Wednesday 16 August 2023
On the role of imaginary numbers in describing/explaining the real world.
Intelligent Design in Imaginary Numbers
The concept of imaginary numbers can seem like an especially esoteric or detached-from-reality notion, perhaps dreamt up in the overactive imaginations of mathematicians. The absurdity of the concept doesn’t improve when one learns that the origin of imaginary numbers comes from trying to take the square root of a negative number.
Positive numbers have a pleasing solidity to them, in that they correspond to countable “things.” Five apples, ten people, and even fractional amounts, such as 2.2 pounds of rice. The number zero becomes less concrete, referring merely to the absence of something — such as, zero cows in the barn. Although most people are more comfortable with negative numbers than imaginary numbers, when we encounter a number like negative five, we have already departed from a simple association of the number with tangible things.
The ancients rejected negative numbers as being without meaning because they could see no way physically to interpret a number that is less than nothing….As late as the sixteenth century we find mathematicians referring to the negative roots of an equation as fictitious or absurd or false.1
Nonetheless, we learn fairly early on in school how to do simple arithmetic with negative numbers. In particular, we learn that the square of any (real) number is positive. So, how could a negative number even have a square root? Surely, the label “imaginary” is appropriately applied to the result of any such endeavor.
Why Bring Up Imaginary Numbers?
I raise the issue simply because imaginary numbers have shown themselves to facilitate, despite their moniker, mathematics without which we would be seriously impeded in our ability to describe and quantify reality. To understand their nature, it will help to introduce complex numbers. If the square root of -1 is designated with the letter i, then a complex number, z, might be written as z = a+bi, where a and b are real numbers. So, a complex number has two parts, a real part, a, and an imaginary part, b.
We are accustomed to real numbers being laid out on a “number line,” typically with the positive numbers to the right of zero and the negative numbers to the left of zero. In this sense, real numbers are one-dimensional, with a number corresponding to every point on the line. Analogously, it can be useful to consider complex numbers as two-dimensional quantities, with a complex number corresponding to every point on a plane.
This is exactly how complex numbers are visualized — as points on a so-called complex plane. The familiar number line makes up the x-axis, or real axis, so any point on this line has its imaginary part b=0. Points in the plane above or below the real axis have a positive or negative imaginary part, respectively. Since complex numbers correspond to the points in a plane and the entire real number axis is just a single line in that plane, it makes sense that learning to work with complex numbers should lead to expanded functionality in mathematics.
And Such Is the Case!
What kind of practical benefit has come to us through the use of complex numbers?
When people first considered taking square roots of negative numbers, they felt very uneasy about the problem….They certainly would not have believed that the new numbers could be of any practical use. Yet complex numbers are of great importance in a variety of applied fields; the electrical engineer would, to say the least, be severely handicapped without them.2
Solving mathematical problems involving fluid flow, oscillatory motion, and quantum mechanics are all facilitated through theorems and procedures for handling functions of complex variables. At this point, someone might object to all this by asserting that the physical world corresponds to real things, so how could invoking imaginary numbers avoid a departure from reality? Remember, however, that the form of a complex number, z = a+bi, contains two real numbers, a and b, and it’s these real numbers that end up corresponding to actual properties of real-world phenomena.
An electrical engineer can analyze an alternating current circuit by assuming a complex-number form of the electrical current. The actual current corresponds to just the imaginary part, but the complex resistance, known as the impedance, has physical meaning in both its real (ohmic resistance) and imaginary parts (capacitive and inductive impedance).
The term, imaginary, for the part of a complex number that lies off the real number axis may contribute to our sense that mathematicians are dabbling in something out of a fairy tale. René Descartes, in 1637, is credited with being the first to assign this label to results involving the square root of a negative number.
Before Descartes’ introduction of this term, the square roots of negative numbers were called sophisticated or subtle.3
I think these earlier labels would’ve actually been more appropriate, considering the enormous benefit complex number theory has given us in mathematical descriptions and calculations of many aspects of the physical world. Exploring the possibilities found in the realm of complex numbers (perhaps not too dissimilar, after all, to an adventure in fairy land) we reach a particularly magical point known as Cauchy’s integral formula. This ultimately useful mathematical result comes from doing calculus with complex numbers.
