PhD Thesis and Beyond

An idea of my PhD thesis is given below. Here I would like to say something about my research activities which started with the thesis. I add a text presented recently at a conference, as indicated in it.

Here we go.

Logic and the empirical science in the South of Brazil

Décio Krause https://sites.google.com/view/krausedecio

Delivered at the Philosophical Schools After 1950

International Conference

(Warsaw, May 9–10, 2022)

Abstract In this talk, I make reference to some works in logic and the foundations of science, mainly quantum theories, which have been developed in the South of Brazil with special emphasis on non-reflexive logic and quasi-set theory.

I am proud of being invited to deliver some words about a line of research developed in the South of Brazil related to the foundations of science, mainly quantum theories. I will not speak of all the people who have addressed such issues, but of my participating group.

I regard Professor Newton da Costa as the main mentor of such kind of research. When in Curitiba, in the late 1950s and the 1960s, he commanded a group of people interested in logic, which remained active until his move to São Paulo, where he started working at Campinas and the São Paulo city. Since then, the research in logic was no more pursued in Curitiba, until I and Adonai Sant’Anna, two former students of Newton in São Paulo, started working in the logic and foundations of quantum mechanics, linked to the department of mathematics of the Federal University of Paraná (Curitiba is the capital of the state of Paraná, one of the three states of the South of Brasil).

Newton’s interest in that time was practically all in logic, mainly paraconsistent logic, as other colleagues have emphasized in this same seminar. But in his book Ensaio sobre os Fundamentos da Lógica (Essay on the Foundations of Logic), in showing that any principle of classical logic can be questioned (`dialectized’, as he preferred to say following Bachelard), in particular the Principle of Identity, he found in Erwin Schrödinger’s ideas a motivation to the elaboration of a system where such a principle does not hold in full. Schrödinger said in his Science and Humanism that the notion of identity doesn’t apply to the elementary particles in quantum physics. The reason is that we cannot discern among the particles of the same kind when joined in a collection. Bosons can partake the same quantum state, and so, although there may be thousands of them, they cannot be discerned in any way by either experimental means, not by the theory; fermions do not share the same state, so some characteristics would discern them, but we are prohibited to know which is which. Taking these ideas, da Costa elaborated a first-order two sorted logical system he termed `Schrödinger logic’ with terms of the first kind and terms of the second kind. Then he postulated that expressions like x=y are not well-formed for terms of the first kind, and consequently the Principle of Identity in the form “Forall x (x=x)” doesn’t hold in general; these logics were later termed `non-reflexive'. Then he said that a standard semantics, that is, a semantics founded in a standard set theory could be provided, but the fundamental thing was that the collection of the objects denoted by the terms of the first kind would be not a set, since a set is a collection of discernible objects by definition. He didn’t provide the details of such semantics, but just commented that a `right’ semantics, one which could be fair to the logic, would use not sets, but a more general notion of a quasi-set, something not existing those days, where the notion of identity would not hold for its elements.

When the book was published, da Costa was already at the University of São Paulo (USP) since 1970, but the idea of developing the Schrödinger logic and the theory of quasi-sets in order to find a semantics for such a system was taken as the motto of my PhD dissertation at USP from 1987 on under his supervision. Since I was linked to the department of mathematics of the Federal University of Paraná, in Curitiba, a small group of people interested in logic was formed with colleagues and students of different areas, but having logic as the main subject. My PhD

thesis was finished in 1990; there, I have extended da Costa’s logic to a higher-order system (simple theory of types) and defined a Henkin semantics for the resulting system, proving a weak completeness theorem relative to a semantics still founded in a standard set theory. In the same work, I started the development of a theory of quasi-sets, a theory where there may exist collections (quasi-sets) with a cardinal but with no associated ordinal since the elements of a quasi-sets may be indiscernible in all aspects. That time, I visited M. L. Dalla Chiara in Florence (1992-1993) and Michel Bitbol in Paris (1993) where I presented my ideas related to quasi-sets.

