Jul 6, 2024 11:54 PM
https://iai.tv/articles/is-mathematical-..._auid=2020
INTRO: Galileo claimed that “mathematics is the language in which the book of nature is written” and no more is this apparent than in our most successful theory of nature, quantum field theory. As the mathematical backbone for The Standard Model, quantum field theory’s numerical predictions have been experimentally verified to the highest precision, however, as Timothy Nguyen argues, predictive success alone is not enough for a fundamental theory of reality and only with the safeguard of rigour can science separate truth from fantasy.
EXCERPTS: So what about our most successful theories of physics, which use mathematics to describe the world, but whose rigour derives from their confirmation through experiments? A priori, one can have a mathematically rigorous theory that has no known instantiation in the real world and conversely one can have a scientific theory deficient in mathematical rigour that nevertheless makes spectacularly successful predictions. A prime example of the latter is the success of Newtonian mechanics, which was not made fully rigorous until the work of Cauchy and others in the 19th century (nearly two centuries after Newton) which put the underlying methods of calculus on firm mathematical foundations.
[...] With regards to rigour, general relativity is a rigorous theory while quantum field theory is not. For general relativity is a classical theory whose basic mathematical objects have long been studied (e.g. manifolds, metric tensors) and whose basic equation (Einstein’s equation) is uncontroversially well-defined. On the other hand, while each of the ingredients of quantum field theory mentioned above have mathematically rigorous formulations, quantum field theory combines them in such a way so as to make use of mathematical objects that are often ill-defined.
[...] My view is that a fundamental theory of nature needs to be mathematically rigorous. This is because a fundamental theory is irreducible: it cannot be explained by a deeper theory. Said another way, theories which are not fundamental are emergent, dealing with approximate, coarse quantities derived from more fundamental quantities.
[...] Once a fundamental theory is formulated in terms of mathematics, then mathematical rigour is the gold standard that both mathematicians and physicists ought to strive for. For it is not enough to rely on the physicist’s rigour of experimental verification: after all, only finitely many experiments can be performed! Without the safety net provided by mathematical rigour, from the infinite sea of predictions made by a theory, one cannot know a priori which ones were arrived at through legitimate means. Only rigour can distinguish valid from invalid operations... (MORE - missing details)
INTRO: Galileo claimed that “mathematics is the language in which the book of nature is written” and no more is this apparent than in our most successful theory of nature, quantum field theory. As the mathematical backbone for The Standard Model, quantum field theory’s numerical predictions have been experimentally verified to the highest precision, however, as Timothy Nguyen argues, predictive success alone is not enough for a fundamental theory of reality and only with the safeguard of rigour can science separate truth from fantasy.
EXCERPTS: So what about our most successful theories of physics, which use mathematics to describe the world, but whose rigour derives from their confirmation through experiments? A priori, one can have a mathematically rigorous theory that has no known instantiation in the real world and conversely one can have a scientific theory deficient in mathematical rigour that nevertheless makes spectacularly successful predictions. A prime example of the latter is the success of Newtonian mechanics, which was not made fully rigorous until the work of Cauchy and others in the 19th century (nearly two centuries after Newton) which put the underlying methods of calculus on firm mathematical foundations.
[...] With regards to rigour, general relativity is a rigorous theory while quantum field theory is not. For general relativity is a classical theory whose basic mathematical objects have long been studied (e.g. manifolds, metric tensors) and whose basic equation (Einstein’s equation) is uncontroversially well-defined. On the other hand, while each of the ingredients of quantum field theory mentioned above have mathematically rigorous formulations, quantum field theory combines them in such a way so as to make use of mathematical objects that are often ill-defined.
[...] My view is that a fundamental theory of nature needs to be mathematically rigorous. This is because a fundamental theory is irreducible: it cannot be explained by a deeper theory. Said another way, theories which are not fundamental are emergent, dealing with approximate, coarse quantities derived from more fundamental quantities.
[...] Once a fundamental theory is formulated in terms of mathematics, then mathematical rigour is the gold standard that both mathematicians and physicists ought to strive for. For it is not enough to rely on the physicist’s rigour of experimental verification: after all, only finitely many experiments can be performed! Without the safety net provided by mathematical rigour, from the infinite sea of predictions made by a theory, one cannot know a priori which ones were arrived at through legitimate means. Only rigour can distinguish valid from invalid operations... (MORE - missing details)