Almost Insurmountable Difficulty
Anyone who has taken a calculus course learns the process of integration for finding the area under the curve described by a mathematical function. One also learns, however, that integrating certain functions poses almost insurmountable difficulty. Cauchy’s integral formula, however, melts the difficulty of doing a wide class of integrals by taking advantage of the two-dimensional number space of complex numbers.
When I’ve taught complex variable theory as part of an advanced mathematics course for physics majors, introducing this topic to students feels something like giving them a secret power to do math. The author of the book already referenced on imaginary numbers describes his first encounter with Cauchy’s formula while studying at Stanford University:
For me, complex function theory was a revelation bordering on a mystical experience….With Cauchy’s theory of complex integration one could almost without effort, calculate the values of a seemingly endless number of incredibly odd, strange, and downright wonderfully mysterious-looking definite real integrals….Such calculations were to me, then, seemingly possible only if one had the powers of a sorcerer.4
One example of a practical application of Cauchy’s formula for complex variables is in solving an integral from the equation of motion for a planet with an elliptical orbit, which leads to the famous result of Kepler’s Third Law of Planetary Motion. A completely different use of complex numbers arises in the research field of computational nanoelectronics. Analyzing electron transmission through nanoscale structures routinely includes considering information revealed by the imaginary part of the complex variable associated with the energy of the electrons. The following statement is from an article on currents circulating around a benzene ring molecular structure.
The positions of the transmission zeros and poles in the complex energy plane, and their possible interference or even complete cancellation of each other, are shown to correlate with the amplitude and resonance structure of the circular transmission resonances.5
What Does All This Have to Do with Intelligent Design?
I think that the discovery of the almost unbelievably practical applications of complex number theory — derived from the bold intellectual extension of real numbers into the two-dimensional realm — is so “wonderfully mysterious” that it seems more consistent with a buried treasure we were intended to find than a prodigious bit of unexpected luck encountered by accidental beings in a happenstance universe.
For centuries, mathematicians discounted the “imaginary” solutions of equations involving the square roots of negative numbers. But when these nonsensical outcomes of mathematical manipulations were tentatively embraced and explored, they revealed a hidden usefulness without which our quantitative formulations of reality would be severely curtailed. If the extension of our familiar one-dimensional number line to a two-dimensional complex number plane proved fruitful, who’s to say that a further extension of numbers to encompass a complete three-dimensional volume might not disclose further unimaginable treasures?6
Notes
Paul J. Nahin, An Imaginary Tale: The Story of √(-1), (Princeton University Press, 1998), 5-6.
Mary L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed. (Hoboken, NJ: John Wiley & Sons, Inc., 2006), 47.
Boas, Mathematical Methods in the Physical Sciences, (2006), 6.
Nahin, An Imaginary Tale, (1998), 188.
Eric R. Hedin, Arkady M. Satanin, and Yong S. Joe, “Circular transmission resonances and magnetic field effects in a ring of quantum dots connected to external leads in the meta-configuration,” Jnl. of Computational Electronics, June 2019, Volume 18, Issue 2, pp 648–659. DOI 10.1007/s10825-018-01291-2.
A speculation on what a 3-dimensional number might correspond to is suggested by the mathematical impossibility of the square root of a negative number, which led to the two-dimensional plane of complex numbers. Another mathematical “impossibility” is the problematic case of division by zero, giving infinity. Perhaps the three-dimensional space above and below the complex number plane is populated by quantities (call them complete numbers) that correspond to complex numbers divided by zero. This hint is not without basis, as complex numbers that lead to infinities in real quantities (such as the transmission amplitude) have turned out to have practical significance (signifying the energy width of the transmission resonance). Furthermore, Cauchy’s formula, referred to above, actually exploits the existence of points in the complex plane that lead to infinites.
Bioelectricity vs. Darwinism.
Bioelectricity Gives Biologists a Jolt
We’ve explored bioelectricity in cells. We’ve looked at bioelectricity within the human body. Now, functional use of “electrical engineering” is being found in the realms between.
Physicists learn about electrostatics, the laws governing stationary charges. Then they learn about electrodynamics, the laws governing moving charges. Biologists are finding that life utilizes both systems of laws at all scales, from within the cell to tissues, organs, and entire organisms. Here are some recent discoveries in the emerging science of bioelectricity.
Electric Transportation
How does that tick jump from its twig onto your clothing as you walk through brush? The answer, says Current Biology, is by hopping on an electrostatic bullet train. A cow or other host animal walking through the bushes carries a net static charge. The tick, regardless of its own charge polarity, is “pulled by these electric fields across air gaps of several body lengths.”
