We now know of a number of ways of developing real analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in his "Reals by Abstraction" program differs by placing additional emphasis upon what I here term Frege's Constraint, that a satisfactory foundation for any branch of mathematics should somehow so explain its basic concepts that their applications are immediate. This paper is concerned with the meaning of and motivation for this constraint. Structuralism has to represent the application of a mathematical theory as always posterior to the understanding of it, turning upon the appreciation of structural affinities between the structure it concerns and a domain to which it is to be applied. There is, therefore, a case that Frege's Constraint has bite whenever there is a standing body of informal mathematical knowledge grounded in direct reflection upon sample, or schematic, applications of the concepts of the theory in question. It is argued that this condition is satisfied by simple arithmetic and geometry, but that in view of the gap between its basic concepts (of continuity and of the nature of the distinctions among the individual reals) and their empirical applications, it is doubtful that Frege's Constraint should be imposed on a neo-Fregean construction of analysis.

This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume's Principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers:

$(\mathrm{DIF})\qquad \mathrm{INT}(a,b) = \mathrm{INT}(c,d)\equiv(a+d)=(b+c).$

$(\mathrm{QUOT}) \qquad \begin{eqnarray}\mathrm{Q}(m,n)=\mathrm{Q}(p,q)\equiv(n=0 \: \& \: q=0) \\ \qquad \vee\: (n\neq 0 \: \& \: q\neq 0 \: \& \: m \cdot q=n \cdot p).\end{eqnarray}$

The development of the real numbers is an adaption of the Dedekind program involving "cuts" of rational numbers. Let $P$ be a property (of rational numbers) and $r$ a rational number. Say that $r$ is an upper bound of $P$, written $P\leq r$, if for any rational number $s$, if $Ps$ then either $s<r$ or $s=r$. In other words, $P\leq r$ if $r$ is greater than or equal to any rational number that $P$ applies to. Consider the Cut Abstraction Principle:

$(\mathrm{CP}) \qquad \forall P \forall Q(C(P)=C(Q) \equiv \forall r(P\leq r \equiv Q\leq r)).$

In other words, the cut of $P$ is identical to the cut of $Q$ if and only if $P$ and $Q$ share all of their upper bounds. The axioms of second-order real analysis can be derived from (CP), just as the axioms of second-order Peano Arithmetic can be derived from Hume's Principle. The paper raises some of the philosophical issues connected with the neo-Fregean program, using the above abstraction principles as case studies.

The historical antecedents of Frege's treatment of binocular vision in "The thought" were the physiological writings of Johannes Mueller, Hermann von Helmholtz, and Emil du Bois-Reymond. In their research on human vision, logic was assigned an unexpected role: it was to be the means by which knowledge of a world extended in three dimensions arises from stimuli that are at best two-dimensional. An examination of this literature yields a richer understanding of Frege's insistence that a proper epistemology requires us to recognize the existence and importance of nonsensible sources of knowledge.

The neo-Fregean program in the philosophy of mathematics seeks a foundation for a substantial part of mathematics in abstraction principles—for example, Hume's Principle: The number of $F$s $=$ the number of $G$s iff the $F$s and $G$s correspond one-one—which can be regarded as implicitly definitional of fundamental mathematical concepts—for example, cardinal number. This paper considers what kind of abstraction principle might serve as the basis for a neo-Fregean set theory. Following a brief review of the main difficulties confronting the most widely discussed proposal to date—replacing Frege's inconsistent Basic Law V by Boolos's New V which restricts concepts whose extensions obey the principle of extensionality to those which are small in the sense of being smaller than the universe—the paper canvasses an alternative way of implementing the limitation of size idea and explores the kind of restrictions which would be required for it to avoid collapse.

Hume's Principle requires the existence of the finite cardinals and their cardinal, but these are the only cardinals the Principle requires. Were the Principle an analysis of the concept of cardinal number, it would already be peculiar that it requires the existence of any cardinals; an analysis of bachelor is not expected to yield unmarried men. But that it requires the existence of some cardinals, the countable ones, but not others, the uncountable, makes it seem invidious; it is as if an analysis of people required that there be men but not women, or whites but not blacks. If we deprive the Principle of existential commitments, it will cease to yield Dedekind's axioms for the natural numbers and so fail a good test of material adequacy. But since there are cardinals no second-order theory guarantees, neither can the Principle be beefed up to require all cardinals.

A few remarks concerning the history of logicism in the twentieth-century and the prospects for neologicism in the twenty-first.