Author: Richard Melson

Essay 36: What We Mean by “Epochal Waters”

We sometimes use the phrase “epochal waters” to refer to the deepest layers of the past which we “swimmers” at the surface of the ocean don’t see or know. “Epochal waters” are latent, currents are closer to the surface.

There’s a similar idea from the French philosopher Michel Foucault who died in 1984. In his The Order of Things, classic from 1966, he talks about the “episteme” (as in epistemology) that frames everything from deep down. (The Greeks distinguished between “techne” (arts, crafts, practical skills and “episteme” (theory, overview).

“In essence, Les mots et les choses (Foucault’s The Order of Things) maintains that every period is characterized by an underground configuration that delineates its culture, a grid of knowledge making possible every scientific discourse, every production of statements. Foucault designates this historical a priori as an episteme, deeply basic to defining and limiting what any period can—or cannot—think.

Each science develops within the framework of an episteme, and therefore is linked in part with other sciences contemporary with it.

(Didier Eribon, Michel Foucault, Harvard University Press,  1991, page 158)

Take a simple example. A discussion comes up about what man is or does or thinks or knows. In today’s episteme or pre-definition, one thinks immediately not of man in terms of language or the invention of gods, but in terms of computational genomics, big data, bipedalism (walking upright on two legs). Its assumed in advance via an invisible episteme, that science and technology. physics, genetics, big data, chemistry and biology hold the answer and the rest is sort of outdated. This feeling is automatic and reflexive like breathing and might be called “mental breathing.”

One’s thoughts are immediately sent in certain directions or grooves, a process  that is automatic and more like a “mental reflex” than a freely chosen “analytical frame.” The thinker has been “trained” in advance and the episteme pre-decides what is thinkable and what is not.

There are deep episteme that underlie all analyses: for example, in the Anglo-American tradition of looking at things, the phrase “human nature” inevitably comes in as a deus ex machina (i.e., sudden way of clinching an argument, the “magic factor” that has been there all along). If you ask why are you suddenly “importing” the concept of “human nature,” the person who uses the phrase has no idea. It’s in the “epochal water” or Foucault’s episteme, and it suddenly swims up from below at the sea floor.

Another quick example: In the Anglo-American mind, there’s a belief from “way down and far away” that failure in life is mostly about individual behavior (laziness, alcoholism, etc.) and personal “stances” while “circum-stances” are an excuse. This way of sequencing acceptable explanations is deeply pre-established in a way that is itself hard to explain. It serves to “frame the picture” in advance. These are all “epochal water“ or episteme phenomena.

Essay 35: Education and the Need for Enchantment

Max Weber (1864-1920) and Émile Durkheim (1858-1917) are considered the two fathers of modern sociology at the highest level.

Weber sees the modern world as the zone of “Ent-zauber-ung:” where ent means removal of, Zauber means magic or enchantment and ung means the process of.

He sees our world as “dis-enchanted.” Everything is scientific or profitable or unwelcome. This makes modern life a productive engine of sorts but extremely desiccated and arid and leads to what Durkheim calls “anomie” (the sense of being adrift, directionless).

We argue in this book that education should be seen as the “last exit” to enchantment before the “grind of life” comes down on the student after the “moratorium” of college.

What is enchantment? Enchantment is that special feeling about something, some topic, field, math problem, painting exhibit, novel, movie, debate, that there’s something there that “makes it all worthwhile” and like a great piece of music, “gets to you” and flies under all cynical radar. The best kind of enchantment can last from age 19-95, if you live that long.

Think of a math or physics problem or novel or painting that gives the student “permanent uplift.”

The pedagogical dimension of enchantment works like this: the student encounters a puzzle or conjecture or story or depiction that constitutes a “healthy obsession.”

After interaction with this phenomenon, he or she can “walk backwards” to the 900-page textbook and go to those pages that are relevant, this making the textbook more like a dictionary that serves as a handy reference book and not as a daunting, exhausting endless “Mt. Everest” of names and equations or faces or dates. The student can “conquer” textbooks by enchantment and only enchantment. Without that engine or motor for the mind and will, one is weighed down and demoralized in advance.

Let’s do two quick examples:

Heraclitus is supposed to have said, “you can’t step into the same river twice.” Zeno says you can’t really cross the street because first you have to reach the midpoint, then the next midnight, and so on forever. You never complete your crossing (see Joseph Mazur’s book, Zeno’s Paradox, from 2008).

