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April 25, 2007

Learning from Our Ancestors

Posted by David Corfield

Back in this post I argued against Bernard Williams’ view of science:

The pursuit of science does not give any great part to its own history, and that it is a significant feature of its practice… Of course, scientific concepts have a history: but on the standard view, though the history of physics may be interesting, it has no effect on the understanding of physics itself. It is merely part of the history of discovery.

Taking mathematics as a science, I took Robert Langlands to be on my side against Williams:

Despite strictures about the flaws of Whig history, the principal purpose for which a mathematician pursues the history of his subject is inevitably to acquire a fresh perception of the basic themes, as direct and immediate as possible, freed of the overlay of succeeding elaborations, of the original insights as well as an understanding of the source of the original difficulties. His notion of basic will certainly reflect his own, and therefore contemporary, concerns.

Now, from the interview I mentioned in the last post it appears that Connes has read Galois’ papers with profit. Meanwhile, John has been encouraging us to better ourselves by reading Felix Klein’s Erlanger program. Something I’d like to hear about are instances where people feel they have gained something by reading works from the nineteenth century or earlier, or histories on those works, especially instances where there has been some element of surprise at how not all that was good about a certain way of thinking has survived to the present day.

Posted at April 25, 2007 11:30 AM UTC

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Re: Learning from Our Ancestors

In the Preface to the marvelous book

  • I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, 1994

great tribute is paid to the work of Arthur Cayley:

“…Moreover, in the process of writing we discovered a beautiful area which had nearly been forgotten so that our work can be regarded as a natural continuation of the classical developments in algebra during the 19th century.

We found that Cayley and other mathematicans of the period understood many of the concepts which today are commonly thought of as modern and quite recent. Thus, in an 1848 note on the resultant, Cayley in fact laid out the foundations of modern homological algebra. We were happy to enter into spiritual contact with this great mathematician.”

The 1848 article is reproduced in an appendix. They also credit Cayley with discovering hyperdeterminants (the multidimensional determinants in their title).

Posted by: Todd Trimble on April 25, 2007 2:14 PM | Permalink | Reply to this

Re: Learning from Our Ancestors

Note also that hyperdeterminants in Cayley were:

“A technically defined extension of the ordinary determinant to ‘higher dimensional’ hypermatrices. Cayley (1845) originally coined the term, but subsequently used it to refer to an algebraic invariant of a multilinear form.”

The “higher dimension” aspect drew in the pre-Coxeter King of Polytopes, Ludwig Schlafli (Alicia Boole Stott being the Queen).

Schläfli, L. “Uber die Resultante eine Systemes mehrerer algebraischer Gleichungen.” Denkschr. Kaiserl. Akad. Wiss., Math.-Naturwiss. Klasse 4, 1852.

A uniqueness result was proved:

Glynn, D. G. “The Modular Counterparts of Cayley’s Hyperdeterminant.” Bull. Austral. Math. Soc. 57, 479-497, 1998.

Above citations thanks to Eric W. Weisstein:

http://mathworld.wolfram.com/Hyperdeterminant.html

Posted by: Jonathan Vos Post on April 25, 2007 11:22 PM | Permalink | Reply to this

Re: Learning from Our Ancestors

On reading Cantor’s work I was surprised by how much of it was metaphysical (and application orientated) rather than (in the modern sense) mathematical, and then thereby encouraged to add to it and to criticise it on metaphysical grounds.

Posted by: Enigman on April 25, 2007 4:11 PM | Permalink | Reply to this

Re: Learning from Our Ancestors

Once, as a student, I was looking at an old textbook on theoretical physics, from the beginning of the 20th century.

Apparently, it was from before the time when linear algebra had been cleaned up to a point that a physics textbook could freely play around with vector spaces.

Instead, that book didn’t know the concept of a vector – or almost knew it, but not quite. Instead of using vectors, the text was full of direction cosines.

This way, many very simple statements, from the present perspective, appeared in a comparatively opaque fashion.

Remarkably, at one point there was a discussion of the classical gravitational field outside of a solid ball.

