StrictlyLogical Posted February 26 Report Share Posted February 26 1 hour ago, Bill Hobba said: I will do a separate post on why I think mathematics is founded in reality Depending on your definition of what constitutes mathematics that statement may be overly broad. Although it sounds absurd, “mathematics” could include both invalid and valid forms, in such a context some mathematics definitely are not founded in reality… and could be floating abstractions at best and wholly based on imagination and irrationality at worst. On the other hand if you propose to define valid mathematics or just “mathematics” as only including that which IS founded in reality then I look forward to your delineation of what falls within and without of that classification. Abstractions are meant to be meaningfully based on and in correspondence to or connected with reality but we all know that anti-concepts, invalid and floating abstractions are all commonly found among the mental contents of a great many people. Quote Link to comment Share on other sites More sharing options...

Boydstun Posted February 26 Report Share Posted February 26 11 hours ago, Bill Hobba said: Godel is not the profound thing it is made out to be. It is logically equivalent to a very practical problem in computer science called the halting problem. As a programmer, I would love a program that could accept my program as input and tell me if it will loop or not. A rather convenient thing to have. If such a program even exists is called the halting problem. The answer is actually no - you can't write such a program. Bummer. But it is nothing esoteric, weird or anything like that - simply a limitation on the tools computer science allows programmers to have. . . . Gödel's incompleteness theorems (and the halting problem) may well be profound, but not in the pedestrian way of profound: taking them to bolster skepticism. Gödel's incompleteness theorems are, at least, splendid limitative theorems about formal systems, as are his completeness proofs. Some more resources: Set Theory, Logic and Their Limitations Perspectives in Computation by Geroch Computability – Turing, Gödel, Church, and Beyond Quote Link to comment Share on other sites More sharing options...

StrictlyLogical Posted February 26 Report Share Posted February 26 (edited) 12 hours ago, StrictlyLogical said: Depending on your definition of what constitutes mathematics that statement may be overly broad. Although it sounds absurd, “mathematics” could include both invalid and valid forms, in such a context some mathematics definitely are not founded in reality… and could be floating abstractions at best and wholly based on imagination and irrationality at worst. On the other hand if you propose to define valid mathematics or just “mathematics” as only including that which IS founded in reality then I look forward to your delineation of what falls within and without of that classification. Abstractions are meant to be meaningfully based on and in correspondence to or connected with reality but we all know that anti-concepts, invalid and floating abstractions are all commonly found among the mental contents of a great many people. @Bill Hobba I should let you know that hearing your interest in these subjects is refreshing. So many mathematicians, physicists, and philosophers stay in their own lanes far too much or hold on to what they have been told by authorities in their field with far too little independent scrutiny. That said, as an analogy I would like to introduce the statement “abstractions are founded in reality” as a generalization which is subject to the same problem. When abstractions are used in a context of referring to reality, with any language, mathematical or not, the system of abstractions should be founded in reality. These sorts of considerations were never really investigated in my training in physics. I would argue the BEST professors admonished us to look at the equations SOLELY as a tool for predicting and quantifying reality… implicitly our way of dealing with reality is not reality… a nice warning about the clear distinction between our abstractions and their referents in reality, without sophisticated explication. Edited February 26 by StrictlyLogical Quote Link to comment Share on other sites More sharing options...

Bill Hobba Posted February 27 Report Share Posted February 27 12 hours ago, Boydstun said: Gödel's incompleteness theorems (and the halting problem) may well be profound, but not in the pedestrian way of profound: taking them to bolster skepticism. That is a better way of expressing the situation than I did. Thanks Bill Quote Link to comment Share on other sites More sharing options...

