Math is Sideways

 

Ananse Wimbledon-Sutcliffe: You're listening to KSLF public radio, and right now, Padme Omicron-Danforth has been working on a story which has taken the world by storm.  As I understand it, Padme, math is sideways.  Is that right?  

Padme Omicron-Danforth: Hi, Ananse.  Yyyyyyes.  Ish.

A W-S: How did that happen?  How did math turn sideways?  What does that even mean?

P O-D: Alright. First of all, math didn't turn sideways.  In fact, this story, unfortunately, has been a little over-hyped.  Someone in the media, and I'm not sure who, came up with the phrase, "Math is sideways," which is just such a great, punchy, three-word slogan, and it just went viral, and now everyone is talking about how math is sideways, and people are touting this discovery as a giant transformation of our understanding of the universe, as big as when we learned, through Einstein's relativity, that geometry is non-Euclidean, and so forth....

A W-S: Okay, so what's really going on here?

P O-D: Well, Ananse, to understand what's really going on, I have to take you and our listeners back to high school, where you probably studied something called "imaginary numbers," though you may not remember....?

A W-S: I do, kind of, remember, but maybe you could give me the quick refresher course?

P O-D: Sure.  So, most of the time, the numbers we use every day are on the number line.  You know?  So on the line there's only two directions you can go: more or less.  Usually the way people write it out, more goes to the right, like 1, 2, 3, 4, 5, and so on, and less goes to the left: 5, 4, 3, 2, 1, 0, -1, -2, -3, and so on, indefinitely.  Are you with me so far?

A W-S: Yup.

P O-D: And in between all these numbers are fractions, like 2 1/2, or decimals, like 5.3, and pi is in here somewhere, between 3 and 4, right?  But they're all on this line.  These are what are known as the "Real numbers."

A W-S: Okay, right....

P O-D: But then mathematicians started thinking: what's the square root of -1?  And that seemed like an impossible question to answer, because if the number were positive, the square of it would be positive, if it were negative, and you multiplied it by itself, the negatives would cancel out, and the answer would be positive, and if it were zero, well, zero squared is still zero.  So whatever the square root of -1 is, it's not positive, it's not negative, and it's not zero, so, it's just nowhere on this number line.

A W-S: So it's impossible.  Can't we just leave it at that?  Why are mathematicians - why can't they just leave a problem alone?

P O-D: Hahaha, well, okay, fair point.  But what mathematicians did is, they said, I know it's impossible with numbers as we know them.  But what if?  What if there were a number where, if you multiplied it by itself, you'd get -1?  Let's imagine that such a number exists.  Let's call it "i".  And it can't be on the number line as we know it, so let's just say it's over here.

A W-S: Off the number line.

P O-D: Off the number line, exactly.  So now, instead of a number line, you have a number space.  A number plane.  And you can have i, or you can have 2i, or 3i - 

A W-S: That means, "3 times i" -

P O-D: Three times i, exactly, and they'd be even further off the number line: 4i, 5i, and so on and on, and in the other direction, -i, -2i, -3i, -4i, and so on....

A W-S: Huh.  Okay.  And 0i?

P O-D: Well, zero times i is just zero.  Because zero times anything is zero.  So now you've got two number lines, which cross at zero.  The number line we're all familiar with - the real number line, -3, -2, -1, 0, 1, 2, 3, 4, 5 - and the imaginary number line, which looks exactly like the real number line, but everything is times i, so -3i, -2i -i, 0, i, 2i, 3i, and so on.  And somewhere in there is 2.65i, and pi i, and everything.  Usually we draw this so that the real number line goes left and right, and the imaginary number line goes up and down, and they cross in the middle at zero.

A W-S: [sighs] Alright.

P O-D: And now, what if you add a number from this line to a number from that line?  Say, you take the 2i from this line, the imaginary number line, and add it to the 3 from this line, the real number line....?

A W-S: I don't know, you tell me.

P O-D: Okay, you get 3+2i.

A W-S: Oh, so it's... just... there's no.... it's just "3+2i".

P O-D: Right, which would be somewhere over... here... 

A W-S: We're on the radio, Padme.

