If imaginary numbers are needed to describe reality, then isn’t materialism dead already?

_{Denyse O'Leary

December 22, 2021

Intelligent Design, Mathematics}

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The researchers stressed, however, that their experiment only rules out theories that forgo imaginary numbers if the reigning conventions of quantum mechanics are correct. Most scientists are very confident that this is the case, but this is an important caveat nonetheless.
The result suggests that the possible ways we can describe the universe with math are actually much more constrained than we might have thought, Renou said.
Ben Turner, “Imaginary numbers could be needed to describe reality, new studies find” at Live Science (December 21, 2021)

Comments

If Imaginary Numbers Are Needed To Describe Reality, Then Isn’t Materialism Dead Already?
Not from my perspective--I agree with JVL's reductio ad absurdum. That several systems of mathematics can be employed to model what we observe from reality is convenient, but not predictive. Plus, that mathematical models typically fail at some extreme and need to be replaced by more complex mathematics is also telling. For example, the inverse square law applied to gravity produces reasonably good results at our typical scales means it's pragmatic. That it fails at larger scales such as with the orbit of mercury in proximity of the sun means that it's not a fundamental reality. -QQuerius_{December 23, 2021
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^^^ Ditto ^^^bornagain77_{December 23, 2021
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Viola Lee:
there is nothing special about complex numbers: their usefulness and even necessity in describing the world (which was the actual point of the article in the OP) doesn’t add any special new element to to the argument that math disproves materialism.
So you keep saying but still cannot form a coherent argument for that.ET_{December 23, 2021
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JVL:
If you can point to a connection between ‘materialism’ and complex numbers and the number line then, by all means, show us.
There isn't any. That's the point. They work because materialism is false, and the universe was intelligently designed using mathematics. JVL:
Imaginary numbers work because the universe was designed?
Not what I said. Try again.
I just want to be clear here . . . the fact that some mathematicians came up with the idea of there being a square root of a negative number is an indication that the universe was designed?
They did not come up with the idea. They discovered the existing concept.ET_{December 23, 2021
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A general idea here, which BA expressed, is that all mathematics disproves materialism because math has an abstract reality. Leaving that point aside, there is nothing special about complex numbers: their usefulness and even necessity in describing the world (which was the actual point of the article in the OP) doesn't add any special new element to to the argument that math disproves materialism. That's in part why I (and Ram, I think) objected to News's headline. And complex numbers are really neat: one of my favorite topics to teach.Viola Lee_{December 23, 2021
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ET: They work because materialism is false, and the universe was intelligently designed using mathematics. Imaginary numbers work because the universe was designed? I just want to be clear here . . . the fact that some mathematicians came up with the idea of there being a square root of a negative number is an indication that the universe was designed? Is that what you think?JVL_{December 23, 2021
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ET: How do you know? If you can point to a connection between 'materialism' and complex numbers and the number line then, by all means, show us.JVL_{December 23, 2021
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Ram:
I do agree materialism is false, but the question of materialism has nothing to do with complex numbers any more than materialism has anything to do with the number line.
