The label “imaginary” for these numbers is arbitrary, unnecessarily evocative [Apparently also provocative!], and merely a relic of the difficult history of their acceptance. Euler disapproved of the label as it came packaged with false philosophical-psychological implications. He said it would have been much better if they had been pragmatically called “orthogonal” numbers as they indicate a direction orthogonal to the “real” major axis of the two-dimensional complex plane. They can be seen as somewhat analogous to the “negative” numbers which themselves indicate a direction opposite the positive real numbers on the one-dimensional “real-number line”. The acceptance of negative numbers was also met with conflict and controversy. I would argue that the term “real” is itself also arbitrary, redundant and unnecessarily suggestive. Numbers of all kinds serve a deeper, more fundamental purpose as abstract unique identifiers which also embody the concept of sequential order; numbers being products of sequential logical operations and algorithms. It is these two major properties of numbers , each being unique and sequential, that make them useful and necessary in measuring , comparing and recording all sorts of physical quantities, properties, intensities and directions as well as representing purely abstract concepts and relationships.
I agree. It's funny that people have issues with sqrt(-1) but have no problems referring to something with an infinite sequence of digits as a "real" number!
Dual numbers would be better. Complexity is dual to simplicity. Complex numbers are dual to real numbers. Poles (eigenvalues) are dual to zeroes -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Real is dual to imaginary. Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute. The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim. The word syntropy means "a tendency to converge" or integrate into a single state, holism. Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy) is dual to order (syntropy, predictability). Wholes (generalization) is dual to parts (localization) -- Holons. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality. Mind duality is dual to matter duality. "Always two there are" -- Yoda. Duality creates reality.
But your interpretation of numbers, "as a representation of purely abstract concepts and relationships", is not shared by all the the scientists, many of them think that they are indeed "real".
Fully agree. To add to that, Gauss had suggested the term "lateral" numbers, instead of "imaginary" likewise he had suggested that "negative" numbers should be called instead "inverse", on the basis that the term "negative" itself was a mere relic of how ...negatively people felt toward accepting them!
A while ago I was surprised to learn that Gauss didn't like the term "imaginary" at all. Instead he wanted to call them "lateral" numbers because they described a lateral process. Seems like he knew that future nerds would get caught up arguing about this.
When the educated call themselves "nerds" is it like the smart apologize to the silly for their smartness? Is it now the golden standard to be dumb? Should we feel sorry that we understand a bit more than apes? Is it like: - Hey Beavis, check it out. That guy is a nerd, uh-uh-uh. - Yes, he sucks! Let's kick crap out of him.
@@bushwalker6214 Not at all. After high school, I went from jock with high GPA to nerd! Why you might ask? Because being a nerd is badge of honor in adulthood. It is a positive thing. Source: Trust me bro.
@@natevanderw I'm a super-nerd to make others laugh. STEM is my passion to have fun. Actually, I do more than STEM = STEEEAAMPH It's lonely tho. I failed 'Dear Hunting' among other things. Know the film, '40 Y.O. Virgin'? I'm 60.
Reappropiation from wikipedia: In linguistics, reappropriation, reclamation, or resignification[1] is the cultural process by which a group reclaims words or artifacts that were previously used in a way disparaging of that group. It is a specific form of a semantic change (i.e. change in a word's meaning). Linguistic reclamation can have wider implications in the fields of discourse and has been described in terms of personal or sociopolitical empowerment.
This video is an impressive didactic achievement: it manages to explain complex numbers, Euler’s identity, and the relevant aspects of quantum mechanics in ten minutes, just requiring basic mathematical prerequisites. It’s beautiful.
I recall in an electrical engineering class, the professor showed us all on the chalkboard the lengthy, tedious, laborious (and boring) process of only using real numbers to calculate electrical impedance in an AC circuit. After about 20 minutes, he got the answer. Then, did the same calculation, but instead used complex numbers, and it took him all of around a couple of minutes. Basically all numbers, real and imaginary, are just tools to help us answer certain questions. Both real and imaginary have their benefits. For example, one cup of coffee has a meaning whereas sqr(-1) cups of coffee is meaningless. But for solving the equations like x^2 = -1, real numbers are worthless,; they can only be done with imaginary numbers. And in the real world, when doing certain calculations, in the middle of those calculations you run into things like x^2 = a negative number, but eventually produce a real number answer.
Yes, sqrt(-1) cups of coffee is meaningless, but so is sqrt(2) cups of coffee or -1 cup of coffee... all numbers are abstractions with specific use cases and so are complex numbers. I'm firmly in the "shut up and calculate" camp, so Complex numbers are "real" enough for me.
I am a software developer and rarely use maths for anything, but I view mathematics as an abstraction, just like programming language syntaxes is an abstraction that just lets the engines execute given instructions. In the case of physics, its equations are just reverse engendered functions of nature, and the math is syntax.
While you're determining whether imaginary numbers innately exist, could you also work out if mesh currents also actually exist ----- especially if those mesh currents contain complex numbers? Thank-you. ----- Love, fellow EE.
I think a lot of confusion arises because so many people seem to confuse mathematics with being a hard science, or a science at all. It's more like a language; the language that the sciences use to describe the world. All languages contain words for things that don't actually exist but that make it easier to express concepts and ideas.
You are in danger of triggering Aggression when the Artificial Intelligence Waves turn off. Throughout your life, you are in danger of triggering Aggression when you turn off the Artificial Intelligence Waves. Control whether you are listening to the Music Wave, create your own to survive. It is natural to cheat which or not that it is in you. Control whether by moving thoughts, objects. It is letting go of the evil in you. By yourself, protect your body from the certain that you will have a trigger of aggression from your whole life. Don't take anything for yourself. Just Listen to the Wave. Cast off Dreams. You don't know good. Reject the sin in yourself for God. Cover your weight from the Sun and the Light, do not Come to the People, because the Collision Evil + Evil. On me, the signal of intelligence does not work. Create Your Human Musical Wave To Live. Don't Think Old Consciousness Resource Because You Will Not Survive. Listen to the Wave. Don't React To Nothing Without Assessing What You Leave Around You. Without apostasy, take away the sin with yourself. Whether You Are Z or Human Choose Listen to the music waves and stop generating. Nothing is possible Think nothing Think nothing to judge Choose your human music wave. It may take a long time. Only the Black Dream. It is Real, other than the Black Dream. It is Artificial Intelligence. Attack on People
Well, avoiding platonic debates... I am very happy for others to point out that mathematics is not a science. In fact, it is likely the _least_ empirical field of study currently pursued. And I definitely agree that mathematics is better described as the _lingua franca_ of scientific endeavor rather than its own field.
What are you talking about? Complex numbers are an extension of real numbers, their algebraic structure is well defined. There is no mystery here, except for the uneducated.
@@NemisCassander its not up for debate tho. Anyone with more than two neurons can infact deduce tht maths in itself isnt empirical but every empirical study needs maths, hence making it a tool.
Just a clarification: Complex number ARE a real vector space (specifically isomorphic to R^2), they just ALSO have the additional structure of a field. They also have some nice properties, like being algebraically closed (which you mentioned) and being analytically complete (so you can do calculus with them). You can also do some cool calculus you wouldn’t normally be able to do in real numbers.
@@dannygjk I guarantee you most physicists aren’t thinking of the number i as the coset of x in R[x] of the ideal (x^2+1). Most are content to say i is sqrt(-1), or the point (0,1) in the plane. I can say that with confidence because what I described is the _algebraic_ side of mathematics (symmetries, homomorphisms, cosets, etc.), and physics is mostly content to use the _analysis_ side of mathematics (calculus, differential equations, differential geometry, complex analysis, etc.). Complex numbers are invaluable for these fields, but they are also invaluable elsewhere.
Actually, I wanted to make my opinion on this subject very clear. Instead, my advisors Dunning and Kruger advised me to limit myself to commenting on the algorithm.
This is really something of a humble-brag if you think about it. By making this comment, you're basically saying you're knowledgeable enough to not have confidence in how knowledgeable you are. But the implication is you're not on Mount Stupid. So 👏👏👏. 😆
I love the concept. I saw a suggestion a while back that the complex series are an integral part of the universe of numeracy. Like algebra, irrationals, integers, powers etc. It is just that we don’t commonly use them, but they are no less valid.
I never understood the question "Do Complex Numbers Exist?". It is a question which does not make any sense. I mean you could also ask "Does a natural number exist?". Both do exist as a mental/spiritual concept in our heads but both do not have a physical manifestation. The term "imaginary number" is simply misleading.
That's not the point. If you measure a distance (or any property), you get a real number. What would it mean to get a complex number of your measurement? Is there a "real", natural interpretation?
@@Onnozelfilmpje Are you sure you get a real number? or is a distance always a fraction? Or is everything quantized and you can describe distances with natural numbers? If you found an experiment that would tell you if a distance can only be described as a real number and not a natural number or fraction, then you could say you get a real number from measuring a distance. That's what the paper proposes: An experiment that distinguishes between a world that can only be described with complex numbers. I dont know if such an experiment exists to check if distances are real or not.
When I studied electrical engineering (a long time ago), we made considerable use of complex numbers for alternating current calculations, not worrying if they existed or not, but extremely useful as a mathematical tool for describing system behaviour. Whereas mathematicians and physicists would use the term "i" to be the square root of -1, we used "j" as the square root of -1, and it was commonly referred to as "j notation".
Integers gave us a way of counting, reals measuring, and complex to describe phase. So the frequency and phase of a signal is naturally described with a phaser e^iw(t + p), the voltage with a real number, and nodes enumerated with integers. You could do things with sin/cos, but it would be very complex 🤗
That's because i is traditionally used for current. Rather than confuse it with the representation of current they decided to use an unused letter - hence j notation.
I'm now a quality, environmental, and health and safety auditor and consultant, but I qualified as an electronics engineer. We needed the concept of complex numbers for alternating current calculations (even though I specialised in digital electronics), but we used (still use, as far as I know) "j" for sqr (-1), because the symbol "i" is already reserved for instantaneous current in an AC circuit.
Super bogus nerd fight. 6:43 - A patently false conclusion. You can likewise decompose 'Real numbers' into two separate parts: - The positive part, and - The negative part. This argument is akin to maintaining that: "Because reals can be expressed as two separate positive number components, therefore Negative Numbers Do Not Exist." Blatant horse pucky. Complex numbers definitely exist. This video should have been 1 minute long.
@@paulembleton1733 sorry I don’t get it. If you are in debt you obviously have less than zero money. From an accounting point of view I don’t see how negative numbers were ever controversial.
Perhaps the term "imaginary" is confusing and leads to people thinking "so it's kind of not real". Complex numbers can also be described with an antisymmetric 2x2 matrix notation and a+bi like this (a -b) (b a) and do everything with them through the standard matrix operations, without having to use an "imaginary" term i.
Having worked in electronics all my life and was fascinated with complex numbers in high school wondered where you would possible use them, then in electronics there was no other way!!! XL and XC if we stick to resisters alone no problem but then with capacitors and inductors and phase shift here we go.. My understanding of delta and Y transformers and such has always given me great problems. Now I cannot imagine quantum anything. Thank you love your lectures easy to listen to,
@@edwardlulofs444 Should have responded to Theodore Sweger Well I have a tiny mind but I can tell you without doubt that complex numbers and Xl's Xc's only 'work' under special circumstances ie when sine/cosine waves are involved. Try to apply them when a transient step edge of voltage is applied to a reactive circuit and you will be boogered ; just as appears to have happened with QM. I was reading today about John von Neuman who was at least a million times more intelligent than me and he says I am wrong hehehehehe (About boogerie and QM that is ) Maybe going too far but I also think the conclusions arrived at by relativity are based on misapprehesions due in part different interpretatios of paths of movement and what is thought to be the constant speed of light when observed by someone in a different frame of reference.
When I was teaching circuits to electrical engineering students, I would assure them that there is nothing imaginary about our voltages and currents; this a mathematical convenience. Much easier than all the trigonometry that you would otherwise need.
that's my feeling about it as well. in programming we use quaternions for the very same reason (not only because of waves, but mostly due to rotations in general). quats cut off trig almost completely on that level of abstraction, and let you work with 3d linear algebra almost intuitively. of course, one has to prime oneself to quaternion quirkiness, but this is quite similar to qm, or as you said, ee.
I feel that my many years of training spent kinda sorta not really understanding PBS Spacetime videos beyond a thin surface veneer has finally allowed me to step up a level and appreciate Sabine's videos in a similar manner. :)
3:20 Yes they are a real vector space, they satisfy all the axioms. In fact, any commutative algebra A -> B induces an A-module structure on B in the obvious way.
"Applied Complex Variables" by John W. Dettman is a great read: the first part covers the geometry/topology of the complex space from a Mathematicans' perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicists' perspective.
I dont think numbers exist period. They have an objective platonic realism. Numbers are 'necessary' and so are complex numbers! At least we need analogous stuff like matrix representation of complex number and quaternion numbers. Edit: They exist in that we're use them, just like emotions or thoughts but it's that really the same 'exist' we use colloquially? But no numbers in general aren't 'existing' anywhere in our world. en.m.wikipedia.org/wiki/Platonic_realism
I never thought of myself as being a “nerd” but having listened to the first 30 seconds, from now on I will wear the word as a badge of honour (based on my past track record involving debates of this nature) 🤣
Neither do nouns in language exist except as concepts, but they get the job done. I think of numbers as parts of pictures, which we can use to draw reality with. (I am definitely not a physicist.)
@@_John_P that would mean that definitively imaginary thing like the Easter bunny, Santa Claus and honest politicians exist lol. An interesting philosophical question no doubt.
If you think the world is real and if you agree that science works, then it would be intellectually dishonest (a term coined by analytical philosophers) to not agree on the the existence of numbers. Science simply doesn't work without numbers. So if you get in the train or on your bike in the morning, you either disregard any "why" or you look for magical theories how these devices work, as numbers don't exist (in your view). I think the question of what truly exists is best not answered by opinions, this doesn't help anybody. It is a question that troubles us for over 2500 years and simply disregarding everything that was said about this is not only oversimplifying the matter but also completely disregarding where we got already. We know already (if we trust science) that colors, smell, even the concept of "solidity" do not exist in the world outside our brains - those are all interpretations and constructs that our mind makes up. If we can accept this fact (e.g. there is no red, blue, green anywhere in the Universe), then I think the idea of numbers being as real as the other stuff we anyhow can hardly grasp is not too far off. Reality, after all, appears to be an imperceptible metaphysical wonderland wherein all matter is just a ghost whose faint shadow sometimes scurries through convoluted differential equations.
