Talk:Elementary proof

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Who uses this concept?

I don't think that it is correct that anyone calls a proof "elementary" just because it does not use the square root of (-1). Nor is it true that an elementary proof is necessarily preferred.

It is not an assumption that (-1) can have a square root. It can be proved. Nor is the square root unique. (-1) has 2 square roots, if it has any.

I suggest killing this article. It just isn't useful. RSchlafly 00:11, 5 February 2007 (EST)

If the test were what is "useful", then most math articles should be deleted!

I'd love to see a proof that the following exists and is unique (plus and minus roots):

\sqrt{-1}

--Aschlafly 20:12, 6 February 2007 (EST)

Existence depends on what system you are operating in. Uniqueness is fairly easy if you are operating in a field. Proposition: in any field F, with c and element of F, the equation x2 = c has at most 2 solutions. Proof: Consider x2 = c. this implies that x2c = 0. We may assume that c has at least one square root (call it c1 / 2). So we have (x + c1 / 2)(xc1 / 2). Now, since fields have no zero divisors (easy excercise), we must have x + c1 / 2 = 0 or xc1 / 2 = 0 which gives us only two choices. Q.E.D. JoshuaZ 20:44, 6 February 2007 (EST)
I have never heard an "elementary proof" used to refer to a proof that "cannot be broken down into smaller proofs", whatever that is supposed to mean, and moreover, this doesn't clearly have anything to do with the other definition about avoiding complex analysis. Moreover, Selberg's Fields was primarily for his other work on the Riemann zeta function (complex analysis! the horror!) and Selberg formula, which indicated the direction of this proof. The proof itself was not the grounds for the medal, as this article indicates. Some editor here seems to have a bizarre fascination with "elementary proofs", but would surely be hard-pressed to find a serious mathematician who rejected residue calculus on the grounds that it is not an elementary method, as that article claims.

Re: "I'd love to see a proof that the following exists and is unique (plus and minus roots)", the questioner might try to learn a bit about fields and algebraic extensions thereof. By "sqrt(-1)" we mean some symbol which is "adjoined" to a field and which satisfies x^2+1=0. Such a thing exists in a field extension (and one can prove this quite rigorously), and is obviously unique there. Any basic algebra text (Artin, Dummit/Foote, etc) will cover the requisite material better than I could hope to do here.

- jlezzy

The "some editor" you refer to is either the owner or myself and neither of us have a bizarre fascination with this subject. Mockful sarcasm doesn't get you far on this site and neither does a woefully inappropriate choice of username. -Foxtrot 00:39, 4 January 2009 (EST)

This poster was indeed rude, but can we get citations for any of this? I can't find any evidence that there are mathematicians who reject the method of residues for being non-elementary.

Definition of Elementary proof

I would like to remove the statement about elementary proofs being those that cannot be broken down into smaller proofs. This statement doesn't make much sense to me. What so we mean by a 'smaller' proof? What do we mean by breaking a proof down into other proofs? Typically the proof of a long theorem is broken down by proving many of the steps as a lemma. This is done to make the proof of the main theorem easier to comprehend. This can be done with any theorem whose proof has more than one step. Any non-trivial theorem is bound to have many steps so I don't think that this is what the author intends. AndyJM 12:05, 9 January 2009 (EST)

AndyJM, I really don't understand what your are finding so baffling about the concept of elementary proofs. The line you removed was an important statement, and as you can see, Andy (the owner) reinstated it with some clearer wording. If you do not have experience with elementary proofs, then please do not remove content from the articles -- you've been banned for such actions before. -Foxtrot 22:00, 18 January 2009 (EST)
Hi Foxtrot. I thought that I had clearly laid out what I find so baffling about one particular statement in this article. Did you read my comment? While I am not delighted with the addition that Schlafly made it is a definite improvement over what had been there. The vast majority (perhaps all) all the proofs that I have published could be classified as elementary according to the first definition, i.e. not using complex analysis. However I have no idea whether or not they would elementary according to the second definition. This suggests to me that the second definition could be a bit suspect or 'loose', so to speak. I am not out to cause trouble here, and despite your beliefs to the contrary I have not made any edits for ideological reasons. In fact to my mind mathematics is refreshingly free of ideology. Dia dhuit (to which the polite response is 'Dia is muire dhuit'). AndyJM 07:22, 19 January 2009 (EST)

