Intelligent Design Mathematics

Sabine Hossenfelder asks, Do complex numbers exist?

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

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

The question which they look at in the new paper is then whether there are ways to entangle particles in the normal, complex quantum mechanics that you cannot build up from particles that are described entirely by real valued functions. Previous calculation showed that this could always be done if the particles came from a single source. But in the new paper they look at particles from two independent sources, and claim that there are cases which you cannot reproduce with real numbers only. They also propose a way to experimentally measure this specific entanglement.

I have to warn you that this paper has not yet been peer reviewed, so maybe someone finds a flaw in their proof. But assuming their result holds up, this means if the experiment which they propose finds the specific entanglement predicted by complex quantum mechanics, then you know you can’t describe observations with real numbers. It would then be fair to say that complex numbers exist. So, this is why it’s cool. They’ve figured out a way to experimentally test if complex numbers exist!

Sabine Hossenfelder, “Do Complex Numbers Exist?” at BackRe(Action)

But, she warns, if they are right,

This conclusion only applies if you want the purely real-valued theory to work the same way as normal quantum mechanics. If you are willing to alter quantum mechanics, so that it becomes even more non-local than it already is, then you can still create the necessary entanglement with real valued numbers.

Sabine Hossenfelder, “Do Complex Numbers Exist?” at BackRe(Action)

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

72 Replies to “Sabine Hossenfelder asks, Do complex numbers exist?

  1. 1
    Viola Lee says:

    I didn’t think this post made sense. Then I looked at the paper and see that she says, ” it’s only if a mathematical structure is actually necessary to describe observations that we can say they “exist” in a scientifically meaningful way.”

    That is certainly different than what the question of whether complex numbers exist usually means. I wonder if what she says is a widely accepted use of ““exist” in a scientifically meaningful way,” or whether this is an idiosyncratic meaning of her own.

  2. 2
    Concealed Citizen says:

    In electrical engineering, for example, complex numbers are used in dealing with with vector rotation and phase relationships. There is absolutely nothing mysterious about them at all.

  3. 3
    polistra says:

    This is only mysterious if you’re starting from the Platonic nonsense that mathematics exists outside the human mind. If you start from reality, complex numbers are just a highly useful notation.

    It’s like asking if plural endings on nouns exist. Suffixes and Inflections are not an intrinsic part of the universe, but they’re a highly useful way of symbolizing a type of numerical relationship.

  4. 4
    bornagain77 says:

    Of related note,

    The argument over whether complex numbers are real or imaginary goes back a long way.

    Carl Friedrich Gauss was the mathematician who first explained the ‘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

    In response to complex numbers, “Descartes had rejected complex roots and coined the derogatory term “imaginary” to describe the square root of negative one,”

    Yet both Leibniz and Gauss rejected the notion that complex numbers were ‘imaginary’.

    Leibniz stated 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.”

    And Gauss went so far as to say that complex magnitudes should be awarded “full civil rights.”

    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.
    http://www.keplersdiscovery.com/ComplexNum.html

    Of note, both Leibniz and Gauss were devout Christians.

    Christian Mathematicians – Leibniz – January 30, 2012
    By Steve Bishop
    Excerpt: Leibniz believed in the God of Christianity and he also had an extraordinarily high esteem for reason and its capabilities. … In 1709 he attempted to improve the ontological argument for God and in 1710 His Theodicy, or “Vindication of the Justice of God“, was published.
    https://godandmath.com/2012/01/30/christian-mathematicians-leibniz/

    Carl Friedrich Gauss
    Excerpt: Carl Friedrich Gauss was a devout Christian who supported monarchy and opposed Napoleon, whom he saw as an outgrowth of the revolution.,,,
    http://www.conservapedia.com/Carl_Friedrich_Gauss

    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. 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:

    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

    In Hossenfelder’s article the argument appears to be that we presently don’t really need complex numbers in order to do our calculations for quantum mechanics,

    However, you can of course take the wave-function and this equation apart into a real and an imaginary part. Indeed, one often does that, if one solves the equation numerically. And I remind you, that both the real and the imaginary part of a complex number are real numbers. Now, if we calculate a prediction for a measurement outcome in quantum mechanics, then that measurement outcome will also always be a real number. So, it looks like you can get rid of the complex numbers in quantum mechanics, by splitting the equation into a real and imaginary part, and that’ll never make a difference for the result of the calculation.

    ,,, And the argument in her article continues with the fact that this proposed experiment will prove that complex numbers are not just a ‘mathematical convenience’ but are irreducibly necessary for our calculations in quantum mechanics, and are therefore to be considered ‘real’.

    in the new paper they look at particles from two independent sources, and claim that there are cases which you cannot reproduce with real numbers only.,,,
    ,,, 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!
    ,,, If you, on the other hand, are in the camp of people who think there’s something wrong with quantum mechanics because it uses complex numbers that we can never measure, then you are now caught between a rock and a hard place. Either embrace complex numbers, or accept that nature is even more non-local than quantum mechanics.

    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.

    Secondly, let me bluntly state the fact that Atheistic Naturalists have no clue why mathematics, (‘real’ numbers or otherwise), should even be able to describe the universe in the first place.

    Both Wigner and Einstein are on record as to regarding the applicability of mathematics to the universe to be a ‘miracle’. Einstein even went so far as to chastise ‘professional atheists’ in the process of calling it a ‘miracle’:

    The Unreasonable Effectiveness of Mathematics in the Natural Sciences – Eugene Wigner – 1960
    Excerpt: ,,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.....igner.html

    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

  5. 5
    bornagain77 says:

    Moreover, regardless of whatever the Atheistic Naturalist’s definition of ‘real’ may be in regards to any mathematics that may describe this universe, Godel, with his incompleteness theorem, has proven that mathematics is necessarily incomplete, and therefore mathematics cannot function as a ‘God substitute’ (as Atheistic Naturalists are apparently falsely presupposing in their arguments,,,, falsely presupposing in their arguments whether or not complex numbers are found to be ‘real’ or “imaginary’.)