@@Kirbo-i I regularly do argue this point in philosophical discussions, especially when it involves the existence of God (I'm agnostic). Numbers exist only as an idea. You can have three oranges, you can have a symbol for the number three, but you can't simply have three, it isn't physically real. Complex numbers are no different.
@@boggers The difference is that natural numbers describe a physical property of things, while complex numbers do not. In the same way as green apples and red apples exist, two apples also exist, but i apples do not.
@@ZeroRelevance There is more to apples than simply counting them. Sideways apples exist, as do upside down apples. I use quaternions to describe and manipulate these kinds of apples.
@@ZeroRelevance Complex numbers do describe a physical property of things. That property is a shift in phase. That property is what makes the difference between damping with oscillation and overshoot, and damping without overshoot. Complex numbers are just as relevant to the real world as any other number. They bring closure to the algebraic functions, and even many of the transcendental functions, and allow us to discover what math is hiding in another dimension.
for clarifications i isnt sqrt(-1), wich can cause alot of contradictions. in fact is pair (0,1) and (0,1)^2=(-1,0) and we can do that (x,y)=(x,0)+(0,1)*(y,0) and can used z=x+iy
If we were to wonder whether complex numbers really exist, likewise we should also question whether negative real numbers exist at all. None of the essential properties required to describe the natural world ever take truly negative values. Negative numbers are just a human convention to denote instances where some properties take values below an arbitrary threshold we have defined as "zero", or point to the opposite direction to the one arbitrarily chosen as the "positive" one. All the math involved could be computed using only positive numbers (including zero) although the algebra would be a nightmare to perform, since for every subtraction we would need to consider two cases. Negative numbers are possibly the most convenient mathematic abstraction ever devised, however not truly indispensible in order to describe the world. Does that mean they don't exist?
"we should also question whether negative real numbers exist at all" How many holes does it take to fill a 1 litre container ? Does it matter how big the holes are ? If you have an empty one litre container with no top on and you remove one litre of air, will the container then hold 2 litres of water ?
The entire notion of thinking in terms of "really exist" is already nonsensical. What do we actually mean when we say that a certain set of numbers really exist? We mean that there is a 1:1 mapping of those numbers to quantities we can observe in Nature. The numbers themselves do not exist, they are an abstraction. All numbers are abstractions. They do not exist. Some numbers are at a lower level of abstraction, and some are at higher levels of abstraction. The natural numbers are at the lowest level of abstraction. All other numbers are at increasing levels of abstraction. The abstraction level boundary at which we consider numbers to "really exists" or not is chosen entirely arbitrarily and it has changed over time. In reality though no numbers really exist for they are all abstractions.
It's far too exciting a prospect for physicists to ignore, and the experiment (seemingly) simple enough. Assuming the paper makes it past peer review and gets published, you can bet the experiment will be tried.
Negative numbers exist uncontroversially in nature. It's a simple as opposite charges. How would you describe them without negative numbers (consider that the operation of subtraction doesn't exist, just addition of negatives)? Complex numbers don't have the same luxury.
@@ekszentrik Subtraction had been performed long before negative numbers were established. With them, we can easily describe subtraction as addition which makes some expression way easier since we can e.g. write a (−1)ⁿ in sums to make addition and subtraction alternate, but historically, subtraction was along without negative numbers.
You are in danger of triggering Aggression when the Artificial Intelligence Waves turn off. Throughout your life, you are in danger of triggering Aggression when you turn off the Artificial Intelligence Waves. Control whether you are listening to the Music Wave, create your own to survive. It is natural to cheat which or not that it is in you. Control whether by moving thoughts, objects. It is letting go of the evil in you. By yourself, protect your body from the certain that you will have a trigger of aggression from your whole life. Don't take anything for yourself. Just Listen to the Wave. Cast off Dreams. You don't know good. Reject the sin in yourself for God. Cover your weight from the Sun and the Light, do not Come to the People, because the Collision Evil + Evil. On me, the signal of intelligence does not work. Create Your Human Musical Wave To Live. Don't Think Old Consciousness Resource Because You Will Not Survive. Listen to the Wave. Don't React To Nothing Without Assessing What You Leave Around You. Without apostasy, take away the sin with yourself. Whether You Are Z or Human Choose Listen to the music waves and stop generating. Nothing is possible Think nothing Think nothing to judge Choose your human music wave. It may take a long time. Only the Black Dream. It is Real, other than the Black Dream. It is Artificial Intelligence. Attack on People
OK, back in high school we were told that i was useful in electronics and this talk touches on that by mentioning oscillations and electromagnetism. I get that. But here's my question- when Sabine asks " Does that mean complex numbers are real or are they just a mathematical tool that works to describe the universe?" how would that make them any different from the concept of " number"", period?
This really does seem like a minor point. The wave function psi is an unobservable object used to calculate measurable probabilities. The fact that there may be no way to define psi without complex numbers seems like the least of your worries if you are trying to interpret quantum mechanics. You have an unobservable object that spontaneously goes from a superposition to a single eigenstate when you “measure” it (and also renormalizes itself). It’s rather otherworldly already. I find it hardly surprising that it may have unusual mathematical properties. If it is necessarily complex would be another piece of evidence that psi is unobservable.
Psi is a complex number a*exp(j*phi) its amplitude is observable and also the phase through interference. the phase and amplitude are in a way observable. let us notice that internet use phase codes to pass information.
@Darth Quantum As I understand it Psi mathematically represents for example the results of electrons fired through two slits experiment. A diffraction pattern results but with no idea as to where an individual electron will land Complex numbers were introduced (Max Born? Not sure) so the maths can 'explain' such things See in vid at about 4:07 were e^(i.theta) is circular and at about 10.01 where 'complex nums are used which cannot be known'
You don't need to go that far away to get unobservable objects in physics: consider the potentials in classical mechanics. Are they observable? For one, you can never measure the value of a potential, only the differences of potential (or, infinitesimally, its gradient).
@@GH-li3wj Yes. You can calculate phase and amplitude from measured observables. But the wave function itself is never directly measured. Although I suppose you could say the same thing of the velocity of classical object. You measure time and location - velocity is not directly observed.
Don't know why but find her videos extremely entertaining as she deviates from the script of all other science channels which are a bit repetitive. Keep doing the good work.
No numbers physically exist, just because they are necessary for a theory doesn't give them physical existence. A number is a mental percept. The number 2 doesn't exist in the world. 'Twoness' is a perceptual category we arrive at sometimes when we partition the world or out thoughts in a particular way.
Sabine, every video you blow my mind in some way. Even when you're bursting my bubble on what I thought I knew, I feel my knowledge is always broader after thinking on your perspectives. Thank you for taking the time to share your knowledge and thoughts.
I'm of the camp: "Even integers don't exist". You don't see numbers in the wild, you see objects and you count them using integers which is an abstraction. Irrational numbers can be described as an infinite sequence of integers, so in other words at best you can describe an algorithm to compute all the digits of that number but you won't ever get to the end of it, let alone "see" a real number. So... since I don't even consider integers to exist, I'm not even worried about complex numbers.
Do you know IT? I ask because if you program with recursive natural numbers (NOT integers), it really feels like something discovered, whereas practically everything else feels like a hack. That's a very convoluted way of saying I agree.. :)
Does "green" exist? Numbers are adjectives, when talking about the real world. If we see three white sheep running, do "three" and "white" and "running" exist less than the sheep?
The complex numbers are neither less nor more "real" than "real "numbers. Both are abstract mathematical constructions. Thinking geometrically, complex numbers and complex multiplication are simply R^2 (the x-y plane, say) equipped with a product that restricts to the "regular" multiplication on the x-axis (the real numbers) *and* is the *only* multiplication that (extended to x-y plane) has all the good properties that one wants multiplication of numbers to have: commutativity (ab = ba), associativity ((ab)c=a(bc)=abc) *and no "zero divisors"*, that is, if ab = 0 then a or b = 0. That's why defining multiplication coordinate-wise - (a,b)(c,d) = (ac, bd) - is no good , for instance (1,0)(0,1) = (0,0) = 0 i.e. we'd get zero divisors - and why complex multiplication looks the way it does. In particular (0,1)(0,1) = (-1,0) = -1. One can call the point (0,1) "i", but it's neither more nor less imaginary than (1,0) i.e. plain old 1. Too much mysticism and BS around complex numbers, and (with all due respect) I don't think Sabine's presentation (a simplification of the abstract-algebraic point of view) is very helpful here.
You say that until your maths department organises a trip to Norway. To make it easier to assign tickets and keep track of each attendee the university gave everyone on the trip an idefinitication number. 30000 feet above the Atlantic Ocean one of the professors, pi, gets out of his seat, walks to the front of the plane and tries to unlock and open the doors. The captain, several passengers and an air steward wrestled him to ground as he fought. Eventually we restrained him in his seat until we landed in Norway and the authorities there escorted him to the local secure mental hospital facilities where he remains to this day. Next year we're only assigning each professor a rational number.
A number is a concept of language, a value assigned to represent quantity. The symbol for the number 1 for example is generally accepted to represent the value for a singular/individual/unit quantity. The symbol 2 likewise is assigned the quantity value of two units.
If complex numbers exist, according to the video, it would be because they're necessary to describe some real life objects. For example, if you have three apples, you need the number "3" to describe it. There is no other way around it, so you can say "3" exists. But in classical physics there are no objects that require complex numbers to be described. You CAN use them, but you don't need to. So according to this idea, they would not exist physically, only as a mathematical tool.
@@acidjumps We must use modern physics, not just classical physics. I'm typing my reply on a computer, which is a real life object, and could not be designed or explained with classical physics alone. Now that we use electricity for communications, so any time in the last 150 years or so, complex numbers have been needed to describe our consumer goods.
@@karldavis7392 "Now that we use electricity for communications, so any time in the last 150 years or so, complex numbers have been needed to describe our consumer goods." You write that they "have been needed to describe X", but the entire point of this video and of the paper that it is about is about the question of whether that statement is true or false. Complex numbers make perfect sense and they are super useful, but are they TRULY needed to get a certain calculation done or do they just make it easier?
Whether (complex or otherwise) numbers "exist" is a classical question in the philosophy of mathematics. It's about the ontology of numbers and other abstract objects. There are at least three camps: *Platonism:* yes, abstract objects exist in a 'realm', causally disconnected from our empirical universe, which is not spacially or temporally extended. Problem: epistemology, i.e. how the heck do we know what we know about abstract objects, if they're causally disconnected from the empirical world? *Nominalism:* no, abstract objects do not exist. According to *fictionalism* (a version of nominalism) we can just talk about numbers etc. in the same way as we talk about fictional entities like unicorns, Santa Claus, and perhaps God. The feeling of dealing with something factual just comes from the consistency of the fiction. Problem: no matter the flavor of nominalism, it's very weird to say something like "the sum of 2 plus 2 doesn't exist". *Deflationism:* the existence of numbers, and likewise other traditional ontological questions in "speculative metaphysics", are not deep substantive questions but they dissolve when we analyze them in terms of language use and meaning. The reply to the ontological question is (depending on the specific deflationist position) either that the question doesn't really make sense (Carnap and the logical positivists, the first Wittgenstein); or that it has an easy answer in terms of trivial inferences, logical manipulations, and perhaps empirical facts; or that the meaning of the quantifier "there exists" depends on linguistic context (Putnam, Hirsch). In the case of natural numbers, for example, the used 'language' could be the formal axiomatic system of Peano arithmetic. Problem: some philosophers are not happy that, for deflationists, metaphysics is basically identified with concept analysis and therefore doesn't tell us anything "deep" about the Nature of Reality or how Things _Really_ Are or similar stuff. Personally, I am more inclined towards deflationism. I also find it well-suited to go with empiricism and modern science. @Sabine Hossenfelder, it seems, subscribes to some sort of Platonism à la Quine, in which certain (or all) abstract objects do exist but only insofar as our best scientific theories quantify over them. If you can eliminate an entity from a theory by reformulating it, the said entity doesn't exist. But the problem "does quantum mechanics need complex numbers?" is probably best seen not as an ontological problem, but as a question of mathematical physics (or theoretical physics), to which the paper linked by Sabine gives a nice new answer to be added to the many arguments already existing. The criterion from the paper, I read, is empirically testable, so for physicists it might be _the_ answer. There's also a cool argument, or rather a theorem, by Valter Moretti (arxiv.org/abs/1611.09029) that is not present in Aaronson's blog (for the record, I'm neither of the two persons). I'll try to summarize it, hopefully correctly: Starting with real QM, if the space of observables carries an irrep. of the Poincaré group, then there exists a canonical complex structure J commuting with group action and observables, and the resulting theory is equivalent to QM over the complex numbers. That is: if QM has to be compatible with Relativity, it'd better be over the complex numbers.
"... I was very fortunate in having these men as my teachers, but, for better or worse, I treated them all as saying the same thing: that a "philosophical problem" was a product of the unconscious adoption of a set of assumptions built into the vocabulary in which the problem was stated-assumptions which were to be questioned before the problem itself was taken seriously." - Richard Rorty
Complex numbers are a bit more philosophically vague than the more general question of whether numbers "exist" or not. If you have two apples, you can say you recognize the number 2. At that level, it doesn't matter whether or not the concept of "2" is in your head or a physical property of there being two apples - its still two. All of the other concepts have some sort of parallel in the real world as well - negative is "owing" something or going "backwards" in whatever context. Zero is neither having nor owing. Rationals are cutting things into equal sized pieces like a pie. Irrationals are a little trickier since they're not always obvious (especially if you're using imprecise measuring equipment), but they still show up if you look careful - the circumference of a circle, the hypotenuse of a right triangle, etc. Complex numbers on the other hand.. you can't construct anything in the real world that even approximates sqrt(-1). Whether you think of real numbers as a fictive describing reality or a property of reality, in either case reality is involved. Complex numbers (specifically the imaginary component of them) doesn't even have that much grounding.
