(a+b^n)/n = x. Therefore, God Exists.

Denis Diderot

                                                                     (Part i)

“ (a+bⁿ)/n = x. Therefore, God exists…” is a joke and a joke with multiple layers. From it we can learn about theology, humor, and even literary history, as I found when trying to track it down to teach in my seventh-grade math class. Let’s try to unpeel those layers.¹

Some readers will know that the line is the climax of a famous story about the philosopher Diderot (“Deedero”) and the mathematician Euler (“Oiler”) at the court of Catherine the Great.

"The following anecdote is found in Thiebault's Souvenirs de Vingt Ans de Sejour a Berlin, published in 1804. Thiebault does not claim personal knowledge of the anecdote, but he vouches for its being received as true all over the north of Europe. Diderot paid a visit to Russia at the invitation of Catherine the Second. At that time he was an atheist or at least talked atheism: it would be easy to prove him one thing or the other from his writings. His lively sallies on this subject much amused the Empress, and all the younger part of her Court. But some of the older courtiers suggested that it was hardly prudent to allow such unreserved exhibitions. The Empress thought so too, but did not like to muzzle her guest by an express prohibition: so a plot was contrived.

The scorner was informed that an eminent mathematician had an algebraical proof of the existence of God, which he would communicate before the whole Court, if agreeable. Diderot gladly consented. The mathematician, who is not named, was Euler. He came to Diderot with the gravest air, and in a tone of perfect conviction said, 'Monsieur! (a+bⁿ)/n = x, donc Dieu existe; repondez!' Diderot, to whom algebra was Hebrew, though this is expressed in a very roundabout way by Thiebault and whom we may suppose to have expected some verbal argument of alleged algebraical closeness, was discon­certed, while peals of laughter sounded on all sides. Next day he asked permission to return to France, which was granted." (from August De Morgan, A Budget of Paradoxes [1872])

In case you stop reading here, take note: do not believe this story and do not retell it unless you read to the end of this essay. It isn’t exactly false, but neither is it exactly true. But let that pass for the moment; we’ll proceed as if it were gospel truth.

We must start with what (a+bⁿ)/n = x means. Literally, it means that for number labels (that is, variables) a, b, n, and x that we have not yet defined, if we take b to the nth power, add it to a, and then divide the whole thing by n, we get x as a result. But whether this is a true or false equation depends on how we set the definitions. If we set a = 1, b = 2, and n = x = 3, then the equation is true; if we change to x = 4, it is false. Thus, we really don’t have enough to go on for this to mean much of anything. Since it doesn’t mean anything once you understand what it’s saying, it can’t be a proof that God exists, though the difficulty of understanding what it’s saying means my seventh graders might accept it as proof that Satan exists.²

Thus, what the story seems to be saying is that when an atheist tries to prove God doesn’t exist, it’s great if you can find some totally false argument to expose his ignorance and make him look silly. Tie him in knots with mathematical notation or abstract verbiage and you’ll win.

Now smokeblowing can be an effective way to win an argument, but it’s not an honest way. It’s not suitable for someone who believes in God to argue like that. Using technical language to obscure the question and show how smart you are does work with a lot of people, but not with wise people. A wise person, whether intelligent or slow-minded, sees through that technique and concludes that the obscurant is using unfair tactics, very likely because he knows he can’t win with genuine arguments. If the obscurant had a good argument and really was smart, he would try to phrase his argument simply enough that it could be followed step by step.

The only way I can see for such a technique to be valid is if the speaker admits at the end that his algebraic argument was a bluff, a joke, and he made it either just for fun or to illustrate the point that someone who is clever with words or mathematics can make nonsense sound valid, something clever atheists like Diderot do. That is, the technique could be used as a counterattack when it is the opponent who is being obscure, jargonish, and slippery.

Or (but this is a weaker argument), the mathematician could be making the point that Diderot, being too stupid or ignorant to understand simple equations, has no business addressing a truly hard topic such as the existence of God. This is a legitimate version of the ad hominem argument, legitimate because it is used to counter your opponents’ authority rather than his arguments and he was trying to persuade by brilliance rather than logic or evidence. It’s like poison gas: only legitimate if the other side used it first.

