Abstract. We compare fifteen systems for the formalizations of mathematics with the computer. We present several tables that list various properties of these programs. The three main dimensions on which we compare these systems are: the size of their library, the strength of their logic and their level of automation. 1

Formal proofs in mathematics and computer science are being studied because these objects can be verified by a very simple computer program. An important open problem is whether these formal proofs can be generated with an effort not much greater than writing a mathematical paper in, say, L A T E X. Modern systems for proof-development make the formalization of reasoning relatively easy. Formalizing computations such that the results can be used in formal proofs is not immediate. In this paper it is shown how to obtain formal proofs of statements like Prime(61) in the context of Peano arithmetic or (x + 1)(x + 1) = x 2 + 2x + 1 in the context of rings. It is hoped that the method will help bridge the gap between the efficient systems of computer algebra and the reliable systems of proof-development. 1. The problem Usual mathematics is informal but precise. One speaks about informal rigor. Formal mathematics on the other hand consists of definitions, statements and proo...

What is the enactive approach to cognition? Over the last 15 years this banner has grown to become a respectable alternative to traditional frameworks in cognitive science. It is at the same time a label with different interpretations and upon which different doubts have been cast. This paper elaborates on the core ideas that define the enactive approach and their implications: autonomy, sensemaking, emergence, embodiment, and experience. These are coherent, radical and very powerful concepts that establish clear methodological guidelines for research. The paper also looks at the problems that arise from taking these ideas seriously. The enactive approach has plenty of room for elaboration in many different areas and many challenges to respond to. In particular, we concentrate on the problems surrounding several theories of value-appraisal and valuegeneration. The enactive view takes the task of understanding meaning and value very seriously and elaborates a proper scientific alternative to reductionist attempts to tackle these issues by functional localization. Another area where the enactive framework can make a significant contribution is social interaction and

There are many examples in the literature that suggest that indistinguishability is intransitive, despite the fact that the indistinguishability relation is typically taken to be an equivalence relation (and thus transitive). It is shown that if the uncertainty perception and the question of when an agent reports that two things are indistinguishable are both carefully modeled, the problems disappear, and indistinguishability can indeed be taken to be an equivalence relation. Moreover, this model also suggests a logic of vagueness that seems to solve many of the problems related to vagueness discussed in the philosophical literature. In particular, it is shown here how the logic can handle the sorites paradox. 1

this paper we will make a similar attempt but now concentrating on the logical aspects of the example. 1. CLASSICAL ENTITIES AND QUANTUM ENTITIES. In the discipline quantum logic the basic structure of research was originally the structure of orthomodular lattice. Quantum-like entities had such a structure of orthomodular lattice for the set of their propositions, while classical entities had a structure of distributive orthomodular lattice (or Boolean algebra) for the set of their propositions. Meanwhile the classification classical entities versus quantum-like (or non-classical entities) has been studied much more in detail, relating the corresponding structures to real physical situations [2]. We have taken a very easy criterion from the results of this research on the possibility of distinguishing between the two kind of entities classical and non-classical. We shall consider an entity (classical or non-classical) to be described by a set M (denoted by m, n, ...) of measurements and a set ## # # ( denoted by p, q, ...) of states. In different approaches different names have been given to these two basis sets (measurements have been called yes-no experiments, questions, propositions, observables and operations, and states have also been called preparations), but mainly these differences will play no role in what we would like to show in this paper. The results can easily be translated in the proper approach. We must remark that when we use the concept 'state', then we mean 'pure state'. The more general situation of mixed states will not be considered here, since our example in any case does not contain mixed states. The following characterization of a classical entity shall be adopted (it is the characterization explicitly used in [3]) : Criterion for classical cha...

It is argued that the natural kinematics of the pilot-wave theory is Aristotelian. Despite appearances, Galilean invariance is not a fundamental symmetry of the low-energy theory. Instead, it is a fictitious symmetry that has been artificially imposed. It is concluded that the search for a Lorentz-invariant

tremendous hustle and bustle: merchants were rushing in and out, groups of people stood talking in every corner, a small orchestra was playing charming melodies, and a plentiful table had been laid, overflowing with all the most delicious dishes from that part of the world. The wise man was talking to everyone and the boy had to wait for two hours before it was his turn. The wise man listened attentively to the reason for his coming, but told him that unfortunately he did not have time at that moment to explain the secret of happiness to him. He suggested he go for a twohour walk around the castle and to come back afterwards. "I would however like to ask you something," said the wise man in conclusion, as he handed the boy a teaspoon from which hung two drops of oil. "I would like to ask you, while you are walking, to hold this spoon so that the drops of oil do not drip o#." The boy began to walk up and down the stairs of the palace with his eyes fixed rigidly on the sp

Studies using the sensory substitution devices designed for visually impaired persons reveal that perceptive activity itself is embodied in a lived body capable of movement and possessing its own spatial dimensions. We have used these prosthetic devices, based on the substitution of the visual sensory input by a tactile sensory input, in order to carry out a systematic study of perception, and in particular of spatial depth. We show that the spatial localisation of a target requires dynamic sensori-motor coupling, and that this activity involves the spatial dimensions of the lived body of the perceiving subject.

. The representation of indiscernibility phenomena by ß-equivalences and of accessibility phenomena by oe-equivalences enables a graph-theoretical formulation of topological notions in the alternative set theory. Generalizing the notion of Hamiltonian graph we will introduce the notion of Hamiltonian embedding and prove that for any finite graph without isolated vertices there is a Hamiltonian embedding into any infinite set connected with respect to some ß- or oe-equivalence. Roughly speaking, in some sense this means that each such an infinite connected set, (in particular, each connected set in a complete metrizable topological space), contains each finite graph inside, and even is exhausted by the images of its edges. Moreover, the main Theorem 3, dealing with the so called deeply connected sets, is in fact a theorem of nonstandard arithmetic. 1. A Brief Outline of Alternative Set Theory The alternative set theory (AST), similarly as nonstandard analysis, enables to treat the in...

Quantum mechanics is the theory used to 'describe' the processes that take place in the micro-world. From the start quantum mechanics has been a 'strange' theory, in the sense that it seemed to contradict in various ways the image of a micro-world consisting of 'objects' moving around in a three dimensional space, and interacting with each other in this three dimensional space. So from the advent of the theory a lot of disagreement existed as to the 'physical meaning' of this quantum theory, and a lot of discussions of a philosophical nature have taken place among the founding fathers. Only however during the last years experiments have been performed that, independently of the strangeness of the quantum theory, confront us directly with the strangeness of the reality of the microworld. We have in mind the experiments on the EPR problem.