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May 13th, 2019 (Permalink)

Still No Vacancy at the Hilbert Hotel

In our last visit to the Hilbert Hotel*, we saw that even though the hotel was full it could still provide a room to a new customer. That's because the hotel has an infinite number of rooms: room 1, room 2, room 3, etc., and there is no highest room number.

The Hilbert Company, Inc., that owns the hotel also owns a railroad line. Mr. Hilbert, the president of the company, decided that a special train was needed to supply an infinite number of guests for his hotel. The railway line uses the same patented Transfinite™ technology as the hotel, so that the Hilbert train can hold an infinitude of passengers. Since it can carry an infinite number of passengers the train is, of course, infinitely long, as is the track that it runs on.

This morning, a train arrived at the small station next to the hotel carrying a full load, but the "No Vacancy" light at the hotel was lit. Now, if the train had been carrying only a finite number of passengers, there would have been no problem. The clerk on duty at the desk knew how to accommodate finite numbers of new guests: supposing that there were n newcomers, all he had to do was move every current guest in room m into room n + m. In other words, the guest in room 1 would be moved to room n + 1, the guest in room 2 into room n + 2, the one in room 3 to room n + 3, and so on. This process would leave the first n rooms in the hotel empty.

However, the clerk had never dealt with an infinite number of new guests. He couldn't very well move the current guest in room m into room m + ∞, since there is no such room! Not knowing what to do with an infinite number of impatient new guests, the clerk called the manager for help.

"No problem!" replied the manager, and he explained to the new clerk how to make an infinite number of new rooms available in the hotel when it was full.

What did the manager tell the clerk?


*See: No Vacancy at the Hilbert Hotel, 4/29/2019

May 2nd, 2019 (Permalink)

A News Weak Poll

The news media have to report news 24 hours a day even when there is no news, so why not manufacture new "news"1 by sponsoring a public opinion poll? The presidential election is still more than a year and a half away but polls about it are already being conducted. Trying to predict the nominees now, let alone the eventual winner, is likely to be about as accurate as predicting what the weather will be the same time next year2.

I don't suppose that Newsweek magazine is worse about promoting such polls than other news media, but I can't resist punning on its name. Here's a recent Newsweek headline:

Is Joe Biden Unbeatable?
Ex-Vice President Opens Huge Poll Lead Over Bernie Sanders―
but There's a Catch3

The answer to the headline question is, of course, "no", as it usually is4, and the "catch" is that Biden's lead is too early to mean much. Despite its title, the body of this article makes this point:

Former Vice President Joe Biden opened up a 14-point lead over his main rival for the Democratic presidential nomination, Bernie Sanders, after Biden officially entered the race last week―but there was a catch. Other polling suggested there was a substantial number of undecided voters waiting to be won over by the various campaigns. Biden’s early advantage…will likely soften over time as other candidates come to the fore in the race. … Lee Miringoff, director of the Marist College Institute for Public Opinion, told Newsweek it is still way too early to crown anyone as a certainty to win. "The political graveyards are full with candidates who got off to an early lead but failed to win the nomination…." Cornell Belcher, a former Democratic National Committee pollster…described public polls as “completely meaningless,” particularly so early in the primary process.

It's nice when a news article hyping a poll tells you that it's junk.

Update (5/11/2019): In more junk news about polls, we're told:

The latest bit of bad news for Beto [O'Rourke] comes in the form of a Monmouth poll of New Hampshire primary voters. He's tied for sixth place, along with Sens. Cory Booker (D., N.J.) and Amy Klobuchar (D., Minn.), winning the support of just 2 percent of poll respondents, which is getting dangerously close to "statistically insignificant" territory. It's also not much better than truly insignificant candidates such as Rep. Tim Ryan (D., Ohio), former Gov. John Hickenlooper (D., Colo.), and random Silicon Valley bro Andrew Yang, who are all polling at 1 percent.5

Speaking of statistical significance and insignificance, what is the margin of error (MoE) for this poll? The report quoted above does not tell us. It used to be standard practice for American newspapers to report certain basic facts about polls, including the MoE, even though this information was sometimes relegated to a separate paragraph at the end of the article.

