## Introduction: getting near the history of fairly modern mathematics

While much of the history of mathematics requires an understanding of the relevant mathematics, just as working on the philosophy of quantum mechanics requires a grasp of the science, there is a large area that needs much less than you might think, and which indeed you might already know. In fact, for the period before, say, 1800, it is more likely that languages will pose a greater problem, because the sources, if not the modern scholarship, are in Chinese, Greek, Arabic and so forth, (not to mention Akkadian and other delights) and cannot be read unless you immerse yourself in them. Post 1800, the predominant languages (by crude page count) are French and German, then Italian, and then perhaps English, ahead of, say, Latin, Spanish, and Russian, but as the years go by the balance shifts markedly towards English. There are increasing numbers of translations, but why be a monoglot? And of course the difficulty of the mathematics generally rises. So the question for you is: is there a Goldilocks zone, where the mathematics is not too hard and the linguistic demands not too fierce? And the answer, happily, is yes. What's more, the subjects are not presently over-worked. In no particular order, they are issues in the overlap of history and philosophy of mathematics, and mathematics and language, the popularisation of mathematics, and institutional questions. (There are others, I'm sure, but these are ones I know a bit about.)

## History and philosophy of mathematics

Most obviously, there is something of a marriage being arranged these days between history and philosophy of mathematics. Even if it turns out to be more of a fling, there is a considerable degree of interest at present in the way mathematics was understood around 1900 and the foundational and philosophical issues it raised (and is raising again). What is at stake, philosophically, is things like fruitfulness (a Fregean theme) explanation, understanding, and purity of method, not the thorny but profound matters in mathematical logic that flourished after Gödel. There are famous texts here that had a significant impact in their day: several essays by Helmholtz, Poincaré and others, including Ernst Mach (all in English translations, a point interesting in itself, as we shall see).

As a way into the philosophy, you can consult

Aspray W. and P. Kitcher 1988 *History and Philosophy of Modern Mathematics*, Minneapolis, University of Minnesota Press.

A classic well worth agreeing and disagreeing with is:

Kitcher, P. 1984: *The nature of mathematical knowledge*, Oxford University Press.

Recently published, and a little more demanding than Aspray and Kitcher, but more up-to-date are the essays in:

Ferreirós, J. and J.J. Gray, 2006 *The Architecture of Modern Mathematics*, Oxford University Press.

A good guide to the logical and philosophical issues that is also open to the history is:

Giaquinto M. 2002: *The Search for Certainty*, Oxford University Press.

Almost its mirror image, a good history that is very clear on the logical and philosophical implications, is:

Ferreirós J. 1999: *Labyrinth of thought: A history of set theory and its role in modern mathematics*, Birkhäuser Verlag, Boston and Basel.

There is also, among books that go deeper into the philosophy of mathematics:

Potter, M. 2000: *Reason's Nearest Kin: Philosophies of arithmetic from Kant to Carnap*, Oxford University Press.

Among anthologies, two stand out:

Ewald W. (ed.) 1996: *From Kant to Hilbert: A source book in the foundations of mathematics*, Oxford University Press, 2 vols.

and

Mancosu, P. 1998: *From Brouwer to Hilbert: The debate on the foundations of mathematics in the 1920s*, Oxford University Press.

A volume that pushed energetically for what it says on the cover is

Tymoczko, T. 1998: *New Directions in the Philosophy of Mathematics*, Basel and Boston, Birkhäuser. Revised and expanded edn. in Princeton University Press, 1998.

On Frege there is a large literature, but ones that are most relevant here include:

Beaney, M. 1996 *Frege: Making Sense*, Duckworth.

Demopoulos, W. 1995: *Frege's Philosophy of Mathematics*, Harvard University Press.

Tappenden, J. 1995 'Extending knowledge and "fruitful concepts": Fregean themes in the foundations of mathematics', *Noûs* 29, no. 4, 427-467 (see also the references in Ferreirós and Gray, 2000)

and, among slightly older texts but notable for its historical detail:

Sluga, H. 1980 *Gottlob Frege*, Routledge, London and New York.

Often opposing views in Frege studies are defended by Dummett and Crispin Wright, but they take us away from history of mathematics and into interesting philosophical questions. A good place to start with this is:

Dummett, M. 1991 *Frege: Philosophy of Mathematics*, Duckworth

and

Hale, Bob and Crispin Wright, 2001, *The Reason's Proper Study: Essays towards a Neo-Fregean Philosophy of Mathematics*, Oxford University Press.

Even more recent mathematics is tackled in Corfield's book:

Corfield, D., 2003 *Towards a philosophy of real mathematics*, Cambridge University Press.

Another work that many people like is:

Torretti, R. 1978: *Philosophy of Geometry from Riemann to Poincaré*, Dordrecht, Reidel.

And by the way, it seems to me that Lakatos's work is ripe for a re-appreciation. One can start with:

Koetsier, T. 1991 *Lakatos' Philosophy of Mathematics, A Historical Approach*, Amsterdam, North-Holland, 1991.