Images in the paper show that a rabbit or cow literally glows with an electric field as it walks through vegetation. “Live ticks are passively attracted by the electric fields of their hosts,” the scientists found through experiment and measurement. It may not be good news for us, but the discovery suggests ways to fight back.
We also find that this electrostatic interaction is not significantly influenced by the polarity of the electric field, revealing that the mechanism of attraction relies upon induction of an electrical polarization within the tick, as opposed to a static charge on its surface. These findings open a new dimension to our understanding of how ticks, and possibly many other terrestrial organisms, find and attach to their hosts or vectors. Furthermore, this discovery may inspire novel solutions for mitigating the notable and often devastating economic, social, and public health impacts of ticks on humans and livestock.
Electromagnetic induction was one of the major discoveries made by the devout scientist Michael Faraday in 1831 (published independently in America the following year by another devout scientist, Joseph Henry). Yet here we see a tiny arachnid making use of electromagnetic induction. We can’t blame the tick for this trick. It doesn’t intentionally carry disease germs. It’s just taking advantage of a transportation system to hitchhike around, the way a cocklebur does when its Velcro-like seeds latch onto the fur of a passing cow. Pretty clever, actually.
Roundworms also know about this trick. In another paper in Current Biology, scientists wondered why dauers [larvae] of the common roundworm C. elegans come equipped with electrical sensors. The answer: “electroreception helps these microscopic worms to attach themselves to insects for transportation.” Leave it to scientists to design clever experiments to test and measure this trick! They charged bumblebees up to 724 thousand volts per meter!
The electric field strength (200 kV/m) required to induce leaping behavior in C. elegans far exceeds the upper limit of those seen in aquatic animals. It is also worth noting that air is a good electrical insulator compared with the aquatic environment, which makes it possible for terrestrial animals to carry significantly more electrostatic charges. Thus, it is highly possible that dauers can electrostatically interact with other animals in nature. To directly test this theory, the authors used bumblebees that are known to be highly electrostatically charged in the wild. These bumblebees were artificially charged by rubbing them against a Canadian goldenrod flower. Further experiments confirmed that the charge of bumblebees obtained in the lab was comparable to those observed in the wild. When the charged bumblebees were put close to the nictating dauers, leaping behavior was detected (Figure 1A). The electric field strength calculated was about 724 kV/m, exceeding the 200 kV/m leaping threshold. Strikingly, as many as 80 dauers were able to leap at the same time (Figure 1B). The leaping distance between dauers and bumblebees was about five times the dauer body length, which is also biologically meaningful.
Plant Electrodynamics in the Venus Flytrap
The involvement of electricity in the well-known traps of Dionaea muscipula, the Venus flytrap, are becoming increasingly appreciated. Researchers at Linköping University in Sweden speak of the flow of electrical signals in these amazing plants — and probably to some extent in all plants.
Most people know that the nervous system in humans and other animals sends electric impulses. But do plants also have electrical signals even though they lack a nervous system? Yes, plants have electrical signals that are generated in response to touch and stress factors, such as wounds caused by herbivores and attacks on their roots. As opposed to animals, who can move out of the way, plants must cope with stress factors where they grow.
For a plant studied by Darwin, it’s remarkable how much remains unknown about electrical propagation in the Venus flytrap. How can a plant, without neurons, conduct electricity? This team found some new things.
Electrical signalling in living organisms is based on a difference in voltage between the inside of cells and the outside environment. This difference in voltage is created when ions, i.e. electrically charged atoms, are moved between the inside and the outside of the cell. When a signal is triggered — for instance by mechanical stimulation in the form of bending a sensory hair — ions flow very fast through the cell membrane. The rapid change in voltage gives rise to an impulse that is propagated.
Their results, published in Science Advances, add to knowledge about plant electrophysiology. They carefully observed the “action potentials” of the traps, and how the signal is propagated in the leaf. Using 30 delicate electrodes arranged in a “neurogrid” attached to the inside of the trap, the team found that the action potential (AP) spreads first, followed by a calcium ion wave. It begins at the trigger hair, as expected, but then propagates radially outward at 2 cm/s across both lobes of the trap without a particular direction.
