Dear Vaughan - You write:
As a case in point, just now I checked a relatively recent Brittanica article on algebra (1987 ed.), which states flatly (p.260a) that "No elementary algebraic proof of [the FTAlg] exists, and the result is not proved here." (Not even "is known" but "exists"; an expository article should not assume that the reader knows the jargon meaning of this term as "exists in the literature".) The authors taking responsibility for this claim were Garrett Birkhoff, Marshall Hall, Pierre Samuel, Peter Hilton, and Paul Cohn. They go into detail to show that z^n = a has n roots, starting with the geometry of addition and multiplication in the Argand diagram, so it's not as if their exposition was at too elementary a level to talk in terms of mapping circles, or that "algebraic" ruled out simple geometric arguments.
I submit their nonexistence claim as prima facie evidence for my claim that the very few who knew this argument weren't even letting the likes of Birkhoff, Hall, etc. in on it, let alone "the rest of us."
I really doubt those authors were unaware of the topological proof of the fundamental theorem of calculus in 1987. After all, it's exercise H.5 in chapter 1 of Spanier's "Algebraic Topology", copyright 1966. This book used to be the canonical textbook on algebraic topology, and Peter Hilton is a darn good algebraic topologist. I think I learned this topological proof sometime in grad school, around 1986. So, I don't think it was any sort of secret by then. I don't know what counts as an "elementary algebraic proof", but people often say that there is no "purely algebraic proof" of the fundamental theorem of calculus. After all, this theorem is about the complex numbers, which are often defined in terms of the real numbers, which are often defined as a topological completion of the rational numbers. I hope this is what the Encyclopedia article was trying to say. There are some so-called "algebraic proofs": http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra that use a bare minimum of topology. These proofs tend to have a purely algebraic core, namely "if odd-degree polynomials and the polynomial x^2 + 1 have roots in some field, this field is algebraically closed". But, they use the intermediate value theorem for continuous functions f: [0,1] -> R to show that C meets these conditions. So, I wouldn't call them "purely algebraic". It's sort of ironic that the so-called "fundamental theorem of algebra" doesn't have a purely algebraic proof. Gauss is famous for having given a proof of the fundamental theorem of algebra in his dissertation back in 1799. On the St. Andrews math history website they write: Gauss's proof of 1799 is topological in nature and has some rather serious gaps. It does not meet our present day standards required for a rigorous proof. They don't say how the proof went. So, I decided to find out! I was hoping I could irritate you by showing that it was just the topological proof you claim is so new. There's a discussion of it here: Hans Willi Siegberg Some Historical Remarks Concerning Degree Theory, American Mathematical Monthly, 88 (1981), 125-139. (Available on JSTOR, or via Google Scholar.) As the title hints, Gauss' proof uses ideas closely related to the winding number. Unfortunately, it's slightly different than the proof you like. The idea is to take a polynomial of degree n, say P: C -> C break it into real and imaginary parts P = U + iV, see where they vanish: S = {z: U(z) = 0} T = {z: V(z) = 0} and show that the intersection of S and T is nonempty. Gauss argues that far from the origin, S and T are smooth curves. Because the leading term of the polynomial dominates the rest, each of these curves intersects any sufficiently large circle transversely at n points. If we go around the circle these intersection points alternate: first a point in S, then one in T, then one in S, and so on. Moreover, the curves I'm talking about can't just disappear as we follow them into the disk, since they separate the region where U (resp. V) is positive from the region where it's negative. They may become singular, or intersect, but they can't just end! "So", S and T must intersect somewhere. This is true, but it takes more topology to prove it rigorously than was available to Gauss. Gauss knew his proof wasn't completely rigorous, so he invented some other arguments. The "winding number" idea you like is lurking in Gauss' third proof, which he wrote up in 1816 - but he only gave this winding number proof explicitly in 1840. According to Siegberg, Indeed, in a lecture "Theorie der imaginaeren Groessen (1840), Gauss mentioned [see Fraenkel, 1922] that his third proof of the fundamental theorem of the algebra originated from his first one, and he gave the function-theoretic argument that the winding number W(P|S, 0) equals n [the degree of the polynomial], whereas the winding number of any map F: (B,S) -> (R^2, R^2 - {0}) vanishes if there is no zero of F in B [see Fraenkel, 1922]. However, this argument cannot be found explicitly in [Gauss, 1816]. So, I guess that except perhaps for Gauss, nobody knew the proof you're talking about until 1840. Best, jb
John Baez wrote:
I really doubt those authors were unaware of the topological proof of the fundamental theorem of calculus in 1987. After all, it's
Right, both my claim and its premises needed a fair bit of tuning (as with my recent question about the quasivariety "groups+free monoids" -- this is a good list to get corrective feedback from). (But a neat piece of historical research there, John.) The issue seems to be coming down to Mike Barr's question, which if I can paraphrase it without changing its intent, was, what is the proper status of an appeal to the very plausible in a proof? My suggestion in my last message to Peter Freyd was that the prover should point out the gap, its cause (lack of a simple proof), and its plausibility notwithstanding. This suggestion raises more questions than it answers. 1. Is a proof with a gap more acceptable for expository purposes when the bridgability of the gap is more plausible? (The case in point being an extreme example.) 2. How is plausibility to be judged? By consensus, or are there objective criteria? 3. It is certainly not necessary to prove A before B merely because B depends on A; indeed one common-sense practice when proving a two-lemma proof is to get the easier lemma out of the way first, even if it depends on the harder one. Is it kosher to truncate such a proof after the first lemma (or in this case the final result), call it an exposition, and point to the literature for the second lemma? Regarding 3, the authors of the Britannica article seemed not to think so, but perhaps this just reflects Garrett Birkhoff's attitude that "I don't consider this algebra, but this doesn't mean that algebraists can't use it" cited by Michael Artin when proving FTAlg in his 1991 book "Algebra". Who on this list considers the fundamental theorem of algebra "not algebra"? These questions are probably more appropriate for a philosophy of mathematics list than this one. What makes FTAlg such an interesting case study for those with something at stake in such questions is that the tensions here are so extreme. The final result (FTAlg) is not at all obvious, whereas the lemma it rests on, whether it be that |P(z)| attains its minimum, or that circles around a hole don't retract, or the intermediate value theorem, or the existence of a root for a real polynomial of odd degree, seems self-evident. Yet the one that is hard to see is easy to prove, while the one that is easy to see is hard to prove. If seeing is believing, what is proof? In the real world, when something is easy to see it is up to the opposition to demonstrate that it is nonetheless false. How did mathematics evolve to play by a different rule book? Vaughan Pratt
A couple of mistakes. I wrote:
I really doubt those authors were unaware of the topological proof of the fundamental theorem of calculus in 1987.
I meant "fundamental theorem of algebra".
Gauss argues that far from the origin, S and T are smooth curves. Because the leading term of the polynomial dominates the rest, each of these curves intersects any sufficiently large circle transversely at n points.
Should be: any sufficiently large circle centered at the origin. Best, jb
On Thu, 30 Mar 2006, Vaughan Pratt wrote:
Regarding 3, the authors of the Britannica article seemed not to think so, but perhaps this just reflects Garrett Birkhoff's attitude that "I don't consider this algebra, but this doesn't mean that algebraists can't use it" cited by Michael Artin when proving FTAlg in his 1991 book "Algebra". Who on this list considers the fundamental theorem of algebra "not algebra"?
