Review. "Square One: the Foundations of Knowledge" by Steve Patterson

2013

Mind, 2023

Mathematical Intelligencer, 2005

Poincaré famously compared the logician's understanding of mathematics to the understanding we would have of chess if we were only to know its rules. "To understand the game," Poincaré wrote, "is wholly another matter; it is to know why the player moves this piece rather than that other which he could have moved without breaking the rules of the game. It is to perceive the inward reason which makes of this series of moves a sort of organized whole." [P, pp. 217-218] The Dutch mathematician L.E.J. Brouwer took a position similar to Poincaré's: genuinely mathematical reasoning is not simply a matter of logical inference. It is, as Poincaré put it, a matter of mathematical insight.

Topoi, 2009

Philosophy is facing a serious crisis, but no one cares. When German Idealism, Existentialism, and Marxism allied with Sociology, Psychoanalysis, Cultural History, and Literature Studies in the early 20th century, all attempts at conducting philosophy in a style similar to that of the scientists got expelled from the High Church of Philosophy. The creation of the Wykeham Professorship in Hermeneutics (formerly: Logic) at Oxford and the Stanford Chair of Textual Non-Presence (formerly: Methodology of Science) are well-known indicators of these, by now, historical developments. The best philosophical work since then is to be found in the history of philosophy-if one is lucky. One cannot help but wondering what turn philosophy would have taken if someone had picked up the revolutionary developments in logic and mathematics in the 1920s and directed them towards philosophy. Maybe there would still be logic courses in philosophy departments? Who knows? Recently a monograph has appeared which seems to ignore the respiratory epicycles of this modern philosophical slumber completely (and rightly so). Rudolf Carnap, a young German philosopher who is working at the University of Vienna, has the chutzpah to conceive a logischer Aufbau der Welt-a logical structure of the world-as if nothing had happened in philosophy in the last eight years or so. Well, not much has happened indeed. If only a book like that had been published back then! Carnap's main aim is to argue for the following thesis: Give him any meaningful sentence from natural language or science; then this sentence can be reformulated in a precisely delineated conceptual framework of primitive expressions, exact definitions, and logical and mathematical rules. In fact, there are several possible choices for such frameworks, each one serving a different purpose, but what all of them have in common is that the hidden logical structure of the original sentence will become completely transparent through reformulation. That the

The Philosophical Quarterly, 1970

2008

Critical Views of Logic, 2023

This volume explores what we call "critical views of logic". Following Frege, logic is often regarded as epistemologically and methodologically fundamental. All disciplines-including mathematics-are answerable to logic rather than vice versa. Critical views of logic disagree with this "logic-first" view. The logical principles that govern some subject matter may depend on the metaphysics of this subject matter or on the semantics of our discourse about it. Challenging the logic-first view According to Frege, logic codifies "the basic laws" of all rational thought, and the laws of logic must therefore be presupposed by all other sciences. What, then, might justify a law of logic? We could of course consult logic itself: As to the question, why and with what right we acknowledge a logical law to be true, logic can respond only by reducing it to other logical laws. Where this is not possible, it can give no answer. (Frege 1893 (2013), p. xvii) But this will not take us very far, for logic too will need some fundamental laws. Frege therefore continues by asking whether there are extralogical considerations to which we can appeal: Stepping outside logic, one can say: our nature and external circumstances force us to judge, and when we judge we cannot discard this law-of identity, for example-but have to acknowledge it if we do not want to lead our thinking into confusion and in the end abandon judgement altogether. I neither want to dispute nor to endorse this opinion, but merely note that what we have here is not a logical conclusion. What is offered here is not a ground of being true but of our taking to be true. (ibid.) Thus, there is no help to be had from extralogical considerations either. Whereas in the Grundlagen (Frege 1884 (1974)) Frege seemed attracted to the idea that logic is constitutive of our thinking or judging, he is now unwilling to endorse this as a reason for the truth of the laws of logic, seeing it only as a reason for our taking the laws of logic to be true. It will also not help to look to other sciences:

Teaching Philosophy, 2013

This book is part of The Beginners Guide series: the dedication reads: "Dedicated to those who do not know-including-those who do not know they do not know." We are tentatively placed on Socratic ground. In the prologue, we are admonished to take our time. The philosopher should be a child of wonder and, "is not a citizen of any community of ideas. That is what makes him a philosopher." Those are words firom the Cambridge philosopher Ludwig Wittgenstein, arguably the greatest philosopher of the twentieth century (xi-xii). The Platonic ground shifts. We are reminded that philosophy is not poetry, fiction, and religion, and that it should be held to the grindstones of arguments and clarity and to the uncovering of assumptions (xiii). How is it possible to be human and to avoid a community of ideas? What an idea. Cave admits a fondness for reasoning and argument, the focus of the Anglo-analytic tradition and for Wittgenstein as the greatest twentieth century thinker. Wittgenstein moved, however, from his attachment of meaning and truth to propositions in the Tractatus, to an interest in puzzles, in language games, and in thought experiments, to matters of how meanings/words are used in the Philosophical Investigations. This is Cave's method with some modification. The question of Wittgenstein's final position is, I believe, still out; was he a mystic trying to say the unsayable or had he just settled in to puzzling and promulgating language games? These questions aside, this book would likely fill the expectations of those non-mystics of the Anglo-analytic persuasion, which still dominate the current philosophical world, for good or for ill, but, in any case, which clearly constitute "a community of ideas." The analytic movement and its Wittgensteitiian progeny, in faimess, do not seem to offer a dogma or a party line beyond the dizziness accompanying, initially, the limitation of meaning to propositional language that is either tme or false and then to opting for methods of verification if propositions are found wanting. Thought experiments and collections of family resemblances in language, both ordinary and extraordinary in a reach for meaning, may then be hauled in to extend meaning. Cave's intentions are modest. Instead of providing a laborious analysis of terms and theories, his approach is "to provide a fiavor of the problems, a flavor that will stimulate thought and encourage further reading" (xiv). Cave begins chapter one, "What is it to be human?," by appealing to the reader's capacity to doubt, to confuse dreams with life or life with dreams, and to recall betrayal. How do we know that we are not dreaming, that we are

Philosophical Books, 2006