Sabine Hossenfelder asks, Do complex numbers exist?

_{Denyse O'Leary

March 6, 2021

Intelligent Design, Mathematics}

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Complex numbers have a real component, say 6, and an imaginary component, say √-1. So it is 6+√-1. In a recent math flap, someone claimed that complex numbers don’t really exist and others claim, yes, they do.

Sabine Hossenfelder thinks they do exist, at least for quantum physics. Discussing the “yes, they do” camp, she explains:

The question which they look at in the new paper is then whether there are ways to entangle particles in the normal, complex quantum mechanics that you cannot build up from particles that are described entirely by real valued functions. Previous calculation showed that this could always be done if the particles came from a single source. But in the new paper they look at particles from two independent sources, and claim that there are cases which you cannot reproduce with real numbers only. They also propose a way to experimentally measure this specific entanglement.
I have to warn you that this paper has not yet been peer reviewed, so maybe someone finds a flaw in their proof. But assuming their result holds up, this means if the experiment which they propose finds the specific entanglement predicted by complex quantum mechanics, then you know you can’t describe observations with real numbers. It would then be fair to say that complex numbers exist. So, this is why it’s cool. They’ve figured out a way to experimentally test if complex numbers exist!
Sabine Hossenfelder, “Do Complex Numbers Exist?” at BackRe(Action)

But, she warns, if they are right,

This conclusion only applies if you want the purely real-valued theory to work the same way as normal quantum mechanics. If you are willing to alter quantum mechanics, so that it becomes even more non-local than it already is, then you can still create the necessary entanglement with real valued numbers.
Sabine Hossenfelder, “Do Complex Numbers Exist?” at BackRe(Action)

The people who don’t think complex numbers really exist would probably not be happy with quantum mechanics being even more non-local without them. But of course, if complex numbers really do exist, then immaterial things really exist. Not a good time to be a hard core materialist.

