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July 28, 2007

Algebra 1 versus Algebra 2

Posted by David Corfield

In Delphi, Colin McLarty performed some myth-busting for us. Many of you will have heard of Paul Gordan’s supposed reaction to a result of David Hilbert in the theory of invariants:

This is not Mathematics, it is Theology!

Often this is taken as one of the reactionary old guard standing in the way of the new algebra. However, Colin does a great job explaining how the true story is far more subtle.

Hilbert in 1888 said he found his proof “with the stimulating help of” this very professor Gordan.

Rather than recapping his argument, we may as wait until it appears. Here I want to know more about what happens next. In particular, I’d like to know whether Gian-Carlo Rota’s distinction between Algebra 1 and Algebra 2 holds water. He does this somewhere in English, Chapter III of Indiscrete Thoughts I believe.

Online, all I can find is in Italian. Here Rota picks out key figures in each:

Algebra 1: algebraic geometry and algebraic number theory, represented by Kronecker, Hilbert, Weil, …

Algebra 2: ‘Combinatoria Algebrica’ - algebraic combinatorics, represented by Boole, Capelli, Young, Gordan, Hall, Birkhoff, …

Does this chime with anyone?

Posted at July 28, 2007 5:45 PM UTC

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Re: Algebra 1 versus Algebra 2

I always that Gordan’s quote was a joke that was being interpreted as uncharitably as possible for propaganda purposes – propaganda for a cause that had long been won, but the quote and its interpretation lingered on.

Posted by: Walt on July 28, 2007 6:46 PM | Permalink | Reply to this

Re: Algebra 1 versus Algebra 2

Colin reports one person at the time, Gerhard Kowalewski, who thought Gordan was actually making a compliment. Far from rejecting Hilbert’s proof,

Gordan consistently supported Hilbert and this proof, except that when Hilbert submitted it to the Mathematische Annalen Gordan felt it was not ready. Gordan wanted a clearer argument and he wanted Hilbert to develop the idea farther first. Hilbert soon did develop his work in just the way Gordan wanted – and proved his famous Nullstellensatz to do it, which became basic to algebraic geometry – but not before getting his 1888 idea into the Annalen!

Posted by: David Corfield on July 29, 2007 11:56 AM | Permalink | Reply to this

On that quote; Re: Algebra 1 versus Algebra 2

Gordon, P

This is not mathematics, it is theology.
[On being exposed to Hilbert’s work in invariant theory.]
Quoted in P. Davis and R. Hersh The Mathematical Experience, Boston: Birkhäuser, 1981.

Googling that quote elicits:

An Informal History of Formal Proofs: From Vigor to Rigor?
Klaus Galda
The Two-Year College Mathematics Journal, Vol. 12, No. 2 (Mar., 1981), pp. 126-140
doi:10.2307/3027375
This article consists of 15 page(s).

On Mon, 16 Aug 1999, cxm7@po.cwru.edu (Colin Mclarty) wrote:

> On Gordan’s supposed quote (which I believe he did say
> in fact) “this is not mathematics it is theology”, the earliest
> source I have found is Max Noether’s obituary of Gordan in
> MATHEMATISCHE ANNALEN 75 (1914) 1-41, page 18….

The story is told by Felix Klein in the _Lectures on the
Development of Mathematics in the 19th Century_, and his language
sounds as though he had personal knowledge of it:

The unusual nature of [Hilbert’s] work produced quite diverse
reactions. I resolved to draw Hilbert to Goettingen at the
first opportunity. Gordan at first reacted negatively, saying
“This is not mathematics, it is theology.” But later he said
“I have become persuaded that even theology has its uses.” In
fact, he himself substantially simplified Hilbert’s argument.

(At the moment, I don’t have the German at hand, but it’s fairly
late in the book, following the discussion of Hilbert’s work.)

William C. Waterhouse
Penn State

The Place of Rigor in Mathematics
E. T. Bell
The American Mathematical Monthly, Vol. 41, No. 10 (Dec., 1934), pp. 599-607
doi:10.2307/2301907
This article consists of 9 page(s).