In both cases, the mathematician would be appealing to the unintelligent and the very intelligent person, as opposed to the person in between who isn’t very intelligent but thinks he is. As St. Paul says in I Corinthians 1:

For the preaching of the cross is to them that perish foolishness; but unto us which are saved it is the power of God. For it is written, I will destroy the wisdom of the wise, and will bring to nothing the understanding of the prudent. Where is the wise? where is the scribe? where is the disputer of this world? hath not God made foolish the wisdom of this world? For after that in the wisdom of God the world by wisdom knew not God, it pleased God by the foolishness of preaching to save them that believe. For the Jews require a sign, and the Greeks seek after wisdom: But we preach Christ crucified, unto the Jews a stumblingblock, and unto the Greeks foolishness.

Of course, this kind of argument is most effective when it’s an intellectual person saying that in some things simple people know best, e.g., Paul, a Roman citizen trained by the famous Jewish rabbi Gamaliel, or Euler, one of the smartest mathematicians of all time. Or Alexander Pope in his 1709 Essay on Criticism,

A little learning is a dangerous thing;
Drink deep, or taste not the Pierian spring:
There shallow draughts intoxicate the brain,
And drinking largely sobers us again,³

where Pope is rephrasing Francis Bacon’s 1601 essay, On Atheism,

God never wrought miracle, to convince atheism, because his ordinary works convince it. It is true, that a little philosophy inclineth man's mind to atheism; but depth in philosophy bringeth men's minds about to religion. For while the mind of man looketh upon second causes scattered, it may sometimes rest in them, and go no further; but when it beholdeth the chain of them, confederate and linked together, it must needs fly to Providence and Deity.

We will return to Bacon towards the end, so don’t forget what he said.

---

Catherine the Great as Princess

                                                                           (Part ii)

This is where my essay would stop if the story were not multi-layered. But consider how the Diderot story is retold fifty years later in Bell’s delightful Men of Mathematics:

"We shall tell once more the famous story of Euler and the atheistic (or perhaps only pantheistic) French philosopher Denis Diderot (1713-1784). Invited by Catherine the Great to visit her Court, Diderot earned his keep by trying to convert the courtiers to atheism. Fed up, Catherine commissioned Euler to muzzle the windy philosopher. This was easy because all mathematics was Chinese to Diderot. De Morgan tells what happened (in his classic Budget of Paradoxes, 1872): Diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of God, and would give it before all the Court, if he desired to hear it. Diderot gladly consented ... Euler advanced towards Diderot, and said gravely, and in a tone of perfect conviction: 'Sir, (a+bⁿ)/n = x, hence God exists; reply!' It sounded like sense to Diderot. Humiliated by the unrestrained laughter which greeted his embarrassed silence, the poor man asked Catherine's permission to return at once to France. She graciously gave it. Not content with this masterpiece, Euler in all seriousness painted his lily with solemn proofs, in deadly earnest, that God exists and that the soul is not a material substance. It is reported that both proofs passed into the treatises on theology of his day." (from E. Bell, Men of Mathematics [1937])

“Earned his keep by trying to convert the courtiers to atheism.”???

“The poor man asked Catherine's permission to return at once to France.”???

“Not content with this masterpiece, Euler in all seriousness painted his lily with solemn proofs, in deadly earnest, that God exists and that the soul is not a material substance. It is reported that both proofs passed into the treatises on theology of his day.”???

All three additions Bell made to De Morgan’s story seem to be invented out of thin air. He made history into a docudrama. Bell’s book is entertaining enough that we should be warned, and he even gives us a warning in his Euler, Analysis Incarnate chapter with reference to another story, “It should be noted that the modern higher criticism which has so been so effective in discrediting all the interesting anecdotes in the history of mathematics has shown that the astronomical problem was in no way rsponsible for the loss of Euler’s eye.” We can only be thankful that he didn’t add a love interest involving the Empress (perhaps via Euclid’s pons asinorum?).⁴ I have to admit, thought, that I prefer Suetonius to Tacitus, and still recommend Men of Mathematics to my students.

This inaccuracy should make us worry. Bell’s retelling of the story is one layer above De Morgan’s, and he invented part of it. What about the layer below De Morgan? De Morgan cites Thiebault, but what did Thiebault actually say? Here it is, in English translation:

From the moment of his arrival Diderot was well received, all his expenses had been paid by the Empress whom he amused immensely by the fecundity and fire of his imagination, by the abundance and singularity of his ideas, and by the zeal, boldness and eloquence with which he publicly upheld atheism. But several of the oder courtiers more experienced and perhaps more easily alarmed, persuaded their autocratic sovereign that teachings of this kind could have unfortunate consequences for the whole court, and especially among the large youthful group, destined for important empire posts, who might em­brace this doctrine with more eagerness than careful scrutiny. The Empress then desired that some restraint be put upon Diderot on this subject, provided that she did not appear to play any part in the matter, and provided that no one should show any undue authority about it. It was therefore announced to the French philosopher one evening, that a Russian philosopher, a learned mathematician and a distinguished member of the Academy, was prepared to prove the existence of God to him, algebraically, and before the whole court. Diderot said that he would be happy to listen to such a demonstration, in the validity of which of course, he did not believe, and so an hour and a day were fixed to convince him.

