How has Bayes’ Rule changed the way I think?

People talk about how Bayes’ Rule is so central to rationality, and I agree. But given that I don’t go around plugging numbers into the equation in my daily life, how does Bayes actually affect my thinking?
A short answer, in my new video below:



(This is basically what the title of this blog was meant to convey — quantifying your uncertainty.)

Colbert Deconstructs Pop Music, Finds Mathematical Nerdiness Within

Stephen Colbert channeling Kurt Godel

And here I thought I didn’t like pop music. Turns out I just hadn’t found the songs that invoke questions about the foundations of logic and mathematics. Fortunately, Stephen Colbert brings our attention to the fascinating – and paradoxical! – pop song “That’s What Makes You Beautiful” by One Direction. Watch Stephen do his thing deconstructing the lyrics with glorious nerdy precision before we take it even further (the good part starts at 1:54 or so):

For those of you who can’t watch the video, here’s the nerdy part, hastily transcribed:

Their song “That’s What Makes You Beautiful” isn’t just catchy, it has a great message. “You don’t know you’re beautiful. That’s what makes you beautiful.”

First of all: great dating advice. Remember girls, low self esteem – very attractive to men. Guys always go for the low hanging fruit, easy pickings.

Second: the lyrics are incredibly complex! You see, the boys are singing “You don’t know you’re beautiful, that’s what makes you beautiful.” But they’ve just told the girl she’s beautiful. So since she now knows it, she’s no longer beautiful!

But – stick with me, stick with me, oh it goes deeper! – but she’s listening to the song, too. So she knows she’s not beautiful. Therefore, following the syllogism of the song, she’s instantly beautiful again!

It’s like an infinite fractal recursion, a flickering quantum state of both hot and not. I mean, this lyric as iterated algorithm could lead to a whole new musical genre. I call it Mobius pop, which would include One Direction and of course the rapper MC Escher.

They say the way to a man’s heart is through his stomach but honestly, talking about recursion, fractals, and flickering quantum states does far more to win my love.  We can find intellectual stimulation in anything!

And there’s more – we can go nerdier!

Stick With Me, Stick With Me, Oh It Goes Deeper

Let’s analyze the dilemma a bit further:

  1. She can’t KNOW she’s beautiful because, as Stephen points out, that leads to a logical contradiction – she would no longer be beautiful.
  2. She can’t KNOW that she isn’t beautiful, because that also leads to a logical contradiction – she would be beautiful again.
  3. It’s impossible for the girl to know that she is or isn’t beautiful, so she has to be uncertain – not knowing either way.
  4. This uncertainty satisfies the requirements: she doesn’t know that she’s beautiful, therefore, she’s definitely beautiful and can’t know it.

It turns out she’s not in a flickering state of hot and not, she’s perpetually hot – but she cannot possibly know it without logical contradiction! From an external perspective, we can recognize it as true. From within her own mind, she can’t – even following the same steps. How weird is that?

Then it hit me: the song lyrics are a great example of a Gödel sentence!

Gödel sentences, from Kurt Gödel’s famous Incompleteness Theorems, are the statements which are true but unprovable within the system.  Gödel demonstrated that every set of mathematical axioms complex enough to stand as a foundation for arithmetic will contain at least one of these statements: something that is obviously true from an outside perspective, but isn’t true by virtue of the axioms.  (He found a way to coherently encode “The axioms do not prove this sentence to be true.”)  This raises the question: what makes a mathematical statement true if not the fact that it can be derived from the axioms?

Gödel’s findings rocked the world of mathematics and have had implications on the philosophy of mind, raising questions like:

  • What does it mean to hold a belief as true?
  • What are our minds doing when we make the leap of insight (if insight it is) that identifies a Gödel sentences as true?
  • How does this set us apart from the algorithmic computers, which are plagued by their own version of Incompleteness, the Halting Problem?

I had no idea pop music was so intelligent!

Was the boy band comparing her, not to a summer’s day, but a turing-complete computer?  Were they glorifying their listeners by reminding us that, according to some interpretations of Incompleteness Theory, we’re more than algorithmic machines?  Were they making a profound statement about mind/matter dualism?

I don’t know, but apparently I should turn on the radio more often.

[For related reading, see various analyses of Mims' "This is Why I'm Hot"]

As they say in the Sirius Cybernetics Corporation: Share and Enjoy!

