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July 16th, 2025 (Permalink)

Striking a False Note

The following passage from this month's New Book¹ caught my eye:

Over the years, supposedly scientific claims about everything from healthy eating to brain function have propagated widely, often from unexpected or unknown sources. For example, it's often claimed that adults should drink eight glasses of water a day, even though this advice doesn't have a clear origin. It may stem from health advice published in 1945, which stated "a suitable allowance of water for adults is 2.5 litres daily in most instances." However, that advice also noted that "most of this quantity is contained in prepared foods."²

This got my attention because I've heard this urban legend since I was a child, but have always been skeptical of it and even researched and written about it³. Despite that, I had never come across the two-and-a-half liters claim. Two-and-a-half liters is about 85 fluid ounces or more⁴, whereas eight eight-ounce glasses is only 64 ounces. While eight twelve-ounce glasses would work with room to spare, two-and-a-half liters doesn't naturally divide into eight glasses. In addition, I've done some research and documented that the eight glasses a day claim goes back to at least 1924. So, the two-and-a-half liters claim is not the source of the eight-and-a-half glasses one.

When reading a non-fiction book, it's good advice to check a few notes at random, assuming that the book actually has notes of some sort. If it doesn't, that's a mark against it since it makes it extremely difficult if not impossible to check the book's factual claims. If one or more references fails such a random check, that's a bad sign: either the author or editors were lazy―or worse. When it comes to misleading notes, keep in mind Goldfinger's Axiom: Once is happenstance, twice is coincidence, the third time it's enemy action⁵.

In this particular case, I was prompted to look for a note only because I was curious about the author's source for this claim. The book does indeed have endnotes―a lot of them in fact. So, the book passes muster on that score. However, the only endnote close to the claim in question is at the end of the entire paragraph quoted from above. That note points to a short article with the title "Accuracy of comparing bone quality to chocolate bars for patient information purposes: observational study"⁶, which sounds like a practical joke. Nonetheless, it was published in the prestigious British Medical Journal (BMJ).

The authors tested twenty candy bars―which was apparently all they could afford on their research budget―by dropping them on the kitchen floor―yes, an actual floor in a kitchen―from increasing heights until they fractured. What happened to the candy after the experiment ended was not reported.

It's an amusing article, and not much more ridiculous than a lot of the seemingly serious research I've documented in this weblog. However, what it has to do with how much water you should drink, I don't know, though I expect the researchers needed some to wash down the candy bars. Moreover, this research is not referred to in the paragraph footnoted, nor is there any other reference to it in the remainder of the book as far as I can tell.

So, the note turned out to be a dead end, but the paragraph directly quoted a source twice, and a search for the two quotes turned up what must be that source⁷. The short article in question debunks seven "medical myths", including that you should drink eight glasses of water a day. It was also published in the BMJ preceding the candy article in the same issue, and the URLs for the two online versions of the articles differ only in the final digit which, presumably, was miscopied.

So, this note was probably the result of a simple typographical error, but it's an amusing example of what can be a serious issue. Readers tend to rely on the notes to support a book's factual claims, but they don't always do so. Naturally enough, most people don't check notes, and even those of us who might occasionally do so don't check them all. Most books that include notes at all often have too many to check; for instance, Proof has over five-hundred. I checked a few other notes in the book without encountering any problems, so I'll put this one down to happenstance.

Notes:

↑ See: New Book: Proof, 7/2/2025.
↑ Adam Kucharski, Proof: The Art and Science of Certainty (2025), p. 222.
↑ See: "Emerging Evidence", 1/9/2023.
↑ "What is 2.5 liters in fluid ounces?", Wolfram Alpha, accessed: 7/15/2025. To be exact, it's 84.5 American fluid ounces but 88 in the United Kingdom―go figure.
↑ See: Ian Fleming, Goldfinger (1959), epigraph.
↑ Phil Jones, Sarah Jones & Debbie Stone, "Accuracy of comparing bone quality to chocolate bars for patient information purposes: observational study", British Medical Journal, 12/22/2007.
↑ Rachel C. Vreeman & Aaron E Carroll, "Medical Myths", British Medical Journal, 12/20/2007.

July 4th, 2025 (Permalink)

The Fourth of July Picnic

At the Logicians' Club July Fourth picnic, hot dogs were served. Three condiments were available for the frankfurters: mustard, ketchup, and pickle relish. Of the forty-one people who attended the picnic―logicians and their families―all used at least one condiment on their hot dogs, except for one little boy who ate his plain. Twenty-six attendees put mustard on their hot dogs, sixteen used ketchup, and eighteen added relish. Five picnickers put both mustard and ketchup on their frankfurters, six had both ketchup and relish, while three had all three condiments.