Bill Hobba Posted February 27 Report Share Posted February 27 10 hours ago, StrictlyLogical said: When abstractions are used in a context of referring to reality, with any language, mathematical or not, the system of abstractions should be founded in reality. It is said by Gleick that truly great physicists (eg Einstein, Landau, Feynman and likely Fermi), have a literally scary ability to see the substance behind the equations mere mortals can't. It's not like the great Von Neumann - a polymath (although as a physicist, he was in the top echelon) whose intellect was mind-boggling. But they somehow saw to the heart of things better - which is not to say Von Neumann was a slouch at it - he was right up there with the best. This was expressed well by Wigner: “I have known a great many intelligent people in my life. I knew Max Planck, Max von Laue, and Wemer Heisenberg. Paul Dirac was my brother-in-Iaw; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me. [...] But Einstein's understanding was deeper than even Jancsi von Neumann's. His mind was both more penetrating and more original than von Neumann's. And that is a very remarkable statement. Einstein took an extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity; and for all of Jancsi's brilliance, he never produced anything so original.” Thanks Bill Quote Link to comment Share on other sites More sharing options...

Bill Hobba Posted February 27 Report Share Posted February 27 Mathematics is grounded in reality simply because history shows that no area of math does not eventually find application. One of the purest of pure mathematicians, Hardy, who defended mathematics purely on the grounds of its beauty, once claimed 'nothing he had ever done had any commercial or military usefulness'. How wrong he was. Just take one area he was interested in, the divergent series. I have a book sitting in front of me called Advanced Engineering Mathematics that has a whole chapter on it because it is so useful in solving differential equations that occur in engineering. Even in physics, calculating the Casimir Force (by one method, anyway) requires summating a divergent series. Hardy's claim is the excrement of the male bovine - with all due respect to Hardy, whose mentoring of Ramanujan was a great service to mathematics. Others have remarked on this - even what looks like the most useless branch of math always seems to find application. The reverse sometimes occurs as well - sloppy math, sometimes used in engineering or physics, can inspire other mathematicians to sort out what is happening. An example is the Dirac Delta function, the solution to what it means, requiring a whole new branch of math called Distribution Theory. This has led to all sorts of views, e.g. those of Penrose, which I once agreed with, but now don't. The most straightforward answer is - mathematics is about the real world. I will give an example. Imaginary numbers were introduced to solve any quadratic equation. But we now know that looking at numbers as part of reality made their existence obvious. Draw a line and mark it with the distance from its start. These are the positive real numbers. Now let's continue the line in the other direction. We can think of -1 as an operator that rotates a point on the line by 180% so that -1 = -1*1 is one rotated 180%. We can extend this further to i being an operator that rotates whatever is after it by 90%. If nothing is after it, take it as 1 follows it. So i rotates the number 1 by 90% anticlockwise (by convention). Hence i^2 = -1. We can think of the imaginary number line as the real line rotated by 90%, (i.e. with i applied to all points on the real line), and you can specify a point in the plane by a number on the real line (a say) and a number on the complex line (b say). This is written as a + i*b or a with b rotated by 90% added to it. In other words, all complex numbers are, is a way to describe points on a plane. You can do it in other ways of course (eg vectors) - but all that shows is there are several ways to skin a cat - each with its own advantages and disadvantages. To see its power, consider the operator f(x) that rotates whatever is after it by the angle x. Well f(x) = f(n*x/n) = f(x/n)^n. But if n is large, the rotation by angle x/n = 1+(i*x/n) to a good approximation, getting better as n becomes larger. So to good approximation f(x) = f(x/n)^n = (1+ i*x/n)^n, and we expect this to be exact as n goes to infinity. But from calculus, we know the e^x = (1+x/n)^n as n goes to infinity. Substituting i*x for x, we have e^ix = (1+ ix/n)^n as n goes to infinity. So you get e^ix as an operator that rotates by an angle x. Take the derivative of e^i*x, and you get i*e^i*x - try it - it's easy. This makes proving the trig identities, derivatives etc., a snap compared to what is usually done. For example, a line of unit length rotated by angle x from the real line is cos(x) + i*sine(x). So e^i*x = cos(x) + i*sine(x). Hence taking the derivative we have cos(x)’ + i*sine(x)’ = i*cos(x) - sine(x) or cos(x)’ = -sine(x) and sine(x)’ = cos(x). Compare that to other proofs; you will see it is much easier. Now try it on e^i*(x+y), and you get the standard trig identities without further ado. IMHO, before doing more advanced trig, calculus etc., you should study complex numbers - things are more manageable. Math is about reality - but different mathematics can describe the same reality, each with its advantages and disadvantages. Thanks Bill 20 views Quote Link to comment Share on other sites More sharing options...