The occasion having arrived, with the whole court pres­ent, that is to say, the men and more particularly the younger members, the Russian philosopher gravely advanced towards the French philosopher, and speaking in a tone of voice to imply his full conviction, said, "Monsieur, (a+bⁿ)/z = x, therefore God exists: answer that!" Diderot was willing to show the futil­ity and stupidity of this so-called proof, but felt in spite of himself, the em­barrassment that one would, on discovering, (among them), their intention of making a game of it, so that he was not disposed to attempt to admonish them for the indignities proposed for him. This adventure made him fearful that there might be others in store for him of a like nature, and so sometime afterwards he expressed his desire to return to France. Then the Empress having declared her willingness to pay all his traveling expenses, he was sent on his journey after having received 50,000 francs.

(from vol. 3 of Mes Souvenirs de Vingt Ans de Sejour a Berlin, Dieudonne Thiebault (1804) ).⁵

What! “Diderot was willing to show the futil­ity and stupidity of this so-called proof?” “This adventure made him fearful that there might be others in store for him of a like nature, and so sometime afterwards he expressed his desire to return to France?” Now, in the original, it seems Diderot knew exactly what was going on, but he was intimidated by how mean to him the court was, and he went home to avoid being bullied again.

It’s also worth pointing out that De Morgan got the formula wrong. Thiebault wrote (a+bⁿ)/z = x, not (a+bⁿ)/n = x. It’s forgiveable to make a copying error like that, which doesn’t matter to the point of the story, but what’s noteworthy is that Bell followed him in the same error, and so, according to the Gillings article I cite in footnote 1, did everyone else who told the story after 1900, including everything you will find on the Web.

The important mistake is not in the algebra, though, but that if we take what Thiebault says at face value, Diderot knew a joke was being played on himbut was too polite to say so. He wasn’t ignorant and defeated— he was polite and defeated.

Still, we haven’t reached the bottom layer. We’ve only reached Dieudonne Thiebault, God-Given though he may be, and that only in English translation.⁶ Very likely the equation (a+bⁿ)/z = x isn’t what Euler said. It’s a nonsense equation, the kind of thing a non-mathematician might make up for a story where an equation of some kind had to be inserted. Thiebault probably heard the story as “some equation”— the gossip probably didn’t specify the exact equation— and put in a concrete example, as legitimate poetic license. Euler would have chosen a more meaningful equation. I have one in particular in mind, to which we shall return later.

It would be useful to know what Thiebault and Diderot thought of each other. Were they enemies? Were they friends? Or allies? Even beyond that, remember that “Thiebault does not claim personal knowledge of the anecdote, but he vouches for its being received as true all over the north of Europe.” People had played “Telephone” with the story even before it got to Thiebault.

It sounds, however, as if Thiebault’s version of the story comes from Diderot. “Diderot was willing to show the futil­ity and stupidity of this so-called proof, but felt in spite of himself, the em­barrassment that one would, on discovering, (among them), their intention of making a game of it, so that he was not disposed to attempt to admonish them for the indignities proposed for him.” Who would know that Diderot felt these things, except for Diderot himself? Suppose the story played out as De Morgan said it did, with Diderot being not polite and defeated, but ignorant and humiliated? Indeed, what if Diderot, instead of being silent, had tried to reply to the equation argument, but failed, and then shut up after ten minutes rather than doing himself more damage., Master publicist as Diderot was (he was very ancien regime that way), he would have done damage control in the rumor mills by spinning the story as we hear it from Thiebault. And I should add something very important: Diderot was not a mathematican of the caliber of Euler, but he did write original articles on the mathematics of curves.⁷ And Diderot was a seasoned debater who would not be easily abashed.⁸

So the evidence really is neither for nor against the De Morgan story as being what really happened, though De Morgan should not have told it as he did, to be sure. When there is no evidence either for or against a fact, you shouldn’t tell it as if it were certain. I wish we knew what happened at Queen Catherine’s court, but we don’t.⁹

