Free The King of Infinite Space by David Berlinski Page B

their own, their sovereign, identity. The positive integers, zero, the negative integers, the fractions, and the real numbers were all in place. They had acquired an indubitable existence in the minds of mathematicians. The system had a kind of abstract integrity. It held together under scrutiny. It was not adventitious.
    T HE REAL NUMBER system represented the confluence of two triumphs: the triumph of arithmetic and the triumph of algebra. The triumph of arithmetic is obvious. The real number system is a system of real numbers . The triumph of algebra, less so. The real numbers satisfy the axioms for an identifiable algebraic structure, what mathematicians call a field . The great achievement of nineteenth- and early twentieth-century mathematics was by a python-like compression of concepts to detach the structure from its examples. Writing in 1910, the German mathematician ErnstSteinitz proposed to make use of fields in an “ abstrakten und algmeinen Weise ”—in an abstract and general sense. A field, he wrote, is a system of elements with two operations: addition and multiplication. That is all that it is. Steinitz then introduced the distinctively new, entirely modern note, the one that marks a decisive promotion of an interesting idea into an independent idea. Never mind the question, the field of what ? The abstract concept of a field is itself at the mittelpunkt of his interests. The examples dwindle away and disappear. The field remains. It becomes itself. 1
    T HE AXIOMS FOR a field bind its various far-flung properties together. 2 Their exposition calls to mind the lawyers in Bleak House rising to make a point.
    â€”A field is a set of elements, M’Lud . . .
    â€”Elements, M’lud , anything really.
    â€”Feel it my duty to add, M’lud , that there are two operations on these elements . . .
    â€”Beg pardon? Any two distinct operations, M’lud .
    â€”Feel it my duty to add that there is 0 somewhere, M’lud . Yes, here it is.
    â€”Do? It does nothing M’lud: a + 0 is always a .
    â€”There is a 1, too, M’lud . Yes, I have it here. Beg pardon? Nothing. It does nothing M’lud : 1 a is always a .
    â€”Feel it my duty to add a word about inverses, M’lud . I have them here.
    â€”Beg pardon? Do? They invert, M’lud . Any element plus its inverse is 0, and any element times its inverse is 1.
    There is no need to pursue this particular courtroom drama beyond the judge’s demand that his attorneys sit down. A field is an abstract object, and so above it all. Still, it is an abstract object whose most compelling example is the ordinary real numbers. An associative law holds force: a + ( b + c ) = ( a + b ) + c . And so does a distributive law: a ( b + c ) = ab + ac . Identities in 0 and 1, and inverses in the negative numbers and fractions, make possible the recovery of subtraction and division. It is, as lawyers say, familiar fare. A last lawyer rises to remind the judge that the real numbers are ordered. It is always one number before the other, or after the other; it is always, as the judge mutters, one thing or another.
    No matter the lawyers, this idea has been a triumph, the second, after the definition of the real numbers themselves. This prompts the obvious question: a triumph over what?
    I N 1899, D AVID Hilbert published a slender treatise titled Grundlagen der Geometrie (The foundations of geometry). Having for many years lost himself in abstractions, a great mathematician had chosen to revisit his roots. Over the next thirty years, Hilbert would revise his book, changing its emphasis slightly, fiddling, never perfectly satisfied. The Grundlagen— the German word has an earthiness lacking in English—is a moving book, at once a gesture of historical respect and an achievement in self-consciousness. In writing about Euclidean geometry, Hilbert was sensitive to the anxieties running through nineteenth-century thought. Well hidden beneath