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Mathematicians disagree on the essential structure of the complex numbers (2024)
- PaulHoule 7mo agoAnother "xyz" domain that doesn't resolve on my network.
- emil-lp 7mo agohttps://web.archive.org/web/20251015174117/https://www.infinitelymore.xyz/p/complex-numbers-essential-structure https://web.archive.org/web/20251015174117/https://www.infin...
- FillMaths 7mo agoThis one has the paywall, but the main site has no paywall currently.
- gnatman 7mo agoYep- there’s some issues representing complex numbers in 3D space. You may want to check out quaternions.
- coldcity_again 7mo agoInstructions unclear, gimbal locked
- francasso 7mo agoThere's no disagreement, the algebraic one is the correct one, obviously. Anyone that says differently is wrong. :)
- emil-lp 7mo agoObviously.
- srean 7mo agoBeing an engineer by training, I never got exposed to much algebra in my courses (beyond the usual high school stuff in high school). In fact did not miss it much either. Tried to learn some algebraic geometry then... oh the horror. For whatever reason, my intuition is very geometric and analytic (in the calculus sense). Even things like counting and combinatorics, they feel weird, like dry flavorless pretzels made of dried husk. Combinatorics is good only when I can use Calculus. Calculus, oh that's different, it's rich savoury umami buttery briskets. Yum. That's not the interesting part. The interesting part is that I thought everyone is the same, like me. It was a big and surprising revelation that people love counting or algebra in just the same way I feel about geometry (not the finite kind) and feel awkward in the kind of mathematics that I like. It's part of the reason I don't at all get the hate that school Calculus gets. It's so intuitive and beautifully geometric, what's not to like. .. that's usually my first reaction. Usually followed by disappointment and sadness -- oh no they are contemplating about throwing such a beautiful part away.
- macromagnon 7mo agoSchool calculus is hated because it's typically taught with epsilon delta proofs which is a formalism that happened later in the history of calculus. It's not that intuitive for beginners, especially students who haven't learn any logic to grok existential/universal quantifiers. Historically, mathematics is usually developed by people with little care for complete rigor, then they erase their tracks to make it look pristine. It's no wonder students are like "who the hell came up with all this". Mathematics definitely has an education problem.
- jjgreen 7mo agoYou can do it with infinitesimals if you like, but the required course in nonstandard analysis to justify it is a bastard.
- jonahx 7mo agoOr you can hand wave a bit and trust intuition. Just like the titans who invented it all did! The obsession with rigor that later developed -- while necessary -- is really an "advanced topic" that shouldn't displace learning the intuition and big picture concepts. I think math up through high school should concentrate on the latter, while still being honest about the hand-waving when it happens.
- Sharlin 7mo ago"The Axiom of Choice is obviously true, the Well-ordering theorem obviously false, and who can tell about Zorn's lemma?" (attributed to Jerry Bona)
- ajb 7mo agoHah. This perspective is how you get an embedding of booleans into the reals in which False is 1 and True is -1 :-) (Yes, mathematicians really use it. It makes parity a simpler polynomial than the normal assignment).
- cperciva 7mo agoThe complex numbers are just elements of R[i]/(i^2+1). I don't even understand how people are able to get this wrong.
- deleted 7mo ago[deleted]
- FillMaths 7mo agoOf course everyone agrees that this is a nice way to construct the complex field. The question is what is the structure you are placing on this construction. Is it just a field? Do you intend to fix R as a distinguished subfield? After all, there are many different copies of R in C, if one has only the field structure. Is i named as a constant, as it seems to be in the construction when you form the polynomials in the symbol i. Do you intend to view this as a topological space? Those further questions is what the discussion is about.
- cperciva 7mo agoI mean, yes of course i is an element in C, because it's a monic polynomial in i. There's no "intend to". The complex numbers are what they are regardless of us; this isn't quantum mechanics where the presence of an observer somehow changes things.
