Math History: The Secret Lives of Numbers (Kitagawa & Revell)

Kate Kitagawa & Timothy Revell. The Secret Lives of Numbers (2003) A history of mathematics taking all currently known mathematical texts into account. The Eurocentric view of mathematical development is shown to be egregiously wrong-headed. Miscellaneous theorems (and some proofs) were discovered or invented in many different places at many different times well before Euclid’s demonstration of the logical coherence of all mathematics. Algorithms for solving trade and other complicated problems ditto. The notation that freed European mathematicians to discover number theory was invented in India, and brought to Europe by Arabs. The need to plan planting and seed-time prompted the study of astronomy, which was perhaps the first science to be mathematised. Either it, or geometry, needed for land surveys. Formal mathematics is at least as old as writing.
     The history of mathematics is not even a winding road; it’s a maze of paths leading in all directions with surprising shortcuts, connections in unexpected places, and backtracking. What’s constant is that whenever possible mathematicians exchanged ideas and knowledge. Powerful rulers recognised the value of mathematics and other knowledge, and sponsored the collection of texts, and their study and collation by the best minds they could attract. And ever and again, barbarians with limited insight into anything beyond their immediate goals of getting treasure and women destroyed those collections. We owe a great debt to the scholars who preserved what knowledge they could and taught their students to do likewise.
     I think that Kitagawa and Revell deprecate Euclid’s achievement. True, pretty well every theorem he proved in his books, and many of the proofs themselves, were known before him. Compilations of all known mathematics were made centuries before him. But he seems to have been the first one to organise all known mathematics into a logical system, in which rules of inference applied to a handful of axioms, carefully defined, would connect all theorems. It is the critique and emulation of his methods that has led to new mathematics.
     I also think that Kitagawa and Revell don’t examine the source of mathematics in ordinary language. As far as I know, it’s possible to express distance, time, size, weight, quantity, similarity and difference, direction, etc, in all human languages. The only variation seems to be in emphasis and detail. Mathematics is the more or less systematic formalisation of these concepts when people found it necessary to do so for some practical purposes involving trade and taxes. (Aside: Where I grew up, distance was expressed as time. A certain relative lived one hour away, for example. That’s an hour’s walk. This may be one reason why I find it easy to accept Einstein’s proposal of a space-time continuum, even though I can’t do the relevant math.)
     A keeper, worth an occasional reread. Breezy style, often cliched, which makes it seem easier to understand the math than it really is. The title is a teaser, possibly intended to attract the unnerdy.***

Drunk or High? (Lapham's Quarterly 06-1, Intoxication)

 LQ 06-1: Intoxication (2013) Not only about alcohol and drugs (though they figure prominently) but also about poisons. For all of our known history, we humans have taken (usually mild) poisons in small doses because they messed up our brains, thus creating an “altered state of consciousness.”  We aren’t the only animals that seek this experience: not only mammals but insects and birds have been observed slurping fermented fruit.
     A wide-ranging selection of first-person reports on the joys of getting drunk or high, praises of the grape and other intoxicants, scoldings for over-indulgence, severe frowning on any kind of intoxicant, etc. But the one question that I would like to see some answer to is hardly touched on: Why do we do it? The evidence suggests that we mostly seek the pleasure drugs provide, and some seek what they believe are transcendental visions of ultimate reality. The hangover or withdrawal are accepted as a (relatively) small price to pay for these treasures.
     Intoxication I think touches on the question of consciousness. The drugs’ effects show that our experience of reality is constructed by the brain. Deflect the brain from normal functioning, and that experience changes. Drugs and alcohol aren’t the only means of doing this, but they are the most reliable. That’s why we seek them out. That’s also why moralists of all stripes condemn them: the easy alteration of the brain’s function shows that the sense of self, the “I”, is a construction of the brain. That implies no independent Self or Soul to transcend this mortal life. It also makes nonsense of the belief that the Self can be somehow “uploaded” into an abiological, possibly electronic, but potentially immortal body (as envisioned by Ray Kurzweil among others).
     The Matrix is real: it resides in three pounds of jelly encased in a bony shell that protects it, mostly. We can mess with the program, and we like doing that. That’s what this collection demonstrates. I’ve been drunk twice in my life, and don’t want to repeat the experience. It’s enough to get a mild buzz, which I can also get without ingesting chemicals. Knowing that my experience of the world around me is not an exact replica is strangely comforting.
     Recommended. ****

Existential Physics (Hossenfelder 2022)

     Sabine Hossenfelder. Existential Physics (2022) Hossenfelder has made a YouTube reputation as a disturber and explainer. Search for her videos; they’re fun and enlightening. She believes that modern physics is in crisis because it claims more insight and understanding than is warranted by experiment and observation, especially when it comes to dark matter and energy. The theories, the sets of interlocking equations, describe what’s measured, but for consistency’s sake, theorists have added entities that haven’t been observed to interact with the entities that we know about. In her Warning, she says, “Science has limits, and yet humanity has always sought meaning beyond those limits.” Quite so.
     Then Hossenfelder goes on to show how science can inform some of the answers to the questions that exceed the limits of science. Science can clarify and disambiguate some of those questions. For example, do we have free will? The scientific answer (summarised) is: “No, if by free will you mean the ability to choose without being subject to the laws of physics.”