The theorem is algebra, but its proof isn't: any proof has to involve some topological input (though that can be reduced to the Intermediate Value Theorem). Vaughan seems to have a problem with the phrase "elementary algebraic proof": of course, not all elementary proofs are algebraic (and not all algebraic proofs are elementary), and it is the word "algebraic" that matters here. Incidentally, I used that Birkhoff quote in the Introduction to "Stone Spaces" (1982). Did Mike Artin get it from me, or did he discover it independently? Even more incidentally, the first published proof of the Fundamental Theorem is not by Gauss. It appears in the only mathematical paper (in Phil. Trans. Roy. Soc. volume 88, 1798) of the Reverend James Wood, who was then a Fellow (and subsequently Master) of St John's College, Cambridge. (His other publications were all theological -- he was a Doctor of Divinity.) Wood's argument is essentially the same as Gauss's second proof (1816); by modern standards, what he writes in the paper doesn't constitute a rigorous proof, but (to quote the late Frank Smithies) "anyone reading Wood's paper must end up with the conviction that there is a proof somewhere there". Peter Johnstone
Let me reiterate this: There can in principle be no purely algebraic proof of the FToA because the reals have no purely algebraic definition. (Unless you define them as a real closed field of transcendence degree c, but that leaves the FToA as a trivial consequence and cannot be what is wanted.) The proof I outlined, which someone showed me 45 years ago, uses only the fact that R is a complete ordered field. Given that that is the analytic definition of R, it is impossible to avoid. That fact is, of course, at the heart of the fact that the circle is not contractible in a punctured plane. Incidentally, even constructivists (well even Errett Bishop, anyway) agree that odd order real polynomials have a real root and that positive numbers have square roots, since there are obvious constructions for these things. Their real line is not complete (it is countable, but the missing numbers are not constructible), but these roots are there anyway. The argument I outlined is elementary, even if not especially easy. First you have to construct the reals, the least elementary part of the argument. Then comes the theorem on symmetric functions. It is not a deep result; it needs a careful proof, but a student can follow it without knowing anything sophisticated. The construction of a splitting field (without getting into UFDs) is a bit tricky. To adjoin a root to an irreducible polynomial p of degree n, you start with a vector space whose basis is called 1, u, u^2,..., u^{n-1} and define a multiplication, by having p(u) = 0. This is analogous to how you get from R to C. Of course, you use the division algorithm to show you get a field. Michael
Yet the one that is hard to see is easy to prove, while the one that is easy to see is hard to prove. Ain't that the truth or as Rene Thom once remarked about one of his assertions Very easy to see, very had to prove jim Vaughan Pratt wrote: [...]
These questions are probably more appropriate for a philosophy of mathematics list than this one. What makes FTAlg such an interesting case study for those with something at stake in such questions is that the tensions here are so extreme. The final result (FTAlg) is not at all obvious, whereas the lemma it rests on, whether it be that |P(z)| attains its minimum, or that circles around a hole don't retract, or the intermediate value theorem, or the existence of a root for a real polynomial of odd degree, seems self-evident. Yet the one that is hard to see is easy to prove, while the one that is easy to see is hard to prove.
If seeing is believing, what is proof? In the real world, when something is easy to see it is up to the opposition to demonstrate that it is nonetheless false. How did mathematics evolve to play by a different rule book?
Vaughan Pratt
Even in my original posting starting this thread I acknowledged that contractibility of the circle was not elementary:
Except, that is, for the theorem that a loop wound d times around the hole in the punctured plane cannot be continuously retracted to a point, which was tacitly smuggled in there. But that statement is less intimidating than anything based on holomorphic functions.
I haven't at any time claimed that it was not necessary to prove this, nor that the proof was easy. What I have been claiming is that the result has a certain self-evident quality to it that, it seemed to me, qualified the argument as at least sufficiently "morally elementary" as to qualify it for inclusion in the Britannic article on algebra. How could the definitive encyclopedia article on algebra not give at least a hint as to why that subject's fundamental theorem was true? However I've been reflecting on just what is behind the very uniform insistence on the distinction between an algebraic proof and an analytic one. Since algebra is descended from analysis, it seems unkind for algebra to deny its parentage in this way. But I see now that this denial is logically necessary. For consider the algebraic plane, the least algebraically closed subfield of the complex plane, consisting of the algebraic numbers. The FTAlg is by definition true there, so it ought to be provable there. One can carry out the same proof, and it all goes through in the same way (using circles of growing algebraic radius, all of which are