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There is a famous legend inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg Academy. The French philosopher Denis Diderot was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments for atheism were influencing members of her court, and so Euler was asked to confront the Frenchman. Diderot was informed that a learned mathematician had produced a proof of the existence of God: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced this non-sequitur: "Sir, $\frac{a+b^{n}}{n}=x$, hence God exists—reply!" Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. However amusing the anecdote may be, it is apocryphal, given that Diderot himself did research in mathematics. The legend was apparently first told by Dieudonné Thiébault with embellishment by Augustus De Morgan

The above is an excerpt from Euler's page on Wikipedia. The article is quite clear - the story is apocryphal. Too bad if it is, but I'm aware that Denis Diderot knew enough mathematics to see through the ruse.

Just wanted to ask a few questions though:

  1. What exactly is the equation in this "proof" of god?
  2. Does it belong to a certain algebraic category?
  3. What would be the general solution for such types of equations?
  4. Can mathematics prove God exists?
  5. Comments of all kinds are welcome.

I plotted the equation $[y = \frac{a + b^x}{x}]$ where a = 2 and b = 3 using Desmos graphing calculator and I got this:

enter image description here

Weird things happen to the graph when I change the signs of a and b.

P.S. I also remember seeing this in a movie or was it my imagination? I can't tell. If someone has any information on a movie in which there's a scene which reenacts this legend, please do let me know.

Muchas gracias in advance and apologies if the question fails to meet the exacting SE standards. I'm hoping that some of you will find this mathematical anecdote interesting and might have insights to share to boot.

Merry Christmas.

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    $\begingroup$ The equation Euler gave is surely nothing more than "symbol-salad" that had no other purpose than to demoralize his opponent. $\endgroup$ Dec 25, 2022 at 22:58
  • $\begingroup$ Danke, very interesting. I thought a symbol salad would be something like $\times a - +$. I can graph $\frac{a + b^n}{n} = x$. $\endgroup$
    – Hudjefa
    Dec 26, 2022 at 6:40
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    $\begingroup$ "I thought a symbol salad would be something like" --- Our symbol salads are something like $\times a - +,$ but Euler's symbol salads exist in higher realms of coherence, and Gauss's symbol salads are probably publishable $\ldots$ :) $\endgroup$ Dec 26, 2022 at 11:47
  • $\begingroup$ @DaveLRenfro, like this awesome compliment I read online - his bad ideas are our good ideas. $\endgroup$
    – Hudjefa
    Dec 27, 2022 at 10:17
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    $\begingroup$ I vote for this question to be closed it is not about history. $\endgroup$
    – Mauricio
    Jan 2 at 15:48

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It seems that $\frac{a+b^n}{n}=x$ is wrongly attributed to that one that used to prove existence of God by Euler, however there is another formula named "Euler's identity", this formal is truly mysterious, since it has all essence of mathematical analysis, and the fact that this formula actually exist and exist exactly in this shape is actually mystical.

Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants.

$$ e^{i\pi}+1=0 $$

Galileo once said, “Mathematics is the language with which God wrote the Universe.”

However, this formula not just essence of math beauty, but it also connects the beauty of mathemathics with core of natural science, Euler identity is often used to describe the electro-dynamics of harmonical oscillation, for electrical current particularly. Matter of the universe is nothing more but just energy and electro-magnetic interaction, that so easy to describe with essence of pure human logic expressed in God's language of math.

Of course any mathematical formula will be pure speculation, and strictly following scientific logic to prove existence of God, it will require more solid evidence, that are falsifiable according to Karl Popper, if you scientist and philosopher you should know that name.

However, if you are believer, like Euler was, who also was one of the greatest mathematician minds by the way, and you believe that this world was created in idealistic way, by some supernatural being, who shared his intelligence and his way of thinking kindly with us, as Bible says that, God used image of himself to create an Adam, first human being. So if you are an intelligent Being and you creating the world, most likely you will invent algebra and mathematical analysis first, and you will invent this formula, and then you will use this formula and math to describe the matter of surrounding universe.