P O-D: Haha!  I know!  Sorry!  I'm pointing to the upper-right quadrant, the region of the positive real numbers and the positive imaginary numbers.  And 3-2i would be here, in the lower-right, with the positive reals and the negative imaginaries, and -3-2i would be down here, in the lower-left - 

A W-S: But...

P O-D: So you have this whole plane, it's called the complex number plane, it's -

A W-S: But Padme, I don't - 

P O-D: I know what you're saying.  But this is all imaginary, right?  We know that it's impossible for a number squared to be -1, so why are we bothering to make this whole complicated thing?  Is this just a thing that mathematicians made up, for fun?  What's the point of all of this?

A W-S: Yes.  Thank you.  

P O-D: Well it turns out that complex numbers - that is, numbers that can be expressed in the form a+bi, a sum of two numbers where "a" is the real component and "bi" is the imaginary component - it turns out that they have all kinds of practical applications.  Engineers use them all the time - especially electrical engineers - and they're very useful in physics - 

A W-S: Even though they're imaginary.

P O-D: Well, yeah, and a lot of mathematicians hate that term "imaginary",  Some of them feel that it's a mistranslation, or a misunderstanding, and these numbers are every bit as real as the so-called real numbers, which, come to think of it, in a certain sense those numbers are imaginary, too.

A W-S: No they're not!

P O-D: Why not?

A W-S: If I have 2 apples, and I get 3 more, now I have 5.  That's real.  That's not imaginary.

P O-D: Yes, but it required human imagination to figure that out.

A W-S: Okay, yes, but... it's true!  I can't decide that 2 + 3 = 4, and have that be true.

P O-D: In other words, what you're saying is that the numbers we are familiar with, the so-called "real numbers" - they're not totally arbitrary.  They follow certain rules, reliable rules.  

A W-S: Yyyyes.

P O-D: Well, it turns out that the imaginary numbers have rules, too, which are very similar, and just as reliable.

A W-S: Hmm.  Okay.  Really?

P O-D: Yes.  I mean, this gets into the whole philosophy of math thing... there are some very deep questions we're talking about here...

A W-S: So, when did this happen?

P O-D: What do you mean?

A W-S: When did people invent imaginary numbers?

P O-D: I would say, discover, rather than invent.  The person who discovered them was Hero of Alexandria, a Greek philosopher who lived in Egypt in the first century CE.

A W-S: Oh.  I thought... oh.

P O-D: Yeah, this isn't a recent thing.  And then they were developed further by Gerolamo Cardano and Rafael Bombelli in the 16th century, and the giants of math, Euler and Gauss, in the 18th, at which point they became mainstream among all mathematicians, and became a foundation for scientific discoveries.  And if they are useful and allow us to create mathematical models of the real world, and they work, well, that's as real as it gets, isn't it?  I mean, which is more important, for something to be real: that you can develop precise mathematical models that produce measurably true results?  Or that it feels right to you, and matches your preconceptions?

A W-S: Well, when you put it that way....

P O-D: But hang on to that feeling of uncertainty that you have.  Because it turns out there's more to this story.

A W-S: First let's take a break.

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A W-S: And we're back.  If you're just joining us, I'm speaking with Padme Omicron-Danforth, who is explaining how math is sideways.  So, how about it, Padme?  How is math sideways?

P O-D: Well, remember when I said that in the complex number plane, every number can be represented by the formula a+bi?

A W-S: Yes.  So this is the, that means, like, on the graph, the plane, that would mean "a" steps to the right, and "b" steps up.  Right?

P O-D: Right.  But what if it were the other way around?  What if it's ai+b?  What if it means "a" steps up, and "b" steps to the right?  In other words, what if we flip-flop the axes?

A W-S: I don't follow you.

P O-D: Well, this is kind of like what you were saying before the break.  If it's all arbitrary, couldn't we set this up a different way?

A W-S: But you said that there were rules....

P O-D: And there are.  Okay, maybe I better back up here.  I said before that there were all kinds of applications for complex numbers.  And several of them are in physics.  In Einstein's theory of Special Relativity....

A W-S: Oh boy, here we go.