Biologists have long known that the trigger hairs must be touched twice within thirty seconds for the trap to close. This trick allows the traps to ignore non-living stimuli, but how that threshold is encoded is not clear. Did I hear codes?
In addition, any combination of hair stimulations induces faster AP propagation during the second stimulation, indicating that the excitability information must be encoded across the entire trap, rather than coupled with the stimulated hair alone. The nature of this information encoding remains unclear.
This is the first time that biologists have applied methods of measuring electrical transmission in plants that have normally been performed on animals, such as on rodent brains. The authors are excited about the possibilities of learning more about plant electricity. What about Darwinism? They apparently have no need of that hypothesis.
More on Microbes
More findings about intercellular electricity in bacteria have come forth. In May, Phys.org published the “First experimental confirmation that some microbes are powered by electricity.” In “electrosynthesis,” bacteria can make alcohol using carbon dioxide and electricity, but how they do it has been unclear. The new research in Germany was “able to confirm experimentally for the first time that the bacteria use electrons from hydrogen and can produce more chemical substances than previously known.” A report on this research at ChemEurope.com says it may lead to harnessing bacteria to make useful chemicals for industry. Feed the microbes hydrogen and watch them run their power plants.
A month earlier, Duke University reported that a “previously unknown intracellular electricity may power biology.” Specifically, that article says that electric fields may underlie the formation of biological condensates that bring interacting molecules together. Read about condensates in my earlier article here.
The future of bioelectricity looks bright. Here, an unexpected series of discoveries opened the door to new ways of looking at biological processes. And with it, as in previous revelations, biologists are finding codes, communication of information, and exquisite engineering.
Conserving function from root to branches of a "tree of life"
Quiz: Is This a Prediction from the Tree of Life?
The three-domain Tree of Life (TOL) geometry below is textbook orthodoxy, due largely to the work of the late Carl Woese on patterns in 16S rRNA.
Okay — here’s a quiz:
On the basis of this TOL geometry, what would you predict as the genome content for the Last Bacterial Common Ancestor (LCBA) — the hypothetical entity occupying the branch near the red circle?
Or, to turn the puzzle around — this is the actual quiz question — if the LCBA existed, what would you expect to find in the “core genomes” of bacterial groups at the tips of the branches within the domain Bacteria? Should every bacterial group conserve the same core set of orthologous genes?
Give your answer, then read on…
Now refer to a new article in Genome Biology, “Reconstruction of the last bacterial common ancestor from 183 pangenomes reveals a versatile ancient core genome” (open access). From the article:
core genomes differed greatly at the gene-level, with no single OG [orthologous group] observed in all 183 core genomes, consistent with previous observations that biochemical functions rather than individual genes tend to be conserved.”
Conservation of function, but not genes, can be understood with an analogy to natural language. Consider two sentences:
The hulking great lorry skirted round an enormous chasm in the roadway.
The massive truck dodged a huge pothole in the street.
The individual words in these sentences — e.g., “lorry” vs. “truck” — are functional synonyms, but not direct variants of each other (such as “color” and “colour”). Varying “lorry” letter by letter will take a LONG way to get to “truck,” with nearly all of the intermediate character strings not referring to large commercial vehicles.
Is conservation of function, without conservation of genes, a TOL prediction? It would seem not.
Physicists still on the quest for JEHOVAH'S Mind?
Barbieri’s Dilemma: Biological Information without Intelligence
Last week, we looked at a most interesting paper, by University of Ferrara theoretical biologist Marcello Barbieri. He was discussing the discomfort biologists feel with the vast amount of information in life forms, which — in the view of many — “does not really belong in science.” The divide, he says, is between biologists who insist that life is chemistry only and those who, like him, see it as chemistry plus information. The problem is obvious: Information is by its nature immaterial. It is measured in bits, not kilograms or joules. It is understood in terms that invoke mathematics and probability more than chemistry and physics.
A physicalist biologist ignores or discounts the role of information. Barbieri wants to show that information is fully compatible with current assumptions in biology. So, in his 2016 paper, he tries his hand at incorporating it into a materialist origin-of-life story:
It comes from the idea that life is artefact-making, that genes and proteins are molecular artefacts manufactured by molecular machines and that artefacts necessarily require sequences and coding rules in addition to the quantities of physics and chemistry. More precisely, it is shown that the production of artefacts requires new observables that are referred to as nominable entities because they can be described only by naming their components in their natural order. From an ontological point of view, in conclusion, information is a nominable entity, a fundamental but not-computable observable.