- FillMaths 7mo agoIt's not about observers, but about mathematical structure and meaning. Without answering the questions, you are being ambiguous as to what the structure of C is. For example, if a particular copy of R is fixed as a subfield, then there are only two automorphisms---the trivial automorphism and complex conjugation, since any automorphism fixing the copy of R would have to be the identity on those reals and thus the rest of it is determined by whether i is fixed or sent to -i. Meanwhile, if you don't fix a particular R subfield, then there is a vast space of further wild automorphisms. So this choice of structure---that is, the answer to the questions I posed---has huge consequences on the automorphism group of your conception. You can't just ignore it and refuse to say what the structure is.
- yaks_hairbrush 7mo agoIt works if you don't care about magnitudes, distances, or angles of complex numbers. Those properties aren't algebraic.
- zeroonetwothree 7mo agoThe whole substack is great, I recommend reading all of it if you are interested in infinity
- emil-lp 7mo agoHe's also very active at Stack Exchange – https://stackexchange.com/users/510234/joel-david-hamkins https://stackexchange.com/users/510234/joel-david-hamkins – https://mathoverflow.net/users/1946/joel-david-hamkins https://mathoverflow.net/users/1946/joel-david-hamkins
- slwvx 7mo agoIs there agreement Gaussian integers? This disagreement seems above the head of non mathematicians, including those (like me) with familiarity with complex numbers
- btilly 7mo agoThere is perfect agreement on the Gaussian integers. The disagreement is on how much detail of the fine structure we care about. It is roughly analogous to asking whether we should care more about how an ellipse is like a circle, or how they are different. One person might care about the rigid definition and declare them to be different. Another notices that if you look at a circle at an angle, you get an ellipse. And then concludes that they are basically the same thing. This seems like a silly thing to argue about. And it is. However in different branches of mathematics, people care about different kinds of mathematical structure. And if you view the complex numbers through the lens of the kind of structure that you pay attention to, then ignore the parts that you aren't paying attention to, your notion of what is "basically the same as the complex numbers" changes. Just like how one of the two people previously viewed an ellipse as basically the same as a circle, because you get one from the other just by looking from an angle. Note that each mathematician here can see the points that the other mathematicians are making. It is just that some points seem more important to you than others. And that importance is tied to what branch of mathematics you are studying.
- lmkg 7mo agoThe Gaussian integers usually aren't considered interesting enough to have disagreements about. They're in a weird spot because the integer restriction is almost contradictory with considering complex numbers: complex numbers are usually considered as how to express solutions to more types of polynomials, which is the opposite direction of excluding fractions from consideration. They're things that can solve (a restricted subset of) square-roots but not division. This is really a disagreement about how to construct the complex numbers from more-fundamental objects. And the question is whether those constructions are equivalent. The author argues that two of those constructions are equivalent to each other, but others are not. A big crux of the issue, which is approachable to non-mathematicians, is whether it i and -i are fundamentally different, because arithmetically you can swap i with -i in all your equations and get the same result.
- ActorNightly 7mo agoHonestly, the rigid conception is the correct one. Im of the view that i as an attribute on a number rather than a number itself, in the same way a negative sign is an attribute. Its basically exists to generalize rotations through multiplication. Instead of taking an x,y vector and multiplying it by a matrix to get rotations, you can use a complex number representation, and multiply it by another complex number to rotate/scale it. If the cartesian magnitude of the second complex number is 1, then you don't get any scaling. So the idea of x/y coordinates is very much baked in to the "imaginary attribute". I feel like the problem is that we just assume that e^(pi*i) = -1 as a given, which makes i "feel" like number, which gives some validity to other interpretations. But I would argue that that equation is not actually valid. It arises from Taylor series equivalence between e, sin and cos, but taylor series is simply an approximation of a function by matching its derivatives around a certain point, namely x=0. And just because you take 2 functions and see that their approximations around a certain point are equal, doesn't mean that the functions are equal. Even more so, that definition completely bypasses what it means to taking derivatives into the imaginary plane. If you try to prove this any other way besides Taylor series expansion, you really cant, because the concept of taking something to the power of "imaginary value" doesn't really have any ties into other definitions. As such, there is nothing really special about e itself either. The only reason its in there is because of a pattern artifact in math - e^x derivative is itself, while cos and sin follow cyclic patterns. If you were to replace e with any other number, note that anything you ever want to do with complex numbers would work out identically - you don't really use the value of e anywhere, all you really care about is r and theta. So if you drop the assumption that i is a number and just treat i as an attribute of a number like a negative sign, complex numbers are basically just 2d numbers written in a special way. And of course, the rotations are easily extended into 3d space through quaternions, which use i j an k much in the same way.