    THE GOD OF THE MATHEMATICIANS – DAVID P. GOLDMAN – August 2010
    Excerpt: we cannot construct an ontology that makes God dispensable. Secularists can dismiss this as a mere exercise within predefined rules of the game of mathematical logic, but that is sour grapes, for it was the secular side that hoped to substitute logic for God in the first place. Gödel’s critique of the continuum hypothesis has the same implication as his incompleteness theorems: Mathematics never will create the sort of closed system that sorts reality into neat boxes.
    http://www.firstthings.com/art.....ematicians

    Taking God Out of the Equation – Biblical Worldview – by Ron Tagliapietra – January 1, 2012
    Excerpt: Kurt Gödel (1906–1978) proved that no logical systems (if they include the counting numbers) can have all three of the following properties.
    1. Validity … all conclusions are reached by valid reasoning.
    2. Consistency … no conclusions contradict any other conclusions.
    3. Completeness … all statements made in the system are either true or false.
    The details filled a book, but the basic concept was simple and elegant. He (Godel) summed it up this way: “Anything you can draw a circle around cannot explain itself without referring to something outside the circle—something you have to assume but cannot prove.” For this reason, his proof is also called the Incompleteness Theorem.
    Kurt Gödel had dropped a bomb on the foundations of mathematics. Math could not play the role of God as infinite and autonomous. It was shocking, though, that logic could prove that mathematics could not be its own ultimate foundation.
    Christians should not have been surprised. The first two conditions are true about math: it is valid and consistent. But only God fulfills the third condition. Only He is complete and therefore self-dependent (autonomous). God alone is “all in all” (1 Corinthians 15:28), “the beginning and the end” (Revelation 22:13). God is the ultimate authority (Hebrews 6:13), and in Christ are hidden all the treasures of wisdom and knowledge (Colossians 2:3).
    https://uncommondescent.com/philosophy/why-god-appears-to-be-a-mathematician/#comment-703707

    Mathematics, contrary to what the vast majority of theoretical physicists apparently believe today, simply never will have the capacity within itself to function as a God substitute.

    As Dr. Bruce Gordon explains, “The world of space, time, matter and energy is dependent on a reality that transcends space, time, matter and energy. This transcendent reality cannot merely be a Platonic realm of mathematical descriptions, for such things are causally inert abstract entities that do not affect the material world,,,
    Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.”

    BRUCE GORDON: Hawking’s irrational arguments – October 2010
    Excerpt: ,,,The physical universe is causally incomplete and therefore neither self-originating nor self-sustaining. The world of space, time, matter and energy is dependent on a reality that transcends space, time, matter and energy.
    This transcendent reality cannot merely be a Platonic realm of mathematical descriptions, for such things are causally inert abstract entities that do not affect the material world,,,
    Rather, the transcendent reality on which our universe depends must be something that can exhibit agency – a mind that can choose among the infinite variety of mathematical descriptions and bring into existence a reality that corresponds to a consistent subset of them. This is what “breathes fire into the equations and makes a universe for them to describe.” Anything else invokes random miracles as an explanatory principle and spells the end of scientific rationality.,,,
    Universes do not “spontaneously create” on the basis of abstract mathematical descriptions, nor does the fantasy of a limitless multiverse trump the explanatory power of transcendent intelligent design. What Mr. Hawking’s contrary assertions show is that mathematical savants can sometimes be metaphysical simpletons. Caveat emptor.
    http://www.washingtontimes.com.....arguments/

    Of supplemental note:

    KEEP IT SIMPLE – 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

    March 2020
    This view the mathematics exists “because they are God’s thoughts” and the Christian view that God created the universe, (i.e. that the universe is contingent upon God), and that the universe has not always existed, (as Aristotle had held), were presuppositions that were necessary for modern science to finally take root in Medieval Christian Europe.
    https://uncommondescent.com/philosophy/edward-feser-on-mathematics-and-the-sense-of-the-divine/#comment-695391

    Verse and Quote;

    John 1:1
    “In the beginning was the Word, and the Word was with God, and the Word was God”

    What is the Logos?
    Logos is a Greek word literally translated as “word, speech, or utterance.” However, in Greek philosophy, Logos refers to divine reason or the power that puts sense into the world making order instead of chaos.,,,
    In the Gospel of John, John writes “In the beginning was the Word (Logos), and the Word was with God, and the Word was God” (John 1:1). John appealed to his readers by saying in essence, “You’ve been thinking, talking, and writing about the Word (divine reason) for centuries and now I will tell you who He is.”
    https://www.compellingtruth.org/what-is-the-Logos.html

    ‘the Word’ in John1:1 is translated from ‘Logos’ in Greek. Logos is also the root word from which we derive our modern word logic
    http://etymonline.com/?term=logic

  6. 6
    Viola Lee says:

    People had this same debate about whether negative numbers were real: some major mathematicians rejected them as nonsense, and it took about a century for them to get fully accepted. But, as Hossenfelder herself point out, complex numbers have been accepted and used for a very long time. That is not the way she is using “exists”, which in the quote I clipped in #1 she puts in quotation marks.

  7. 7
    Fasteddious says:

    Complex numbers are combinations of real numbers and imaginary numbers. I wonder how many in this discussion want imaginary numbers to really “exist” for ideological reasons? E.g. if imaginary numbers exist then other imaginary things might also exist, like unicorns, fairies, magic, macroevolution, and so on.

  8. 8
    Viola Lee says:

    re 7: I hope you know that imaginary numbers exist in exactly the same way that all other kinds of numbers exist. The word “imaginary” was coined by a skeptic when they first were developed, and the name stuck, but exactly the same kind of negative arguments (see what I did there?) were made about negative numbers, and even zero, at other points in time.

  9. 9
    Seversky says:

    This raises the obvious question of what it means to say something “exists”.

    If all that is required for something to exist is that a conscious mind perceives or is aware of it then numbers – real or imaginary – or London or the United States or the Earth or the Sun all exist. But the so does Middle Earth or Gandalf or the Star Wars galaxy or Narnia.

    On the other hand, is there a Universe of phenomena out there which exist whether or not there is a conscious mind to be aware of them? Is there an objective reality as well as a subjective reality and where is the boundary between the two?

    As WJM correctly points out, all our experience of anything happens in our conscious mind. There seems to be no way for us to step “outside” of our mental world to see if there is actually anything beyond. I believe in a mental model theory in which our conscious world is an imperfect model of what is actually out there built from data abstracted from that external reality through our physical sensory systems. The problem is that, as far as I can see, there is no way to prove it. As he says, by Occam’s Razor, his theory is preferable on the grounds that it is more parsimonious. All I can say is that I prefer mine because it fits better with what I experience. But I could be completely wrong.

  10. 10
    bornagain77 says:

    Given the naturalistic premises that dominate science nowadays, the whole controversy over whether the square root of negative 1 is real or imaginary is fairly humorous.

    In the naturalist’s and/or materialist’s denial of the primacy of conscious for any conception of reality that we may have for the universe,

    “The principal argument against materialism is not that illustrated in the last two sections: that it is incompatible with quantum theory. The principal argument is that thought processes and consciousness are the primary concepts, that our knowledge of the external world is the content of our consciousness and that the consciousness, therefore, cannot be denied. On the contrary, logically, the external world could be denied—though it is not very practical to do so. In the words of Niels Bohr, “The word consciousness, applied to ourselves as well as to others, is indispensable when dealing with the human situation.” In view of all this, one may well wonder how materialism, the doctrine that “life could be explained by sophisticated combinations of physical and chemical laws,” could so long be accepted by the majority of scientists.”
    – Eugene Wigner, Remarks on the Mind-Body Question, pp 167-177.

