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The Math of Love: Calculating the Odds of Finding Your Soulmate

The science of why there are roughly 871 special someones for you out there.

Since the dawn of recorded history, poets and philosophers have pondered the nature of love and, in recent times, so have scientists. But can the concrete lens of science really be applied to something as seemingly abstract and amorphous as amore? Joe Hanson, mastermind of the wonderful science-plus compendium It’s Okay To Be Smart, has a new online show in partnership with PBS and the latest episode explores what the search for extraterrestrial life can teach us about our odds of finding that much-romanticized human soulmate, using the Fermi paradox, the Drake equation, and a lesson in love from Carl Sagan — who, with his timelessly magnificent Golden Record love story, should know a thing or two about the wisdom of the heart.

Joe ends with a beautiful quote from Sagan’s 1985 debut novel, Contact:

For small creatures such as we the vastness is bearable only through love.

BP

Mathemusician Vi Hart Explains Space-Time with a Music Box and a Möbius Strip

The fabric of the universe via backwards Bach.

If mathemusician Vi Hart — who for the past three years has been bringing whimsy to math with her mind-bending, playful, and illuminating stop-motion musical doodles — isn’t already your hero, she should be, and likely will be. (Cue in the GRAMMYs newly announced search for great music teachers.) In her latest gem, Hart uses music notation, a Möbius strip, and backwards Bach to explain space-time:

Music has two recognizable dimensions — one is time, and the other is pitch-space. … There [are] a few things to notice about written music: Firstly, that it is not music — you can’t listen to this. … It’s not music — it’s music notation, and you can only interpret it into the beautiful music it represents.

Also see Hart on the science of sound, frequency and pitch, and her blend of Victorian literature and higher mathematics to explain multiple dimensions.

For a decidedly less whimsical but enormously illuminating deeper dive, see these 7 essential books on time and watch Michio Kaku’s BBC documentary on the subject, then learn how to listen to music.

BP

Geometrical Psychology: Mathematical Models of Consciousness by the 19th-Century Psychologist Benjamin Betts

“Imagination receives the stream of Consciousness, and holds apart and compares the different experiences.”

Geometrical Psychology: Mathematical Models of Consciousness by the 19th-Century Psychologist Benjamin Betts

“What makes a mathematician is not technical skill or encyclopedic knowledge,” mathematician Paul Lockhard observed, “but insatiable curiosity and a desire for simple beauty.” But what if this mathematical curiosity and desire for beauty were applied to questions that have perplexed scientists and philosophers for millennia — questions about consciousness, what it is, how it works, and how it shapes our lives? That is precisely what the New Zealand psychologist Benjamin Betts set out to do in the 19th century with his unusual diagrams of consciousness, collected in the 1887 volume Geometrical Psychology, or, The Science of Representation (free ebook) — a predecessor to Julian Hibbard’s geometric diagrams of love.

Educated in England and trained in architecture and decorative art, Betts spent time in India and the Far East immersed in the Eastern philosophical traditions before settling in Auckland to work as a “trigonometrical computer” for the Civil Service department of the local government. He became intensely interested in developing a “science of representation” that would trace the successive stages of the evolution of human consciousness. After completing his first set of diagrams, he sent them to the great Victorian art critic John Ruskin. Ruskin dismissed them. Betts persisted. He next shared the diagrams with his sister, who was close with Mary Everest Boole — the widow of the great English mathematician George Boole. Fascinated more by the mathematical than the metaphysical aspect of the diagrams, she entered into lengthy correspondence with Betts, cited his diagrams in her 1884 reference book on symbolic methods of study, and showed them to her circle of scientific friends, including the president of the Royal Society, as well as to a number of artists.

In his letters to Boole, Betts describes the ultimate objective of his project — not a scientific objective but a spiritual one, aiming to serve as a clarifying force for the human spirit:

When you are thoroughly able to understand these diagrams and the truths they inculcate, when you look at any forms of humanity it will not be at their outward appearance, neither at their hapless struggles after vanities, but at their unhappy Ideal, which is giving them such trouble, and which they would almost fain be rid of that they might eat and sleep undisturbed.

[…]

It is not true that there is an Earth in space with individuals wandering about on it ; it is not true mathematically, but each carries its own world with it, and if there is any ground of relation between my world and your world or other worlds, that ground exists in you or me and not in the world, except only as it is a part of each independently.