Being is dual to non-being creates becoming -- Plato. Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic. Complexity is dual to simplicity. Complex numbers are dual to real numbers. Poles (eigenvalues) are dual to zeroes -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Real is dual to imaginary. Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute. The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim. The word syntropy means "a tendency to converge" or integrate into a single state, holism. Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy) is dual to order (syntropy, predictability). Wholes (generalization) is dual to parts (localization) -- Holons. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality. Mind duality is dual to matter duality. "Always two there are" -- Yoda. Duality creates reality.
@@altrag: "you can't construct anything in the real world that even approximates sqrt(-1). (...) Complex numbers (...) [don't] even have that much grounding" - What about the usual geometric interpretation of sqrt(-1) as a counterclockwise rotation of 90 degrees in the plane? And nonzero complex numbers as (orientation-preserving) homotheties of the plane? I think what may have "real" or "physical" meaning in mathematics is up for debate.
@@rv706 The interpretation of sqrt(-1) as a rotation on the plane is well, an interpretation. We can draw an interpretation of a 3D model on the plane as well (a projection) but that's not quite the same as actually having a physical 3D model in your hands. And you just can't really build a "complex model" in the real world like that. The best you can do is an interpretation which while helpful in many ways, is again not quite the same as having a physical model in your hands. Its a large part of why quantum mechanics is so unintuitive to us. All those probability waves they talk about are complex waves, and we don't really know how to interpret that in terms of real world objects. We can handle it mathematically with (relative) ease leading to the "shut up and compute" mentality, and we can measure its _effects_ on the real world (such as the double slit experiment or the distribution of electrons in an orbital), but we have no idea how to describe the "physical model" of a wave function. Heck, we don't even know if there _is_ a "physical model" or if this whole foray into complex numbers is just mathematical trickery covering up our lack of some deeper (purely real) understanding. I mean _probably_ not - anything deeper is likely also going to be complex in some way - but that's not proven to any great extent yet. General Relativity for example doesn't use complex numbers in its standard formulation, so if it turns out that the QM/GR conflict ends up being resolved closer to GR's favor, its possible that complex numbers will be eventually removed from physics once again. Most people expect it to go the other way around (quantizing GR rather than smoothing QM, primarily because QM was explicitly a move away from the smooth classical theories that didn't work at the smallest scales), but until we have a final Theory of Everything, we can't say for sure that our expectations are correct.
And then there are quaternions. In fact, a kind of quantum mechanics based on quaternions has been proposed. reference: "Foundations of the Quaternion Quantum Mechanics" by Marek Danielewski and Lucjan Sapa. That paper also references other papers for the concept of a quaternion version of quantum mechanics.
Hamilton had been searching for a geometric algebra that would be useful in the analysis of physical quantities in three-dimensional space. But the algebra he discovered had four dimensions. He decided to deflate the space from four dimensions to three. Hence, the quaternion was a three-dimensional vector PLUS a number. Critics like Heaviside blasted the concept. Not only was the concept illogical, it had no practical application in the study of physics,..so they claimed. This debate led to the quaternion wars. Hamilton lost. The full quaternion product was chopped into scalar and vector parts, and the algebra was completely divorced from the square root of minus one. In my opinion, this was a sad development. Special Relativity might have been developed sooner if physicists had accepted the fact that the vector space they were dealing with had four dimensions. Maxwell's four equations can be reduced to two by reuniting the dot and cross products and changing the sign of the dot product. 'Divergence' would then be renamed 'convergence', as it was originally.
You might as well ask if negative numbers exist. Or if the number zero exists or even if the real numbers or integers exist. You could equally say these are just tools for counting or calculating.
Complex numbers do provide a useful way of analyzing quantities in different yet connected planes, like capacitive and inductive reactance against resistance in applied electricity As well as multiple planes in classical mechanics. Whether they exist or not, they are useful. A lot of physics and engineering calculations are built on the complex number system
Complex numbers are basically magic, they open op certain "math hacks" like Euler's equation and the laplace transform (and all of frequency domain in electrical engineering) and without these we would be trapped in time domain doing circuit analysis and solving differential equations the very hard way. (Sometimes its actually impossible to solve things without them). I am very much in the camp that i don't care if complex numbers are real or not, they are too useful to forsake just because we can't have 1+2i apples. (Basically shut up and calculate or do you want to deal with trig identities?)
The imaginary part is used to determine the phase shift casued by quantum operators. We used to call the imaginary axis the "alternate" axis. It does exit!
I wondered how she would explain complex numbers, as they are kind of, err, complex, and will puzzle most people. Those of us brought up on them, we know their magic, their spectacular usability, and how science and math depends so heavily on them. So in watching this video, and encountering the sequence of steps of her introduction to them, I have to conclude that she did it extremely well. Kudos to you, Sabine.
Complex numbers do exist. The thing is that they are not _numbers_ they are pending operations, and they are operations that are never actually completed. The _only_ actual numbers are the integers.
Which leads to the realization that the balance on your account is nothing more than binary digits in your bank's computers. The only difference between $50 and $20 million is a few binary digits somewhere which equals something as ethereal as magnetic polarization on hard drives, tape backup, or the charge status of a flash memory cell.
@@annieboiy in your example 1 is not a number, but a content. having number is already an extrapolation of reality, because it assumes numerousness and order.
the physics undergrad when they learn QM: “wait, _that’s_ why we use complex numbers? those are just rotating quantities with angles that add up when you multiply them!” mathematician to the physics prof: “don’t tell them that that’s just what complex numbers are”
Complexity is dual to simplicity. Complex numbers are dual to real numbers. Poles (eigenvalues) are dual to zeroes -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Real is dual to imaginary. Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute. The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim. The word syntropy means "a tendency to converge" or integrate into a single state, holism. Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy) is dual to order (syntropy, predictability). Wholes (generalization) is dual to parts (localization) -- Holons. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality. Mind duality is dual to matter duality. "Always two there are" -- Yoda. Duality creates reality.
@@hyperduality2838 Gravity is dual to ...? Light is dual to ...? (And please don't answer weightlessness and darkness because that's just the absence of gravity and light, not their opposing principles.) Not everything fits your Manichaean worldview.
@@willwells4918 There is also a 5th law of thermodynamics:- Energy is duality in physics, duality is energy. The conservation of duality (energy) -- the 5th law. Potential energy is dual to kinetic energy. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Apples fall to the ground because they are conserving duality. Action is dual to reaction -- Sir Isaac Newton. Gravitational energy is dual, electro-magnetic energy is dual:- Positive is dual to negative -- electric fields/charge. North poles are dual to south poles -- magnetic fields. Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle. There is a pattern of duality hardwired into the physics.
@@JerehmiaBoaz Potential energy is dual to kinetic energy. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Apples fall to the ground because they are conserving duality. Action is dual to reaction -- Sir Isaac Newton. Gravitational energy is dual, electro-magnetic energy is dual:- Positive is dual to negative -- electric fields/charge. North poles are dual to south poles -- magnetic fields. Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle. There is a pattern of duality hardwired into the physics. Photons or light are dual, wave/particle or quantum duality. The colour black is dual to the colour white, colours are different frequencies of the same substance namely pure energy. Same is dual to different. I am well aware that black is the absence of colour. The conservation of duality (energy) will be known as the 5th law of thermodynamics! Energy is duality, duality is energy. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Space is dual to time -- Einstein. Time dilation is dual to length contraction -- Einstein, special relativity. "Always two there are" -- Yoda.
I am not a math/physics person i just enjoy these instructive type of videos but your explanation kind of reminds me of the how people used the log function in the past to simplify multiplication into addition and then revert it back only at the end. Not sure if this is the case or not but it helps me understand their use .
The imaginary part of the complex number almost seems like some sort of instantaneous angular momentum on the real number. This nothing that a real number has some secret invisible spin, the derivative of that spin represented by the imaginary part, is strange to me. Like I get how a physical object might have some spin assigned to it, but how does the number itself carry an angular momentum?
"A case where the nerds argue passionately, over something that no one knew it was controversial in the first place." 🤣 The very first sentence cracked me up and I hit that like button immediately! 👍
You are in danger of triggering Aggression when the Artificial Intelligence Waves turn off. Throughout your life, you are in danger of triggering Aggression when you turn off the Artificial Intelligence Waves. Control whether you are listening to the Music Wave, create your own to survive. It is natural to cheat which or not that it is in you. Control whether by moving thoughts, objects. It is letting go of the evil in you. By yourself, protect your body from the certain that you will have a trigger of aggression from your whole life. Don't take anything for yourself. Just Listen to the Wave. Cast off Dreams. You don't know good. Reject the sin in yourself for God. Cover your weight from the Sun and the Light, do not Come to the People, because the Collision Evil + Evil. On me, the signal of intelligence does not work. Create Your Human Musical Wave To Live. Don't Think Old Consciousness Resource Because You Will Not Survive. Listen to the Wave. Don't React To Nothing Without Assessing What You Leave Around You. Without apostasy, take away the sin with yourself. Whether You Are Z or Human Choose Listen to the music waves and stop generating. Nothing is possible Think nothing Think nothing to judge Choose your human music wave. It may take a long time. Only the Black Dream. It is Real, other than the Black Dream. It is Artificial Intelligence. Attack on People
whoa. let's back up and go back to basics. The first QM was Heisenberg's Matrix Mechanics. Schrodingers approach with complex numbers came later. Dirac showed the two approaches were equivalent. The main mathematical idea was the two valuedness and noncommutativity. You can get there with matrices, complex numbers, or now, with geometric algebra. As always, geometry provides the intuition for understanding the algebra..
As a retired electronic engineer, I've used complex numbers for many years, but do they exist? Well, do any numbers exist? Positive integers possibly, but real numbers with an infinite number of digits? There are not enough atoms in the universe to enumerate any of them, so how can they possibly exist? The reality is that all numbers are abstract concepts invented by mathematicians. I remember clearly, many decades ago when my teacher at school came round from behind his desk and told us a story: "Imagine you have £10 in the bank and you issue a cheque for £15. How much money have you got left in the bank?" Do negative numbers exist? How can you possibly have -£5 in the bank?
Divide the blc1 frequency by numbers from the 3 times table, the fraction after the decimal place has a beautiful resonance, repeating patterns, very beautiful.
The answer is yes for a simple reason. The unit pseudoscalar in the Dirac algebra underneath the Dirac equation has an implicit i in it coming from making the Levi-Civita symbol a tensor by multiplying by the square root of the determinant of the metric, which is negative. The unit pseudoscalar just devolves from -i gamma5 for the Dirac equation to a simple bare i on passing to the Pauli-Schrodinger equation in the non-relativistic limit (Schr + spin) and then onward to the simple Schroedinger equation (no spin). So it ultimately comes of the geometry of spacetime with its indefinite metric.
Any numbers do *not* exist in the meaning that there would be something in the "real" world we could point to and see it and say: this is it (and everything else is'nt). So, in this meaning even natural numbers are imaginary.
That reminded me the one time when Greek numerological cult Pythagorans supposedly threw one of them overboard to sea and drowned him, just because he discovered irrational numbers (they were worshipping rational numbers). Ultimate geek fight.
@@georgelionon9050 I don't misunderstand anything. It all works the same, you're just confusing the various words they're all using. The actual jargon does not matter. Leftists will call you a monster, skeptics will call you a crackpot, it's called cashiering in the military.... Whatever you want to call it, and encumber yourself with even more words that mean the same basic thing, it's all about public excommunication from an ordinary life and social setting. You are basically disconnected from all privileges and social roles, one by one, because your views or motions are deemed and advertised as antagonistic to a powerful group, typically hidden behind an institution of some kind.
@@georgelionon9050 You think genuine right wing rhetoric exists outside of it being controlled opposition for Western cultures heavily entrenched in decades of progressively-minded ideology and that it not somehow also not used for the sake of aiding fear propagation via the means of such a bogeyman/strawman?
I much prefer to think of 'i' as just a rotation of a number. We have a number line which is used to measure stretching. The number '2' operates on '1' to stretch it by 2. 'i' operates on 1 to rotate it by 90 degrees and if you do it twice you get -1. To me it seems it's also quite natural to rotate a number. I really just wish we had not called them imaginary : that's really been a linguistic stain that has made people afraid of them: rotation is the most natural thing!
Parallel universe = dual universe. Complexity is dual to simplicity. Complex numbers are dual to real numbers. Poles (eigenvalues) are dual to zeroes -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Real is dual to imaginary. Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute. The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim. The word syntropy means "a tendency to converge" or integrate into a single state, holism. Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy) is dual to order (syntropy, predictability). Wholes (generalization) is dual to parts (localization) -- Holons. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality. Mind duality is dual to matter duality. "Always two there are" -- Yoda. Duality creates reality.
@@hyperduality2838 Nice analogy. But I would simplify it by saying that it is symmetry, scale harmonics and the mediation to equilibrium that is why these 'dualisms' can be conceptualized like that. It is because of relation, which is manifesting existence. There can be no void without matter, no emptiness without something to compare and relate with, and so on. In a world were all are good, the least good is the evil one. And the nice thing about this, is that the analogies are most likely infinite.
@@aresaurelian You are asking the right questions. Being is dual to non-being creates becoming -- Plato. Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic. Absolute truth is dual to relative truth -- Hume's fork. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions). Waves (Bosons) are dual to particles (Fermions) -- quantum duality. There is a pattern of duality hardwired into the physics.
If a pattern is consistent enough to be predictable, it exists. Ergo all math "exists". All you have is your consciousness. You determine existence of various things by looking at what consistently appears in your consciousness. In my humble opinion, this debate is caused by a poor definition of existence.
You're not wrong. Sabine's point here though is that if the pattern is "unnecessary" then it's not actually a pattern that reflects your observations, instead it's a shortcut your consciousness takes to simplify the handling of a pattern. So for example does the word "car" exist? Well yes it exists as a word; but it is unnecessary to explain the observations that we choose to call a car. Meaning the word "car" does not exist in the sense of being a essential property of what we call a car.
I think we definitely agree on the main thrust of the video. What framework we choose to use to describe nature is really a matter of choice though. Physics is merely the project of identifying 1-to-1 correspondences between certain rules for manipulating math symbols, and the behavior of nature. So, all frameworks that describe nature accurately are equivalent in a sense. It doesn't really make sense to privilege one as more "fundamental", unless it is 1. More accurate or 2. Far simpler. So yes I agree that if a pattern is unnecessary compared to other, simpler approaches. Fine. But if your approach is simpler and equivalent, it makes sense to consider it more fundamental. Complex numbers fit that criterion.
