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!

“Stand back everyone, I’ve been trained for situations like this…”

Last night my friends and I ended up talking about real-life situations in which our math skills serendipitously came in handy. And I got to reminisce about my one exciting “Thank goodness I paid attention in math class!” moment:

I was a high school kid, working as a summer intern at the Corcoran Gallery of Art (this was back in my “I want to be a museum curator” phase). In the sales office one Friday afternoon, I overheard a conversation between my two bosses:

Boss 1: “The new ticket collector didn’t keep track of child and adult ticket sales separately this week. All she sent us is the total number of tickets and total revenue. 953 tickets, $9,050 revenue. But the accountant wants us to record child and adult sales separately.”

Boss 2: “Sigh. All right, I guess we’ll have to go get the pile of ticket stubs and sort them all. What a pain in the ass…”

Me (gasps, runs over): “Wait! We don’t need to sort ticket stubs! We already have all the information we need to solve this!”

Boss 1: “We do? How?”

Me: “We need to set up a system of equations! Okay, let’s call A the number of adult tickets and C the number of child tickets. How much does each one cost?”

Boss 1: “Adult tickets are $10, child tickets are $6.”

Me: “Okay! So we know that the total number of tickets is 953 so we can write A + C = 953. And we know the total revenue is $9050 so we can write $10A + $6C = $9050. So we have two equations,  two variables:

A + C = 953
10A + 6C = 9050

And now we just solve:
10 (953 – C) + 6C = 9050
4C = 480
C = 120. Therefore A = 953 – 120 = 833.
So that’s the answer — we sold 120 child tickets, and 833 adult tickets.”

My bosses were delighted with the “cool trick” I had used. And I like to think my 7th-grade math teacher would have been tickled pink if she’d seen that go down. How often do you get to actually apply your word-problem skills in real life?

Nothing quite that math-textbook perfect has happened since. Though I keep hoping that someday I’ll overhear someone saying, “My friend and I wanted to meet up for lunch tomorrow, so we agreed to each leave our apartments at noon and walk towards each other’s place until we met. He walks at a rate of 4 mph and I walk at a rate of 3 mph. If only there was some way to figure out where and when we would meet so that I could make a lunch reservation…”

All Wikipedia Roads Were Forced to Philosophy

Does everything boil down to philosophy? A case could be made that it’s really all about math and science. Or perhaps breasts. In the alt-text of Wednesday’s XKCD comic, a specific challenge was made: “Wikipedia trivia: if you take any article, click on the first link in the article text not in parentheses or italics, and then repeat, you will eventually end up at ‘Philosophy’.”

Game on. I already had a tab open to the Wikipedia page for “Where Mathematics Comes From” and decided to see how long it took:

Where Mathematics Comes From

  1. George Lakoff
  2. Cognitive linguistics
  3. Linguistics
  4. Human
  5. Taxonomy
  6. Science
  7. Knowledge
  8. Fact
  9. Information
  10. Sequence
  11. Mathematics
  12. Quantity
  13. Property (Philosophy)
  14. Modern Philosophy
  15. Philosophy

Ok, maybe that one was too easy. Let’s use my go-to example: Waffles:


  1. Batter (cooking)
  2. Flour
  3. Powder
  4. Solid
  5. State of matter
  6. Phase (matter)
  7. Outline of physical science
  8. Natural science
  9. Science
  10. Knowledge
  11. Fact
  12. Information
  13. Sequence
  14. Mathematics
  15. Quantity
  16. Property (Philosophy)
  17. Modern Philosophy
  18. Philosophy

Well son of a gun. I’ve tried it with ‘Mongoose’ (11 clicks), Baltimore Ravens football player ‘Ed Reed’ (16 clicks), and ‘Lord of the Rings’ (22 clicks). All led back to Philosophy.

Well, we don’t END at philosophy – we could keep going. It turns out that (as of writing this) there’s a 19-step loop including philosophy, science, mathematics, and mammory gland.

We could just as easily say that all paths lead to science! Or math! Or breasts!

However, before you get too excited, it turns out there’s some mischief afoot.

First, it’s been two days since the XKCD comic went up, and considering how malleable Wikipedia is, some things have been changed. I was suspicious that Quantity’s first link went to Property (Philosophy) so I checked the history page:

# (cur | prev) 09:54, 25 May 2011 (talk) (14,042 bytes) (Edited for xkcd)

# (cur | prev) 09:28, 25 May 2011 (talk) (14,004 bytes) (Undid revision 430815864 by Antony-22 (talk) see today’s xkcd, without the “property” link, it breaks the “all pages eventually end up at philosophy ” game. The link should be there)

I actually found a small loop that Male leads to Gender, which leads back to Male. I expect the Male one will be “fixed” at some point (a phrase you don’t hear outside the veterinarian very often).

The philosophy topic has been in the XKCD forums last Sunday, and the idea was around for longer than that. Tricky editing has been going on toward this goal for a while.

I’d heard that philosophy leads to reason, which leads to rationality, which leads back to philosophy. That’s been changed since Wednesday, and I wonder if a deliberate effort moved the path from rationality to breasts.

