Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient fascination arises from the (...) sense that mathematics explores something ‘out there’. This is illustrated by recent discussions by distinguished contemporary mathematicians. The Enlightenment strand often uses Kant's argot: ‘absolute necessity’, ‘apodictic certainty’ and ‘a priori’ judgement or knowledge. The experience of being compelled by proof, the sense that something must be true, that a result is certain, generates the philosophy. It also creates the illusion that mathematics is certain. Kant's leading question, ‘How is pure mathematics possible?’, is easily misunderstood because the modern distinction between pure and applied is an artefact of the 19th century. As Russell put it, the issue is to explain ‘the apparent power of anticipating facts about things of which we have no experience’. More generally the question is, how is it that pure mathematics is so rich in applications? Some six types of application are distinguished, each of which engenders its own philosophical problems which are descendants of the Enlightenment, and which differ from those descended from the Ancient strand. (shrink)

Ernest Nagel Lecture, Columbia University, Sept. 27, 2007.

Consciousness is a mongrel concept: there are a number of very different "consciousnesses." Phenomenal consciousness is experience; the phenomenally conscious aspect of a state is what it is like to be in that state. The mark of access-consciousness, by contrast, is availability for use in reasoning and rationally guiding speech and action. These concepts are often partly or totally conflated, with bad results. This target article uses as an example a form of reasoning about a function of "consciousness" based on (...) the phenomenon of blindsight. Some information about stimuli in the blind field is represented in the brains of blindsight patients, as shown by their correct "guesses," but they cannot harness this information in the service of action, and this is said to show that a function of phenomenal consciousness is somehow to enable information represented in the brain to guide action. But stimuli in the blind field are BOTH access-unconscious and phenomenally unconscious. The fallacy is: an obvious function of the machinery of access-consciousness is illicitly transferred to phenomenal consciousness. (shrink)

Presenting a look at the human mind's capacity while criticizing artificial intelligence, the author makes suggestions about classical and quantum physics and ..

In his book Shadows of the Mind: A search for the missing science of con- sciousness [SM below], Roger Penrose has turned in another bravura perfor- mance, the kind we have come to expect ever since The Emperor’s New Mind [ENM ] appeared. In the service of advancing his deep convictions and daring conjectures about the nature of human thought and consciousness, Penrose has once more drawn a wide swath through such topics as logic, computa- tion, artificial intelligence, quantum physics (...) and the neuro-physiology of the brain, and has produced along the way many gems of exposition of difficult mathematical and scientific ideas, without condescension, yet which should be broadly appealing. 1 While the aims and a number of the topics in SM are the same as in ENM, the focus now is much more on the two axes that Pen- rose grinds in earnest. Namely, in the first part of SM he argues anew and at great length against computational models of the mind and more specifi- cally against any account of mathematical thought in computational terms. Then in the second part, he argues that there must be a scientific account of consciousness but that will require a (still to be found) non-computational extension or modification of present-day quantum physics. (shrink)

In his stimulating book SHADOWS OF THE MIND, Roger Penrose presents arguments, based on Gödel's theorem, for the conclusion that human thought is uncomputable. There are actually two separate arguments in Penrose's book. The second has been widely ignored, but seems to me to be much more interesting and novel than the first. I will address both forms of the argument in some detail. Toward the end, I will also comment on Penrose's proposals for a "new science of consciousness".

Paradoxically, explorers of the territory of consciousness seem to be studying consciousness out of existence, from inside the field of "consciousness studies". How? Through their love of the phenomenon/process, they have developed powerful single models or lenses through which to understand consciousness. But in doing so, they also seek to destroy the other /equally useful/ lenses. Our opportunity lies in halting the vendettas and cross-speakings/cross-fire. The imploration is to stop the dichotomous thinking and pernicious reification of single models, and instead (...) search for divisions of labor, complementarities, and legitimate redescriptions among the various extant models. In other words, what would happen if we reimagined the conceptual classifications of the various models of consciousness, classifications based on general philosophical dichotomies (e.g., representational/non-representational and individualist/non-individualist), as a variety of compatible and even complementary perspectives on the same complex phenomenon and process? What would happen if rather than dig in our heels vis-à-vis our favorite theory of consciousness, at the exclusion of all the others, we saw our perceived enemy as an actual, indeed necessary, friend-in-waiting? What would it take to see a battlefield as a collaborative opportunity, to see a promising pluralism rather than an endless state space of conflict? (shrink)