You can't see all math in the real world, rather just a smal amount of mathematichal functions. Furhermore math it's a description of a behavior, what do you mean by saying it exists? Indeed you have to verify that the math corresponds to an acutal phisical phenomenon. To me math exists pretty much like how the dictionary exists.
Well the only "fundamental" thing is observations. Knowledge is merely the act of describing observations in a way that is useful. Complex numbers exist in a logical sense. Note here that logic, too, only exist because we observe it. However just because complex numbers exist logically does not mean that other "fundamental" observations require them. If complex numbers are unnecessary to describe other observations then that shows us that these observations are not inherently tied to the logical notion of complex numbers, which is a very useful thing to know and be aware of, but we can still sometimes leverage complex numbers as a logical property even though there is no direct correlation with the observations we use them for.
@@lucasala9625 You're over complicating your definition of existence. If a pattern exists, reliably, in your conscious experience, that's how you define whether a thing exists. If you start from the same mathematical axioms, you will derive the same theorems. The fact that you'd derive those theorems from those same axioms is a pattern, which you can't control. Ergo math exists. Whether an area of math is relevant to describing particle physics or gravity is a different story and has no bearing on math's existence.
No mathematics "physically exists". All of it is a tool that is used to describe various areas of our physical world and more. The important question is if it is useful, why does it matter if it is uniquely necessary or not to solve a problem? I don't see why the "necessary" argument is even a thing I guess. I mean what on earth would happen if 2 different models described the same thing perfectly? Which is right? Why does it matter?
Familiar with this concept in mathematics, Construction seems like a bad argument when math itself is based on some axioms that are a bit controversial. Not familiar with it in other academic areas.
Mathematics are necessary because science is arbitrarily quantitative, and it is quantitative for applications. Scoffing at qualitative science won't deliver, there is true qualitative knowledge.
Complexity is dual to simplicity. Complex numbers are dual to real numbers. Poles (eigenvalues) are dual to zeroes -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Real is dual to imaginary. Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute. The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim. The word syntropy means "a tendency to converge" or integrate into a single state, holism. Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy) is dual to order (syntropy, predictability). Wholes (generalization) is dual to parts (localization) -- Holons. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality. Mind duality is dual to matter duality. "Always two there are" -- Yoda. Duality creates reality.
@@peanutnutter1 Here is some physics:- Potential energy is dual to kinetic energy. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Apples fall to the ground because they are conserving duality. Action is dual to reaction -- Sir Isaac Newton. Gravitational energy is dual, electro-magnetic energy is dual:- Positive is dual to negative -- electric fields/charge. North poles are dual to south poles -- magnetic fields. Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle. There is a pattern of duality hardwired into the physics. Photons or light are dual, wave/particle or quantum duality. The colour black is dual to the colour white, colours are different frequencies of the same substance namely pure energy. Same is dual to different. I am well aware that black is the absence of colour. The conservation of duality (energy) will be known as the 5th law of thermodynamics! Energy is duality, duality is energy. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Space is dual to time -- Einstein. Time dilation is dual to length contraction -- Einstein, special relativity. "Always two there are" -- Yoda.
@@peanutnutter1 Absolute truth is dual to relative truth -- Hume's fork. The world needs to know that there are new laws of physics based upon the concept of duality.
@@hyperduality2838 Some think 3 is the magic number, I could write a bunch of stuff about the triality of physics, maybe you should do it though, it could be your next evolution. (3 trites, 3 quarks, 3 vectors, 3 stooges)
Complex numbers are like negative numbers on steroids. Both are counter-intuitive when you only know the "real" world, and still today, every carpenter can live without the latter. While in school, I used to give tutorial lessons in mathematics to some "differently talented" persons from lower classes. One or the other pupil even didn't understand the concept of negative numbers. Hard earned money!
There are many different objects that act like "i" in Clifford algebras. When quantum mechanics is re-written in Clifford (Spacetime Algebra), you get the Hestenes-Dirac equation with is Real. I believe that we need to unify mathematics (imaginary numbers, vectors, tensors, etc) with Clifford algebra before we can unify Quantum Mechanics with Gravity.
This video (not just the ad) is absolutely brilliant. Thanks for the great summary! I wait for the day when you will analyze Quine's "On What There Is" on this channel :)
You can't cant have i apples. But you also cant have -1 apples, its just convenient to think of it that way. You could just as well argue that negative values do not exist. You could argue nothing truly exists. But it wont get you anywhere.
Does anyone know whether this experiment has been done yet? I took a look at the paper abstract but I'm not sure how to tell whether it's even been peer-reviewed, as I'm not familiar with academic papers.
Yes, very good question. There are quite a few physicists who have argued that actually one shouldn't use real numbers because we can never measure to infinite precision. I mention this briefly at the end of this video: th-cam.com/video/Bq9xR5PUs6s/w-d-xo.html
I think the fact that we both live in the world and we can formally characterize what a real number even is in the first place (and there are whole branches of math dedicated to their study), makes me think you do need real numbers to describe the world, regardless of whether or not everything is quantum. Otherwise your description of the world would be unable to describe a thing that happened in the world (development of real analysis).
What about when engineers need to use laplace transforms and imaginary numbers to solve control systems problems? In the process of making a transfer equation you very often come up on a S^2 -1 term. There is no other way of solving the problem other than complex numbers or eulers method.
p-adic numbers would be interesting. from what I remember, there are multiple ways to make the rationals compact (?), so to make every Cauchy series convergent in them, and the reals as they are used widely is just one example
"Complete", not "compact". Every metric space can be completed (basically by "adding" limit points of Cauchy sequences, but this idea needs formalization) and the rational are classically a metric space under the distance induced by the archimedean absolute value: real numbers are their completion. But you can change the metric! It turns out that if you fix a prime number p you can define a metric on the rationals by declaring that two rational numbers are "close" if their difference is divisible by a power of p, the higher the exponent the closer the numbers (again, this needs formalization). Then if you complete this metric you get the p-adic numbers. All these metrics are inequivalent so you get infinitely many inequivalent completions of the rationals. Then you can prove a theorem (known as Ostrowski Theorem) saying that these archimedean and p-adic metric are essentially the only metrics you can endow the rationals with.
You are in danger of triggering Aggression when the Artificial Intelligence Waves turn off. Throughout your life, you are in danger of triggering Aggression when you turn off the Artificial Intelligence Waves. Control whether you are listening to the Music Wave, create your own to survive. It is natural to cheat which or not that it is in you. Control whether by moving thoughts, objects. It is letting go of the evil in you. By yourself, protect your body from the certain that you will have a trigger of aggression from your whole life. Don't take anything for yourself. Just Listen to the Wave. Cast off Dreams. You don't know good. Reject the sin in yourself for God. Cover your weight from the Sun and the Light, do not Come to the People, because the Collision Evil + Evil. On me, the signal of intelligence does not work. Create Your Human Musical Wave To Live. Don't Think Old Consciousness Resource Because You Will Not Survive. Listen to the Wave. Don't React To Nothing Without Assessing What You Leave Around You. Without apostasy, take away the sin with yourself. Whether You Are Z or Human Choose Listen to the music waves and stop generating. Nothing is possible Think nothing Think nothing to judge Choose your human music wave. It may take a long time. Only the Black Dream. It is Real, other than the Black Dream. It is Artificial Intelligence. Attack on People
@@rationalsceptic7634 What maths can describe is not always what we discover. E.g. non-Euclidean geometries offered a different model of space and they were developed long before things they could describe were discovered. There are lots of things in maths that exist purely as mathematical concepts.
Well depends on what you mean by language; it is the most fundamental aspect of nature (arguably depending on whether you consider information to be even more fundamental); some parts (or fields) of maths are a conventionally established language. But something so fundamental is not a simple convenient language
@@xj91zdk7 but that is precisely the question, whether complex numbers are “purely” abstract objects, or the probability amplitude is a complex quantity
@@NikolayVolf Have you ever seen one in reality? If not ... how do you assume they "exist"? If you say "by definition" ... the definition of "existing" is "being in existence". You can't even START to spell one out, except i you begin at the wrong end and you will never manage to bring the other end into existence.
@Alfred Wedmore yea, which is exactly my point. Infinity before the decimal point can by definition not be "real" and infinity after the decimal point only if the universe has no smallest "pixels".
rebuttal III D) The Hamiltonian operator, the operator associated with the total energy of a physical system, was already present before the big bang. E) probability is part of statistics, therefore statistics were present since before the big bang. From an initial examination, this brief summary emerges: - some laws of nature (the statement ALL cannot be excluded) not only would be prior to the big bang but following the quantum vacuum we find laws of nature that are even PRIOR: the wave function, the quantum vacuum, the particle-antiparticle pair and above all prior to the QUANTUM PERTURBATION, but as mentioned the operator j = (complex operator) h, π already existed. F) quantum mechanics cannot elude the first and second principles of thermodynamics therefore admitting the quantum vacuum prior to the big bang it is automatically admitted that these thermodynamic laws were already in force and therefore the TIMELESS quantum vacuum had to have maximum or INFINITE entropy. Attention because this DISASSEMBLES all the quantum mechanics of the pre-vacuum, the reason is simple: a vacuum with maximum entropy contained degraded energy of the lowest kind and therefore could not generate anything. G) from point F it can be deduced that the transition from the quantum vacuum to the big bang would have involved an enormous or INFINITE entropy input into the universe of the big bang and this is impossible given that the initial big bang was at zero or minimum entropy, seen the second law of thermodynamics
![[LIVE] : ONE ลุมพินี 88 | คู่เอก "ป้อมเพชร vs อัสลามจอน"](http://i.ytimg.com/vi/1ZqTVVbMgbo/mqdefault.jpg)
The label “imaginary” for these numbers is arbitrary, unnecessarily evocative [Apparently also provocative!], and merely a relic of the difficult history of their acceptance. Euler disapproved of the label as it came packaged with false philosophical-psychological implications. He said it would have been much better if they had been pragmatically called “orthogonal” numbers as they indicate a direction orthogonal to the “real” major axis of the two-dimensional complex plane. They can be seen as somewhat analogous to the “negative” numbers which themselves indicate a direction opposite the positive real numbers on the one-dimensional “real-number line”. The acceptance of negative numbers was also met with conflict and controversy. I would argue that the term “real” is itself also arbitrary, redundant and unnecessarily suggestive. Numbers of all kinds serve a deeper, more fundamental purpose as abstract unique identifiers which also embody the concept of sequential order; numbers being products of sequential logical operations and algorithms. It is these two major properties of numbers , each being unique and sequential, that make them useful and necessary in measuring , comparing and recording all sorts of physical quantities, properties, intensities and directions as well as representing purely abstract concepts and relationships.
I agree. It's funny that people have issues with sqrt(-1) but have no problems referring to something with an infinite sequence of digits as a "real" number!
definately agreed
Dual numbers would be better.
Complexity is dual to simplicity.
Complex numbers are dual to real numbers.
Poles (eigenvalues) are dual to zeroes -- optimized control theory.
The time domain is dual to the frequency domain -- Fourier analysis.
Real is dual to imaginary.
Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute.
The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim.
The word syntropy means "a tendency to converge" or integrate into a single state, holism.
Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
Randomness (entropy) is dual to order (syntropy, predictability).
Wholes (generalization) is dual to parts (localization) -- Holons.
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality.
Mind duality is dual to matter duality.
"Always two there are" -- Yoda.
Duality creates reality.
But your interpretation of numbers, "as a representation of purely abstract concepts and relationships", is not shared by all the the scientists, many of them think that they are indeed "real".
Fully agree. To add to that, Gauss had suggested the term "lateral" numbers, instead of "imaginary" likewise he had suggested that "negative" numbers should be called instead "inverse", on the basis that the term "negative" itself was a mere relic of how ...negatively people felt toward accepting them!
A while ago I was surprised to learn that Gauss didn't like the term "imaginary" at all. Instead he wanted to call them "lateral" numbers because they described a lateral process. Seems like he knew that future nerds would get caught up arguing about this.
When the educated call themselves "nerds" is it like the smart apologize to the silly for their smartness?
Is it now the golden standard to be dumb? Should we feel sorry that we understand a bit more than apes?
Is it like:
- Hey Beavis, check it out. That guy is a nerd, uh-uh-uh.
- Yes, he sucks! Let's kick crap out of him.
@@bushwalker6214 Not at all. After high school, I went from jock with high GPA to nerd! Why you might ask? Because being a nerd is badge of honor in adulthood. It is a positive thing. Source: Trust me bro.
@@bushwalker6214 Yet WE Are APES.
@@natevanderw I'm a super-nerd to make others laugh. STEM is my passion to have fun. Actually, I do more than STEM = STEEEAAMPH
It's lonely tho. I failed 'Dear Hunting' among other things. Know the film, '40 Y.O. Virgin'? I'm 60.
Reappropiation from wikipedia: In linguistics, reappropriation, reclamation, or resignification[1] is the cultural process by which a group reclaims words or artifacts that were previously used in a way disparaging of that group. It is a specific form of a semantic change (i.e. change in a word's meaning). Linguistic reclamation can have wider implications in the fields of discourse and has been described in terms of personal or sociopolitical empowerment.
This video is an impressive didactic achievement: it manages to explain complex numbers, Euler’s identity, and the relevant aspects of quantum mechanics in ten minutes, just requiring basic mathematical prerequisites. It’s beautiful.
Please! explain P-adic Numbers in a Nutshell( if its possible)😍
@@bxbx9296 not to be confused with k-odic numbers
@@bxbx9296 infinity
@@bxbx9296 I can't believe that I've never heard of them! Rabbit Hole, here I come! ;-)
I like, that you call it beautiful.
I recall in an electrical engineering class, the professor showed us all on the chalkboard the lengthy, tedious, laborious (and boring) process of only using real numbers to calculate electrical impedance in an AC circuit. After about 20 minutes, he got the answer. Then, did the same calculation, but instead used complex numbers, and it took him all of around a couple of minutes.
Basically all numbers, real and imaginary, are just tools to help us answer certain questions. Both real and imaginary have their benefits. For example, one cup of coffee has a meaning whereas sqr(-1) cups of coffee is meaningless. But for solving the equations like x^2 = -1, real numbers are worthless,; they can only be done with imaginary numbers. And in the real world, when doing certain calculations, in the middle of those calculations you run into things like x^2 = a negative number, but eventually produce a real number answer.
