Recursivity

I suppose it's not so remarkable that creationists can't do mathematics. After all, almost by definition, they don't understand evolution, so that alone should suggest some sort of cognitive deficit. What surprises me is that even creationists with math or related degrees often have problems with basic mathematics.

I wrote before about Marvin Bittinger, a mathematician who made up an entirely bogus "time principle" to estimate probabilities of events. And about Kirk Durston, who speaks confidently about infinity, but gets nearly everything wrong.

And here's yet another example: creationist Jonathan Bartlett, who is director of something called the Blyth Institute (which, mysteriously, lists no actual people associated with it, and seems to consist entirely of Jonathan Bartlett himself), has recently published a post about mathematics, in which he makes a number of very dubious assertions. I'll just mention two.

First, Bartlett calls polynomials the "standard algebraic functions". This is definitely nonstandard terminology, and not anything a mathematician would say. For mathematicians, an "algebraic function" is one that satisfies the analogue of an algebraic equation. For example, consider the function f(x) defined by f^2 + f + x = 0. The function (-1 + sqrt(1-4x))/2 satisfies this equation, and hence it would be called algebraic.

Second, Bartlett claims that "every calculus student learns a method for writing sine and cosine" in terms of polynomials, even though he also states this is "impossible". How can one resolve this contradiction? Easy! He explains that "If, however, we allow ourselves an infinite number of polynomial terms, we can indeed write sine and cosine in terms of polynomial functions".

This reminds me of the old joke about Lincoln: "In discussing the question, he used to liken the case to that of the boy who, when asked how many legs his calf would have if he called its tail a leg, replied, "Five," to which the prompt response was made that calling the tail a leg would not make it a leg."

If one allows "an infinite number of polynomial terms", then the result is not a polynomial! How hard can this be to understand? Such a thing is called a "power series"; it is not the same as a polynomial at all. Mathematicians even use a different notation to distinguish between these. Polynomials over a field F in one variable are written using the symbol F[x]; power series are written as F[[x]].

Moral of the story: don't learn mathematics from creationists.

P.S. Another example of Bartlett getting basic things wrong is here.

I will now tell you my Jerry Garcia story. To appreciate it, you must remember that Jerry was missing part of a finger on one hand.

I was having lunch with a close friend in a crappy Mexican place on Telegraph Ave in Berkeley, California; it must have been about 1983. The restaurant was called La Villa Hermosa, and is long gone. (There is a photo of it here.)

Sitting at the next table was a bearded man who looked familiar. I studied him carefully, while eating my refried beans. Eventually I figured it out. I nudged my mathematician friend gently under the table and said softly, "Hey, that's Jerry Garcia over there."

She looked over doubtfully, and said, "That's not Jerry Garcia."

I insisted, "Yes, it is."

So my friend, who was never one to observe social niceties despite being only a little more than five feet tall, stood up, walked over, put her hands on her hips and demanded of him, "Are you Jerry Garcia?"

He looked at her, held up one hand (clearly missing part of a finger), and said, "No, Jerry Garcia is missing a finger on the other hand."

She came back to my table, satisfied, and announced smugly, "See? I told you so. That wasn't him. Jerry Garcia is missing a finger on the other hand."

I swear it's true!

As academics age, some of them get cranky. I don't mean "cranky" in the sense of ill-tempered, although that's also true. I mean "cranky" in the sense of "being a crank", that is, being "a person who is obsessed by fringe ideas and beliefs". I've written about this before.

Some of them become 9/11-truthers. Some of them get cranky about anthropogenic global warming. One became cranky on the subject of Turing's proof of unsolvability of the halting problem.

One of the most popular crank topics is evolution, and that's the subject of today's blog. Yes, it's David Gelernter again. Prof. Gelernter, who teaches computer science at Yale, recently wrote a review for the far-right Claremont Review of Books entitled "Giving up Darwin". All the warning signs are there:

• Gelernter is not a biologist and (to the best of my knowledge) has no advanced formal training in biology. That's typical: the crank rarely gets cranky in subjects of his own competence. (I say "his" because cranks are almost always male.)
• Gelernter has basically done almost nothing in his own field for the last 20 years (according to DBLP, he's published only two papers in CS since 1998). That's also typical: intellectually-fulfilled academics are usually happy to contribute more to their own fields of competence, and don't have the time for bizarre detours into other fields.
• Gelernter is also a devout theist, and has written books praising the wisdom of his particular religious sect. Nearly all the intellectual opposition to evolution comes from theists, who "find in the theory of evolution a disturbing and mysterious challenge to their values" (to quote Anthony West).
• Gelernter pals around with other anti-evolution cranks, like Stephen Meyer and David Berlinski.
• Gelernter, like most anti-evolutionists, is politically conservative and is obsessed with what he feels are the intellectual failings of liberals.
• Gelernter's review was not published in a science journal, but in a politics journal run by a far-right think tank.
• His review cites no scientific publications at all, and makes claims like "Many biologists agree" and "Most biologists think" without giving any supporting citations.

