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Travel might alert us to the beauty of mathematics.

Einstein's Field Equations

When I'm traveling, my iPad browser accumulates tabs saved for later research. Here is a collection of things thus marked, a way of postponing the actual work and clearing away the tabs.

Some call it the most beautiful equation. Picture links to an overview.

The first tab points to Einstein's Field Equations. It's admittedly intimidating, but there are a few easy things to understand about the most famous theory in physics: no computational skills required. Notice how simple it looks in the illustration. You've heard, no doubt, Einstein's insight was realizing the mysterious force of gravity is not a force, but a trick of geometry, the curvature of space. His famous field equation, EFE, equates curvature and mass/energy. Aside from his amazing physical insight, and his mathematical intuition that guided him to the curved geometry of Riemann, and in particular to the multi-dimensioned (indexed) vectors (called tensors), he also invented his own notational shorthand to express his Einstein Tensor (the R in the above equation indexed with mu and nu, Greek letters).

Picture links to Einstein's Notebook.

To sense the beauty of the above equation consider two aspects, its simplicity and its conceptual structure. First, this expression of physical reality is like Picasso giving us a dove with a few blue strokes. Those few symbols represent ten formidable looking partial differential equations. To solve them, or even represent them in a form allowing arithmetic estimations, has occupied armies of physicists for over a century. Second, the equation represents the relation between space curvature, matter and energy in an almost tactile way. You can point to the parts that are space and mass.

Picture links to Choptuik's Computational methods.

Over a century after Einstein's eureka moment, we have solutions to his equations for only a few, highly restricted, cases. Two camps have arisen: the "solvers" and the "estimators." All results require stating the equations in "well posed" formulations, so actual numbers can be applied to get actual answers. The solvers claim great success with many special cases, but none of any practical value. For practical problems the estimators use numerous computer-driven numerical analyses. Einstein has provided the fuse for an explosion of PhD activity.

Picture links to Schrodinger's equation.

Quantum Theory

The two great scientific theories of the twentieth century, relativity and quantum theory, have Albert Einstein in their genesis. Quantum theory was taken over by others, notably Erwin Schrodinger, who developed his famous wave function, and speculated on cats. He gave us an enduring mystery.

Picture links to Professor Adam's online course.

College freshmen read of Heisenberg's uncertainty principle, an artifact of the quantum world, and shrug. There are things we don't know. So what? There are lots of things a freshman doesn't know. The subtelties manifest slowly. If part of reality is unknowable, reality is unknowable: the source of magical thinking.

Picture links to a discussion of Paul Dirac.

There can be a feeling of magic with quantum theory. Something grand but highly counter-intuitive is happening. Matter at its tiniest, by which we mean the building blocks of our visible world, seems not to exist until we look at it. Then those tiniest bits of stuff seem either hard little grains or insubstantial light waves, depending on "how" we look. Our only means of predicting where they will be a nanosecond from now is by using something called Schrodinger's wave function. It gives probabilities of something being here or there or anywhere. Uncertainty tells us we cannot know both where they are and where they're going. Yet, we can pick up a bunch of them, in the form of a brick, say, and fling them across the room. Each little bit, we are told, flies off hither and yon, willy nilly, but collectively they break the window every time.

Picture links to a discussion of the wave function.

All that seems pretty peculiar, but stranger yet, quantum mechanics, using Schrodinger's calculations, produces amazing engineering feats. Suppose bridges were built by estimating the probability of them crossing the river up here, or down there, or maybe a thousand miles away. The intuitively opaque quantum theory has been rivetingly successful in manipulating the brick-hard world we inhabit. The word "weird" is applied to this state of things, and it probably means in this context what it always means. There is something we're not seeing.

Quantum theory, plus general relativity, enables modern technology. It's definitely something to understand better.

Picture links to a discussion of Hyperreal Numbers.

