THere [sic] can even be uncertainty in mathematics. For example, mathematicians in the 1700’s kept finding paradoxes in mathematics, which you would have thought was well-defined. For example, what is the answer to this infinite sum: 1+ (-1) + 1 + (-1) …? If we group them in pairs, then the first pair =>0, so the sum is: 0+0+0… = 0. But if we skip the first term and group it in pairs, we get 1 + 0+0+0… = 1. So which is it?

Mathematicians call these “ill-posed” problems and argue that ambiguity in posing the question causes the ambiguity in the result. If we replace the numbers with variables, do some algebra on the sum, we find the answer. It’s not 0 and it’s not 1, it’s 1/2. By the 1800’s a whole field of convergence criteria for infinite sums was well-developed, and the field of “number theory” extended these results for non-integers etc. The point is that a topic we thought we had mastered in first grade–the number line–turned out to be full of subtleties and complications.

Nearly every statement of Sheldon here is wrong. And not just wrong -- wildly wrong, as in "I have absolutely no idea of what I'm talking about" wrong.

1. Uncertainty in mathematics has nothing to do with the kinds of "infinite sums" Sheldon cites. "Uncertainty" can refer to, for example, the theory of fuzzy sets, or the theory of undecidability. Neither involves infinite sums like 1 + (-1) + 1 + (-1) ... .

2. Ill-posed problems have nothing to do with the kind of infinite series Sheldon cites. An ill-posed problem is one where the solution depends strongly on initial conditions. The problem with the infinite series is solely one of giving a rigorous interpretation of the symbol "...", which was achieved using the theory of limits.

3. The claim about replacing the numbers with "variables" and doing "algebra" is incorrect. For example if you replace 1 by "x" then the expression x + (-x) + x + (-x) + ... suffers from exactly the same sort of imprecision as the original. To get the 1/2 that Sheldon cites, one needs to replace the original sum with 1/x - 1/x^2 + 1/x^3 - ..., then sum the series (using the definition of limit from analysis, not algebra) to get x/(1+x) in a certain range of convergence that does not include x=1, and then make the substitution x = 1.

4. Number theory has virtually nothing to do with infinite sums of the kind Sheldon cites -- it is the study of properties of integers -- and has nothing to do with extending results on infinite series to "non-integers etc."

It takes real talent to be this clueless.