# Real Numbers, Generalizations of the Reals and Theories of Continua.

This is a fascinating collection, an intellectual feast for anyone interested in the continuum - whether from a philosophical or historical or purely mathematical point of view. It consists of eleven pieces, of which three are valuable historical documents dating from the first decade of this century. The remaining eight (including the editor's introduction) were written or adapted especially for this volume. All but one of these eleven pieces have a common underlying theme; they all reflect on one central question: is a non-Archimedean continuum acceptable in geometry or in arithmetic/analysis?

The one exception is quite massive, being by far the longest piece in the book; it is the contribution of Douglas S. Bridges, an eminent disciple of Errett Bishop, and provides a wide-ranging constructivist (primarily Bishopian) view of the real number system. This paper, whose concerns are very different from those of all the others, detracts from the collection's unity; but as it has considerable intrinsic merit, it would be churlish to complain about its inclusion.

Archimedes' axiom - the fifth in his The Sphere and the Cylinder - states (in modern terminology) that if AB and CD are any rectilinear segments, then there exists a positive integer n and a point B[prime], collinear with AB, such that AB goes into AB[prime] exactly n times and A C goes into AB[prime] at least once. In the arithmetic of ordered number systems, the counterpart of this axiom states that if is a positive number and b is any number, then there exists a positive integer n such that na [greater than or equal to] b. A non-Archimedean continuum, in which the axiom is not satisfied, necessarily contains - for any given unit of measurement - magnitudes that are actually infinite (greater than n units for any n) and those that are actually infinitesimal (smaller than nth of a unit for any positive n).

The nineteenth-century arithmetization of analysis, culminating in the work of Dedekind and Cantor, produced the familiar arithmetical continuum, the field of real numbers, which is of course Archimedean. This conception of the continuum soon became the dominant orthodoxy. It still is. (This applies even to the constructivist school. Indeed, the notion of the real continuum expounded by Bridges in the volume under review is formally identical to Cantor's. The difference lies 'only' in the underlying logic.)

But just as the Dedekind-Cantor view of the continuum was winning its great victories in mathematical public opinion, it was challenged by Giuseppe Veronese (1854-1917), one of the founders of modern geometry. His critique consisted of two main distinct elements. First, he maintained that the continuum cannot be an aggregate of points, which are themselves indivisible. Second, he pointed out that non-Archimedean continua cannot be excluded on empirical or logical grounds: non-Archimedean geometry, in which Archimedes' axiom is denied, is just as legitimate as non-Euclidean geometry, in which Euclid's fifth postulate is denied. Part of Veronese's massive treatise, Fondamenti di Geometria (1891), is devoted to the purely synthetic development of precisely such a geometry.

Veronese was thus a founding father of modern non-Archimedean mathematics. But his name and the importance of his contribution have fallen into almost total oblivion. His own work is written in a difficult and often confused style; and its content has been superseded or overshadowed by the work of his famous student, Tullio Levi-Civita and, following him, that of Arthur Schoenflies, Hans Hahn, and others. (Thus, for example, in the 1994 Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences edited by Ivor Grattan-Guinness, Veronese is mentioned just three times, and only in passing. More surprisingly, the Archimedean axiom is not mentioned at all, not even in connection with Hilbert's work on the foundations of geometry, in which he constructs a model for non-Archimedean geometry!)

One of the main virtues of Ehrlich's collection is that it salvages this largely submerged early phase of modern non-Archimedean mathematics. In a sense, the whole book (with the aforementioned single exception) is a kind of homage to Veronese's profound insight. Arguably the most valuable of the three historical documents is Veronese's On Non-Archimedean Geometry, an Invited Address to the 1908 International Congress of Mathematics. In addition, there is an interesting new paper by Gordon Fisher on Veronese's life and work, to which is appended a translation of a brief but important passage from the latter's opus magnum.

Another fascinating historical document is E. W. Hobson's 1902 Address to the London Mathematical Society upon his retirement as the Society's president, On the Infinite and the Infinitesimal. In a period when English - that is, predominantly Cambridge - pure mathematicians were mostly unaware of the revolutionary achievements of Dedekind and Cantor, Hobson (followed by Hardy) was one of the few who brought these developments to the attention of his colleagues. Judging from this Address, he was not only well informed, but also a perceptive judge of those continental ideas. While committed to the new Dedekind-Cantor orthodoxy, Hobson was aware of Veronese's critique, and considered it sufficiently weighty to deserve careful attention.

The third historical piece in the collection is Poincare's 1902 review of Hilbert's Grundlagen der Geometrie. In admiration of Hilbert's work, Poincare's well-known hostility to the formalist viewpoint is mellowed:

This notion may seem artificial and puerile; and it is needless to point out how disastrous it would be in teaching and how hurtful in mental development; how deadening it would be for investigators, whose originality it would nip in the bud. But, as used by Professor Hilbert, it explains and justifies itself, if one remembers the end pursued (p. 150).

The inclusion of Poincare's review in the present collection is justified mainly by the fact that he - unlike some later commentators - is fully aware of the importance of Hilbert's construction of a model for non-Archimedean geometry, and in fact discusses it in some detail.