I don't think that anyone uses the term "elementary proof" as it is defined here. Here is more typical usage:

Keep in mind that the word ‘elementary’ is being used in many dif-ferent ways in this discussion. In informal mathematical usage, ‘elemen-tary’ is often used to mean ‘simple’ or ‘easy to understand’. In the Sel-berg/Nathanson sense it is used to mean ‘avoiding the use of infinitary (or analytic) methods’. The fact that Selberg’s proof of Dirichlet’s theorem is harder to understand than the standard analytic one shows that these two senses can be at odds! I have already noted that the use of the word ‘elementary’ in the phrase ‘elementary arithmetic’ is due to the fact that the axiom system is closely related to the class of ‘elementary functions’ in the sense of Kalmar. To complicate matters even further, among logicians the word ‘elementary’ is often used to mean ‘first-order’. [1]

As you can see, none of this matches what is in the article. I suggest that the article be deleted. RSchlafly 15:07, 30 January 2009 (EST)

It certainly seems reasonable that not using AoC would be desirable for calling a proof elementary. However, in my experience, the notion of an elementary proof is closer to "not using mathematics from unrelated areas" than it is in avoiding specific techniques or axioms. In number theory, this usually entails avoiding the methods of complex analysis. But to me, it is perfectly natural to say "To avoid this, we next give an elementary proof, which uses Zorn's lemma in a typical way" in a paper on algebra, where the application is indeed typical. That excerpt is from "Some ring theoretic equivalents to the Axiom of Choice", easily googleable, and I find several other examples of authors citing AoC in "elementary" proofs. It's easy to ignore this as the work of sloppy authors and/or liberals, but clearly the usage defined in this CP is at the very least not universal, and indeed I have never encountered it. Might I modify the page to indicate that some authors use a different characterization of elementary proofs, and can we find a source that actually uses a definition resembling that here? --MarkGall 01:23, 25 August 2009 (EDT)