Question: How many people had both mustard and relish but no ketchup on their hot dogs?

Disclaimer: The above puzzle is a work of fiction: no one would ever put both mustard and ketchup on a hot dog.

July 2nd, 2025 (Permalink)

New Book: Proof

Quote: "Life is full of situations that can reveal remarkably large gaps in our understanding of what is true and why it's true. This is a book about those gaps. It is the story of the ideas that have helped scientists and societies discern between truth and falsehoods, improving decision-making and reducing dangerous errors. From medieval juries to modern scientific revolutions, it is about the methods people have used to accumulate evidence, negotiate uncertainty, and converge on proof. And, crucially, what happens when those methods fail."¹

Title: Proof

Subtitles: This book has different subtitles for its United Kingdom and United States editions:

UK: The Uncertain Science of Certainty²
Comment: This is certainly an intriguing subtitle, but I'm uncertain what it means. The science of certainty would seem to refer to formal logic, but what's uncertain about it?
US: The Art and Science of Certainty
Comment: What is meant by "the art…of certainty"? The invocation of certainty, like the word "proof", suggests the logical and mathematical, as opposed to more informal notions, of "proof". I'll have more to say about these ideas in the General Comments, below.

I wonder why it was thought advisable to have different subtitles for the US and UK. All of this, of course, may have little to do with the book or its author, since the subtitles may have been selected by the book's publisher. Was there some reason to think that the second subtitle would sell better here in the US than the first, or that the first would sell better in the UK?

Author: Adam Kucharski

Comment: Please don't hold it against him, or against me, but I'd never heard of Kucharski prior to this book. Based on the author's short biography at the end of the book, he is a mathematician and epidemiologist, and has written two previous books―The Perfect Bet and The Rules of Contagion―neither of which I've read.

Date: 2025

Summary: So far, I've read only the Introduction to the book, which lacks a summary of its structure, and the chapter titles are not very revealing, so I'm guessing as to the topics covered. Beside the Introduction, there are eight chapters. I suppose that the first chapter is an introductory one; the second appears to concern "proof" in the fullest sense, that is, in logic and mathematics; the third, in contrast, seems to deal with the weaker sense of "proof" used in the law and legal trials; the fourth would seem to concern the sense of "proof" in the statistical trials of medical and social research; the penultimate chapter may have something to do with computers; and the last may be a summing up since there is no afterword.

Given that Kucharski is an epidemiologist, I was curious whether he discusses the events of the last few years in the book. None of the chapter titles suggest a concern with the pandemic, though the fourth is a likely suspect. Thankfully, the index indicates that, unlike Marty Makary, author of a previous New Book³, Kucharski does treat the pandemic at some length.

The Blurbs: The book is positively blurbed by Tim Harford, author of The Data Detective, and Alex Bellos, who wrote Can You Solve My Problems?, both of which I've read.

General Comments: "Proof" has both a strong and a weak sense:

Strong: The strong sense of proof is that which occurs in deductive logical and mathematical contexts. In this sense, a proof is an argument that's valid. This means that if the premisses of the proof are true then the conclusion must also be true. It does not mean that the conclusion of the proof is true, let alone necessarily so, since one or more of the premisses might not be true. It would appear to be this sense of "proof" that's discussed in chapter four of the new book.
Weak: A proof in this weaker sense is one that makes its conclusion more likely to be true than false, but not necessarily so. This sense is weaker than the strong one in that the premisses of such a proof could be true and the conclusion false. For instance, the legal standard of "beyond a reasonable doubt" is not a proof in the strong sense since it does not mean beyond any doubt whatsoever, and it's possible that an innocent person might be convicted of a crime based on evidence that meets the standard. Similarly, the evidence supporting a theory in empirical science never amounts to a proof in the strong sense as there is always a possibility that new evidence will refute it. This sense would seem to be the subject of chapters three and four.

I plan to read this book and may have more to say about it in the near future.

Disclaimer: I haven't read this book yet, so can't review or recommend it, but its topic is right in the wheelhouse of my bailiwick, so I'm interested in it and thought that readers might be as well.

Notes:

↑ "Introduction", p. 3.
↑ For the UK cover with subtitle, see: "Proof: The Uncertain Science of Certainty", Adam Kucharski, accessed: 7/1/2025.
↑ New Book, 3/27/2025.

June 27th, 2025 (Permalink)

Innumeracy in the News

During a congressional oversight hearing on the Department of Government Efficiency (DOGE), the large poster was displayed that you can see in the above screenshot taken from video of the meeting¹. As you can see, the percentages shown for the three categories, which appear to be mutually exclusive, add up to 110%. Of course, all percentages of mutually exclusive categories out of some whole should total no more than 100%², since 100% of the whole is all of it. So, something went wrong with the numbers.

