# At Mind Matters News: Recent research: Imaginary numbers are part of the real world

According to new research, if we try to leave them out of quantum mechanics, our description of nature becomes faulty:

“Without imaginary numbers, the device on which you are reading this article probably wouldn’t even work. As science writer Padavic-Callaghan points out, When you set out to try and capture a quantum state in the language of mathematics, these seemingly impossible square roots of negative numbers are an integral part of your vocabulary. Eliminating imaginary numbers would highly limit how accurate of a statement you could make. The discovery and development of quantum mechanics upgraded imaginary numbers from a problem seeking a solution to a solution that had just been matched with its problem. As the physicist and Nobel laureate Roger Penrose noted in the documentary series Why Are We Here? (2017): ‘[Imaginary numbers] were there all the time. They’ve been there since the beginning of time. These numbers are embedded in the way the world works at the smallest and, if you like, most basic level.’ – Karmela Padavic-Callaghan, “Imaginary Numbers Are Real” at Aeon (July 14, 2022)”

News, “Recent research: Imaginary numbers are part of the real world” at Mind Matters News (July 17, 2022)

Takehome: Perhaps the universe is bound to seem mysterious, in part because it is not wholly material. It keeps the rules but the rules are not always what we expect.

You may also wish to read: You may also wish to read: Yes, you can manipulate infinity in math. The hyperreals are bigger (and smaller) than your average number — and better! (Jonathan Bartlett)

## 3 Replies to “At Mind Matters News: Recent research: Imaginary numbers are part of the real world”

1. 1
bornagain77 says:

As to the Aeon headline; “Imaginary numbers are real”,

The finding that “Imaginary numbers are real” makes what was already an intractable problem for Atheistic Materialists that much worse still.

To start off with, Atheistic Materialists simply have no clue why mathematics should even be applicable to the universe in the first place.

This is because mathematics is profoundly immaterial in its foundational essence,

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

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/

In fact, no less than Eugene Wigner and Albert Einstein are both on record as to regarding the applicability of mathematics to the universe to be a miracle.

Eugene Wigner, (who received a Nobel prize for his work in the foundations of quantum mechanics, and after rightly calling into question the ability of Darwin’s 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.
https://web.njit.edu/~akansu/PAPERS/The%20Unreasonable%20Effectiveness%20of%20Mathematics%20(EP%20Wigner).pdf

Likewise, Albert Einstein himself is also 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’.

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

And the last time I checked, miracles are considered to be the sole province of God:

definition – Miracle
– a surprising and welcome event that is not explicable by natural or scientific laws and is therefore considered to be the work of a divine agency.
“the miracle of rising from the grave”

What is so interesting in particular about “Imaginary Numbers” being found to be “Real” is that imaginary numbers, particularly the square root of negative one (i), is essential for understanding the ‘wave packet’ in quantum mechanics prior to measurement, and/or prior to the ‘collapse of the wave function’, (but is not required after the ‘collapse of the wave function)

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

Moreover, to make the dilemma even more vexing for Atheistic Materialists, the wave function, prior to collapse of the wave function, is mathematically described by an ‘infinite dimensional’ Hilbert space,

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,
https://web.njit.edu/~akansu/PAPERS/The%20Unreasonable%20Effectiveness%20of%20Mathematics%20(EP%20Wigner).pdf

Wave function
Excerpt “wave functions form an abstract vector space”,,, This vector space is infinite-dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function.
http://en.wikipedia.org/wiki/W.....ctor_space

Why do we need infinite-dimensional Hilbert spaces in physics?
You need an infinite dimensional Hilbert space to represent a wavefunction of any continuous observable (like position for example).
https://physics.stackexchange.com/questions/149786/why-do-we-need-infinite-dimensional-hilbert-spaces-in-physics

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 Carrol; in Through the Looking Glass.) No physicist today understands why this is possible..

2. 2
bornagain77 says:

Moreover, this ‘infinite dimensional’ Hilbert space takes an infinite amount of information to describe properly,

Explaining Information Transfer in Quantum Teleportation: Armond Duwell †‡ University of Pittsburgh
Excerpt: In contrast to a classical bit, the description of a (quantum) qubit requires an infinite amount of information. The amount of information is infinite because two real numbers are required in the expansion of the state vector of a two state quantum system (Jozsa 1997, 1)
http://www.cas.umt.edu/phil/fa.....lPSA2K.pdf

Quantum Computing – Stanford Encyclopedia
Excerpt: Theoretically, a single qubit can store an infinite amount of information, yet when measured (and thus collapsing the superposition of the Quantum Wave state) it yields only the classical result (0 or 1),,,
http://plato.stanford.edu/entr.....tcomp/#2.1

WHAT SCIENTIFIC IDEA IS READY FOR RETIREMENT? Infinity – Max Tegmark
Excerpt: real numbers with their infinitely many decimals have infested almost every nook and cranny of physics, from the strengths of electromagnetic fields to the wave functions of quantum mechanics: we describe even a single bit of quantum information (a qubit) using two real numbers involving infinitely many decimals.
https://www.edge.org/response-detail/25344

To point out the obvious, the ‘infinite dimensional’ Hilbert space corresponds to the Theistic attribute of omnipresence. Whereas the infinite information that is required to describe the ‘infinite dimensional’ wave function prior to collapse corresponds to the Theistic attribute of omniscience.