- nyeah 7mo agoTo be clear, this "disagreement" is about arbitrary naming conventions which can be chosen as needed for the problem at hand. It doesn't make any difference to results.
- heinrichhartman 7mo agoAgreed. To me it looks like the entire discussion is just bike-shedding.
- gowld 7mo agoIt's math. Bikeshedding is the goal.
- mmooss 7mo agoNames, language, and concepts are essential to and have powerful effects on our understanding of anything, and knowledge of mathematics is much more than the results. Arguably, the results are only tests of what's really important, our understanding.
- sunshowers 7mo agoI'm not a professional, but to me it's clear that whether i and -i are "the same" or "different" is actually quite important.
- kergonath 7mo agoA bit like +0 and -0? It makes sense in some contexts, and none in others.
- grumbelbart 7mo agoThey would never be the same. It's just that everything still works the same if you switch out every i with -i (and thus every -i with i).
- alexey-salmin 7mo ago
- phkahler 7mo agoTo the ones objecting to "choosing a value of i" I might argue that no such choice is made. i is the square root of -1 and there is only one value of i. When we write -i that is shorthand for (-1)i. Remember the complex numbers are represented by a+bi where a and b are real numbers and i is the square root of -1. We don't bifurcate i into two distinct numbers because the minus sign is associated with b which is one of the real numbers. There is a one-to-one mapping between the complex numbers and these ordered pairs of reals.
- FillMaths 7mo agoYou say that i is "the square root of -1", but which one is it? There are two. This is the point in the essay---we cannot tell the difference between i and -i unless we have already agreed on a choice of which square root of -1 we are going to call i. Only then does the other one become -i. How do we know that my i is the same as your i rather than your -i? To fix the coordinate structure of the complex numbers (a,b) is in effect to have made a choice of a particular i, and this is one of the perspectives discussed in the essay. But it is not the only perspective, since with that perspective complex conjugation should not count as an automorphism, as it doesn't respect the choice of i.
- phkahler 7mo agoThere are 2 square roots of 9, they are 3 and -3. Likewise there are two square roots of -1 which are i and -i. How are people trying to argue that there are two different things called i? We don't ask which 3 right? My argument is that there is only 1 value of i, and the distinction between -i and i is the same as (-1)i and (1)i, which is the same as -3 vs 3. There is only one i. If there are in fact two i's then there are 4 square roots of -1.
- czgnome 7mo agoYour view of the complex numbers is the rigid one. Now suppose you are given a set with two binary operations defined in such a way that the operations behave well with each other. That is you have a ring. Suppose that by some process you are able to conclude that your ring is algebraically equivalent to the complex numbers. How do you know which of your elements in your ring is “i”? There will be two elements that behave like “i” in all algebraic aspects. So you can’t say that this one is “i” and this one is “-i” in a non arbitrary fashion.
- d--b 7mo agoWhoever coined the terms ‘complex numbers’ with a ‘real part’ and ‘imaginary part’ really screwed a lot of people..
- cess11 7mo agoHow come? They are part real numbers, what would you call the other part?
- maxbond 7mo agoWe could've called the imaginaries "orthogonals", "perpendiculars", "complications", "atypicals", there's a million other options. I like the idea that a number is complex because it has a "complicated component".