Your approach is the same as Wikipedia's, and unfortunately it does not withstand scrutiny. The objection to non-elementary proofs is not that they use math "from unrelated areas." There is no coherent objection to that.
The real objection is that non-elementary proofs rely on additional assumptions. For complex analysis, that additional assumption is that there is a unique square root of negative 1, which can be manipulated algebraically. Maybe true, maybe false, and an elementary proof is desirable because it does not rely on that assumption. The Axiom of Choice also includes a questionable assumption.--Andy Schlafly 09:50, 25 August 2009 (EDT)
I am not questioning that there is a valid objection to using proofs that require additional assumptions, for example choice. This is a very reasonable view. I do question whether such proofs are actually called "elementary" by mathematicians. The case of applying complex analysis to number theory is non-elementary by either proposed definition. I have given one example in the literature which clearly contradicts the characterization of choice as necessarily non-elementary, and can easily find more. Can you please cite someone who uses your definition of an elementary proof? I'm willing to believe that this usage is in circulation, but this is the first I've run into it. --11:00, 25 August 2009 (EDT)
A Fields Medal was given for an elementary proof a few decades ago. Enough said? Isolated sloppiness or denial since then is no reason to reject the underlying logic behind the value of elementary proofs.
Most academic scientists today claim that man-made global warming is settled science, and that government can stop it. Such consensus does not make that claim true or logical.
Your definition of an elementary proof does not withstand scrutiny for the reasons I stated, which you did not address.--Andy Schlafly 11:46, 25 August 2009 (EDT)
I don't wish to argue about what would be the best definition of an elementary proof, or the importance elementary methods, which I appreciate. I just want to point out that the definition of "elementary proof" given here is not actually how the term is used in the mathematical literature, and I have supported this with a citation. As such I think this page needs some adjustment -- it gives a definition of "elementary proof" which is used, as far as I can tell until you provide a citation, by no one except you. It's quite possible that this would be a superior usage of the term, and I have no problem with the ideas on the page as it stands. I would only ask to add a disclaimer that when mathematicians use the term "elementary proof", this is not always what they mean (as indicated by the AoC example above). --MarkGall 11:58, 25 August 2009 (EDT)
Mark, the entry does describe the traditional and only meaningful definition of "elementary proof." Your definition, which is also used by Wikipedia, is non-substantive and cannot explain why the prestigious Fields Medal was given for it. Nor does your definition explain why Paul Erdos, perhaps the most prolific mathematician of the 20th century, was such a stickler in preferring elementary proofs.
Perhaps there are some new mathematicians who are in denial about the importance of elementary proofs. There are also new physicists who deny the importance of falsifiability. But self-described experts do not get to be the only watchdogs for their fields. Some accountability can and should come from outside the fields of expertise.--Andy Schlafly 12:23, 25 August 2009 (EDT)
If that is indeed the traditional definition (i.e., if the traditional definition would necessarily exclude proofs using AoC), you should have no difficulty providing a relevant citation, and I won't argue any more! It is true that proofs involving choice are generally dispreferred, but as far as I know they have never been called "non-elementary". I have given a citation which contradicts your characterization, and can produce more -- please provide just one in favor of it! If your definition were indeed in use at all, I would expect that searching for "Tychonoff theorem non-elementary" would find some page describing this result as "non-elementary" since it is one of the most famous proofs involving choice. However, it does not, but it does find an "elementary" proof of the theorem (employing choice, of course).
Of course some accountability from non-mathematicians is important, but this discussion isn't about whether proofs using AoC should be recognized as such and pointed out (such oversight is of course welcome). The question is whether the term "elementary proof" excludes all proofs using choice, and it seems to me that it does not.
Historically this term originated to describe proofs in number theory which did not use complex analysis, a usage which now falls under either umbrella. Selberg's Fields Medal was for a variety of contributions (like the Nobel prize, it is given not for a single result, but for overall contributions to mathematics), the primary of which (according to the official citation) was his extension of Brun's work on the zeta function (showing that a positive proportion of zeroes lie on the critical line), decidedly non-elementary work! --MarkGall 12:55, 25 August 2009 (EDT)
Mark, Conservapedia does not simply cut and paste quotes from others, many of whom has obvious self-serving biases. The logic underlying the concept of an "elementary proof" is clear and cannot seriously be disputed. You're not addressing the substance of the significance of an "elementary proof." Once you address that, it will become clear that proofs relying on the Axiom of Choice are not elementary proofs. It doesn't matter whether any math professors admit that or not. No math professor will publicly admit that the best mathematicians are men rather than women, but that denial does not alter the obvious truth.--Andy Schlafly 13:23, 25 August 2009 (EDT)
This article defines a common mathematical term in a way in which it is not used by anyone and seems never to have been used. I infer that your last post cedes that mathematicians don't use it, though I still welcome a citation on the historical point, which should be easy to come by if the usage is indeed "traditional" as you have claimed. I'm not disputing the logic of what you call an "elementary proof", which is clear. I agree with you about the significance of proofs such as you describe, and am not addressing "the substance of the significance of an elementary proof'" only because it's not relevant to the point at hand: what I'm disputing the use of that term, which has a well-established meaning distinct from what you intend and is therefore inappropriate. What use is an encyclopedia article that simply makes up new definitions for well-known terms, even if these new definitions have a clear and important philosophy? This serves no purpose but to confuse readers trying to understand how the term is used and hurt the credibility of CP math generally. Why not invent a new phrase to describe proofs such as those you discuss, whose philosophy I agree is valuable? --MarkGall 13:51, 25 August 2009 (EDT)
Mark, you're no longer being substantive in responding to my comments. This entry defines "elementary proof" in the only meaningful way, and then applies it in a straightforward manner that cannot be logically disputed.
Academic mathematicians, as in other academic fields, are in denial about many things: gender differences, religious truth, the value of self-criticism, and the bankruptcy of liberal politics. Wikipedia denied the concept of "elementary proof" entirely until I listed it as an example of its Bias in Wikipedia, and now Wikipedians deny the significance of the concept. Feel free to continue to your discussion there, but I'm going to stick with logic and truth here. Thanks.--Andy Schlafly 14:20, 25 August 2009 (EDT)

I don't understand these significant advantages. No proof ever assumes that there is a unique, algebraically manipulable square root of negative one. There are two such roots, and there is no need to assume anything because the proofs are given in field theory textbooks. Elementary proofs are not necessarily any harder to simplify. A sophicated non-elementary proof might be only 2 lines long, and thus impossible to simplify except by removing reliance on other work. The axiom of choice is not any less rigorous than any other axiom. The whole article is nonsensical. You seem to want to suggest that some proofs are more rigorous than other proofs, but that is just not true at all. RSchlafly 22:27, 26 August 2009 (EDT)