Such ancient paradoxes are still perplexing. Great thinkers like Bertrand Russell, Whitehead, Frege, et al wrestled with them many decades.

It’s also puzzling that certain math or logic questions open up “oceans” of analysis. Why might that be? Is that enchanting or depressing?

The last chapter of Tolstoy’s War and Peace masterpiece is a set of reflections on history itself. It’s very enchanting as he wrestles with this “caprice machine” called history.

Enchantment gives you the first steps towards what we call “pre-understanding,” a prerequisite for all deep study.

Essay 34: Arguments Without End: Are They Good or Bad?

The Dutch historian Pieter Geyl (died in 1966) coined the phrase “argument without end” to get at the constant reappearance of old arguments or viewpoints. One gets the impression that arguments are either persistent or perhaps permanent. One simplistic example could be argument about socialism: Sweden is “good,” but Venezuela (or Cuba) is bad. This book takes the view that “arguments without end” are not the end of knowledge but rather a potential beginning: it could be that some issues cannot be captured by one school of thought: the awarding of the 1974 Nobel Prize to both Hayek (“the right”) and Myrdal (“the left”) is an example of this need for hybridity. Both Hayek and Myrdal are each seeing something valid and it’s a “fool’s errand” to decide who is “eternally” correct.

Let’s apply this thinking to a deep “argument without end” within and about history.

Michel Foucault (died in 1984) following Nietzsche, argues that history seems “linear” but is more random and non-linear than the “linear” historians see or admit.

There’s an aphorism in Nietzsche, (from his The Dawn) which Foucault uses…history is made by the “iron-hand of necessity shaking the dice-box of chance.”

In other words the world we know, traveling somehow from the assassination of Kennedy (November 2, 1963) to the impeachment hearings of Trump in October 2019, must be thought of as a kind of “random walk” behind which are trends, cycles, so that one gets a fusion of structure and surprise. If you emphasize surprise you’re closer to Foucault than to those narrative historians who think they can show you the exact threads which connect “then and now.”

Here’s an example of such a historian, the celebrated G.R. Elton of England, whose classic The Tudor Revolution in Government is a masterpiece of orthodox analysis. The book centers on the administrative revolution in the 1530s in England which implied, says Elton, “As regards political and social structure, the sixteenth century produced something quite new in England—the self-contained sovereign state in which no power on earth could challenge the supremacy of statute made by the crown in parliament.”

“In this revolution, in this making of a new kind of state productive of a new kind of society, administrative reforms played their part. It is against this background of controlled upheaval that they must be seen and understood.”

(Elton, The Tudor Revolution in Government, Cambridge University press, 1966, page 426/427)

Orthodox historians see history as a “nail-down-able” system of storylines and the controlled upheavals have a direction (teleology) which allows you to use—in this case the 1530s in England—as a beginning, an origin, a “datum line,” and once you have this clear starting point you can follow the story to now and include comparative developments in France or Germany or China.

The orthodox “explain strategy” starts with an origin, a “starting gate” like a horse-race.

The Foucault–Nietzsche view is that these starting points are not entirely useless but in the end don’t help you because history is in the end governed by “the dice-box of chance” even if it is held by “the iron-hand of necessity.” History is more “upheaval” than “control” more surprise than structure. “Determinism” such as perhaps based by pinning down a starting point from which one can “build out,” is a wish-dream since history is nonlinear and nondeterministic. Even Elton’s phrase “controlled upheaval” is full of questions and problems.

Modern “complexity theory” in mathematics tries to get at these differences analytically. A “meta-intelligent” student would go from this historians’ “argument without end” to the analysis of complexity in math as a way of rounding out the exploration.

An “argument without end” can thus be useful if the student does not insist on some final “apodictic” or certain-forever answer.

Essay 33: How to Jump From a Field to a Larger Understanding: The Example of Globalization

In a university, one is trained to “inspect” fields. That produces what might be called a “monographic” intelligence.

Our purpose is to show and promote something ancillary to this, what might be called a “circum-spective” intelligence (i.e., using the specialized knowledge as one “brick” in a larger structure of understanding).

