April 10th, 2019 was my 70th birthday. On that day, the world got an extraordinary gift, the first image of a black hole. I was hoping for Sagittarius A*, the center of our galaxy, but it is M87, one of the largest found. In the 1980s these were called Quasars or, originally, Messier Objects after Charles Messier in 1771, the M in M87. That these were both black holes was then still speculation.
In 1983, I read about these objects. Enthralled, I searched the clear dark summer nights of Catalina, AZ for the center of the Milky Way. Sagittarius, half-man, archer. I found it, a non-descript smudge in the heavens.
My obsession then was not just black holes but singularity and how it expresses itself in mathematics and real life. Divide a number by zero on a calculator and "Cannot Divide By Zero" appears. In the old calculators, you would get nonsense or a lock-up. Dividing by zero is a singularity, a black hole where arithmetic fails. In the 18th century, this worried many mathematicians. Russel and Whitehead spent decades trying to resolve this and failing. Then, in 1930, three years before Hitler took power, Kurt Godel proved that any mathematics as complex as arithmetic has to contain singularities. Godel was not Jewish but his association with "Jewish Science" condemned him. After the assassination of one of his Jewish colleagues by right-wing students in Vienna in 1936, he fled Europe. In 1940, he settled at Princeton with Einstein and others.
In 1915, Karl Schwarzschild found exact solutions to Einstein’s equations for general relativity. In 1919, Einstein realized that black holes were possible but did not think they existed. A singularity, in physics, is surrounded by a region, a disk, where measurement could no longer occur, a black hole. The singularity itself arose from the geometric structure of gravity, a point where this structure fails. A mathematical proof of the incompleteness of mathematics is a real entity that represents the incompleteness of our physical world.
Other singularities work in the physical. It is impossible to describe rotations in 3D space without a singularity, a divide by zero error. One example of this is gimbal lock, in which a gyroscope rotating in more than two rotational axes at once will lock up. This happened on Apollo 13 where a gimbal-locked gyroscope heated up and exploded a leaking oxygen tank. There is a solution. Discovered in 1843 by William Hamilton while gazing at the Dublin skyline from the Broome Bridge. Called quaternions, they are a 4D solution (with two imaginary axes) to a 3D problem. Science ignored his solution in favor of a simpler, but flawed one using Euler angles. In the ’80s, computer gamers wanting to model the random spin of rocks in space (Astroids!) rediscovered quaternions. Turns out this is a computationally superior solution. Our smartphones use quaternions to calculate screen rotation.
In 1967, a researcher in Montreal was experimenting on himself, generating electrical shocks to his heart. One shock knocked the dynamics of his heart into what is a singularity in the underlying equations [1] and killed him. In 1984, I did a performance piece about this at the old Dinnerware Art gallery on Congress St in downtown Tucson. John played along on keyboards and Greg videotaped it. A mixed bag, vaguely about our anxieties of the techno future:
“Give a Rob a Job!”
Advertising:
“WarfareIn Spray” Near th club Under th freeway Around th mall Kills ugly poor on contact
WarfareIn Spray Cleans yr city Kills th poor!
I asked the audience to write down their vision of the future. My favorite:
“Bodiless lightbulbs take up less room on the bus.”
Then I talked about my obsession with singularity and that person in Montreal. My conclusion:
“We have an emptiness in our hearts.”
In the mid-1970s Steven Hawking and Jacob Berkenstein [2][3] formulated the theory that black holes radiated a small amount of energy. The larger the black hole, the smaller this energy is, smaller than the background noise of the Universe. Only very tiny black holes can be successfully measured and none have been found. Attempts by CERN to create tiny black holes have failed. Enter analog black holes. Analog black holes mimic gravitational black holes except the singularity did not occur because of gravity, but some analog. This analog experiment was attempted first in 2010 with optical light pulses and then again this summer (2019) with sound. Imagine a sonic black hole! Both experiments produced possible Hawking Radiation and the second one confirmed that the radiation was thermal, a key conjecture for the Hawking paradox. This paradox states that if the radiation is thermal, it can contain no information and thus violate a basic tenet of quantum theory. Whether analog black holes are equivalent to gravitational black holes is a big question but again, more singularities and paradoxes.
What is the nature of time? Time to humans is different from the time of physics. To us, time is one-way, the arrow of time. Physics demands a negative time, a timeline. Once again a paradox. Physics starts at the Big Bang, so how can a continuous line just begin? Sounds more like the human concept of time. Notice also The Big Bang, another singularity. Hawking, of course, has something to say about this [4]. Human time starts as a singularity but physical time, a different thing altogether resides fully contained within the Universe because the Universe is both finite and unbounded. Think of a great circle on a surface like a shuttlecock, which expands outward and then inward to a rounded pole. Think of the singularities at the poles as the Big Bang and the Big Crunch and human time’s arrow going from Bang to Crunch, the Universe first expanding and then contracting. Still singularities with all their strangeness and just one of a raft of theories. Recent data suggest an ever-expanding Universe. Hawking, of course, we should always take it seriously.
