Category Archives: mathematics

Review 218: Why does E=MC2?

LL 218 - Why Does EMC2Why Does E=mc2? by Brian Cox and Jeff Forshaw

Way back in the mists of ancient time, when I was a college drama student, we all went down to New York City to see Tom Stoppard’s Arcadia. I don’t remember much of it now, but I remember it had to do with math and fractals, time and space, and that when the play was over and we all went outside for a smoke, I had a moment of what could only be called sublime clarity. I stood out there with my cigarette, staring off into the middle distance, and – for just an ever-so-brief time – I understood everything.

Not the play, mind you. Everything. It all made sense. It was nothing I could have put into words or explained in any period of time shorter than a lifetime, but it all worked. It all fit together, and I knew what the universe was and what my place in it was. It’s probably how Fenchurch felt in The Hitchhiker’s Guide right before the Earth was demolished.

It... it all makes sense now.

It… it all makes sense now.

And that feeling was wonderful.

It passed, though, because no one can be allowed to hold on to that kind of clarity of understanding. We’d never get anything done. By the time I got on the bus, I was trying to claw my way back to it, understanding but not caring that this was a place you couldn’t find the same way twice. The fine, crystalline perfection of the universe had once again been hidden from my mind, and all that was left was the memory of what it had felt like to know that everything was as it should be.

Reading this book was kind of the opposite of that experience. On every page, I knew that if I would be able to hold on to these ideas just a moment longer, if I could just put the pieces together a little faster, then I would have true understanding of the elegant beauty of creation. But I couldn’t, and I was left with the feeling that it was my own shortcomings that were at fault, rather than those of the authors.

Cox and Forshaw have set a very interesting challenge for themselves in this book. They want to explain one of the most famous equations in human history, and to do it in such a way that the non-scientist reader can understand not only what it means, but where it came from and what its implications are. This is no mean feat, of course, on any front. For all its simplicity, E=mc2 contains within it some of the most important and fundamental understandings about how the universe works. To truly understand this equation is to understand time and space, matter and energy, existence in four dimensions and at scales both vast and tiny.

They begin with what looks like a very simple question: where are we? Galileo pondered this question for a while, and came up with an answer that was probably both enlightening and horrifying for his time.

We don’t know.

Very helpful, thank you.

Very helpful, thank you.

Oh sure, we can know where we are in relation to something else – between a pair of arbitrarily numbered latitude and longitude lines, for example, or at a position around the star that we orbit. But a moment’s thought reveals that we still need to explain where the reference point is, and that we can only explain that in relation to something else, which can only be positioned by yet another relative measurement. In other words, there is no such things as an absolute location in space. There is no universal “there” there by which we can understand the position of anything.

Man, that must’ve freaked him out.

The next insight is that if there is no absolute place, then there also cannot be any absolute motion. As I type this, I am sitting in my comfy chair. As far as I’m concerned, I’m motionless. But I’m not. To an alien on the moon, I’m moving with the Earth’s rotation, whipping past at a breakneck pace of about 1,600 kilometers per hour. On top of that, the Earth is moving around the sun at over 100,000 km/h. which is in turn dragging the whole solar system around the center of the galaxy at roughly 220 kilometers per second, and the galaxy itself is moving through intergalactic space at over 600 km/s, and space itself is expanding at what can only be Ludicrous Speed.

So questions that seem like they should be simple turn out to be really hard to answer. But what comes next is even worse: if there is no absolute place or motion, then what about time? How can we have an immutable, fixed time if there is no such thing as an immutable, fixed place?

Look, I don't know how to make it any simpler than this.

Look, I don’t know how to make it any simpler than this.

Cox and Forshaw proceed to lead us by the hand through the discoveries and realizations of scientists such as Faraday and Maxwell, with a little bit of Pythagoras and Galileo, before bringing us to Einstein and beyond. Through the use of lightspeed trains, mirror-clocks, and a whole variety of illustrative analogies, they take us step-by-step through the process of moving from our understanding of three-dimensional semi-Euclidean space to a four-dimensional spacetime. They guide us through physics and geometry, on scales both large and small, and show not just what E=mc2 means, but how Einstein got to it, and how we’ve proven it so far.