The Nature and Growth of Modern Mathematics
By Edna Ernestine Kramer

Italian Studies in the Philosophy of Science
By Maria Luisa Dalla Chiara

Posted by: Jonathan Vos Post on July 28, 2007 8:44 PM | Permalink | Reply to this

Re: Algebra 1 versus Algebra 2

David asks:

In particular, I’d like to know whether Gian-Carlo Rota’s distinction between Algebra 1 and Algebra 2 holds water. He does this somewhere in English, Chapter III of Indiscrete Thoughts I believe.

Right, it’s in Chapter III of Part I, in the section titled “Bottom Lines”. I’ll copy out some relevant passages (pp. 52-54):

Those who have reached a certain age remember the visceral and widespread hatred of lattice theory from around 1940 to 1979; this has not completely disappeared. Such an intense and unusual disliking for an entire field cannot be simply attributed to personality clashes. It is more likely to be explained by pinpointing certain abysmal differences among bottom lines of the mathematicians of the time. If we begin such a search, we are likely to conclude that the field normally classified as algebra really consists of two quite separate fields. Let us call them Algebra One and Algebra Two for want of a better language.

Algebra One is the algebra whose bottom lines are algebraic geometry or algebraic number theory. Algebra One has by far a better pedigree than Algebra Two, and has reached a high degree of sophistication and breadth. Commutative algebra, homological algebra, and the more recent speculations with categories and topoi are exquisite products of Algebra One. It is not infrequent to meet two specialists in Algebra One who cannot talk to each other since the subject is so vast. Despite repeated and dire predictions of its demise, Algebra One keeps going strong.

Algebra Two has had a more accidented history. It can be traced back to George Boole, who was the initiator of three well-known branches of Algebra Two: Boolean algebra, the operational calculus that views the derivative as the operator DD, on which Boole wrote two books of great beauty, and finally, invariant theory, which Boole initiated by remarking the invariance of the discriminant of a quadratic form.

Roughly speaking, between 1850 and 1950 Algebra Two was preferred by the British and Italians, whereas Algebra One was once a German and lately a French preserve. Capelli and Young’s bottom lines were firmly in Algebra Two, whereas Kronecker, Hecke, and Emil Artin are champions of Algebra One.

In the beginning Algebra Two was largely cultivated by invariant theorists. Their objective was to develop a notation suitable to describe geometric phenomena which is independent of the choice of a coordinate system. In pursuing this objective, the invariant theorists of the nineteenth century were led to develop explicit algorithms and combinatorial methods. The first combinatorialists, MacMahon, Hammond, Brioschi, Trudi, Sylvester, were invariant theorists. One of the first papers in graph theory, in which the Petersen graph is introduced, was motivated by a problem in invariant theory. Clifford’s ideal for invariant theory was to reduce the computation of invariants to the theory of graphs.

The best known representative of Algebra Two in the nineteenth century is Paul Gordan. He was a German, perhaps the exception that tests our rule. He contributed a constructive proof of the finite generation of the ring of invariants of binary forms which has never been improved upon, and which foreshadows current techniques of Hopf algebra. He also published in 1870 the fundamental results of linear programming, a discovery for which he has never been given proper credit. Despite his achievements, Paul Gordan was never fully accepted by specialists in Algebra One. “Er war ein Algorithmiker!” said Hilbert when Gordan died.

I’m skeptical that any tight distinction can be made between Algebra One and Algebra Two; there is way too much spillover amongst the various branches that Rota names. My guess is that Rota is pointing out general historical tendencies: perhaps Algebra Two would be that tendency to see algebra as so many calculi of symbolic manipulation (Boolean algebra, Gentzen sequent calculus, Young diagrams, Gröbner bases…), suitable for algorithmic or machine implementation. A few paragraphs later he sums up its bottom line as “algebraic combinatorics”. Whereas Algebra One might be that tendency to view algebra primarily as the dual counterpart of “geometry” – here conceiving geometry broadly in terms of varieties and schemes. (Obviously referring to Grothendieck’s foundations for algebraic geometry, which includes algebraic number theory. But the idea of thinking of number fields in geometric terms is of course much older [think divisors, class groups].)

