Takeuti's proof theory in the context of the Kyoto School

Gaisi Takeuti (1926–2017) is one of the most distinguished logicians in proof theory after Hilbert and Gentzen. He extensively extended Hilbert's program in the sense that he formulated Gentzen's sequent calculus, conjectured that cut-elimination holds for it (Takeuti's conjecture), and obtained several stunning results in the 1950–60s towards the solution of his conjecture. Though he has been known chiefly as a great mathematician, he wrote many papers in English and Japanese where he expressed his philosophical thoughts. In particular, he used several keywords such as "active intuition" and "self-reflection" from Nishida's philosophy. In this paper, we aim to describe a general outline of our project to investigate Takeuti's philosophy of mathematics. In particular, after reviewing Takeuti's proof-theoretic results briefly, we describe some key elements in Takeuti's texts. By explaining these texts, we point out the connection between Takeuti's proof theory and Nishida's philosophy and explain the future goals of our project

On the relationship between plane and solid geometry

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned area

Imagination in mathematics

This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras

Idéaux de preuve : explication et pureté

Why do mathematics often give several proofs of the same theorem? This is the question raised in this article, introducing the notion of an epistemic ideal and discussing two such ideals, the explanatoriness and purity of proof

Spiking LCA in a Neural Circuit with Dictionary Learning and Synaptic Normalization

The Locally Competitive Algorithm (LCA) [17, 18] was put forward as a model of primary visual cortex [14, 17] and has been used extensively as a sparse coding algorithm for multivariate data. LCA has seen implementations on neuromorphic processors, including IBM’s TrueNorth processor [10], and Intel’s neuromorphic research processor, Loihi, which show that it can be very efficient with respect to the power resources it consumes [8]. When combined with dictionary learning [13], the LCA algorithm encounters synaptic instability [24], where, as a synapse’s strength grows, its activity increases, further enhancing synaptic strength, leading to a runaway condition, where synapses become saturated [3, 15]. A number of approaches have been suggested to stabilize this phenomenon [1, 2, 5, 7, 12]. Previous work demonstrated that, by extending the cost function used to generate LCA updates, synaptic normalization could be achieved, eliminating synaptic runaway [7]. It was also shown that the resulting algorithm could be implemented in a firing rate model [7]. Here, we implement a probabilistic approximation to this firing rate model as a spiking LCA algorithm that includes dictionary learning and synaptic normalization. The algorithm is based on a synfire-gated synfire chain-based information control network in concert with Hebbian synapses [16, 19]. We show that this algorithm results in correct classification on numeric data taken from the MNIST datase

On the alleged simplicity of impure proof

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim

Perinatal paracetamol exposure in mice does not affect the development of allergic airways disease in early life

Background Current data concerning maternal paracetamol intake during pregnancy, or intake during infancy and risk of wheezing or asthma in childhood is inconclusive based on epidemiological studies. We have investigated whether there is a causal link between maternal paracetamol intake during pregnancy and lactation and the development of house dust mite (HDM) induced allergic airways disease (AAD) in offspring using a neonatal mouse model. Methods Pregnant mice were administered paracetamol or saline by oral gavage from the day of mating throughout pregnancy and/or lactation. Subsequently, their pups were exposed to intranasal HDM or saline from day 3 of life for up to 6 weeks. Assessments of airway hyper-responsiveness, inflammation and remodelling were made at weaning (3 weeks) and 6 weeks of age. Results Maternal paracetamol exposure either during pregnancy and/or lactation did not affect development of AAD in offspring at weaning or at 6 weeks. There were no effects of maternal paracetamol at any time point on airway remodelling or IgE levels. Conclusions Maternal paracetamol did not enhance HDM induced AAD in offspring. Our mechanistic data do not support the hypothesis that prenatal paracetamol exposure increases the risk of childhood asthma