- cess11 7mo agoComplex means that it is composed of parts, contrary to simplex, i.e. single or simple.
- srean 7mo agoTwisted ? Rotated ?
- d--b 7mo agoI mean that they're not really numbers. Usually they explain it something like: oh, at first people didn't know what 2-5 added up to, but then we invented negative numbers. Well, complex numbers are that but for square roots of negative numbers. But that's a completely misleading way to explain these things. Complex numbers aren't numbers aren't numbers really.
- cess11 7mo agoI mean, yeah, they aren't real numbers, they are composed of a real number and another one that is the multiplicative of the square root of -1. Hence they're called complex, i.e. composed of parts. If the square root of -1 is not a number, what is it? How come you can do arithmetic with it?
- yifanl 7mo agoNotably, neither `1 + i > 1 - i` or `1 + i < 1 - i` are correct statements, and obviously `1 + i = 1 - i` is absurd.
- chongli 7mo agoWhat do > and < mean in the context of an infinite 2D plane?
- yifanl 7mo agoTypically, the order of complex numbers is done by projecting C onto R, i.e. by taking the absolute value.
- chongli 7mo agoYes I’m aware. It’s a work around but doesn’t give you a sensible ordering the way most people expect, i.e: -2 > 1 (in C) Which is why I prefer to leave <,> undefined in C and just take the magnitude if I want to compare complex numbers.
- layer8 7mo agoOne is above the plane and the other is below it. ;)
- bell-cot 7mo agoIn a word - "true". In more words - it's interesting, but messy: https://en.wikipedia.org/wiki/Partial_order https://en.wikipedia.org/wiki/Partial_order https://en.wikipedia.org/wiki/Ordered_field https://en.wikipedia.org/wiki/Ordered_field > The complex numbers also cannot be turned into an ordered field, as −1 is a square of the imaginary unit i.
- mmooss 7mo agoKnowledge is the output of a person and their expertise and perspective, irreducibly. In this case, they seem to know something of what they're talking about: > Starting 2022, I am now the John Cardinal O’Hara Professor of Logic at the University of Notre Dame. > From 2018 to 2022, I was Professor of Logic at Oxford University and the Sir Peter Strawson Fellow at University College Oxford. Also interesting: > I am active on MathOverflow, and my contributions there (see my profile) have earned the top-rated reputation score. https://jdh.hamkins.org/about/ https://jdh.hamkins.org/about/
- nigelvr 7mo agoThe link is about set theory, but others may find this interesting which discusses division algebras https://nigelvr.github.io/post-4.html https://nigelvr.github.io/post-4.html Basically C comes up in the chain R \subset C \subset H (quaternions) \subset O (octonions) by the so-called Cayley-Dickson construction. There is a lot of structure.
- zarzavat 7mo agoThe way I think of complex numbers is as linear transformations. Not points but functions on points that rotate and scale. The complex numbers are a particular set of 2x2 matrices, where complex multiplication is matrix multiplication, i.e. function composition. Complex conjugation is matrix transposition. When you think of things this way all the complex matrices and hermitian matrices in physics make a lot more sense. Which group do I fall into?
- bheadmaster 7mo agoMy biggest pet peeve in complex analysis is the concept of multi-value functions. Functions are defined as relations on two sets such that each element in the first set is in relation to at most one element in the second set. And suddenly we abandon that very definitions without ever changing the notation! Complex logarithms suddenly have infinitely many values! And yet we say complex expressions are equal to something. Madness.
- alexey-salmin 7mo agoIdk, to me it feels much much better than just picking one root when defining the inverse function. This desire to absolutely pick one when from the purely mathematical perspective they're all equal is both ugly and harmful (as in complicates things down the line).
- bheadmaster 7mo agoWell, yeah, the alternative is also bad. But couldn't we just switch the nomenclature? Instead of an oxymoronic concept of "multivalue function", we could just call it "relation of complex equivalence" or something of sorts.