Animated by his conviction that “Love is the Substance of all things” and Wisdom is Love’s counterpart in the evolution of human consciousness, Betts drew on his time in the East to position his diagrams as an integration of Eastern and Western thought necessary for the advancement of wisdom and love in the service of attaining higher consciousness:

There appears to me to be a fundamental antithesis between Eastern and Western Thought. This would only be carrying out the necessary conditions of all existence, without which Existence would lapse in Being Western Thought has sprung from the Hebrew “I am,” crude and arbitrary at its first promulgation, but subdued, and humanised, and spiritualised in its latest announcement, so that now this Western idea is taking root as a demand for harmony, and is breaking out on all sides as emotional activity, and is even getting quite unanswerable in its demand, but let but the Lily show herself and you will find a wonderful change come over modern history. A sudden breeze springing up, our ship shall again obey her helm and spring forward toward her horizon.

Eventually, Betts enlisted the help of a woman by the name of Louisa S. Cook to edit what became the published volume. Primitive and metaphysically clouded as they may be, his diagrams endure as a visionary early attempt to map human consciousness at the improbable intersection of mathematics and moral philosophy, long before the birth of neuroscience and even before the dawn of modern psychology as we know it.

Cook writes in the introduction:

Mr. Betts has spent more than twenty years in studying the evolution of Man. He contemplates Man, not from the physical, but from the metaphysical point of view ; thus the evolution of Man is for him the evolution of human consciousness. He attempts to represent the successive stages of this evolution by means of symbolical mathematical forms. These forms represent the course of development of human consciousness from the animal basis, the pure sense-consciousness, to the spiritual or divine consciousness; both which extremes are not man — the one underlying, the other transcending the limits of human evolution.

Mr. Betts felt that consciousness is the only fact that we can study directly, since all other objects of knowledge must be perceived through consciousness.

Mathematical form, he considers, is the first reflection and most pure image of our subjective activity. Then follows number, having a close relation to linear conception. Hence mathematical form with number supplies the fittest symbols for what Mr. Betts calls “The Science of Representation,” the orderly representation by a system of symbolisation of the spiritual evolution of life, plane after place. “Number,” Philo said, “is the mediator between the corporeal and the incorporeal.”

Cook notes the form of Betts’s forms:

The symbolic forms which Mr. Betts has evolved through his system of Representation resemble, when developed in two dimensions, conventionalised but very scientifically and beautifully conventionalised leaf-outlines. When in more than two dimensions they approximate to the forms of flowers and crystals.

These mathematical curves might serve as a truer and more scientific basis of classification for Botany than de Candolle’s system or any other yet employed, many so-called amorphous developments of the Flora being readily reducible to law according to this method. For instance, the simple corollas, the horn-shaped corollas, and the bi-axial corollas would supply three main classes of flower forms, each of which might be divided into various distinct sub-classes.

The fact that he has accidentally portrayed plant-forms when he was studying human evolution is an assurance to Mr. Betts of the fitness of the symbols he has developed, as it affords presumptive evidence that the laws he is studying intuitively admit of universal application.

She considers Betts’s model of the imagination:

After the repeated recurrence of any sensation, though slightly varying in form, the individual develops the consciousness of its identity, and he begins to form an image or idea, both of the subjective sensation and of the accompanying objective perception, which he can retain in his mind though the sense affection of which it is the counterpart is transitory. Mr. Betts calls this power of ideation Imagination, using it in the literal sense of the word. As a prism receives a beam of light and deflects the rays, holding them apart so that the colours of the spectrum are separated and distinguished, so Imagination receives the stream of Consciousness, and holds apart and compares the different experiences.

Comparison is represented in the diagrams by the angle-; Consciousness from one-dimensional becomes two-dimensional, the line is expanded to a surface.

[…]

And since every idea is dual — e.g., the positive idea of light brings with it the negative complementary idea of darkness — of a colour, its complementary colour — therefore the positive representative line on the right hand of the diagram is duplicated by a counterpart line on the left. The sensation of the present moment is not yet reflected as an idea, nor distinguished by comparison. In the diagram it is the apex of the form. When more than two senses occupy Consciousness the lines representing them are arranged radially round the centre. Although the distinction must then be represented by a smaller angle, it does not follow that it is less in amount, as the form itself of Consciousness has become enlarged. At the same time it is quite possible that when the number of modes of manifestation is very limited the sensations are more vivid, and consequently the distinctions more marked, than when more modes of consciousness are differentiated.

Whether abstracting something as complex as consciousness in such concrete physical forms is a reasonable proposition remains a question for the metaphysicians — but the forms themselves are, unequivocally, pure visual mesmerism.

Betts’s diagrams are now in the public domain and the book is available for free in multiple digital formats from Open Library.

BP

Measurement: Exploring the Whimsy of Math through Playful Patterns, Shape and Motion

“What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for simple beauty.”