I think Rota has a tendency to exaggerate or overreach here and there to make a rhetorical point, and it’s not clear how well this separation of Algebras would hold up under fire. Nor is it clear to me that Hilbert was actually dissing Gordan when he said that thing after Gordan died. (By the way, who repeatedly predicts the demise of Algebra One? And what fundamental results of linear programming did Gordan discover?)

Posted by: Todd Trimble on July 30, 2007 2:53 AM | Permalink | Reply to this

Re: Algebra 1 versus Algebra 2

I agree with Todd. I think this distinction might have seemed reasonable during the Grothendieck revolution, or during the counter-revolution that followed, but nowadays it seems like trying to separate points on a continuum.

Regarding Gordon’s theology comment, when I first heard it years ago, I also interpreted it as the dying cry of the old guard. But in the past few years, I’ve come to view it with sympathy. I find it really helpful to think of the axiom of choice as theology in the sense that if you use it, no one will ever be able to prove you wrong, and sometimes it might help you get through the day, but that relying on it keeps you from seeing the true nature of things. I just think that mathematics at its best is about truly real things, like non-real zeros of zeta functions, and not about truly imaginary things like algebraic bases of (real, say) function spaces. For the same reason, I prefer topos-theoretic approaches to the foundations algebraic geometry to the point-set-theoretic approaches, which rely of the theological position that rings have sufficiently many prime ideals. Again, it’s not so much that I disagree, I just think it distracts from the true issues.

You’d think I’d have the same problem with the axiom of infinity. I’m not sure why I don’t.

Posted by: James on July 30, 2007 1:00 PM | Permalink | Reply to this

Re: Algebra 1 versus Algebra 2

I just think that mathematics at its best is about truly real things.

We chatted about a certain sense of ‘real’ at Urs’ old blog. You can follow a link to Cherednik’s views on the real/imaginary divide.

Posted by: David Corfield on July 30, 2007 2:03 PM | Permalink | Reply to this

Re: Algebra 1 versus Algebra 2

It seems that Cherednik means something completely different. His imaginary component appears to grow with abstraction, whereas my imaginary component is essentially discrete and grows with the set-theoretic axioms you use.

Johnstone once used the phrase “recondite axioms of set theory”, which I like.

Posted by: James on July 30, 2007 10:53 PM | Permalink | Reply to this

Re: Algebra 1 versus Algebra 2

According to Rota,

Er war ein Algorithmiker!” said Hilbert when Gordan died.

Was it Hilbert? A quick Google search suggests that it was Gordan’s friend Max Noether who said that, and who I think could be considered a fellow Algorithmiker. (That would make it less likely that the term was meant disparagingly, I think.)

James writes

I find it really helpful to think of the axiom of choice as theology in the sense that if you use it, no one will ever be able to prove you wrong, and sometimes it might help you get through the day, but that relying on it keeps you from seeing the true nature of things. I just think that mathematics at its best is about truly real things, like non-real zeros of zeta functions, and not about truly imaginary things like algebraic bases of (real, say) function spaces.

Yes, absolutely. I also feel somewhat uncomfortable invoking such strong existential principles, except when absolutely necessary. The ‘imaginary forms’ will simply not exist in most universes (e.g., the axiom of choice fails to hold in most toposes); ‘real forms’ are much more likely to survive transportation to other contexts.

I suppose that as computers enter our mathematical lives more and more, those forms which exist only in black boxes, such as Hamel bases, will more and more be put aside in favor of ‘real forms’ that one can get work with explicitly.

Posted by: Todd Trimble on July 30, 2007 7:29 PM | Permalink | Reply to this

Re: Algebra 1 versus Algebra 2

Yes, it was Noether.

McLarty writes:

Certainly “[Gordan] was an algorithmiker” (Noether, 1914, p. 37). But there is no evidence that he rejected other mathematics. And algorithm then did not mean what it does now. Meyer (1892, p. 187) aptly calls Gordan’s method an algorismus meaning a framework for formal calculation. It is not a specific calculational routine and so not an “algorithm” in our sense today.

Posted by: David Corfield on July 30, 2007 7:56 PM | Permalink | Reply to this

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