    “No, I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates consciousness.”
    – Max Planck (1858–1947), one of the primary founders of quantum theory, The Observer, London, January 25, 1931

    “Consciousness cannot be accounted for in physical terms. For consciousness is absolutely fundamental. It cannot be accounted for in terms of anything else.”
    – Schroedinger, Erwin. 1984. “General Scientific and Popular Papers,” in Collected Papers, Vol. 4. Vienna: Austrian Academy of Sciences. Friedr. Vieweg & Sohn, Braunschweig/Wiesbaden. p. 334.?

    In the naturalist’s and/or materialist’s denial of the primacy of conscious for any conception of reality that we may have for the universe, the entire naturalist’s and/or materialist’s conception of reality collapses into self-refuting incoherency.

    Basically, because of reductive materialism (and/or methodological naturalism), the atheistic materialist (who believes Darwinian evolution to be true) is forced to claim that he is merely a ‘neuronal illusion’ (Coyne, Dennett, etc..), who has the illusion of free will (Harris), who has unreliable, (i.e. illusory), beliefs about reality (Plantinga), who has illusory perceptions of reality (Hoffman), who, since he has no real time empirical evidence substantiating his grandiose claims, must make up illusory “just so stories” with the illusory, and impotent, ‘designer substitute’ of natural selection (Behe, Gould, Sternberg), so as to ‘explain away’ the appearance (i.e. the illusion) of design (Crick, Dawkins), and who also must make up illusory meanings and purposes for his life since the hopelessness of the nihilism inherent in his atheistic worldview is simply too much for him to bear (Weikart), and who must also hold morality to be subjective and illusory since he has rejected God (Craig, Kreeft). Who, since beauty cannot be grounded within his materialistic worldview, must also hold beauty itself to be illusory (Darwin).
    Bottom line, nothing is truly real in the atheist’s worldview, least of all, beauty, morality, meaning and purposes for life.,,,
    Darwinian Materialism and/or Methodological Naturalism vs. Reality – video
    https://www.youtube.com/watch?v=CaksmYceRXM

    Thus, given that the methodological naturalists end up claiming that practically everything, (that everyone, (including atheists themselves), regard as being real), is merely illusory, then it is quite humorous that people who are, (I assume), by and large committed to a naturalistic conception of reality, are debating whether complex numbers should be regarded as real or should be regarded as imaginary.

    As Viola lee pointed out, exactly the same kind of arguments were made about negative numbers, and even zero, at other points in time.

    It seems that there is an underlying naturalistic bias in mathematics to regard the counting numbers, which can be assigned to things we see, as real, and to regard any other numbers as merely imaginary.

    But alas, as Shakespeare once quipped, “There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy.”

    “There are more things in heaven and earth, Horatio,
    Than are dreamt of in your philosophy.”
    – Hamlet (1.5.167-8), Hamlet to Horatio

    And so it is with mathematics, i.e. “There are more things in mathematics, Horatio, Than are dreamt of in your naturalistic philosophy.”

    Naturalists, by their continual dismissing of certain ‘unnatural’ concepts in mathematics as being merely imaginary, simply do not expect ‘unnatural’ mathematical descriptions to be forthcoming for the universe.

    Yet, ‘unnatural’ mathematics underly our most ‘perfect’ theories in science.

    Our best theories in science are all based on ‘unnatural’ higher dimensional mathematics and geometry, instead of being based on ‘natural’ Euclidian geometry and ‘classical’ mathematics, (as would be presupposed by naturalists who hold that this 3-D physical realm is the only realm of existence),

    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

    Of note: In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form.

    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

    Four-dimensional space – with 4-D animation:
    Excerpt: The idea of adding a fourth dimension began with Joseph-Louis Lagrange in the mid 1700s and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann.,,,
    Higher dimensional spaces have since become one of the foundations for formally expressing modern mathematics and physics. Large parts of these topics could not exist in their current forms without the use of such spaces.,,,
    Einstein’s concept of spacetime uses such a 4D space, though it has a Minkowski structure that is a bit more complicated than Euclidean 4D space.
    https://en.wikipedia.org/wiki/Four-dimensional_space

    Spacetime
    Excerpt: In 1908, Hermann Minkowski—once one of the math professors of a young Einstein in Zurich—presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. A key feature of this interpretation is the definition of a spacetime interval that combines distance and time. Although measurements of distance and time between events differ for measurements made in different reference frames, the spacetime interval is independent of the inertial frame of reference in which they are recorded.
    Minkowski’s geometric interpretation of relativity was to prove vital to Einstein’s development of his 1915 general theory of relativity, wherein he showed that spacetime becomes curved in the presence of mass or energy.,,,
    Einstein, for his part, was initially dismissive of Minkowski’s geometric interpretation of special relativity, regarding it as überflüssige Gelehrsamkeit (superfluous learnedness). However, in order to complete his search for general relativity that started in 1907, the geometric interpretation of relativity proved to be vital, and in 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity.[10]:151–152 Since there are other types of spacetime, such as the curved spacetime of general relativity, the spacetime of special relativity is today known as Minkowski spacetime.
    https://en.wikipedia.org/wiki/Spacetime

    also see the higher dimensional ‘amplituhedron” that lies behind Quantum Electrodynamics

    Of related interest, up until a few years ago, Anton Zeilinger featured this following video on his outreach page,

    Dr. Quantum (in) Flatland – video
    https://www.youtube.com/watch?v=sEVEKL1Fbx0

    Now if anything should be considered ‘unnatural’ by an atheistic naturalist that video is it 🙂

    It is also interesting to note that, whereas atheists have no observational evidence whatsoever that the Multiverses that they have postulated to try to ‘explain. away’ the fine tuning of the universe are real, (nor do Atheists have any evidence whatsoever that the ‘parallel universes’ that they postulated to try to ‘explain away’ quantum wave collapse are real), Christians, on the other hand, can appeal directly to Special Relativity, General Relativity, and Quantum Mechanics, (i.e. our most precisely tested theories ever in the history of science), to support their belief that God upholds this universe in its continual existence, as well as to support their belief in a heavenly dimension and in a hellish dimension.”
    https://uncommondescent.com/intelligent-design/closer-to-truth-are-there-really-extra-dimensions/#comment-722947

    So in conclusion, it is humorous to see naturalists debating what is real and what is imaginary in mathematics since their own worldview can’t even properly distinguish between what is real and what is illusory in real life. And since their naturalistic worldview certainly didn’t anticipate the ‘unnatural’ higher dimensional mathematics that ended up accurately describing this universe

    To repeat Shakespeare,

    “There are more things in heaven and earth, Horatio,
    Than are dreamt of in your philosophy.”
    – Hamlet (1.5.167-8), Hamlet to Horatio

    Verse:

    John 3:12
    I have spoken to you of earthly things and you do not believe; how then will you believe if I speak of heavenly things?