- prmph 7mo agoJust think of it as a function that returns an array or a set: it still one value in a sense
- articulatepang 7mo agoYou can think of it as returning an equivalence class if you like. Then it's single-valued. More explicitly, it returns an equivalence class whose members are complex numbers that differ by integer multiples of 2*pi*i. When it's important to distinguish members of the class, we speak of branches of the logarithm. Also note the very cool and fun topology connection here. The keyword to search for is Riemann surface.
- bheadmaster 7mo ago
- clintonc 7mo agoI have a Ph.D. in a field of mathematics in which complex numbers are fundamental, but I have a real philosophical problem with complex numbers. In particular, they arose historically as a tool for solving polynomial equations. Is this the shadow of something natural that we just couldn't see, or just a convenience? As the "evidence" piles up, in further mathematics, physics, and the interactions of the two, I still never got to the point at the core where I thought complex numbers were a certain fundamental concept, or just a convenient tool for expressing and calculating a variety of things. It's more than just a coincidence, for sure, but the philosophical part of my mind is not at ease with it. I doubt anyone could make a reply to this comment that would make me feel any better about it. Indeed, I believe real numbers to be completely natural, but far greater mathematicians than I found them objectionable only a hundred years ago, and demonstrated that mathematics is rich and nuanced even when you assume that they don't exist in the form we think of them today.
- jgrahamc 7mo agoI don't know if this will help, but I believe that all of mathematics arises from an underlying fundamental structure to the universe and that this results in it both being "discoverable" (rather than invented) and "useful" (as in helpful for describing, expressing and calculating things).
- HackerNewt-doms 7mo agoWhy do you believe that the same mathematical properties hold everywhere in the universe?
- billforsternz 7mo agoNot the person you're replying too, but ... because it would be weird if they didn't.
- bolangi 7mo agoThere are legitimate questions if physical constants are constant everywhere in the universe, and also whether they are constant over time. Just because we conceive something "should" be a certain way doesn't make it true. The zero and negative numbers were also weird yet valid. How is the structure of mathematics different from fundamental constants, which we also cannot prove are invariant.
- topaz0 7mo agoMost commenters are talking about the first part of the post, which lays out how you might construct the complex numbers if you're interested in different properties of them. I think the last bit is the real interesting substance, which is about how to think about things like this in general (namely through structuralism), and why the observations of the first half should not be taken as an argument against structuralism. Very interesting and well written.
- Traster 7mo agoIt is very re-assuring to know, on a post where I can essentially not even speak the language (despite a masters in engineering) HN is still just discussing the first paragraph of the post.
- deleted 7mo ago[deleted]
- loglog 7mo agoReal men know that infinite sets are just a tool for proving statements in Peano arithmetic, and complex numbers must be endowed with the standard metric structure, as God intended, since otherwise we cannot use them to approximate IEEE 754 floats.
- Syzygies 7mo agoI began studying 3-manifolds after coming up with a novel way I preferred to draw their presentations. All approaches are formally equivalent, but they impose different cognitive loads in practice. My approach was trivially equivalent to triangulations, or spines, or Heegaard splittings, or ... but I found myself far more nimbly able to "see" 3-manifolds my way. I showed various colleagues. Each one would ask me to demonstrate the equivalence to their preferred presentation, then assure me "nothing to see here, move along!" that I should instead stick to their convention. Then I met with Bill Thurston, the most influential topologist of our lifetimes. He had me quickly describe the equivalence between my form and every other known form, effectively adding my node to a complete graph of equivalences he had in his muscle memory. He then suggested some generalizations, and proposed that circle packings would prove to be important to me. Some mathematicians are smart enough to see no distinction between any of the ways to describe the essential structure of a mathematical object. They see the object.