Easy Math Puzzle – Or is it?

How good are you at basic math? Can you solve this simple logic puzzle? Here, give it a go and let me know how long it took you to answer:

Got it yet?

It looks easier than it is. The options are presented beautifully to cause maximum mental confusion.

As my dad put it, the answer depends on the answer. If the answer is 60%, it’s 25%. If the answer is 25% it’s 50%. If the answer is 50% it’s 25%. There’s an endless loop with no correct answer.

Don’t lose sleep, I “found” an answer, it was hidden: [edited for clarity]

Yes, I photoshopped this. I’m either cheating or engaging in outside-the-box thinking. Sometimes it’s tough to tell the difference.

My preferred set of answers would be:

  • A) 25%
  • B) 50%
  • C) 75%
  • D) 50%

Though I’m tempted to throw a “0%” in for good measure…

(Puzzle via PostSecret by way of Spencer of Ask a Mathematician/Ask a Physicist)

[Edited for clarity]

Spinoza, Godel, and Theories of Everything

On the latest episode of Rationally Speaking, Massimo and I have an entertaining discussion with Rebecca Goldstein, philosopher, author, and recipient of the prestigious MacArthur “genius” grant. There’s a pleasing symmetry to her published oeuvre. Her nonfiction books, about people like philospher Baruch Spinoza and mathematician Kurt Godel, have the aesthetic sensibilities of novels, while her novels (most recently, “36 Arguments for the Existence of God: A Work of Fiction”) have the kind of weighty philosophical discussions one typically finds in non-fiction.

It’s a wide-ranging and fun conversation. My main complaint is just over her treatment of Spinoza. Basically, people say he “believed God was nature.” That always made me roll my eyes, because it’s not making a claim about the world, it’s merely redefining the word “God” to mean “nature,” for no good reason. I voice this complaint to Rebecca during the show and she defends Spinoza; you can see what you think of her response, but I felt it to be weak; it sounded like she was just pointing out some dubious similarities between nature and the typical conception of God.

Nevertheless! It’s certainly worth a listen:

Lies and Debunked Legends about the Golden Ratio

In my eyes, there’s a general pecking order for named mathematical constants. Pi is at the top, e gets a good amount of attention, and Tau, like a third-party candidate, sits by itself on the fringes while its supporters tell anyone who’ll listen that it’s a credible alternative to Pi. But somewhere in the middle is Phi, also known as the Golden Ratio. It’s no superstar, but it gets its fair share of credit in geometry and culture.

I was first introduced to Phi as a kid by watching the charming video Donald in Mathmagic Land. One of the things I remembered over the years is that the Greeks used the Golden Ratio in their paintings and architecture, particularly the Parthenon. Thanks to the power of the internet, I can share this piece of my childhood with you:

How brilliant and advanced of the Greeks, right? But there’s one problem…

It’s probably not true. My faith was first shaken reading Keith Devlin’s The Unfinished Game, where he entertained a quick digression:

Two other beliefs about this particular number [Phi] are often mentioned in magazines and books: that the ancient Greeks believed it was the proportion of the rectangle the eye finds most pleasing and that they accordingly incorporated the rectangle in many of their buildings, including the famous Parthenon. These two equally persistent beliefs are likewise assuredly false and, in any case, are completely without any evidence. For one thing, tests have shown that human beings who claim to have a preference at all vary in the rectangle they find most pleasing, both from person to person and often the same person in different circumstances. Also, since the golden ratio is actually not a ratio of two whole numbers, it is impossible to construct (by measurement) a rectangle having that proportion, even in theory.

What?! Donald, I trusted you! It was tempting to tell myself that the Greeks could have found ways to approximate the ratio, and that this is just one source, and I’ve heard it so many times it must be true, and la la la I don’t want Donald to have lied to me.

But I looked into it a bit more, checking out what Mario Livio had to say about it in his book The Golden Ratio. He acknowledges that it’s a very common belief, but ultimately backed Devlin up:

The appearance of the Golden Ratio in the Parthenon was seriously questioned by University of Maine mathematician George Markowsky in his 1992 College Mathematics Journal article “Misconceptions about the Golden Ratio.” Markowsky first points out that invariably, parts of the Parthenon (e.g. the edges of the pedestal [in a provided figure]) actually fall outside the sketched Golden Rectangle, a fact totally ignored by all the Golden Ratio enthusiasts. More important, the dimensions of the Parthenon vary from source to source, probably because different reference points are used in the measurements… I am not convinced that the Parthenon has anything to do with the Golden Ratio.