You're denying the significance of elementary proofs, for which a Fields Medal has been awarded and to which Paul Erdos and many number theorists have been dedicated. But your denial does not address the substance. An elementary proof does rely on fewer assumptions than a non-elementary proof. There's no way around that fact, and it is significant.
The view that "all proofs are equivalent no matter what theories they use" may be appealing to some, but it does not withstand scrutiny. Proofs relying on fewer assumptions are obviously preferable.--Andy Schlafly 22:44, 26 August 2009 (EDT)
Selberg did get a Fields Medal for an elementary proof of the prime number theorem, but I am not sure that anyone says that Selberg's proof is any better than any other. All of those proofs rest on the same set of assumptions. Can you give an example of an elementary proof that is better than a non-elementary proof because it relies on fewer assumptions? I agree that a proof not using the axiom of choice might be preferable, but can you give another example? RSchlafly 02:43, 27 August 2009 (EDT)

The notion that complex analysis depends on a purported uniqueness of the square root of -1 needs to be put to rest. There is no assumption that the square root of -1 has to be unique. And a good thing, too, because it isn't unique. No one denies that -1, like all nonzero numbers, has two square roots, that are negatives of each other. All nonzero numbers also have three cube roots, related by de Moivre's formula, and so on. Complex analysis does not rely on any such uniqueness. The complex numbers (or complex plane, or Argand diagram) is constructed from a special formal element "i", which simply serves as the second basis element of the vector space being constructed. The "square root of -1" and "i" are not the same thing. There are two square roots of -1, but only one "i". The square root of -1 doesn't even exist until the complex plane has been constructed, so it couldn't have been used in the construction.

The complex numbers, so constructed, are every bit as rigorous as the rational numbers, the real numbers, Hilbert spaces, Frechet spaces, Lie groups, symplectic groups, etc.

If there were any ambiguity in the interpretation of the "imaginary part" of a complex number, many fields of mathematics, physics, and engineering would fall apart. To pick just a few examples, the field of linear circuits in electrical engineering depends on the Laplace transform, which would not be well-defined if the imaginary unit "i" were not well-defined. Also, there could be no agreement on what the Schordinger equation of quantum mechanics means, or how the various quantum mechanical operators work. The Fourier transform would not be well-defined.

There may well be reasons why one would prefer a theorem of number theory that does not involve complex analysis, but such a proof's lack of rigor is not one of them.

We need to work out a careful survey of the things that make some proofs more complicated than other proofs of the same theorem, and how the word "elementary" is used to describe this, and what the various ramifications of this usage are. I'm willing to work on this. PatrickD 23:06, 26 August 2009 (EDT)

Beyond specific examples (and I haven't much to add to Patrick's exposition), Mr. Schlafly's general point that "Proofs relying on fewer assumptions are obviously preferable" is an important one, and I think that most mathematicians would agree with this. However, as Patrick explains, the use of complex analysis is not an additional assumption in any sense, and it is constructed and verified with complete rigor from nothing beyond ZF. So I'm mostly interested in AoC as it pertains to the notion of the elementary proof -- as RSchlafly had added, this seems to be by far the most important case where we find proofs with extra assumptions. More later on the issue of analytic number theory.
Proofs avoiding choice are to be preferred for their weaker assumptions, as noted: for example, the Schroeder-Bernstein theorem is a theorem for which there are two proofs: a quick trick relying on choice, and a slightly more nuanced proof without it. In my experience (both in studying and teaching), the latter is generally taught despite its relative difficulty. My only encounter with the former was as a homework problem introducing the application of AoC. My issue with this page is that the term "elementary proof" is not actually used to describe such proofs. There are other ways to characterize such proofs formally, but I'm not sure of any commonly-used term for them. I'd suggest that discussion of related issues ought to be incorporated into the article on AoC, and that this article be changed to reflect what the term actually means.
I'd also reiterate that Selberg's Fields was awarded largely on the grounds of his contributions to the study of the zeta function, so to imagine that it was awarded for minimizing the use of complex analysis is fantasy. His proof was important, however, for the new methods that it made available to number theorists. Erdos's dedication to "elementary proofs" arose not from doubts about the validity of complex analysis (in the case of PNT and other problems in number theory), but rather from the intrinsic aesthetic appeal of proofs which don't make use of unnecessarily powerful methods. A proof which does not crack a nut with a sledgehammer (as older proofs of PNT did) is more beautiful than one that does, and Erdos deeply valued "beautiful" proofs. It is these proofs, which expose the profound and elegant nature of the theorems they establish, that are to be found "in the book".
An unrelated issue is that there seem to be two notions of elementary proof used in this article, and I'm not sure how to reconcile them. Selberg's proof of PNT is magnificently complicated, though it avoids the methods of analytic number theory. It's not a proof that I would say "cannot be improved by expressing it in simpler form" -- indeed the analytic proofs are far simpler. It's true that at heart these have a different method, but it's my understanding that a touch of complex analysis can simplify the Selberg argument while leaving its fundamental insights intact. --MarkGall 00:24, 27 August 2009 (EDT)
The talk-to-substance ratio is very high in the above replies, and that's not a good sign. Complex analysis does rely on an assumption that there is a unique, algebraically manipulable square root of negative one. It's a unique pair, obviously, and it's a sign of desperation that some above looked for a way to deny this.
Proofs that rely on this assumption are obviously less favored than proofs that do not, and hence the important concept of an elementary proof. Deny its importance all you want, but the beauty of logic is that it doesn't matter who accepts or denies it. The logic remains as truthful regardless.--Andy Schlafly 11:26, 27 August 2009 (EDT)