Boydstun Posted February 28 Report Share Posted February 28 (edited) On 2/27/2023 at 12:44 AM, Bill Hobba said: Mathematics is grounded in reality simply because history shows that no area of math does not eventually find application. One of the purest of pure mathematicians, Hardy, who defended mathematics purely on the grounds of its beauty, once claimed 'nothing he had ever done had any commercial or military usefulness'. How wrong he was. Just take one area he was interested in, the divergent series. I have a book sitting in front of me called Advanced Engineering Mathematics that has a whole chapter on it because it is so useful in solving differential equations that occur in engineering. Even in physics, calculating the Casimir Force (by one method, anyway) requires summating a divergent series. Hardy's claim is the excrement of the male bovine - with all due respect to Hardy, whose mentoring of Ramanujan was a great service to mathematics. Others have remarked on this - even what looks like the most useless branch of math always seems to find application. The reverse sometimes occurs as well - sloppy math, sometimes used in engineering or physics, can inspire other mathematicians to sort out what is happening. An example is the Dirac Delta function, the solution to what it means, requiring a whole new branch of math called Distribution Theory. This has led to all sorts of views, e.g. those of Penrose, which I once agreed with, but now don't. The most straightforward answer is - mathematics is about the real world. I will give an example. Imaginary numbers were introduced to solve any quadratic equation. But we now know that looking at numbers as part of reality made their existence obvious. Draw a line and mark it with the distance from its start. These are the positive real numbers. Now let's continue the line in the other direction. We can think of -1 as an operator that rotates a point on the line by 180% so that -1 = -1*1 is one rotated 180%. We can extend this further to i being an operator that rotates whatever is after it by 90%. If nothing is after it, take it as 1 follows it. So i rotates the number 1 by 90% anticlockwise (by convention). Hence i^2 = -1. We can think of the imaginary number line as the real line rotated by 90%, (i.e. with i applied to all points on the real line), and you can specify a point in the plane by a number on the real line (a say) and a number on the complex line (b say). This is written as a + i*b or a with b rotated by 90% added to it. In other words, all complex numbers are, is a way to describe points on a plane. You can do it in other ways of course (eg vectors) - but all that shows is there are several ways to skin a cat - each with its own advantages and disadvantages. To see its power, consider the operator f(x) that rotates whatever is after it by the angle x. Well f(x) = f(n*x/n) = f(x/n)^n. But if n is large, the rotation by angle x/n = 1+(i*x/n) to a good approximation, getting better as n becomes larger. So to good approximation f(x) = f(x/n)^n = (1+ i*x/n)^n, and we expect this to be exact as n goes to infinity. But from calculus, we know the e^x = (1+x/n)^n as n goes to infinity. Substituting i*x for x, we have e^ix = (1+ ix/n)^n as n goes to infinity. So you get e^ix as an operator that rotates by an angle x. Take the derivative of e^i*x, and you get i*e^i*x - try it - it's easy. This makes proving the trig identities, derivatives etc., a snap compared to what is usually done. For example, a line of unit length rotated by angle x from the real line is cos(x) + i*sine(x). So e^i*x = cos(x) + i*sine(x). Hence taking the derivative we have cos(x)’ + i*sine(x)’ = i*cos(x) - sine(x) or cos(x)’ = -sine(x) and sine(x)’ = cos(x). Compare that to other proofs; you will see it is much easier. Now try it on e^i*(x+y), and