So, not surprisingly, the porcine Gelernter makes a fool of himself in his review, which resembles a "greatest hits" of creationist misconceptions and lies:

• In the Cambrian explosion "a striking variety of new organisms—including the first-ever animals—pop up suddenly in the fossil record". Debunked here.
• "most species enter the evolutionary order fully formed and then depart unchanged". What could it possibly mean for a species to appear not "fully formed"?
• "no predecessors to the celebrity organisms of the Cambrian explosion": actually, some believe the Ediacaran biota were some of the ancestors of those of the Cambrian explosion, but you won't find the word "Ediacaran" anywhere in Gelernter's review.
• the 10^-77 figure of creationist Doug Axe for the improbability of obtaining a stable protein (Debunked here.)
• the false claims of Stephen Meyer about "functionally specified digital information" (debunked here and here, among other places)

And there are lots of other problems in the review. Gelernter shows no sign of having read about, much less understood, basic facets of modern evolutionary biology, such as evo-devo and gene duplication, which are critical to understanding how it works.

Altogether, yet another embarrassing performance for Prof. Gelernter. And a cautionary note for aging professors: before you start attacking another field, make a little more effort learning about it. Unless you enjoy being a crank.

Michael Egnor, the man for which the term "egnorance" was coined, is at it again, sneering at experts while demonstrating he knows little about linguistics, philosophy, or ethology.

In this piece he makes a number of claims that are either flatly false, or contradict what we know, or are given without any justification at all. Why he thinks this kind of pompous tripe will convince anyone is beyond me. Maybe, in their jobs, neurosurgeons get accustomed to making pronouncements that everyone else accepts without questioning.

I lack the time to do a complete fisking here, but I'll mention a few of his bogus claims.

1. "The accepted definition of reason is simple and straightforward: it is the power to think abstractly, without concrete particulars."

Whenever Egnor talks about something being "accepted" or "simple and straightforward", you can be pretty sure that the opposite is the case. Anyone who wants to check Egnor's claim can just go to the Oxford English Dictionary and type in "reason". There are three senses for the word, two as a noun and one as a verb. The uses as a noun include 17 different subdefinitions and another 15 or so different usages in phrases. The uses as a verb include 8 different subdefinitions. The word "abstract" appears nowhere in any of these subdefinitions (it does appear in two citations, but not in the sense Egnor refers to). So Egnor is wrong twice: the "accepted definition" of the word is neither simple nor straightforward, and the meaning Egnor claims is not an "accepted" one.

2. "Only man thinks abstractly; that is the ability to reason. No animal, no matter how clever, can think abstractly or reason."

Egnor's made this claim before, and it was refuted before. He just repeats it here, with no evidence, without addressing previous objections.

Of course, if you understand the theory of evolution, you realize his claim is likely to be utter nonsense. Abstract thinking is not a black-white thing; it's a range of capabilities that, even among people, we see a huge variation in. Any capability with huge variation is subject to selection, and so it can evolve. Since people are descended from earlier ape-like creatures, it is quite believable that non-human animals would also display the ability for abstract thought, in varying degrees. And they do! Ethologists, who actually study this kind of thing, disagree with Egnor. (Also see baboons and crows, to name just a couple more examples.)

3. "Reason is an immaterial power of the mind—it is abstracted from particular things, and cannot logically be produced by a material thing."

This is vintage Egnor -- a flat assertion, made with no evidence, and contradicting what we know about (for example) machine learning. Machines can abstract from specific cases to more general concepts; that is exactly what is done routinely in machine learning. (To cite just one example, see here.)

Egnor offers no rationale for why reason has to be "immaterial", and when he says something is "logical", you can be pretty sure there's no actual logic involved.

4. "This immaterial power of the soul is precisely what makes man qualitatively different from every other living thing. And I am not “forced to lean on supernaturalism” by pointing this out. I’m merely making an observation that’s obvious to all. Man, and man alone, has the power to reason."

Souls don't exist; there's no evidence for them. There's no evidence for "immaterial powers". Egnor's claim is disputed by many, and it's a plain lie to say it's "obvious to all".

5. "We routinely ask questions that entail reasoning. Animals never do."

How does Egnor know animals never do this? He never says.

As we know from the example of Ben Carson, it is perfectly possible for a neurosurgeon to be good at their job, but incompetent when it comes to anything else. Egnor is yet another data point.

Here.

I wrote before about Inference, a weird "journal" that bills itself as an "International Review of Science", but has published some very questionable pieces by some very questionable people.