Here few quotes about the beauty of Euler's Identity by recognized professors of mathematics:

https://en.wikipedia.org/wiki/Euler%27s_identity

(rev: https://en.wikipedia.org/w/index.php?title=Euler%27s_identity&oldid=1189364066)

Stanford University mathematics professor Keith Devlin has said, "like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence".[7] And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler's formula and its applications in Fourier analysis, describes Euler's identity as being "of exquisite beauty".[8]

Mathematics writer Constance Reid has opined that Euler's identity is "the most famous formula in all mathematics".[9] And Benjamin Peirce, a 19th-century American philosopher, mathematician, and professor at Harvard University, after proving Euler's identity during a lecture, stated that the identity "is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth".[10]


Something I can add from myself, there is also one psychological mystery connected with this formula, identity as often it is referenced, it is not just mathematical definition, it also has certain psychological definition. So if you see this beauty in that equation, you identify your self with Euler, like professor Keith Devlin and Constance Reid, and because Euler was a really great mathematician mind it is really honor. In this moment you are sharing the intelligence of the God, that God kindly shared his intelligence with us in the moment of creation, this is the place where you can really see that God actually exists.

So as you can see here, three things are connected here: Mathematics(our logic), Science(our world's matter) and Psychology(our soul). Too many just for one silly formula, right?

https://en.wikipedia.org/wiki/Identification_(psychology)

This way to prove the existence of God, it is not truly scientific, it is speculative but I think it has right to exist. It is giving a hope to all of us, because otherwise it is very sad to be alone without any meaning on this planet lost somewhere in the universe.

"The most incomprehensible thing about the universe is that it is comprehensible." - Albert Einstein

Need to mention that A. Einstein publicly asked not to use any of his quotes to support any religion.

https://en.m.wikipedia.org/wiki/Religious_and_philosophical_views_of_Albert_Einstein


For all the sceptics, who tend to follow strict science approach, chinese room is known paradox, that shows weakness of scientific approach, science can't disprove existent of God, the best science can do is to say that there is no scientific tools to answer this question. Saying that there is no God, it is not scientific statement, it rather another form of religion and believe in God unexistence. You can believe in existence or you can believe in unexistence of God. However many greatest mind noticed some mystical mystery in this world with unusually strong comprehensiveness property of that world that supposed to be chaotic. And Euler with his identity come very close to idealistic beginning of the world and comprehensiveness.

"consciousness" have exactly same properties form scientific perspective like "God", but obviously "consciousness" is real, so strict scientific approach regarding unexistence of God is destroyed. https://en.wikipedia.org/wiki/Chinese_room

Simply saying, if this world ever had a purpose or this world created by intelligent being - the only evidence that we can find internally, is to see an order and beauty, in that place where we suppose to see a chaos. And obviously Euler found an order in the place where everyone were expecting to see a chaos, maybe it just mind game of our beautiful humans mind, or maybe it really artifact of rational creation. Here is another quote: “is absolutely paradoxical; we cannot understand it, and we don't know what it means” - Benjamin Peirce, I am totally agree with Benjamin Pierce.

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    $\begingroup$ I do not see what these more or less random thoughts and quotes have to do with the question, besides mentioning Euler and God. $\endgroup$ Jan 2 at 6:20
  • $\begingroup$ @TorstenSchoeneberg Do you think Euler And many other scientist like Einstein were silly when they were talking about comprehensives of this world? The truth is that world is not entirely chaotic as it supposed to be per materialist study. One problem that science can come any close, and it beat the ground of any atheistic or materialistic study is question about consciousness. I guess you would be interested to learn about Chinese room experiment: en.wikipedia.org/wiki/Chinese_room $\endgroup$ Jan 4 at 0:22
  • $\begingroup$ I do not think those people were silly, I am aware of those matters and that thought experiment, I just think none of that is more than cursorily related to the question. Even if it were, I think these matters are far too complex and subtle to be addressed with some context-less quotes and links from diverse sources. $\endgroup$ Jan 4 at 3:20
  • $\begingroup$ @TorstenSchoeneberg I agree that trying to look outside of material universe is complex, and more than that mathematics is complex science, not everyone can see same beauty in this formula like Euler see, you need to be intelligent enough and have certain knowledge in math. Being intelligent enough, it is very subjective, it is not scientific way to prove, however weird coincidence that identity has psychological definitions, to understand Euler’s identity, you need to identify yourself with Euler, how such coincidence could happen. If you are believer, it is easy to accept, it is another sign $\endgroup$ Jan 4 at 4:30

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