you get the standard trig identities without further ado. IMHO, before doing more advanced trig, calculus etc., you should study complex numbers - things are more manageable. Math is about reality - but different mathematics can describe the same reality, each with its advantages and disadvantages. Thanks Bill 20 views Bill, I’ve gathered that in the history of mathematized physics right up to the present, people invoke some sort of intellectual sense of when some implication of the mathematics characterizing some physical relations would be something not plausibly physical and should not be the mathematical characterization without mitigation (I'm thinking of infinities [and renormalization] and spacetimes rejected as not plausibly real.) This intellectual sense is fallible, as I imagine the history of resistance to characterizations of physical reality using complex numbers would show. (I keep in mind too that for physical outcomes in QM that wave function gets conjugated to yield real-number values for physical detectability of outcomes.) To be sure, the applicability of higher mathematics in physics, indeed by now the indispensability of it for further advance in physics, has seemed amazing. Additional applicability of complex numbers and thought about their mathematical character is put forth here by an applied mathematician who is an Objectivist. It seems to me too strong, however, to say that history has shown that there is no area of mathematics that does not eventually show physical application. I wonder if topological spaces that are not Hausdorff have found a job in characterizing something physical. Or if any mathematics that has its only proof by using Zorn’s lemma has found physical work. If not, we might say that it is a reasonable conjecture (not guaranteed), based on history, to suppose that there is some physical applicability of those things that we simply have not discovered so far. An additional tie between physics and mathematics is the history of how much mathematics has been invented/discovered on account of some specific need(s) for it in mathematical characterization of some physical realm. It seems to me that all the amazing ties of mathematics to physics, and to engineering, support the idea that mathematics is grounded, or at least partly grounded, in physical reality. However, I think there are other aspects of a grounding account that need to get specified in order make a dispositive case that mathematics is grounded ultimately in physical reality. We need a plausible specification of what sorts of things perceived in the physical world are mathematical sorts of things, we need a specification of our means of such perceptions and how it differs from the sort of perception that gets us started towards physics,* and we need a specification of how those different sorts of perceptual starts are joined to the different sorts of method we use in discovering higher mathematics and in making scientific discovery of more and more of physical reality. *My perceptual discernment that in the case of a music staff and in the case of the fingers of my right hand the number of spaces between the staff lines and between my fingers is one less than the number of lines or number of fingers is a different sort of discernment than the perceptual discernment that keeping a tight grip with both hands is a good idea for safety when using an axe or baseball bat. And perceptually discerning that the number of spaces between longitude lines on the globe in the office equals the number of those lines seems quite a different sort of perceptual discernment than discerning that, having removed the globe from its stand, it is not a good ball for dribbling. Edited February 28 by Boydstun Bill Hobba 1 Quote Link to comment Share on other sites More sharing options...