In this case, if you want to know the MoE, you have to go to the Monmouth University Polling Institute's report6. There, you can find out that the MoE for this question is ±5.1 percentage points, which is a larger than usual MoE because only democrats were polled.

To say that something is "statistically significant" simply means that it is not likely to be the result of random chance7. The MoE is a way of indicating how large a difference between statistics must be to be unlikely to be the result of chance. In this case, the difference between two candidates needs to be greater than 5.1 percentage points to be statistically significant.

Logically, "statistically insignificant" must mean that such a difference may well be the result of random chance. Given its MoE, any difference between two candidates that's less than five percentage points is a statistically insignificant difference. So, I'm not sure what the author of the quoted article means in saying that candidates polling only 1% are "truly insignificant". If those polling at 1% are already in "'statistically insignificant' territory", then O'Rourke at 2% is in there with them.

The only stand-outs in the poll are Joe Biden, who is the front-runner at 36%, and Bernie Sanders, who at 18% is a distant second. However, all of the other candidates are grouped together in a pack in the single digits, with Pete Buttigieg leading the pack at 9%. While slightly greater than the MoE, the difference between O'Rourke and Buttigieg is not significant8.

However, as mentioned in the original entry above, it's so early in the race that there's plenty of time for O'Rourke to gain ground, and just as much time for Biden to lose it. The New Hampshire primary, which this poll is supposed to be forecasting, isn't until February of next year9.


  1. See: Junk Headline, 4/22/2019.
  2. Stephanie Slade, "I'm Betting the 2020 Election Polls Are Junk", Reason, 4/4/2019.
  3. Shane Croucher, "Is Joe Biden Unbeatable? Ex-Vice President Opens Huge Poll Lead Over Bernie Sanders―but There's a Catch", Newsweek, 4/30/2019.
  4. This is an old journalistic rule of thumb, namely, that the answer to any headline in the form of a question is: "no".
  5. Andrew Stiles, "Beto O’Rourke Approaching ‘Statistically Insignificant’ Territory in New Hampshire Poll", The Washington Free Beacon, 5/9/2019.
  6. "Trump Bigger Factor than Obama for 2020 Dem Primary Voters", Monmouth University Polling Institute, 5/9/2019.
  7. See: Victor Cohn, News & Numbers: A Guide to Reporting Statistical Claims and Controversies in Health and Other Fields (1989), p. 176.
  8. I used the following calculator to figure this: "Ballot Lead Calculator", American Research Group, Inc., accessed: 5/11/2019.
  9. See: Jennifer Rubin, "What’s happening in New Hampshire?", The Washington Post, 5/9/2019. Rubin discusses the same poll as that discussed above and begins by noting: "Current polls for the 2020 Democratic primary aren’t predictive."

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April 29th, 2019 (Permalink)

No Vacancy at the Hilbert Hotel

The "No Vacancy" sign was on at the Hilbert Hotel because every room in the hotel had at least one guest staying in it. There's nothing particularly odd about a full hotel, but the Hilbert is an unusual one because it has an infinite number of rooms. In other words, there is a room 1, a room 2, a room 3, and so on, ad infinitum. This night, every room in the hotel was occupied which, of course, means that there were an infinite number of guests, but what's the point of having an infinite hotel if you can't accommodate an infinite number of guests?

Unfortunately for the clerk on duty, a man walked in the front door just before midnight and approached the desk.

"I need a room for the night," he told the clerk.

"I'm sorry, sir," the clerk responded, "but the hotel is full."

"What!" the man answered, "this is the Hilbert, an hotel with an infinite number of rooms, and you mean to tell me there's no vacancy?"

"That's correct, sir."

"What's the point of an infinite hotel if you still can't get a room?"

"I'm sorry, sir, but the only way that we could get a room for you tonight would disturb an infinite number of our guests. They would be very angry; in fact, infinitely so. If you'll just return at ten o'clock tomorrow morning, I'm sure that we can find you a room."

The man stormed out of the hotel in a huff. However, the next morning he returned at exactly ten o'clock and was given an empty room, despite the fact that no one had checked out in the meantime. Moreover, no guests had had to double up, that is, no one who had had a separate room had moved into anyone else's room, thus freeing up a room.