In a footnote to the rather simple computation, the author remarked that, allegedly, Newton’s publication of his Principia was held back for many years because he wasn’t able to prove his conjecture that this field is the same as that of a point source of the same mass.

The footnote concluded with an emphasis of how important it is, in theoretical physics and in math, to use the right symbolic language. If only Newton had bothered to invent something better than his fluxion notation for doing calculus, he would have easily been able to do this calculation in two minutes, as every student today does.

I found that interesting. An author who could see of every linear equation only a faint shadow of direction cosines was marvelling at the power of notational elegance.

Posted by: urs on April 25, 2007 8:14 PM | Permalink | Reply to this

Re: Learning from Our Ancestors

That’s an amusing story!

Posted by: Bruce Bartlett on April 25, 2007 8:27 PM | Permalink | Reply to this

Re: Learning from Our Ancestors

A bit of history from a physicist who lurks in the corners of the cafe. The vector mechanics we teach to freshmen was originated by Boltzmann in Germany and Gibbs at Yale (primarily), from roughly 1850’s to about 1910.

Newton’s fluxions are really “finite differentials” as the notion of limits was not refined until the mid-ninteenth century. He basically thought in terms of an elliptical orbit of a planet (say) as a big polygon your senses approximated this with a smooth curve. Laplace and McLaren turned this on its ear at the end of the eighteenth century and said the motion was a smooth curve which we dealt with as a polygon (Newton’s method for integration like they teach in computer programming today). One of the ironies of all this is that Newton only lacked the notion of smallest possible units for his “fluxions of impulse” (which changed the momentum and made the trajectory change from one side of the polygonal orbit to the next) or he could have arrived at what we today would call a quantum gravity theory in the 17th century!!!

Posted by: Steven Maxson on April 26, 2007 3:06 PM | Permalink | Reply to this

Re: Learning from Our Ancestors

Thanks for these facts!

One quibble, though, if you don’t mind:

One of the ironies of all this is that Newton only lacked the notion of smallest possible units for his “fluxions of impulse” (which changed the momentum and made the trajectory change from one side of the polygonal orbit to the next) or he could have arrived at what we today would call a quantum gravity theory in the 17th century!!!

I think we should really, at best, call that instead the quantum mechanics in a fixed gravitational field. That’s quite different from having the gravitational field itself quantized.

Likewise, the discussion of the quantum mechanics of the hydrogen atom (formally the same problem) is not quantum electrodynamics.

A few years ago there was an experiment at the Laue-Langevin reactor in Grenoble which measured quantum bound states of very cold neutrons that were simply bouncing on the floor, i.e. bound by a linear (in first and only practical approximation) gravitational potential.

While theoretically rather unexciting (performing the experiment itself was possibly a major feat, though), this attracted a lot of attention in the public, because lots of press releases (see any of this) referred to this experiment as having detected quantum gravity effects.

But it didn’t. It didn’t say any more about quantum gravity than the observation of the orange glow of a street light says about quantum electrodynamics.

Okay, but could Newton have found quantum mechanics, then, by simply trusting that his fluxions came as integer multiplies of a fixed quantum of fluxion?

Almost, I think. He might at least have been lead to the old Bohr-Sommerfeld quantization rules.

Posted by: urs on April 26, 2007 4:09 PM | Permalink | Reply to this

Re: Learning from Our Ancestors

The older I get, the more I get interested in the history of math and physics. I hope it’s not just because my past is getting longer than my future. I think it’s because I keep learning new stuff from the past.

One big lesson is that: what seems obvious now was not obvious before… so what seems nonobvious now may seem obvious later!

For example, when Newton’s Principia came out, it caused a big stir, but very few people could follow it — classical mechanics and calculus were comprehensible only to a few top experts. Now they’re routinely taught in good high schools.

(The Principia is still very hard to read.)

This gives me hope that differential geometry, topology, category theory and so on will eventually be taught in good high schools. Which is important, because otherwise people will eventually be quite old by the time they reach the frontiers of knowledge.

(Of course, life spans are increasing, too…)

Another big lesson is: what smart people once did often seems strange or silly now — so what smart people are doing now may seem strange or silly in the future.