     For choosing is a brain-function, and brains function according the laws of physics. This fact has funked recent philosophers, who see no way out of the answer. But there is one: when we deliberately choose we figure alternatives, and weigh their desirability. We may choose differently than we chose in the past or will choose in the future. We will often choose differently than others choose. Thus, while our choices may not be freely willed, neither are they automatic. We aren’t automatons; we are agents. But we can’t choose without preferring one alternative to the others. Since our preferences are shaped by our genetics and our experience, in that sense, the choice is not “free”. However, we can choose to change our preferences. Odd, that. Is the choice to change our preference free or not? At the neurological level, I think no. At the psychological level, I think yes. And then there's  the spoiler question: How would you distinguish between free and determined choice?
     Hossenfelder does fall into what I think is the common philosophical error of physicists: She believes that physics reveals reality as it really is. Or at least that it is closer to doing so than the messier, less abstract sciences such as chemistry, biology, psychology, and so on. She reminds us that every "emergent property" that biology describes can be explained by chemistry and physics, and that everything that chemistry describes can be explained by physics. Neurology is solving some of the puzzles of psychology by showing how brain function varies with different behaviours, and emotions. Or at least suggesting how to reframe the puzzles.
     In short, she says, no so-called “emergent” phenomenon has (so far) been found to be inexplicable by the lower level from it which supposedly emerged. AFAIK, she’s right. But since the more abstract theories are derived from and explain the less abstract ones, that’s not, I think, a surprise.
     As I see it, physics describes the structure of reality. Einstein’s space-time makes this absolutely clear: What we observe depends on where in space-time we are relative to other entities, and how we are moving relative to other entities. General relativity (GR) describes how one observer’s worldview (measurements) is precisely transformable into another observer’s worldview: a clock runs fast from one POV, runs slow from another, and we can calculate exactly how much the measurements differ. (These calculations are necessary for GPS systems to function.)
     Quantum mechanics (QM) shows that what we observe depends on the event’s context: Electrons behave like particles in some contexts, and like waves in others. Or better, wave equations describe some electron behaviours, and particle equations describe others. None describe electrons. And those equations are the best descriptions we have, so far. There are probably better ones “out there”, and maybe they’ll be discovered. But not in my lifetime, I think. Bummer.

     The fact that GR and QM cannot (at present) be reconciled should not surprise us either, I think. Both are highly abstract descriptions of what’s common and different in our perceptions of reality. Our experience of reality is a simulation created by our brains. We can compare each other’s perceptions, and note whether we perceive the same differences and similarities. That’s the beginning of science, and it’s already at least one level of abstraction away from the simulation which is our experience of the world around us. But that simulation is itself an abstraction, constructed (computed?) by our brains. It’s sufficiently accurate that we can navigate the world, get our food, find our mates, etc. It must be structurally similar to reality, else we could not survive. It may make sense to say that the topology of our experience (the simulation) must be similar enough to the topology of reality to enable our survival. I don’t know enough about topology or brain function to be able to say. I also haven’t a clue how the brain’s simulation becomes what “I” experience. I suspect it’s because “I” is part of the simulation, probably the essential part, but how would one test that notion?
     I enjoyed this book, because (as the above may show) it prompted rethinking many of my ideas. I will read it again. Hossenfelder is an excellent explainer.
     Recommended. ****

Magic Shows (Lapham's Quartely 05-3 2012)

 LQ 05-3: Magic Shows (2012) Magicians take advantage of the way the brain constructs a simulation of reality. The brain does a good enough job that most of us survive long enough to succeed at whatever ambitions we may have. But the simulation is flawed in fundamental ways, and the magicians know how to exploit those errors to make us believe we see what doesn’t happen while we miss what actually happens.
     There have always been people who believe that magic is real, that the illusions crafted by the magicians aren’t illusions at all. Some scallywags have exploited this desire to believe in mastery of supernatural powers, which has always damaged people’s wallets, and too often their health as well.
     Of course, “magic” has many senses. The common core means something like “actually or apparently breaking the laws that govern reality as we know it.” There’s always the whiff of the supernatural about magic. At one extreme, people may believe that evil entities use magic to mislead or harm. At the other extreme, all apparently inexplicable phenomena are dismissed as fraud or delusion. Superstition and cynicism are close cousins.
     Me, I enjoy a good magic show. I also have more or less serious doubts about the claims for sightings of the Loch Ness monster, or abductions by aliens arriving on Earth in flying saucers. As for spoon bending, while I don’t know how to do this trick, other people do, and psychic energy isn’t part of it.
     Magic is designed to elicit wonder, which makes it close cousin to science and art. It requires skills useful in both. As entertainment, it satisfies. As a reminder that the universe is largely inexplicable, it may rekindle wonder and gratitude at the gift of life, and a brain able to both explain the mystery at the heart of reality, and know where that explanation shades into confabulation.
     Recommended. ****

The Disordered Cosmos (Chanda Prescod-Weinstein 2021)

Chanda Prescod-Weinstein. The Disordered Cosmos. (2021) A mix of science, history of science, memoir, sociology, and psychology. Prescod-Weinstein’s thesis is that while Western science has given us unimaginable insight into the structure of the cosmos, it has also ignored, deprecated or suppressed the contributions of women and Indigenous people. Worse, it has often dismissed their contributions as mere superstition.
     I found this book both exhilarating and painful to read. Exhilarating because of her skill in explaining the abstruse and esoteric concepts that are the core of modern physics. Painful because of what it cost her to achieve these insights. She played the academic game, and achieved academic career success. She’s now using her position to try to change the culture of science. I hope she succeeds.
     One thing her book confirms: Modern physics is mostly about how we cannot know what we would like to know. The equations describe mysteries so well that we can predict the interactions we will measure in our experiments and observations. But exactly what the interacting entities are is unknown, and likely unknowable. Given invisible matter and contestable energy, we may never know more than a tiny fraction of the cosmos, and understand less. Perhaps we are limited by our very nature: we are stardust, electromagnetic entities.
     Read this book. ****