How did the hotel manage this remarkable feat? And why did the clerk refuse to do it the previous night?


April 24th, 2019 (Permalink)

What's New?

I've added a new entry to the "Familiar Contextomies" page. This one quotes the current president out of context. See: Familiar Contextomies: Donald Trump.

April 22nd, 2019 (Permalink)

Junk Headline

What's wrong with the following headline?

Poll: 39 percent say Mueller failed to prove Trump campaign did not collude with Russia1

Let's start with the poll question. Why was this question asked when it was? The poll2 ended before the report on the Mueller investigation was released, so no member of the general public had been able to read it at the time of polling3. Why ask people for their opinions on the results of an investigation before they can have read the report?

Here's another recent headline that's relevant:

Who Really Cares About Mueller Report? New Poll Finds One-Third Of Voters Don't Even Know Who William Barr Is4

This is a report on the same poll as the previous headline, but focusing on a different question that shows just how little many of those polled know.

Here's the poll question whose results are reported in the first headline: "Do you think the Mueller investigation proves Donald Trump’s campaign did not collude with the Russians during the 2016 campaign, or does it not prove that?5" This is a confusing question given that the choices offered for answering it are: "Yes", "No", and "Don't know".

At this point, we need to look at some erotetic logic, that is, the logic of questions and answers. I'll try to keep it as brief and non-technical as possible, but some technical language is unavoidable. To start with, the poll question is a disjunctive one, that is, one of the form: "Is P or Q the case?" Disjunctive questions are not yes-or-no questions since to answer "yes" would merely affirm that one of the disjuncts is the case; answering "no" rejects both disjuncts, thus denying the presupposition of the question.

For example, suppose you feel a draft and ask: "Is the door or the window open?" An answer of simply "yes" is likely to be annoying since all that the answer tells you is that one or the other is indeed open, but you want to know which. In contrast, if the answer is "no", you'd assume that neither door nor window was open. Instead, a direct answer to a disjunctive question is any one of the disjuncts, so to answer the question directly is to say either: "The door is open" or "the window is open"6.

An additional point about the poll question is that the second disjunct is the negation of the first, so that the question has the more specific logical form: "Is P or not-P the case?" To answer this simply "yes" or "no" would be either to truly affirm a tautology or falsely deny it.

To adapt the previous example, suppose that you ask: "Is the door open or is it closed?" If the answer comes back: "yes", you'll probably be angry at the answerer since, again, you want to know which. In these sort of situations, people usually assume that a "yes" answer means that the first, affirmative disjunct is the case, and a "no" means that the second, negative one is.

I assume that most of those who answered the poll question assumed that a "yes" answer affirmed the first disjunct and a "no" affirmed the second, which is equivalent to denying the first. The 39% referred to in the first headline, above, are those that answered "no" to the poll question5.

So, probably most of those who took the poll were able to decipher this question, despite its logical problems. Nonetheless, potential confusion could have been easily avoided simply by rewording the three possible answers to something like: "Yes, it proved no collusion", "No, it did not prove no collusion", and "I don't know".

I did a double take when I first read the headline because of its two negations: "not" and "failed". They are not a double negation that simply cancels out, as the negation of "failed" has wide scope and "not" has narrow scope. In other words, "not" negates only the predicate "colludes with Russia" whereas "failed" negates the claim that "Mueller proved Trump campaign did not collude with Russia".

As a general rule, one should use as few negations as possible―either one or none―since multiple negations are hard to cognitively process. One way to eliminate the narrow scope negation is realize that to prove no collusion is to exonerate, so the headline could have been reworded as follows:

Poll: 39 percent say Mueller failed to exonerate the Trump campaign of collusion with Russia

This is easier to understand, but yet another problem lurks here, namely, the word "failed". This word can't be blamed on an editor, since the article under the headline begins: "Almost 4 in 10 Americans think special counsel Robert Mueller's investigation failed to prove President Trump's campaign did not collude with Russia….1" Nor are the pollsters to blame for the occurrence of the word "fail" in the headline since, as we have seen, the poll question did not use that word.