Let’s do an example:

Think of all the descriptions and analyses of something called globalization. An objective evaluation of the literature on this show two analyses that stand out and tower above the rest:

  1. Prof. Jeffrey Williamson and Kevin O’Rourke, Globalization and History, MIT, 1999 (this is a “quantitative history” or “cliometric” study and a classic).
  2. Elhanan Helpman, Understanding Global Trade, Harvard, 2011 (a masterpiece of trade-based analysis)

Both of these authors are Harvard professors in economics and deserve the high regard that such books have won them. In these two books, there are several technical disagreements of which the deepest is that Williamson focuses on the emergence of one world market price (say for wheat) and argues that this “price convergence” is the best measure of globalization. Thus at a certain point wheat of a certain kind (hard, durum, etc.) was price at the same world rate whether the wheat came from Kansas, Canada, Argentina or Ukraine. The world is then a global price-making market mechanism. This price convergence then extends to their kinds of prices as globalization processes deepen. Williamson explicitly considers other approaches to globalization such as trade share of GDP as confusing.

Helpman, on the other hand, uses export plus imports/over GDP as his measure, clearly disagreeing with the Williamson approach of prices and not trade shares.

Interestingly, both scholars conclude that something we call globalization begins to “show up in the data” in the 1820s. Thus, Marco Polo-type stories are colorful and “multinational” but have little to do with actual (i.e., data based) globalization as we see it, looking backwards from 2020.

Both of these books are classic works and show the intricacies and utility of the “cliometric” approach (i.e., explaining the past quantitatively, using data from economics).

However, there’s a deep perspectival omission in both works:

As the novels of Balzac (1799-1850) show there begins to “co-evolve” with this globalization story a parallel story of global colonization and empire-building by the European powers. Algeria is seized in 1830 and culminates in the brutal Algerian War of 1954-1962. Without de Gaulle‘s supreme prestige as the savior of France, the French would have gone to a destructive civil war and the defeat in French Indochina at Dien Ben Phu. 1954 almost lead to endless strife based on events on the other side of the world.  Balzac’s novels are often set in the 1820s and mention a deepening involvement of France with North African empire-building.

This culminates in Maupassant’s novel Bel-Ami from 1885, which centers on the inexorable rise of an unethical “manipulation machine,” who returns from North Africa as a penniless soldier and after many twists and turns makes several killings in North Africa in various shady schemes which he gets wind off via his journalism contacts.  In other words, the rise of Western industrial technology (from railroads to cars to planes) conquers the world one way while the European colonial armies conquer the world another way.

The peoples like the Vietnamese and Algerians “see” the world in colonial terms with colonialism backstopped by industrial technology. Their quest for dignity begins with this analysis and not with the analysis which says industry and science are primary and colonialism a footnote.

It is this fundamental clash of historical interpretations on a worldwide scale which bedevils the changing relationship between the West and the non-West and is more profound than the econometric differences between a Prof. Williamson and a Professor Helpman.

By seeing how these layers and stories are “entwined” gives you a deeper and wide-angle vision which one field—economics or cliometrics—can’t offer because it is one brick or building block in a larger story. Fields have to be “opened up” in this way, which is the mission of this book.

Essay 32: Movies Novels and Songs as an “Open University”

This essay is a continuation of the previous one and will show you how a novel gives you a backdoor or side window into education.

The novel Howards End (1910) by E.M. Forster has two foci that it orbits, in a kind of ellipse, like a planet. One focus is social mores of different strata of society, the buccaneering money families (Wilcoxes), the artistic and culture (based on inheritances such as the Schlegels) and the “people of the abyss,” the marginal insurance clerk Leonard Bast. The other focus is money and wealth.

Henry Wilcox warns the Schlegel sisters (Margaret and Helen) that the insurance company, the Porphyrion Fire insurance Company their new friend (the poor clerk Leonard Bast) works for might go bankrupt or “smash” in British lingo:

“The Porphyrion’s a bad, bad concern.—Now don’t say I said so. It’s outside the Tariff Ring.”

“I thought an insurance company never smashed,” was Helen’s contribution. “Don’t the others always run in and save them?”

“You’re thinking of re-insurance,” said Mr. Wilcox mildly. “It is exactly there that the Porphyrion is weak. It has tried to undercut, has been badly hit by a long series  of small fires, and it hasn’t been able to reinsure. I’m afraid that public companies don’t save one another for love.”

This advice turns out to be fear-mongering and Leonard Bast quits his job at the Porphyrion and can’t find new work and is desperate since he’s part of the ‘people of the abyss.’”

