The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview
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ematics matter. He therefore sought its origins in the deepest recesses of the human cognitive soul. The view, I believe, can be roughly described as follows. It is one thing for a proposition to be plausible or what we might call merely believable, and quite another thing for it to be incorporatable into a set of epistemic cares and concerns around which a human cognitive life might be centred. This is related to the question of epistemic agency; that is, the question of when a belief becomes the belief of an agent. The answer, I think, is that a belief becomes a belief of a given agent when but only when it comes to occupy a place of centrality in her human epistemic life--when, that is, it is either linked to the deepest parts of her individuality, or when it is seen as belonging to that common set of cognitive matters that are determinative of human epistemic life generally and which thus make one's epistemic life a distinctively human epistemic life.

I believe that Brouwer thought of things in this way. He was a kind of mathematical 'existentialist' in that he sought a conception of mathematical knowledge that makes it basic to our human existence. For this to be so, he believed, mathematical knowledge must be intimately related to our most basic knowledge of ourselves--that is, to that 'existential' awareness that we have of ourselves as willing, acting beings. Such views are, to be sure, extraordinary when viewed in the light cast by the bland and vapid views that have dominated the philosophy of mathematics of the past few decades. They are not, however, for all that, either silly or out of place.

These remarks, of course, only skip across the surface of what are very deep waters. Still, I hope, they may serve to indicate in general terms where I see the deepest differences between finitist and intuitionist epistemology and, in particular, between their respective conceptions of the exhibition condition as lying. They connect, in the end, I believe, with matters as basic as those which concern the relation of the individual to society and our private to our public selves.45


Brouwer presented some of his arguments in favour of rejecting spatial intuition (loc. cit., 63-4) as part of a critique of Russell's views of space. Contrary to Russell, he claimed that space is but the 'form of externality' in our sense experience and need not be admitted as a necessary condition of that experience: 'empirical space is an arbitrary creation to enable us . . . with the aid of mathematical induction to

A more detailed discussion of some of the issues touched on in the concluding section may be found in the author's 'Peace, Justice and Computability: Leibniz' Program and Church's Theorem'.


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