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Boydstun Posted March 20 Report Share Posted March 20 "For years, it was generally accepted that real quantum theory was experimentally indistinguishable from complex quantum theory. In other words, in quantum theory, complex numbers would only be convenient, but not necessary, to make sense of quantum experiments. Next we prove this conclusion wrong." (12/15/21). In the April 2023 issue of Scientific American, there is an article by the researchers getting the result that the use of complex numbers in standard quantum theory is not really just a convenience, but that some results of standard quantum theory cannot be also obtained in any alternative formulation using only real numbers. In other words, if I'm getting this right, characterization of quantum mechanics using complex numbers (or hypercomplex numbers?), is the only correct mathematical characterization of the physics. That is, purely real-number characterization—however complicated—of quantum mechanics is inadequate as a characterization of the physics. Quote Link to comment Share on other sites More sharing options...

StrictlyLogical Posted March 20 Report Share Posted March 20 (edited) 12 hours ago, Boydstun said: In other words, if I'm getting this right, characterization of quantum mechanics using complex numbers (or hypercomplex numbers?), is the only correct mathematical characterization of the physics. That is, purely real-number characterization—however complicated—of quantum mechanics is inadequate as a characterization of the physics. That understanding would be incorrect. Particularly that last sentence. Anything characterizable in complex numbers must be characterizable in real numbers, because every complex number is characterizable in real numbers. As for things, any complex attribute of a thing can be represented by two non-complex attributes associated with the thing. In fact the necessity of characterizing an attribute as (specifically) complex is nothing more and nothing less than having to use more than one (specifically two) real number to characterize that attribute. Every operator and mathematical calculation in the complex plane deals simultaneously with two real quantities, phase and magnitude or alternatively real and imaginary parts. A complex number IS two things, and of a necessity is b0th reducible analytically into those two things and cannot be constituted by less than those two things. It can never be a "simple" number, after all it is a complex number, and it has two absolutely independent components, and cannot be thought of as having any less than two components, and therefore it IS two things. So what is the error? Reification. The fact that an attribute (in a particular framework or theory of reality), to reflect reality has two things associated with it, and that the operators must take into account both, say phase and magnitude, means that a thing has a two-attribute attribute. The reification is an erroneous identification of this two attribute attribute WITH the abstractions we use to work with them... it comes from the way we understand and express and work with this two attribute attribute, namely with complex numbers. Moreover, the simplicity with which we deal with the calculations of the two attribute attribute as if it were one attribute (because they always go together)... every phase must have a magnitude... leads one to believe the formalism as expressed is the only way to express that formalism. That when you change the expressions one has changed the formalism... and that "QM based on real numbers" means somehow trying to use real numbers in a framework built for using complex numbers... The idea of a hypothetical "real quantum state" is nonsensical. Why? Because the referent of the modifier term "real" is purely mathematical or abstract, and the referent of the term "quantum state" is supposed to be an entity of reality. This should be a huge clue to how the authors are thinking...or how they are not being careful about what they are talking about, i.e. what refers to abstractions and what refers to reality. The foundation of QM on the idea of representing reality as states in a vector space, and whose inner product corresponds with probabilities of outcomes. We assign states in the same direction when probability is 1 and assign states as orthogonal vectors when probabilities are 0. We have operators to rotate those vectors modeled on causation and interaction. A nice little game no? It turns out that correspondence between these vectors we have concocted to real world outcomes, requires the use of complex coefficients and operators... but what has that done to the formalism? All that has cone is doubled the degrees of freedom. Sure one could not write QM in the standard formulation, the standard way with real numbers, the correspondence between it and reality requires complex numbers but that does not mean one could not rewrite the entire thing, vector spaces and all using real numbers. Edited March 20 by StrictlyLogical Quote Link to comment Share on other sites More sharing options...

tadmjones Posted March 20 Report Share Posted March 20 As a non mathematical layperson, it 'feels' like all the maths can describe how much and illustrate a lot of the 'how' precisely by delineating the 'much' , but can not answer any 'why' and especially when 'why' s are not appropriate in a query. StrictlyLogical 1 Quote Link to comment Share on other sites More sharing options...

Doug Morris Posted March 20 Report Share Posted March 20 Sure, we could rewrite the whole thing using only real numbers, but how much additional complication would that create? The more additional complication, the more Occam's razor would say that is not the way to go. Quote Link to comment Share on other sites More sharing options...

StrictlyLogical Posted March 20 Report Share Posted March 20 33 minutes ago, Doug Morris said: Sure, we could rewrite the whole thing using only real numbers, but how much additional complication would that create? The more additional complication, the more Occam's razor would say that is not the way to go. No one is proposing that, indeed it would be unnecessary. The potential or purported import of the paper is its implication that *which particular kinds* of abstractions we can choose to use in the context, are somehow actually limited. Quote Link to comment Share on other sites More sharing options...