---

Leonhard Euler

                                                              (Part iii)

I said earlier that I had a guess about what equation Euler really used. You readers who are mathematicians: it isn’t what you are thinking of. You are thinking of Euler’s most famous and mind-boggling equation, Euler’s Identity,

                                                           eπi = -1,

where e is "Euler's number", which is about 2.72; π is the ratio of a circle’s circumference to its diameter, which is about 3.14, and i is the square root of -1. Euler’s identity is awe inspiring, because it relates four special, specific, numbers to each other exactly. A 1990 poll of readers of The Mathematical Intelligencer rated it the most beautiful theorem in all of mathematics.

I agree. But I find Euler’s Identity deficient as a proof of God, because it actually comes from a way to define what it means to write e^ix for some variable x. Since i is the square root of negative 1, it is called an “imaginary number”, and it isn’t clear what rules it should follow for exponentiation— there is no real-world way to do it. What makes sense, though, is to define imaginary exponentiation so that calculus works the same way there as with the “real numbers”, which means making sure that the slope of the line y = e^ix is iy (that is, ie^ix), since the slope of e^nx is ne^nx for real numbers. It turns out that so doing requires defining e^ix in terms of trigonometry functions, as¹⁰ e^ix = cos(x) + i sin(x). Setting x = π we get e^ix = cos(π) - i sin(π), and since cos(π) = -1 and sin(π) = 0, this means e^ix = -1 - 0i = -1. (That’s more math than most readers want, but I trust your eyes know when to glaze over, and when not to.) This goes to show that Euler’s Identity, remarkable though it is, is based on forcing definitions so artificial imaginary numbers fit in with the rest of mathematics.

What, then is my candidate equation for Euler vs. Diderot? It comes from a math problem posed in 1650 about the value of a certain infinite sum (“The Basel Problem”). Euler solved the problem in 1734 while in his 20’s by showing that

\(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \ldots = \frac{\pi^2}{6}\)

This is is plausible, since it says that 1 + 1/4 + 1/9 + 1/16 plus extra terms (which is a little more than 1.4) equals a sixth of about 3², which is 1.5. But how do π and 6 fit in there so exactly? The number π is about the ratio of the circumference of a circle to the diameter, and the infinite sum is about squares. They shouldn’t match up. And 6 is a rather special number too— it isn’t like 2, 3, or 4, which in math often show up in odd places here and there.¹¹

I’ll repeat the passage from Francis Bacon’s, On Atheism.

God never wrought miracle, to convince atheism, because His ordinary works convince it. It is true, that a little philosophy inclineth man's mind to atheism; but depth in philosophy bringeth men's minds about to religion. For while the mind of man looketh upon second causes scattered, it may sometimes rest in them, and go no further; but when it beholdeth the chain of them, confederate and linked together, it must needs fly to Providence and Deity.

Euler’s Basel Problem Equation is my candidate for the “ordinary work” that made Euler “fly to Providence and Deity”. I am not alone in my awe at its neatness. I learned about it recently, while leafing through Jay Cummings’s Real Analysis: A Long-Form Textbook (2019, 2nd ed.). Here is what he says on p. 129, which can conclude our excursion into the history of mathematical theology.

“The number π is defined by circles. What the hell do sums of inverse squares have to do with circles? And why squared? It seems utterly unreasonable that a π should be involved in the answer, and I hope Euler fell off his seat when he discovered it. I wouldn’t say there’s too much direct evidence that there’s a God. But. . . I think we need to put this one in God’s column.”

1. I take all quotes from the excellent article, “The So-Called Euler-Diderot Incident,” R. J. Gillings, American Mathematical Monthly, 77-80 (1954).

2

 Pop quiz: (A punishment for reading footnotes? Sorry. Shouldn't I reward you, diligent reader? But this quiz will be good for you, so really it is a reward. ) 

         If the equation is true, is it false that a + bn/n = x? 

                                                                                               Hint: order of operations 

                                                                                                Answers are in footnote 12.

