At Mind Matters News: Why would a purely physical universe need imaginary numbers?

_{Denyse O'Leary

February 17, 2022

Intelligent Design, Mathematics}

Share: Facebook; Twitter; LinkedIn; Flipboard; Print; Email

Our computers and the entire modern world depend on them, says science writer Michael Brooks in an excerpt from his new book:

In an excerpt from his new book, The Art of More: How Mathematics Created Civilization, science writer Michael Brooks offers the intriguing idea that the modern world arose from imaginary numbers:

But what does his claim that the numbers are “not some deep mystery about the universe” leave us? Recent studies have shown that imaginary numbers — which we can’t really represent by objects, the way we can represent natural numbers by objects — are needed to
describe reality. Quantum mechanics pioneers did not like them and worked out ways around them:

In fact, even the founders of quantum mechanics themselves thought that the implications of having complex numbers in their equations was disquieting. In a letter to his friend Hendrik Lorentz, physicist Erwin Schrödinger — the first person to introduce complex numbers into quantum theory, with his quantum wave function (ψ) — wrote, “What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. Ψ is surely fundamentally a real function.”
Ben Turner, “Imaginary numbers could be needed to describe reality, new studies find” at LiveScience (December 10, 2021)

But recent studies in science journals Nature and Physical Review Letters have shown, via a simple experiment, that the mathematics of our universe requires imaginary numbers.
News, “Why would a purely physical universe need imaginary numbers?” at Mind Matters News (February 16, 2022)

Takehome: The most reasonable explanation is that the universe, while physical, is also an idea, one that cannot be reduced to its physical features alone.

You may also wish to read:

Why the unknowable number exists but is uncomputable. Sensing that a computer program is “elegant” requires discernment. Proving mathematically that it is elegant is, Chaitin shows, impossible. Gregory Chaitin walks readers through his proof of unknowability, which is based on the Law of Non-contradiction.

Most real numbers are not real, or not in the way you think. Most real numbers contain an encoding of all of the books in the US Library of Congress. The infinite only exists as an idea in our minds. Therefore, curiously, most real numbers are not real. (Robert J. Marks)

and

Can we add new numbers to mathematics? We can work with hyperreal numbers using conventional methods. Surprisingly, yes. It began when the guy who discovered irrational numbers was—we are told—tossed into the sea. (Jonathan Bartlett)

Comments

Yes to Jerry's comment at 15, although I think both statements are related because one of the things driving our need to describe reality is the need to have useful descriptions. However it is often stated that we also have aesthetic needs to have elegant descriptions as well as useful ones. But no matter what you emphasize, describing reality is a human enterprise separate from reality being itself, so to speak, irrespective of our attempts to describe it.Viola Lee_{February 17, 2022
February
02
Feb
17
17
2022
08:17 AM
8
08
17
AM
PDT}

Because probably few people know what a Hilbert space is:
In mathematics, Hilbert spaces (named for David Hilbert) allow generalizing the methods of linear algebra and calculus from the finite-dimensional Euclidean spaces to spaces that may not have a finite dimension. A Hilbert space is a vector space equipped with an inner product which allows defining a distance function so that it becomes a complete metric space. They serve as a first template for extending the differential and integral calculus that is normally done in Rn, though this can be done more generally using normed spaces. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.
Those who are not fans of infinity will no doubt get their knickers in a twist. Also, math is not a spectator sport.JVL_{February 17, 2022
February
02
Feb
17
17
2022
08:13 AM
8
08
13
AM
PDT}

Polistra: We don’t need imaginary numbers. Everything can be described and calculated without them. They’re just a convenient tool for clearer symbolism and clearer thinking. Exactly so. Or, as Jerry put it: what we find useful to describe reality.JVL_{February 17, 2022
February
02
Feb
17
17
2022
08:08 AM
8
08
08
AM
PDT}

what we need to describe reality
Or should it be?
what we find useful to describe reality
jerry_{February 17, 2022
February
02
Feb
17
17
2022
07:54 AM
7
07
54
AM
PDT}

Re the OP and headline: there is a difference between what reality needs (which is a category error: reality has no needs, it just is) and what we need to describe reality. We should keep that in mind.Viola Lee_{February 17, 2022
February
02
Feb
17
17
2022
07:11 AM
7
07
11
AM
PDT}