Omnipotent, Omniscient and Omnipresent God: Definition
Excerpt: Omnipotence, Omniscience, and Omnipresence
Omnipotence means all-powerful. Monotheistic theologians regard God as having supreme power. This means God can do what he wants. It means he is not subject to physical limitations like man is. Being omnipotent, God has power over wind, water, gravity, physics, etc. God’s power is infinite, or limitless.

Omniscience means all-knowing. God is all all-knowing in the sense that he is aware of the past, present, and future. Nothing takes him by surprise. His knowledge is total. He knows all that there is to know and all that can be known.

Omnipresence means all-present. This term means that God is capable of being everywhere at the same time. It means his divine presence encompasses the whole of the universe. There is no location where he does not inhabit. This should not be confused with pantheism, which suggests that God is synonymous with the universe itself; instead, omnipresence indicates that God is distinct from the universe, but inhabits the entirety of it. He is everywhere at once.

In short, the finding that “Imaginary numbers are real” makes what was already an intractable problem for Atheistic Materialists that much worse still since it is rather stunning confirmation of the Christian’s contention, (via Neoplatonic philosophy and Augustinian theology), that the (higher dimensional) mathematics that are found to describe this universe really are “God’s thoughts”. Just as was originally held by the Christian founders of modern science.

“O, Almighty God, I am thinking Thy thoughts after Thee!”
– Johannes Kepler – (stated shortly after elucidating the mathematical laws of planetary motion)

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.

Of supplemental note, not only do Atheistic Materialists have no realistic clue why mathematics should even be applicable to the universe in the first place, they also have no clue how we can comprehend this immaterial world of mathematics.

In 2014, a group of prominent Darwinists, who are leading experts in this area of research, authored a paper in which they honestly admitted that they have, “essentially no explanation of how and why our linguistic computations and representations evolved.”

Leading Evolutionary Scientists Admit We Have No Evolutionary Explanation of Human Language – December 19, 2014
Excerpt: Understanding the evolution of language requires evidence regarding origins and processes that led to change. In the last 40 years, there has been an explosion of research on this problem as well as a sense that considerable progress has been made. We argue instead that the richness of ideas is accompanied by a poverty of evidence, with essentially no explanation of how and why our linguistic computations and representations evolved.,,,
(Marc Hauser, Charles Yang, Robert Berwick, Ian Tattersall, Michael J. Ryan, Jeffrey Watumull, Noam Chomsky and Richard C. Lewontin, “The mystery of language evolution,” Frontiers in Psychology, Vol 5:401 (May 7, 2014).)
Casey Luskin added: “It’s difficult to imagine much stronger words from a more prestigious collection of experts.”
http://www.evolutionnews.org/2.....92141.html

No less than Alfred Wallace, Darwin’s contemporary, considered our ability ‘do math’, among other things, to be proof that man must have an immaterial soul.

“Nothing in evolution can account for the soul of man. The difference between man and the other animals is unbridgeable. Mathematics is alone sufficient to prove in man the possession of a faculty unexistent in other creatures. Then you have music and the artistic faculty. No, the soul was a separate creation.”
Alfred Russel Wallace – 1910 – The World of Life – interview
https://evolutionnews.org/2010/08/alfred_russel_wallace_co-disco/

Of note:

“Human beings think abstractly, and nonhuman animals do not. Human beings have the power to contemplate universals, which are concepts that have no material instantiation. Human beings think about mathematics, literature, art, language, justice, mercy, and an endless library of abstract concepts.,,,
It is in our ability to think abstractly that we differ from apes. It is a radical difference — an immeasurable qualitative difference, not a quantitative difference.
We are more different from apes than apes are from viruses. Our difference is a metaphysical chasm.”
– Michael Egnor
https://evolutionnews.org/2015/11/the_fundamental_2/

3. 3
kairosfocus says:

H’mm, how about the 2-d vector approach?

Identify O, an origin and zero-point in a planar space. [ O ]
Extend Ox, a vector to x, defining a polar axis, abbreviate x
Define a/c rot by right angle j*x, giving an Oy axis
Now do j*j*x, we get -Ox, or -x
Substitute, j*j* –> j^2
So, j^2x = -x; or,
j^2 = -1
We have a natural interpretation of j as sqrt[-1]
Once we bring in rotating vectors