So, was the Golden Ratio used in the Parthenon’s design? It is difficult to say for sure… However, this is far less certain than many books would like us to believe and is not particularly well supported by the actual dimensions of the Parthenon. [emphasis mine]

Alas, claims about the Greeks using Phi in their architecture seem overrated. Some sites bring you celebrity gossip, we bring gossip about celebrated mathematical constants. Welcome to Measure of Doubt!

Watching the video again, I can’t tell exactly how they decided where to overlay the Golden Rectangles. How much of the pedestal do we include in the rectangle? How much of the pillar? Does the waist start here, or there? It seems a bit arbitrary, as though we’re experiencing pareidolia and seeing the Golden Rectangle in everything.

Talk about disillusionment.

Self-Referential Haikus and Nerdy Math Shirts

I don’t always buy t-shirts. But when I do, I tend to make them really nerdy ones. ThinkGeek is a good source, but Snorg Tees might be my new favorite.

Self-reference, like this sentence, is hilarious.

But you can never have just one haiku. When they get out in public, they have a tendency to spawn as people are inspired to create their own. Here was my contribution to the arts:

Haiku are easy
but the ones I write devolve
into self-reference.

To which a friend responded,

Reference. Syllables?
If reference is two, I’m good.
Three? Then I am screwed.

If self-reference isn’t your cup of tea, SnorgTees also has a couple great math shirts:

That’s right: I keep it real. After I posted the picture to Facebook, a cousin commented:

It might be real…but it’s not natural.

and my dad chimed in with the brilliant:

Aren’t you the negative one.

I love my family and friends.

The darker the night, the brighter the stars?

“The darker the night, the brighter the stars” always struck me as a bit of empty cliche, the sort of thing you say when you want to console someone, or yourself, and you’re not inclined to look too hard at what you really mean. Not that it’s inherently ridiculous that your periods of pleasure might be sweeter if you have previously tasted pain. That’s quite plausible, I think. What made me roll my eyes was the implication that periods of suffering could actually make you better off, overall. That was the part that seemed like an obvious ex post facto rationalization to me. Surely the utility you gain from appreciating the good times more couldn’t possibly be outweighed by the utility you lose from the suffering itself!

Or could it? I decided to settle the question by modeling the functional relationship between suffering and happiness, making a few basic simplifying assumptions. It should look something roughly like this:

Total Happiness = [(1-S) * f(S)] – S

S = % of life spent in suffering
(1-S) = % of life spent in pleasure
f(S) = some function of S

As you can see, f(S) acts as a multiplier on pleasure, so the amount of time you’ve spent in suffering affects how much happiness you get out of your time spent in pleasure. I didn’t want to assume too much about that function, but I think it’s reasonable to say the following:

  • f(S) is positive — more suffering means you get more happiness out of your pleasure
  • f(0) = 1, because if you have zero suffering, there’s no multiplier effect (and multiplying your pleasure by 1 leaves it unchanged).

… I also made one more assumption which is probably not as realistic as those two:

  •  f(S) is linear.**

Under those assumptions, f(S) can be written as:
f(S) = aS + 1

Now we can ask the question: what percent suffering (S) should we pick to maximize our total happiness? The standard way to answer “optimizing” questions like that is to take the derivative of the quantity we’re trying to maximize (in this case, Total Happiness) with respect to the variable we’re trying to choose the value of (in this case, S), and set that derivative to zero. Here, that works out to:

f’(S) – Sf’(S) – f(S) – 1 = 0

And since we’ve worked out that f(S) = aS + 1, we know that f’(S) = a, and we can plug both of those expressions into the equation above:

a – Sa – aS – 1 – 1 = 0
a – 2aS = 2
-2aS = 2 – a
2aS = a -2
S = (a – 2) / 2a

That means that the ideal value of S (i.e., the ideal % of your life spent suffering, in order to maximize your total happiness) is equal to (a – 2)/2a, where a tells you how strongly suffering magnifies your pleasure.

