Gaisi Takeuti extended Gentzen's work to higher-order case in 1950's–1960's and proved the consistency of impredicative subsystems of analysis. He has been chiefly known as a successor of Hilbert's school, but we pointed out in the previous paper that Takeuti's aimed to investigate the relationships between "minds" by carrying out his proof-theoretic project rather than proving the "reliability" of such impredicative subsystems of analysis. Moreover, as briefly explained there, his philosophical ideas can be traced back to Nishida's philosophy in Kyoto's (...) school. For the proving the consistency of such systems, it is crucial to prove the well-foundedness of ordinals called "ordinal diagrams" developed for it. Takeuti presented such arguments several times in order to show that they are admitted in his stand point. As a starting point of investigating his finitist stand point, we formulate the system of ordinal notations up to ε0 and reconstruct the well-foundedness arguments of them. (shrink)

Can static pictures depict motion and temporal properties? This is an open question that is becoming increasingly discussed in both aesthetics and the philosophy of mind. Theorists working on this issue have mainly focused on static pictures of dynamic scenes and streaky images – such as futurists’ paintings or long-exposure photographs. And yet, we could ask: if there is some success in creating an illusory impression of movement in a static image - as is the case in optical illusions of (...) movement, such as Bridget Riley’s Fall (1963: Tate, London), or Kitaoka’s Rotating Snakes Illusion - is this to say that such static images depict movement? As far as I know, no one working on the depiction of motion has specifically and systematically tackled motion-based illusions nor tried to answer this question. This paper considers two cases of optical illusions of movement and concludes that one of them is involved in the depiction of movement. While the declared goal of my analysis is to answer a quite circumscribed question, it is also the occasion to tackle motion-based illusions tout court - to account for the complex visual experiences they elicit and related phenomenology. Moreover, this account has interesting consequences for theorizing depiction and pictorial experience in general. In particular, it constitutes a counterexample to resemblance theories of depiction. (shrink)