How do you know?ET_{December 23, 2021
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VL, I think the TL;DR version of what BA77 is arguing is that relationship of mathematics to what we call the physical world can no longer be characterized as descriptive; it can only be described as causal. If it were not causal, there is no reason to expect that everything we experience as the physical world should be constrained to that which is describable by mathematics - especially not if higher-dimensional mathematics are required in that description. BA77 can correct me if I've misunderstood him. There comes a point where the happy coincidence of apparent correspondence ends and causality becomes the only rational explanation. Mathematics appears to be governing, or causing, how physical reality works, not vice-versa. This goes back to the old saw about so-called "physical laws" and "forces," and materialists mistaking their descriptions for causes. Materialists make this conceptual error repeatedly. Gravity, for instance, is not a cause even though it is invoked as such; it is a description of behavior. No materialist can explain how the behavior is instantiated or why, under their view, it should exist at all. Natural laws and forces appear to be mathematically caused, the behavior of physical reality somehow constrained into following abstract, mathematical principles at every level. There is no material explanation for how the behavior of matter is governed/caused by abstract rules and principles. The only rational explanation I can see is that what we call physical reality is the product of a higher, real, abstract dimension that causes it.William J Murray_{December 23, 2021
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Viola Lee states, "Also, whether complex numbers are necessary to describe some phenomena has no bearing on whether materialism is true or not." Well actually, although VL is trying to pooh pooh this finding and say that there is "Nothing to see here. Please disperse.",,,
"Nothing to see here. Please disperse" https://tenor.com/view/leslie-nielsen-nothing-to-see-here-disperse-crowd-disperse-disaster-gif-12767181
,,, although VL is trying pooh pooh this finding, the fact of the matter is that it is not just complex numbers that question whether materialism can be true or not. The entire field of mathematics, ('real' numbers, complex 'imaginary' numbers, and classes, sets, functions, and etc.. etc..), all question whether materialism can be true or not. The irresolvable dilemma for materialists is that mathematics, ALL of mathematics, is profoundly immaterial in its foundational essence. As M. Anthony Mills explains, "The trouble is that numbers — along with other mathematical entities such as classes, sets, and functions — are indispensable for modern science. And yet — here’s the rub — these “abstract objects” are not material. Thus, one cannot take science as the only sure guide to reality and at the same time discount disbelief in all immaterial realities."
What Does It Mean to Say That Science & Religion Conflict? - M. Anthony Mills - April 16, 2018 Excerpt: In fact, more problematic for the materialist than the non-existence of persons is the existence of mathematics. Why? Although a committed materialist might be perfectly willing to accept that you do not really exist, he will have a harder time accepting that numbers do not exist. The trouble is that numbers — along with other mathematical entities such as classes, sets, and functions — are indispensable for modern science. And yet — here’s the rub — these “abstract objects” are not material. Thus, one cannot take science as the only sure guide to reality and at the same time discount disbelief in all immaterial realities. https://www.realclearreligion.org/articles/2018/04/16/what_does_it_mean_to_say_that_science_and_religion_conflict.html
And as Michael Egnor explains, "what is natural and physical about imaginary numbers, infinite series, irrational numbers, and the mathematics of more than three spatial dimensions? Mathematics is entirely about concepts, which have no precise instantiation in nature,,,"
Naturalism and Self-Refutation – Michael Egnor – January 31, 2018 Excerpt: Mathematics is certainly something we do. Is mathematics “included in the space-time continuum [with] basic elements … described by physics”?,,, What is the physics behind the Pythagorean theorem? After all, no actual triangle is perfect, and thus no actual triangle in nature has sides such that the Pythagorean theorem holds. There is no real triangle in which the sum of the squares of the sides exactly equals the square of the hypotenuse. That holds true for all of geometry. Geometry is about concepts, not about anything in the natural world or about anything that can be described by physics. What is the “physics” of the fact that the area of a circle is pi multiplied by the square of the radius? And of course what is natural and physical about imaginary numbers, infinite series, irrational numbers, and the mathematics of more than three spatial dimensions? Mathematics is entirely about concepts, which have no precise instantiation in nature,,, https://evolutionnews.org/2018/01/naturalism-and-self-refutation/
To clearly illustrate just how problematic mathematics is for atheistic materialists, both Eugene Wigner and Albert Einstein are on record as to regarding it as a 'miracle' that mathematics should even be applicable to the universe in the first place. Eugene Wigner, (after rightly calling into question the ability of natural selection to produce our 'reasoning power'), stated that, "It is difficult to avoid the impression that a miracle confronts us here,,, and "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.,,"
The Unreasonable Effectiveness of Mathematics in the Natural Sciences - Eugene Wigner - 1960 ? Excerpt: ,, The great mathematician fully, almost ruthlessly, exploits the domain of permissible reasoning and skirts the impermissible. That his recklessness does not lead him into a morass of contradictions is a miracle in itself: certainly it is hard to believe that our reasoning power was brought, by Darwin’s process of natural selection, to the perfection which it seems to possess.,,, It is difficult to avoid the impression that a miracle confronts us here, quite comparable in its striking nature to the miracle that the human mind can string a thousand arguments together without getting itself into contradictions, or to the two miracles of the existence of laws of nature and of the human mind's capacity to divine them.,,, The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
Likewise, Albert Einstein is also on record as to regarding the applicability of mathematics to the universe as a 'miracle'. Einstein even went so far as to chastise 'professional atheists' in the process of calling it a 'miracle'.
