## Analogous spaces

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Today's xkcd:

Mouseover title: "If you draw a diagonal line from lower left to upper right, that's the ICP 'Miracles' axis."

[The ICP reference, if you don't get it…]

For Linguistics, the x-axis might be labelled "How difficult it is to frame the question in a scientifically coherent way" ("Difficulty" for short), and the upper right corner (the "danger zone") would be populated with "Innateness" and "Language and Thought".

## David L said,

July 11, 2017 @ 8:25 am

I disagree. The math needed to understand quantum mechanics, at least at a basic level, is not that hard — differential equations plus a few other bits and pieces. What's difficult is the conceptual understanding, with wavefunctions and probabilities and so on.

The mathematics of general relativity is more intricate, I would say, but conceptually the idea of curved spacetime is easier to grasp.

That was my experience, anyway. Of course, once you get into more advanced particle physics all bets are off.

## MattF said,

July 11, 2017 @ 9:44 am

@David L

I disagree with your disagreement. For example, non-locality and 'entanglement' in QM is a fairly basic feature of the mathematics of multi-particle wavefunctions, but is considered by many to be a deep mystery. So, which is it? Basic or mysterious?

I agree that single-particle QM is… relatively straightforward, if that's what you mean by 'basic'.

## Mara K said,

July 11, 2017 @ 10:21 am

@David Scientists and the public understand what is difficult differently–for the public, it's not about math at all. Relativity is easily explained by, for example, talking about how fast a clock runs under a particular acceleration, or the moderately frightening "guillotines on a train tunnel" analogy. We've sent humans into space, we can understand the twin paradox. We've seen kids on a trampoline, we can understand curved spacetime.

The problem with explaining quantum mechanics using similar analogies is that they really don't map onto what's really going on. Light is made of tiny ball bearings…that simultaneously diffract like water waves? You can't know where something is and how fast it's going at the same time? But…my car has GPS! And how can a thing be spinning but not actually spinning? The macro analogies create more confusion than they resolve, such that the only way a non-mathematician has of understanding quantum mechanics is to enter a Daoist-like transcendent state of rejecting the concept of duality altogether. And that's where the woo comes from: the people who really do understand it start to sound like Laozi, so people who want to get rich off pretending to be Laozi feel comfortable ripping them off.

## A. said,

July 11, 2017 @ 10:43 am

@David L: I agree about non-relativistic quantum mechanics, but quantum field theory should be at least as far right as fluid dynamics.

## David L said,

July 11, 2017 @ 10:47 am

MattF and Mara K: I didn't make myself clear, I think. I agree that quantum mechanics is hard to understand at a conceptual level — waves vs particles, entanglement, all that stuff. My point is that the difficulty of understanding has no connection, IMO, to the sophistication of the mathematics involved. Or to put it the other way around, you can master all the intricacies of quantum mechanics in a mathematical way (Hilbert spaces, matrix mechanics) and still be befuddled by the conceptual difficulties.

The XKCD cartoon seems to imply that if you learn enough math you will "understand the answers" to the philosophical questions. Not true at all.

## Michael said,

July 11, 2017 @ 10:50 am

Apparently Chaos Theory is now so unfashionable that it doesn't even get a mouseover mention.

## Mara K said,

July 11, 2017 @ 11:27 am

@Michael is that because nobody liked the Butterfly Effect movie? Can a bad enough, publicized-enough movie kill public interest in a science topic?

## J.W. Brewer said,

July 11, 2017 @ 12:38 pm

For the linguistics analogy, it might be better to think of what sort of popular-amongst-enthusiastic-amateurs-outside-the-academy topics fit into the "danger zone" quadrant (e.g. "what other languages are related to Basque?" or "what's up with Koko the Gorilla"?) and then reverse-engineer the quality to be measured by the x-axis that would best account for the actual content of the zone.

## Jamie said,

July 11, 2017 @ 12:49 pm

@David L, I *think* that is exactly the point that Randall is making: the danger zone exists because people get carried away with the philosophical "explanations" (analogies) instead of understanding the math.

