Evandro Agazzi

Philosophy of mathematics today

General Philosophical Perspectives.- Logic, Mathematics, Ontology.- From Certainty to Fallibility in Mathematics?.- Moderate Mathematical Fictionism.- Language and Coding-Dependency of Results in Logic and Mathematics.- What is a Profound Result in Mathematics?.- The Hylemorphic Schema in Mathematics.- Foundational Approaches.- Categorical Foundations of the Protean Character of Mathematics.- Category Theory and Structuralism in Mathematics: Syntactical Considerations.- Reflection in Set Theory. The Bernays-Levy Axiom System.- Structuralism and the Concept of Set.- Aspects of Mathematical Experience.- Logicism Revisited in the Propositional Fragment of Le?niewskis Ontology.- The Applicability of Mathematics.- The Relation of Mathematics to the Other Sciences.- Mathematics and Physics.- The Mathematical Overdetermination of Physics.- Godels Incompleteness Theorem and Quantum Thermodynamic Limits.- Mathematical Models in Biology.- The Natural Numbers as a Universal Library.- Mathematical Symmetry Principles in the Scientific World View.- Historical Considerations.- Mathematics and Logics. Hungarian Traditions and the Philosophy of Non-Classical Logic.- Umfangslogik, Inhaltslogik, Theorematic Reasoning.

Probability in the sciences

1: Logical, Methodological and Philosophical Aspects of Probability.- Probability: A Composite Concept.- Two Faces and three Masks of Probability.- Ambiguous Uses of Probability.- Some Logical Distinctions Exploited by Differing Analyses of Pascalian Probability.- Probability and Confirmation.- Chance, Cause and the State-Space Approach.- World as System Self-synthesized by Quantum Networking.- A Brief Note on the Relationship between Probability, Selective Strategies and Possible Models.- 2: Probability, Statistics and Information.- Critical Replications for Statistical Design.- The Contribution of A.N. Kolmogorov to the Notion of Entropy.- The Probability of Singular Events.- Probability, Randomness and Information.- 3: Probability in the Natural Sciences.- Probability, Organization and Evolution in Biochemistry.- Relativity and Probability, Classical and Quantal.- Probabilistic Ontology and Space-Time: Updating an Historical Debate.- Probability and the Mystery of Quantum Mechanics.- Probability and Determinism in Quantum Theory.- Index of Names.- Index of Subjects.

The Concept of Empirical Data

The fact of necessarily including ‘data’ may be considered as the fundamental difference between empirical theories and formal ones. From a methodological viewpoint, this fact may be conceived as a rather radical difference in the way the two different classes of theories fulfill the essential condition of possessing some ‘immediate truth’. Formal theories are characterized by the fact that the immediate truth of some of their sentences is ‘stated’ by the theory itself, whereas in the case of empirical theories such a truth is considered as something which is ‘found’, which comes from outside the theory; moreover, the theory is thought to construct its internal truth on the essential condition of keeping faithful to this external truth and of becoming able, in a way, to include it. In other words: every scientific theory has the problem of ascertaining the truth of its accepted sentences and this may be done rather often by generating it out of the truth of previously accepted statements, but this in turn is only possible if there are sentences which possess their own truth intrinsically: formal theories may be qualified as those which simply ‘single out’ some of their sentences as being endowed with such a truth, while empirical theories must learn from outside which are their immediately true sentences.

Commensurability, incommensurability, and cumulativity in scientific knowledge

Until the middle of the present century it was a commonly accepted opinion that theory change in science was the expression of cumulative progress consisting in the acquisition of new truths and the elimination of old errors. Logical empiricists developed this idea through a deductive model, saying that a theory T′ superseding a theory T must be able logically to explain whatever T explained and something more as well. Popper too shared this model, but stressed that T′ explains the old known facts in its own new way. The further pursual of this line quickly led to the thesis of the non-comparability or incommensurability of theories: if T and T′ are different, then the very concepts which have the same denomination in both actually have different meanings; in such a way any sentence whatever has different meanings in T and in T′ and cannot serve to compare them. owing to this, the deductive model was abandoned as a tool for understanding theory change and scientific progress, and other models were proposed by people such as Lakatos, Kuhn, Feyerabend, Sneed and Stegmüller. The common feature of all these new positions may be seen in the claim that no possibility exists of interpreting theory change in terms of the cumulative acquisition of truth. It seems to us that the older and the newer positions are one-sided, and, in order to eliminate their respective shortcomings, we propose to interpret theory change in a new way.The starting point consists in recognizing that every scientific discipline singles out its specific domain of objects by selecting a few specific predicates for its discourse. Some of these predicates must be operational (that is, directly bound to testing operations) and they determine the objects of the theory concerned. In the case of a transition from T to T′, we must consider whether or not the operational predicates remain unchanged, in the sense of being still related to the same operations. If they do not change in their relation to operations, then T and T′ are comparable (and may sometimes appear as compatible, sometimes as incompatible). If the operational predicates are not all identical in T and T′, the two theories show a rather high degree of incommensurability, and this happens because they do not refer to the same objects. Theory change means in this case change of objects. But now we can see that even incommensurability is compatible with progress conceived as the accumulation of truth. Indeed, T and T′ remain true about their respective objects (T′ does not disprove T), and the global amount of truth acquired is increased.In other words, scientific progress does not consist in a purely logical relationship between theories, and moreover it is not linear. Yet it exists and may even be interpreted as an accumulation of truth, provided we do not forget that every scientific theory is true only about its own specific objects.It may be pointed out that the solution advocated here relies upon a limitation of the theory-ladeness of scientific concepts, which involves a reconsideration of their semantic status and a new approach to the question of ‘theoretical concepts’. First of all, the feature of being theoretical is attributed to a concept not absolutely, but relatively, yet in a sense different from Sneedss: indeed every theory is basically characterized by its ‘operational’ concepts, and the non-operational are said to be ‘theoretical’, this distinction clearly depending on every particular theory. For the operational concepts it happens that their mean-