dense in their complex completion to a connected circle) right up to the last step when we claim that the wildly growing loop that is the image of the tamely growing circle must eventually collide with the origin, d times in fact for a degree d polynomial. And indeed it does, all d times, exactly as with the complex numbers, and with the same roots (the coefficients of the polynomial necessarily being algebraic in this domain). But now analysis has nothing to do with it, since these circles and their image loops while dense are totally disconnected. For all we know the origin could have missed the loop by going through any of its uncountably many gaps. Indeed the loop has measure zero, so the chances of the origin colliding with it even once are less than Buckley's. But with aim that would be the envy of any sniper the origin hit the loop with every one of its d shots. And how do we know this? Using analysis. The consensus would seem to be that there is no other way. Logic alone cannot help. If that's the case, then without analysis there is no algebraic plane. Without the huge continuum to support it, that tiny countable set would not exist! It is ironic that a theorem of algebra about an algebraic domain that itself has no element of analysis to it, being just the algebraic closure of the rationals, a small and totally disconnected space, should require analysis, the parent of algebra, for its proof. The fundamental theorem of algebra is like a student calling home for more money. It takes a continuum to raise an algebraic number. Vaughan Pratt
For the bookworms among the readers of this FToA thread, let me offer four older references to undergraduate-accessible expositions of proofs along the lines already mentioned: First, in Birkhoff & Mac Lane (my own undergraduate algebra text), Section 3 of Chapter V of the 1953 ("revised") edition offers a proof along winding number lines on pp. 107-109. Next, in the 1975 MIR English edition of Kurosh's Higher Algebra (described as the "second printing"), section 23 of Chapter 5 offers a proof relying on the D'Alembert Lemma (on pp. 142-151). In the same Kurosh volume, moreover, section 55 of Chapter 11 offers a proof along symmetric function lines on pp. 337-340. Finally, one may find the Artinian proof in the real-closed fields section of van der Waerden's pre-WWII classic, Modern[e] Algebra. I refrain from citing other textbooks, and I remark that numberings (of pages, sections, chapters) may differ in other editions. Cheers, -- Fred Prof. Peter Johnstone wrote:
On Thu, 30 Mar 2006, Vaughan Pratt wrote:
Regarding 3, the authors of the Britannica article seemed not to think so, but perhaps this just reflects Garrett Birkhoff's attitude that "I don't consider this algebra, but this doesn't mean that algebraists can't use it" cited by Michael Artin when proving FTAlg in his 1991 book "Algebra". Who on this list considers the fundamental theorem of algebra "not algebra"?
The theorem is algebra, but its proof isn't: any proof has to involve some topological input (though that can be reduced to the Intermediate Value Theorem). Vaughan seems to have a problem with the phrase "elementary algebraic proof": of course, not all elementary proofs are algebraic (and not all algebraic proofs are elementary), and it is the word "algebraic" that matters here.
Incidentally, I used that Birkhoff quote in the Introduction to "Stone Spaces" (1982). Did Mike Artin get it from me, or did he discover it independently?
Even more incidentally, the first published proof of the Fundamental Theorem is not by Gauss. It appears in the only mathematical paper (in Phil. Trans. Roy. Soc. volume 88, 1798) of the Reverend James Wood, who was then a Fellow (and subsequently Master) of St John's College, Cambridge. (His other publications were all theological -- he was a Doctor of Divinity.) Wood's argument is essentially the same as Gauss's second proof (1816); by modern standards, what he writes in the paper doesn't constitute a rigorous proof, but (to quote the late Frank Smithies) "anyone reading Wood's paper must end up with the conviction that there is a proof somewhere there".
Peter Johnstone
Fred E.J. Linton wrote:
First, in Birkhoff & Mac Lane (my own undergraduate algebra text), Section 3 of Chapter V of the 1953 ("revised") edition offers a proof along winding number lines on pp. 107-109.
Thanks, Fred, I wish I'd noticed that before. I have the sixth printing (1948) of the 1941 edition, which says, "Many proofs...are known; ...we have selected one whose non-algebraic part is *especially plausible intuitively*." (My emphasis.) Then they give the proof "I like". To administer one more lash to this dead horse, the wording in the Britannica article implies that the absence of an elementary algebraic argument was the reason for omission of a proof of FTAlg. Whence the change of heart about arguments that are "especially plausible intuitively?" If they're good enough for an algebra text they should be even more acceptable for an encyclopaedia article. Vaughan Pratt
Hello, Vaughan Pratt wrote:
3. It is certainly not necessary to prove A before B merely because B depends on A; indeed one common-sense practice when proving a two-lemma proof is to get the easier lemma out of the way first, even if it depends on the harder one. Is it kosher to truncate such a proof after the first lemma (or in this case the final result), call it an exposition, and point to the literature for the second lemma?