Yes, it’s true that the first link on an article is likely to be broad and trend toward science/philosophy, but this isn’t unguided evolution. This is intelligently designed.

Quality Webcomic Nerdiness

I love it when comics are both intelligent and fun. One of the first things I do when I wake up is read webcomics – it makes for a good transition to consciousness. And Today’s Saturday Morning Breakfast Cereal is particularly relevant to last month’s discussion:

First thought: that could totally have been my high school.
Actual first thought: Blurg… (read: Where’s my coffee)
Second thought: Wait a minute, 0^0 is a complicated question! It’s not necessarily 1!

But before I got too far up on my high horse about knowing that 0^0 is poorly defined, I looked at the scroll-over bonus panel:

And this is why I love SMBC.

Math education: you’re doing it wrong

Recent discussion about the problems with our educational system reminded me about the story of why my friend J almost didn’t make it into his high school honors math class. Now, to clarify, J is easily one of the smartest people I know. But he is also a smartass, and thirteen-year-old J was certainly no different.

On the entrance exam for his honors math class, several of the problems asked you to fill in the next number in the sequence, such as: 2, 4, 8, 16, _?_. Obviously, whoever wrote the exam wanted you to complete that sequence with “32,” because the pattern they’re thinking of is powers of 2. For n = 1, 2, 3, 4, 5, the formula  2n = 2, 4, 8, 16, 32. But J didn’t write “32.” He wrote “π.”

When his teacher marked that problem wrong (as well as all of the other sequence questions, which J had answered in similar fashion), J explained that there are literally an infinite number of numbers that could complete that sequence, because there are an infinite number of curves which go through the points (1, 2), (2, 4), (3, 8), and (4, 16). Sure, he said, one of those curves is the obvious one which also goes through (5, 32), but you can also derive a curve which goes through (5, π). He showed her an example:

As you can see if you try plugging in the numbers 1, 2, 3, 4, and 5, to the equation above, you get the sequence 2, 4, 8, 16, π. Here are the two curves plotted on a graph, both the “correct” curve and J’s smartass curve (hat tip to the mathematician at www.askamathematician.com for graphing this for me in Mathematica):

Anyway, after thirteen-year-old J explained the math behind his unconventional, but admittedly accurate, answer to the original problem, his teacher replied, “Oh come on, you knew what it was asking for!” and refused to give him any credit. I can’t think of a better illustration of the triumph of the stick-to-the-book method of teaching over kids’ innate creativity… or of the triumph of math education over actual math skills.

What is 0^0? And is math true, or just useful?

When you hear mathematicians talk about “searching” for a proof or having “discovered” a new theorem, the implication is that math is something that exists out there in the world, like nature, and that we gradually learn more about it. In other words, mathematical questions are objectively true or false, independent of us, and it’s up to us to discover the answer. That’s a very popular way to think about math, and a very intuitive one.

The alternate view, however, is that math is something we invent, and that math has the form it does because we decided that form would be useful to us, not because we discovered it to be true. Skeptical? Consider imaginary numbers: The square root of X is the number which, when you square it, yields X. And there’s no real number which, when you square it, yields -1. But mathematicians realized centuries ago that it would be useful to be able to use square roots of negative numbers in their formulas, so they decided to define an imaginary number, “i,” to mean “the square root of -1.” So this seems like a clear example in which a mathematical concept was invented, rather than discovered, and in which our system of math has a certain form simply because we decided it would be useful to define it that way, not because that’s how things “really are.”

This is too large of a debate to resolve in one blog post, but I do want to bring up one interesting case study I came across that points in favor of the “math is invented” side of the debate. My friends over at the popular blog Ask a Mathematician, Ask a Physicist did a great post a while ago addressing one of their readers’ questions: What is 0^0?

The reason this question is a head-scratcher is that our rules about how exponents work seem to yield two contradictory answers. On the one hand, we have a rule that zero raised to any power equals zero. But on the other hand, we have a rule that anything raised to the power of zero equals one. So which is it? Does 0^0 = 0 or does 0^0 = 1?

Well, I asked Google and according to their super-official calculator, the answer is unambiguous:

Indeed, the Mathematician at AAMAAP confirms, mathematicians in practice act as if 0^0 = 1. But why? Because it’s more convenient, basically. If we let 0^0=0, there are certain important theorems, like the Binomial Theorem, that would need to be rewritten in more complicated and clunky ways. Note that it’s not even the case that letting 0^0=0 would contradict our theorems (if so, we could perhaps view that as a disproof of the statement 0^0=0). It’s just that it would make our theorems less elegant. Says the mathematician:

“There are some further reasons why using 0^0 = 1 is preferable, but they boil down to that choice being more useful than the alternative choices, leading to simpler theorems, or feeling more “natural” to mathematicians. The choice is not “right”, it is merely nice.”

This is what happens when you fish for a pep talk from a mathematician

ME (whining at being unable to accomplish something): “Ugh, I feel so useless and pointless.”

MATHEMATICIAN: “Aw, you’re not pointless! You have lots of points. An infinite number, in fact.”

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