Of the remaining contributions to the book, not mentioned so far, I find three of particular interest. First, J. H. Conway presents a brief glimpse into his astonishing system of surreal numbers. For those not familiar with Conway's work (mainly his On Numbers and Games, Academic Press, 1976), the tantalizing curiosity aroused by his present paper is partly relieved by the editor's paper, 'All Numbers Great and Small', in which he provides a somewhat different route to the surreals, analogous to the conventional definition of the von Neumann ordinals. Indeed, the ordinals can be regarded as a species of surreal numbers, which therefore constitute a proper class. Briefly, the field of surreals is the richest possible, absolutely saturated, real-closed ordered field. It includes as substructures not only the ordinals, but any other totally ordered structure that can reasonably be regarded as a number system. Thus it is the maximal non-Archimedean continuum.

H. Jerome Keisler's 'The Hyperreal Line' is a compact, lucid, informative presentation of Robinson's nonstandard enlargement of the real number system. (To be precise, Keisler imposes on the enlarged system a stronger saturation condition than the one used by Robinson. This is vital for certain post-Robinsonian constructs, such as the Loeb measure, explained in this paper.) 'Hyperreals' is Keisler's term for members of the enlarged system, among which are the ordinary (standard) reals. The field of hyperreals is non-Archimedean; nevertheless, in some sense (which can be made precise) it has exactly the same formal properties as the original field of standard reals. This makes it possible to obtain beautifully natural nonstandard characterizations of standard concepts such as continuity and compactness, as well as easier proofs of various theorems of standard analysis, using infinitesimals in rigorous versions of old-style pre-Weierstrass arguments. Also, it is possible to use non-standard entities in new and natural constructions of useful standard entities. An example of such a construction - using the Loeb measure to obtain the Lebesgue measure on the standard real line - is briefly explained by Keisler. The paper ends with some remarks on profound philosophical issues, concerning the uniqueness and reality (in either the mathematical or physical sense) of the real and hyperreal lines. One of his conclusions is identical to a view advocated by Veronese: our experience of physical continua cannot be sufficiently fine or precise to guide our intuition as to whether an Archimedean or non-Archimedean mathematical continuum is a better model of physical reality.

Personally, I regret one omission in Keisler's paper. A beautiful example of a nonstandard construction of a standard entity is the Robinsonian construction of the real line itself. Starting with the field of rational numbers, one obtains by enlargement the field of hyperrationals. The finite hyperrationals constitute a valuation ring in the latter field, and the non-invertible members of the ring constitute (as in every valuation ring) a maximal ideal - in this case, they are the infinitesimal hyperrationals. Forming the quotient field of the ring modulo the maximal ideal (that is, applying the canonical place map), one obtains a new field. It is not difficult to show that this new field is a - and hence the - completion of the original field of rationals. Thus we have obtained the reals. Mention of this construction would have been apposite, particularly in view of Conway's grumbles, seconded by the editor, about the difficulties arising in the construction of the reals by Dedekind cuts: defining multiplication among cuts is somewhat messy. Conway goes so far as to suggest - with tongue firmly in cheek, to be sure - that the reals ought perhaps to be introduced via his surreals. Well, one of the many virtues of the Robinsonian construction is that the reals are born as a ready-made ordered field, with their order relation and arithmetical operations already given.

Returning to the collection as a whole, it seems to me that it would have been useful to include a piece on synthetic differential geometry via category theory. This would tie up with Veronese's first critique of the orthodox continuum: the latter is a set of points, whereas Veronese's view, mentioned above, was that the continuum is not intuited as an aggregate of indivisible elements. In this regard, Veronese is at one with a long and distinguished line of thinkers (including both Aristotle and Kandt and, more recently, Charles S. Peirce, Hermann Weyl, Brouwer, and Rene Thom). However, while Veronese was able to back up his second critique - against the unquestioning acceptance of the Archimedean axiom - by a mathematical theory, a non-Archimedean geometry, he was unable to give mathematical form to his intuition of the continuum as made up of intervals rather than points. He therefore found himself forced to concede that while '[t]he [intuitive] rectilinear continuum is never composed of its points but of segments', nevertheless: 'mathematically . . . a determination of the continuum by means of a well-defined ordered system of points is sufficient' (pp. 142-3). Now, synthetic differential geometry has succeded where previous attempts to construct a really smooth non-punctuate continuum had failed.

Ehrlich does mention, in passing, a work on this subject (I. Moerdijk and G. Reyes, Models for Smooth Infinitesimal Analysis, Springer-Verlag, 1991) but has not included anything about it in the present collection. In my view, at least a short piece (something like John L. Bell's paper 'Infinitesimals and the Continuum' in The Mathematical Intelligencer, 17, 1995, pp. 55-7) would have been valuable, and well in line with the Veronesean theme of the book.

Finally, a couple of minor gripes. First, there are many misprints. For the most part, they are trivial irritants but in a few places it seems that pieces of text have got lost, impeding comprehension. Second, the editor could in my view have done more to enhance the formal unity of the collection, for example by imposing a unified system of references and reference lists. In one place (fn. 4 on p. 182) the editor gives detailed bibliographic references to a certain paper by Poincare. Amazingly, he omits to mention that the paper in question appears in this very book.
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