Talk:Euler's identity

At the end of The Feynman Lectures on Physics, vol. 1, chapter 22, he says: "We summarize with this, the most remarkable formula in mathematics: e^iθ = cos θ + i sin θ. This is our jewel." I think the Feynman quote belongs on the Euler's formula page instead. Feynman also doesn't say in this book that it is remarkable for those reasons. Not that they aren't true; they just weren't said by Feynman.

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I removed the following paragraph twice:

There has been substantial debate in the philosophy of mathematics on the "real meaning" or "deep meaning" or even sacred geometry reflected by the Identity's relationship of key constants and operations (multiplication, exponentiation, addition, equality). Some assert that it describes cognitive properties of an embodied mind - and advocate a cognitive science of mathematics. At other extremes, some assert it represents rational social consensus of mathematicians, or is simply a fundamental fact of the physical universe, and that algebra itself is a natural consequence of its structure. If so, the formula would be more than simply remarkable - it would be 'divine'.

There has not been any substantial debate about sacred geometry related to this identity in the philosophy of mathematics. If I have missed the relevant literature, please point me to books, articles, conference presentations etc.

have you read Tymoczko, 1998? "The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind." ([Thomas Tymoczko]?)

The way that traditional cultures refer to this "external mathematical reality" is with "sacred geometry" - whether or not mathematicians call it that.

Of course that is the central question of the philosophy of mathematics. I asked specifically about references relating Euler's identity to the concept of "sacred geometry", and I am still waiting. I dispute the claim that "sacred geometry" is a commonly used term; EB doesn't list it at all. AxelBoldt

do a google. you'll find a fair bit. The idea is somewhat contrary to Christian dogma, and occurs in Buddhism and Judaism and certain Hermetic beliefs - sometimes in Christian dogma it is associated with Satan, i.e. the pentagram, etc. One of the major sources of anti-semitism, actually, was the belief that Kabbalic rituals were "Satanic".

I just Googled for < "sacred geometry" Euler >. None of the resulting pages made any connection between the two, I'm afraid. Matthew Woodcraft

Furthermore, the paragraph presents the issue as "some assert..." — "at the other extreme....", as if those two were the only positions on the question, while in fact many other popular positions are left out.

not much room... philosophy of mathematics gave some room to this.

Well, then put a link to that page here and be done with it. AxelBoldt

ok, but the "remarkable" nature of the identity was here before I edited it, and another paragraph to establish that this "remarkable" nature may have some other origins is important.

Algebra cannot be a natural consequence of this equation, because the equation records a fact about the complex numbers, while in algebra many

there can be no such thing as "a fact about the complex numbers" since the complex numbers, and complex analysis, is a notational convenience to begin with. Your concept of reality is wrong. Fix it. ;-)

You seem to think that the questions of the philosophy of mathematics have been finally answered by your little pet theory; you're wrong. There will never be consensus on those questions. You also don't seem to understand that there can be facts about notational conveniences, and that notational conveniences are part of reality. AxelBoldt

no, there can't be facts about notation conveniences in the Popperian sense, as they are only falsifiable w.r.t. the rest of the notation - at best this is internal consistency. And no, notational conveniences are not part of "reality", they are part of colonialism or a certain paradigm of science at best. And no, again, there is no claim that the questions have been "answered by my little pet theory", as the theory that mathematics arises from the mind is very old, and the theory of mind arising in cognitive science is very deep... so it is *your* "little pet theory" that is under discussion, and its irrelevance in the face of cognitive science and philosophy of mathematics combined. As to your prediction that there will "never be consensus", that could be established merely by killing all over-educated people. To disprove this thesis, of course, you must kill them all yourself. Which brings us to the question of reasonable method...

In other words, you believe that colonialism is not part of reality. Can I quote you on that, 24? AxelBoldt, Sunday, March 31, 2002

other fields, rings and groups are studied which have nothing whatsoever to do with the complex numbers and with Euler's identity. The "divine"

that's foolish. How can fields, rings, and groups be totally independent of the operations of addition, multiplication, exponentation, and especially equality and equivalence? Euler's identity summarizes exactly these issues, and it is the way complex numbers "disappear" in the identity's resolution that makes it interesting. Also, fields rings and groups were more or less an invention of Galois - prior to that, Euler's identity summarized what was known. Suggestion, read cognitive science of mathematics and the references.

Euler's indentity summarizes issues about addition, multiplication, exponentiation and equality of complex numbers. Just because we use the word "addition" in every abelian group doesn't mean that those additions share all properties of complex addition. Euler's identity says precisely nothing about the multiplication in the monster group. It cannot even be interpreted in any way in that context, because there's no exponential map and no addition and no zero element in that context. AxelBoldt

why is *complex addition* the standard meaning? It isn't required for Euler's identity in particular, as the "e to the i pi" isn't a complex value according to Euler's formula but rather is "equal to minus one".

But i is a complex number, and the exponential function e^x is a function defined on the complex plane. Formulas don't just sit there, they are valid in a certain context. The context in which Euler's identity is valid is the complex number field.

The "monster group" is a post-Eulerism that wouldn't exist if not for Galois's theory, which is not necessarily a guide to mathematics pre-Euler. I think the naive terms "plus" or "times" meant less to Euler than Galois... who may well have overly generalized them.