- kmill 7mo agoWould you mind sharing your representation? :-)
- mlochbaum 7mo agoI was interested in how it would make sense to define complex numbers without fixing the reals, but I'm not terribly convinced by the method here. It seemed kind of suspect that you'd reduce the complex numbers purely to its field properties of addition and multiplication when these aren't enough to get from the rationals to the reals (some limit-like construction is needed; the article uses Dedekind cuts later on). Anyway, the "algebraic conception" is defined as "up to isomorphism, the unique algebraically closed field of characteristic zero and size continuum", that is, you just declare it has the same size as the reals. And of course now you have no way to tell where π is, since it has no algebraic relation to the distinguished numbers 0 and 1. If I'm reading right, this can be done with any uncountable cardinality with uniqueness up to isomorphism. It's interesting that algebraic closure is enough to get you this far, but with the arbitrary choice of cardinality and all these "wild automorphisms", doesn't this construction just seem... defective? It feels a bit like the article's trying to extend some legitimate debate about whether fixing i versus -i is natural to push this other definition as an equal contender, but there's hardly any support offered. I expect the last-place 28% poll showing, if it does reflect serious mathematicians at all, is those who treat the topological structure as a given or didn't think much about the implications of leaving it out.
- deleted 7mo ago[deleted]
- mlochbaum 7mo agoMore on not being able to find π, as I'm piecing it together: given only the field structure, you can't construct an equation identifying π or even narrowing it down, because if π is the only free variable then it will work out to finding roots of a polynomial (you only have field operations!) and π is transcendental so that polynomial can only be 0 (if you're allowed to use not-equals instead of equals, of course you can specify that π isn't in various sets of algebraic numbers). With other free variables, because the field's algebraically closed, you can fix π to whatever transcendental you like and still solve for the remaining variables. So it's something like, the rationals plus a continuum's worth of arbitrary field extensions? Not terribly surprising that all instances of this are isomorphic as fields but it's starting to feel about as useful as claiming the real numbers are "up to set isomorphism, the unique set whose cardinality matches the power set of the natural numbers", like, of course it's got automorphisms, you didn't finish defining it.
- mebassett 7mo agothe title is a bit clickbait - mathematicians don't disagree, all the "conceptions" the article proposes agree with each other. It also seems to conflate the algebraic closure of Q (which would contain the sqrt of -1) and all of the complex numbers by insisting that the former has "size continuum". Once you have "size continuum" then you need some completion to the reals. anyhow. I'm a bit of an odd one in that I have no problems with imaginary numbers but the reals always seemed a bit unreal to me. that's the real controversy, actually. you can start looking up definable numbers and constructivist mathematics, but that gets to be more philosophy than maths imho.
- TimorousBestie 7mo ago> But in fact, I claim, the smooth conception and the analytic conception are equivalent—they arise from the same underlying structure. Conjugation isn’t complex-analytic, so the symmetry of i -> -i is broken at that level. Complex manifolds have to explicitly carry around their almost-complex structure largely for this reason.
- brcmthrowaway 7mo agoWhat does Terry Tao think?
- Traster 7mo agoDoes anyone have any tips on how I would fundamentally understand this article without just going back to school and getting a degree in mathematics? This is the sort of article where my attempts to understand a term only ever increase the number of terms I don't understand.
- gowld 7mo agoStudy analysis and algebra from books and videos on your own?
- Paracompact 7mo agoPhD math guy here. Personally, I don't think this is a great article for a lay person to approach or appreciate. You might be able and interested to follow along with the various constructions of C, but without appreciation for formal logic and category theory it'll seem like distinctions without a difference (and to working mathematicians, they are). The model theory I found quite interesting, but was high effort even for me. And the philosophical points are also rather specific.