  11. 11
    Viola Lee says:

    BA, I don’t think anyone in this conversation is “debating whether complex numbers should be regarded as real or should be regarded as imaginary.” Hossenfelder has offered what I called an idiosyncratic definition of what “exists” means in terms of numbers, but everyone else that has commented understands that the nature of imaginary numbers is no different than that of those we call real numbers. It’s interesting how a comment can send you down your own self-motivated anti-naturalist rabbit hole without really engaging the conversation at hand.

  12. 12
    bornagain77 says:

    Whatever VL. I wasn’t even talking to you.. My comment was in regards to the character of the debate between real and imaginary in the naturalism in general and mathematics in particular and my comment stands on its own merits in regards to that topic, (as I made clear in my comment). i.e. The main point is simply that the ‘unnatural’ mathematics that ended up describing this universe is unexpected given naturalistic premises, and that our mathematical descriptions of the universe fit far better with a Theistic presupposition and conception of nature.

    Given that Theism holds that there is life after death, then that is, or should be, VERY good news for you. And In so far as my comment even concerned you personally, that fairly straight forward conclusion that I drew should have made you feel happy instead of defensive.

  13. 13
    jerry says:

    There have been discussions about numbers before very recently. The only numbers that exist in our universe are positive integers that enumerate individual entities.

    All else are mental constructs. They are extremely useful for making our lives better but don’t represent anything real. No real numbers. No rational numbers, no negative numbers, no zero. No infinity. Just positive integers. No lines let alone straight lines, no geometric shapes, no angles, And definitely no imaginary numbers.

  14. 14
    Viola Lee says:

    I see, BA, although that is sort of what I meant by saying that your post was a “self-motivated anti-naturalist rabbit hole.” It wasn’t in response to this conversation or even the OP.

  15. 15
    bornagain77 says:

    Well VL, I don’t comment to please your whims about what and how I should comment. I expect you feel pretty much the exact same way about my whims about what and how you comment.

    Let’s try to keep that mutual respect shall we?

    As to my supposedly “anti-naturalist rabbit hole” that you have denigrated my comment with twice now, since that ‘rabbit hole’ analogy reflects a “Alice in Wonderland’ fantasy land, perhaps you can tell me exactly why the real imaginary ‘rabbit hole’ actually lies with naturalism itself and not with my Christian conception of nature?? and thus why you chose to use that false fantasy land analogy on my comment instead of using it on naturalism itself where it rightfully belongs?

    After all, naturalism is the dominate philosophy in science, and on college campuses, today.,,, It is not as if my comment is completely irrelevant to science and society at large.

    to repeat,

    Basically, because of reductive materialism (and/or methodological naturalism), the atheistic materialist (who believes Darwinian evolution to be true) is forced to claim that he is merely a ‘neuronal illusion’ (Coyne, Dennett, etc..), who has the illusion of free will (Harris), who has unreliable, (i.e. illusory), beliefs about reality (Plantinga), who has illusory perceptions of reality (Hoffman), who, since he has no real time empirical evidence substantiating his grandiose claims, must make up illusory “just so stories” with the illusory, and impotent, ‘designer substitute’ of natural selection (Behe, Gould, Sternberg), so as to ‘explain away’ the appearance (i.e. the illusion) of design (Crick, Dawkins), and who also must make up illusory meanings and purposes for his life since the hopelessness of the nihilism inherent in his atheistic worldview is simply too much for him to bear (Weikart), and who must also hold morality to be subjective and illusory since he has rejected God (Craig, Kreeft). Who, since beauty cannot be grounded within his materialistic worldview, must also hold beauty itself to be illusory (Darwin).
    Bottom line, nothing is truly real in the atheist’s worldview, least of all, beauty, morality, meaning and purposes for life.,,,
    Darwinian Materialism and/or Methodological Naturalism vs. Reality – video
    https://www.youtube.com/watch?v=CaksmYceRXM

    It is certainly not much of a stretch, (if any stretch at all), for me to note that the rabbit hole in “Alice in Wonderland’ has more going for it that is real than naturalism has going for it that is real.

    In other words, although the Darwinian Atheist and/or Methodological Naturalist may firmly believe that he is on the terra firma of science (in his appeal, even demand, for naturalistic explanations over and above God as a viable explanation), the fact of the matter is that, when examining the details of his materialistic/naturalistic worldview, it is found that Darwinists/Atheists themselves are adrift in an ocean of fantasy and imagination with no discernible anchor for reality to grab on to.

    It would be hard to fathom a worldview more antagonistic to modern science, indeed more antagonistic to reality itself, than Atheistic materialism and/or methodological naturalism have turned out to be.

    2 Corinthians 10:5
    Casting down imaginations, and every high thing that exalteth itself against the knowledge of God, and bringing into captivity every thought to the obedience of Christ;

  16. 16
    Viola Lee says:

    Rabbit hole was probably not an apt metaphor. Tangent might have been better.

  17. 17
    kairosfocus says:

    F/N: The rotating vectors view removes the needless mystification. take some x on the horizontal axis, a vector runs from origin O>x. Let an operator j*() rotate O>x through a right angle anticlockwise, up the vertical or Y axis. Apply J*() again, we have a further rotation yielding O>-x, so j*(j*((x)) = -x, or simplifying, j*j*x = – x, We then transfer j*^2 = -1. Simplify again j^2 = -1. So, j is “obviously” sqrt(-1). KF

  18. 18
    Concealed Citizen says:

    Jerry: The only numbers that exist in our universe are positive integers that enumerate individual entities.

    You mean I can’t have 3.5 bags of sugar?

  19. 19
    jerry says:

    You mean I can’t have 3.5 bags of sugar?

    Nope.

    A half bag is a different entity from a whole bag. And each bag is essentially a large number of individual sugar molecules. What is called a half bag is a smaller number of sugar molecules.

    But the concept is very useful for living. I’m not denying that. So is calculus and geometry and all mathematics. Our modern world is built using these ideas. But the actual world is not continuous.

    The idea of a half is a mental construct. A positive number is a mental construct too that shows a relationship between one set of entities and another set.

    This same discussion was part of another thread in recent months. I’ll see if I can find it. Also the discussion included imaginary numbers.

  20. 20
    Ralph Dave Westfall says:

    Hammers, drills, saws etc. are tools. They can be used to achieve objectives. So are mathematical methods and constants. Without them it would be impossible or very difficult to do many things eg. send a spacecraft to Mars.

    Therefore mathematical methods and constants, and other abstractions, exist in a meaningful sense. It’s just a somewhat different sense than that of physical object tools.

  21. 21
    kairosfocus says:

    Jerry,

    The idea of a half is a mental construct. A positive number is a mental construct too that shows a relationship between one set of entities and another set.

    Clearly, we speak in terms of wholes and fractions, but is this merely a notion we impose?

    Why would it have any utility?

    Ponder a parallelogram on a base b, with height h. Run verticals from the base to the level of the top. Notice how congruent triangles appear, so that if we slice off from one end we can move to the other to form a rectangle. This rectangle of dimensions b * h, patently, has area equal to the area of the parallelogram.

    This shows a natural context in which equality appears. Ask any tiler.

