From Barry Greenstein’s insightful poker book, Ace on the River:

Someone shows you a coin with a head and a tail on it. You watch him flip it ten times and all ten times it comes up heads. What is the probability that it will come up heads on the eleventh flip?

A novice gambler would tell you, “Tails is more likely than heads, since things have to even out and tails is due to come up.”

A math student would tell you, “We can’t predict the future from the past. The odds are still even.”

A professional gambler would say, “There must be something wrong with the coin or the way it is being flipped. I wouldn’t bet with the guy flipping it, but I’d bet someone else that heads will come up again.”

Yes I know the math student would really say “individual trials are uncorrelated,” not “we can’t predict the future from the past.” The lesson still holds.

Happy Labor Day, everyone.

Yes, I know, I’m not very good at this hiatus thing. But there is important news that needs to be promulgated widely — the news of calculus. No more will innocent citizens cower in fear at the thought of derivatives and integrals, or flash back in horror to the days of terror and confusion in high-school math class. Because now there is a cure for these maladies — The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse.

Yes, you read that subtitle correctly. Let’s be clear: this book is probably not for you. That’s because you, I have no doubt, already love calculus. You carry a table of integrals in your back pocket, and you practice substituting variables to while away the time in the DMV. This isn’t the book for people who already appreciate the austere beauty of a differential equation, or even for people who want to study up for their AP exam.

No, this is the book for people who hate math. It’s for people who look at you funny and turn away at parties when you mention that you enjoy science. It’s for your older relatives who think you’re crazy for appreciating all that technical stuff, or your nieces and nephews who haven’t yet been captivated by the beauty of mathematics. The Calculus Diaries is the book for people who need to be convinced that math isn’t an intimidating chore — that it can be fun.

Know anybody like that? Any gift-giving holidays coming up?

Now it’s true, I know the author. In fact, I appear as a character in the book (to a certain degree of comic effect). I’m the one who gets soaked when we ride Splash Mountain at Disneyland, but also the one who maximizes his winnings at craps by clever betting in Vegas. You get the idea: this isn’t a textbook, it’s a tour through the real world (and occasional fantasy worlds), pointing out that math is all around us, and that perceiving it is kind of cool.

When you understand math, how you think about the world changes. Every day, we all change position by accumulating velocity, or do informal optimization problems when making a decision. But most people don’t know about the wonderful insights that math can add to these processes. You know, because you are a mathphile. But you are outnumbered by the mathphobes. You have a secret that they don’t know, but now there’s a way to share it. What are you waiting for?

Ah, not this one again. The folks at Iglu Cruises have put together a helpful infographic to explain various features of the Gulf of Mexico oil spill (via Deep Sea News). Here’s the bit where they compare the recent spill (which, by the way, is still ongoing at a fantastic rate) to previous oil spills. Click for full resolution.

Doesn’t make the current fiasco seem so bad, does it? That little blob on the left looks a lot smaller than the blob right next to it, representing Saddam Hussein’s dump of oil into the Persian Gulf during the first Gulf War. In fact, when you think about it, it looks a lot smaller. Which is weird, when you look at the numbers and see that the current spill is 38 million gallons (as of May 27), while the Iraqi spill was 520 million gallons, a factor of about 14 times bigger. The blob representing Iraq’s spill seems a lot more than 14 times the size of the blob for the current spill. You don’t think — no, they couldn’t have done that. Could they?

Yes, they did. When measure the diameter of the circle representing the Iraqi spill, I get about 360 pixels (in the high-res version), while the smaller spill is about 26 pixels — a factor of about 14 larger. But that’s the diameter, not the area. The area of a circle, as many of us learned when we were little, is proportional to the square of its radius: A = π r². The radius is just half the diameter, so the area is proportional to the diameter squared, not to the diameter. In other words, that big blob is about (14)² = 196 times the area of the little one, when it should be only 14 times bigger.