In that sense, this book is a great success. The popular vision of Einstein is that he came up with Relativity because he was bored at work, and that it popped into his head fully formed. But without the work of countless scientists before him, Einstein wouldn’t have had a place to start. E=mc2 is built on the foundations of meticulous science, and is supported by a logical structure that is both elegant and simple. What’s more, his theories of relativity have been tested again and again in all kinds of ways, and they have stood up to those tests. And not for lack of trying, mind you – there isn’t a physicist alive who wouldn’t be thrilled to prove Einstein wrong and propose a more accurate version of reality. But so far, it seems to be the best explanation there is.

For all their care and meticulousness, however, the book is still a bit tough to get through. One of the things that got in my way was how they constantly apologized for using math. I understand why they did it – a lot of adults have a hate-fear relationship with math, and especially equations that start using letters. Math still has an element of mystery and wizardry about it, at least if you’re not very proficient in it, and I get that they didn’t want to scare off any math-phobics from their book.

Wau!

Wau!

But at the same time, I think I would rather they had said, “Okay, follow along with us – it’s about to get MATHTASTIC!” Well, maybe not those words, but I started to get a little tired of being talked to like a timid child as the book went on. They said over and over that I could skip the math parts if I wanted to, and sure enough that’s what I ended up doing. But I think I would have come out of this book with a much better sense of understanding and accomplishment if Cox and Forshaw had said, “We’re going to do math and you’re going to understand it.” As it is, they talked to me like I was a slightly dim child, and I still didn’t fully understand. So what, then, does that say about me?

That I seriously overthink things, for a start.

In any case, even if I didn’t get the math, and didn’t see where all their conclusions came from – especially when they started going over the Master Equation of particle physics near the end – I at least came away with a better understanding of both the chain of reasoning that led to E=mc2 and the ramifications it has on our understanding of the universe. I don’t understand everything this time, at least not yet, but I know more. And that will have to do.

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“Science at its best is driven by inquiring minds afforded the freedom to dream, coupled with the technical ability and discipline to think.”
– Brian Cox & Jeff Forshaw, Why Does E=mc2?

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Filed under astronomy, astrophysics, Brian Cox, Jeff Forshaw, mathematics, physics, science

Review 125: Logicomix

Logicomix: An Epic Search for Truth by Apostolos Doxiadis and Christos Papadimitriou

I have a question for you. It’s a simple-sounding question, but hard to answer, so I really want you to put a good amount of thought into it before you do. Okay? Yes, I’m still in Teacher-mode, but that’s not important right now. My question is this:

What is truth?

Good luck with that whole "free will" thing.

It’s one of those unanswerable questions that has bugged us ever since we started being able to ask unanswerable questions. Along with “Why is there evil in the world?” and “Do we have free will or are our lives pre-determined from the beginning?” or “What’s the deal with that Justin Bieber kid? I mean really?” this question is one that people either ignore or obsess over.

Didn’t think I could do a pop-culture reference like that, did you? Shows how much you know….

This graphic novel is about one man’s pursuit of this question, and the ways in which it nearly destroyed his life. The man was Bertrand Russell, and we follow his life from his childhood to late adulthood as he searches for an unshakable foundation to mathematics and logic, and thus an absolute truth that he could rely on.

Bertrand Russell does not find the truth. He teaches it to come when it is called.

As a child, Russell lived with the question of why things are the way they are, and got no good answers from his domineering grandmother. It wasn’t until his introduction to geometry and the wonder of mathematical proofs that he could finally say there was something about which he could be absolutely sure in the universe. Mathematics, he thought, would be the answer to everything. Pure, unsullied and utterly, utterly reliable.