P O-D: Yes.  In Einstein's Special Relativity, the way he set up his mathematical model of the universe, it's a complex coordinate plane, but here there are 3 real dimensions - length, width, height - and one dimension of time, which for him was imaginary.

A W-S: Time is imaginary?

P O-D: Yes, time is imaginary.

A W-S: It doesn't feel imaginary.

P O-D: Again, we're using the mathematical meaning of "imaginary," which is something very real.  It's that axis: i, 2i, 3i, and so on.  So now we're talking about 4 dimensional spacetime, where 3 of the dimensions - the spatial dimensions - are real, and one dimension, the dimension of time, is imaginary.  And in fact, since Einstein, other physicists have made this even more complicated, so now we think there are 10 or 11 real, spatial dimensions, and one imaginary dimension of time.

A W-S: This is getting way over my head.

P O-D: And just about everyone's!  So to keep things simple, when people think about relativity, they usually simplify it to just one real spatial dimension and one imaginary time dimension.  So we're back to that 2-d complex coordinate plane we were discussing before.

A W-S: Okay....

P O-D: But, again, who's to say which axis is real, and which axis is imaginary?  Since Einstein, we've been assuming that the spatial dimension is real, and the time dimension is imaginary, but what if we switch them?  What if we flip-flop the axes of our coordinate system here, so that time is the real dimension and space is the imaginary dimension?  That's what is meant by math being sideways.  What used to be left and right is now up and down, and what used to be up and down is now left and right.

A W-S: Uhhhh....

P O-D: Now scientists had been toying with this idea for decades, but it never worked.  The math just wasn't there.  They had given up on the idea, and decided it was just plain wrong.  That is, until a young man at UC Santa Cruz named Jonah Kernow came up with a new mathematical way of tackling this.  Even he didn't really take it seriously, though, at first, but pretty soon more physicists and mathematicians were developing the mathematical framework he'd come up with, but it was still a pretty fringe, or I should say, niche concept in the world of physics, not really explored or even taken seriously by most scientists.  And then, last year, a team of experimental physicists in South Korea, at Sungkyunkwan University, led by Park Hee-Young, actually devised an experiment to test whether this way of understanding relativity would actually work.  They worked together with some Swiss, French, and American physicists, and did this experiment, which involved going deep, deep under the surface of the Earth in an abandoned coal mine....

A W-S: Really?

P O-D: Yes, really.  And to just about everyone surprise, the experiment seems to have been a success.  Now, the authors of the paper, which just came out, stress that these are preliminary findings, and they need to be corroborated by a lot more research, but the publication of this paper has set the world of physics - and math, for that matter - on fire.  And soon the press got ahold of it, and now everyone is saying that math is sideways.

A W-S: So, what would it mean if math is sideways?  I still don't really understand.

P O-D: Yes, and I should say that the math here is extremely complicated.  I've been trying to wrap my head around it, and to be honest, I'm not there myself.  Part of the issue here is what is known as "k-symmetry," and, I should say, first of all, what this experiment showed, or preliminarily indicated, was not that math is sideways, necessarily - it's not showing that the old model, where time is imaginary and space is real, is wrong.  It's just showing that it works both ways - where time is the sole imaginary dimension, or where all of the spatial dimensions, 10, 11, 12, however many there are - where they're all imaginary, and time is real.  Both models work - the traditional model and Kernow's model.  So this allows physicists to flip back and forth, between the two models, which allows them to solve a lot of problems that were mathematically intractable before.  And this could help us solving a whole lot of problems.  In fact, immediately after the South Korean, well, the international team's paper came out - like, within a few days after that, somewhere around 50 more papers were published, building off of their work.  Now I don't know if that means that this was a watershed, or that we have an over-publication crisis, but... 

A W-S: [laughs]

P O-D: ...But it's definitely a big deal.  And it's spilled out into the larger culture.  This is partly what I love about it.  Now kids learning math in school... it's raising those kinds of questions that you raised in the previous segment, those deep, philosophical questions.  What is math?  What is real?  You know, so I love it that we're having this discussion.  Those big questions never really went away, but now they've been reinvigorated. 

A W-S: Well, thank you for your reporting, Padme.  We'll definitely keep tracking this story as it develops.