Comments

abstract spaces pervade our world
Above from https://uncommondescent.com/intelligent-design/while-in-quarantine-from-the-plague-newton-transformed-the-way-we-calculate-pi/#comment-726832 Yet you cannot point to one. They are mental only.
they ground our ability to infer that material ruler edges, ball bearings etc are imperfect relative to the geometric ideal.
I believe you just agreed with me.jerry_{March 26, 2021
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Today is Pi day. Will it go on and on?jerry_{March 14, 2021
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JVL, pardon but you are not sole or primary audience. In context, reality of 1/2 is on the table and Mr Carlin was brought in, to which I responded on substantial import. Breaking a crumb of cookie will indeed transform it into two smaller crumbs, typically of unequal size. He is correct that this is not two half-a-crumbs. However, joke notwithstanding, were the breaking an even one the new crumbs would stand in half-ness of volume and presumably weight relative to the original . . . cookies are inhomogeneous. Halving is an operation, involving even splitting (and extending into other fractions as well as division in general). Halving is legitimate, half-ness is a quantitative relationship, and using the general unit, 1, stands at 0.5 using decimals, on the reals line. Were the original crumb to weigh say 0.9 oz, the assignment of unit to this leads to the new ones weighing 0.45 oz each. Grams could be used, and would be consistent via a conversion factor. Half-ness is a relation, it is part of the structure and quantity of this or any possible world, it emerges in many practical circumstances such as the halving crease impressed into many pills. KFkairosfocus_{March 11, 2021
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What did the 0 say to the 8? Nice belt! Never have a conversation with pi. Pi just goes on and on and never stops.jerry_{March 11, 2021
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Jerry: You are confusing what is useful with what is actually happening. You don't seem to have a sense of humour either. Oh well, I tried. And, by the way, like Kairosfocus you can be condescending. If that works for you then keep doing it. But I will probably respond a bit less often.JVL_{March 11, 2021
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Carlin failed to note the next bit, breaking evenly and that the two new crumbs are half-size.
Two things relevant to crumbs. Actually three. Not 2 1/2. 1) The new so called half crumbs are entirely different entities. If you came across them independently of the breaking them apart, one would just say there are two small crumbs. 2) they will never be exactly equal in mass or number of molecules. So not really halves. If they were, the most scientific description would be that there are two small crumbs of exactly equal size. 3) Of course we need a definition of just what a crumb is. When does a small piece of a cake become a crumb? That’s the more important question. We don’t want to be accused of just giving someone a crumb.
half of a unit, is a real number
Yes but it is not a half. It’s actually a positive integer. The whole was say 500,000 various molecules that made up the string. The so called half is just 250,000 of the same molecule combination. Actually it would probably be physically impossible to ever get that exact. You are confusing what is useful with what is actually happening. I’m not assaulting math. Just the opposite. I’m trying to make it more understandable.jerry_{March 11, 2021
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Kairosfocus: Carlin failed to note the next bit, Do you have a sense of humour at all? And, again, please don't be condescending. I know very well what 'halving' means.JVL_{March 11, 2021
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JVL, Carlin failed to note the next bit, breaking evenly and that the two new crumbs are half-size. A 1 yard string bent double and cut has become two half-yard strings. Halving is an operation, dividing into two even parts and there is a halfness that appears on a scale. 0.5 or equivalent, half of a unit, is a real number. KFkairosfocus_{March 11, 2021
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If you break a crumb in half, you don't get two half-a-crumbs, you get two crumbs.
. -- George CarlinJVL_{March 11, 2021
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Interesting thing. I was editing previous comment and my edit disappeared. I thought there was another malfunction in the WP. Then I realized my 20 minutes had expired. Everything just vanished. —————— A number is just a word or concept we assign to a collection of entities. All the collections of entities that have a 1 to 1 correspondence with each other are given the same number or word as a descriptor. There is no need for each entity to be identical to the others. They can be but it’s not necessary. We can have a collection of 5 pennies or we can have a collection of 5 tools all different. Each has the number 5 assigned. If you take one of the entities and some how separate it into two different parts there are now more entities in the collection. For example suppose we have 5 packets of sugar and tear one apart. We now have 6 entities, 4 whole packets and two partial packets If you then remove one of the partial packets from the collection there will be the same number of entities as before. One entity will be different from before. There will be 4 whole packets and one partial packet. In no way am I advocating using this understanding in every day use. The concept of a half or any other fraction is extremely useful for life. There is just a difference between what is useful and what is actually happening.jerry_{March 11, 2021
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half is not a number? Did you mean that literally?
Yes.
Or did you mean that halving is an operation leading to the rational 1/2?