This lesson makes me feel less compelled to follow every trend that comes along.

There are also lots of specific technical facts one can learn from studying the history of science — and I could talk about these for hours — but in a way these are less important.

Posted by: John Baez on April 25, 2007 9:33 PM | Permalink | Reply to this

Re: Learning from Our Ancestors

hope that differential geometry, topology, category theory and so on will eventually be taught in good high schools.

Are there high schools with e.g. real combinatorics courses, on the level of two or three chapters of Concrete Mathematics? That seems a lot more plausible than the universe you envision.

Posted by: Allen Knutson on April 26, 2007 5:44 AM | Permalink | Reply to this

Re: Learning from Our Ancestors

Vladimir Arnold has interesting papers where he revisits ancient authors and finds ideas and problems left unexplored, for example this on Poincaré, and with Vasiliev they wrote in the Notices of the AMS in 1989 the piece Newton’s Principia read 300 years later (that’s Notices AMS 36(9):1148-1154 and addendum 37(2):144). There’s probably more in the book on open problems.

Posted by: thomas1111 on April 25, 2007 9:40 PM | Permalink | Reply to this

Re: Learning from Our Ancestors

Thanks everyone for these comments.

To these I can add that on my old blog ‘dt’ wrote:

It’s worth pointing out that, from time to time, modern mathematicians find it extremely fruitful to read the classics from eras with inferior epistemic resources. Geometric invariant theory certainly owes a lot to Mumford reading Hilbert. Intersection homology owes a lot to MacPherson reading Poincare. Manjul Bhargava of course read Gauss.

I also mention there that Gian-Carlo Rota once described reading nineteenth century mathematical texts as like entering a hothouse full of exotic plants whose existence you had never suspected.

Robert Hermann wrote (see end of this)

In reading Lie’s work in preparation for my commentary on these translations, I was overwhelmed by the richness and beauty of the geometric ideas flowing from Lie’s work. Only a small part of this has been absorbed into mainstream mathematics. He thought and wrote in grandiose terms, in a style that has now gone out of fashion, and that would be censored by our scientific journals! The papers translated here and in the succeeding volumes of our translations present Lie in his wildest and greatest form.

I seem to recall he also says somewhere that he periodically returns to Darboux’s works, finding he can understand more each time.

I wonder if Pierre Cartier read Euler to write Mathemagics (A Tribute to L. Euler and R. Feynman)

I suppose one might still argue that there’s an essential difference between mathematics and philosophy along the following lines. One can practice mathematics without returning to earlier texts, even though many good mathematicians have profited greatly by doing so, whereas one should not practice philosophy without having read Plato and Aristotle.

I’m not so sure this points to an essential difference.

Posted by: David Corfield on April 26, 2007 8:19 AM | Permalink | Reply to this

Book on Lie’s original paper

I happily tracked down Robert Hermann’s comments you gave above (I think the link is faulty - should it be this one?)

My investigations led me to the following book :

Sophus Lie’s 1880 Transformation Group Paper, translated by Michael Ackerman with comments by Robert Hermann, available here at Google Books.

It’s got a preface by the man himself - Sophus Lie - which I’m sure you’ll find really interesting. Here is the beginning :

As an introduction to the present work it seems to me to be fitting to preface it with some general remarks.

In the course of time the position of analysis with respect to geometry and to the various branches of applied mathematics has passed through many extensive changes. During the past century the mutual stimulation of the seperate disciplines was greatly advanced; and in this, first one and then the other discipline was dominant. This is in the nature of things and in itself is not to be regretted. But in our century, mathematics has split up into many very extensive areas, and this division has often led to the representatives of one area to misjudge the importance of others, so that, to the detriment of their own discipline, fruitful ideas from the outside have been ignored.

Permit me to elucidate these remarks by recalling briefly some of the phases of the development of mathematics.

What’s nice about this book (and its sucessors) is that you have the original papers, together with introductions about how they look in modern notation, and frequent comments in the text. There’s even a section about how Lie uses categories occasionally! Apparantly he makes often makes use of the category C XC_X associated to a smooth manifold XX, whose objects are open sets and morphisms are (I’m not sure).