Later, a young colleague of mine, Adonai Sant’Anna started working also with da Costa at USP but dealt with other issues in the foundations of physics; ever since then, Adonai is a member of the department of mathematics of the Federal University of Paraná. Anyway, we have joined efforts in some works with the application of the theory of quasi-sets to the foundations of quantum mechanics. We realized in a series of papers that an important postulate, the Postulate of Indistinguishability, which says that any permutation of indiscernible particles does not produce differences in the expected values of any measurement on the system, is no more necessary, resulting in a theorem from the starting assumption that the notion of identity doesn’t hold for these entities, so following Schrödinger and da Costa’s claim.

Later, in 1995, I started working in a more general system encompassing intensionalities, since I was motivated by Maria Luisa Dalla Chiara and Giuliano Toraldo di Francia’s ideas that ``microphysics is a world of intensions”. Then I developed a higher-order modal logic (in Daniel Gallin’s style) and constructed a semantics for such a system in the theory of quasi-sets and proved a Henkin completeness theorem too. The interesting results are not just logical, but we can make sense of the claim that some intensions, such as `to be an electron of a certain collection with this and that properties’ can have alternative extensions, so getting a completely different idea that contrasts the standard doctrine that an extension can have distinct intensions; here the opposite holds, sine more that one collection (quasi-set) of electrons can satisfy the intension.

Newton da Costa has collaborated in all these words, being receptive to discussions and ideas. I reput to him the correct motivations and incentives. In the late 1980s, da Costa introduced me to the British philosopher of physics Steven French, who was in Campinas (state of São Paulo) at that time. The interests were linked; Steven was publishing a series of papers which became references to the field, on the philosophical aspects of quantum indistinguishability, Postulate of Indistinguishability and the like. Later Steven moved to Leeds where I visited him in 1995-1996. His philosophy and the mathematics I have developed found a complement to one another, and we published a book in 2006 on the subject, so as did other papers on related issues. The book was analysed in a section of the American Philosophical Association (Pacific Division) in Vancouver, 2010 by Bas Van Fraassen, Don Howard, Otávio Bueno and others and gained many positive reviews.

In 2000 I moved to the Federal University of Santa Catarina, another South state. There I found a quite good place to work and continued working with da Costa, who also moved there in 2003, so as with some students and now of the staff of the department of philosophy, such as Jonas R. A. Arenhart. Several works were done since then and a book on the foundations of science was published in 2017. At the Federal University of Santa Catarina, da Costa started a semanal workshop on the foundations of logic, mathematics and, of course, physis, so continuing his influence and teachings in the South. The seminars were stopped in 2020 due to the COVID pandemia.

Today the theory of quasi-sets has been improved by Eliza Wajch, from the Siedlce University, Poland who has practically reconstructed the theory by admitting quasi-classes and improving it in several aspects. The work is in construction.

The philosophical aspects of the lack of identity are being discussed in several papers and conferences in several places in the world. Some Argentian physicists and philosophers have joined the group, contributing in much with the development of the ideas and in their application to physics; for instance, Graciela Domenech (Un. Of Bueno Aires), Federico Holik (Un. of La Plata), Olimpia Lombardi (Un. Buenos Aires), Christian de Ronde (Un. Buenos Aires), and many others.

A few references are given below, where further ones can be found.

References

Da Costa, N. C. A. (1980), Ensaio sobre os Fundamentos da Lógica. São Paulo: Hucitec

da Costa, N. C. A. and Krause, D. (1994), Schrödinger logic. Studia Logica 53 (4): 533-50.

da Costa, N. C. A. & Krause, D. (1997), An intensional Schrödinger logic. Notre Dame Journal of Formal Logic 38 (2): 179-194.

Domenech, G., Holik, F. And Krause, D. (2008), Q-Spaces and the Foundations of Quantum Mechanics. Foundations of Physics 38: 969-994.

French, S. And Krause, D. (2006), Identity In Physics: A Historical, Philosophical, and Formal Analysis. Oxford: Oxford Un. Press.

Howard, D., van Fraassen, B., Bueno, O., Castellani, E., Crosilla, L., French, S. And Krause, D. (2011), ook Symposium: The physics and metaphysics of identity and individuality Steven French and Décio Krause: Identity in physics: A historical, philosophical, and formal analysis. Metascience 20: 225-251.