    Next, go back to the parm. Cut it in half along a diagonal. This demonstrates how the area of a triangle must be 1/2 * b * h.

    As a close corollary, any triangle can be doubled into a parm by producing parallels to two adjacent sides on the opposite vertex, to construct a congruent triangle with the third side as common diagonal.

    Half-ness here appears as a necessary and embedded feature of the world. Indeed, any possible world will have such an abstract plane associated, once we go from N to Z,Q,R,C. For C, the j* rotation operation suffices to generate the plane and a coordinate system, just relabel as x and y axes.

    Similarly, stretch out, peg then fold a rope in two. Naturally, we see a half of a continuum, which can be then put in our plane.

    Similarly, for simple uniformly accelerated kinematics, construct a graph with time as horizontal axis, t, initial velocity u, onward velocity v, with velocities on the vertical axis. Let initial time be zero. Produce, again, a line from u at t = 0 to the vertical line at some given t, where velocity is v. We see that the area under velocity vs time is a triangle on top of a rectangle of area u * t. The area of the whole can be readily shown to be average velocity (u + v)/2 times t. From this we get that

    v^2 = u^2 + 2*a*x.

    We already saw the halving in the average. But there is something much deeper here, if we introduce mass, recognise NL2 that F = m*a, and rearrange:

    m *[ 2ax] = m*[v^2 – u^2]

    Regrouping LHS,

    2* [ma]*x = m*[v^2 – u^2], or,

    2* F*x = m*[v^2 – u^2]

    Again, we can reduce:

    F*x = 1/2*m*[v^2 – u^2]

    F*x is the work by force F moving its application point along its direction, through displacement x. This is equal to the change in something measured by halving m*v^2, a term once known as vis viva To see what this is about, simply set u = 0, so we set initial value to at rest..

    Of course 1/2*m*v^2 is now known as kinetic energy, the energy of translational motion.

    This is a key initial glance at a major principle of physics, energy and its conservation.

    Notice, 1/2 appears there, naturally.

    And, connected to tiling.

    All of this shows us that Q is naturally present in any possible world. Beyond Q, we can see that any r in R is the sum of a whole number part and a fractional part. For a rational, and with a decimal, binary or similar place value system, we have a repeating cycle beyond a certain point [which may be zero] as we see from long division. For an irrational, the cycle never occurs, i.e. we are at the surreal construction of R in w steps, i.e. convergent power series of fractional parts to get pi, e, sqrt2, most logs, etc.

    From R mileposted by N, we readily see C, R* etc.

    We are back to, a core of math — structure, quantities, sets and relationships, is embedded in any possible world as part of its framework tracing to its distinct identity. This is the root of Wigner’s wonder.

    KF

  22. 22
    kairosfocus says:

    PS: Ponder what is required to cut a gear with a natural, counting number of even, evenly meshing teeth on a circular disk of metal, given that c = 2 * pi * r for a circle. Of course, this then extends to a gear train. Gears, of course, are at the heart of power trains. Irrationals lie in the heart of technology.

  23. 23
    jerry says:

    Clearly, we speak in terms of wholes and fractions, but is this merely a notion we impose?

    Yes, it is a notion we impose.

    Don’t you read anything I say. I probably at one time knew more about math than anyone on this site. Not now, since I haven’t studied it in years. But I understand the usefulness of it and that usefulness has been life changing for the world. You confuse it’s usefulness with the reality of what it is.

    But the math constructs are only in your head not in the real world. They most definitely have application to the real world. But yet you write paragraphs and paragraphs that are irrelevant. The real world consists only of individual entities. So a half is just a way of enumerating a smaller number of these entities. Extremely useful.

    So all your equations are irrelevant. All your examples of geometric figures are irrelevant. None actually exist in the real world. Yes shapes resembling them exists and the logic of math in your mind help tremendously with their use. There are no circles, there are no lines. There is no pi. There are no angles.

    There are objects, the accumulation of zillions of microscopic entities that we manipulate to look perfect to the eye. We then talk about the precision of these objects. But all are imperfect when looked at finely. This imperfection does stop the object from being extremely useful.

    They are only in your mind. You apparently are confusing a desired result in the real world with a mental construct that helps achieve that result. We live in a discontinuous world. The mathematics you keep on bringing up assumes a continuous one.

    Let’s not get Into whether your mental images are real or not. They are real. But like the weightless elephant or frictionless surface or science fiction character they do not have exact real world counterparts. Even the positive numbers that enumerate individual entities are essentially mental constructs expressing relationships between sets of separate entities.

  24. 24
    kairosfocus says:

    Jerry, I showed cases where fractions naturally emerge in pivotal contexts. KF

  25. 25
    jerry says:

    where fractions naturally emerge in pivotal contexts

    In your mind. In the real wot;d. It will show up as a smaller real number.

  26. 26
    JVL says:

    Jerry: I probably at one time knew more about math than anyone on this site.

    Which parts did you study? Discrete? Continuous?

    It will show up as a smaller real number.

    Hmmmm . . . 2.5 is a real number and it has fractional components . . .

  27. 27
    Viola Lee says:

    Hmmm. I’m curious, Jerry. You might be right, but what level of college math did you have, or have you studied otherwise? Differential equations? Algebraic theory? Non-euclidean geometry? Fractals and chaos theory? The history of math?

  28. 28
    Viola Lee says:

    Jerry writes, “But the math constructs are only in your head not in the real world.”
    Yes, and the natural numbers are also. Yes, they are built from our experience of individual items, just as our concept of line is built from a stretched string.

    I agree that all of math consists of concepts. But I don’t agree the counting numbers are fundamentally different in that regard just because they are one the foundational concepts.

  29. 29
    jerry says:

    what level of college math did you have, or have you studied otherwise?

    I had almost the equivalent of a masters degree when I graduated from college having taken graduate level courses at U of Pennsylvania while an undergraduate at another college in Philadelphia area. This enabled me to get a perfect score on math GRE. Actually I got 72 of 75 correct. Did not answer 3 questions on statistics as we were told to leave questions blank we did not know since a wrong answer subtracted from you score. I had arrogantly avoided statistics since I didn’t consider it math. I had no idea what a mean or standard deviation was.

    I then spent a year at Duke University on a fellowship. I left to go into the Navy for a more interesting life experience. (I actually did see the world, at least 5 continents while in Navy). I couldn’t see myself spending the rest of my life with the type of people I found in high level math.

    In college I wrote a paper titled “What is a number?” This was for a philosophy course. That is the basis of my point of view which I later confirmed with other math professors, one a world expert on Number Theory.

    So in no way do I denigrate math or it’s usefulness.

    Aside: at Duke I knew how to prove by heart the theorem referred to as the Law of Large Numbers, the basis for statistics.

  30. 30
    JVL says:

    Jerry: Aside: at Duke I knew how to prove by heart the theorem referred to as the Law of Large Numbers, the basis for statistics.