Boydstun Posted March 20 Report Share Posted March 20 (edited) Quote In its Hilbert space formulation, quantum theory is defined in terms of the following postulates^{5,6}. (1) For every physical system S, there corresponds a Hilbert space ℋ_{S} and its state is represented by a normalized vector ϕ in ℋ_{S}, that is, 〈φ|φ〉=1. (2) A measurement Π in S corresponds to an ensemble {Π_{r}}_{r} of projection operators, indexed by the measurement result r and acting on ℋ_{S}, with ∑𝑟Π𝑟=𝕀𝑆. (3) Born rule: if we measure Π when system S is in state ϕ, the probability of obtaining result r is given by Pr(𝑟)=〈φ|Π𝑟|φ〉. (4) The Hilbert space ℋ_{ST} corresponding to the composition of two systems S and T is ℋ_{S} ⊗ ℋ_{T}. The operators used to describe measurements or transformations in system S act trivially on ℋ_{T} and vice versa. Similarly, the state representing two independent preparations of the two systems is the tensor product of the two preparations. This last postulate has a key role in our discussions: we remark that it even holds beyond quantum theory, specifically for space-like separated systems in some axiomatizations of quantum field theory^{7,8,9,10} . As originally introduced by Dirac and von Neumann^{1,2}, the Hilbert spaces ℋ_{S} in postulate (1) are traditionally taken to be complex. We call the resulting postulate (1_{¢}). The theory specified by postulates (1_{¢}) and (2)–(4) is the standard formulation of quantum theory in terms of complex Hilbert spaces and tensor products. For brevity, we will refer to it simply as ‘complex quantum theory’. Contrary to classical physics, complex numbers (in particular, complex Hilbert spaces) are thus an essential element of the very definition of complex quantum theory. Owing to the controversy surrounding their irruption in mathematics and their almost total absence in classical physics, the occurrence of complex numbers in quantum theory worried some of its founders, for whom a formulation in terms of real operators seemed much more natural (“What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. Ψ is surely fundamentally a real function.” (Letter from Schrödinger to Lorentz, 6 June 1926; ref. ^{3})). This is precisely the question we address in this work: whether complex numbers can be replaced by real numbers in the Hilbert space formulation of quantum theory without limiting its predictions. The resulting ‘real quantum theory’, which has appeared in the literature under various names^{11,12}, obeys the same postulates (2)–(4) but assumes real Hilbert spaces ℋ_{S} in postulate (1), a modified postulate that we denote by (1ℝ). If real quantum theory led to the same predictions as complex quantum theory, then complex numbers would just be, as in classical physics, a convenient tool to simplify computations but not an essential part of the theory. However, we show that this is not the case: the measurement statistics generated in certain finite-dimensional quantum experiments involving causally independent measurements and state preparations do not admit a real quantum representation, even if we allow the corresponding real Hilbert spaces to be infinite dimensional. Our main result applies to the standard Hilbert space formulation of quantum theory, through axioms (1)–(4). It is noted, though, that there are alternative formulations able to recover the predictions of complex quantum theory, for example, in terms of path integrals^{13}, ordinary probabilities^{14}, Wigner functions^{15} or Bohmian mechanics^{16}. For some formulations, for example, refs. ^{17,18}, real vectors and real operators play the role of physical states and physical measurements respectively, but the Hilbert space of a composed system is not a tensor product. Although we briefly discuss some of these formulations in, we do not consider them here because they all violate at least one of the postulates (1ℝ) and (2)–(4). Our results imply that this violation is in fact necessary for any such model. Edited March 20 by Boydstun Quote Link to comment Share on other sites More sharing options...

Boydstun Posted March 20 Report Share Posted March 20 (edited) 6 hours ago, StrictlyLogical said: That understanding would be incorrect. Particularly that last sentence. Anything characterizable in complex numbers must be characterizable in real numbers, because every complex number is characterizable in real numbers. . . . So mathematically, complex analysis is really equivalent to real analysis? A complex number consists of a real number plus an imaginary number, where the imaginary number has a real coefficient. But using the real number 3 as a coefficient in 3i does not turn 3i into a real number does it? Edited March 20 by Boydstun Quote Link to comment Share on other sites More sharing options...