Later on Henry Wilcox (Anthony Hopkins in the film) reverse himself and says that now the Porphyrion is ‘safe as houses.’”

The Tariff Ring referred to above has nothing to do with the tariffs we think of in 2020, duties on Chinese or Canadian or European goods.

The Tariff Ring refers to a consortium of insurance companies which agree not to undersell each other. You can think of this as price-fixing if you like or perhaps as price-stabilization whereby the insurance companies are creating “reinsurance” by these means of dealing with one another in this block or consortium.

The Tariff refers to the price of the policy, the rate, the premium, to carry the insurance. The word tariff has several meanings and one must not confuse these tariffs (cost of insurance policy for the insured) with import duties, whether for protectionist or revenue motives.

A current leading global practitioner of modern reinsurance is Munich RE. You should go to their website and use this essay as a way to link to this whole world of insurance and reinsurance.

This shows you how a novel or movie serves as an “open university” if you get into this flexibly “circum-spective” frame of mind (i.e., real and deep learning without “silos”).

Essay 31: Learning to Use Movies & Books or Songs As Off-Campus Universities

Howards End is a 1910 novelistic masterpiece by the great British writer E.M. Forster. It became an excellent Merchant-Ivory movie. The story is set in Edwardian England (1901-1910) and shows the interaction of three families, each representing a section of British society:

  1. The Wilcoxes (Anthony Hopkins is Henry Wilcox) who represent cut-throat new money, with “Thatcherite” views.
  2. The Schlegels (Emma Thompson is Margaret Schlegel) represent the “culture vultures” who are always talking about music, literature, poetry, high-toned pursuits.
  3. The Basts (Leonard Bast is played by Samuel West) who is a lowly clerk in the Porphyrion Fire Insurance Company and part of what one writer (Jack London) called “people of the abyss.” (their job and wage insecurity is all-consuming).

The story involves the storm created by the interaction of these families and the destruction of the marginal clerk Leonard Bast. Surprisingly, the character Margaret Schlegel of the “artiste” family, talks about the centrality of money in their lives:

“You and I and the Wilcoxes stand upon money as upon islands. It is so firm beneath our feet that we forget its very existence. It’s only when we see someone near us tottering that we realize all that an independent income means. Last night, when we were talking up here round the fire, I began to think that the very soul of the world is economic, and that the lowest abyss is not the absence of love, but the absence of coin.”

“…we ought to remember, when we are tempted to criticize others, that we are standing on these islands, and that most of the others are down below the surface of the sea.”

“I stand each year upon six hundred pounds, and Helen [her sister] upon the same, and Tibby [their brother] will stand upon eight, and as fast as our pounds crumble away into the sea they are renewed—from the sea, yes, from the sea.

“And all our thoughts are the thoughts of the six-hundred-pounders, and all our speeches; and because we don’t want to steal umbrellas ourselves, we forget that below the sea people do want to steal them, and do steal them sometimes, and that what’s a joke up here is a down there reality.”

There are not only these terrifying lines of cleavage between rich and poor but these cleavages extend to sex and romance as well.

Leonard Bast is seduced by Helen Schlegel (Margaret’s bohemian rebellious sister) who becomes pregnant by him. Paul Wilcox (Henry’s explosive son) beats Bast to death which then brings the social “storm” of the Wilcoxes, Schlegels and Basts, to a climax or denouement.

A close reader of English literature will perhaps have noticed that already in Jane Austen novels, set in a time one hundred years prior to Howards End there is constant talk of annuities. 

Already in Jane Austen “the name of the game” is Caribbean sugar while in Howards End the “game” is African rubber (and invested in English real estate), where the Wilcox fortune comes from (think of the Firestone company in Liberia as a classic example of the rise of the rubber business, with the need for rubber for tires.)

Thus we have in these novels an intermingling of families, overseas sources of wealth (West/non-West economic relations), rigid social rituals. (a romance between Helen Schlegel and Leonard Bast, the marginal clerk, is a no-no punished by death.)

Decades before E.M. Forster, there was the novel by Charles Dickens, Dombey and Son which “orbits” money.

Someone once made the comment that Marx and Freud destroyed the Victorian taboos on money and sex and that comment is confirmed largely by such novels as Howards End with the very surprising utterances of Margaret Schlegel (from the family of “culture vultures”), on money and its centrality in their lives and mentalities.

Think of such novels as an “intuitive kind of open university.”

