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January 26th, 2014 (Permalink)


Don't trust online reviews!
A fifth are left by people who have NEVER tried the product

Newspaper headlines and the titles of articles in magazines are usually written by editors, rather than the authors of the entitled story. This is why we shouldn't blame writer Victoria Woollaston for the above headline to her story yesterday in England's Daily Mail Online newspaper. As an exercise in critical thinking, first read the article―see the Source, below; read the whole thing, it's short!―and see if you can see what's wrong with the headline above. Only then click on the link below to see if you're right.

Why is this headline wrong?

Source: Victoria Woollaston, "Don't trust online reviews! A fifth are left by people who have NEVER tried the product", Daily Mail Online, 1/24/2014

January 21st, 2014 (Permalink)

In the Mail: Philosophy of Pseudoscience

I mentioned the new anthology Philosophy of Pseudoscience last October―see the Resource, below―and I will post a review of it in the near future.

Update (1/11/2017): Make that the not-so-near future.

Source: Massimo Pigliucci & Maarten Boudry (editors), Philosophy of Pseudoscience: Reconsidering the Demarcation Problem (The University of Chicago Press, 2013)

Resource: New Book: Philosophy of Pseudoscience, 10/14/2013

January 19th, 2014 (Permalink)

An Infinitely Paradoxical Proof

Consider the following infinite series.

1 + (-1) + 1 + (-1) + 1 + (-1) + …

What does it add up to? Notice that the series can be written in the following way:

1 - 1 + 1 - 1 + 1 - 1 + …

Furthermore, let's group the members of the series as follows:

(1 - 1) + (1 - 1) + (1 - 1) + …

If we carry out the subtractions in parentheses, we get:

0 + 0 + 0 + …

Clearly, then, the sum of this series is zero. However, going back to the original way of writing the series with negative numbers, there's another way of grouping its members:

1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + …

If we carry out the additions in parentheses, we get:

1 + 0 + 0 + 0 + …

Clearly, then, the sum of this series is one. Therefore, 0 = 1.

What's wrong with this "proof"?


January 15th, 2014 (Permalink)

How to Misread a Poll

Here's Slate's David Weigel yesterday, in an article titled "Poll: Christie's GOP Approval Numbers Up Since Start of Scandal":

[Christie's]…chief political adviser…[said] that Christie was getting "positive, proactive" feedback from Republicans. You laugh…but then you look at the Monmouth poll, the only one taken in New Jersey since the scandal began.
"…Republicans are sticking by Christie, giving him an 89% approval rating which is in line with the 85% GOP support he received last month."
…The Republican uptick is a better omen for Christie than anything else in the poll.

These claims should set off some warning bells for the following reasons:

So, even without checking the polling report itself to see what the actual MoE is, you can tell that Weigel is probably making a big deal about nothing. I'll leave it as an exercise for the reader to check that report―see Source 1, below. It's a worthwhile exercise in not trusting everything you read in the media, not even what you read in The Fallacy Files!


  1. "New Jersey Riveted to Bridgegate", Monmouth University/Asbury Park Press Poll, 1/13/2014 (PDF). Update (3/16/2016): This source seems to be no longer available, so you'll just have to take my word for it after all.
  2. David Weigel, "Poll: Christie's GOP Approval Numbers Up Since Start of Scandal", Slate, 1/14/2014

Resource: How to Read a Poll

January 14th, 2014 (Permalink)

A Doublespeak Dictionary Puzzle

Can you figure out what an aerodynamic personnel decelerator is?


Source: Hugh Rawson, Dictionary of Euphemisms and Other Doubletalk, Revised Edition (1995)

January 6th, 2014 (Permalink)

This looks like a job for Fallacy Man!

Actually, that's not the real Fallacy Man in the Source, below. That's obviously Zorro with an "F" on the front of his shirt for some reason. Also, you can tell he's not the real thing because he calls the appeal to nature "the naturalistic fallacy", among other mistakes. It's the Masked Man Fallacy! However, it's funny and the Fallacy Fallacy at the end is a nice touch. Check it out.

Source: "The Adventures of Fallacy Man", Existential Comics

Via: Corey Mohler, "Have You Met Fallacy Man? Here’s How to Defeat Him.", Slate, 1/6/2014

Why the headline is wrong: The article is a report on a survey done by the YouGov organization―see the Source, below. The survey asked various questions about whether those surveyed pay attention to online reviews of products, and whether they ever reviewed products themselves. Based on this kind of information, how could one determine that what the headline claims is true? Instead, here's the headline of YouGov's own article reporting the survey results:

21% of Americans who have left reviews, reviewed products without buying or trying them

So, the "fifth" in question is not the proportion of online reviews left by people who have not tried the product, but the proportion of people who have reviewed products who have left such a review. These are different things, and there's no good reason to think that these proportions should be the same. For instance, it's possible that a fifth of reviewers at least once review a product that they haven't tried, but that most of the reviews they leave are of products they have used. Thus, the proportion of such fraudulent reviews might be much smaller than a fifth of all reviews. In contrast, it's also possible that a small proportion of dishonest reviewers leave a much larger proportion of all reviews than a fifth, assuming that they work much harder to do so than honest reviewers.

So, the survey result concerns the proportion of reviewers who have left dishonest reviews, which the headline reports as the proportion of reviews that are dishonest. Clearly, the survey result doesn't support the alarming headline warning the reader not to trust online reviews.

Source: Zaraida Diaz, "21% of Americans who have left reviews, reviewed products without buying or trying them", YouGov, 1/22/2014

Solution to an Infinitely Paradoxical Proof:

The series of additions―let's call it "S", for short―in the "proof" is infinite, so adding it up would take an infinite amount of time. More importantly, the usual laws of arithmetic, such as the associative law used in the "proof", are defined only for finite sums. Now, some infinite series converge on a certain value, for instance:

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …

If you start to carry out the additions in this series, you get the following series of sums:

3/4, 7/8, 15/16, 31/32, …

The series approaches closer and closer to the number 1 as you carry out each addition, so it's reasonable to consider this and other such infinite series―they are called "convergent"―to have "sums", namely, the numbers that they converge upon. Moreover, such "sums" have uses in physics, and even in philosophy for explaining Zeno's paradoxes. However, it should be kept distinctly in mind that this is an extension or generalization of the sense of "sum" that applies to finite additions, and it doesn't immediately follow that the laws of arithmetic apply even to convergent series.

However, S is not a convergent series, rather it is "divergent": if you do partial sums you will see that they switch back and forth between 0 and 1, but S never appears to converge on either―such series are said to "oscillate". Now, while the paradoxical "proof" does not prove that 0=1, it does show that the associative law for addition does not apply to series such as S.

Is there any sense in which you can "sum" a divergent series? The answer is "yes", keeping in mind that we're getting farther and farther away from familiar addition. For instance, in the case of series such as S which oscillate between two values, we can take the "sum" of the series to be the average of the two values, so that the sum of S is 1/2 rather than 0 or 1.

However, careful mathematicians always refer to such "sums" as "Cesàro sums" or "Ramanujan sums" or whatever, since there are different methods. There's nothing mathematically wrong with such methods, but it should be kept in mind that "summing" an infinite series in these senses is not the same thing as summing a finite one, and you can't just use the usual laws of arithmetic without further ado.


Answer to a Doublespeak Dictionary Puzzle: This is military jargon for a parachute.

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