Indeed, should we expect non-mathematicians "believe in" results like the Jordan curve theorem, which is very easy to see and very hard to prove? Maybe they do not appreciate a proof of something that looks completely obvious. Or, another veryeasy thing: If somebody walks from A to B and somebody walks fron B to A at the same time on the same route, they will me meet somewhere, even if they don't walk with constant speed or if they stop somewhwere. Rigorously, this is essentiaally equivalent to the intermediate value theorem, which is not too hard to prove, but nevertheless not trivial. I think there are better subjects to illustrate what mathematics is about. Some months about I was asked to contribute to a calender, in which people working in different scientific subjects give an insight in their work and their subjects. I contributed the party theorem that a (simple) finite graph with at n>1 vertices has two vertices of the same dergree. The proof just uses the pidgeon-hole principle and the observation that a vertex of degree zero and a vetex of degree n-1 cannot both exist. If course, I avoided mathematical terminology not known to the general public and spoke of guests of a party, where some shake hands with each other and some don't. I think this example may give a flavour of what a proof is, but does not bother them with formalisms which are not necessay here (but somewhere else). Of course, category theoty usrather abstract and can hardly be explained to non-mathematians. One remark ti the fundamental theory of algebra. From Harald Holman I learned a proof based on a simple ideas, which can be made rigorous quite easily: Since a non- constant complex polynomial is large outside a sufficiently large circle, its modulus must have a (local am global) minimum inside the circle (by compactness); we can shift it into zero (by translation) and assume the minimum is attained in zero. Since the polynomial is not constant, it is of the form a_0+a_m*z^m+higher terms, a_m different from 0. If a_0=0, the polynomial has a zero in 0, and we are done, so assume that a_0 is not 0. Then there exists a complex number w with w^m=- a_m/a_0; this follows from the polar coordinate representation, which is taught in calculus courses. Then for sufficienly every positive real h<1, the modulus of a_0+a_m*(hw)^m is smaller than the modulus of a_0; if we choose h small enough, the modulus of value of the polynomial at hw is also smaller than a_0, because the higher terms can be neglected. This contradicts our assumption that the modulus attains a minimum in 0. Greetings Reinhard
Vaughan writes:
However I've been reflecting on just what is behind the very uniform insistence on the distinction between an algebraic proof and an analytic one. Since algebra is descended from analysis, it seems unkind for algebra to deny its parentage in this way.
I thought people knew how to add before they knew how to take limits. :-)
But I see now that this denial is logically necessary. For consider the algebraic plane, the least algebraically closed subfield of the complex plane, consisting of the algebraic numbers. The FTAlg is by definition true there, so it ought to be provable there.
Hmm. How do you propose to show there *exists* an algebraically closed subfield of the complex numbers? I would do it using the fundamental theorem of algebra - the usual one, for the complex numbers. Unless you have some other way, I don't understand how you hope to circumvent the use of analysis by introducing such an entity. Indeed, the usual proof that the real numbers contains a square root of 2 uses the completeness of the real numbers, which also counts as "analysis".
It is ironic that a theorem of algebra about an algebraic domain that itself has no element of analysis to it, being just the algebraic closure of the rationals, a small and totally disconnected space, should require analysis, the parent of algebra, for its proof.
That the rational numbers has an algebraic closure is a purely algebraic result, with no mention of topology in either the statement or proof. That the complex numbers is algebraically closed is not an algebraic result: it has topology built into the statement, and also the proof(s). That the algebraic closure of the rationals embeds in the complex numbers has topology in the statement - and I bet also in every proof. Best, jb
Perhaps the moral is not to bother with the Britannica. Wikipedia has several proofs including the winding number argument and the one I outlined using the symmetric function argument. Then a couple of analytic ones. Of course, Wiki has no size limitations. Perhaps we have flogged this particular horse enough. On Sun, 2 Apr 2006, Vaughan Pratt wrote:
Fred E.J. Linton wrote:
First, in Birkhoff & Mac Lane (my own undergraduate algebra text), Section 3 of Chapter V of the 1953 ("revised") edition offers a proof along winding number lines on pp. 107-109.
Thanks, Fred, I wish I'd noticed that before. I have the sixth printing (1948) of the 1941 edition, which says, "Many proofs...are known; ...we have selected one whose non-algebraic part is *especially plausible intuitively*." (My emphasis.) Then they give the proof "I like".
To administer one more lash to this dead horse, the wording in the Britannica article implies that the absence of an elementary algebraic argument was the reason for omission of a proof of FTAlg. Whence the change of heart about arguments that are "especially plausible intuitively?" If they're good enough for an algebra text they should be even more acceptable for an encyclopaedia article.
Vaughan Pratt
participants (7)
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Fred E.J. Linton -
jim stasheff -
John Baez -
Michael Barr -
Prof. Peter Johnstone -
Reinhard Boerger -
Vaughan Pratt