- Nevermark 7mo agoThe square root of any number x is ±y, where +y = (+1)*y = y, and -y = (-1)*y. So we define i as conforming to ±i = sqrt(-1). The element i itself has no need for a sign, so no sign needs to be chosen. Yet having defined i, we know that that i = (+1)*i = +i, by multiplicative identity. We now have an unsigned base element for complex numbers i, derived uniquely from the expansion of <R,0,1,+,*> into its own natural closure. We don't have to ask if i = +i, because it does by definition of the multiplicative identity. TLDR: Any square root of -1 reduced to a single value, involves a choice, but the definition of unsigned i does not require a choice. It is a unique, unsigned element. And as a result, there is only a unique automorphism, the identity automorphism.
- riemannzeta 7mo agoI really know almost nothing about complex analysis, but this sure feels like what physicists call observational entropy applied to mathematics: what counts as "order" in ℂ depends on the resolution of your observational apparatus. The algebraic conception, with its wild automorphisms, exhibits a kind of multiplicative chaos — small changes in perspective (which automorphism you apply) cascade into radically different views of the structure. Transcendental numbers are all automorphic with each other; the structure cannot distinguish e from π. Meanwhile, the analytic/smooth conception, by fixing the topology, tames this chaos into something with only two symmetries. The topology acts as a damping mechanism, converting multiplicative sensitivity into additive stability. I'll just add to that that if transformers are implementing a renormalization group flow, than the models' failure on the automorphism question is predictable: systems trained on compressed representations of mathematical knowledge will default to the conception with the lowest "synchronization" cost — the one most commonly used in practice. https://www.symmetrybroken.com/transformer-as-renormalization-group-flow/ https://www.symmetrybroken.com/transformer-as-renormalizatio...
- SpaceManNabs 7mo agoI thought i understood complex numbers and accepted them until I did countour integration for the first time. Ever since then I have been deeply unsettled. I started questioning taking integrals to (+/-) infinity, and so I became unsettled with R too. If C exists to fix R, then why does R even exist? Why does R need to be fixed? Why does the use of the upper or lower plane for counter integration not matter? I can do mathematically why, but why do we have a choice? This blog post really articulated stuff formally that I have been bothered by for years.
- einpoklum 7mo agoI found the article mildly interesting "light reading", until I got to this part: > I was astounded to see that the Google AI overview in effect takes a stand amongst three conceptions Uh oh. Hype alert. Should we continue reading? ... [a few moments later] ... Oh, ok, the answer is yes. That was a bit of pandering but the author goes on to discuss how mathematicians think if this issue. Also, don't miss this gem of a pun : > Choosing the square root of -1 is a mathematical sin :-)
- tliltocatl 7mo agoAs a non-mathematican, I found that trying to introduce C as a closure of R (i. e. analytically in author's terms) invariably triggers confusion and "hey, why do mathematicians keep changing rules on the fly, they just told me square of minus one doesn't exist". And in terms of practical applications it doesn't seem particularly useful on the first glance (who cares about solving cubics algebraically? The formula is too unwieldy anyway.) Most applications tends to start in the coordinate view and go from there. And it does introduce a nasty sharp edge to cut oneself on (i vs -i), but then for instance physics is full of such edges: direction of pseudo-vectors, sign of voltage on loads sources, holes in dimensional analysis (VA vs W, Ohm/square), the list could go on. And nobody really care.
- throwaway_2494 7mo ago> "hey, why do mathematicians keep changing rules on the fly, they just told me square of minus one doesn't exist Mathematicians aren’t chasing numerical solutions, they’re chasing structure. ℂ isn’t just about solving cubics, it’s about eliminating holes in algebra so the theory behaves uniformly and is easier to build upon. And as for "changing rules" they haven't changed, they have broadened the field (literally) over which the old rules applied in a clever way to remove a restriction.
- klysm 7mo agoIdk if this perspective is naive, but complex numbers to me are most motivated by spinning things.
- aspendougy 7mo agoFor what it's worth, Errett Bishop, the famous constructivist did not have this kind of existential issue with the complex numbers, commenting that the Reals were inadequate for some things. I really liked the trig cos isin connection in High School
- aspendougy 7mo agoThe famous constructivist Errett Bishop did not have this sort of existential issue with the Complex Numbers, only saying the Reals were inadequate for some things.