StrictlyLogical Posted March 20 Report Share Posted March 20 (edited) 30 minutes ago, Boydstun said: So mathematically, complex analysis is really equivalent to real analysis? A complex number consists of a real number plus an imaginary number, where the imaginary number has a real coefficient. But using the real number 3 as a coefficient in 3i does not turn 3i into a real number does it? "Analysis" is a complicated formal construct. That which one builds to surround and is supported by real numbers is not the same as that which one builds to surround and is supported by complex numbers. So no, complex analysis is not "really equivalent" to real analysis. These are two different games played in two different arenas... we can and did make them so. Now the idea of analysis is like the idea of a mathematical expression, how it is used or what symbols are there may be different but what it represents or refers to, is not the same as the expression used. The identity operator helps us understand that although the expressions are not the same what they refer to are one in the same, an identity. So I do not discount the possibility that some form of complicated real number based formalism, using Euler relations etc. cannot create something like an answer, i.e. refer to some quantity, which complex analysis also refers to. When used as a coefficient, a real number can be interpreted as a kind of scaling or "quantity of" operator, so taking i, as operated on by 3 you have three of them, so no 3 times i does not remove the i. i however is like a 90 degree rotation operator in the 2d plane we use to arrange our complicated useful contrivances we call complex numbers. EDIT: The last paragraph seems very important to limit and understand the import of their findings. Edited March 20 by StrictlyLogical Quote Link to comment Share on other sites More sharing options...

StrictlyLogical Posted March 20 Report Share Posted March 20 2 hours ago, StrictlyLogical said: "Analysis" is a complicated formal construct. That which one builds to surround and is supported by real numbers is not the same as that which one builds to surround and is supported by complex numbers. So no, complex analysis is not "really equivalent" to real analysis. These are two different games played in two different arenas... we can and did make them so. Now the idea of analysis is like the idea of a mathematical expression, how it is used or what symbols are there may be different but what it represents or refers to, is not the same as the expression used. The identity operator helps us understand that although the expressions are not the same what they refer to are one in the same, an identity. So I do not discount the possibility that some form of complicated real number based formalism, using Euler relations etc. cannot create something like an answer, i.e. refer to some quantity, which complex analysis also refers to. When used as a coefficient, a real number can be interpreted as a kind of scaling or "quantity of" operator, so taking i, as operated on by 3 you have three of them, so no 3 times i does not remove the i. i however is like a 90 degree rotation operator in the 2d plane we use to arrange our complicated useful contrivances we call complex numbers. EDIT: The last paragraph seems very important to limit and understand the import of their findings. Real numbers characterize and are abstraction we find useful for quantifying things. Imaginary numbers, as directly as I can relate them to things, characterize and are abstractions we find useful for (some specific) relationships between things. Quote Link to comment Share on other sites More sharing options...

InfraBeat Posted March 21 Report Share Posted March 21 (edited) 5 hours ago, Boydstun said: So mathematically, complex analysis is really equivalent to real analysis? No. Real analysis is the study of the real number system, which is a complete ordered field. Complex analysis is the study of the complex number system, which is a field but not an ordered field. (As characterizations of those areas of studies, those are gross simplifications, but they are a start in noting the difference.) Of course, the theorems regarding both are axiomatized by set theory. 5 hours ago, Boydstun said: using the real number 3 as a coefficient in 3i does not turn 3i into a real number does it? No, it does not. My remarks in this post are about the mathematics. I'm not commenting in this post on philosophy. I have not studied complex analysis very much, but the most basic definitions and theorems are simple. A very good treatment is found, for example, in Rudin's 'Principles Of Mathematical Analysis', which is one of the classic textbooks. A complex number is, by definition, an ordered pair of real numbers. In other words: Definition: x is a complex number if and only if x is an ordered pair of real numbers. Notation: < > for order pair + for real addition (we'll also define (complex)+ for complex addition) * for real multiplication (we'll also define (complex)* for complex multiplication) - for real subtraction and for the 'negative' symbol Definitions and theorems: df x is a complex number if and only if there exist real numbers a and b such that x = <a b> th <a b> = <c d> if and only if (a=c & b=d) df <a b> (complex)+ <c d> = <a+c b+d> To simplify the notation, we can drop '(complex)' and use just '+' for complex addition, so that both real and complex addition use '+' though officially they are different operations. df <a b> (complex)* <c d> = <(a*c - b*d) (a*d + b*c)> To simplify the notation, we can drop '(complex)' and use just '*' for complex multipication, so that both real and complex mulitipication use '*' though officially they are different operations. df i = <0 1> th i*i = <-1 0> th <a b> = <a 0> + <0 b> th <a b> = <a 0> + (<b 0> * i) (so a is called the 'real part' and b is called the 'imaginary part') Edited March 21 by InfraBeat Quote Link to comment Share on other sites More sharing options...