It might seem like this conclusion is unhelpful, since we don’t know what a is. But there is something interesting we can deduce from the result of all our hard work! Check out what happens when a gets really small or really large. As a approaches 0, the ideal S approaches negative infinity – obviously, it’s impossible to spend a negative percentage of your life suffering, but that just means you want as little suffering as possible. Not too surprising, so far; the lower a is, the less benefit you get from suffering, so the less suffering you want.

But here’s the cool part — as a approaches infinity, the ideal S approaches 1/2. That means that you never want to suffer more than half of your life, no matter how much of a multiplier effect you get from suffering – even if an hour of suffering would make your next hour of pleasure insanely wonderful, you still wouldn’t ever want to spend more time suffering than reaping the benefits of that suffering. Or, to put it in more familiar terms: Darker nights may make stars seem brighter, but you still always want your sky to be at least half-filled with stars.

* You’ll also notice I’m making two unrealistic assumptions here:

(1) I’m assuming there are only two possible states, suffering and pleasure, and that you can’t have different degrees of either one – there’s only one level of suffering and one level of pleasure.

(2) I’m ignoring the fact that it matters when the suffering occurs – e.g., if all your suffering occurs at the end of your life, there’s no way it could retroactively make you enjoy your earlier times of pleasure more. It would probably be more realistic to say that whatever the ideal amount of suffering is in your life, you would want to sprinkle it evenly throughout life because your pleasures will be boosted most strongly if you’ve suffered at least a little bit recently.

** Linearity is a decent starting point, and worth investigating, but I suspect it would be more realistic, if much more complicated, to assume that f(S) is concave, i.e., that greater amounts of suffering continue to increase the benefit you get from pleasure, but by smaller and smaller amounts.

Tales of Badass Mathematicians: Cardano

Giorlamo Cardano (Sep 24, 1501 - Sep 21, 1576): Annoying, Arrogant, Brilliant, Badass.

When people think of excitement, intrigue, and violence, they rarely think of mathematicians. That’s because they haven’t heard enough about Girolamo Cardano: 16th century Italian mathematician, physician, inventor, and general badass. This situation must be remedied.

Cardano was one of the first mathematicians to publish an autobiography, and it’s well-deserved. Not only did he have academic accomplishments, he lead a fascinating life. I was reading about how he published the first mathematical examination of probability theory when this passage in Keith Devlin’s The Unfinished Game caught my eye:

“Throughout his life, Cardano was a compulsive gambler who needed every bit of help he could find at the gambling tables, from mathematics or any other source. (And he did find other sources of help. Once, when he suspected he was being cheated at cards, he took out the knife he always carried with him and slashed his opponent’s face.)”

Let’s just say that Cardano wouldn’t have stood idly by as Roman soldiers disturbed his circles. He wasn’t particularly strong, but according to his autobiography he trained persistently and became quite the swordsman. He also boasts, “Another feat I acquired was how to snatch an unsheathed dagger, myself unarmed, from the one who held it.” Not a mathematician to mess with.

Cardano was also a talented physician. Despite his abilities, the College of Physicians in Milan rejected him – ostensibly due to his illegimate birth, but probably because he had an annoying personality (something Cardano admits to). That didn’t stop Cardano – though it wasn’t allowed, he treated patients on the side and developed a reputation as one of the best. Even as his fame grew, he couldn’t help but make enemies.

“With a client list that soon included wealthy people of influence in Milan – including some members of the college – it was surely only a matter of time before the college would be forced to admit him. But then, in 1536, still fuming at his continuing exclusion, he killed his chances by publishing a book attacking not only the college members’ medical ability but their character as well.”

Oh, as we used to say in middle school, snap. (Actually, even calling them “artificial” and “insipid” didn’t prevent Cardano from getting into the College – Devlin goes on to say that they admitted him a couple years later under pressure from supporters.)

The drama goes on and on. He got into a feud with Tartaglia, another mathematician, over whether he had promised to keep Tartaglia’s method for solving cubic equations secret. His eldest son was convicted of poisoning his cheating wife, and Cardano wasn’t able to save him from torture and execution. Then his younger son got into gambling debt and stole from Cardano, who sadly turned him over to the authorities to be banished.