So are singularities real? A much deeper question: Is mathematics real? Are mathematical entities themselves out of the realm of human affairs as ideal representations of reality itself. This was Plato’s concept and that of current-day Neo-Platonists. This view is generally that of a great deal of scientists and mathematicians. In the last 20 years, cognitive scientists and neuroscientists have challenged mathematics as reality. Mathematics to them is a human endeavor that grew out of the evolution of the human brain and the constraints the human brain places on thought. Would other sentient species have the same mathematics? What would octopus mathematics be? A being made of smoke? Could such sentience exist that does not compartmentalize? Does this compartmentalization define our geometry, our difficulty with defining flow?
So what about the data on how we perceive numbers? [5] Babies and some animals have the genetic ability to count small natural numbers and even add very simple sums. Beyond simple addition, it becomes a processing problem as wrong answers rise exponentially. Even older cultures have words for a couple of natural numbers: one, two, three, and many. A number sense is inherent but complex arithmetic and mathematics are an evolved and evolving part of human culture. Another astounding finding is that babies have an inherent sense of physics! By this, I mean the idea of boundary and continuous motion. In hindsight, I can see where this comes from. Humans have been a migrating and moving species since the beginning and all biological forms need to maneuver within the confines of their environments. Mathematics, like language, is a human invention. I believe that the question we have to ask is, “If we rewound human evolution would we discover the same mathematical objects?” If the answer is yes then does this make these objects some universal outside human reality? Can we even answer this question?
Is mathematics a language? Galileo, Gibbs, and other mathematicians have said yes. History is replete with instances of symbolic changes leading to breakthroughs, one important one the introduction of Arabic numerals to Europe by Fibonacci of Pisa in 1202. Some talk about mathematics as a universal language, again, Neo-Platonism creeping in.
Noam Chomsky [6] points out that without language a person cannot do mathematics. It cannot be purely symbolic. There are some purely symbolic proofs but not many of them. So why does the ability to do mathematics exist in so many different languages? First, as I have mentioned before, different cultures and different mathematicians have discovered ways of symbolizing mathematical objects that have influenced mathematics and have become part of the mathematical lexicon. Secondly, in Chomsky’s current ideas [7] on the evolution of language, he divides human language into two parts: external language, a linear spoken language of which there are thousands, and internal language, a non-linear human thought process. Perhaps mathematics is something that has evolved out of this internal language, thus giving it at least a human universality. Mathematics, like language, could be a spandrel [8], a byproduct that evolved out of whatever adaptive value fixed the trait of internal human thought. Chomsky speculates that the adaptive value might be an improvement in our geospatial sense.
What of its use in science, the so-called applied mathematics? Some would rightly say that mathematics is a model, an abstraction of reality but not reality. Still, the implications of a mathematical model extrapolating into things we really had no idea about is astounding. Without Einstein’s predictions, based on a mathematical model, of gravitational waves, why would we have looked for them at all? Chomsky argues that [6], in a Cartesian sense of perceived reality, there are many things that are beyond our understanding. An example he gives is gravity. Newton discovered it and Einstein geometrically explained it as space-time. What exactly is it? Maybe we cannot know in any way except by science: experiment, modeling, and abstraction. Why is this so? I would reject the magical notion that the Universe wants to be what we want it to be. But black holes? We have limits to our cognition. These limits expand using abstraction. Still, thinking about this can give me chills down the spine. The only thing I can do is paraphrase Einstein, “spooky”
One mathematical way out of singularity is to move the problem into a generalized abstraction where it disappears. This is how the problem of 3-dimensional rotation is resolved with 4-dimensional quaternions. All solutions to quadratic equations require the two complex planes and the imaginary but very useful square root of the negative one. Another example is the attempt of string theory to extend the theory of gravity and describe it at the quantum level. These can be a way around singularities although Godel’s Theorem says this new abstraction requires its own singularities.
What is mathematics? I agree with the notion that it is an evolved abstraction and a product of our unique human facilities. It does allow us to model basic truths about our reality. As humans, our minds can feel split from our bodies but we also feel this split between “I” and others, and between “I” and everything else. We have this genetically baked-in sense of basic numbers and a basic sense of physical space. A connection to a world we have to be a part of. With our unique ability to abstract, model, and expand these abstractions we have evolved mathematics and science.
All humans must deal with their own singularities, that of birth and death. At this time of my life spirals towards my own event horizon, time dilating, weeks flipping by like days. What does science say about this? Not much. Ceasing to exist, is not very satisfying, but if that is it, I accept my human fate. I don’t have a whole lot of choice.
1. “Geometry of Biological Time” – Authur T. Winfree
2. “Black Holes and Entropy” – Jacob Berkenstein
3. “Black Hole Explosions” – S. W. Hawking
4. “The Beginning of Time” – S. W. Hawking
5. “The Number Sense” - Stanislas Debaene
6. “Why Only Us?” – Noam Chomsky
7. “What Kind of Creatures are We?” – Noam Chomsky
8. “The Spandrels of San Marco and the Panglossian Paradigm: A Critique of the Adaptationist Programme” – Stephen Jay Gould and Richard C. Lewontin
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