Essay 30: Magic or Sacred Geography as a Kind of Silent Education

Magic and its clone, sacred geography, are all around us and are crucial organizing principles for the way people think. Such emotions are an overlay over all formal education.

For Communists, the grave of Marx in Highgate Cemetery in England is sacred ground.  For some German soldiers after WWII who committed suicide on the steps in Feldherrnhalle (“Field Marshall’s Hall”—a display in Munich in Odeonsplatz of large statues of famous military leaders in German history), these statues and their place in Munich “means” something magical or sacred to them. North Koreans have Paektu Mountain which Kim recently ascended in a ritual addressed to the North Koreans. Think of Camelot or Lourdes.  Think of sites such as the Lincoln Memorial or Notre-Dame Cathedral in Paris or “holy sites” in Jerusalem or Mecca.

Think of magical and sacred things like the three imperial regalia in Japan which Emperor Hirohito fixated on at the end of WWII.

How does this kind of thinking permeate our lives in a way that nobody quite sees clearly:

The left in American politics (Bernie Sanders, Jeffrey Sachs, et al) explicitly or implicitly, think of Scandinavian social and economic systems as a kind of “magic geography” (i.e., defects and problems are not “welcomed” in their idealized visions). On the right, there’s a Singapore paradise of the imagination (what magic geography is) whereas Boris Johnson of England sees a “high wage, low tax” investment utopia which serve as a marvelous locale for the founding of both new businesses and new families. In this vision, men found businesses and women found or establish families, so everybody’s happy.

These competing visions can be traced back as far as you like, but we point to 1974 when Hayek (“the right”) and Myrdal (“the left”) shared the Nobel Prize for Economics.

Professor Niall Ferguson, the conservative Harvard (now Stanford) financial historian (you may have seen his The Ascent of Money PBS mini-series), had a program years ago on educational TV where he walked around places in Chile that he presented as a pension “heaven.” Chile is now kind of falling apart with street riots convulsing Santiago. (Ferguson’s ideal Milton Friedman of Chicago was the main advisor of General Pinochet after the 1973 Chilean coup.

What none of these people see is that social reality is complex and highly changeable and that “magicalizing” one place or system (Sweden, Hong Kong, Singapore, Chile) won’t work because successes that look solid or eternal are often caused by all kinds of “conjunctural” (i.e., of the moment) factors which don’t last “forever.”

Thus, much less “dogmatism” is called for so that one is not “swept along” by “ideological foolishness.” embedded in “magical geography” or its clone “sacred geography.”

Essay 29: Overview Strengthening: Using Meta-Intelligence to Orient Oneself

Let’s do an exercise in “overview building,” one of the key purposes of an education that is meaningful and not just an obstacle course or a maze governed by exams.

Think of the PBS miniseries of a few years ago: Guns, Germs, and Steel. It was a decent overview of history based on Jared Diamond’s famous book by the same name. Guns presumably encapsulates coercion, whether by armies, police or criminals. Germs stand for illness and disease in a germ-based view of sicknesses. Steel is the modern building material that allows skyscrapers, etc.

The viewer of the series might have felt a bit uncomfortable since humans with their “rage to belong, and rage to believe” (to paraphrase William James) make ideas into beliefs and doctrines and are not reducible to material factors like the Jared Diamond trio of guns, germs and steel.

The brilliant 20th century sociologist Ernest Gellner had a more inclusive encapsulation of human experience or world history when he entitled his own book of this type, Plough, Book, and Sword.

“Plough” gives you settled agriculture (farming, roughly speaking) (i.e., food production not based on hunting). Gellner’s “Book” gives you ideas, doctrines, bibles, epics, stories, collections of documents and speeches, record-keeping beyond clay tablets, and so on. “Sword” gives you coercion à la Jared Diamond’s “guns.”

While Diamond omits “books.” Gellner omits “germs.” Books implies some level of reading and writing.

Ask yourself if you have an improved trio of mega-dimensions to explain human historical pathways or perhaps a quartet. Another trio might be: “Gods, Flags and Families.”

Tolstoy emphasizes the capriciousness of world history and accidents or “random walks” might then well be a plausible candidate for inclusion in our own version of a Diamond/Gellner-type title, as accident or accidentality or randomness.

Essay 28: The World Understood As “Grievance Machine: Missing Education in Symbolic Wounds”

Standard textbooks on history always give a narrative of kings and battles, routs and rallies. Some mention various vested interests.