You cannot "fail" at something unless you try to do it. For instance, I did not fail to climb Mount Everest since I never even tried. What reason do we have for thinking that Mueller tried to exonerate the Trump campaign? Since when was it his job to prove that Trump did not collude with Russia? Did the writer of the article presume that the Trump campaign was guilty of collusion and that Mueller's job was to prove its innocence? What happened to the presumption of innocence7?

These considerations lead to a final revision of the offending headline:

Poll: 39 percent say Mueller did not exonerate the Trump campaign of collusion with Russia

The news media like polls because they are manufactured news: instead of waiting for something to happen, just commission a poll and no matter what the results you get instant news. In this case, the poll question was premature, badly worded, and the article shouldn't have focused on it. Moreover, the article gives the false impression that the purpose of the Mueller investigation was to exonerate the Trump campaign.

"Fake news" is "news" that reports things that aren't true, so this isn't fake news. Rather, it's junk news―the journalistic equivalent of junk food: fast, cheap, and low in nutrition.


  1. Chris Mills Rodrigo, "Poll: 39 percent say Mueller failed to prove Trump campaign did not collude with Russia", The Hill, 4/18/2019.
  2. "Fox News Poll", Fox News, 4/18/2019.
  3. The first sentence of the story under the headline reads, in part: "…according to a Fox News poll released a day before a redacted version of the report was slated to be made public."
  4. Daniel Moritz-Rabson, "Who Really Cares About Mueller Report? New Poll Finds One-Third Of Voters Don't Even Know Who William Barr Is", Newsweek, 4/18/2019. More precisely: 30%; see question 13 in the poll report, linked to in note 2, above.
  5. See question 23 in the poll report, linked to in note 2, above.
  6. Unless, of course, the correct answer is: "I don't know." If the question is based on an incorrect presupposition, the answer is: "Neither".
  7. The presumption of innocence is not just a legal or ethical presumption but a logical one as well: the burden of proof is not on President Trump or his administration to prove that it did not collaborate with Russia, but on those who make that accusation.

April 15th, 2019 (Permalink)

Charts & Graphs: The IRS Baked Two Pies for Tax Day

The two pie charts below appear near the end of the booklet of instructions1 for filling out and filing the 1040 tax form put out by the Internal Revenue Service that is due today. The charts serve no purpose in helping figure taxes, and most taxpayers probably ignore the page where the charts occur in the rush and hassle of preparing a return. Instead, the charts give information on what percentage of the government's income comes from income taxes, and what those taxes pay for. Income & Outlays

These charts are three-dimensional pie charts which, as I've pointed out previously2, can be misleading. These are particularly bad examples of this type of chart since the angle from which the "pies" are portrayed is quite acute. This means that the areas of the pies are distorted, so that some look larger in comparison to others than they should. Furthermore, these are "deep dish" pies with thick edges that make the wedges facing the viewer appear to be larger than similarly sized wedges at the back of the pies, whose edges cannot be seen.

For instance, in the "Income" chart, the wedge for "Corporate income taxes" looks almost the same size as that next to it for "Borrowing to cover deficit", despite the fact that the former is only 7% of the pie while the latter is 17%. Similarly, the wedge for "Social security, Medicare,…" behind them represents almost a third of income, but appears to the eye to be about a quarter.

In the "Outlays" chart, the wedge facing the viewer representing "National defense,…" appears to be considerably more than 20% of the pie. In contrast, the largest segment, labelled "Social security, Medicare,…", is over twice the percentage of that for "National defense,…", but doesn't appear to be twice the size.

I don't suppose that these charts were intentionally constructed to mislead, especially since the percentage for each segment is included next to its label. However, there's not much point in baking a pie chart if you have to read a bunch of numbers in order not to be fooled. Instead, a couple of tables listing the parts and their percentages would have conveyed the same information with no risk of misleading the reader. If we must be served with pies, then the angle from which we view them should be close to 90°.


  1. P. 112. See also: "Major Categories of Federal Income and Outlays for Fiscal Year 2017", Internal Revenue Service, accessed: 4/14/2019.
  2. Charts & Graphs: Three-Dimensional Pie, 5/5/2013.

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