German philosophy in the late 19th Century fought a turf war with psychology (well described in Kusch, *Psychologism*) but one can see an interesting story there by considering, as many people did at the time, what the relationship is between mathematics and logic (defined, for the moment, as clear and correct reasoning). Now is that conception of logic captured by the very elementary symbolism of Boole, or indeed the more sophisticated, but not overwhelming, symbolism of Charles Sanders Peirce (a very interesting character, much studied but by no means exhausted)? Come to that, what is (or was) the proper relation of logic to philosophy? Asking these questions around 1900 took people into a vigorous debate between Kantians and Leibnizians. Here you will find Russell, of course, and Cassirer, and any number of other figures, including Frege. Again, what is involved is the proper understanding and meaning of some novel mathematical ideas, and those ideas in themselves are often not as hard to understand as the debates they generated. Throw in the question of the relationship of any or all of the above to language, and you not only have a debate going back to the 17th Century in Europe, but one that reaches to the role of artificial languages such as Esperanto. That involved quite a number of mathematicians, and even the Cycling Club of France. And the old saw that mathematics is a language was turned around most remarkably by LEJ Brouwer, the arch-exponent of Intuitionism. He is the subject of a good biography (volume two is due out shortly in English):

van Dalen, D. 1999 *Mystic, geometer, and intuitionist: the life of L.E.J. Brouwer*, Vol.1, The dawning revolution, Oxford: Clarendon Press,

and there is a good book on the huge fuss his philosophy of mathematics caused

Hesseling, D. 2003: *Gnomes in the Fog*, Basel and Boston, Birkhäuser.

The literature on Russell is huge, of course, but I happen to like:

Griffin, N. 1991 *Russell's idealist apprenticeship*, Oxford: Clarendon.

On Cassirer (another Open Court author, by the way) you can start with:

Friedman, M. 2000 *A parting of the ways: Carnap, Cassirer, and Heidegger*, Chicago, Ill.: Open Court, 2000.

A very interesting pair of books is:

Cassirer, E. 1923 *Substance and Function, and Einstein's Theory of Relativity*, translations of 'Substanzbegriff und Funktionsbegriff' and 'Zur Einstein'schen Relativitätstheorie'. Open Court, Dover reprint 1953.

There is a book that should have opened up the topic of the Leibniz revival, but to my taste it stops too soon – there is work to be done! It is:

Peckhaus, V. 1997 *Logik, Mathesis universalis und allgemeine Wissenschaft: Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrhundert*, Berlin: Akademie Verlag, 1997.

Many of these issues are tied to the theme of Modernism in mathematics, for which the founding text is:

Mehrtens, H. 1990 *Moderne Sprache Mathematik: eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme*, Frankfurt: Suhrkamp.

## Popularisations

Many of the original works were published by the Open Court Press and in due course became Dover books, but to my knowledge little has been done on Paul Carus, the creator of Open Court and editor of *The Monist*. (And I am told there is an unexplored archive.) While it is true that some of these themes are well worked over and an original angle may not be easy to find, there were a number of French and American popular journals that got quite agitated about the nature of mathematics around 1900, and these debates are quite accessible. They concern such things as the truth of mathematics, the paradoxes of set theory and their implications for the foundations of mathematics, and the fourth dimension (think HG Wells, Abbott's *Flatland*, Zöllner and the medium Henry Slade).

## Religion

Mention of Carus suggests another possible line of enquiry not entirely particular to him: mathematics and religion. Another figure here, although more of a philosopher, is Josiah Royce. Possible topics here include the philosophy of monism, which comes in various flavours from the pantheistic version of Carus to the more overtly anti-Catholic kind advocated by Haeckel, about whom there is an enormous literature. For what were taken to be dangerous speculations about the truth of mathematics and its implications for other big truths, see:

Richards, J. 1988 *Mathematical visions: the pursuit of geometry in Victorian England*, Academic Press.

## Education

There were extensive surveys by mathematicians of the state of school and university education in the period 1900-1914. A lot of it is in German, but there may be good sources in other languages too, and the differences are quite wide and interesting. Institutional history is not what excites me, and I believe the literature is patchy, but some of it is good. In a different vein, there is the excellent study by Warwick of the mathematical sciences at Cambridge:

Warwick, A. 2003 *Masters of theory: Cambridge and the rise of mathematical physics*, University of Chicago Press.

Some French stories are told in

Gispert, H., 1991 'La France mathématique', *Cahiers d'histoire et de philosophie des sciences* 34, 13-180

an American one with strong German overtones in

Parshall, K.H. and D.E. Rowe, 1994 *The Emergence of the American Mathematical Research Community; J.J. Sylvester, Felix Klein, and E.H. Moore*, American and London Mathematical Societies, 8, Providence, Rhode Island.

## The history of logic

The history of logic is not a Goldilocks zone for me, covering as it does a period almost as extensive as the history of mathematics and being at least as difficult in the modern period. There may be Goldilocks zones in it, and certainly there have been some fine books recently, but this would be a matter for careful further exploration.

## And mathematics?

I must end with a plug for a more dangerous life: if you actually have some mathematics there is much to be done.