Errors

This page is frankly bizarre. I can't imagine a working mathematician agreeing with much of it in its current state.

1. The assertion that an elementary proof "uses only real numbers or real analysis rather than the use of complex analysis" agrees with the cited MathWorld stub, but that stub misrepresents its source, Hoffman's book on Erdos. The original source says, "In this context, elementary means that the proof of the formula relies on a restricted set of numbers, the so-called real numbers...", the context being Erdos avoiding the "heavy machinery" including complex analysis employed by Hardy and Ramanujan in an asymptotic estimate of the partition function. An elementary proof is not in general characterized by the avoidance of complex analysis but rather by the avoidance of any "heavy machinery" whatsoever (like, say, the Hahn-Banach theorem, which in its real formulation is very much a part of real analysis). A decent test for the elementary-ness of a proof is whether a bright high-schooler could follow it. The definition needs to be re-written and a better source found. It should also be noted that the definition is inherently vague and different people disagree on the finer points.
2. The assertion that non-elementary proofs rely "on less rigorous axioms, such as the Axiom of Choice" is misleading. The axiom of choice is not "less rigorous" than, say, the axiom of infinity--it simply asserts an assumption; rigor does not enter in. The axiom of choice is somewhat controversial, though; "less controversial axioms" in the above would be true.
3. "...but in 1949 and 1950 an elementary proof by Paul Erdos and Atle Selberg earned Selberg the highest prize in math" -- it's misleading to imply Selberg's elementary proof was the only thing that got him his Fields medal. "helped earn" would be correct and would avoid a digression into discussing Selberg's work.
4. "avoiding the assumption that there is a unique, algebraically manipulable square root of negative one" -- this highlights a fundamental misunderstanding which may underlie my second point. The existence of complex numbers is not an "assumption" in the sense mathematicians use. Typically, one assumes some basic facts about set theory (say the ZF axioms, which includes the axiom of infinity I mentioned before) and uses those to build up algebraic objects. The existence of complex numbers satisfying the usual properties is then a *consequence* of these background assumptions (which are also used in elementary proofs) rather than an assumption itself. The use of complex analysis does not entail the use of a new assumption [nitpick: the axiom of dependent choice is typically added to the usual ZF axioms to do complex analysis, though that's rather uncontroversial and is also used in real analysis, almost always implicitly because it's so intuitive]. Again, the point of an elementary proof is that it uses no "heavy machinery"; I've never actually seen an elementary proof that used fewer assumptions than a non-elementary counterpart even though this page mentions that phenomenon several times. An example (keeping in mind the proper mathematician's idea of an "assumption") would be helpful.
5. The article does not discuss this, but elementary proofs are not always preferred. Elementary proofs tend to be very long and rather messy--a famous solution to the Basel problem comes to mind as an example. If one uses non-elementary Fourier analysis, one can solve the problem in less than half a page by computing the Fourier coefficients of f(x) = x and applying Parseval's theorem. The elementary alternative I'm aware of fiddles around with trigonometric identities and polynomial arithmetic for quite a while, is tedious, and is actually much harder to follow. The formulation I'm aware of also uses complex numbers, though not complex analysis.

