this paper, it will turn out logicians have universally missed the true, exceedingly simple feature of ordinary first-order logic that makes it incapable of accommodating its own truth predicate. (See Section 4 below.) This defect will also be shown to be easy to overcome without transcending the first-order level. This eliminates once and for all the need of set theory for the purposes of a metatheory of logic.

I argue that Brouwer’s general philosophy cannot account for itself, and, a fortiori, cannot lend justification to mathematical principles derived from it. Thus it cannot ground intuitionism, the job Brouwer had intended it to do. The strategy is to ask whether that philosophy actually allows for the kind of knowledge that such an account of itself would amount to. Brouwer tried to go ‘from philosophy to mathematics ’ and grounded his intuitionistic mathematics in a more general philosophy. 1 This background philosophy can be characterized as a transcendental one. That is, it purports to explain how a non-mundane subject builds up its world in consciousness. It is a radical transcendental philosophy in that this ‘world ’ does not contain just physical objects but everything, including abstract objects and the mundane subject (the subject as part of the world). From the empirical point of view, such a non-mundane subject is an idealized one. Like fellow transcendentalists

Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl’s third claim is wrong, by his own standards. To explain this thesis, let me first introduce the term ‘revisionism’. It is understood here, following Crispin Wright, as the term that applies to ‘any philosophical standpoint which reserves the potential right to sanction or modify pure mathematical practice ’ [Wright 1980, p.117]. I want to make a distinction between weak and strong revisionism. The point of reference is the actual practice of mathematics. Weak revisionism then potentially sanctions a subset of this practice, while strong revisionism potentially not only limits but extends it, in different directions. In strong revisionism, certain combinations of limitation and extension may lead to a mathematics that is no longer compatible with the unrevised one. ‘May lead’, not ‘necessarily leads’: it is all a matter of reserving rights; whether there is occasion to exercise them is a further question. To illustrate these categories, let me give examples of non-revisionism, weak revisionism, and strong revisionism. Non-revisionism can be found in Wittgenstein’s Philosophische Untersuchungen, where philosophy can neither change nor ground mathematics: Die Philosophie darf den tatsächlichen Gebrauch der Sprache in keiner Weise antasten, sie kann ihn am Ende also nur beschreiben. Denn sie kann ihn auch nicht begründen. Sie läßt alles wie es ist. Sie läßt auch die Mathematik wie sie ist, und keine mathematische