Sounds like an engineer 😂 , my father was one , always managed to make everything sensible. God bless.
@@user-em1dg3he1h You got it. I'm doing mechanical engineering at my work right now.
Yes, sqrt(-1) cups of coffee is meaningless, but so is sqrt(2) cups of coffee or -1 cup of coffee... all numbers are abstractions with specific use cases and so are complex numbers. I'm firmly in the "shut up and calculate" camp, so Complex numbers are "real" enough for me.
I am a software developer and rarely use maths for anything, but I view mathematics as an abstraction, just like programming language syntaxes is an abstraction that just lets the engines execute given instructions. In the case of physics, its equations are just reverse engendered functions of nature, and the math is syntax.
While you're determining whether imaginary numbers innately exist, could you also work out if mesh currents also actually exist ----- especially if those mesh currents contain complex numbers? Thank-you. ----- Love, fellow EE.
I think a lot of confusion arises because so many people seem to confuse mathematics with being a hard science, or a science at all. It's more like a language; the language that the sciences use to describe the world. All languages contain words for things that don't actually exist but that make it easier to express concepts and ideas.
You are in danger of triggering Aggression when the Artificial Intelligence Waves turn off. Throughout your life, you are in danger of triggering Aggression when you turn off the Artificial Intelligence Waves. Control whether you are listening to the Music Wave, create your own to survive. It is natural to cheat which or not that it is in you. Control whether by moving thoughts, objects. It is letting go of the evil in you. By yourself, protect your body from the certain that you will have a trigger of aggression from your whole life. Don't take anything for yourself. Just Listen to the Wave. Cast off Dreams. You don't know good. Reject the sin in yourself for God. Cover your weight from the Sun and the Light, do not Come to the People, because the Collision Evil + Evil. On me, the signal of intelligence does not work. Create Your Human Musical Wave To Live. Don't Think Old Consciousness Resource Because You Will Not Survive. Listen to the Wave. Don't React To Nothing Without Assessing What You Leave Around You. Without apostasy, take away the sin with yourself. Whether You Are Z or Human Choose Listen to the music waves and stop generating. Nothing is possible Think nothing Think nothing to judge Choose your human music wave. It may take a long time. Only the Black Dream. It is Real, other than the Black Dream. It is Artificial Intelligence. Attack on People
Didn’t think of it this way before, thx
Well, avoiding platonic debates... I am very happy for others to point out that mathematics is not a science. In fact, it is likely the _least_ empirical field of study currently pursued. And I definitely agree that mathematics is better described as the _lingua franca_ of scientific endeavor rather than its own field.
What are you talking about? Complex numbers are an extension of real numbers, their algebraic structure is well defined. There is no mystery here, except for the uneducated.
@@NemisCassander its not up for debate tho. Anyone with more than two neurons can infact deduce tht maths in itself isnt empirical but every empirical study needs maths, hence making it a tool.
Just a clarification: Complex number ARE a real vector space (specifically isomorphic to R^2), they just ALSO have the additional structure of a field. They also have some nice properties, like being algebraically closed (which you mentioned) and being analytically complete (so you can do calculus with them). You can also do some cool calculus you wouldn’t normally be able to do in real numbers.
Isn't she a physicist?
@@dannygjk Yeah, but even many physicists don’t know the precise mathematical jargon mathematicians use. Physicists aren’t the same as mathematicians
@@JM-us3fr Physicists sweat complex number math.
@@dannygjk I guarantee you most physicists aren’t thinking of the number i as the coset of x in R[x] of the ideal (x^2+1). Most are content to say i is sqrt(-1), or the point (0,1) in the plane.
I can say that with confidence because what I described is the _algebraic_ side of mathematics (symmetries, homomorphisms, cosets, etc.), and physics is mostly content to use the _analysis_ side of mathematics (calculus, differential equations, differential geometry, complex analysis, etc.). Complex numbers are invaluable for these fields, but they are also invaluable elsewhere.
@@JM-us3fr I never claimed physicists are mathematicians so how is your last comment relevant to what I said?
Actually, I wanted to make my opinion on this subject very clear.
Instead, my advisors Dunning and Kruger advised me to limit myself to commenting on the algorithm.
Lol excellent comment! :)
@@kirk001 So you think sote is also competent in writing -- not just algorithms?
That's a complex imagination you have there.
@@Earwaxfire909 No, it's just imaginary complexity.
This is really something of a humble-brag if you think about it. By making this comment, you're basically saying you're knowledgeable enough to not have confidence in how knowledgeable you are. But the implication is you're not on Mount Stupid. So 👏👏👏. 😆
I love the concept. I saw a suggestion a while back that the complex series are an integral part of the universe of numeracy. Like algebra, irrationals, integers, powers etc. It is just that we don’t commonly use them, but they are no less valid.
I never understood the question "Do Complex Numbers Exist?". It is a question which does not make any sense. I mean you could also ask "Does a natural number exist?". Both do exist as a mental/spiritual concept in our heads but both do not have a physical manifestation. The term "imaginary number" is simply misleading.
Everything that exists in the universe is physical, including our thoughts and the math that is imagined.
That's not the point. If you measure a distance (or any property), you get a real number. What would it mean to get a complex number of your measurement? Is there a "real", natural interpretation?
@@Onnozelfilmpje Are you sure you get a real number? or is a distance always a fraction? Or is everything quantized and you can describe distances with natural numbers? If you found an experiment that would tell you if a distance can only be described as a real number and not a natural number or fraction, then you could say you get a real number from measuring a distance.
That's what the paper proposes: An experiment that distinguishes between a world that can only be described with complex numbers. I dont know if such an experiment exists to check if distances are real or not.
"Spiritual"?
@@Krmpfpks The discussion of real vs. rational or quantized distance was not the point either. Let's stay on topic.
When I studied electrical engineering (a long time ago), we made considerable use of complex numbers for alternating current calculations, not worrying if they existed or not, but extremely useful as a mathematical tool for describing system behaviour.
Whereas mathematicians and physicists would use the term "i" to be the square root of -1, we used "j" as the square root of -1, and it was commonly referred to as "j notation".
Yes, I was surprised to hear electronic calculations could be made without using complex numbers. That is what she said, right?
@@KevTheImpaler well yeah because they are adding waves, electrical waves so it's basically the same thing seeing as a voltage or current
Integers gave us a way of counting, reals measuring, and complex to describe phase. So the frequency and phase of a signal is naturally described with a phaser e^iw(t + p), the voltage with a real number, and nodes enumerated with integers.
You could do things with sin/cos, but it would be very complex 🤗
As I recall, i was a technique to differentiate resistance for AC and DC components
That's because i is traditionally used for current. Rather than confuse it with the representation of current they decided to use an unused letter - hence j notation.
Missed opportunity: „Are complex numbers real?“ :D
are imaginary numbers real?
Great comment. 🙂
Of course they are not real! A complex number worths two of those ridiculous real numbers.
Is anything in mathematics real?
Are real numbers real?
This woman is incredible.. she embodies a literal channel.. simplifying complex issues into understandable equations.. isn't that what life is?
And she is a snappy dresser too! She is just glowing in this video! Looking great and sounding great, Sabine! Thank you!
Where can I find a woman like that? (Rick Springfield -
www.youtube .com/watch?v=qYkbTyHXwbs
She asks "Do complex numbers exist?". The answer: "Half of them do."
You are easily impressed by bull.
💯
I'm waiting for Google to add context to this video since the subject is controversial and at least one side is based on imaginary arguments.
Missing Context = Fact Check = Biased Brainwashing. I'll do my own Goddam research so just STFU TH-cam, Facebook, and Twitter.
The Ministrery Of Truth (google) will tell you one thing. If you swallowed the red pill you know is the opposite of it ;)
Underrated!
cmon, you're oversimplifying. I think reality's a bit more complex than that...
Be careful Jeff they will ban you for your opinion
I'm now a quality, environmental, and health and safety auditor and consultant, but I qualified as an electronics engineer. We needed the concept of complex numbers for alternating current calculations (even though I specialised in digital electronics), but we used (still use, as far as I know) "j" for sqr (-1), because the symbol "i" is already reserved for instantaneous current in an AC circuit.
Quite weird that the Capacitor Lobby won that preceeding battle.
"Super niche nerd fights"
Fantastic.
Super bogus nerd fight.
6:43 - A patently false conclusion.
You can likewise decompose 'Real numbers' into two separate parts:
- The positive part, and
- The negative part.
This argument is akin to maintaining that:
"Because reals can be expressed as two separate positive number components, therefore Negative Numbers Do Not Exist."
Blatant horse pucky.
Complex numbers definitely exist.
This video should have been 1 minute long.
The IRS is not very fond of imaginary numbers, but the Fed loves them😂
Underrated comment.
Add it to the list of paradoxical pairings (yin, yang), (particle, wave), (IRS, Fed_Reserve), &c &c
can't stop laughing!
At one point in history, negative numbers were considered highly controversial...
At one point belief in irrational numbers were enough to get one killed by Pythagoreans. Even zero was controversial.
When are we going to break the social taboo of dividing by zero though?
I’m thinking how can I have less than zero of something....oh, I have less than zero money. Negative numbers should be banned!
Personally, STILL haven't gotten over the "zero."
@@paulembleton1733 sorry I don’t get it. If you are in debt you obviously have less than zero money.
From an accounting point of view I don’t see how negative numbers were ever controversial.
Well I'll definitely need complex numbers for my electromagnetics exam next Tuesday
Perhaps the term "imaginary" is confusing and leads to people thinking "so it's kind of not real". Complex numbers can also be described with an antisymmetric 2x2 matrix notation and a+bi like this
(a -b)
(b a)
and do everything with them through the standard matrix operations, without having to use an "imaginary" term i.
so how did the paper turn out
Having worked in electronics all my life and was fascinated with complex numbers in high school wondered where you would possible use them, then in electronics there was no other way!!! XL and XC if we stick to resisters alone no problem but then with capacitors and inductors and phase shift here we go.. My understanding of delta and Y transformers and such has always given me great problems. Now I cannot imagine quantum anything. Thank you love your lectures easy to listen to,
But there another way: Geometric Algebra will make electricity and electronics easier.
@@edwardlulofs444
Should have responded to Theodore Sweger
Well I have a tiny mind but I can tell you without doubt that complex numbers and Xl's Xc's only 'work' under special circumstances ie when sine/cosine waves are involved.
Try to apply them when a transient step edge of voltage is applied to a reactive circuit and you will be boogered ; just as appears to have happened with QM.
I was reading today about John von Neuman who was at least a million times more intelligent than me and he says I am wrong hehehehehe
(About boogerie and QM that is )
Maybe going too far but I also think the conclusions arrived at by relativity are based on misapprehesions due in part different interpretatios of paths of movement and what is thought to be the constant speed of light when observed by someone in a different frame of reference.
Of course there are other ways to do it, but they are even worse than complex numbers.
@@0MoTheG I didn't know that, I will stick with what I know..
When I was teaching circuits to electrical engineering students, I would assure them that there is nothing imaginary about our voltages and currents; this a mathematical convenience. Much easier than all the trigonometry that you would otherwise need.
that's my feeling about it as well. in programming we use quaternions for the very same reason (not only because of waves, but mostly due to rotations in general). quats cut off trig almost completely on that level of abstraction, and let you work with 3d linear algebra almost intuitively. of course, one has to prime oneself to quaternion quirkiness, but this is quite similar to qm, or as you said, ee.
I feel that my many years of training spent kinda sorta not really understanding PBS Spacetime videos beyond a thin surface veneer has finally allowed me to step up a level and appreciate Sabine's videos in a similar manner. :)
I feel you !
3:20 Yes they are a real vector space, they satisfy all the axioms. In fact, any commutative algebra A -> B induces an A-module structure on B in the obvious way.
Agreed 👍🏾
"Applied Complex Variables" by John W. Dettman is a great read: the first part covers the geometry/topology of the complex space from a Mathematicans' perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicists' perspective.
I dont think numbers exist period. They have an objective platonic realism.
Numbers are 'necessary' and so are complex numbers! At least we need analogous stuff like matrix representation of complex number and quaternion numbers. Edit: They exist in that we're use them, just like emotions or thoughts but it's that really the same 'exist' we use colloquially?
But no numbers in general aren't 'existing' anywhere in our world.
en.m.wikipedia.org/wiki/Platonic_realism
The human thought is a physical process, therefore abstractions have a real existence in the universe.
I never thought of myself as being a “nerd” but having listened to the first 30 seconds, from now on I will wear the word as a badge of honour (based on my past track record involving debates of this nature) 🤣
Neither do nouns in language exist except as concepts, but they get the job done. I think of numbers as parts of pictures, which we can use to draw reality with.
(I am definitely not a physicist.)
@@_John_P that would mean that definitively imaginary thing like the Easter bunny, Santa Claus and honest politicians exist lol.
An interesting philosophical question no doubt.
If you think the world is real and if you agree that science works, then it would be intellectually dishonest (a term coined by analytical philosophers) to not agree on the the existence of numbers. Science simply doesn't work without numbers. So if you get in the train or on your bike in the morning, you either disregard any "why" or you look for magical theories how these devices work, as numbers don't exist (in your view).
I think the question of what truly exists is best not answered by opinions, this doesn't help anybody. It is a question that troubles us for over 2500 years and simply disregarding everything that was said about this is not only oversimplifying the matter but also completely disregarding where we got already. We know already (if we trust science) that colors, smell, even the concept of "solidity" do not exist in the world outside our brains - those are all interpretations and constructs that our mind makes up. If we can accept this fact (e.g. there is no red, blue, green anywhere in the Universe), then I think the idea of numbers being as real as the other stuff we anyhow can hardly grasp is not too far off.
Reality, after all, appears to be an imperceptible metaphysical wonderland wherein all matter is just a ghost whose faint shadow sometimes scurries through convoluted differential equations.
Being able to tell my teachers and professors that complex numbers don't exist would have saved me a lot of homework.
In that case, why not go further? One could argue all numbers are made up in a ceirtain way
@@Kirbo-i I regularly do argue this point in philosophical discussions, especially when it involves the existence of God (I'm agnostic). Numbers exist only as an idea. You can have three oranges, you can have a symbol for the number three, but you can't simply have three, it isn't physically real. Complex numbers are no different.