I remember reading on some other blog about this same mistake being made in a completely different context, but I have no recollection of where. (Update: it was at Good Math, Bad Math, sensibly enough.) Probably won’t be the last time.

In today’s link roundup, Uncertain Chad points to a new digital library of mathematical function. As a huge, huge fan of Abramowitz & Stegun (which can now be downloaded as a PDF, if you would rather not have it sitting majestically on your shelf), I am thrilled.

Sometimes you’ll be happily calculating along, and wind up with an equation you wouldn’t want to meet at night in a dark alley. But, a quick flip through Abramowitz & Stegun frequently turns up your nemesis, along with handy tricks for disarming it. Additional satisfaction comes when you write the paper, and get to throw off lines like “The solutions to equation 4 are confluent hypergeometric functions (of the first kind)”.

However, it will be hard top my amusement when I once discovered that the solutions to my problem were closely related to Anger functions.

I can only hope that Dr. Hate and Professor Loathing someday derive equally useful function forms.

Here’s a fun logic puzzle (see also here; originally found here). There’s a family resemblance to the Monty Hall problem, but the basic ideas are pretty distinct.

An eccentric benefactor holds two envelopes, and explains to you that they each contain money; one has two times as much cash as the other one. You are encouraged to open one, and you find $4,000 inside. Now your benefactor — who is a bit eccentric, remember — offers you a deal: you can either keep the $4,000, or you can trade for the other envelope. Which do you choose?

If you’re a tiny bit mathematically inclined, but don’t think too hard about it, it’s easy to jump to the conclusion that you should definitely switch. After all, there seems to be a 50% chance that the other envelope contains $2,000, and a 50% chance that it contains $8,000. So your expected value from switching is the average of what you will gain — ($2,000 + $8,000)/2 = $5,000 — minus the $4,000 you lose, for a net gain of $1,000. Pretty easy choice, right?

A moment’s reflection reveals a puzzle. The logic that convinces you to switch would have worked perfectly well no matter what had been in the first envelope you opened. But that original choice was complete arbitrary — you had an equal chance to choose either of the envelopes. So how could it always be right to switch after the choice was made, even though there is no Monty Hall figure who has given you new inside information?

(more…)

I did my graduate work at the University of Chicago, and lived in Hyde Park. On occasion I would take the bus (the #6 Jeffery Express) to downtown. Although the buses were scheduled to run every 15 minutes, I would invariably end up waiting a half hour. Sometimes more. Often in the freezing cold, or the sweltering heat. Most infuriatingly, when the bus finally arrived, there was always another one immediately behind it! The buses inevitably came in pairs. Sometimes even in triples or quads.

Let’s assume that the buses are supposed to arrive every 15 minutes. If the buses adhered to their schedule, and I showed up at a random time, I should generally have to wait roughly half the mean bus arrival time: 7.5 minutes. If the buses were totally random, then I would have to wait the average time between bus arrivals: 15 minutes (if you haven’t thought about this before, this statement should sound crazy; perhaps I’ll do a future post on it). So the question is: why did I always end up waiting roughly 30 minutes or more?

I always assumed that the Universe was conspiring against me. This is a common feeling in graduate school. However….

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As I reached the end of what would be called high school in the US, I was certain that I wanted more than anything to become a mathematician. Soon afterwards, as a beginning maths undergraduate at Cambridge, I had become even more committed to the subject, after having spent some time reading about the history of the subject, and becoming enthralled by the lives and contributions of some of the great mathematicians. Among these, I found myself personally drawn to Bertrand Russell, partly because I was more interested in philosophy then than I am these days, but mostly because of the sheer grandness of the vision embodied in the Principia Mathematica – the opus co-authored by Russell with Alfred North Whitehead – and it’s later challenge from Gödel.