Later, in what Devlin suspects was a deliberate attempt to gain notoriety, Cardano provoked the Catholic Church by publishing a horoscope for Jesus Christ and writing a book praising anti-Christian Nero. He was convicted of heresy (To add to the intrigue, Wikipedia says that “Apparently, his own son contributed to the prosecution, bribed by Tartaglia.”) After serving a few months in prison and making up with the Pope, he spent the last few years of his life writing an autobiography.

Even his death had style. Cardano died on September 21th, 1576 – the exact date he had predicted years ago. It’s believed that he committed suicide just to make sure he got the date right. What a way to go.

De Finetti’s Game: How to Quantify Belief

What do people really mean when they say they’re “sure” of something? Everyday language is terrible at describing actual levels of confidence – it lumps together different degrees of belief into vague groups which don’t always match from person to person. When one friend tells you she’s “pretty sure” we should turn left and another says he’s “fairly certain” we should turn right, it would be useful to know how confident they each are.

Sometimes it’s enough to hear your landlord say she’s pretty sure you’ll get towed from that parking space – you’d move your car. But when you’re basing an important decision on another person’s advice, it would be better describe confidence on an objective, numeric scale. It’s not necessarily easy to quantify a feeling, but there’s a method that can help.

Bruno de Finetti, a 20th-century Italian mathematician, came up with a creative idea called de Finetti’s Game to help connect the feeling of confidence to a percent (hat tip Keith Devlin in The Unfinished Game). It works like this:

Suppose you’re half a mile into a road trip when your friend tells you that he’s “pretty sure” he locked the door. Do you go back? When you ask him for a specific number, he replies breezily that he’s 95% sure. Use that number as a starting point and begin the thought experiment.

In the experiment, you show your friend a bag with 95 red and 5 blue marbles. You then offer him a choice: he can either pick a marble at random and, if it’s red, win $1 million. Or he can go back and verify that the door is locked and, if it is, get $1 million.

If your friend would choose to draw a marble from the bag, he preferred the 95% chance to win. His real confidence of locking the door must be somewhere below that. So you play another round – this time with 80 red and 20 blue marbles. If he would rather check the door this time, his confidence is higher than 80% and perhaps you try a 87/13 split next round.

And so on. You keep offering different deals in order to hone in on the level where he feels equally comfortable selecting a random marble and checking the door. That’s his real level of confidence.

The thought experiment should guide people through the tricky process of connecting their feeling of confidence to a corresponding percent. The answer will still be somewhat fuzzy – after all, we’re still relying on a feeling that one option is better than another.

It’s important to remember that the game doesn’t tell us how likely we are to BE right. It only tells us about our confidence – which can be misplaced. From cognitive dissonance to confirmation bias there are countless psychological influences messing up the calibration between our confidence level and our chance of being right. But the more we pay attention to the impact of those biases, the more we can do to compensate. It’s a good practice (though pretty rare) to stop and think, “Have I really been as accurate as I would expect, given how confident I feel?”

I love the idea of measuring people’s confidence (and not just because I can rephrase it as measuring their doubt). I just love being able to quantify things! We can quantify exactly how much a new piece of evidence is likely to affect jurors, how much a person’s suit affects their persuasive impact, or how much confidence affects our openness to new ideas.

We could even use de Finetti’s Game to watch the inner workings of our minds doing Bayesian updating. Maybe I’ll try it out on myself to see how confident I feel that the Ravens will win the Superbowl this year before and after the Week 1 game against the rival Pittsburgh Steelers. I expect that my feeling of confidence won’t shift quite in accordance with what the Bayesian analysis tells me a fully rational person would believe. It’ll be fun to see just how irrational I am!

Happy Tau Day!

I almost missed the chance to promote Tau Day! Many of you probably know about Pi Day, held on March 14th. At my high school we used to bring in pies to the math room and eat them at 1:59PM in a glorious (and delicious) celebration of mathematics. But the inimitable Vi Hart lobs an objection: using Pi often doesn’t make as much sense as using Tau, the ratio of the circumference of a circle over its RADIUS.

Thus, we need a new day in celebration of the more-useful Tau:

Seeing as Tau is approximately 6.28 and today is June 28th, have yourself a great Tau Day and enjoy two pi(e)s! While you’re eating, you can go check out more of Vi Hart’s work – she does a fantastic job showing how much fun math can be. We need more voices like hers, and I’ll be sure to post more of her videos!


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