I'd also like to say that Aschlafly should probably avoid editing this page. The quote, "For complex analysis, that additional assumption is that there is a unique square root of negative 1, which can be manipulated algebraically. Maybe true, maybe false" shows a fundamental misunderstanding of the role of assumptions in mathematics and also shows his ignorance about the subject. The construction of complex numbers from real numbers is quite simple: mod the ring of one-variable real polynomials by the ideal generated by (x^2 + 1); this is the standard splitting field construction in this special case. There is no additional assumption necessary, just more machinery (rings, ideals, quotients, etc.).

I haven't edited the article since (from this talk page) doing so seems rather controversial.

Elproof 19:45, 7 April 2012 (EDT)

These are nitpicky points, and I don't see why any of them would require any changes to the entry. I address each point:
(1) Your comment complains about the definition, but does not offer a better one. Instead, your comment says the definition is "vague", but then uses even vaguer terminology like "heavy machinery" or whether "a bright high-schooler could follow it."
(2) Your comment that "less rigorous" should be replaced by "less controversial" would, again, be a step in the wrong direction. "Controversy" does not delineate limits on truth. Jesus Christ was no less truthful simply because he was (and is) controversial.
(3) This point is silly and does not merit a response.
(4) It's obviously not the number of assumptions that matter, but the lack of rigor in the assumptions made. Assuming that there is a unique square root of -1 is a highly non-rigorous, even implausible, assumption underlying complex analysis.
(5) "but elementary proofs are not always preferred ..." by whom? One doesn't allow a fox to guard a henhouse, and neither mathematicians nor politicians should be setting their own rules as they go, or else before long all sorts of nonsensical "proofs" might be asserted by some academics at liberal institutions.--Andy Schlafly 00:46, 8 April 2012 (EDT)
There is no point in arguing with you. You seem to have already permanently made up your mind. I will be leaving Conservapedia after finishing this post; please don't cry. I would thank you to read the rest, though, since I read yours.
Your response is unfortunately what I expected--prompt, ignorant of mathematical convention, argumentative, mathematically incorrect, rude, off-topic, political, religious, incomplete, and badly reasoned. It's characterized by the misapplication of decent general principles, often due to faulty reasoning by analogy (apparently a favorite of yours?). You are a high-functioning but certainly insane person. That would be fine, except that you're also productive, verbose, and superficially authoratative, so a small fraction of the population listens to you. You are unfortunately intelligent: if you were a little stupider, you'd have no audience, and if you were a little smarter, you'd see your own insanity.
I had hoped to edit the article to correct some obvious mistakes, but on a larger scale I had hoped that Conservapedia was interested in correct articles vetted by experts, at least on what I expected to be an uncontroversial topic. That hope, as I predicted initially, has been dashed. I wanted to test Conservapedia for myself and now have. Thankfully, I can soon leave this dark corner of the internet.
I have a request. Please, delete this article. It's not a terribly important term and the article is short. Its content does not conform to Conservapedia's guidelines either: "Conservapedia articles [sic] tone, style, and content should be written with an American, conservative and/or Christian orientation or focus" (taken from the Quick Reference page). The current page has none of those focuses since it mentions nothing about region, politics, or religion. Many mathematics articles likely fail in the same way, which makes me curious why you allow them on Conservapedia at all. Perhaps all Mathematics-category articles should be deleted?
There are other reasons to delete this article. It has generated an order of magnitude more talk than content which wastes your time. This article also makes Conservapedia look bad compared to Wikipedia since the Wikipedia version is longer, more detailed, a little better sourced, and has no talk on its talk page whatsoever, let alone numerous unfruitful debates (which, frankly, make you look terrible; even your own people repeatedly point out your ignorance throughout this page--why they don't simply leave is beyond me).
Please feel free to ban me and/or delete this account. I won't be back so it doesn't matter. I do wish you a happy life, Andy, though I also wish it were in seclusion so that your brand of crazy couldn't infect anyone else.
P.S. I'm gay. I read your article on homosexuality, since some of the worst atrocities ever occurred because of close-mindedness. It makes some good points: gay men have high STD rates and often pursue very unsafe sex; some men can convert from homosexuality to heterosexuality behaviorally; "homophobe" is thrown around too frequently. However, there is no discussion whatsoever of loving long-term gay relationships. Reading the article one gets the impression that being gay means taking meth and attending a circuit party every night until you die of AIDS after beating up your boyfriend and infecting your wife; the best parts of homosexuality are entirely ignored. Do not simply ignore difficult points. You've done the same thing repeatedly on this page by not responding to RSchlafly's questions, and you didn't respond to my own point that you should refrain from editing an article you are so inexpert in. You also did the same with a group of your own students in an episode discussing grading irregularities I stumbled across--you even called their points "nitpicky" just like you did with me. Enough. You're a fool and your web site is mostly garbage; neither of you are worth any more of my time. Elproof 08:38, 10 April 2012 (EDT)

