Tag Archives: math

What’s in a number?

23 Apr

http://www.lastwordonnothing.com/2012/04/12/whats-in-a-number/

 

What’s In a Number

By Richard Panek | April 12, 2012 |

 

“Since there is an infinite number of alternative universes, there must be one in which there isn’t an infinite number of alternative universes. Perhaps this is it.”

 

No, that speculation didn’t come from the “Ask Mr. Cosmology” mailbag. It’s from a reader of New Scientist, courtesy of LWON’s own Sally, who is an editor at the magazine. She forwarded it to me because, she said, “it kind of made my head asplode.” After receiving reassurances from her that her head hadn’t actually spontaneously detonated—this is, after all, someone who is capable of falling into the Thames without any help—I sat and thought and tried to find the flaw in the logic.

 

The speculation has a logical basis in the current standard cosmological model. According to quantum theory, virtual particles should be popping into and out of existence all the time—and are, as experiments have repeatedly shown over the past six decades. In that case, the universe could be the product of one such quantum pop.

If it is, then it could have gone through a process that physicists call a “phase transition” and that everyone else calls “the thing that happens when water turns into ice or vice versa.” At the age of a trillionth of a trillionth of a trillionth of one second—that’s a 1 followed by 36 zeros, or 1036—the universe would have expanded ten septillion-fold—or to 1025 times its previous size. And it would have done so over the course of 1/1035 seconds.

And if inflation can pop one quantum universe into existence, then why not many? In fact, according to quantum theory, it should. It would, if inflation actually happened.

The inflationary universe. Also, Sally’s head.

 

The case for inflation isn’t airtight, but with every fresh observation of the Cosmic Microwave Background—the remnant echo of the Big Bang, loosely speaking—the evidence has looked better and better. Over the past decade, consensus has coalesced: We very likely did come from a quantum pop. In that case, our inflationary bubble would be one of an ensemble of 10500 universes. The number isn’t quite infinity, as the New Scientist reader suggests, but who’s counting?

Still, let’s say the number of universes is infinite. In that case, the reader’s argument goes like so:

A. The number of universes is infinite.

B. A universe exists in which the number of universes is not infinite.

C. This might be it.

When the argument is stated this starkly, the flaw in the logic becomes pretty clear. B contradicts A. “The number of universes is infinite” and “the number of universes is not infinite” can’t both be true. The contradiction, however, is obscured by the inclusion of “A universe exists in which.” The implication is that there’s something special about universes, something that, for instance, doorknobs don’t have. “Since there is an infinite number of doorknobs, there must be one for which there isn’t an infinite number of doorknobs” wouldn’t make anyone’s head asplode, except perhaps in bewilderment.

So what’s so special about universes that the existence of an infinite number of them would, for physics-savvy readers, somehow seem to suggest the necessary existence of one that allows the impossible?

I suspect the answer is quantum probability. According to quantum theory, everything is a matter of probability; therefore anything is possible. Anything. The probability that a butterfly will give birth to a dragon or that I will one day fall into the Thames is vanishingly small—but, technically, it’s not zero. Same with the emergence of a universe, or a cornucopia of universes, from nothing. The laws of physics allow it.

And that’s the implicit, but missing, “something special” in premise B: the laws of physics. As in “A universe exists in which the laws of physics require the number of universes to not be infinite.” What prompted the New Scientist reader, and what posed a threat to Sally’s noggin, was an unthinking assumption: that “the laws of physics”—in particular quantum theory—are part of the argument.

It’s a tempting assumption. According to current cosmological thinking, if an infinite ensemble of (or 10500, anyway) universes exists, then presumably each could come equipped with its own laws of physics. So couldn’t our universe be the one in which the laws of physics require that other universes don’t exist?

Yes—but only if our laws of physics have something to do with the other universes. We all, however, went our separate ways 13.7 billion years ago. Our laws of physics affect what happens within our universe, but there’s no reason to think they would influence the multiverse at large. Doorknobs, after all, don’t dictate the laws of physics.

Still, if they did, then maybe we could reframe the New Scientist‘s reader’s comment:

“Since there is an infinite number of alternative universes, there must be one in which there is just one alternative universe. Perhaps this is it.”

“Since there is an infinite number of alternative universes, there must be one in which there are two alternative universes. Perhaps this is it.”

“Since there is an infinite number of alternative universes, there must be one in which there are 2,125,179,218 alternative universes. Perhaps this is it.”

Memo to New Scientist staff: You can remove your plastic ponchos now.

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Nerd Alert – Science Tattoos

1 Nov

Yet another bit of awesomeness from the Discovery blogs.

26 pages of pure dedication to science:

http://blogs.discovermagazine.com/loom/science-tattoo-emporium/?nggpage=1

I’m working on the design for my first and after seeing all of these I’m thinking I might make some additions… or maybe just wait til next time and become that person who just can’t stop.