Krause, D. (1992), On a quasi-set theory. Notre Dame J. Formal Logic 33 (3): 402-411. Krause, D. And Arenhart, J. R. B. (2017), The Logical Foundation of Scientific Theories:

Languages, Structures, and Models. London: Routledge.

Krause, D., Arenhart, J. R. A. And Bueno, O. (2022), The non-individuals interpretation of quantum mechanics. In Freire Jr. et al. (eds.), The Oxford Handbook of the History of Quantum Interpretations. Oxford Un. Press, Cap.46.

Sant’Anna, A. S. And Krause, D. (1997), Hidden variables and indistinguishable particles. Foundations of Physics Letters 10: 409-426.

Schrödinger, E. (2004), Nature and the Greeks and Science and Humanism. Cambridge: Cambridge Un. Press.

PhD Thesis:

Krause, Décio (1990), Non-reflexivity, Indistinguishability, and Weyl’s Aggregates (in Portuguese), Department of Philosophy, University of São Paulo (USP).

Supervisor: Newton C. A. da Costa

I have considered the suggestion and the motivations given by Newton da Costa for the development of a "theory of quasi-sets", proposed as a way  to build a semantics for a first-order logical system he sketched in his book Ensaio sobre os Fundamentos da Lógica  (S.Paulo: Hucitec & EdUSP, 1980), called by him "Schrödinger logics". These logics were proposed to show that even the Principle of Identity (in this case, in its first-order formulation, namely, "any object is identical to itself") can be logically violated. His thesis (in the book) is that there is  no logical principle or assumption that cannot be questioned by a reasonable logical system. For instance, the celebrated Principle of Non-Contradiction is questioned in paraconsistent logics (which da Costa is one of the main proposers); the Principle of the Excluded Middle is questioned in  paracomplete logics, such as many valued logics or intuitionistic logic. In order to question the principle of identity, da Costa took a motivation from Erwin Schrödinger's claim that the notion of identity, or sameness, does not make sense for elementary particles in quantum mechanics. He sketched a first-order two sorted logic in which the expression x = y is not a formula for x and y being entities of one of the sorts,  but guessed that a "right" semantics could not be defined in a standard set theory, since in these theories the notion of identity holds universally, being in need a "quasi"-set theory to fulfill the requirements that, for some entities, the standard notion of identity should not hold. I have considered the idea as relevant not only from the purely logical point of view, but also for the physical foundational point of view.  If identity does not hold, we cannot say that an object is or not identical to itself, but in some way we sometimes need to count them as more than one. This is the first idea about the possibility a collection (a quasi-set)  having a cardinal but without an associated ordinal, as classical set theory tells us. It became clear to me that this could not be  just a mathematical joke. I was entering in deep philosophy, in metaphysical considerations but, apparently, I had a strong support in quantum mechanics; bosons can be in the same quantum state, sharing all their properties, and even so they are not reduced to just one entity, and they cannot be counted. So I needed to know more about quantum theory. I have studied the basics of this discipline, and for that I have had the help of many people, such as Professor Christiano Graff from the department of physics of the Federal University of Paraná, from whom I have got many and many patient teachings.  In my thesis, which was basically in logic, I extended da Costa's first-order system to higher-order Schrödinger Logics (simple theory of types) so that Leibniz's Principle of the Identity of Indiscernibles can be fully formulated and discussed, and have proven a weak completeness theorem (in the Henkin's style) for such logics. I have also presented the first version of a theory of quasi-sets, aiming to solve the problem of the semantics for Schrödinger logics, something I have done later. 

My examiners were the following professors and doctors; I owe special thanks to Zara and to Iole, who have discussed a lot with me.

Prof. Newton C. A. da Costa, Department of Philosophy, USP - Supervisor

Prof. Jacob Zimbarg, Department of Mathematics, USP

Prof. Antonio Mario Sette, Department of Mathematics, UNICAMP

Dr. Iole de Freitas Druck, Department of Mathematics, USP

Dr. Zara Issa Abud, Department of Mathematics, USP

I've got a 10,0 from all examiners.