On the Rational Order of the World: a Letter to Maurice Solovine – Albert Einstein – March 30, 1952 Excerpt: “You find it strange that I consider the comprehensibility of the world (to the extent that we are authorized to speak of such a comprehensibility) as a miracle or as an eternal mystery. Well, a priori, one should expect a chaotic world, which cannot be grasped by the mind in any way .. the kind of order created by Newton’s theory of gravitation, for example, is wholly different. Even if a man proposes the axioms of the theory, the success of such a project presupposes a high degree of ordering of the objective world, and this could not be expected a priori. That is the ‘miracle’ which is constantly reinforced as our knowledge expands. There lies the weakness of positivists and professional atheists who are elated because they feel that they have not only successfully rid the world of gods but “bared the miracles.” -Albert Einstein http://inters.org/Einstein-Letter-Solovine
Thus. although Viola Lee may try to pooh pooh this present finding on 'imaginary' complex numbers as no big deal and having "no bearing on whether materialism is true or not", the fact of the matter is that atheistic materialism and the entire field of 'immaterial' mathematics are completely at odds with each other, even diametrically opposed to each other. The fact that 'higher dimensional', and 'imaginary', complex numbers are presently found to be irreducible in our quantum mechanical description of the universe just makes what was already a bad situation for atheistic materialists exponentially worse. You see, although all of the original theories of modern science, such as Newtonian Mechanics, were all based on 3-Dimensional Euclidean geometry, all of our present, and most accurate, theories in science, (relativity, quantum mechanics, quantum electrodynamics, etc..) are all based on 'higher dimensional mathematics, As Eugene Wigner explained, "We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity.,,, The two theories operate with different mathematical concepts - the four dimensional Riemann space and the infinite dimensional Hilbert space, respectively."
The Unreasonable Effectiveness of Mathematics in the Natural Sciences – Eugene Wigner – 1960 Excerpt: We now have, in physics, two theories of great power and interest: the theory of quantum phenomena and the theory of relativity.,,, The two theories operate with different mathematical concepts - the four dimensional Riemann space and the infinite dimensional Hilbert space, respectively. So far, the two theories could not be united, that is, no mathematical formulation exists to which both of these theories are approximations. All physicists believe that a union of the two theories is inherently possible and that we shall find it. Nevertheless, it is possible also to imagine that no union of the two theories can be found.,,, https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf?
Thus in conclusion, while atheistic materialists have no clue why the universe should even be 'miraculously' described by mathematics in the first place, the fact that 'higher dimensional' mathematics, instead of 3-Dimensional Euclidean mathematics, are used in all of our best theories in science makes what was already a bad situation for atheistic materialists exponentially worse. Simply put, the Christian Theist, since he holds that God created this universe from a higher heavenly dimension, is very comfortable with the fact, (and even 'expects'), that the universe would be described by higher dimensional mathematics. Whereas on the other hand, and once again, atheistic materialists simply have no clue why the universe should even be 'miraculously' described by mathematics in the first place. Of supplemental note: besides this present experiment confirming the 'reality' of complex 'imaginary' numbers, Near Death Experiencers also testify to the physical reality of a 'higher heavenly dimension' above this 3-Dimensional temporal realm:
December 2021 - Thus in conclusion Einstein himself may not have personally believed in life after death, (nor in a personal God), but Special Relativity itself contradicts Einstein and offers stunning confirmation that Near Death Testimonies are accurate ‘physical’ descriptions of what happens after death, i.e. going to a ‘higher timeless/eternal dimension’, i.e. heavenly dimension, that exists above this temporal realm. https://uncommondescent.com/intelligent-design/at-mind-matters-news-einstein-believed-in-spinozas-god-who-is-that-god/#comment-741908
Also of note:
December 2021 - And while most people, (who are at least semi-literate in science), are aware of the fact that all of our best theories in science are all based on ‘higher dimensional’ mathematics, (and are not based on 3-D Euclidean mathematics as Newton’s theory was), most people are unaware of the fact that life itself is based on ‘higher dimensional’ mathematics, and/or principles, and that life is not based on 3-D Euclidean mathematics, and/or principles, as is presupposed within Darwinian materialism. https://uncommondescent.com/intelligent-design/at-mind-matters-news-can-higher-dimensions-help-us-understand-biblical-miracles/#comment-742961
Quote and Verse:
“There are more things in heaven and earth, Horatio, Than are dreamt of in your (materialistic 3-D) philosophy.” – Hamlet (1.5.167-8), Hamlet to Horatio (paraphrased) Mark 8:37 Is anything worth more than your soul?