I’d often heard that Lie’s ideas were overlooked by many of the people who subsequently worked on “Lie groups”, only to be rediscovered in modern times. For instance, apparantly the idea of the moment map in symplectic geometry dates back to him.

Posted by: Bruce Bartlett on April 26, 2007 1:35 PM | Permalink | Reply to this

Re: Book on Lie’s original paper

If I recall correctly, Anders Kock somewhere cites Sophus Lie as saying in a private note that he is using synthetic differential geometric reasoning (apparently he really used the word “synthetic”) when privately thinking about a problem, but that due to the impossibility of communicating this kind of reasoning always hides it in his publications.

Posted by: urs on April 26, 2007 1:49 PM | Permalink | Reply to this

Re: Book on Lie’s original paper

Thanks for spotting the empty link. And thanks for pointing out the Google Books entry. I hadn’t realised there was so much there, like Klein’s Development of Mathematics in the Nineteenth Century.

There’s even a section about how Lie uses categories occasionally!

I wonder if that isn’t a case of excessive anachronism.

Posted by: David Corfield on April 26, 2007 2:23 PM | Permalink | Reply to this

Re: Book on Lie’s original paper

Indeed that was a bit “anachronistic”. By the way, I wonder if anyone else here at the n-cafe is, like me, a bit uncomfortable with this Google Books phenomenon. Apparently many books, like the ones we’ve just found, are available in their entirety at Google Books.

From the wikipedia entry on this, it seems that Google have a policy of “freely copying any work until notified by the copyright holder to stop”. This seems dodgy, to say the least.

I love the idea of the knowledge in books becoming easily avaliable to all. I’m just uncomfortable that this process is being carried out by a private for-profit company. It seems unethical to me :

1. Someone writes a book, with a lot of blood, sweat and tears.

2. The book gets published.

3. Google scans the book in, without the consent of anybody.

4. Scanned portions of this book (often the entire book) go on the Google web-site, thus providing Google with income, which the original author didn’t seem to have a say in.

Moreover, I’m uncomfortable that many of our esteemed academic institutions have been “hoodwinked” by this scheme, for want of a more polite word. For instance, the University of Michigan, Harvard (Widener Library), Stanford (Green Library), Oxford (Bodleian Library), and the New York Public Library. In all these cases it seems there are Google employees working night and day, scanning in all the books.

Again, I love the idea of knowledge being freely and widely available, and I use Google all the time. But I would much prefer it if it were libraries and government organizations who were the “custodians of human knowledge”, rather than for-profit companies.

Posted by: bart on April 26, 2007 3:40 PM | Permalink | Reply to this

Re: Book on Lie’s original paper

It is well known that Lie conjectured that there are four infinite series (ABCD) of simple finite-dimensional Lie algebras over C, and that Cartan and Killing showed that there are also five exceptions (E6 E7 E8 F4 G2). Lie also conjectured that there are four series in the infinite-dimensional polynomial case (WSHK), something that E Cartan proved in 1909. This result was virtually forgotten until the mid 1960s, and is still much less known than the ABCDEFG classification, despite the physical importance of these infinite-dimensional algebras.

Which shows that even important results can be forgotten.

Posted by: Thomas Larsson on April 26, 2007 4:06 PM | Permalink | Reply to this

Re: Book on Lie’s original paper

Amusingly, Killing showed there were six exceptional simple Lie algebras, including two 52-dimensional ones. Cartan noticed that those two were isomorphic.

Just an example of the fractal texture of history: whatever you say, someone can come along and add more details.

Posted by: John Baez on April 26, 2007 5:40 PM | Permalink | Reply to this

Re: Learning from Our Ancestors

So Euler is read today. And this was encouraged by André Weil:

Apparently, Weil claimed that he was trying to convince the mathematical community that the students of mathematics would profit much more from a study of Euler’s “Introductio in analysis infinitorum”, rather than of the available modern textbooks…

Posted by: David Corfield on June 26, 2007 5:34 PM | Permalink | Reply to this

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