Examples:

Zach writes: “It is a half sleeve up my upper right arm based around an image taken by one of the CERN bubble chambers. It is based on this image. I first saw that image my freshman year of college. It had the sublime, simple beauty that only something made of math and science can have. It stuck with me for 8 more years before I actually decided to get it etched into me. Oddly enough, on Valentine’s Day. I guess it was my Valentine’s to physics and science. Oh, and when people ask who drew it, I always respond ‘God.'”

 

 

Drew, an oceanography graduate student, writes:
“This, on my leg, is the incompressible form of the conservation of mass equation in a fluid, also known as the continuity equation. When people ask what it means, I say it defines flow. Sometimes I say it means you should have studied more physics, but that is only when I am feeling like being funny. What it means in more detail is that, for an incompressible fluid, the partial derivative of the velocity of the fluid in the three spatial dimensions must sum to zero. It therefore concisely states the fundamental nature of a fluid.”My advisor took this picture, and I swear he is obsessed (in a good way) with this tattoo. He is giving a talk at Woods Hole next week as he is the recipient of an award, and he is planning to show off ‘how quantitative scripps students are’ which i think is hilarious and only slightly mortifying. Speaking of mortifying, it is slightly mortifying to be sending this email at all–I have to admit I am a little embarrassed. It is definitely the most vain thing i have done today. I do have an ulterior motive which I have no problem admitting: I want to stake a claim on this particular piece. I guess it might be a little lame to want to claim ownership over something so silly but there it is and I guess at least I can admit it.”

 

 

Marc Morency, Quartermaster 1st class, USN, writes: “While I am by no means a scientist, I have been fortunate enough to be paid by the government to get ships from pt. A to pt. B serving in the US Navy as a Quartermaster. I was drawn to the navigation when I joined. In my opinion, it is the only job in the military that is both a science and an art Celestial navigation has been something I have become profoundly interested in since I joined ten years ago. In this age of GPS, it is, in my opinion more important now than ever for Navigators to remain proficient in the old ways to fix a ship’s position using a sextant and trigonometry. My tattoo is the visual depiction of how to plot a line of position from a celestial body using the altitude intercept method, a method which has been time tested for more than a century. For me it serves as a reminder that while technology improves, the sea remains an unpredictable place and it is up to the older generation to teach the younger the old school ways of doing business.”

 

 

Ariel writes, “I am happy to see that other science dorks like myself have inked up our passions. This is the molecular representation of glutamic acid, the amino acid associated with the Umami flavor, the proverbial fifth taste. I am a former chef turned public health major and fell in love with the elegance of chemical compounds but never forgot my unctuous roots.

The beauty of numbers and the elements of math

17 Jun

Most people really hate numbers and math, but there’s some real good shit in the nuts and bolts of it all.
These columns might present a different way of thinking for people who were taught in the “standard” way… or by a teacher who just didn’t know how to make the subject really available for consumption.

http://topics.nytimes.com/top/opinion/series/steven_strogatz_on_the_elements_of_math/index.html

From the author:
I have a friend who gets a tremendous kick out of science, even though he’s an artist. Whenever we get together all he wants to do is chat about the latest thing in evolution or quantum mechanics. But when it comes to math, he feels at sea, and it saddens him. The strange symbols keep him out. He says he doesn’t even know how to pronounce them.
In fact, his alienation runs a lot deeper. He’s not sure what mathematicians do all day, or what they mean when they say a proof is elegant. Sometimes we joke that I just should sit him down and teach him everything, starting with 1 + 1 = 2 and going as far as we can.
Crazy as it sounds, over the next several weeks I’m going to try to do something close to that. I’ll be writing about the elements of mathematics, from pre-school to grad school, for anyone out there who’d like to have a second chance at the subject — but this time from an adult perspective. It’s not intended to be remedial. The goal is to give you a better feeling for what math is all about and why it’s so enthralling to those who get it.

On to the articles:

From Fish to Infinity
A debut column on math features an introduction to numbers, from upsides (they’re efficient) to down (they’re ethereal).

Rock Groups
Treating numbers concretely — think rocks, for instance — can make calculations less baffling.


The Enemy of my Enemy
The disturbing concept of subtraction, and how we deal with the fact that negative numbers are so . . . negative.

Division and its DiscontentsThis week, division — where many students hit the mathematical wall — is made less confusing.

The Joy of X
The series moves on to high school math, specifically algebra and formulas.

Finding Your RootsComplex numbers, a hybrid of the imaginary and the real, are the pinnacle of number systems.

Square Dancing
Geometry, intuition and the long road from Pythagoras to Einstein.

Think Globally
Differential geometry can show us the shortest route between two points.

Power ToolsIn math, the function of functions is to transform.

Take it to the LimitArchimedes recognized the power of the infinite, and in the process laid the groundwork for calculus.


Change We Can Believe In
Differential calculus can show you the best path from A to B, and Michael Jordan’s dunks can help explain why that is.

It Slices, It Dices
The integral, perhaps mathematics’ most graceful sign, is a foundation of calculus.

Chances Are
The improbable thrills of probability theory.

Group Think
Group theory, one of the most versatile parts of math, bridges the arts and sciences.

Hilbert Hotel
An exploration of infinity as this math series, not being infinite, comes to an end.

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