bornagain77_{December 23, 2021
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ET: Right. They work because materialism is false, and the universe was intelligently designed using mathematics. I do agree materialism is false, but the question of materialism has nothing to do with complex numbers any more than materialism has anything to do with the number line. Both are the same level of "mystery." --Ramram_{December 22, 2021
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This is a bizarre topic/question. The Schrödinger equation has been shown to be _derivable_. And it is derivable from the most basic assumptions: 1. That observations are real numbers. 2. That there exist both space and time symmetries in the laws of nature. See https://henry.pha.jhu.edu/quantum.html. The derivation ends up needing complex numbers. Btw, there is nothing "imaginary" about complex numbers. Complex numbers are not "a square root of -1". Complex numbers are the ordered pairs of real numbers, with certain rules defined upon those pairs. One of the consequences of those rules is that there exists such a pair of real numbers (0, 1) that (0, 1) * (0, 1) = -1, where "*" is the specific multiplication rule defined for these types of pairs.Eugene_{December 22, 2021
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Ram:
All physics, modern electronics design, etc, rely on complex numbers, and we know exactly why they work. No mystery. Period.
Right. They work because materialism is false, and the universe was intelligently designed using mathematics.ET_{December 22, 2021
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Viola Lee:
Also, whether complex numbers are necessary to describe some phenomena has no bearing on whether materialism is true or not.
How do you know?ET_{December 22, 2021
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All physics, modern electronics design, etc, rely on complex numbers, and we know exactly why they work. No mystery. Period. Substitute the word imaginary/complex with "lateral" and watch a Youtube or two about the subject. There is not one darn thing mysterious about any of this. For anyone interested, get to know what phase relationships means. Bad OP, Denise. Sorry (but not sorry.) --Ramram_{December 22, 2021
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The "Is" in the headline makes no sense. Also, whether complex numbers are necessary to describe some phenomena has no bearing on whether materialism is true or not. Sometimes I'm just baffled about the extraneous implications that News puts in the headlines. Often the stories she highlights are interesting, but then she has to add some extra snark that is just not relevant.Viola Lee_{December 22, 2021
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As the "The Mathematics Of Higher Dimensionality" video that I referenced made clear, Gauss’s work on complex numbers, like the square root of negative one, extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that would be impossible with only real numbers. And this ‘higher dimensional number line’, particularly this understanding gained for the ‘higher dimensionality’ of the square root of negative one (i), is essential for understanding the ‘wave packet’ in quantum mechanics prior to measurement. (as the present experiment in the OP has now demonstrated).
Why do you need imaginary numbers (the square root of negative one) to describe Quantum Mechanics? “Quantum theory needs existence of an x such that x^2= -1. The reason for this is that orthogonal function spaces, of dimension greater than 2, cannot exist otherwise. In fact the only place where i (the square root of negative one) is needed is in the wave packet prior to measurement. Even the Canonical Commutation Relation doesn’t need it. And nor do the eigenvalue equations. In those, any general scalar will do. But in the wave packet, you need an i.” – Steve Faulkner – Philosophy of Science, Logic, Epistemology https://www.researchgate.net/post/Why_do_you_need_imaginary_numbers_to_describe_Quantum_Mechanics2
Also of related note to quantum mechanics, and the 'higher dimensional' mathematics used to describe quantum mechanics, is the 'infinite dimensional' Hilbert space. If the 'imaginary' complex number of the square root of negative one ruffled naturalistic presuppositions by being found to 'real' instead of being merely imaginary, then the infinite dimensional Hilbert space that is used within quantum mechanics ought to really give them fits. As the following article states, "The role of Hilbert spaces in quantum mechanics, then, is much more profound than the descriptive role of a single concept. An entire formalism-the Hilbert space formalism-is matched with nature. Information about nature is being “read off” the details of the formalism. (Imagine reading off details about elementary particles from the rules of chess-castling. en passant-a la Lewis Carro;; in Through the Looking Glass.) No physicist today understands why this is possible."