BARBIERI MARCELLO 2016 WHAT IS INFORMATION? PHIL. TRANS. R. SOC. A.3742015006020150060
HTTP://DOI.ORG/10.1098/RSTA.2015.0060, 13 MARCH 2016
Wait. Artifacts, are made by intelligent, purposeful agents. Almost all artifacts are, of course, developed by humans, using abstract thinking. Some other animals use artifacts in the form of simple tools. But these life forms are not prebiotic chemistry; they are already highly developed life forms.
That Raises a Question
If the James Webb Space Telescope required many human intelligent agents, why should we simply accept that the development of early cells, also complex, could be managed by chemistry alone — when such a development has never happened in nature since? That is, there is no spontaneous generation, as far as know.
Barbieri uses terms like “manufactured” and “naming” freely but they are only meaningful if we suppose intelligent agents. Some may blame the limits of human language for that. But there is a reason why language features those limits: When writing for a serious purpose, we don’t attribute decisions that clearly require intelligence to non-intelligent agents.
A Speculative History
Barbieri sketches a speculative history of the molecular machines that, he argues, preceded life:
The origin of protein life, on the other hand, was a much more complex affair, because proteins cannot be copied and their reproduction required molecular machines that employ a code, machines that have been referred to as codemakers. The evolution of the molecular machines, in short, started with bondmakers, went on to copymakers and finally gave origin to codemakers.
BARBIERI, 2016
It’s an interesting story. It sounds a bit like the history of a human industry. Which again raises the problem: If these makers were themselves unintelligent and non-purposeful, some other entity must have been using them as instruments. Complex, specified artefacts don’t just “happen” to get built. When Barbieri tells us that “The divide between life and matter is real because matter is made of spontaneous objects whereas life is made of manufactured objects,” he sounds like an intelligent design theorist. Unpopular but right.
The resemblance becomes even clearer when he offers,
Both the sequence of nucleotides in a gene and the sequence of letters in a book are carrying information: hereditary information in genes and syntactic information in language. In both cases, the information is digital (because it is made of discrete units) and linear (because the units are arranged in a linear order).
A book? Yes. He cites photos and music as well. Then, “Finally, we can represent letters, numbers, pixels, musical notes and many other symbols with the characters of computer language… ”
Indeed We Can!
Immaterial ideas can be represented in any number of ways. But now, as to the origin of immaterial ideas…
Barbieri, of course, would not want to associate himself with intelligent design theory! Instead, he cites in his defense eminent biologist Ernst Mayr (1904–2005): “There is nothing in the inanimate world that has a genetic program which stores information with a history of three thousand million years!” Also, information theorist Hubert Yockey (1916–2016) who “tirelessly pointed out that no amount of chemical evolution can cross the barrier that divides the analogue world of chemistry from the digital world of life, and concluded from this that the origin of life cannot have been the result of chemical evolution.”
He senses that there is something missing from Yockey’s summation:
At this point, one would expect to hear from Yockey how did linear and digital sequences appear on Earth, but he did not face that issue. He claimed instead that the origin of life is unknowable, in the same sense that there are propositions of logic that are undecidable. This amounts to saying that we do not know how linear and digital entities came into being; all we can say is that they were not the result of spontaneous chemical reactions. The information paradigm, in other words, has not been able to prove its ontological claim, and that is why the chemical paradigm has not been abandoned.
BARBIERI, 2016
Exposing the Problem
In doing so, Barbieri exposes the problem: “Life is chemistry” is an accepted dogmatic proposition that flies in the face of the evidence of large amounts of information in life that did not get there only by chemistry.
But defenders of the evidence for information, like Barbieri, are stymied. Complex, specified information does not originate without underlying intelligence. But the defenders do not wish to acknowledge that intelligence and anyway, they wouldn’t be allowed to. They would be speaking an unspeakable truth and would lose their membership in the establishment.
Therefore, it is said, they have not proved their case. Either the information does not exist or anyway, it can be treated as if it did not. And all is well.
Note: As it happens, intelligent design theorists consider Yockey’s work a “primary contribution to the ID movement,” though an unintentional one, to be sure.
Next and last in this three-part series: Can information be separated from intelligence? Barbieri’s dilemma seems to be that he can’t give information its rightful place in life without acknowledging truths he cannot afford.
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