## peterv said,

July 11, 2017 @ 1:07 pm

David L:

The manipulation of mathematical statements involving probabilities is not that difficult, and can be taught to high school students. Understanding the associated concepts – the semantics of probability statements – is altogether harder, as is shown by the fact that we still have no agreed semantics for probability statements. a mere 350 years after first developing a syntax for these statements.

This is an area led by statistical physicists in the first quarter of the twentieth century, and by AI people in the last quarter. Only in the middle years did statistics, the discipline that purports to study uncertainty, lead this.

## David L said,

July 11, 2017 @ 1:19 pm

For whatever reason, I don't seem to be getting my point across:

@Jamie:

people get carried away with the philosophical "explanations" (analogies) instead of understanding the math. But (1) you can understand at least some of the philosophical problems of qm with only a modest grasp of math, and (2) you can master the math as deeply as possible and still stumble over the philosophical problems.@peterv: I entirely agree that probabilistic statements can be explained in terms accessible to a high school student. You don't need a whole lot of math to work through the Bell inequalities, for example. But understanding what that means in terms of the physics implied — non-locality, most importantly — is much harder. No one has a good grasp of it, as far as I know.

Anyway, perhaps I am failing to see the point of the cartoon.

## Rubrick said,

July 11, 2017 @ 1:36 pm

I wonder how Randall generated the graph, since it's fundamentally impossible to simultaneously determine how physically exciting and how mathematically difficult a subject is.

## Mark P said,

July 11, 2017 @ 1:52 pm

@David L. I think it depends on what you mean by "understand." I have seen several stories on various TV shows where someone talks about the Schrödinger's Cat thought experiment and it is pretty clear they don't really understand the issue. Or, at least they understand it in the sense that they once heard it explained and can nearly repeat the explanation. I also once saw an educational show where a well-known physicist said that although he could easily (!) do the math in one of the more abstruse areas of physics, he was not sure that he could say that he really understood in any kind of intuitive, physical sense what was actually going on.

I pretty much agree with Jamie. It's not necessary to quantify exactly how much math is necessary in the various areas. What is important is that not understanding the math in quantum mechanics leads people to silly ideas. Not so much in fluid dynamics.

## bks said,

July 11, 2017 @ 4:06 pm

"Physicists are not terribly comfortable with finding themselves inside their theories. Most hope that consciousness and the brain can be kept out of quantum theory, and perhaps vice versa."

http://www.bbc.com/earth/story/20170215-the-strange-link-between-the-human-mind-and-quantum-physics

## Bob Moore said,

July 11, 2017 @ 5:01 pm

The position of special relativity in the diagram explains why it used to be the first half of second semester freshman physics at MIT. (Alas, no more apparently.) The goal was to give students something cool to study while those who had not learned calculus to MIT's standards in high school caught up to where the needed to be to tackle topics further to the right in the xkcd diagram.

## Brett said,

July 11, 2017 @ 9:16 pm

@Bob Moore: When was 8.02 (and for that matter, 18.01 and 18.02) taught the way you describe?

## tangent said,

July 12, 2017 @ 1:30 am

I agree with David L.: knowing the math for basic QM is not too hard, but it does not buy you understanding.

The math of superposition is easy, it's just algebraic addition. The math of "collapse" is easy, it's taking the lengths of some 2-D vectors. It's all high-school math, truly.

But when do you apply that "collapse" math, versus not? The math doesn't tell you, it's basically empirical, and when your teacher tells you to.

Why does collapse appear to exist at all? Some people understand quantum decoherence theory, they assure me, but that's much much harder stuff.

Why do we observe collapse happen onto unit vectors that make sense to us? A different basis would work just as well as far as the math cares. Again, there's theory about this, but it's much harder than the basic math.

For decades of practicing QM, there was zero workable theory of these things, is my understanding; Nobel Prize-winning physicists just shrugged and turned the math crank how it seemed to work. They knew the math but nobody in the world understood it.

## tangent said,

July 12, 2017 @ 1:43 am

But I also agree with Mark P that this isn't why people actually go off the rails with QM. Nobody has woo-woo ideas relating to preferred eigenbases.