The Applicability of Mathematics as a Philosophical Problem – Mark Steiner – (page 44) Excerpt: Let us now recapitulate: beginning with the concept of a Hilbert space, a certain kind of (usually infinite-dimensional) vector space, and the formal requirement that a unit vector on the space represents all possible information can be gleaned. First, the space cannot be a real vector space; the usual formalism is, therefore, based on a complex Hilbert space. With this formalism the Heisenberg uncertainty principle follows directly. So does the quantization of angular momentum, including the so called “space quantization”. So does the prediction that “electron spin” cannot be due to spatial rotation. And so do the selection rules for the spectrum of hydrogen, based on the “nonphysical” concept of parity. The role of Hilbert spaces in quantum mechanics, then, is much more profound than the descriptive role of a single concept. An entire formalism-the Hilbert space formalism-is matched with nature. Information about nature is being “read off” the details of the formalism. (Imagine reading off details about elementary particles from the rules of chess-castling. en passant-a la Lewis Carro;; in Through the Looking Glass.) No physicist today understands why this is possible.. https://books.google.com/books?id=GKBwKCma1HsC&pg=PA44
Of course much more could be said about the fact that higher dimensional mathematics are essential, (even 'real'), for all of our best theories in science, (i.e. quantum mechanics, Relativity, etc..),
Euler’s Formula Proof & The Beauty of Complex Numbers - Oct. 2021 The use of complex numbers comes from an assumption that seemed useless at the time. Why work with a set of numbers that aren’t real? Since the discovery of complex numbers, they are at least somewhat real. The applications of complex numbers in the real world are endless, control theory, signal analysis, relativity, and fluid dynamics all use complex numbers. Not to mention the advent of the complex plane, https://medium.com/intuition/eulers-formula-proof-the-beauty-of-complex-numbers-a4c8eb9f10d8
,,, although much more could be said, suffice it for now to say that the fact that higher dimensional 'imaginary' numbers are now found to 'really exist', (Hossenfelder), is rather stunning confirmation of the contention, via Neoplatonic philosophy and Augustinian theology, that the (higher dimensional) mathematics that describe this universe really are "God’s thoughts"
Keep It Simple - - by Edward Feser - April 2020 Excerpt: Mathematics appears to describe a realm of entities with quasi-divine attributes. The series of natural numbers is infinite. That one and one equal two and two and two equal four could not have been otherwise. Such mathematical truths never begin being true or cease being true; they hold eternally and immutably. The lines, planes, and figures studied by the geometer have a kind of perfection that the objects of our experience lack. Mathematical objects seem immaterial and known by pure reason rather than through the senses. Given the centrality of mathematics to scientific explanation, it seems in some way to be a cause of the natural world and its order. How can the mathematical realm be so apparently godlike? The traditional answer, originating in Neoplatonic philosophy and Augustinian theology, is that our knowledge of the mathematical realm is precisely knowledge, albeit inchoate, of the divine mind. Mathematical truths exhibit infinity, necessity, eternity, immutability, perfection, and immateriality because they are God’s thoughts, and they have such explanatory power in scientific theorizing because they are part of the blueprint implemented by God in creating the world. For some thinkers in this tradition, mathematics thus provides the starting point for an argument for the existence of God qua supreme intellect. https://www.firstthings.com/article/2020/04/keep-it-simple
Verse:
Psalm 115:2-3 Why should the nations say, “Where is their God?” Our God is in heaven; He does as He pleases.
bornagain77_{December 22, 2021
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Back in March, when Sabine Hossenfelder first announced this proposed experiment on whether 'imaginary' complex numbers were 'real' or not, I stated, "First off, let me state that I firmly believe that the proposed experiment will be successful. Quantum Mechanics has a very long history of shattering ‘naturalistic’ assumptions about locality and realism."
