The debate on scientific realism has raged among philosophers of science for decades. The scientific realist's claim that science aims to give us a literally true description of the way things are, has come under severe scrutiny and attack by Bas van Fraassen's constructive empiricism. All science aims at is to save the observable phenomena, according to van Fraassen. Scientific realists have faced since a main sceptical challenge: the burden is on them to prove that the entities postulated by our scientific theories are real and that science is still in the ‘truth’ business.

One of the hallmarks of Kantian philosophy, especially in connection with its characterization of scientific knowledge, is the importance of unity, a theme that is also the driving force behind a good deal of contemporary high energy physics. There are a variety of ways that unity figures in modern science—there is unity of method where the same kinds of mathematical techniques are used in different sciences, like physics and biology; the search for unified theories like the unification of electromagnetism and optics by Maxwell; and, more recently, the project of grand unification or the quest for a theory of everything which involves a reduction of the four fundamental forces (gravity, electromagnetism, weak and strong) under the umbrella of a single theory. In this latter case it is thought that when energies are high enough, the forces (interactions), while very different in strength, range and the types of particles on which they act, become one and the same force. The fact that these interactions are known to have many underlying mathematical features in common suggests that they can all be described by a unified field theory. Such a theory describes elementary particles in terms of force fields which further unifies all the interactions by treating particles and interactions in a technically and conceptually similar way. It is this theoretical framework that allows for the prediction that measurements made at a certain energy level will supposedly indicate that there is only one type of force. In other words, not only is there an ontological reduction of the forces themselves but the mathematical framework used to describe the fields associated with these forces facilitates their description in a unified theory. Specific types of symmetries serve an important function in establishing these kinds of unity, not only in the construction of quantum field theories but also in the classification of particles; classifications that can lead to new predictions and new ways of understanding properties like quantum numbers. Hence, in order to address issues about unification and reduction in contemporary physics we must also address the way that symmetries facilitate these processes.

Eugene Wigner's several general discussions of symmetry and invariance principles are among the canonical texts of contemporary philosophy of physics. Wigner spoke from a position of authority, having pioneered (and won the Nobel prize in 1963) for recognition of the importance of symmetry principles from nuclear to molecular physics. But perhaps recent commentators have not sufficiently stressed that Wigner always took care to situate the notion of invariance principles with respect to two others, initial conditions (or events) and laws of nature. Wigner's first such general consideration of invariance principles, an address presented at Einstein's 70th birthday celebration, held in Princeton on 19 March 1949, began by laying out just this distinction, and in a way that seems to suggest that the three notions arise through abstraction in an analysis of the general problem of cognition in the natural sciences:

The subject of this paper is objectivity from Kant's point of view: or better, my own perspective on Kant's perspective on objectivity. More precisely, I want to draw attention to some aspects of the latter, which I believe are too narrow and must be widened before we can benefit from a Kantian approach today.

Kant's original version of transcendental philosophy took both Euclidean geometry and the Newtonian laws of motion to be synthetic a priori constitutive principles—which, from Kant's point of view, function as necessary presuppositions for applying our fundamental concepts of space, time, matter, and motion to our sensible experience of the natural world. Although Kant had very good reasons to view the principles in question as having such a constitutively a priori role, we now know, in the wake of Einstein's work, that they are not in fact a priori in the stronger sense of being fixed necessary conditions for all human experience in general, eternally valid once and for all. And it is for precisely this reason that Kant's original version of transcendental philosophy must now be either rejected entirely or (at least) radically reconceived. Most philosophy of science since Einstein has taken the former route: the dominant view in logical empiricism, for example, was that the Kantian synthetic a priori had to be rejected once and for all in the light of the general theory of relativity.

There is evidence in Kant of the idea that concepts of particular numbers, such as the number 5, are derived from the representation of units, and in particular pure units, that is, units that are qualitatively indistinguishable. Frege, in contrast, rejects any attempt to derive concepts of number from the representation of units. In the Foundations of Arithmetic, he softens up his reader for his groundbreaking and unintuitive analysis of number by attacking alternative views, and he devotes the majority of this attack to the units view, with particular attention to pure units. Since Frege, the units view has been all but abandoned. Nevertheless, the idea that concepts of number are derived from the representation of units has a long history, beginning with the ancient Greeks, and was prevalent among Frege's contemporaries. I am not interested in resurrecting the units view or in righting wrongs in Frege's criticisms of his contemporaries. Rather, I am interested in the program of deriving concepts of number from pure units and its history from Kant to Frege. An examination of that history helps us understand the units view in a way that Frege's criticisms do not, and in the process uncovers important features of both Kant's and Frege's views. I will argue that, although they had deep differences, Kant and Frege share assumptions about what such a view would require and about the limits of conceptual representation. I will also argue that they would have rejected the accounts given by some of Frege's contemporaries for the same reasons. Despite these agreements, however, there is evidence that Kant thinks that space and time play a role in overcoming the limitations of conceptual representation, while Frege argues that they do not.

Kant says patently conflicting things about infinity and our grasp of it. Infinite space is a good case in point. In his solution to the First Antinomy, he denies that we can grasp the spatial universe as infinite, and therefore that this universe can be infinite; while in the Aesthetic he says just the opposite: ‘Space is represented as a given infinite magnitude’(A25/B39). And he rests these upon consistently opposite grounds. In the Antinomy we are told that we can have no intuitive grasp of an infinite space, and in the Aesthetic he says that our grasp of infinite space is precisely intuitive.