    That’s interesting . . . which version: strong or weak?

  31. 31
    jerry says:

    which version: strong or weak?

    Have no idea. It was in a thick book called Analysis. I believe the proper name of the theorem was the Central Limit Theorem. Not sure, it’s been awhile. Since have lost the book.

    I found it ironic that a year after I took the GRE exam and couldn’t answer any questions on statistics, I was able to prove a major theorem in the topic I had avoided.

    Later took several statistics courses. And recommend that this be taught in high school instead of calculus.

  32. 32
    Viola Lee says:

    re Jerry at 29. Out of 800, my GRE’s were 790 in both math and verbal.

    I’m not sure knowing how to prove a theorem by heart is a strong indicator of “knowing math”.

    Pretty sure you’re wrong about statistics! 🙂

    I’m also pretty sure your position on the philosophy of numbers is not common in the philosophy of math.

  33. 33
    kairosfocus says:

    Jerry,

    As this will show https://web.maths.unsw.edu.au/~jim/forrestarmstrong.pdf there is a significant philosophical debate and the typical view of practitioners is mathematical platonism which is not the same as Plato’s views.

    What I have raised in the already linked, is that we can see that once a distinct world exists, it structurally embeds the characteristic that there is distinction so we can see an empty partition say W = {A|~A} where we got to W from near neighbour W’ by augmenting it with some distinct thing A.

    So, W = {A|W’}, where A is a simple unit and W’ a complex one generally speaking. We have a diversity of units, so we have duality too: the QUANTITIES denoted by 0, 1, 2 are present automatically. By von Neumann, we have the onward succession, so N. From N we have Z by additive inverses, x + (-x) = 0. Ratios of integers in Z give Q, thence R, C, R* etc. These are quantities with structural relationships, e.g. integers and reals, complexes are vectors.

    That is the sense in which I have noted that we have here universally present structures and quantities antecedent to names for same and organised study of same. I call these as abstract, existing entities embedded in any distinct possible world.

    We may then proceed to recognise and study.

    That is the sense in which I see a dual definition of math: [the study of] the logic of structure and quantity, adapting prof Neiderreiter.

    The study is culturally framed, but certain core structures and quantities are embedded in what it takes to have any distinct possible world.

    KF

  34. 34
    jerry says:

    I’m not sure knowing how to prove a theorem by heart is a strong indicator of “knowing math”

    Who said it was?

    Pretty sure you’re wrong about statistics!

    What am I wrong about?

    I’m also pretty sure your position on the philosophy of numbers is not common in the philosophy of math.

    What’s wrong with it? I discussed it with a world renown number theorist and he agreed. It is basic common sense.

    Doesn’t undermine anything in math. This and other discussion came out of asserting imaginary things as real. First, infinity and here imaginary numbers.

  35. 35
    JVL says:

    Jerry: Have no idea. It was in a thick book called Analysis. I believe the proper name of the theorem was the Central Limit Theorem. Not sure, it’s been awhile. Since have lost the book.

    Analysis is the area of mathematics that includes Calculus. The Central Limit theorem is a slightly different animal than the so called Law of Large Numbers.

    I found it ironic that a year after I took the GRE exam and couldn’t answer any questions on statistics, I was able to prove a major theorem in the topic I had avoided.

    Actually, statistics sits on a bed of calculus/analysis so I don’t think it’s incredibly ironic. The 400-level stats course I took was all calculus.

    Later took several statistics courses. And recommend that this be taught in high school instead of calculus.

    In fact, many US community colleges started allowing basic level stats classes instead of algebra for meeting their minimum math requirement. It certainly is more useful for most people than algebra.

    I discussed it with a world renown number theorist and he agreed.

    Do you remember their name? Just curious.

    This and other discussion came out of asserting imaginary things as real. First, infinity and here imaginary numbers.

    But . . . are imaginary numbers really IMAGINARY? We think so because we’ve been taught that the product of two positives is positive, the product of two negatives is positive and the product of a positive and a negative is negative. Therefore, you can’t have the square root of a negative. BUT . . . it’s all a matter of definition isn’t it?

    Long time ago -1 was considered less than -100. Really. And, if you look at their magnitudes that is correct.

    Imaginary/complex numbers are useful, can be used to model real world things, follow set rules, etc. Just like real numbers.

  36. 36
    JVL says:

    Viola Lee: Out of 800, my GRE’s were 790 in both math and verbal.

    Impressive! My memory of the maths GRE is that it was pretty nasty. Not Putnam level nasty but not something I wanted to repeat. At my university we did zero prep for it; they just wanted us to take it. Partially I think that way they could weed out people from dubious undergraduate programmes.

  37. 37
    Viola Lee says:

    I remember well that I felt very clear-headed that day, and that I got fairly immediate understandings of the problems. It was fun.

  38. 38
    kairosfocus says:

    JVl, complex numbers are 2-d vectors represented algebraically. The rotation operator approach demystifies. KF

  39. 39
    JVL says:

    Kairosfocus: JVl, complex numbers are 2-d vectors represented algebraically. The rotation operator approach demystifies.

    That’s one use/interpretation. I’m not mystified, are you? You always seem to assume the rest of us don’t quite ‘get it’ when we do.

  40. 40
    Viola Lee says:

    Ditto, JVL. Anyone who has studied complex numbers knows that representing them as 2-d vectors in the complex plane is one of the neat things about them. Also writing them in the form e^(ix). KF seems to see significances sometimes that he somehow thinks others are missing. (Or in the case of e^(i*pi) = -1, avoiding for ideological reasons.)

  41. 41
    kairosfocus says:

    JVL, I do NOT have in mind people as sophisticated as you are. I do have vividly in mind my 6th form classmate who threatened to study Math to expose the cheat involved in such dubious novelties. Had the rotating vectors approach been taken, I think this would not have been a problem. I think the very term imaginary is itself part of the problem. Seeing numbers with directions as vectors and introducing a second dimension through rotation makes things a lot easier. There was no similar eruption when ijk vectors were introduced. The Q-word* was not said, very wisely, methinks. KF

    *Quaternions.

  42. 42
    kairosfocus says:

    PS: Did you notice how you chose to state the Euler identity in a form that is quite unusual? That’s a clue on your other complaint. There is no more specifically mathematical content in adding 1 to lhs and rhs,

    0 = 1 + e^i*pi

    However, this draws out an infinitely precise integration of five key numbers and linked operations, involving significant domains. That is, it opens up insights that the form you gave does not. That also happens when one moves from f’ to dy/dx.

  43. 43
    Viola Lee says:

    I agree 100%, KF, that imaginary is a very bad term and has misled people, and being able to visualize geometrically in the complex plane helps show that they are no different in important ways than real numbers.

    As to Euler’s identity, we’ve discussed before, but I brought it up, so I’ll repeat. e^(i*pi) is a specific value for the function e^(ix) = cos x + i sin x, which are two different ways to express complex numbers in the complex plane. Let x = pi and you get e^(i*pi) = -1. That is straightforward, and there is nothing unusual about it. Rewriting it as e^(i*pi) + 1 = 0 is neat, and I used to point this out to my students, but it does not “open up insights” that the other doesn’t.