Unique square root of -1 necessary for complex analysis

Dear Mr. Schlafly,

Where are you getting your information that complex analysis requires there to be a unique square root of −1? I am currently taking a complex analysis course, and I understand that there are two square roots of −1, namely i and −i.1 I don't see how this undermines complex analysis. It may help to examine how complex numbers are derived in the first place. (The interested reader may consult Complex Function Theory, 2nd ed., by Sarason, sections I.1 and I.2.)

  1. We define a set \mathbb{C} to consist of all ordered pairs of real numbers.
  2. We define addition of elements of \mathbb{C} as follows:
    (a,b) + (c,d): = (a + c,b + d)
  3. We define multiplication of elements of \mathbb{C} as follows:
    (a, b) \cdot (c, d) := (ac - bd, ad + bc)
  4. We verify the following about + :
    1. It is commutative
    2. It is assoiciatve
    3. There exists an element of \mathbb{C}, namely 0: = (0,0), such that for all a \in \mathbb{C}, a + 0 = a = 0 + a
    4. For every element a \in \mathbb{C} there exists an additive inverse a such that a + ( − a) = 0 = ( − a) + a
    5. Therefore, \mathbb{C} is an abelian group under addition.
  5. We verify the following about \cdot:
    1. It is commutative
    2. It is associative
    3. It has the distributive property over addition
    4. There exists an element of \mathbb{C}, namely 1: = (1,), such that for all a \in \mathbb{C}, a \cdot 1 = a = 1 \cdot a
    5. For every nonzero element a \in \mathbb{C} there exists a multiplicative inverse (reciprocal) a − 1 such that a \cdot a^{-1} = 1 = a^{-1} \cdot a
    6. Therefore, \mathbb{C} is a field.
  6. We consider the subset S = \{(a, 0)|a \in \mathbb{R}\} \subseteq \mathbb{C} We note that there is a natural isomorphism from S to \mathbb{R}. Thus, we can associate elements of \mathbb{C} with zero second coordinate with real numbers.
  7. As notation, we write the ordered pair (a,b) as a + bi. This is just notation. We note that it does not constitute an abuse of notation since (a,b) = (a,0) + (0,b), so the complex number notated by a + bi is really the sum of the complex numbers a + 0i and 0 + bi.
  8. We can also use natural shorthand for the a + bi notation. In particular, we can write i to represent the element (0, 1) \in \mathbb{C}. Note that i^2 = (0, 1)\cdot(0, 1) = (-1, 0) = -1.

This is how complex numbers are typically derived. (One could of course, define them to be the cosets in the quotient field \mathbb{R}[x]/(x^2 + 1); this is clearly equivalent to the derivation above.) Please let me know where you see a fault in this derivation.

1Incidentally, Galois theory tells us that there cannot be a unique nonreal solution to a polynomial with real coefficients. Considering \mathbb{C} as a field extension of \mathbb{R}, we note that the Galois group of \mathbb{C}/\mathbb{R}, which consists of all automorphisms of \mathbb{C} that fix elements of \mathbb{R}, has precisely two members: the identity function and the function taking complex numbers to their complex conjugate (which we will denote φ. Suppose that f(x) = a_0 + a_1 x + \cdots + a_n x^n is a polynomial with real coefficients that has a root r. Then, 0 = a_0 + a_1 r + \cdots + a_n x^n. We apply the aforementioned φ to both sides; the application of this function leaves 0 and all the coefficients a_0, \ldots, a_n unchanged. What we get is 0 = a_0 + a_1 \bar{r} + \cdots + a_n \bar{r}^n. This means that \bar{r}, the complex conjugate of r, is also a root of f. Thus, any polynomial in real coefficients with a nonreal root r also has \bar{r} as a root. In particular, complex square roots of − 1 are roots of the polynomial x2 + 1, and since we know this polynomial has no real solutions, the two roots guaranteed by the fundamental theorem of algebra must be a complex conjugate pair. GregG 01:13, 8 April 2012 (EDT)