^^^^^:
LIVING IN A QUANTUM WORLD - Vlatko Vedral - 2011 Excerpt: experiments now leave very little room for such processes to operate. The division between the quantum and classical worlds appears not to be fundamental. It is just a question of experimental ingenuity, and few physicists now think that classical physics will ever really make a comeback at any scale.,,, Thus, the fact that quantum mechanics applies on all scales forces us to confront the theory’s deepest mysteries. We cannot simply write them off as mere details that matter only on the very smallest scales. For instance, space and time are two of the most fundamental classical concepts, but according to quantum mechanics they are secondary. The entanglements are primary. They interconnect quantum systems without reference to space and time. If there were a dividing line between the quantum and the classical worlds, we could use the space and time of the classical world to provide a framework for describing quantum processes. But without such a dividing line—and, indeed, without a truly classical world—we lose this framework. We must explain space and time (4D space-time) as somehow emerging from fundamentally spaceless and timeless physics. http://phy.ntnu.edu.tw/~chchang/Notes10b/0611038.pdf Looking Beyond Space and Time to Cope With Quantum Theory - (Oct. 28, 2012) Excerpt: To derive their inequality, which sets up a measurement of entanglement between four particles, the researchers considered what behaviours are possible for four particles that are connected by influences that stay hidden and that travel at some arbitrary finite speed. Mathematically (and mind-bogglingly), these constraints define an 80-dimensional object. The testable hidden influence inequality is the boundary of the shadow this 80-dimensional shape casts in 44 dimensions. The researchers showed that quantum predictions can lie outside this boundary, which means they are going against one of the assumptions. Outside the boundary, either the influences can't stay hidden, or they must have infinite speed.,,, The remaining option is to accept that (quantum) influences must be infinitely fast,,, "Our result gives weight to the idea that quantum correlations somehow arise from outside spacetime, in the sense that no story in space and time can describe them," says Nicolas Gisin, Professor at the University of Geneva, Switzerland,,, http://www.sciencedaily.com/releases/2012/10/121028142217.htm?
bornagain77_{February 17, 2022
February
02
Feb
17
17
2022
06:19 AM
6
06
19
AM
PDT}

Off-topic: Sorry, but I'm looking for a place to post this interesting article: https://www.sciencealert.com/time-dilation-has-been-measured-at-the-smallest-scale-ever
Atomic Clocks Experiment Reveals Time Dilation at The Smallest Scale Ever DAVID NIELD 17 FEBRUARY 2022 In his theory of general relativity, Einstein predicted something called time dilation: the notion that two clocks under two different gravitational pulls will always tick at different speeds. The effect has been observed in many experiments since, but now scientists have recorded it at the smallest scale seen so far.
And
"This is a completely new ballgame, a new regime where quantum mechanics in curved space-time can be explored," says Ye. "If we could measure the redshift 10 times even better than this, we will be able to see the atoms' whole matter waves across the curvature of space-time. Being able to measure the time difference on such a minute scale could enable us to discover, for example, that gravity disrupts quantum coherence, which could be at the bottom of why our macroscale world is classical.”
[my emphasis]Viola Lee_{February 17, 2022
February
02
Feb
17
17
2022
05:40 AM
5
05
40
AM
PDT}

We don't need imaginary numbers. Everything can be described and calculated without them. They're just a convenient tool for clearer symbolism and clearer thinking.polistra_{February 17, 2022
February
02
Feb
17
17
2022
05:25 AM
5
05
25
AM
PDT}

Also, wrt the claim that God "created matter:" where? Where did God create "matter?" We can find any here. Where is it? When quantum physics experimentation killed materialism because it found that "matter" did not exist anywhere it looked, it also - unfortunately for many here - killed traditional dualism and any form of realism. All we are left with is some form of ontological idealism.William J Murray_{February 17, 2022
February
02
Feb
17
17
2022
04:27 AM
4
04
27
AM
PDT}

LCD, "Matter" and "the idea of matter" are two entirely different kinds of things. A material building is not an idea of a material building.William J Murray_{February 17, 2022
February
02
Feb
17
17
2022
04:15 AM
4
04
15
AM
PDT}

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/faculty/duwell/DuwellPSA2K.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/entries/qt-quantcomp/#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
As is fairly obvious, the 'infinite dimensional' Hilbert space corresponds to the Theistic attribute of omnipresence. And the infinite information 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. https://study.com/academy/lesson/omnipotent-omniscient-and-omnipresent-god-definition-lesson-quiz.html
In essence, the wave function is, basically, mathematically described as being one of "God's thoughts' prior to its collapse to its finite 'material' state. Which 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.
bornagain77_{February 17, 2022
February
02
Feb
17
17
2022
04:07 AM
4
04
07
AM
PDT}