    In fact the original function e^(ix) = cos x + i sin x is really the insightful expression, because it shows how imaginary, exponential, and trigonometric numbers are tied together, rather than just showing one result.

  44. 44
    JVL says:

    Viola Lee: In fact the original function e^(ix) = cos x + i sin x is really the insightful expression, because it shows how imaginary, exponential, and trigonometric numbers are tied together, rather than just showing one result.

    Yes! I quite agree. Plugging in a particular value of x does give an interesting and beautiful result but that’s not the real bridging concept. It reminds me of the difference between treasure hunting and real archaeology: spectacular finds are cool and fun but the real point of excavation is much broader. I think it’s very easy to fall into the trap of thinking that one or two events or equations or works are tentpoles for a whole area of knowledge. The Mona Lisa is a lovely painting but putting it in context, comparing it to other works being done at about the same time all over Italy (and Europe) shows that it’s an example of a popular trend or movement. Besides, Raphael was a better artist. 🙂

    Mathematics is like a very, very large wood, focusing on particular trees loses some of the important perspective that gives you the real grasp of the breath and depth and development.

    I too enjoy lists like: a history of this or that in 100 pieces or works but they are no substitute for real study and understanding. In fact, it is only through first knowing the broader landscape that a real and true appreciation of the finer works can be achieved.

    Additionally, I can’t help but point out that there is a big difference between e^i*pi and e^(i*pi). If you’re going to speak the language you have to do it correctly or you risk being misinterpreted.

  45. 45
    Steve Alten2 says:

    Throughout my life I have had to use algebra, geometry, calculus, statistics, etc. But I regret to say that I never developed a passion for it like JVL and Viola Lee obviously have. I guess we each have our own passions.

  46. 46
    Concealed Citizen says:

    Jerry: A half bag is a different entity from a whole bag.

    Each of the whole bags are “different entities” from each other. So what?

    Saying, “I want 3 1/2 bags of sugar” actually has meaning. Nobody misunderstands this in the modern world. Or, “do you want a scoop of ice cream?” “Naw, just give me half.”

    “Half” has relative meaning. Meaning is what matters. It’s a real concept.

  47. 47
    jerry says:

    Half” has relative meaning. Meaning is what matters. It’s a real concept.

    Never said it didn’t. It’s just not a number.

    You are pointing to a group of four things. Three of which you are calling a bag. And one you are calling a half bag or a partial bag. You are actually using the word “half” here as if it were number.but is just a word describing a different entity. Extremely useful and effective communication.

    I suggest you read what I actually say.

  48. 48
    JVL says:

    Jerry is reminding me of something one of my professors said when I was in graduate school: apparently you can pick a smallest number/value (as long as it is really small) and still get most of our practical/applied mathematics from that.

    If you did that then you wouldn’t need fractions because everything would just be so many times the smallest value of whatever. Nothing would actually be continuous, just a huge bunch of very tiny steps or chunks or whatever. You wouldn’t get ‘infinity’ either . . . I don’t think.

    It kind of does my head in because I’ve learned to think of functions and such as continuous. Some math topics (like Graph Theory or Number Theory or Set Theory) are already pretty discrete so they wouldn’t be affected until you get to things like the Prime Number Theorem or countably infinite.

    Anyway, the formulations we use now have turned out to be very useful and practical so I doubt there’s a great push to change them.

    I am still curious which Number Theorist Jerry talked to. Some of them are pretty interesting people.

  49. 49
    Viola Lee says:

    The ancient Babylonians used base 60 because most of the fractional parts they needed were whole numbers, as they didn’t know how to express fractions.

    The other reason they used sixty is because six equilateral triangles fit inside a circle, and if the angles are divided into 60 parts, you get 360 parts for the whole circle, which they thought coincided perfectly with the number of days in a year. That’s why there are 360° in a circle.

    You may know all that, but it’s an interesting historical tidbit.

  50. 50
    JVL says:

    Another did you know: in Italy the surveyors don’t use degrees or radians they use gradians. There are 400 of them in a circle so a right angle is 100 gradians.

    For those of you who were wondering what the ‘GRAD’ setting on your calculator was for.

  51. 51
    Viola Lee says:

    Grads make more sense than degrees from the point of view of being consistent with our decimal system. And I know you know this, but most calculators use DRG for degrees, radians, and grads. I almost enjoyed introducing my students to radians as the “natural” way to measures angles and arcs, similar to how e is the natural base, as opposed to 10.

    More trivia: dividing a degree into 60 minutes per degree and 60 seconds per minute harks back to this base 60 deal. I think that some are just using decimal parts of a degree these days, but old habits die hard. For instance, if I google “how wide is the moon” I get answers both in decimal parts of a degree and in arc minutes.

    All fun stuff to know.

  52. 52
    jerry says:

    The ancient Babylonians used base 60 because most of the fractional parts they needed were whole numbers, as they didn’t know how to express fractions.

    I too read someplace that one of the reasons they used 360 was this number represented the days in a year. Obviously off a little but it expressed an orderliness to what they were trying to understand.

    It was then appleid to the sky.

    Then someone, the Egyptians?, divided the day into 24 parts and sky likewise was divided. Then they divided these parts into smaller parts and they were called minutes meaning small parts (by the Romans.) Then they divided the minutes again to be more precise and they were called second minutes or second small part and then just seconds. This was then applied to time as well as the sky and is now applied to the earth.

  53. 53
    Viola Lee says:

    The minutes and seconds were from the Babylonians, base 60. The Egyptians used base 12.

    Here’s a good article about the whole subject: Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?

    However, it does say that the reason the Babylonians used base 60, other than fractional convenience, is unknown, and I don’t think that’s true: I think the part about 360 days in a year is quite well established.

  54. 54
    JVL says:

    So, Jerry, who was the Number Theorist you spoke to all those years ago? I was thinking it would be interesting to see what area of research he worked on.

  55. 55
    jerry says:

    who was the Number Theorist

    It will have to remain a mystery. I’m not going to reveal a private conversation with someone. Choose to not believe it happened if you wish.

    Anyway, the formulations we use now have turned out to be very useful and practical so I doubt there’s a great push to change them.

    I doubt any sane person would want to change mathematics to reflect the particle nature of our universe. It is too successful. Infinity, real numbers, rational numbers, negative numbers, zero, imaginary numbers, geometric shapes continuums are extremely useful. There might be some applications that might be appropriate but I certainly don’t know of any.

  56. 56
    JVL says:

    Jerry: It will have to remain a mystery. I’m not going to reveal a private conversation with someone. Choose to not believe it happened if you wish.

    It’s your call, clearly. I’m purely interested in an academic sense AND I believe it did happen. That’s why I’m interested. I’d be interested in continuing the conversation.

