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"In Mathematical Language": On Mathematical Foundations of Quantum Foundations.

Arkady Plotnitsky1

  • 1Literature, Theory, and Cultural Studies, and Philosophy and Literature Program, Purdue University, West Lafayette, IN 47907, USA.

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|November 27, 2024
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Summary
This summary is machine-generated.

Modern physics advances through new mathematical structures. Quantum theory uniquely uses mathematical postulates for probabilistic predictions, redefining scientific realism without explaining physical phenomena.

Keywords:
Galois theorycontinuitydiscontinuitymathematical complexityreality without realism (RWR) renormalization

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Area of Science:

  • Physics
  • Philosophy of Science

Background:

  • Modern physics, particularly since the 16th century, has advanced through the development of novel mathematical structures.
  • The nature of scientific advancement in physics is intrinsically linked to the invention of new mathematical frameworks.

Purpose of the Study:

  • To argue that the advancement of modern physics is defined by the invention of new mathematical structures.
  • To explore how quantum theory, specifically quantum mechanics and quantum field theory, imbues this thesis with new meaning through unique physical and mathematical features.
  • To introduce the mathematical complexity principle and offer a new perspective on continuity and discontinuity in quantum physics.

Main Methods:

  • Analysis of the historical development of mathematical structures in physics.
  • Examination of the foundational principles of quantum mechanics and quantum field theory.
  • Exploration of "reality without realism" (RWR) interpretations of quantum theory.
  • Introduction and application of the mathematical complexity principle.

Main Results:

  • Quantum theory distinguishes itself by defining phenomena through purely physical features and employing purely mathematical postulates.
  • Quantum mechanics and quantum field theory connect to phenomena via probabilities, without representing their physical causation, aligning with RWR interpretations.
  • A new perspective on the continuity-discontinuity problem in quantum physics is offered, focusing on probabilistic predictions and the exclusion of representational accounts.

Conclusions:

  • The advancement of physics is fundamentally tied to mathematical innovation.
  • Quantum theory represents a paradigm shift, utilizing mathematical structures to predict phenomena probabilistically without recourse to mechanistic explanations.
  • The mathematical complexity principle offers a novel framework for understanding quantum phenomena and their relationship to mathematical formalisms.