It's always one of two specific areas: 1) "the consciousness of the observer" as magic force, or 2) many-worlds. These play into preexisting philosophical ideas, so people take QM as license.

## Rose Eneri said,

July 12, 2017 @ 8:36 am

@ Rubrick – LOL

One issue I have with QM is the terminology. I wish physicists always would have coined new words for new concepts instead of using existing words (color, spin, flavor). This leads to the mind trying to force the word meaning onto the totally unrelated concept.

I wish our education system would offer/require credit courses in science for non-majors that minimize the math but give a layman's understanding of the concepts, which is better than nothing.

## Rube said,

July 12, 2017 @ 9:24 am

@Rose Eneri: I'm not sure that a layman's understanding is "better than nothing". Without the maths, it's really easy to think you know way more than you do, and end up with some really bizarre ideas.

## Bob Moore said,

July 12, 2017 @ 2:23 pm

@Brett: I was a freshman in 1966-67. 18.01 was single-variable calculus, 18.02 was multi-variable calculus 8.01 was Newtonian dynamics, 8.02 was special relativity followed by oscillations and waves. The oscillations and waves part of 8.02 required some multi-variable calculus, and at that time only about 1/3 of incoming freshman placed out of 18.01; hence the delay in presenting it until everyone was well into 18.02 or beyond. I am reasonably sure that this structure continued into the 1970s, since I think I would have heard about it if it changed while I was still at MIT.

## Jamie said,

July 12, 2017 @ 5:25 pm

@Rube: exactly.

## DWalker07 said,

July 13, 2017 @ 2:35 pm

For me, the mysterious part comes when we are told (in layman's terms) that two entangled particles have a spin, but that spin is indeterminate until one of them is measured. Not just "unknown to us", as in "the particles have a definite up-or-down spin (or other property) and we just don't know it until we measure it" but specifically "the spin is nether up nor down until one of them is measured".

Which, of COURSE, leads to confusion because once you measure one particle, the entangled particle instantly has whatever the quantum mechanics says it should have. Questions arise like "how does the other particle know that you measured the first particle".

Being a scientifically-minded computer programmer with moderate math skills, I have yet to find a convincing explanation why a particle doesn't just HAVE a property that we don't KNOW until it's measured.

Maybe that's true (in some sense) but the math to prove THAT part is hard. However, all of the woo-woo with QM seems to be "take this part on face value, and then ponder the weirdness that results". Well, I can't get past "two entangled particles don't have a spin until you measure one of them".

## Mark P said,

July 14, 2017 @ 11:04 am

@DWalker07, I think this is exactly what the cartoon is getting at. I am not a quantum physicist, so what I know about it is probably about what you know about it. The problem arises when a physicist, who, after all, is human and wants to understand things in a way they can relate to, tries to explain the physics in everyday, intuitive ways. The fact that those explanations seem contrary to our everyday experience seems to imply that there is an underlying reality that is at odds with what we experience every day. We do not experience a cat that is both alive and dead (or neither alive nor dead). In our experience, a dead cat died exactly when it died and it has been dead ever since. We don't see it both alive and dead, nor do we see it in separate worlds, one in which it is alive and one in which it is dead. In short, it seems that the implications of quantum mechanics simply do not translate into something that we understand in everyday terms.

## Mark P said,

July 14, 2017 @ 11:04 am

@DWalker07, I think this is exactly what the cartoon is getting at. I am not a quantum physicist, so what I know about it is probably about what you know about it. The problem arises when a physicist, who, after all, is human and wants to understand things in a way they can relate to, tries to explain the physics in everyday, intuitive ways. The fact that those explanations seem contrary to our everyday experience seems to imply that there is an underlying reality that is at odds with what we experience every day. We do not experience a cat that is both alive and dead (or neither alive nor dead). In our experience, a dead cat died exactly when it died and it has been dead ever since. We don't see it both alive and dead, nor do we see it in separate worlds, one in which it is alive and one in which it is dead. In short, it seems that the implications of quantum mechanics simply do not translate into something that we understand in everyday terms.