The Challenge of Providing Renal Replacement Therapy in Developing Countries: The Latin American Perspective

The costs of health care place developing countries under enormous economic pressure. Latin America is a region characterized by wide ethnic and per capita gross domestic product variations among different countries. Chronic kidney failure prevalence and incidence, as well as provision of renal replacement therapy (RRT), have increased in all Latin American countries over the last 20 years. From an ethical point of view, life-sustaining therapies such as RRT should be available to all patients with chronic kidney disease who might benefit. However, even among Latin American countries with similar per capita incomes and health care expenditures, only some have been able to achieve universal access to RRT. This indicates that it is not just a problem of wealth or distribution of scarce health care resources, but one of social justice. Strategies to increase the availability of RRT and renal palliative-supportive care, as well as implementation of interventions to prevent chronic kidney disease development and progression, are needed in Latin America and other developing countries.

What is Complexity

To measure the amount of time that a program takes, we can simply use a clock and measure execution of a program from start to finish. In practice, this is only true way get a sense of what the real execution time of a program, independent of any mathematical models we develop about our program. Such a mathematical model is necessarily incomplete, it cannot capture all the effects that go into the performance of a program, so it is an approximation of the true run time of a program. However, even though this is the case, mathematical models are usually preferable to work with than with the physical process of time itself. This is due to a variety of reasons:

The Universe as a Scientific and Philosophical Problem

Cosmology used to be a classical subdomain of traditional philosophy: according to a well known partition of the whole of reality into three great spheres (Man, Nature and God), it became customary at a certain time to articulate the so-called systematic philosophy into three branches: anthropology or psychology (the philosophy of Man), natural philosophy or cosmology (the philosophy of Nature), and theology (the philosophy of God).1 However, this partition became controversial with the development of modern science, since a certain effect of this development was the conviction that at least some of these domains of inquiry had been removed from the competence of philosophy and passed on to the competence of science. This was the case, in particular, with Nature: in fact, as is well known, the very notion of natural philosophy rapidly changed its meaning, in the sense that what became actually understood under this term was the new physical science initiated by Galileo and Newton. Of course, at least until the end of the 18th century things were not perceived in this light: those people whom we now qualify as scientists were considered — and considered themselves — to be natural philosophers; they were supposed to cultivate philosophy (or at least a part of it), and neither to develop an investigation of Nature which would stand besides (or even at variance with) the philosophical investigation of Nature, nor to promote a new kind of inquiry, which should replace or delegitimate the philosophical study of Nature. Only in the 19th century, and in particular with the positivistic movement, did “science” and “philosophy” become charged with very distinct and even almost opposite meanings.2

Do Experiments Depend on Theories or Theories on Experiments

The constitution of every scientific discipline seems to follow a well known pattern, which was first outlined in the speculations of the founders of modern science, such as Bacon and Galilei, and which has undergone many refinements up to the philosophy of science of our century. According to this pattern, scientific inquiry begins with a systematic collection of empirical evidence, which we may call the level of experience, followed by the elaboration of a set of hypotheses, the level of theories, which are then submitted to test by means of experiments. Once these three aspects have reached a stage of mutual logical consistency, we usually say that a satisfactory “scientific theory” is available concerning our domain of investigation. While disagreement still exists regarding the transition from experience to hypotheses (e.g. some maintain that it occurs by induction, while others claim that it depends on bold conjectures), a more general agreement exists regarding the second point: experiments are designed as possible logical consequences of the hypotheses constituting the theory and they are able either to validate or invalidate those hypotheses. Disagreements emerge again regarding the third step, as a variety of positions exists concerning whether the negative outcome of experiments really entails the rejection of the theory. It is this last disagreement in particular that invites us to analyze the mutual dependence of experiments and theories.

Reductionism as Negation of the Scientific Spirit

The polemic accent implicit in the title of this essay must be understood as a statement addressed against an absolutization or excess, namely, the absolutization of a methodological approach which, in other respects, is fully legitimate and correct in itself, provided one remains conscious of its conditions of application and limits. This approach is the perspective of reduction, of which≪reductionism≫ (as in general for every ≪-ism≫) results in unjustified dogmatic generalization. But, because one may speak of reduction according to different meanings1, it would be a difficult task to define reductionism by relying upon the various meanings of the concept of reduction. It is, therefore, easier to characterize the concept of reductionism directly, for it corresponds to an intellectual attitude, a genuine metaphysical vision, which is more fundamental and more ancient than the methodological approaches one could class under the epistemological notion of ≪reduction≫ and which, at bottom, is responsible for the very absolutization of which we have just spoken.

Waves, Particles, and Complementarity

It is shown that the complementarity principle does not really eliminate the conceptual difficulties of the coexistence of wave-like and particle-like features in quantum phenomena. In order to overcome them a purely formal consideration of quantum sentences must first be applied, which provides a consistent implicit contextual definition of the concepts involved. This is enough for providing exact meanings of these concepts and for eliminating contradictions. At the same time it opens the possibility of constructing models of the microworld that may be endowed with physical meaning though not being intuitive.