What went wrong? According to the Representative who displayed the poster, Democrat Melanie Stansbury³, the percentages on the placard were taken from a poll conducted by Quinnipiac University two and a half weeks ago. However, according the poll itself⁴, 57% of those responding rated DOGE's work as either "not so good"―12%―or "poor"―45%. So, the extra ten percentage points were not from Quinnipiac but, presumably, from someone in Stansbury's office. Moreover, at the very least, someone on her staff should have noticed the discrepancy and checked it with the poll before displaying it.

I mention the party of the representative not to further humiliate it or her, but because it reveals an underlying bias. The Democratic Party opposes the efforts of DOGE to cut government spending, so that pointing out the apparent unpopularity of those efforts supports its position. It's unlikely that such a mistake in the opposite direction―that is, one that undermined their position―would have been displayed on a large placard during a public meeting, since someone would have caught such an error beforehand. The actual poll numbers support Stansbury's claim that DOGE is unpopular but, by falsely exaggerating them, Stansbury undermined her own position and made herself and her party look foolish.

This is just a reminder that one should be even more careful not to make mistakes that support one's own position as those that undermine it, not just because we want to get at the truth, but also because the result of carelessness can hurt one's cause.

Notes:

↑ See: Breaking News: Marjorie Taylor Greene Battles Democrats In Fiery DOGE Hearing, Forbes, 6/24/2025.
↑ The one exception is due to rounding: if the percentages are rounded, the total may add up to slightly more than 100%, but this should be noted somewhere.
↑ Joe DePaolo, Watch: Democrat Trots Out Placard in House Hearing With a Glaring Math Error―Promptly Gets Dunked On By Marjorie Taylor Greene, Mediaite, 6/24/2025.
↑ See: Majority Of Voters Oppose GOP Budget Bill, With Just 67% Of Republicans In Support, Quinnipiac University National Poll Finds; Trump Job Approval: 38%, His Handling Of Russia-Ukraine War Lowest Among List Of Issues, Quinnipiac University, 6/11/2025.

June 22nd, 2025 (Permalink)

A Chess Tournament at the Logicians' Club

Tonight is the Logicians' Club annual chess tournament. I'm not much of a chess player so I won't be participating, but I got stuck with some of the preparations. Eight members of the club have entered the tournament: Andy, Bernie, Cathy, Debby, Ellie, Freddy, Gertie, and Hank.

You might wonder how so many members of such a small club were available for the tournament. The answer is that three branches of the club from nearby towns are included: A-town, B-ville, and C-city. The players from A-town are Andy, Ellie and Hank; from B-ville they are Bernie, Cathy and Freddy; and Debby and Gertie are from C-city. Since they have many opportunities to play against one another, no players from the same town are matched against each other in the tournament.

The eight players have been assigned to three teams named Cantor, Frege, and Russell. Cantor consists of Cathy and Hank; Frege of Andy, Debby and Freddy; and Russell of Bernie, Ellie and Gertie. In the tournament, players from the same team will not be matched against each other.

For some mysterious reason, Professor Knight^*, who is also not playing in the tournament, was given the task of drawing up the seating chart for the competition. There are four tables in the large meeting room where the tournament is to be held―known simply as tables one, two, three, and four―and each pair of players were assigned a table at which they would play their match.

Unsurprisingly, Professor Knight managed to lose the seating chart, so I asked him what he remembered about the seating.

"Let's see," he said, scratching his beard and looking up at the ceiling, "I seem to remember that Debby was at table one―or was it three? Well, it was one of those two. And I recall that Cathy was definitely not at the fourth table, but I can't narrow it down any further. Andy was also not at table four, but he wasn't at the first table, either. Hank was at table three or maybe table one, I'm not sure which, but he wasn't at the same table as Debby. Ellie was also at the third table, or perhaps it was the second one. Hmm. Anyway, that's all I remember, but I'm sure you can figure it out." He smiled and walked away. I just sighed.

Is the professor right that the seating arrangements can be reconstructed from his fragmentary memories? Can you help figure out at which table each tournament participant is supposed to sit?

Solution
Tables	Player	Player
1	Bernie	Hank
2	Andy	Cathy
3	Debby	Ellie
4	Freddy	Gertie

^* ↑ If you don't know Professor Knight, see:

Christmas at the New Logicians' Club, 12/25/2021
The Absent-Minded Professors, 2/12/2022

Disclaimer: This puzzle is a work of fiction, Professor Knight is not a real person, and there is no Logicians' Club though there ought to be.