Well despite Michael Brooks's complacency about mathematics describing the universe, 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, (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. http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
Likewise, Albert Einstein is also on record as to regarding the applicability of mathematics to the universe as a ‘miracle’. Einstein even went so far as to chastise ‘professional atheists’ in the process of calling it a ‘miracle’.
On the Rational Order of the World: a Letter to Maurice Solovine – Albert Einstein – March 30, 1952 Excerpt: “You find it strange that I consider the comprehensibility of the world (to the extent that we are authorized to speak of such a comprehensibility) as a miracle or as an eternal mystery. Well, a priori, one should expect a chaotic world, which cannot be grasped by the mind in any way .. the kind of order created by Newton’s theory of gravitation, for example, is wholly different. Even if a man proposes the axioms of the theory, the success of such a project presupposes a high degree of ordering of the objective world, and this could not be expected a priori. That is the ‘miracle’ which is constantly reinforced as our knowledge expands. There lies the weakness of positivists and professional atheists who are elated because they feel that they have not only successfully rid the world of gods but “bared the miracles.” -Albert Einstein http://inters.org/Einstein-Letter-Solovine
And the last time I checked, miracles are considered to be the work of God. Moreover, what is so 'non-complacent' and interesting about the use of imaginary numbers in quantum mechanics 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 are 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, the wave function, prior to collapse of the wave function, is mathematically required to be described by a '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, http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html 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/Wave_function#Wave_functions_as_an_abstract_vector_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.. https://books.google.com/books?id=GKBwKCma1HsC&pg=PA44
bornagain77_{February 17, 2022
February
02
Feb
17
17
2022
04:06 AM
4
04
06
AM
PDT}

doesn’t mean that matter itself is an idea
:lol: Before anything existed ,indeed the matter was just an idea in the mind of God then that idea is transformed in real matter and we humans have thoughts about that matter because we see it.Lieutenant Commander Data_{February 17, 2022
February
02
Feb
17
17
2022
04:02 AM
4
04
02
AM
PDT}

Now, if by physical News does not mean "matter," then yes, it can make sense to say that the physical world is an idea - not "also" an idea. (Cue Bernardo Kastrup's book, The Idea of the World.) If what we experience as the physical world is the product of the processing of information into mental experience via mental rules, then of course it makes sense that the physical world we experience exhibits those rules, such as logic, geometry and mathematics. But, that's an entirely different thing than saying "matter exists" and that matter is also an idea.William J Murray_{February 17, 2022
February
02
Feb
17
17
2022
03:44 AM
3
03
44
AM
PDT}

That "matter" (whatever that's supposed to be, since we have yet to find any) can be used to transmit an idea from one mind to another doesn't mean that matter itself is an idea. To say that "matter" (if that is what News is using the term "physical" to mean) is also an idea, she might as well be saying that matter is also not-matter - a logical self-contradiction.William J Murray_{February 17, 2022
February
02
Feb
17
17
2022
03:21 AM
3
03
21
AM
PDT}

F/N: Complex numbers address rotations and oscillations so their presence is in effect saying waves and associated energy are at the core of physical processes. I have argued that the rotating vector view is perhaps a clearer way to conceive of complex numbers. Polynomials and x axis cuts are not the fundamental view of reality. Exponentials including complex exponentials, which take in sine and cos etc, are powerful characteristic functions for many domains. I almost used the half german word there but view it as excessively mystifying. KFkairosfocus_{February 17, 2022
February
02
Feb
17
17
2022
03:14 AM
3
03
14
AM
PDT}

WJM, matter is often infused with linguistic [math is inter alia a language] information. Computers are classic familiar cases but also see D/RNA and proteins as opposed to random folds such as prions. Classically, mind is immaterial and infuses body, too. KFkairosfocus_{February 17, 2022
February
02
Feb
17
17
2022
03:09 AM
3
03
09
AM
PDT}

The most reasonable explanation is that the universe, while physical, is also an idea, one that cannot be reduced to its physical features alone.
That depends on what you mean by the term "physical." If by "physical" you mean "material," then no, it does not make sense to say that something material "is also" an idea. Ideas exist in mind. Matter does not exist in mind.William J Murray_{February 17, 2022
February
02
Feb
17
17
2022
02:55 AM
2
02
55
AM
PDT}

Prev 1 … 9 10 11

You must be logged in to post a comment.

Leave a Reply