3. I, myself, had to look up “Pierian spring”. Wikipedia says, “The name of the spring comes from the Pierides, the gaggle of girls (daughters of King Pierus) who sought a contest with the Muses. When they lost, they were turned into magpies. Ovid tells this tale…

These Emathian sisters laughed to scorn our threatening words; and as they tried to speak, and made great clamour, and with shameless hands made threatening gestures, suddenly stiff quills sprouted from out their finger-nails, and plumes spread over their stretched arms; and they could see the mouth of each companion growing out into a rigid beak. – And thus new birds were added to the forest. – While they made complaint, these Magpies that defile our groves, moving their stretched-out arms, began to float, suspended in the air. And since that time their ancient eloquence, their screaming notes, their tiresome zeal of speech have all remained.

Thus, Pope is very obliquely saying that atheists are raucous, screeching, magpies. Is there a name for this kind of insult, which is so indirect that its target can’t even tell it’s an insult?

4. You may have noticed that the mathematician is here not named as Euler by Thiebault. De Morgan surmises that it was Euler. Gillings, who is otherwise highly critical of De Morgan, concedes that “Thiebault's phrase, ‘a Russian philosopher, a learned mathematician and a distinguished member of the academy,’ seems rather definitely to refer to the Swiss Mathematician Euler.” Euler was notably pious, and would have become a Calvinist pastor like his father except friends insisted he really had to make full use of his math talent. Also, he liked to talk metaphysics but was notably clumsy at it, at least according to his other patron, Frederick the Great of Prussia (Euler only associated with Greats, it seems). The identity of the mathematician is a point worth worrying about, but I think it’s fair enough to deduce that it really was Euler— an example of excusable additional inference about a story.

5. Gillings tells us that the 1806 English translation of Thiebault’s book in two volumes in Philadelphia (no doubt pirated) omits the Diderot story. In the 1860 edition, it is on pp. 8-9 of Part 3, Chapter VII (“Voyageurs a Berlin”). Here is the 1804 story from pp. 141-143.

6. Not quite everyone makes that mistake. I found one page on the web that makes a different mistake, giving (a + bⁿ)/x = x instead of (a + bⁿ)/n = x.

7. If anyone has scanned in the original French version, please send it to me and I’ll put it in a footnote here and see if the English translation is faithful. I’ll probably buy the Amazon/Walmart reprint and do that myself, but I might not get round to it. June 22: I got a copy and found it on the web; see the screenshots above. The rest of Thiebault looks worth reading too. He knows how to tell a story, and your French doesn’t have to be very good to understand it.

8. “The Mathematical Writings of Diderot,” Isis, Lester Krakeur, Raymond Krueger, and Diderot 33: 219-232 (1941).

9. The day I published this on Substack, I sent it to Joseph Epstein, whom I have long admired for his erudite and charming essays on belles lettres (f you like this essay of mine, you’ll like his essays even better). He emailed me right back with a copy of his essay, “A Philosophe in Full: Denis Diderot's Enlightenment,” (Claremont Review of Books, Fall 2019), which says,

The salonnière Madame Geoffrin, who eventually outlawed Diderot from her salon, reported to a friend that “[h]e is always like a man in a dream, and who believes everything he has dreamed.” The playwright and literary critic Jean-Françoise de La Harpe found Diderot altogether too delighted with his own conversation and, in Zaretzsky’s paraphrase, “his own most ardent and attentive listener.” This view seems to have been partially seconded by Diderot, who of himself said, “I’m high-minded and, on occasion, come across great and powerful ideas that I convey in a striking fashion.” Note the “on occasion.”

Once arrived in Saint Petersburg, Diderot met each afternoon from 3:00 to 5:00 with Catherine. She was initially much taken with him. “Diderot’s imagination, I find,” she told Voltaire, “is inexhaustible. I place him among the most extraordinary men who have ever lived.”

It is hard to believe Diderot would ever be at a loss for words.

10. We could discuss further what the writer of history should do in the conflict between drama and literal truth. I say “literal truth” because sometimes poetic license makes the description better convey reality than would the bare facts. It’s a little like how a statement can be false but reliable (the earthquake hit at 2:05 p.m., when it was actually 2:13 p.m.) versus true but misleading, the style of 2023’s corporate media (Smith shot Jones— while not mentioning that Jones was lunging at Smith with a knife until paragraph 14). “When the legend becomes fact, print the legend,” said the editor in The Man Who Shot Liberty Valance, modified by novelist Larry McMurtry to “When you have to choose between history and legend, print the legend.”

11. If f(x) = e^ix = cos(x) + i sin(x), then the derivative is df(x)/dx = -sin(x) + icos(x), which equals i²sin(x) + i cos (x), which equals i [cos (x) + i sin(x)], which is i [e^ix].