StrictlyLogical Posted March 21 Report Share Posted March 21 (edited) 2 hours ago, StrictlyLogical said: Real numbers characterize and are abstraction we find useful for quantifying things. Imaginary numbers, as directly as I can relate them to things, characterize and are abstractions we find useful for (some specific) relationships between things. and that includes relationships between "portions" of the same thing, so to speak. From what I recall in QM probabilities there is never any "absolute" phase, but relative phase is ubiquitous and the foundation for determining interference, and plays much in what the results of the inner product look like (i.e. probabilities). So real numbers we use to quantifying things but "i" and the like are useful for relating things, in particular phase differences. Edited March 21 by StrictlyLogical Quote Link to comment Share on other sites More sharing options...

Boydstun Posted March 21 Report Share Posted March 21 12 hours ago, StrictlyLogical said: and that includes relationships between "portions" of the same thing, so to speak. From what I recall in QM probabilities there is never any "absolute" phase, but relative phase is ubiquitous and the foundation for determining interference, and plays much in what the results of the inner product look like (i.e. probabilities). So real numbers we use to quantifying things but "i" and the like are useful for relating things, in particular phase differences. Interesting. Newton maintained that numbers as in 7 feet are really ratios, which would be a relation. I've not had time to chase it down, but do you recall an imaginary term that falls out of a classical E-M radiation equation which turns out to correspond to quantity of the radiation absorbed in a medium? Quote Link to comment Share on other sites More sharing options...

Boydstun Posted March 21 Report Share Posted March 21 (edited) InfraBeat, Welcome to Objectivism Online! Thank you for the helpful informative post! Edited March 21 by Boydstun Quote Link to comment Share on other sites More sharing options...

Boydstun Posted March 21 Report Share Posted March 21 Real and Complex Analysis (especially over at 9:00 forward) Quote Link to comment Share on other sites More sharing options...

StrictlyLogical Posted March 21 Report Share Posted March 21 59 minutes ago, Boydstun said: Interesting. Newton maintained that numbers as in 7 feet are really ratios, which would be a relation. I've not had time to chase it down, but do you recall an imaginary term that falls out of a classical E-M radiation equation which turns out to correspond to quantity of the radiation absorbed in a medium? The point was not that real numbers cannot be used for relations... that is obvious and you know it. I'll restrain my snippiness to that comment. The point is that in reality you do not literally have "i" quantity of some entity, as such. "i" can be used to help calculate quantities or to represent/characterize (through its odd mathematical qualities) relationships between real things. Do not get me wrong, I am not guilty of reification of real numbers either, they are abstractions we use to directly quantify things in reality, but they are no more real than "i"... merely the directness of their application, and their function differs. Quote Link to comment Share on other sites More sharing options...

StrictlyLogical Posted March 21 Report Share Posted March 21 1 hour ago, Boydstun said: I've not had time to chase it down, but do you recall an imaginary term that falls out of a classical E-M radiation equation which turns out to correspond to quantity of the radiation absorbed in a medium? Ah yes, this is the convention of using a complex refractive index in calculations to take into account absorption. Quite a convenient use of complex numbers, relating incident light to absorption of light in the material as a function of depth. Boydstun 1 Quote Link to comment Share on other sites More sharing options...

StrictlyLogical Posted March 21 Report Share Posted March 21 2 hours ago, Boydstun said: Real and Complex Analysis (especially over at 9:00 forward) Did you learn anything from this video? Quote Link to comment Share on other sites More sharing options...

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