Math, as mathemagician Vi Hart’s stop-motion doodles, photographer Robin Moore’s string portraits, and artist Anatolii Fomenko’s stunning black-and-white illustrations have previously shown us, can be the conduit of great fun and great beauty. In Measurement (public library), mathematician Paul Lockhart invites us to “make patterns of shape and motion, and then [try] to understand our patterns and measure them.” (Because, lest we forget, we have a natural penchant for patterns.) What results as we step away from physical reality and immerse ourselves in the imaginary — and imaginative — world of mathematical reality is a thing of infinite beauty and infinite fascination.

Lockhart writes in the introduction:

Physical reality is a disaster. It’s way too complicated, and nothing is at all what it appears to be. Objects expand and contract with temperature, atoms fly on and off. In particular, nothing can truly be measured. A blade of grass has no actual length. Any measurement made in the universe is necessarily a rough approximation. It’s not bad; it’s just the nature of the place. The smallest speck is not a point, and the thinnest wire is not a line.

Mathematical reality, on the other hand, is imaginary. It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I can never hold a circle in my hand, but I can hold one in my mind.

[…]

The point is I get to have them both — physical reality and mathematical reality. Both are beautiful and interesting… The former is important to me because I am in i, the latter because it is in me.

Lockhart underpins this excitement with a fair warning:

Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively.

And yet, he’s quick to reassure that the gold standard of math, not unlike that of science, isn’t the answer but the driver of curiosity:

What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for simple beauty.

But what makes Lockhart particularly compelling is his ability to relate mathematics to parallel concepts from disciplines and aspects of life that are more familiar, more comfortable, more ingrained in our everyday understanding of the world — for instance, in comparing math to storytelling:

A mathematical argument [is] otherwise known as a proof. A proof is simply a story. The characters are the elements of the problem, and the plot is up to you. The goal, as in any literary fiction, is to write a story that is compelling as a narrative. In the case of mathematics, this means that the plot not only has to make logical sense but also be simple and elegant. No one likes a meandering, complicated quagmire of a proof. We want to follow along rationally to be sure, but we also want to be charmed and swept off our feet aesthetically. A proof should be lovely as well as logical.

In a nod to the value of “useless” knowledge and figuring things out, Lockhart argues:

People don’t do mathematics because it’s useful. They do it because it’s interesting … The point of a measurement problem is not what the measurement is; it’s how to figure out what it is.

He ultimately offers several pieces of advice on engaging with math:

  1. The best problems are your own. Mathematical reality is yours — it’s in your head for you to explore any time you feel like it… Don’t be afraid that you can’t answer your own questions — that’s the natural state of the mathematician.
  2. Collaborate. Work together and share the joys and frustrations. It’s a lot like playing music together.
  3. Improve your proofs. Just because you have an explanation doesn’t mean it’s the best explanation. Can you eliminate any unnecessary clutter or complexity? Can you find an entirely different approach that gives you deeper insight? Prove, prove, and prove again. Painters, sculptors, and poets do the same thing.
  4. Let a problem take you where it takes you. If you come across a river in the jungle, follow it!
  5. Critique your work. Subject your arguments to scathing criticism by yourself and others. That’s what all artists do, especially mathematicians… For a piece of mathematics to fully qualify as such, it has to stand up to two very different kids of criticism: it must be logically sound and convincing as a rational argument, and it must also be elegant, revelatory, and emotionally satisfying. [But don’t] worry about trying to hold yourself to some impossibly high standard of aesthetic excellence.

Lockhart elaborates on the latter point with a poignant reflection that applies to math just as much as it does to life itself:

Part of the problem is that we are so concerned with our ideas being simple and beautiful that when we do have a pretty idea, we want so much to believe it. We want it to be true so badly that we don’t always give it the careful scrutiny that we should read. It’s the mathematical version of ‘rapture of the deep.’ Divers see beautiful sights that they forget to come up for air. Well, logic is our air, and careful reasoning is how we breathe.

Indeed, much of math sounds an awful lot like the art of living itself: Take for instance, that same old fear of failure that often stands in the way of creativity, which also holds us back from immersing ourselves in the art of figuring things out:

The important thing is not to be afraid. So you try some crazy idea, and it doesn’t work. That puts you in some pretty good company! Archimedes, Gauss, you and I — we’re all groping our way through mathematical reality, trying to understand what is going on, making guesses, trying out ideas, mostly failing. And then every once in a while, you succeed… And that feeling of unlocking an eternal mystery is what keeps you going back to the jungle to get scratched up all over again.

Here’s a little teaser for the whimsical jungle of mathematical reality and the logical aesthetic of math:

Measurement, from Harvard University Press, comes seven years after Lockhart’s exquisite critique of math’s tragic fate in contemporary education, A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.

BP

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