From: mk_thisisit
The relationship between mathematics and physics is a complex and often debated topic, with perspectives ranging from mathematics being a mere description of reality to being more fundamental than reality itself. This article explores various viewpoints on this interplay and implications, drawing from discussions between physicists and mathematicians.
Mathematics as the Language of Reality
Some argue that the universe’s facts are governed by precise laws, described in the language of mathematics [00:00:30]. This perspective suggests that mathematics is the language of precision, and anything precise can be described mathematically [00:08:17]. Theories currently in use temporarily describe themselves in the language of known mathematics [00:00:00], [00:12:32].
However, a contrasting view posits that all mathematics is an “unsuccessful escape from the imitation of reality” [00:00:04], or “an unintentional imitation of reality” [00:01:15]. Another perspective suggests that mathematics is “more real than reality,” with reality being a “fictional model” [00:00:08], [00:05:30].
Empirical Roots of Mathematics
It is argued that mathematics, even in its most abstract forms, has its source in empirical observations [00:02:07], [00:17:12]. The very logic upon which all mathematics is based is considered an empirical theory [00:02:21]. The existence of a one-way causal sequence in reality, influenced by time being one-dimensional, shapes our logic and, consequently, mathematics [00:02:38], [00:02:44], [00:05:50]. If time had two dimensions, logical sequences might not exist, and mathematics would be “completely different” [00:03:08], [00:03:15].
Limitations and Challenges
While mathematics is often seen as the language of physics, it is not without its limitations.
Unpredictability in Physics
Physics encounters phenomena that mathematics struggles to predict precisely. For instance, in quantum mechanics, there is no mathematical law predicting the fate of a single electron; only statistical behavior can be described by probability theory [00:08:36], [00:09:02], [00:21:21]. Recent Nobel Prize-winning experiments have shown the absence of a “local and deterministic model” for a single electron’s behavior, indicating that conventional mathematics might not fully capture this reality [00:09:29], [00:09:37].
Furthermore, classical mechanics provides examples where mathematics is “completely helpless.” For potentials that do not meet the Lipschitz condition, a ball placed on a hill could spontaneously start moving right or left with no mathematical prediction possible, not even a probability [00:25:10], [00:26:01], [00:26:21]. This highlights situations where the mathematical theory itself might “fail” in describing physical reality [00:28:17].
Internal Contradictions
All mathematical formulations of physical theories contain some internal contradiction [00:12:10]. For example, general relativity predicts singularities in black holes, where mathematics ceases to function effectively [00:12:13], [00:29:14]. Similarly, classical mechanics, classical electromagnetism, and quantum mechanics are considered “internally contradictory” [00:12:53]. Mathematicians, however, view these as incomplete descriptions rather than failures, seeing a missing point in spacetime as acceptable [00:29:24].
Interdependence and Mutual Inspiration
Despite challenges, physics and mathematics are deeply intertwined. Many theories, such as Einstein’s theory of gravity, would not exist without mathematical foundations like Riemann’s differential geometry [00:06:30]. Calculus, a fundamental tool in physics, was invented by a physicist [00:06:50].
Figures like Newton and Penrose exemplify individuals who were both natural philosophers/physicists and mathematicians, blurring the lines between the disciplines [00:07:32], [00:07:38]. The collaboration between mathematicians and physicists, such as Einstein’s work with Minkowski, highlights this specific “friendship” [00:07:49], [00:07:56].
Physics greatly benefits from mathematical discoveries, with many surprising applications. String theories, for example, are difficult to distinguish from pure mathematics [00:18:35]. Mathematicians are often inspired by physics, just as physicists are inspired by mathematics, creating a “intertwined” relationship [00:17:47], [00:18:21].
The Future: Quantum Gravity and Beyond
A major pursuit in modern physics is the unification of quantum physics and classical world with a quantum theory of gravity [00:19:04], [00:29:43]. While many theories of quantum gravity exist, the current limitation is the lack of empirical observations to determine which, if any, is true [00:19:36], [00:19:51]. There are ongoing efforts to design experiments to test if gravity is quantum at all, which might become feasible within decades [00:32:27], [00:32:41]. Roger Penrose, for instance, believes that quantum theory will ultimately fail and give way to a non-quantum theory in a borderline case with gravity [00:33:04].
Mathematical discoveries continue to drive progress. The Riemann hypothesis, the Borel hypothesis in geometry, and Lang’s geometric program in representation theory are key areas where mathematicians seek deeper understanding [00:33:55]. These areas, like Lang’s program, connect different mathematical theories and sometimes intersect with physics in interesting ways [00:34:20].
An example of physics inspiring mathematics is the theory of holography. The entropy of a black hole depends on its surface area, not volume, suggesting a discrepancy in the description of reality’s dimensions [00:34:45]. This led to the holographic theory, which connects theories in different dimensions (e.g., quantum gravity in one dimension corresponding to a theory on a surface in another dimension) [00:35:16], [00:35:23].
Philosophical Underpinnings
Roger Penrose describes three fundamental existences: mathematics (Platonic ideas), physics (laws governing elementary particles and interactions), and mental existence (human consciousness) [00:13:48]. This division somewhat reflects the relationship between the two fields [00:15:37].
The role of philosophy in physics is debated. Some physicists find little direct help from philosophy in their work, considering it more intellectual entertainment than a tool for scientific progress [00:37:18], [00:37:45]. However, the development of logic, formally within philosophy departments, is acknowledged as a dynamic and interesting field, even if its direct application in physics is not yet clear [00:38:40], [00:39:02]. Philosophers are seen as adept at identifying interesting questions, though their ability to answer them definitively is questioned [00:42:31], [00:42:47].