[Paper Review] Mathematics and "The Trouble with Physics", How Deep We Have to Go ?

Elemér E Rosinger|ArXiv.org|Jul 8, 2007

Mathematical and Theoretical Analysis14 references16 citations

TL;DR

This paper argues that theoretical physics stands to gain significantly from deeper engagement with modern mathematics, particularly through embracing the creative freedom inherent in mathematical theory-building—such as in set theory and category theory—rather than being constrained by outdated conceptions of 'physical' concepts. It proposes that foundational paradoxes in physics, like nonlocality and self-reference, point to the need for a third realm beyond Descartes' res extensa and res cogitans, suggesting that 'thinking' itself may be physical, and that physicists should treat conceptual innovation with the same rigor and openness as mathematicians do.

ABSTRACT

The parts contributed by the author in recent discussions with several physicists and mathematicians are reviewed, as they have been occasioned by the 2006 book "The Trouble with Physics", of Lee Smolin. Some of the issues addressed are the possible and not yet sufficiently explored relationship between modern Mathematics and theoretical Physics, as well as the way physicists may benefit from becoming more aware of what at present appear to be certain less than fortunate yet essential differences between modern Mathematics and theoretical Physics, as far as the significant freedom of introducing new fundamental concepts, structures and theories in the former is concerned. A number of modern mathematical concepts and structures are suggested for consideration by physicists, when dealing with foundational issues in present day theoretical Physics. Since here discussions with several persons are reviewed, certain issues may be brought up more than one time. For such repetitions the author ask for the kind understanding of the reader.

Motivation & Objective

• To challenge the assumption that theoretical physics must be limited to concepts deemed 'physical' in a narrow, historically conditioned sense.
• To argue that physicists can benefit from the creative freedom inherent in modern mathematics, such as in set theory and category theory, to develop radically new foundational theories.
• To highlight that paradoxes in physics—like quantum nonlocality and self-reference—indicate the need for a conceptual framework beyond classical Cartesian dualism.
• To advocate for a shift in physics toward treating 'thinking' and conceptual structures as potentially physical, thus expanding the scope of theoretical inquiry.

Proposed method

• Analyzing historical and conceptual parallels between the development of mathematics (e.g., Cantor’s set theory, Eilenberg and Mac Lane’s category theory) and theoretical physics.
• Drawing analogies between the role of foundational mathematical innovation and the potential for similar breakthroughs in physics.
• Examining paradoxes such as the Liar Paradox and Russell’s Paradox to illustrate the necessity of distinguishing between statements and their interpretations in foundational physics.
• Using thought experiments involving quantum entanglement to argue for a third realm—beyond 'res extensa' and 'res cogitans'—where instant knowledge or nonlocal correlations might be physically realized.
• Reinterpreting the nature of information and thinking in physics by extending the idea that 'information is physical' to include 'thinking is physical'.
• Proposing that physicists should adopt the mathematical mindset of introducing new fundamental concepts freely, without prior justification through empirical correspondence.

Experimental results

Research questions

• RQ1 How might modern mathematical structures like category theory or infinite-dimensional spaces enrich foundational theories in theoretical physics?
• RQ2 What conceptual framework could accommodate nonlocal quantum correlations without violating relativity, if not through classical spacetime or mental cognition alone?
• RQ3 To what extent can self-referential or paradoxical structures in physics—such as those in quantum measurement or relativity—be resolved by distinguishing between statement and interpretation?
• RQ4 Why has physics historically avoided conceptual innovation comparable to that in modern mathematics, and what are the consequences of this avoidance?
• RQ5 Could the act of thinking, as a physical process, be a legitimate and necessary component of a complete physical theory?

Key findings

• The absence of a definitive, comprehensive definition of 'physical' in physics is not a flaw but a feature, allowing for the continuous expansion of physics into new conceptual domains.
• Historically, many mathematical concepts—such as complex numbers and infinite-dimensional spaces—have become foundational in physics, demonstrating that 'physical' concepts are not inherently limited to empirical or geometric intuition.
• The freedom to introduce new mathematical theories, as seen in Cantor’s set theory and Eilenberg and Mac Lane’s category theory, is unique to modern mathematics and offers a model for conceptual innovation in physics.
• Quantum nonlocality suggests that instant knowledge of distant systems occurs not in 'res extensa' nor 'res cogitans', but in a third, as-yet-undefined realm, challenging classical metaphysical dualism.
• The Liar Paradox and its modern variants reveal a fundamental distinction between a statement and its interpretation, a distinction that must be respected in foundational physics to avoid inconsistency.
• Physics may be in a 'naive' stage analogous to pre-paradoxical set theory, where foundational issues involving self-reference and conceptual structure are avoided rather than confronted.