I have no idea what you mean. Since 1/2 doesn’t exist. The misunderstanding arises with division. Addition, subtraction and multiplication of entities provides no issues. Suppose you have 12 quarters and want to give three to each of 4 children. Each will get 3 quarters. With no problem. If you have 10 quarters then the first three will get 3 quarters but there will not be enough to give the fourth child 3 quarters since only 1 remains. 3 1/3 quarters does not exist. But 3 groups of 3 exist and one group of 1 exist. It’s not hard to understand.jerry_{March 11, 2021
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VL, the proof goes through the steps you outline. These lead to the complex exponential result you cite. Generally, a step of algebra is taken yielding a surprising connexion of five famous and pivotal numbers in the history of numbers and wider history of ideas. It also has connexions to the coherence of core mathematics, post Godel. KFkairosfocus_{March 11, 2021
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Jerry, half is not a number? Did you mean that literally? Or did you mean that halving is an operation leading to the rational 1/2? KFkairosfocus_{March 11, 2021
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Viola Lee: I wonder if that idea appears in the philosophy of math literature anywhere,. That's why I'm interested in who he was talking to. I don't understand the need for secrecy; I've never known a mathematician, let alone a world famous one, who was ashamed of their ideas. I really don't think that the mathematician in question would care at all if we knew what he had said. In fact, he'd probably thing we were too stupid to appreciate it!! But, it's Jerry's tale to tell if he wants to. And he doesn't seem interested. Which is a shame since this seems like a good forum to present unusual ideas.JVL_{March 10, 2021
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Actually I do find Jerry's idea unusual that counting numbers exist in some way that all other mathematical concepts don't. I wonder if that idea appears in the philosophy of math literature anywhere,.Viola Lee_{March 10, 2021
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Jerry: It will have to remain a mystery. I’m not going to reveal a private conversation with someone. Choose to not believe it happened if you wish. Now that I'm thinking about it . . . I find your response a bit odd since none of the ideas you mentioned are 'heretical' or even that unusual. I'm not sure what you're afraid of exposing. But, as I said, it's your call.JVL_{March 10, 2021
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Jerry: It will have to remain a mystery. I’m not going to reveal a private conversation with someone. Choose to not believe it happened if you wish. It's your call, clearly. I'm purely interested in an academic sense AND I believe it did happen. That's why I'm interested. I'd be interested in continuing the conversation.JVL_{March 10, 2021
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who was the Number Theorist
It will have to remain a mystery. I’m not going to reveal a private conversation with someone. Choose to not believe it happened if you wish.
Anyway, the formulations we use now have turned out to be very useful and practical so I doubt there’s a great push to change them.
I doubt any sane person would want to change mathematics to reflect the particle nature of our universe. It is too successful. Infinity, real numbers, rational numbers, negative numbers, zero, imaginary numbers, geometric shapes continuums are extremely useful. There might be some applications that might be appropriate but I certainly don’t know of any.jerry_{March 10, 2021
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So, Jerry, who was the Number Theorist you spoke to all those years ago? I was thinking it would be interesting to see what area of research he worked on.JVL_{March 10, 2021
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The minutes and seconds were from the Babylonians, base 60. The Egyptians used base 12. Here's a good article about the whole subject: Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day? However, it does say that the reason the Babylonians used base 60, other than fractional convenience, is unknown, and I don't think that's true: I think the part about 360 days in a year is quite well established.Viola Lee_{March 10, 2021
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The ancient Babylonians used base 60 because most of the fractional parts they needed were whole numbers, as they didn’t know how to express fractions.
I too read someplace that one of the reasons they used 360 was this number represented the days in a year. Obviously off a little but it expressed an orderliness to what they were trying to understand. It was then appleid to the sky. Then someone, the Egyptians?, divided the day into 24 parts and sky likewise was divided. Then they divided these parts into smaller parts and they were called minutes meaning small parts (by the Romans.) Then they divided the minutes again to be more precise and they were called second minutes or second small part and then just seconds. This was then applied to time as well as the sky and is now applied to the earth.jerry_{March 10, 2021
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Grads make more sense than degrees from the point of view of being consistent with our decimal system. And I know you know this, but most calculators use DRG for degrees, radians, and grads. I almost enjoyed introducing my students to radians as the "natural" way to measures angles and arcs, similar to how e is the natural base, as opposed to 10. More trivia: dividing a degree into 60 minutes per degree and 60 seconds per minute harks back to this base 60 deal. I think that some are just using decimal parts of a degree these days, but old habits die hard. For instance, if I google "how wide is the moon" I get answers both in decimal parts of a degree and in arc minutes. All fun stuff to know.Viola Lee_{March 10, 2021