But there was a flaw in math – the Axioms. Mathematics in the 19th century was a direct descendant of Euclid’s work, and rested on a series of axioms in order to function. An axiom, then, is something that is assumed to be true so that you can go on to prove other things. For example, if you have a line, and a point not on that line, there can be only one line drawn through that point that is parallel to the first. Why is this true? Well… it just is. If you have to prove that, then you have to prove a thousand other things first, and you never end up being able to prove the thing you were trying to prove in the first place. It was like, he thought, the cosmological model of the world on the back of a turtle. Which stood on another turtle. Which stood on another, and another – turtles, all the way down.

The bottom turtle's name is "Jeff." (art courtesy of Kenneth Rougeau)

That didn’t satisfy young Russell, and he went off in search of the floor upon which the last turtle stood, as it were – new mathematics that would be able to define the foundations of math, and thereby give a concrete understanding of the universe. Along the way, his desire to apply the certainty of math to human thought and interaction led him to the discipline of logic, a strange chimera of mathematics and philosophy. By becoming a logician, he thought he might finally be able to pin down some absolute truths about not only abstract math but human nature itself.

Of course, he failed. Spectacularly. Broken marriages, broken friendships, ill health – his obsession with an absolute truth to the universe nearly destroyed everything he had. Fortunately for him, Russell pulled back from the abyss before it could swallow him whole, and became one of the early 20th century’s greatest philosophers in the process. His failure to find an ultimate foundation for logic and math was not entirely without fruit – thanks to work by Russell and others, these disciplines were pushed forward in ways that made our modern lives possible. New ways of understanding the universe, from the unfathomable depths of infinity to the simplicity of 1+1=2, everything was open to examination in those days. Because of men like Bertrand Russell, humanity advanced in great leaps and bounds.

1+1=2. Seriously. No more arguments.

In the end, it’s a compelling book. I read and re-read it, convinced each time that there was something else I had missed. I was very often right. Doxiadis and Papadimitriou have put together a compelling tale of a man often overlooked by the general public, and they did so in a medium that’s close to my heart – the graphic novel. The art, done by Alecos Papadatos and Annie Di Donna, is wonderful. It has a simplicity that belies the complexity of its topic, and shows an excellent sense of storytelling. Hats off to the two of them, without a doubt.

This book, it should be noted, is not a primer on logic. If you’re looking to know how logic works, or you want to know a bit more about higher mathematics and how to do them, then you’d best look for another book. As the authors tell us right in the beginning, this book is a story, a great tragedy that owes its inspiration to the ancient productions of the Greeks. It’s the story of a man who pitted himself against the universe and lost, but who did so in such a way that he – and the world – came out better for it. The book ends with a scene from The Oresteia, a classic Greek drama about another man who found himself in a no-win situation with no absolutes to rest upon.

"Stop asking me to prove beauty, dammit!"

Much like Orestes, when faced with two choices that could lead to his destruction, the only way forward for Russell was to compromise and to move forward. By doing so, he not only became a happier man, but became involved with humanity again, as a philosopher, a teacher, and an anti-war activist.

In the end, this book is about the compromises we all have to make as human beings. The world may be a logical place, but we are not. There is a limit to our logical understanding of ourselves, and sooner or later we have to accept that and deal with people as people, rather than as problems to be solved and equations to be balanced. Bertrand Russell’s quest, as interpreted by this novel, is an example of how far we can push the need to know exactly what’s at the bottom of it all. The fact that the foundations of our world appear to be unprovable and unknowable is, ultimately, unimportant. What is important is that we are here, now, and we need to make sense of our own lives.

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“The demand for certainty is one which is natural to man, but is nevertheless an intellectual vice. So long as men are not trained to withhold judgment in the absence of evidence, they will be led astray by cocksure prophets, and it is likely that their leaders will be either ignorant fanatics or dishonest charlatans. To endure uncertainty is difficult, but so are most of the other virtues.”
– Bertrand Russell, Unpopular Essays – Philosophy for Laymen
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Logicomix on Wikipedia
Apostolos Doxiadis on Wikipedia
Christos Papadimitriou on Wikipedia
Bertrand Russell on Wikipedia
Logicomix on Amazon.com

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Filed under Apostolos Doxiadis, Bertrand Russell, biography, Christos Papadimitriou, graphic novel, logic, mathematics, nonfiction, quest