- inquirerGeneral 7mo ago[dead]
- tsoukase 7mo ago"God made the integers, all else is the work of man", Kronecker, 1886. Complex numbers are just a fancy way to represent 2-dimensional numbers that are convenient in geometry through the sq(-1)=π/2 rotation. They are nothing special and they could just be a matrix. Out of my head I think about complex integers. Also there are higher-order complex numbers which are defined by sq(sq(-1)) etc. In Greek complex numbers are called "mongrel". Both names are bad, I would just use 2-dimensional numbers.
- stared 7mo agoIn short: Reposting my comment from https://news.ycombinator.com/item?id=46775758 https://news.ycombinator.com/item?id=46775758 thread: scaling -> real numbers 2d rotations and scaling -> complex numbers 3d rotations and scaling -> quaternions In the case of quaternions, there is called double-covering, which turns out (rather than being an artefact), play fundamental role in particle physics.
- StopDisinfo910 7mo agoI am confused by both the article and the discussion and I don't mean confused by the discussion on the complex which is all fairly clear but by this very weird idea of essential structure whatever that's supposed to mean. I'm wondering if there is something cultural here (I'm French) and linked to the central place algebra has in our curriculum or if I'm just reading a lot of IA generated convoluted discussion. To me, the question doesn't even make sense. There is no disagreement here and I don't understand what is an essential structure. All the properties presented on the complex are consistent. They all exist. There is nothing ever essential about mathematics. I mean, it's just mathematics. As long as you are internally consistent with your axioms, things just are. It's like asking if water is a liquid or a collection of molecules. There is nothing to disagree about here.
- qubex 7mo agoTired: De Moivre’s Theorem Wired: The complex number-line Expired: The complex plane
- OldRonin 7mo agoThe field C is a tool, just as with any other abstraction in math. Accountants don't need it, so ignore it. In abstract algebra they arise - and so do many other abstractions, such as quaternions. Quaternions can be regarded as a further development of the concept of number. Note that every such development gives up some property its predecessors had. In the case of quaternions, commutativity. If a field of application finds a mathematical invention useful, it gets used. Often, too, it's a matter of simplicity. Use mathematical invention X in applied field Y and it simplifies the work. Abstain from using invention X, and then the work could still be done in a less simple way. That's all there is to it.
- DoctorOetker 7mo agoFor me personally, I prefer to perceive as more fundamental those constructions that straightforwardly generate the most with the least of novelty. For example one may be introduced to the real numbers, and later to the complex numbers and then later perhaps the quaternions and the octonions, etc. in a haphazard disconnected way. Given just the real numbers and "geometric algebra" (i.e. Grassmann algebras), this generates basically all these structures for different "settings". I hence view geometric algebra as a lower level and more fundamental than complex numbers specifically. A more interesting question (if we had to limit discussion to complex numbers) would be the following: which representations of complex numbers are known, what are their advantages and disadvantages, and can novel representation of complex numbers be devised that display less of the disadvantages? For example, we have -just to list a few- the following representations: 1. cartesian representation of complex numbers: a + bi : the complex numbers permit a straightforward single-valued addition and single-valued multiplication, but only multivalued integer-powered-roots. addition and multiplication change smoothly with smooth changes in the input values, taking N-th roots do not, unless you use multivalued roots but then you don't have single valued roots. 2. polar representation of complex numbers: r(cos(theta)+i sin(theta)): this representatin admits smooth and single valued multiplication and N-th roots, but no longer permits smooth and single valued additions! Please follow up with your favourite representations for complex numbers, and expound the pro's and con's of that representation. For example can you generate a representation where addition and N-th roots are smooth and single valued, but multiplications are not? Can you prove that any representation for complex numbers must suffer this dilemma in representation, or can you devise a representation where all 3 addition, multiplication, roots are smooth and single valued?