  57. 57
    JVL says:

    Jerry: It will have to remain a mystery. I’m not going to reveal a private conversation with someone. Choose to not believe it happened if you wish.

    Now that I’m thinking about it . . . I find your response a bit odd since none of the ideas you mentioned are ‘heretical’ or even that unusual. I’m not sure what you’re afraid of exposing. But, as I said, it’s your call.

  58. 58
    Viola Lee says:

    Actually I do find Jerry’s idea unusual that counting numbers exist in some way that all other mathematical concepts don’t. I wonder if that idea appears in the philosophy of math literature anywhere,.

  59. 59
    JVL says:

    Viola Lee: I wonder if that idea appears in the philosophy of math literature anywhere,.

    That’s why I’m interested in who he was talking to. I don’t understand the need for secrecy; I’ve never known a mathematician, let alone a world famous one, who was ashamed of their ideas. I really don’t think that the mathematician in question would care at all if we knew what he had said. In fact, he’d probably thing we were too stupid to appreciate it!!

    But, it’s Jerry‘s tale to tell if he wants to. And he doesn’t seem interested. Which is a shame since this seems like a good forum to present unusual ideas.

  60. 60
    kairosfocus says:

    Jerry, half is not a number? Did you mean that literally? Or did you mean that halving is an operation leading to the rational 1/2? KF

  61. 61
    kairosfocus says:

    VL, the proof goes through the steps you outline. These lead to the complex exponential result you cite. Generally, a step of algebra is taken yielding a surprising connexion of five famous and pivotal numbers in the history of numbers and wider history of ideas. It also has connexions to the coherence of core mathematics, post Godel. KF

  62. 62
    jerry says:

    half is not a number? Did you mean that literally?

    Yes.

    Or did you mean that halving is an operation leading to the rational 1/2?

    I have no idea what you mean. Since 1/2 doesn’t exist.

    The misunderstanding arises with division. Addition, subtraction and multiplication of entities provides no issues. Suppose you have 12 quarters and want to give three to each of 4 children. Each will get 3 quarters. With no problem.

    If you have 10 quarters then the first three will get 3 quarters but there will not be enough to give the fourth child 3 quarters since only 1 remains. 3 1/3 quarters does not exist. But 3 groups of 3 exist and one group of 1 exist.

    It’s not hard to understand.

  63. 63
    jerry says:

    Interesting thing. I was editing previous comment and my edit disappeared. I thought there was another malfunction in the WP. Then I realized my 20 minutes had expired.

    Everything just vanished.
    ——————

    A number is just a word or concept we assign to a collection of entities. All the collections of entities that have a 1 to 1 correspondence with each other are given the same number or word as a descriptor. There is no need for each entity to be identical to the others. They can be but it’s not necessary. We can have a collection of 5 pennies or we can have a collection of 5 tools all different. Each has the number 5 assigned.

    If you take one of the entities and some how separate it into two different parts there are now more entities in the collection. For example suppose we have 5 packets of sugar and tear one apart. We now have 6 entities, 4 whole packets and two partial packets If you then remove one of the partial packets from the collection there will be the same number of entities as before. One entity will be different from before. There will be 4 whole packets and one partial packet.

    In no way am I advocating using this understanding in every day use. The concept of a half or any other fraction is extremely useful for life. There is just a difference between what is useful and what is actually happening.

  64. 64
    JVL says:

    If you break a crumb in half, you don’t get two half-a-crumbs, you get two crumbs.

    . — George Carlin

  65. 65
    kairosfocus says:

    JVL, Carlin failed to note the next bit, breaking evenly and that the two new crumbs are half-size. A 1 yard string bent double and cut has become two half-yard strings. Halving is an operation, dividing into two even parts and there is a halfness that appears on a scale. 0.5 or equivalent, half of a unit, is a real number. KF

  66. 66
    JVL says:

    Kairosfocus: Carlin failed to note the next bit,

    Do you have a sense of humour at all?

    And, again, please don’t be condescending. I know very well what ‘halving’ means.

  67. 67
    jerry says:

    Carlin failed to note the next bit, breaking evenly and that the two new crumbs are half-size.

    Two things relevant to crumbs. Actually three. Not 2 1/2.

    1) The new so called half crumbs are entirely different entities. If you came across them independently of the breaking them apart, one would just say there are two small crumbs.

    2) they will never be exactly equal in mass or number of molecules. So not really halves. If they were, the most scientific description would be that there are two small crumbs of exactly equal size.

    3) Of course we need a definition of just what a crumb is. When does a small piece of a cake become a crumb? That’s the more important question. We don’t want to be accused of just giving someone a crumb.

    half of a unit, is a real number

    Yes but it is not a half. It’s actually a positive integer. The whole was say 500,000 various molecules that made up the string. The so called half is just 250,000 of the same molecule combination.

    Actually it would probably be physically impossible to ever get that exact.

    You are confusing what is useful with what is actually happening.

    I’m not assaulting math. Just the opposite. I’m trying to make it more understandable.

  68. 68
    JVL says:

    Jerry: You are confusing what is useful with what is actually happening.

    You don’t seem to have a sense of humour either.

    Oh well, I tried.

    And, by the way, like Kairosfocus you can be condescending. If that works for you then keep doing it. But I will probably respond a bit less often.

  69. 69
    jerry says:

    What did the 0 say to the 8?

    Nice belt!

    Never have a conversation with pi. Pi just goes on and on and never stops.

  70. 70
    kairosfocus says:

    JVL, pardon but you are not sole or primary audience. In context, reality of 1/2 is on the table and Mr Carlin was brought in, to which I responded on substantial import. Breaking a crumb of cookie will indeed transform it into two smaller crumbs, typically of unequal size. He is correct that this is not two half-a-crumbs. However, joke notwithstanding, were the breaking an even one the new crumbs would stand in half-ness of volume and presumably weight relative to the original . . . cookies are inhomogeneous. Halving is an operation, involving even splitting (and extending into other fractions as well as division in general). Halving is legitimate, half-ness is a quantitative relationship, and using the general unit, 1, stands at 0.5 using decimals, on the reals line. Were the original crumb to weigh say 0.9 oz, the assignment of unit to this leads to the new ones weighing 0.45 oz each. Grams could be used, and would be consistent via a conversion factor. Half-ness is a relation, it is part of the structure and quantity of this or any possible world, it emerges in many practical circumstances such as the halving crease impressed into many pills. KF

  71. 71
    jerry says:

    Today is Pi day. Will it go on and on?

  72. 72
    jerry says:

    abstract spaces pervade our world

    Above from

    https://uncommondescent.com/intelligent-design/while-in-quarantine-from-the-plague-newton-transformed-the-way-we-calculate-pi/#comment-726832

    Yet you cannot point to one. They are mental only.

    they ground our ability to infer that material ruler edges, ball bearings etc are imperfect relative to the geometric ideal.

    I believe you just agreed with me.

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