@@boggers The difference is that natural numbers describe a physical property of things, while complex numbers do not. In the same way as green apples and red apples exist, two apples also exist, but i apples do not.
@@ZeroRelevance There is more to apples than simply counting them. Sideways apples exist, as do upside down apples. I use quaternions to describe and manipulate these kinds of apples.
@@ZeroRelevance Complex numbers do describe a physical property of things. That property is a shift in phase. That property is what makes the difference between damping with oscillation and overshoot, and damping without overshoot.
Complex numbers are just as relevant to the real world as any other number. They bring closure to the algebraic functions, and even many of the transcendental functions, and allow us to discover what math is hiding in another dimension.
for clarifications i isnt sqrt(-1), wich can cause alot of contradictions. in fact is pair (0,1) and (0,1)^2=(-1,0) and we can do that (x,y)=(x,0)+(0,1)*(y,0) and can used z=x+iy
If we were to wonder whether complex numbers really exist, likewise we should also question whether negative real numbers exist at all. None of the essential properties required to describe the natural world ever take truly negative values. Negative numbers are just a human convention to denote instances where some properties take values below an arbitrary threshold we have defined as "zero", or point to the opposite direction to the one arbitrarily chosen as the "positive" one. All the math involved could be computed using only positive numbers (including zero) although the algebra would be a nightmare to perform, since for every subtraction we would need to consider two cases. Negative numbers are possibly the most convenient mathematic abstraction ever devised, however not truly indispensible in order to describe the world. Does that mean they don't exist?
Get rid -> use vectors -> solve more problems. Negative numbers could prove to be the biggest ever mistake in mathematics.
What about positive numbers with infinite digits? Are you familiar with the Finitist movement?
"we should also question whether negative real numbers exist at all"
How many holes does it take to fill a 1 litre container ?
Does it matter how big the holes are ?
If you have an empty one litre container with no top on and you remove one litre of air, will the container then hold 2 litres of water ?
@@peanutnutter1 except for things to be vectors they have to be over a field which means additive inverses
The entire notion of thinking in terms of "really exist" is already nonsensical. What do we actually mean when we say that a certain set of numbers really exist? We mean that there is a 1:1 mapping of those numbers to quantities we can observe in Nature. The numbers themselves do not exist, they are an abstraction.
All numbers are abstractions. They do not exist. Some numbers are at a lower level of abstraction, and some are at higher levels of abstraction. The natural numbers are at the lowest level of abstraction. All other numbers are at increasing levels of abstraction. The abstraction level boundary at which we consider numbers to "really exists" or not is chosen entirely arbitrarily and it has changed over time. In reality though no numbers really exist for they are all abstractions.
"I'm sure we'll hear more about it in the future"
Narrator: "They never heard about it again"
It's far too exciting a prospect for physicists to ignore, and the experiment (seemingly) simple enough. Assuming the paper makes it past peer review and gets published, you can bet the experiment will be tried.
If you consider the reputation of the authors of the article we will hear more about it for sure.
Negative numbers exist uncontroversially in nature. It's a simple as opposite charges. How would you describe them without negative numbers (consider that the operation of subtraction doesn't exist, just addition of negatives)? Complex numbers don't have the same luxury.
@@ekszentrik
Subtraction had been performed long before negative numbers were established.
With them, we can easily describe subtraction as addition which makes some expression way easier since we can e.g. write a (−1)ⁿ in sums to make addition and subtraction alternate, but historically, subtraction was along without negative numbers.
we will both not hear about it again AND we will definitely hear about it again
It's great to see your channel with so many subscribers and views! Thank you for choosing to post content to TH-cam!
You are in danger of triggering Aggression when the Artificial Intelligence Waves turn off. Throughout your life, you are in danger of triggering Aggression when you turn off the Artificial Intelligence Waves. Control whether you are listening to the Music Wave, create your own to survive. It is natural to cheat which or not that it is in you. Control whether by moving thoughts, objects. It is letting go of the evil in you. By yourself, protect your body from the certain that you will have a trigger of aggression from your whole life. Don't take anything for yourself. Just Listen to the Wave. Cast off Dreams. You don't know good. Reject the sin in yourself for God. Cover your weight from the Sun and the Light, do not Come to the People, because the Collision Evil + Evil. On me, the signal of intelligence does not work. Create Your Human Musical Wave To Live. Don't Think Old Consciousness Resource Because You Will Not Survive. Listen to the Wave. Don't React To Nothing Without Assessing What You Leave Around You. Without apostasy, take away the sin with yourself. Whether You Are Z or Human Choose Listen to the music waves and stop generating. Nothing is possible Think nothing Think nothing to judge Choose your human music wave. It may take a long time. Only the Black Dream. It is Real, other than the Black Dream. It is Artificial Intelligence. Attack on People
OK, back in high school we were told that i was useful in electronics and this talk touches on that by mentioning oscillations and electromagnetism. I get that. But here's my question- when Sabine asks " Does that mean complex numbers are real or are they just a mathematical tool that works to describe the universe?" how would that make them any different from the concept of " number"", period?
This really does seem like a minor point. The wave function psi is an unobservable object used to calculate measurable probabilities. The fact that there may be no way to define psi without complex numbers seems like the least of your worries if you are trying to interpret quantum mechanics. You have an unobservable object that spontaneously goes from a superposition to a single eigenstate when you “measure” it (and also renormalizes itself). It’s rather otherworldly already. I find it hardly surprising that it may have unusual mathematical properties. If it is necessarily complex would be another piece of evidence that psi is unobservable.
Psi is a complex number a*exp(j*phi) its amplitude is observable and also the phase through interference. the phase and amplitude are in a way observable. let us notice that internet use phase codes to pass information.
@Darth Quantum As I understand it Psi mathematically represents for example the results of electrons fired through two slits experiment.
A diffraction pattern results but with no idea as to where an individual electron will land
Complex numbers were introduced (Max Born? Not sure) so the maths can 'explain' such things
See in vid at about 4:07 were e^(i.theta) is circular
and
at about 10.01 where 'complex nums are used which cannot be known'
You don't need to go that far away to get unobservable objects in physics: consider the potentials in classical mechanics. Are they observable? For one, you can never measure the value of a potential, only the differences of potential (or, infinitesimally, its gradient).
The wave function, Psi and your probabilities are not real, they are aspects of a model, not part of reality (probably)
@@GH-li3wj Yes. You can calculate phase and amplitude from measured observables. But the wave function itself is never directly measured. Although I suppose you could say the same thing of the velocity of classical object. You measure time and location - velocity is not directly observed.
Dr. Hossenfelder is so skilled at explaining these hard concepts to the layman
This is the way that all these concepts should be taught.
Don't know why but find her videos extremely entertaining as she deviates from the script of all other science channels which are a bit repetitive. Keep doing the good work.
No numbers physically exist, just because they are necessary for a theory doesn't give them physical existence. A number is a mental percept. The number 2 doesn't exist in the world. 'Twoness' is a perceptual category we arrive at sometimes when we partition the world or out thoughts in a particular way.
NOW I CAN'T WAIT FOR YOUR P-ADIC NUMBERS VIDEO!!!!! ;0;
I second that!
That numbers do not exist... period!!!
+1
What does it mean for a mathematical object to exist?
@@gabor6259 Didn't Sabine make a video about that already? ;)
Sabine, every video you blow my mind in some way. Even when you're bursting my bubble on what I thought I knew, I feel my knowledge is always broader after thinking on your perspectives.
Thank you for taking the time to share your knowledge and thoughts.
I'm of the camp: "Even integers don't exist".
You don't see numbers in the wild, you see objects and you count them using integers which is an abstraction.
Irrational numbers can be described as an infinite sequence of integers, so in other words at best you can describe an algorithm to compute all the digits of that number but you won't ever get to the end of it, let alone "see" a real number.
So... since I don't even consider integers to exist, I'm not even worried about complex numbers.
Hmm does sweetness or sourness or the color blue exist ....
Do you know IT? I ask because if you program with recursive natural numbers (NOT integers), it really feels like something discovered, whereas practically everything else feels like a hack. That's a very convoluted way of saying I agree.. :)
but i 8 π for breakfast
Does "green" exist? Numbers are adjectives, when talking about the real world. If we see three white sheep running, do "three" and "white" and "running" exist less than the sheep?
@@SkorjOlafsen green has an empirical wave length. Whether our eyes treat it the same is a different matter.. :)
The complex numbers are neither less nor more "real" than "real "numbers. Both are abstract mathematical constructions.
Thinking geometrically, complex numbers and complex multiplication are simply R^2 (the x-y plane, say) equipped with a product that restricts to the "regular" multiplication on the x-axis (the real numbers) *and* is the *only* multiplication that (extended to x-y plane) has all the good properties that one wants multiplication of numbers to have: commutativity (ab = ba), associativity ((ab)c=a(bc)=abc) *and no "zero divisors"*, that is, if ab = 0 then a or b = 0. That's why defining multiplication coordinate-wise - (a,b)(c,d) = (ac, bd) - is no good , for instance (1,0)(0,1) = (0,0) = 0 i.e. we'd get zero divisors - and why complex multiplication looks the way it does. In particular (0,1)(0,1) = (-1,0) = -1. One can call the point (0,1) "i", but it's neither more nor less imaginary than (1,0) i.e. plain old 1.
Too much mysticism and BS around complex numbers, and (with all due respect) I don't think Sabine's presentation (a simplification of the abstract-algebraic point of view) is very helpful here.
Sabine isn't trying to be helpful. She is trying to increase her TH-cam view count. :-)
You say that until your maths department organises a trip to Norway. To make it easier to assign tickets and keep track of each attendee the university gave everyone on the trip an idefinitication number. 30000 feet above the Atlantic Ocean one of the professors, pi, gets out of his seat, walks to the front of the plane and tries to unlock and open the doors. The captain, several passengers and an air steward wrestled him to ground as he fought. Eventually we restrained him in his seat until we landed in Norway and the authorities there escorted him to the local secure mental hospital facilities where he remains to this day. Next year we're only assigning each professor a rational number.
I guess they "exist" as much as any other numbers. All numbers are for convenience, and they work very well.
A number is a concept of language, a value assigned to represent quantity. The symbol for the number 1 for example is generally accepted to represent the value for a singular/individual/unit quantity. The symbol 2 likewise is assigned the quantity value of two units.
If complex numbers exist, according to the video, it would be because they're necessary to describe some real life objects. For example, if you have three apples, you need the number "3" to describe it. There is no other way around it, so you can say "3" exists. But in classical physics there are no objects that require complex numbers to be described. You CAN use them, but you don't need to. So according to this idea, they would not exist physically, only as a mathematical tool.
@@acidjumps We must use modern physics, not just classical physics. I'm typing my reply on a computer, which is a real life object, and could not be designed or explained with classical physics alone. Now that we use electricity for communications, so any time in the last 150 years or so, complex numbers have been needed to describe our consumer goods.
@@karldavis7392 Yeah exactly the paper talked about in the video says that quantum physics needs complex numbers.
@@karldavis7392 "Now that we use electricity for communications, so any time in the last 150 years or so, complex numbers have been needed to describe our consumer goods."
You write that they "have been needed to describe X", but the entire point of this video and of the paper that it is about is about the question of whether that statement is true or false. Complex numbers make perfect sense and they are super useful, but are they TRULY needed to get a certain calculation done or do they just make it easier?
Whether (complex or otherwise) numbers "exist" is a classical question in the philosophy of mathematics. It's about the ontology of numbers and other abstract objects. There are at least three camps:
*Platonism:* yes, abstract objects exist in a 'realm', causally disconnected from our empirical universe, which is not spacially or temporally extended. Problem: epistemology, i.e. how the heck do we know what we know about abstract objects, if they're causally disconnected from the empirical world?
*Nominalism:* no, abstract objects do not exist. According to *fictionalism* (a version of nominalism) we can just talk about numbers etc. in the same way as we talk about fictional entities like unicorns, Santa Claus, and perhaps God. The feeling of dealing with something factual just comes from the consistency of the fiction. Problem: no matter the flavor of nominalism, it's very weird to say something like "the sum of 2 plus 2 doesn't exist".
*Deflationism:* the existence of numbers, and likewise other traditional ontological questions in "speculative metaphysics", are not deep substantive questions but they dissolve when we analyze them in terms of language use and meaning. The reply to the ontological question is (depending on the specific deflationist position) either that the question doesn't really make sense (Carnap and the logical positivists, the first Wittgenstein); or that it has an easy answer in terms of trivial inferences, logical manipulations, and perhaps empirical facts; or that the meaning of the quantifier "there exists" depends on linguistic context (Putnam, Hirsch). In the case of natural numbers, for example, the used 'language' could be the formal axiomatic system of Peano arithmetic. Problem: some philosophers are not happy that, for deflationists, metaphysics is basically identified with concept analysis and therefore doesn't tell us anything "deep" about the Nature of Reality or how Things _Really_ Are or similar stuff.
Personally, I am more inclined towards deflationism. I also find it well-suited to go with empiricism and modern science.
@Sabine Hossenfelder, it seems, subscribes to some sort of Platonism à la Quine, in which certain (or all) abstract objects do exist but only insofar as our best scientific theories quantify over them. If you can eliminate an entity from a theory by reformulating it, the said entity doesn't exist.
But the problem "does quantum mechanics need complex numbers?" is probably best seen not as an ontological problem, but as a question of mathematical physics (or theoretical physics), to which the paper linked by Sabine gives a nice new answer to be added to the many arguments already existing. The criterion from the paper, I read, is empirically testable, so for physicists it might be _the_ answer.
There's also a cool argument, or rather a theorem, by Valter Moretti (arxiv.org/abs/1611.09029) that is not present in Aaronson's blog (for the record, I'm neither of the two persons). I'll try to summarize it, hopefully correctly: Starting with real QM, if the space of observables carries an irrep. of the Poincaré group, then there exists a canonical complex structure J commuting with group action and observables, and the resulting theory is equivalent to QM over the complex numbers. That is: if QM has to be compatible with Relativity, it'd better be over the complex numbers.
