Manuel Bremer

I consider here several versions of finitism or conceptions that try to work around postulating sets of infinite size. Restricting oneself to the so-called potential infinite seems to rest either on temporal readings of infinity (or infinite series) or on anti-realistic background assumptions. Both these motivations may be considered problematic. Quine’s virtual set theory points out where strong assumptions of infinity enter into number theory, but is implicitly committed to infinity anyway. The approaches centring on the indefinitely large and the use of schemata would provide a work-around to circumvent usage of actual infinities if we had a clear understanding of how schemata work and where to draw the conceptual line between the indefinitely large and the infinite. Neither of this seems to be clear enough. Versions of strict finitism in contrast provide a clear picture of a (realistic) finite number theory. One can recapture standard arithmetic without being committed to actual infinities. The major problem of them is their usage of a paraconsistent logic with an accompanying theory of inconsistent objects. If we are, however, already using a paraconsistent approach for other reasons (in semantics, epistemology or set theory), we get finitism for free. This strengthens the case for paraconsistency.

The book aims to set out in which respects concepts are properly studied in philosophy, what methodological role the study of concepts has in philoso-phy's study of the world. Many of the considerations in this book nowadays are placed under the headline ‘metaphilosophy’. In contrast to paradigmatic ordinary language philosophy the book endorses a representationalist theory of meaning and concepts, thus agreeing with many of its critics in philosophy and the cognitive sciences. In contrast to many of these critics and supposedly the majority of cognitive scientists it endorses the viability of conceptual analysis as one method of philosophy. The book reflects on Frege's theory of concepts, because Frege's theory of sentential unity has barely been super-seded, and the problems arising from Frege's understanding of concepts are still alive. The central part of the book starts by reconsidering the approach and the idea of ordinary language philosophy. Although ordinary language philosophy cannot be the whole of analytic philosophy a proper understanding of conceptual analysis turns out to be one part of analytic philosophy.

In our dealings with our pets, and larger animals in general, at least most of us see them as conscious beings. We say “the cat feels pain” ascribing sensation. We notice “My cat wants to get in the kitchen because she thinks there is some cheese left” ascribing beliefs and desires. Explanations likes these can be employed on a variety of occasions, and usually we are content with what they say. We seem to understand why our cat is doing what she does. On the other hand the employment of human categories to animals seems to be problematic. Reflecting on the details of human beliefs, for example, casts serious doubt on whether the cat is able to believe anything at all. Clever as they are none of my cats has the concept of kitchen, because a kitchen is – roughly – a functional part of an artificial dwelling (a house). Since cats do not built houses and do not prepare their food, the place the cat is walking into cannot be conceptualized by the cat as kitchen. Even the concept of cheese in its generality (made out of goat or cow milk, or camel milk …) is beyond the cat. So when we say what our cat believes and wishes we use our concept, in fact express a belief that a human being might have in a similar situation. What, then, does the cat have? Does a cat have beliefs? Or at least something like beliefs?