"First off, let me state that I firmly believe that the proposed experiment will be successful. Quantum Mechanics has a very long history of shattering ‘naturalistic’ assumptions about locality and realism." https://uncommondescent.com/intelligent-design/sabine-hossenfelder-asks-do-complex-numbers-exist/#comment-725634
Moreover, when Hossenfelder announced this proposed experiment, she herself stated, if the experiment was successful then, "It would then be fair to say that complex (imaginary) numbers exist."
",,, assuming their result holds up, this means if the experiment which they propose finds the specific entanglement predicted by complex quantum mechanics, then you know you can’t describe observations with real numbers. It would then be fair to say that complex numbers exist. So, this is why it’s cool. They’ve figured out a way to experimentally test if complex numbers exist!" - Sabine Hossenfelder https://uncommondescent.com/intelligent-design/sabine-hossenfelder-asks-do-complex-numbers-exist/
Moreover, even though Schrödinger himself found the use of complex (imaginary) numbers in quantum theory to be "disquieting",,,
Imaginary numbers could be needed to describe reality, new studies find - Dec. 21, 2021 Excerpt: In fact, even the founders of quantum mechanics themselves thought that the implications of having complex numbers in their equations was disquieting. In a letter to his friend Hendrik Lorentz, physicist Erwin Schrödinger — the first person to introduce complex numbers into quantum theory, with his quantum wave function (?) — wrote, "What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. ? (the wave function) is surely fundamentally a real function." https://www.livescience.com/imaginary-numbers-needed-to-describe-reality
,,, even though Schrödinger himself found the use of complex (imaginary) numbers in quantum theory to be "disquieting", the argument over whether complex numbers are imaginary or real goes back a long way. Long before quantum theory came along and Schrödinger was upset by that they were "fundamentally a real function." Carl Friedrich Gauss was the mathematician who first clearly explained the higher ‘dimensional extension’ of complex numbers over and above the real number line,,,
The Mathematics Of Higher Dimensionality – Gauss & Riemann – video https://www.youtube.com/watch?v=mxy3JhPRlV0
Yet prior to Gauss's work, (and of all people), “Descartes had rejected complex roots and coined the derogatory term “imaginary” to describe the square root of negative one,”
Descartes coined the term imaginary: “For any equation one can imagine as many roots [as its degree would suggest], but in many cases no quantity exists which corresponds to what one imagines.” https://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf
Yet both Gauss and Leibniz rejected Descartes's claim that complex numbers were merely ‘imaginary’. Gauss argued that complex (imaginary) numbers were just as real as 'real' numbers are.
Carl Friedrich Gauss (1777-1855) Gauss introduced the term complex number “If this subjet has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1 and ??1, instead of being called positive, negative and imaginary (or worse still, impossible) unity, been given the names say,of direct, inverse and lateral unity, there would hardly have been any scope for such obscurity.” https://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf
And Leibniz went so far as to state that ”The divine spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and non-being, which we call the imaginary root of negative unity.”
“The Divine Spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and not-being, which we call the imaginary root of negative unity.” — - Gottfried Leibniz https://www2.math.upenn.edu/~deturck/isp/april-22.pdf
And the following article observes that "In the language of Plato’s allegory of the cave, complex numbers represent “forms” from a higher dimension casting “shadows” on the real number line."
Complex Magnitudes Excerpt: Descartes had rejected complex roots and coined the derogatory term “imaginary” to describe the square root of negative one, , but Leibniz thought that “The divine spirit found a sublime outlet in that wonder of analysis, that portent of the ideal world, that amphibian between being and non-being, which we call the imaginary root of negative unity.” Gauss invented the “complex plane” (shown below) to represent these quantities. He suggested that complex magnitudes be called “lateral” instead of “imaginary” magnitudes since they represent a dimensional extension of the continuum. Gauss also proposed that complex magnitudes be awarded “full civil rights.” In the language of Plato’s allegory of the cave, complex numbers represent “forms” from a higher dimension casting “shadows” on the real number line. - per Kepler's discovery
bornagain77_{December 22, 2021
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We don't need any numbers to describe reality in a useful way. Carpenters and masons and astronomers were using diagrams and templates long before number symbols came along. Graphs are better than numbers.polistra_{December 22, 2021
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