Hyperreal Numbers

When we took calculus around 1960, there was a tension between the inventors, Newton & Leibniz, and mathematical rigor. To get the velocity of an object at a particular instant, we started with the average velocity. We divided the distance traveled by the time in transit, and imagined squeezing the time interval ever smaller around our selected instant. We were told Newton imagined the interval becoming infinitesimally small. Then the instructor waved a chalk-dusted hand, and told us there was no such thing as an infinitesimal. We were to ignore Newton.

Picture links to an introduciton of Hyperreals.

The instructor introduced a thing called a limit. The limit also squeezed the interval down to zero, but it had a rigorous looking mathematical definition involving the Greek lower case letters epsilon and delta. Its convoluted logic may as well have been entirely in Greek. We were told we would not be able to understand the definition of a limit until sometime during our sophomore year. The homework didn't require understanding. We slogged on, awaiting that stroke of enlightenment.

Edwin Hewitt, Godfather of Hyperreals.

As predicted, the epsilons and deltas finally grew sensible and we made peace with limit theory. But now, decades later, we discover it was all a misunderstanding. The theory of infinitesimals is not, after all, a medieval leftover from the days of alchemy. By 1960 mathematical thought was already developing the non-standard analysis of hyperreal numbers including infinities and infinitesimals. Six years after we were given our epsilon/delta dosage a mathematician named Edward Robinson published a rigorous foundational development of these exotic creatures.

What Makes a Theory of Infinitesimals Useful?

Math teachers continue pushing the elegant logic of limit theory. It is mechanically reassuring to imagine limits doing the needed squeezing-to-nothing as the chalk flies, the arms wave, and derivatives reverse into integrals and vice versa. Still, it's delightful to find young people discovering the comfort and utility of Newton's infinitesimals. Math students have little opportunity to rebell. Robinson's axiomatic underpinnings of Newton's heresy provides a tiny wedge.

Picture links to PDF version of Daniel Fleisch's book.


Your exploration of reality may have followed 20th century excursions around relativity and quantum theory where lurks a murky thing called a tensor. You might have wondered, "What are all those subscripts and superscripts, and what is a tensor?"

Attempts at an easy answer yield vague, descriptive treatments rather than a clear definition. That's because tensor analysis, and its special notation are highly complex. It demands concentrated effort. But, effort at what?

Picture links to tensor product discussion.

Tensors are a generalization of vectors. They have a singlular property that endears them to abstract thinkers: an equation written in tensor form is valid in any coordinate system. Tensors are a way to sanitize our concepts of local bias.

The name "tensor" was first used in 1846 by William Hamilton for something else. Tensor calculus was conceived by Gregorio Ricci-Curbastro around 1890 and published by him and Tullio Levi-Civita in 1900. The name "tensor" was slapped on in 1898 by Woldemar Voigt.

Einstein made the first major use of tensors. His problem in general relativity was defining curvilinear space-time in a way that gave the same results for all observers (coordinate systems). The laws of physics, he insisted, must be the same for everyone, everywhere. That's what tensors do. They bulletproof the calculations against changes in coordinate system.

Picture links to tensor analysis and nomenclature paper.

The references on this page will give a superficial tensor insight. The indexing conventions mask a staggering series of calculations and require practice. Some of the computer programs that apply tensors will help. They're everywhere these days.

By the way, don't despair. Einstein struggled with tensors after picking them up from Marcel Grossman. Tullio Levi-Civita very kindly helped him correct some errors. Einstein thanked him, "I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot." –Albert Einstein

If Einstein was on foot, how do the rest of us travel?

Link to biography of HH.

Helen Hunt (HH)

When history grabs someone they vanish into an obelisk. Helen Fiske (1830-1885), Emily Dickinson's contemporary, was thus grabbed. Some considered Helen the leading American poet. That arbiter of American letters, Ralph Waldo Emerson, featured her in his 1874 "Parnassus" anthology and told friends she was the "leading poet on the continent." Her sonnet, "Thought," he said, captures "the uncontrollableness of thought by will." In our time Sylvia Plath was an admirer.

Link to Dickinson and HH.

Emerson championed iconoclastic versifier Walt Whitman's "Leaves of Grass." Hunt and Dickinson, longtime friends, heeded the advice of their mentor, Thomas Wentworth Higginson, for "revision and care and conformity to rules of grammar and traditional rhyme schemes." They were Whitman's opposite.