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Another did you know: in Italy the surveyors don't use degrees or radians they use gradians. There are 400 of them in a circle so a right angle is 100 gradians. For those of you who were wondering what the 'GRAD' setting on your calculator was for.JVL_{March 10, 2021
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The ancient Babylonians used base 60 because most of the fractional parts they needed were whole numbers, as they didn't know how to express fractions. The other reason they used sixty is because six equilateral triangles fit inside a circle, and if the angles are divided into 60 parts, you get 360 parts for the whole circle, which they thought coincided perfectly with the number of days in a year. That's why there are 360° in a circle. You may know all that, but it's an interesting historical tidbit.Viola Lee_{March 10, 2021
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Jerry is reminding me of something one of my professors said when I was in graduate school: apparently you can pick a smallest number/value (as long as it is really small) and still get most of our practical/applied mathematics from that. If you did that then you wouldn't need fractions because everything would just be so many times the smallest value of whatever. Nothing would actually be continuous, just a huge bunch of very tiny steps or chunks or whatever. You wouldn't get 'infinity' either . . . I don't think. It kind of does my head in because I've learned to think of functions and such as continuous. Some math topics (like Graph Theory or Number Theory or Set Theory) are already pretty discrete so they wouldn't be affected until you get to things like the Prime Number Theorem or countably infinite. Anyway, the formulations we use now have turned out to be very useful and practical so I doubt there's a great push to change them. I am still curious which Number Theorist Jerry talked to. Some of them are pretty interesting people.JVL_{March 10, 2021
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Half” has relative meaning. Meaning is what matters. It’s a real concept.
Never said it didn’t. It’s just not a number. You are pointing to a group of four things. Three of which you are calling a bag. And one you are calling a half bag or a partial bag. You are actually using the word “half” here as if it were number.but is just a word describing a different entity. Extremely useful and effective communication. I suggest you read what I actually say.jerry_{March 10, 2021
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Jerry: A half bag is a different entity from a whole bag. Each of the whole bags are "different entities" from each other. So what? Saying, "I want 3 1/2 bags of sugar" actually has meaning. Nobody misunderstands this in the modern world. Or, "do you want a scoop of ice cream?" "Naw, just give me half." "Half" has relative meaning. Meaning is what matters. It's a real concept.Concealed Citizen_{March 9, 2021
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Throughout my life I have had to use algebra, geometry, calculus, statistics, etc. But I regret to say that I never developed a passion for it like JVL and Viola Lee obviously have. I guess we each have our own passions.Steve Alten2_{March 9, 2021
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Viola Lee: In fact the original function e^(ix) = cos x + i sin x is really the insightful expression, because it shows how imaginary, exponential, and trigonometric numbers are tied together, rather than just showing one result. Yes! I quite agree. Plugging in a particular value of x does give an interesting and beautiful result but that's not the real bridging concept. It reminds me of the difference between treasure hunting and real archaeology: spectacular finds are cool and fun but the real point of excavation is much broader. I think it's very easy to fall into the trap of thinking that one or two events or equations or works are tentpoles for a whole area of knowledge. The Mona Lisa is a lovely painting but putting it in context, comparing it to other works being done at about the same time all over Italy (and Europe) shows that it's an example of a popular trend or movement. Besides, Raphael was a better artist. :-) Mathematics is like a very, very large wood, focusing on particular trees loses some of the important perspective that gives you the real grasp of the breath and depth and development. I too enjoy lists like: a history of this or that in 100 pieces or works but they are no substitute for real study and understanding. In fact, it is only through first knowing the broader landscape that a real and true appreciation of the finer works can be achieved. Additionally, I can't help but point out that there is a big difference between e^i*pi and e^(i*pi). If you're going to speak the language you have to do it correctly or you risk being misinterpreted.JVL_{March 9, 2021
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I agree 100%, KF, that imaginary is a very bad term and has misled people, and being able to visualize geometrically in the complex plane helps show that they are no different in important ways than real numbers. As to Euler's identity, we've discussed before, but I brought it up, so I'll repeat. e^(i*pi) is a specific value for the function e^(ix) = cos x + i sin x, which are two different ways to express complex numbers in the complex plane. Let x = pi and you get e^(i*pi) = -1. That is straightforward, and there is nothing unusual about it. Rewriting it as e^(i*pi) + 1 = 0 is neat, and I used to point this out to my students, but it does not "open up insights" that the other doesn't. In fact the original function e^(ix) = cos x + i sin x is really the insightful expression, because it shows how imaginary, exponential, and trigonometric numbers are tied together, rather than just showing one result.Viola Lee_{March 9, 2021
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