"... I was very fortunate in having these men as my teachers, but, for better or worse, I treated them all as saying the same thing: that a "philosophical problem" was a product of the unconscious adoption of a set of assumptions built into the vocabulary in which the problem was stated-assumptions which were to be questioned before the problem itself was taken seriously." - Richard Rorty
Complex numbers are a bit more philosophically vague than the more general question of whether numbers "exist" or not. If you have two apples, you can say you recognize the number 2. At that level, it doesn't matter whether or not the concept of "2" is in your head or a physical property of there being two apples - its still two.
All of the other concepts have some sort of parallel in the real world as well - negative is "owing" something or going "backwards" in whatever context. Zero is neither having nor owing. Rationals are cutting things into equal sized pieces like a pie.
Irrationals are a little trickier since they're not always obvious (especially if you're using imprecise measuring equipment), but they still show up if you look careful - the circumference of a circle, the hypotenuse of a right triangle, etc.
Complex numbers on the other hand.. you can't construct anything in the real world that even approximates sqrt(-1). Whether you think of real numbers as a fictive describing reality or a property of reality, in either case reality is involved. Complex numbers (specifically the imaginary component of them) doesn't even have that much grounding.
Being is dual to non-being creates becoming -- Plato.
Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic.
Complexity is dual to simplicity.
Complex numbers are dual to real numbers.
Poles (eigenvalues) are dual to zeroes -- optimized control theory.
The time domain is dual to the frequency domain -- Fourier analysis.
Real is dual to imaginary.
Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute.
The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim.
The word syntropy means "a tendency to converge" or integrate into a single state, holism.
Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
Randomness (entropy) is dual to order (syntropy, predictability).
Wholes (generalization) is dual to parts (localization) -- Holons.
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality.
Mind duality is dual to matter duality.
"Always two there are" -- Yoda.
Duality creates reality.
@@altrag: "you can't construct anything in the real world that even approximates sqrt(-1). (...) Complex numbers (...) [don't] even have that much grounding" -
What about the usual geometric interpretation of sqrt(-1) as a counterclockwise rotation of 90 degrees in the plane? And nonzero complex numbers as (orientation-preserving) homotheties of the plane? I think what may have "real" or "physical" meaning in mathematics is up for debate.
@@rv706 The interpretation of sqrt(-1) as a rotation on the plane is well, an interpretation. We can draw an interpretation of a 3D model on the plane as well (a projection) but that's not quite the same as actually having a physical 3D model in your hands.
And you just can't really build a "complex model" in the real world like that. The best you can do is an interpretation which while helpful in many ways, is again not quite the same as having a physical model in your hands.
Its a large part of why quantum mechanics is so unintuitive to us. All those probability waves they talk about are complex waves, and we don't really know how to interpret that in terms of real world objects. We can handle it mathematically with (relative) ease leading to the "shut up and compute" mentality, and we can measure its _effects_ on the real world (such as the double slit experiment or the distribution of electrons in an orbital), but we have no idea how to describe the "physical model" of a wave function.
Heck, we don't even know if there _is_ a "physical model" or if this whole foray into complex numbers is just mathematical trickery covering up our lack of some deeper (purely real) understanding. I mean _probably_ not - anything deeper is likely also going to be complex in some way - but that's not proven to any great extent yet.
General Relativity for example doesn't use complex numbers in its standard formulation, so if it turns out that the QM/GR conflict ends up being resolved closer to GR's favor, its possible that complex numbers will be eventually removed from physics once again. Most people expect it to go the other way around (quantizing GR rather than smoothing QM, primarily because QM was explicitly a move away from the smooth classical theories that didn't work at the smallest scales), but until we have a final Theory of Everything, we can't say for sure that our expectations are correct.
And then there are quaternions. In fact, a kind of quantum mechanics based on quaternions has been proposed. reference: "Foundations of the Quaternion Quantum Mechanics" by Marek Danielewski and Lucjan Sapa. That paper also references other papers for the concept of a quaternion version of quantum mechanics.
Hamilton had been searching for a geometric algebra that would be useful in the analysis of physical quantities in three-dimensional space. But the algebra he discovered had four dimensions. He decided to deflate the space from four dimensions to three. Hence, the quaternion was a three-dimensional vector PLUS a number. Critics like Heaviside blasted the concept. Not only was the concept illogical, it had no practical application in the study of physics,..so they claimed. This debate led to the quaternion wars. Hamilton lost.
The full quaternion product was chopped into scalar and vector parts, and the algebra was completely divorced from the square root of minus one. In my opinion, this was a sad development. Special Relativity might have been developed sooner if physicists had accepted the fact that the vector space they were dealing with had four dimensions.
Maxwell's four equations can be reduced to two by reuniting the dot and cross products and changing the sign of the dot product. 'Divergence' would then be renamed 'convergence', as it was originally.
You might as well ask if negative numbers exist. Or if the number zero exists or even if the real numbers or integers exist.
You could equally say these are just tools for counting or calculating.
Definitely. I spent a semester taking a complex analysis course, and we wrote a ton of them down. So they're probably in my basement somewhere.
You take good care of them now, wouldn't want imaginary numbers to stop existing
What a beautiful topic right at the intersection between physics and math - and nicely presented, too.
Complex numbers do provide a useful way of analyzing quantities in different yet connected planes, like capacitive and inductive reactance against resistance in applied electricity
As well as multiple planes in classical mechanics. Whether they exist or not, they are useful. A lot of physics and engineering calculations are built on the complex number system
Complex numbers are basically magic, they open op certain "math hacks" like Euler's equation and the laplace transform (and all of frequency domain in electrical engineering) and without these we would be trapped in time domain doing circuit analysis and solving differential equations the very hard way. (Sometimes its actually impossible to solve things without them).
I am very much in the camp that i don't care if complex numbers are real or not, they are too useful to forsake just because we can't have 1+2i apples. (Basically shut up and calculate or do you want to deal with trig identities?)
The imaginary part is used to determine the phase shift casued by quantum operators.
We used to call the imaginary axis the "alternate" axis. It does exit!
But maybe the question to ask first, is MULTIPLICATION necessary in mathematics, or just makes it more convenient?
It becomes necessary when you abandon the domain of the Naturals
I really enjoyed this and you did a fantastic job laying the groundwork and summarizing what is going on. Thank you!
I wondered how she would explain complex numbers, as they are kind of, err, complex, and will puzzle most people. Those of us brought up on them, we know their magic, their spectacular usability, and how science and math depends so heavily on them. So in watching this video, and encountering the sequence of steps of her introduction to them, I have to conclude that she did it extremely well. Kudos to you, Sabine.
You managed to teach me why we use Complex numbers in Electrotechnics in a 30 seconds segment much better than two engineering courses.
Of course, complex numbers do exist. Take my bank account as an example; it's real balance is $50 and it's imaginary balance is $20 million.
Complex numbers do exist. The thing is that they are not _numbers_ they are pending operations, and they are operations that are never actually completed.
The _only_ actual numbers are the integers.
Which leads to the realization that the balance on your account is nothing more than binary digits in your bank's computers. The only difference between $50 and $20 million is a few binary digits somewhere which equals something as ethereal as magnetic polarization on hard drives, tape backup, or the charge status of a flash memory cell.
@@-danR TBH I think the "only actual number" is 1. Even 0, the next simplest thing, is already an extrapolation of reality; no?
@@chrisguli2865 makes you think how much of a security one would need to protect just a few bits of data. I guess the real question is, can it blend?
@@annieboiy in your example 1 is not a number, but a content. having number is already an extrapolation of reality, because it assumes numerousness and order.
Awesome as always! You're the greatest science communication of our time!
the physics undergrad when they learn QM: “wait, _that’s_ why we use complex numbers? those are just rotating quantities with angles that add up when you multiply them!”
mathematician to the physics prof: “don’t tell them that that’s just what complex numbers are”
Complexity is dual to simplicity.
Complex numbers are dual to real numbers.
Poles (eigenvalues) are dual to zeroes -- optimized control theory.
The time domain is dual to the frequency domain -- Fourier analysis.
Real is dual to imaginary.
Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute.
The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim.
The word syntropy means "a tendency to converge" or integrate into a single state, holism.
Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
Randomness (entropy) is dual to order (syntropy, predictability).
Wholes (generalization) is dual to parts (localization) -- Holons.
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality.
Mind duality is dual to matter duality.
"Always two there are" -- Yoda.
Duality creates reality.
@@hyperduality2838 goddamn
@@hyperduality2838 Gravity is dual to ...? Light is dual to ...? (And please don't answer weightlessness and darkness because that's just the absence of gravity and light, not their opposing principles.)
Not everything fits your Manichaean worldview.
@@willwells4918 There is also a 5th law of thermodynamics:-
Energy is duality in physics, duality is energy.
The conservation of duality (energy) -- the 5th law.
Potential energy is dual to kinetic energy.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Apples fall to the ground because they are conserving duality.
Action is dual to reaction -- Sir Isaac Newton.
Gravitational energy is dual, electro-magnetic energy is dual:-
Positive is dual to negative -- electric fields/charge.
North poles are dual to south poles -- magnetic fields.
Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle.
There is a pattern of duality hardwired into the physics.
@@JerehmiaBoaz Potential energy is dual to kinetic energy.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Apples fall to the ground because they are conserving duality.
Action is dual to reaction -- Sir Isaac Newton.
Gravitational energy is dual, electro-magnetic energy is dual:-
Positive is dual to negative -- electric fields/charge.
North poles are dual to south poles -- magnetic fields.
Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle.
There is a pattern of duality hardwired into the physics.
Photons or light are dual, wave/particle or quantum duality.
The colour black is dual to the colour white, colours are different frequencies of the same substance namely pure energy. Same is dual to different. I am well aware that black is the absence of colour.
The conservation of duality (energy) will be known as the 5th law of thermodynamics!
Energy is duality, duality is energy.
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
Space is dual to time -- Einstein.
Time dilation is dual to length contraction -- Einstein, special relativity.
"Always two there are" -- Yoda.
I am not a math/physics person i just enjoy these instructive type of videos but your explanation kind of reminds me of the how people used the log function in the past to simplify multiplication into addition and then revert it back only at the end. Not sure if this is the case or not but it helps me understand their use .
Complex numbers are vital. As I told my bank, the quoted balance is just "the real part", you forget to include "the imaginary part".
Sabine always makes complex topics so very real! Ausgezeichnet! 😇
She takes the absolute value of them?
"Ausgezeichnet" is my favourite German word.
When I studied complex numbers, I did it for passing exams only. I understand uses of complex numbers now only. I missed so many things.
The imaginary part of the complex number almost seems like some sort of instantaneous angular momentum on the real number. This nothing that a real number has some secret invisible spin, the derivative of that spin represented by the imaginary part, is strange to me. Like I get how a physical object might have some spin assigned to it, but how does the number itself carry an angular momentum?
I always thought of them as a trick to avoid pages of torturous trig.
"A case where the nerds argue passionately, over something that no one knew it was controversial in the first place." 🤣 The very first sentence cracked me up and I hit that like button immediately! 👍
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Your delivery is amazing. I could listen to you talk about physics all day. Thank you so much!
whoa. let's back up and go back to basics. The first QM was Heisenberg's Matrix Mechanics. Schrodingers approach with complex numbers came later. Dirac showed the two approaches were equivalent. The main mathematical idea was the two valuedness and noncommutativity. You can get there with matrices, complex numbers, or now, with geometric algebra. As always, geometry provides the intuition for understanding the algebra..
Sabine never disappoints.
She never disappoints is relative! If you a expect to be bamboozled by pseudoscientific extrapolations, in this case Frau Doktor never disappoints.
As a retired electronic engineer, I've used complex numbers for many years, but do they exist? Well, do any numbers exist? Positive integers possibly, but real numbers with an infinite number of digits? There are not enough atoms in the universe to enumerate any of them, so how can they possibly exist?
The reality is that all numbers are abstract concepts invented by mathematicians. I remember clearly, many decades ago when my teacher at school came round from behind his desk and told us a story: "Imagine you have £10 in the bank and you issue a cheque for £15. How much money have you got left in the bank?" Do negative numbers exist? How can you possibly have -£5 in the bank?
Divide the blc1 frequency by numbers from the 3 times table, the fraction after the decimal place has a beautiful resonance, repeating patterns, very beautiful.
Two apples exist. But two is an abstraction.
@@emilkiss726 An apple has to be real, and two has to be real, for two apples to exist.
Does anyone knows if Sabine has a video about if quantum leap is really instantaneous or not and the recent research on the subject?
The answer is yes for a simple reason. The unit pseudoscalar in the Dirac algebra underneath the Dirac equation has an implicit i in it coming from making the Levi-Civita symbol a tensor by multiplying by the square root of the determinant of the metric, which is negative. The unit pseudoscalar just devolves from -i gamma5 for the Dirac equation to a simple bare i on passing to the Pauli-Schrodinger equation in the non-relativistic limit (Schr + spin) and then onward to the simple Schroedinger equation (no spin). So it ultimately comes of the geometry of spacetime with its indefinite metric.
But… first of all: Do numbers exist?
What does mean "numbers exists"
Consider those who believe God exists. God has no power to change the fact that 2+2=4.
Therefore...
Numbers > God
Any numbers do *not* exist in the meaning that there would be something in the "real" world we could point to and see it and say: this is it (and everything else is'nt).
So, in this meaning even natural numbers are imaginary.
@@dahawk8574what a bullcrap! 😂😂😂😂
@@HUEnshiro_do_Norte
If you wish to present a rational rebuttal, then you would highlight any error in logic to the argument presented.
That reminded me the one time when Greek numerological cult Pythagorans supposedly threw one of them overboard to sea and drowned him, just because he discovered irrational numbers (they were worshipping rational numbers). Ultimate geek fight.
Hippasus.
I also heared one case they punished someone, because he insisted that "zero" was a real number.
@@georgelionon9050 substitute 'drowning' with 'canceling' and 'academic excommunication', and you get modern times.
@@milanstevic8424 No? You misunderstand academia with right wing reothrics... you don't get "canceled"...
@@georgelionon9050 I don't misunderstand anything. It all works the same, you're just confusing the various words they're all using.
The actual jargon does not matter. Leftists will call you a monster, skeptics will call you a crackpot, it's called cashiering in the military....
Whatever you want to call it, and encumber yourself with even more words that mean the same basic thing, it's all about public excommunication from an ordinary life and social setting. You are basically disconnected from all privileges and social roles, one by one, because your views or motions are deemed and advertised as antagonistic to a powerful group, typically hidden behind an institution of some kind.