HH (Helen's pseudonym), was orphaned by sixteen. She met Edward Bissell Hunt at a ball hosted by Emily Dickinson in 1851. The son of New York's governor, a West Point graduate, and a scientist, Hunt seemed to Emily Dickinson the most interesting man she ever met. HH married him. Their first baby died of a brain tumor. Hunt died from inhaling poison gas after a failed test of a submarine weapon he'd designed. Their second child died of diphtheria the day before Lincoln's assassination.

Link to Jackson family.

Bronchitis took HH to healthful western climates. In 1875 she married William Sharpless Jackson, a founder of Colorado Springs, progenitor of the Colorado Jacksons, banker and business executive. A lecture by Chief Standing Bear on the mistreatment of the Ponca Indians inspired HH. She toured California, advocated fairness toward native Americans and penned her most famous novel. "Ramona" became her obelisk.

It has sold over 600,000 copies in numerous languages, boasts film, radio, and stage versions, and an annual "Ramona" pageant. Popularity peaked in the 1940s. "Ramona's" cultural appropriation and "Noble Savage" themes have become troubling in our times.

Current events signify either trends or fads. Whitman famously told us he would wait somewhere for us. HH did not wait. Her fame outlasted her 1885 death, meeting indifference recently, but perhaps, someday, we will wait for her.

Picture links to an interview of Frenkel.

Betti Numbers

In Edward Frenkel's marvelous book, Love and Math, he describes his undergraduate work with the famous Russian mathematician, Varchenko. Frenkel had to find the "Betti" numbers of a mathematical group. It's an engaging story even without knowing what a Betti number is, but the question, once asked, provides insight into the genius of young Frenkel.

Group Theory is part of modern algebra, and the braid groups Varchenko was considering appear in various branches of science and mathematics. Betti numbers were named by Henri Poincare after Enrico Betti who helped invent the field of topology. They characterize the structure of topological spaces, and are slippery critters to understand but have amazingly wide usefulness.

Picture links to a discussion of topology.

Topology is the branch of mathematics studying those properties of space preserved by continuous deformation (stretching, twisting, crumpling and bending, but not tearing or gluing). Young topology students are easy to spot as the ones staring blankly into space, holding a coffee cup, suspecting everything is a closed surface with a hole in it.

Characterizing topological spaces leads to mathematical groups, a set of things (for example, integers), that when added together by special rules, get results also in the group. Groups of topological spaces can be organized with careful rules of addition. Topological groups determine equality (homology), based on the structure of the space (remember the coffee cup that equals a sphere with a hole in it).

Picture links to a description of Betti numbers.

Subgroups of a given size can be taken from the main group of spaces, and homology groups can be formed that discard (cancel out) repetitions of elements from the main group homologous to the subgroup. Betti numbers count the number of elements in such homology groups, and there is one Betti number for each size of subgroup.

That's all pretty opaque, but this is a complex subject. It can be understood better, with some thought and practice with examples. While most of us are still pondering that coffee cup, Edward Frenkel was busy reading Varchenko's description of how to calculate Betti numbers for a braid group of a certain size, so he could calculate the Betti number for braid-group subgroups. His book is a delight because he knows a thing or two.

By the way, Betti numbers have found their way onto the Internet, and you can discover numerous applications.

Picture links to Maximilian's Western Travels: 1832.


Madog, ab Owain Gwynedd, son of Owain Gwynedd, King of the Welsh province of Gwynedd, sailed to America in 1170 and established a colony near Mobile Bay. The transplanted Welsh folk intermarried with the locals, and eventually migrated north to become a Welsh-speaking tribe of American Indians. Maybe not. A recent reference to the matter on the website of the University of Southern California at Santa Barbara, tells us about some recent linguistic scholarship that provides a "convincing argument" for the Welsh origins of some American Indian languages. The article ends with a note: "For more on this subject, click here."