So who cares about the subset of mathematics that was known at Euler's times? It has nothing to do with the discussion. You claim that Euler's identity underlies all of algebra, and the Monster group (and countless other examples) disprove that claim. AxelBoldt

connection is completely out of place and does also not relate to what was said earlier: if Euler's identity were just a social consensus, or a property of human cognition, then it would exactly not be divine. AxelBoldt, Sunday, March 31, 2002

and if it were *neither* of those, it *would* be 'divine' in the same sense as the Planck length, etc,. - something part of the fundamental structure of the universe, unchangeable, etc.

there is no need to use the loaded term "divine" for "unchangeable". Furthermore, again you are simplifying matters: Euler's identity would not have to be a fundamental structure of the universe; Platonists would argue that it necessarily holds in any possible universe. AxelBoldt

fair enough... although a god or "divine" concept can be bound by a universe, and in Plato's time, to an even smaller entity. Although you are definitely splitting hairs here, as the difference between "the universe" and "any possible universe" is a distinction that not all theories of note recognize... why should there be more than one universe? There is value in deliberately loading the term, as it makes a connection to theology, where such matters have been more thoroughly discussed...

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"The formula is a consequence of (or, viewed alternatively by some theories in the philosophy of mathematics, assumed in) Euler's formula " -- really? -- Tarquin 10:50 Jan 5, 2003 (UTC)

No, not really, but in the wonderful mind of user:24, which you can also see at work on this very talk page. AxelBoldt 02:05 Jan 8, 2003 (UTC)

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Am I not correct in saying that "Euler's Identity" is shortform for "Euler is Identity", whilst "Eulers Identity" would be the correct way of putting it? I remember a very good educational video on Channel 4 (UK) back when I lived over there that explained the eccentricities of the apostrophy - it had a very addictive little tune that I haven't been able to get out of my head in the 15 or so years since I saw it.. but I'm sidetracking: This must be wrong, right?

    --Smári 19:01, 1 Mar 2004 (UTC)

You are incorrect. See: Apostrophe (punctuation) - Bevo 20:23, 1 Mar 2004 (UTC)

Perfect. Then no need to worry. :) --Smári McCarthy 00:44, 2 Mar 2004 (UTC)

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The last revision by 63.189.8.249 states: "It was however known long before to Chinese mathematicians." -- This is highly unlikely, as the concept of imaginary numbers was unknown to Chinese mathematicians at that time. (It may be a case of confusion about Chinese remainder theorem.) I have removed it for now, until a source is cited in favor of it. --Autrijus 18:06, 2004 Aug 15 (UTC)

I would have to concur with the removal until a valid source/evidence can be found supporting the statement. - Taxman 18:32, Aug 16, 2004 (UTC)

Contents

1 Counter-intuitive? 2 Pi not constant? 3 true by definition? 4 perceptions... 5 External links

Counter-intuitive?

I may only be speaking for myself, but I would find that <math> \lim_{x \to \infty} e^{-x} = 0 <math> is more counter-intuitive than <math>e^{\pi} = 23.14069... \,\!<math> when considering Euler's Identity.

• Erm. Why? As x grows large, you will have the reciprocal of a very large number... why is it a stretch to see this as approaching zero? DocSigma 05:02, 6 Feb 2005 (UTC)

I also consider this remark on the main page (e^\pi vs e^i\pi) quite ... useless, say. The simple insertion of "-" would also change the result, by twice the order of magnitude (speaking of ratios). (Funny... twice the order of magnitude, by putting the square of the exponent....)

I think it would be quite justified to suppress this annotation, — MFH: Talk 19:24, 10 May 2005 (UTC)

I agree. -- Aleph4 16:47, 13 May 2005 (UTC)

I was struck by the non sequitur about "counter-intuitive" myself. There is no viewpoint so naive as to make this property remarkable. I support removing the comment. 66.214.64.122 23:23, 29 May 2005 (UTC)

Pi not constant?

The current version of the article says <math>\pi<math> is a constant in a world which is Euclidean, or on small scales of non-Euclidean geometry otherwise, the ratio of the length of the circumference of circle to its diameter would not be a universal constant, i.e. the same for all circumferences).

While the "otherwise" part of the sentence is true, it does not talk about <math>\pi<math> at all. You could as well define <math>\pi<math> to be the infinite string 3.14159..., and then claim that <math>\pi<math> is not constant if you use hexadecimal numbers. The string 3.14... does not define <math>\pi<math> if you interpret it in hexadecimal, and the expression "ratio of circumference..." does not define <math>\pi<math> if you talk about circles in a non-Euclidean plane.

-- Aleph4 16:47, 13 May 2005 (UTC)

true by definition?

Euler's identity is not "true by definition", because e^{(i pi)} is not "defined" to be -1.

it just happens that if you define e^z as the sum of the infinite series 1 + z + z²/2! + ... , and if you define sin(x) as the sum of the infinite series x - x³/3! + ... , and similarly cos(x) as 1 - z²/2! + ... (this is only one possibility of defining sin and cos on the complex numbers; other equivalent definitions are possible), then

e^{i z} = cos(z) + i sin(z)

is a consequence of that definition, and Euler's identity is again a corollary to that formula.

perceptions...

The first comment on perceptions, referring to 0, is currently not justified, as the formula is written as ...=-1 and not ...+1=0. (Only quite implicitely the 0 is present through the definition of "-".) — MFH: Talk 19:30, 10 May 2005 (UTC)

External links

On the page of Ian Henderson's "proof for the Layman", someone "complains" that this page is not even cited in the discussion. Let's fix this, by remarking that with the power series "formula" for e^x, sin x, cos x, Euler's formula and thus Euler's identity are trivial. All the geometric preliminaries do not contribute to this, they are only used for sin(pi)=0, but using for this the geometric definition of sin, while equality with the power series is not shown.

The other external link to a proof (.../jos/...) (which in fact concerns rather Euler's formula and thus should be moved there) deserves about the same critics (with algebraic instead of geometric preliminaries), IMHO, the crucial step being the comparision of the derivatives in zero on the last page (implicitely (via Taylor series) admitting that both sides are analytic). — MFH: Talk 21:25, 10 May 2005 (UTC)