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 conesnsus 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
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
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
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, adn 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
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