Nevertheless, as one gets older, reads more, and hopefully gains a more sophisticated knowledge of the subject, one’s tastes tend to change somewhat. In my case, a gradual shift in my interests from pure to applied mathematics, and finally to theoretical physics opened up an increasing range of giants to understand and respect. Somehow though, I have always retained a soft spot for Russell; perhaps because of his atheism, perhaps because of his breadth, but more so I think, because when I think of him I can quite viscerally recall the way reading about him made me feel about mathematics.

Because of this, while I have never become an avid reader of graphic novels, I’m hoping to get hold of a copy of Apostolos Doxiadis’ Logicomix, which I learned about via The Guardian, and which the web site describes as

Covering a span of sixty years, the graphic novel Logicomix was inspired by the epic story of the quest for the Foundations of Mathematics.

This was a heroic intellectual adventure most of whose protagonists paid the price of knowledge with extreme personal suffering and even insanity. The book tells its tale in an engaging way, at the same time complex and accessible. It grounds the philosophical struggles on the undercurrent of personal emotional turmoil, as well as the momentous historical events and ideological battles which gave rise to them.

The role of narrator is given to the most eloquent and spirited of the story’s protagonists, the great logician, philosopher and pacifist Bertrand Russell. It is through his eyes that the plights of such great thinkers as Frege, Hilbert, Poincaré, Wittgenstein and Gödel come to life, and through his own passionate involvement in the quest that the various narrative strands come together.

The web site contains a few samples of what to expect, plus a nice summary of the cast of characters. To a graphic novel newbie like myself, it isn’t obvious what to expect from a telling of this kind of sweeping academic story in such a format. But the team involved looks promising, and I’m sufficiently fascinated by the subject matter that I’m really looking forward to getting a look at Logicomix.

Physicists and mathematicians who think about higher-dimensional spaces are, if they allow their interest to somehow become public knowledge, inevitably asked: “How can you visualize more than three dimensions of space?” There are at least three correct answers: (1) You can’t. (2) You don’t have to; manipulating abstract symbols is enough to help you figure things out. (3) There are tricks to help you pseudo-visualize higher-dimensional objects by cleverly projecting them into three dimensions; see here and here.

But really, why can’t we visualize things in more than three dimensions of space? Could a Flatlander, living in a world with only two spatial dimensions, learn to visualize our three-dimensional world? Could we somehow, through practice or direct intervention in the brain, train ourselves to truly visualize more dimensions?

I can think of a couple of explanations why it’s so hard, with different ramifications. One would be simply that our imaginations aren’t good enough to project our consciousness into a constructed world so very different from our own. Could you, for example, really imagine what it’s like to live in two dimensions? Sure, you can visualize Flatland from the outside, but what about asking what it’s like to really be a Flatlander? The best I can do is to imagine a line, flickering with colors, surrounded by darkness on either side. But the darkness is still there, in my imagination.

The other possible explanation is that the process of visualization takes up a three-dimensional space in our actual brain, preventing us from “tuning a dimensionality knob” on our imaginations. The truth is certainly more complicated than that (and I’m not experts, so anyone who is should chime in); the visual cortex itself is effectively two-dimensional, but somehow our brain reconstructs a three-dimensional image of the space around us.

Maybe this could be a new tantric discipline: visualization in higher dimensions. Or maybe the Maharishi already offers a course?

Randall Monroe (of xkcd fame) has taken on Verizon:

(via failblog)

I sometimes forget that we don’t all read the same blogs, and that it’s good to recommend some of the fun stuff out there on the internets. So let me give a shout-out to Matt Springer at Built on Facts, who had the brilliant idea of discussing a different function every Sunday. Functions are one of those things that are as necessary to math and science as breathing, but which don’t necessarily percolate into the wider world. And he (quite correctly, I think) interprets his self-imposed mandate fairly liberally, taking the time to talk about various issues in middle-level mathematics. Here are some selections from Matt’s series:

• Exponential
• Arctangent
• Witch of Agnesi
• Stirling’s approximation
• Continuous but almost-nowhere differentiable

Consider this an open thread to recommend other stuff we should all be reading. Or your favorite functions.