The n-Category Café

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.

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

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 $D$, 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?)