Please

By "unique", it includes unique pair. Also, it looks like you double-posted above and please feel free to delete any duplicate. I look forward to reviewing your analysis further on Resurrection Sunday.--Andy Schlafly 01:19, 8 April 2012 (EDT)
Complex analysis assumes, for example, not only that the square root of -1 exists, but that the cube of the square root of -1 is always equal to the negative of itself.--Andy Schlafly 11:24, 8 April 2012 (EDT)
Why do you say "assumes"? All of those properties are provable from either of the above two definitions of the complex numbers. RSchlafly 12:20, 8 April 2012 (EDT)
Really, that one is obvious: x²+1=0 ⇒ x³ = x(x²+1-1)=x(-1)=-x No further assumptions are needed, only basic algebra.... AugustO 13:05, 8 April 2012 (EDT)
If something could be "defined to exist," then we could simply define "God to exist, let's call Him g," and that would end any debate about the matter.--Andy Schlafly 17:39, 8 April 2012 (EDT)
That is how all of mathematics is. Objects are defined to exist. Then they are proved to be well-defined. It is the same for all other numbers, functions, spaces, and other mathematical objects. RSchlafly 18:08, 8 April 2012 (EDT)
Defining the square root of -1 to exist is different from other mathematical definitions, because the square root of -1 is known not to exist. It is analogous to defining the square root of 2 to be rational without the immediate contradictions.
But if mathematicians insist on the rigor of this approach, then why not simply define God to exist? There is no contradiction that results from that definition.--Andy Schlafly 18:24, 8 April 2012 (EDT)
Now you're just being silly. Maths is not religion and maths is not a physical science. In mathematics one can define any structure one likes, logical consistency is all that is required. In that sense, complex numbers are perfectly well defined. Mathematical structures may or may not be useful for describing reality, but that has to be checked by comparing predictions from mathematical theories to experimental results or observations. Even then, mathematical structures are not (physical) reality, they are tools to describe reality. Complex numbers have proved to be extremely useful in many fields of science and engineering. Aren't you an electrical engineer or something like that? Did you not find complex numbers useful? --FrederickT3 19:20, 8 April 2012 (EDT)
There are people who define God to be the universe, or physical law, or the first cause, or the essence of goodness, or something similar. To a mathematician, such a definition could indeed serve to define God and to prove that he exists. RSchlafly 20:38, 8 April 2012 (EDT)
Responding to Frederick, I would never doubt that complex numbers can be useful. So are approximations, estimates, predictions about the future, intuition, and prayer. The issue is whether a mathematical proof that resorts to use of complex numbers is as rigorous as an elementary proof. The answer is obviously "no".--Andy Schlafly 01:17, 10 April 2012 (EDT)
Obvious? On the contrary, no mathematical proof is more rigorous than any other mathematical proof. A proof is either rigorous or it is not, whether it uses complex numbers or any other mathematical construct. RSchlafly 01:52, 10 April 2012 (EDT)
That's unnecessary semantics. Replace "rigorous" with whatever substitute you prefer, but a proof that relies on fewer or less doubtful assumptions is more [rigorous/robust/credible/advantageous] than one that relies on many or more doubtful assumptions. For example, a proof that relies on the Axiom of Choice is not as good as one that solves the same problem without resorting to that axiom.--Andy Schlafly 11:19, 10 April 2012 (EDT)
How about intuitive? That seems to be the main concern here, that imaginary numbers are not something we can easily wrap our heads around. JustinD 15:23, 10 April 2012 (EDT)

Slightly off-topic, but you might be interested in physical evidence for complex numbers. According to this new paper [2], imaginary numbers are the most crucial difference between classical and quantum mechanics. RSchlafly 20:50, 9 April 2012 (EDT)

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