Books never seem to grasp the deep truth that history everywhere is a national and personal “dignity quest” while food and job/wage insecurity are certainly co-factors that cannot be ignored.  Slogans like “Make America Great Again” play to this hunger for another “championship season.”

Amartya Sen of Harvard gives us a useful overview of this “dignity story” when he writes:

“The devastating effects of humiliation on human lives can hardly be exaggerated. The historical ills of the slave trade and colonization (and the racial insults that were added to physical and social injury) have been seen as ‘the war against Africa’ by the Independent Commission on Africa, which identifies Africa’s principal task today as ‘winning the war against humiliation’ (the chosen title of its report). As the Commission argues, the subjugation and denigration of Africa over the last few centuries have left a massively negative legacy against which the people of the continent have to battle. That legacy includes not only the devastation of old institutions and the forgone opportunity to build new ones, but also the destruction of social confidence, in which so much else depends.

“Africa which gave birth to the human race and was responsible for many of the pioneering developments in the growth of world civilization, was turned into a continent of European domination and the hunting ground for slaves to be transported like animals to the New World.”

(Amartya Sen, Identity and Violence: The Illusion of Destiny, 2006, Allen Lane Penguin Books, page 86.)

This has also led to the proliferation of counter-narratives which are perhaps too “Afrocentric” and thus distort things going the other way.

The 20th century psychologist Bruno Bettelheim and others use the phrase “symbolic wounds” and it must be seen that such wounds lurk everywhere in human society and memory and world history, past and present, which cannot be even remotely grasped without acknowledging the centrality of these scars.

The contemporary French thinker Pierre Nora has tried to explore these nightmare-memories in the collective mind and should be considered in this humiliation/dignity epic tale that governs much of human “thinking.”

Essay 27: Extracting Signals from Noise

We wish to help students “parachute into” or sneak up on mathematics before any “rocket science” (i.e., focus on high school and don’t get lost in the weeds).

Think of square roots. The square root of 4 is 2, of 9 is three of sixteen is 4.

But the moment someone asks:  what about the square root of 17, you will find that crystalline simplicity and obviousness are long since gone.

If you “keep your nerve” and calmly get into the “complexity jumps” you may well find it enchanting that numerical understanding has to be coaxed forth and doesn’t offer itself up readily.  But why would the physical universe, if it is really mathematical in its very “fabric,” be so ready to “jump away from you” in its complexity?

Go back to the square root of 17. Suppose you disallow logarithms and log tables. Suppose you say that “approximation theory” is too approximate. There’s no HP Scientific Calculator. One insight you might need is that 17 is “not far” from 16 so that the square root must be 4 plus a little. Call this “little” x.

Then (4+x) (4+x) = 17. You solve for x with the quadratic formula you had in high school. If you don’t remember the formula or cannot derive it, you’d have to look it up which might be disallowed in this “game.”

If you stand back, “meta-intelligently” (i.e., asking, “what does this tell me?”), you wonder whether the universe and its math fabric are an endless “onion” of such layers and complexity and not “boil-down-able.”

Another such example is Grandi’s series.

Grandi’s series and its trickiness:

In 1703, the mathematician Luigi Guido Grandi studied the addition: 1 – 1 + 1 – 1 + … ( 1-1, infinitely many, always +1 and –1).  You find if you group the numbers in certain valid and legitimate way, you could different results.  How can that be?

In mathematics, the infinite series, 1 − 1 + 1 − 1 + ⋯, can also be written:

Grandi’s series

It is sometimes called Grandi’s series, after Italian mathematician, philosopher, and priest Luigi Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it lacks a sum in the usual sense. (On the other hand, its Cesàro sum is ½.)

One obvious method to attack the series (

One obvious method to attack the series (1 − 1 + 1 − 1 + ⋯) is to treat it like a telescoping series and perform the subtractions in place:

(1 − 1) + (1 − 1) + (1 − 1) + … = 0 + 0 + 0 + … = 0

On the other hand, a similar bracketing procedure leads to the apparently contradictory result:

1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + … = 1 + 0 + 0 + 0 + … = 1

Thus, by applying parentheses to Grandi’s series in different ways, one can obtain either 0 or 1 as a “value.” (Variations of this idea, called the Eilenberg–Mazur swindle, are sometimes used in knot theory and algebra.)