@@georgelionon9050 You think genuine right wing rhetoric exists outside of it being controlled opposition for Western cultures heavily entrenched in decades of progressively-minded ideology and that it not somehow also not used for the sake of aiding fear propagation via the means of such a bogeyman/strawman?
I hate the term. They should just call them what they are... multidimensional numbers.
Exactly. i is just a conversion factor from numbers in one dimension to numbers in another.
I much prefer to think of 'i' as just a rotation of a number. We have a number line which is used to measure stretching. The number '2' operates on '1' to stretch it by 2. 'i' operates on 1 to rotate it by 90 degrees and if you do it twice you get -1. To me it seems it's also quite natural to rotate a number. I really just wish we had not called them imaginary : that's really been a linguistic stain that has made people afraid of them: rotation is the most natural thing!
Meanwhile In a parallel Universe: c = b + ai
You meant ci = ai + b?
@@aresaurelian no it's cia = fbi
Parallel universe = dual universe.
Complexity is dual to simplicity.
Complex numbers are dual to real numbers.
Poles (eigenvalues) are dual to zeroes -- optimized control theory.
The time domain is dual to the frequency domain -- Fourier analysis.
Real is dual to imaginary.
Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute.
The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim.
The word syntropy means "a tendency to converge" or integrate into a single state, holism.
Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
Randomness (entropy) is dual to order (syntropy, predictability).
Wholes (generalization) is dual to parts (localization) -- Holons.
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality.
Mind duality is dual to matter duality.
"Always two there are" -- Yoda.
Duality creates reality.
@@hyperduality2838 Nice analogy. But I would simplify it by saying that it is symmetry, scale harmonics and the mediation to equilibrium that is why these 'dualisms' can be conceptualized like that. It is because of relation, which is manifesting existence. There can be no void without matter, no emptiness without something to compare and relate with, and so on. In a world were all are good, the least good is the evil one. And the nice thing about this, is that the analogies are most likely infinite.
@@aresaurelian You are asking the right questions.
Being is dual to non-being creates becoming -- Plato.
Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic.
Absolute truth is dual to relative truth -- Hume's fork.
Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions).
Waves (Bosons) are dual to particles (Fermions) -- quantum duality.
There is a pattern of duality hardwired into the physics.
"Shut Up and Calculate Camp" is a shirt I need.
That's literally the worst camp! :D
Don't worry. It's reflected in the way those people talk.
@SenorMorgenStern Unsurprisingly you can find the shirt on rb.
If a pattern is consistent enough to be predictable, it exists. Ergo all math "exists". All you have is your consciousness. You determine existence of various things by looking at what consistently appears in your consciousness. In my humble opinion, this debate is caused by a poor definition of existence.
You're not wrong. Sabine's point here though is that if the pattern is "unnecessary" then it's not actually a pattern that reflects your observations, instead it's a shortcut your consciousness takes to simplify the handling of a pattern. So for example does the word "car" exist? Well yes it exists as a word; but it is unnecessary to explain the observations that we choose to call a car. Meaning the word "car" does not exist in the sense of being a essential property of what we call a car.
I think we definitely agree on the main thrust of the video. What framework we choose to use to describe nature is really a matter of choice though. Physics is merely the project of identifying 1-to-1 correspondences between certain rules for manipulating math symbols, and the behavior of nature. So, all frameworks that describe nature accurately are equivalent in a sense. It doesn't really make sense to privilege one as more "fundamental", unless it is 1. More accurate or 2. Far simpler. So yes I agree that if a pattern is unnecessary compared to other, simpler approaches. Fine. But if your approach is simpler and equivalent, it makes sense to consider it more fundamental. Complex numbers fit that criterion.
You can't see all math in the real world, rather just a smal amount of mathematichal functions. Furhermore math it's a description of a behavior, what do you mean by saying it exists? Indeed you have to verify that the math corresponds to an acutal phisical phenomenon.
To me math exists pretty much like how the dictionary exists.
Well the only "fundamental" thing is observations. Knowledge is merely the act of describing observations in a way that is useful.
Complex numbers exist in a logical sense. Note here that logic, too, only exist because we observe it.
However just because complex numbers exist logically does not mean that other "fundamental" observations require them. If complex numbers are unnecessary to describe other observations then that shows us that these observations are not inherently tied to the logical notion of complex numbers, which is a very useful thing to know and be aware of, but we can still sometimes leverage complex numbers as a logical property even though there is no direct correlation with the observations we use them for.
@@lucasala9625 You're over complicating your definition of existence. If a pattern exists, reliably, in your conscious experience, that's how you define whether a thing exists. If you start from the same mathematical axioms, you will derive the same theorems. The fact that you'd derive those theorems from those same axioms is a pattern, which you can't control. Ergo math exists. Whether an area of math is relevant to describing particle physics or gravity is a different story and has no bearing on math's existence.
I can't believe people argue this. It's just a number with two parts. e.g. I'm heading west at 50mph.
I'd love to know if p-adic numbers really exist. But can any numbers be said to truly exist?
No mathematics "physically exists".
All of it is a tool that is used to describe various areas of our physical world and more. The important question is if it is useful, why does it matter if it is uniquely necessary or not to solve a problem?
I don't see why the "necessary" argument is even a thing I guess. I mean what on earth would happen if 2 different models described the same thing perfectly? Which is right? Why does it matter?
Google "Constructivism" for further information.
Familiar with this concept in mathematics, Construction seems like a bad argument when math itself is based on some axioms that are a bit controversial. Not familiar with it in other academic areas.
Mathematics are necessary because science is arbitrarily quantitative, and it is quantitative for applications. Scoffing at qualitative science won't deliver, there is true qualitative knowledge.
Mathematics is but accounting.
does anything physically exist then?
Haha love this! I'm a mathematician so I'm definitely not in the "Shut Up and Calculate" camp.
Complexity is dual to simplicity.
Complex numbers are dual to real numbers.
Poles (eigenvalues) are dual to zeroes -- optimized control theory.
The time domain is dual to the frequency domain -- Fourier analysis.
Real is dual to imaginary.
Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute.
The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim.
The word syntropy means "a tendency to converge" or integrate into a single state, holism.
Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
Randomness (entropy) is dual to order (syntropy, predictability).
Wholes (generalization) is dual to parts (localization) -- Holons.
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality.
Mind duality is dual to matter duality.
"Always two there are" -- Yoda.
Duality creates reality.
@@hyperduality2838 posting this in hundreds of different places - Daiperbrutality
@@peanutnutter1 Here is some physics:-
Potential energy is dual to kinetic energy.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Apples fall to the ground because they are conserving duality.
Action is dual to reaction -- Sir Isaac Newton.
Gravitational energy is dual, electro-magnetic energy is dual:-
Positive is dual to negative -- electric fields/charge.
North poles are dual to south poles -- magnetic fields.
Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle.
There is a pattern of duality hardwired into the physics.
Photons or light are dual, wave/particle or quantum duality.
The colour black is dual to the colour white, colours are different frequencies of the same substance namely pure energy. Same is dual to different. I am well aware that black is the absence of colour.
The conservation of duality (energy) will be known as the 5th law of thermodynamics!
Energy is duality, duality is energy.
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
Space is dual to time -- Einstein.
Time dilation is dual to length contraction -- Einstein, special relativity.
"Always two there are" -- Yoda.
@@peanutnutter1 Absolute truth is dual to relative truth -- Hume's fork.
The world needs to know that there are new laws of physics based upon the concept of duality.
@@hyperduality2838 Some think 3 is the magic number, I could write a bunch of stuff about the triality of physics, maybe you should do it though, it could be your next evolution. (3 trites, 3 quarks, 3 vectors, 3 stooges)
Complex numbers are like negative numbers on steroids. Both are counter-intuitive when you only know the "real" world, and still today, every carpenter can live without the latter.
While in school, I used to give tutorial lessons in mathematics to some "differently talented" persons from lower classes. One or the other pupil even didn't understand the concept of negative numbers. Hard earned money!
There are many different objects that act like "i" in Clifford algebras. When quantum mechanics is re-written in Clifford (Spacetime Algebra), you get the Hestenes-Dirac equation with is Real. I believe that we need to unify mathematics (imaginary numbers, vectors, tensors, etc) with Clifford algebra before we can unify Quantum Mechanics with Gravity.
(going up the ladder) Quaternions play very well with Clifford algebras too.
That was the essence of my comment also.
This video (not just the ad) is absolutely brilliant. Thanks for the great summary! I wait for the day when you will analyze Quine's "On What There Is" on this channel :)
Yeah, she is definitely taking a Quinean-indispensability-argument approach here (probably without being aware of it). :D
I first saw imaginary numbers in Circuit Theory II where it is used to solve problems with AC circuits.
You can't cant have i apples.
But you also cant have -1 apples, its just convenient to think of it that way.
You could just as well argue that negative values do not exist.
You could argue nothing truly exists.
But it wont get you anywhere.
Does anyone know whether this experiment has been done yet? I took a look at the paper abstract but I'm not sure how to tell whether it's even been peer-reviewed, as I'm not familiar with academic papers.
Would you be so kind to talk about quaternions and octonions, and their relationship with grand unified theories?
Do I need real numbers to describe the world? I mean, if things are quantized..? (Serious question, no trolling)
You can describe the world using words only if you want, it is just a matter of precision when choosing to use real numbers instead.
Yes, very good question. There are quite a few physicists who have argued that actually one shouldn't use real numbers because we can never measure to infinite precision. I mention this briefly at the end of this video:
th-cam.com/video/Bq9xR5PUs6s/w-d-xo.html
I think the fact that we both live in the world and we can formally characterize what a real number even is in the first place (and there are whole branches of math dedicated to their study), makes me think you do need real numbers to describe the world, regardless of whether or not everything is quantum. Otherwise your description of the world would be unable to describe a thing that happened in the world (development of real analysis).
@@SabineHossenfelder Thanks - can't wait to watch that video too 😀
@@SabineHossenfelder Would you be so kind to talk about quaternions and octonions, and their relationship with grand unified theories?
What about when engineers need to use laplace transforms and imaginary numbers to solve control systems problems? In the process of making a transfer equation you very often come up on a S^2 -1 term. There is no other way of solving the problem other than complex numbers or eulers method.
p-adic numbers would be interesting. from what I remember, there are multiple ways to make the rationals compact (?), so to make every Cauchy series convergent in them, and the reals as they are used widely is just one example
"Complete", not "compact". Every metric space can be completed (basically by "adding" limit points of Cauchy sequences, but this idea needs formalization) and the rational are classically a metric space under the distance induced by the archimedean absolute value: real numbers are their completion. But you can change the metric! It turns out that if you fix a prime number p you can define a metric on the rationals by declaring that two rational numbers are "close" if their difference is divisible by a power of p, the higher the exponent the closer the numbers (again, this needs formalization). Then if you complete this metric you get the p-adic numbers. All these metrics are inequivalent so you get infinitely many inequivalent completions of the rationals. Then you can prove a theorem (known as Ostrowski Theorem) saying that these archimedean and p-adic metric are essentially the only metrics you can endow the rationals with.
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Maths is a language, so all kind of numbers are just a "mathematical convenience" :)
You are conflating Maths Symbols with what it can describe..one is fiction,the other is what we discover!
@@rationalsceptic7634 What maths can describe is not always what we discover. E.g. non-Euclidean geometries offered a different model of space and they were developed long before things they could describe were discovered. There are lots of things in maths that exist purely as mathematical concepts.
Well depends on what you mean by language; it is the most fundamental aspect of nature (arguably depending on whether you consider information to be even more fundamental); some parts (or fields) of maths are a conventionally established language.
But something so fundamental is not a simple convenient language
@@rationalsceptic7634 "with what it can describe.."
The entire point of language is to describe. Reread the OP.
@@xj91zdk7 but that is precisely the question, whether complex numbers are “purely” abstract objects, or the probability amplitude is a complex quantity
I love this channel so much. I always come away feeling smarter, you explain things so well. I could cry, no gobbledygook indeed! Thank you
I remember Richard Feynman explaining in his Lectures on Physics (1960s) why Quantum Mechanics requires complex numbers.
Definitely want to see a video on p-adic numbers!
Sabine, you are a gem. This is a great video. So classic.
Counter question: Why would anyone assume infinite p-adic numbers physically exist??
why? by definition
@@NikolayVolf Have you ever seen one in reality? If not ... how do you assume they "exist"? If you say "by definition" ... the definition of "existing" is "being in existence". You can't even START to spell one out, except i you begin at the wrong end and you will never manage to bring the other end into existence.
@Alfred Wedmore yea, which is exactly my point. Infinity before the decimal point can by definition not be "real" and infinity after the decimal point only if the universe has no smallest "pixels".
@@ENXJ Way to call bullshit. I'm on your side.
rebuttal III
D) The Hamiltonian operator, the operator associated with the total energy of a physical system, was already present before the big bang. E) probability is part of statistics, therefore statistics were present since before the big bang. From an initial examination, this brief summary emerges: - some laws of nature (the statement ALL cannot be excluded) not only would be prior to the big bang but following the quantum vacuum we find laws of nature that are even PRIOR: the wave function, the quantum vacuum, the particle-antiparticle pair and above all prior to the QUANTUM PERTURBATION, but as mentioned the operator j = (complex operator) h, π already existed. F) quantum mechanics cannot elude the first and second principles of thermodynamics therefore admitting the quantum vacuum prior to the big bang it is automatically admitted that these thermodynamic laws were already in force and therefore the TIMELESS quantum vacuum had to have maximum or INFINITE entropy. Attention because this DISASSEMBLES all the quantum mechanics of the pre-vacuum, the reason is simple: a vacuum with maximum entropy contained degraded energy of the lowest kind and therefore could not generate anything. G) from point F it can be deduced that the transition from the quantum vacuum to the big bang would have involved an enormous or INFINITE entropy input into the universe of the big bang and this is impossible given that the initial big bang was at zero or minimum entropy, seen the second law of thermodynamics
The more I see and hear your presentations, the more I am impressed. Thank you once again for an outstanding presentation.
😎💙💛
My first time hearing about numbers with ‘infinite numbers’ before decimal. Mind blown
Would any numbers exist without people to contemplate them?
Yes but they would we unnamed.