Identity, Individuality and Indistinguishability in Physics and Mathematics

Discussions on the concepts of identity and indistinguishability have long been foundational to our civilization. The origins of these debates can be traced back, at least, to the ancient Greeks, who first pondered the paradox of Theseus' ship. Originally, this paradox centered around the question of whether a ship that had all its components replaced one by one would still be considered the same. More broadly, what is at stake is the very nature of individuality and what distinguishes one entity from another. At the heart of this discussion lies the issue of how properties relate to identity: Can two entities share all their properties and yet remain distinct? The quest for a satisfactory answer has been a recurring theme in philosophical debates.

These questions have played a crucial role throughout the history of philosophy. But their significance extends beyond abstract debates and holds substantial implications for the sciences. Quantum theory, in particular, provides a striking example, where experimental situations arise where elementary particles, such as photons, cannot be distinguished by any means. This intriguing feature prompted Nobel Prize winner E. Schrödinger to assert that "… an elementary particle is not an individual; it cannot be identified, it lacks ‘sameness’". According to the standard formulation of quantum mechanics, there exist situations in which particles of the same kind become completely indiscernible. Although seemingly strange and far from our everyday intuitions, quantum indistinguishability is thought to be at the basis of the development of cutting-edge quantum technologies nowadays. Thus, the discussions surrounding the identity and indistinguishability of quanta create a bridge between philosophy, fundamental theories of matter, and the advancement of modern technologies.

The intriguing features of quantum systems prompted Nobel Prize winner E. Schrödinger to assert that "… an elementary particle is not an individual; it cannot be identified, it lacks ‘sameness’". Nowadays, quantum indistinguishability is though to be the basis of the development of cutting edge quantum technologies.

Identity-related problems are not confined to physics alone. Mathematics has witnessed a remarkable surge of research aimed at redefining the foundations of the discipline by surpassing the conventional notion of identity. In some areas, to assert that two objects are simply identical or different turns out to be too strict to be sufficiently useful. The same applies to strong forms of equivalence, such as those used in some standard caterory theory frameworks. As problems get more sophisticated, mathematicians often look for subtler notions of sameness that are rich enough to keep track of the information contained in the relations between the mathematical objects under consideration. These notions play a pivotal role in various areas of study, such as that of equivalence in infinity categories. Similarly, homotopy type theory introduces new ways of identifying objects through a revision of what it means for two objects to be equal, paving the way for a univalent foundation of mathematics. The growing interest in these topics gives a fresh perspective to the debate on the role of identity in the sciences. Furthermore, it is worth noting that the very notion of quantum indistinguishability has served as inspiration for non-standard logics and set-theoretical frameworks.

All the above mentioned developments have had a strong impact in the philosophy of science community. The astonishing progress in physics and mathematics during the 20st and 21st centuries has compelled philosophers to reconsider the foundational aspects of both disciplines and extract the implications for general metaphysics as well. After all, many of the problems posed by the ancient Greeks are still relevant today, and research in these areas may lead to future developments as we strive to fully understand quantum indistinguishability and its potential technological applications. 

Our theme issue of Philosophical Transactions A brings together researchers from diverse fields to explore various aspects of the application of identity, individuality, and indistinguishability in the sciences. In addition to research articles presenting novel results, the issue also features review and discussion papers. Readers will find self-contained introductions to the problem of quantum indistinguishability, along with its philosophical and technological implications. The problem of quantum indistinguishability is discussed with much detail, including very different positions with regard to the thesis that quantum systems of the same kind are utterly indiscernible. In particular, the conceptual and technical problems related to the entangement of indistinguishable particles is considered by several contributors, since it has been a very active field of research in quantum physics during the last decades. The issue also includes philosophical articles dealing with the history of identity and the problem of how identity ought to be understood in the foundations of mathematics, taking into account recent developments. Overall, the issue gives an up-to-date interdisciplinary overview of where we stand nowadays with regard to the understanding of identity and individuality in the physical and mathematical sciences. Its aim is to stimulate progress in this exciting field of research and facilitate further theoretical discussions among researchers from different disciplines.

It is important to acknowledge that this special issue was made possible through the collaborative efforts of contributors, reviewers, and the editorial team at Philosophical Transactions A, led by Alice Power and her team. It is my hope that the contents of this special issue will serve as an invitation for readers to immerse themselves in the perplexing problems surrounding identity and indistinguishability in the scientific developments of the 21st century.

About the Guest Editors

Dr Federico Holik is a Research Fellow of the National Scientific and Technical Research Council in Argentina. His research focuses on the foundations of quantum mechanics, quantum information theory, the interpretation of quantum probabilities, and the study of the logical, algebraic and geometrical aspects of the quantum formalism.

Dr Gabriel Catren (PhD in theoretical physics, University of Buenos Aires;  PhD in philosophy, University of Paris VIII) is a permanent researcher in philosophy of physics and mathematics at the French National Centre for Scientific Research  (CNRS) and he works at the Institute SPHERE - Sciences - Philosophie - Histoire (Université Paris Cité).

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