Back when they were hiding their editorial board, I deduced that David Berlinski was involved with it somehow, and my deduction was later proved correct.

Now a real investigative journalist has taken the job of looking further into this bizarre venture. It's physicist Adam Becker, and he's published his exposé in Undark.

Turns out that lots of people, when they find out the kind of stuff that Inference publishes, decide not to get involved with them, despite the large amounts they're paying for pieces. And it also turns out that Peter Thiel is one of the big funders. You know, the same Peter Thiel who has donated to far-right politicians like Ted Cruz, Rand Paul, and Dana Rohrabacher.

I wonder if Becker's piece will convince legit academics, such as Andrew Yao, that they don't want to have anything to do with Inference.

Canada and Norway are engaged in a Moose War.

I have to say, Norway is definitely winning. Their moose makes ours look pitiful by comparison.

...is a moose map.

See?

(Hat tip: J. C.)

"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.

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

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.

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.

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.

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.

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.

This American president's library contains many books on religion, but it also contains a copy of Darwin's Origin of Species.

Which American president is it?

That wretched hive of scum and villainy, the Discovery Institute, has announced that its nefarious tentacles have snagged a new venture: a situation comedy called the "Walter Bradley Center for Natural and Artificial Intelligence".

Walter Bradley, as you may recall, is the engineering professor and creationist who, despite having no advanced training in biology, wrote a laughably bad book on abiogenesis. Naming the "center" after him is very appropriate, as he's never worked in artificial intelligence and, according to DBLP, has no scientific publications on the topic.

And who was at the kick-off for the "center"? Why, the illustrious Robert J. Marks II (who, after nearly four years, still cannot answer a question about information theory), William Dembski (who once published a calculation error that resulted in a mistake of 65 orders of magnitude), George Montañez, and (wait for it) ... Michael Egnor.

Needless to say, none of these people have any really serious connection to the mainstream of artificial intelligence. Egnor has published exactly 0 papers on the topic (or any computer science topic), according to DBLP. Dembski has a total of six entries in DBLP, some of which have a vague, tangential relationship to AI, but none have been cited by other published papers more than a handful of times (other than self-citations and citations from creationists). Marks has some serious academic credentials, but in a different area. In the past, he published mostly on topics like signal processing, amplifiers, antennas, information theory, and networks; lately, however, he's branched out into publishing embarrassingly naive papers on evolution. As far as I can tell, he's published only a small handful of papers that could, generously speaking, be considered as mainstream artificial intelligence, none of which seem to have had much impact. Montañez is perhaps the exception: he's a young Ph. D. who works in machine learning, among other things. He has one laughably bad paper in AI, about the Turing test, in an AI conference, and another one in AAAI 2015, plus a handful in somewhat-related areas.

In contrast, take a look at the DBLP record for my colleague Peter van Beek, who is recognized as a serious AI researcher. See the difference?

Starting a center on artificial intelligence with nobody on board who would be recognized as a serious, established researcher in artificial intelligence? That's comedy gold. Congrats, Discovery Institute!

I wrote a joke that was once voted the funniest religious joke of all time. This what I looked like in high school. Who am I?

Snowmobilers worked together to free a stuck moose last month in Newfoundland.

"We knew the moose was stuck really good," one said.

And a trucker saved a moose in British Columbia.

After I retire, that's the job for me!

Hat tip: A. L.

If I live long enough, I'm planning to write a book entitled "The 100 Stupidest Things Anyone Ever Said About Minds, Brains, Consciousness, and Computers". Indeed, I've been collecting items for this book for some time. Here's my latest addition: Michael S. Gazzaniga, a famous cognitive neuroscientist who should know better, writes:

Perhaps the most surprising discovery for me is that I now think we humans will never build a machine that mimics our personal consciousness. Inanimate silicon-based machines work one way, and living carbon-based systems work another. One works with a deterministic set of instructions, and the other through symbols that inherently carry some degree of uncertainty.

If you accept that the brain functions computationally (and I think the evidence for it is very strong) then this is, of course, utter nonsense. It was the great insight of Alan Turing that computing does not depend in any significant way on the underlying substrate where the computing is being done. Whether the computer is silicon-based or carbon-based is totally irrelevant. This is the kind of thing that is taught in any third-year university course on the theory of computation.

The claim is wrong in other ways. It is not the case that "silicon-based machines" must work with a "deterministic set of instructions". Some computers today have access to (at least in our current physical understanding) a source of truly random numbers, in the form of radioactive decay. Furthermore, even the most well-engineered computing machines sometimes make mistakes. Soft errors can be caused, for example, by cosmic rays or radioactive decay.

Furthermore, Dr. Gazzaniga doesn't seem to recognize that if "some degree of uncertainty" is useful, this is something we can simulate with a program!