It's all right for American intellectuals to poke fun at the Welsh like this, but where did such a story come from? Granted, Queen Elizabeth used the legends of Welsh discovery to claim the entire North American continent as English, but she didn't invent the story. In his 1790 analysis, and its follow-on a few years later, John Williams claims the first publication of Madog stories dates to 1584. In 1893 Thomas Stephens finds the story in Welsh verbal traditions as early as the twelfth century.

Picture links to a biography of Madog.

Modern bloggers like the story. The Maritime Heritage Project weighs in. Linguists are unimpressed. Alabama, home of the putative landing site, is agnostic. The Filson Historical Society (of Kentucky and the Ohio Valley Region) has a parochial interest, since The Falls of Ohio have entered the legends as home of the Welsh descendants. They are skeptical but receptive. Not adding to the credibility of the legends is the infiltration of nut-country-like theories of ancient races, and Roman coins.

Picture links to the work of Maredudd ap Rhys.

That's about it. Are we dealing with an April fools' joke? Or, is there substance behind the legend? Remember another story of rhapsodic origin, later written down, that was long considered apocryphal until Heinrich Schliemann dug up the ancient city of Troy. Modern DNA analysis might find evidence of Welsh-American Indians, but hasn't yet. Linguistics evidence hasn't ended the speculations. It's tempting to accept the legends of an ancient voyage, while reserving judgment on tales of American settlement. The mystery involves accounting for all the archeological finds woven into the story. Should America be renamed Owainland?

Picture links to a Traubel bio.

Horace Traubel

Horace Traubel (1858-1919) was adopted by Walt Whitman as his Boswell, the man who tagged after Samuel Johnson and preserved enough quotes from the esteemed man to fill up a biography. Boswell published the life of Samuel Johnson in an acclaimed book that ranges between one and six volumes, depending on the erudition of the publisher. Traubel's effort occupies 5000 pages and nine volumes, the last two of which were published in 1996, 77 years after Traubel's death, and 104 after Whitman's.

Whereas Whitman soared with the timeless spirit of humanity, Traubel mucked in the trenches of social skirmish, befriending such fellow soldiers as Eugene Debs, and Emma Goldman. He tried but failed to get Whitman's endorsement of his politics, apparently not realizing Whitman was sailing in a different sky.

Modern reviewers of Whitman's life and works feel obligated to fault him for something. Who among us is blameless? They pick at his ego, a concept he embodied even if its name had not become the de rigor vogue word of savant thinkers, and tsk-tsk his inconsistencies, as if he were unaware of them. Traubel's work, for all of his heavy lifting in the culture wars, shimmers from the poet's image in a dazzling arc of ageless wisdom.

In our era Mr. Whitman's sexual life gets a good going over. We can learn as much as we want to about his assignations with Traubel and others. It may well be that these "modern" preoccupations with intimate details are a passing fad, or perhaps they are us catching up with Mr. Whitman, who, recall, is somewhere waiting for us.

Picture links to Traubel's nine volumes.

It would be well for someone, some day, to create actual theater from the theater of Whitman's life. Disregarding a few venues and drama programs using Whitman's name, there seem to be no dramatic depictions of the man. That can't be from an excess of caution about putting an actor into a transcendent role. Yahweh, and even Christ, have had that ignominy added to their suffering. If someone is inspired to find Whitman waiting for them on stage, Traubel's nine volumes are waiting for their exploitation.

Traubel's collection can be overwhelming if approaced as a textbook, to be mastered and collated into our internal knowledge. There are a few attempts at abridgement, and the most recent was just published for the Whitman bicentennial. Brenda Wineapple has whittled the nine volumes down into 196 pages. It gets good reviews.

Picture links to a discussion of the wave function.

Picture links to a discussion of the wave function.

While collecting a few thoughts on the subjects mentioned above, we've come across a few Internet Jewels worth remembering.

Blog of Florin Moldoveanu of the University of Maryland, College Park.
Blog of Sean Carroll of Caltech.

The Blog of Physicist Sabine Hossenfelder of the Frankfurt Institute For Advanced Studies.
The Blog of independent scholar and enactor, Rob Velella.

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