From: mk_thisisit
The role and perception of mathematics in understanding the universe is a complex and often debated topic among physicists and mathematicians.
Mathematics as a Language for Physics
Current scientific theories temporarily describe themselves in the language of mathematics [00:00:00]. The universe’s facts are governed by precise laws, which are described in the language of mathematics [00:00:30]. Physics uses mathematics because it is the language of precision; anything that happens precisely is described mathematically [00:08:15]. Many fundamental things in physics are owed to mathematics [00:11:16].
However, the statement that mathematics is the language of physics is not entirely true [00:12:01]. While physicists use mathematics, it may stem from the fact that physics is governed by precise laws and mathematics offers precision [00:08:07].
Mathematics’ Relationship with Reality
One perspective posits that mathematics is an “unsuccessful escape from the imitation of reality” [00:00:04]. Another, contrasting view, suggests that mathematics is “more real than reality,” with reality itself being a “fictional model” [00:00:08].
Despite its utility, it is argued that the fundamental reason mathematics describes reality so well is “untrue,” as many examples show it describes physical reality “terribly” and “does not fit at all” [00:01:41]. Mathematics’ approximate description of some theories might stem from mathematicians’ inability to create truly abstract concepts, instead generalizing from elementary empirical observations [00:01:56]. The very logic underlying mathematics is considered an empirical theory, influenced by the one-dimensional nature of time in our world [00:02:21]. If time had more dimensions, logical sequences might not exist in their known form, and mathematics would be “completely different” [00:03:08]. This suggests that the structure of mathematics is “formatted in a specific way” by the physical world’s properties [00:03:17].
Limitations and Internal Contradictions
All mathematical formulations of physical theories contain internal contradictions [00:12:10]. For example, general relativity includes singularities within black holes, points where mathematics “stops working” [00:12:13]. The assertion that mathematics is a good description of reality is a hypothesis based on current theories only approximately describing themselves in known mathematical language, with “no guarantee” this will always be the case [00:12:24]. All known theories—classical electronics, classical mechanics, and quantum mechanics—are internally contradictory [00:12:53].
Unpredictability in Physics
A significant limitation of traditional mathematical operations in physics arises in quantum mechanics, where mathematics has “nothing to say” about fundamentally unpredictable principles [00:00:13]. For instance, there is no mathematical law predicting the fate of a single electron [00:08:52]. While probability theory can describe statistics of many events [00:08:59], it cannot predict a single one [00:09:54]. Recent Nobel Prize-winning experiments demonstrated there is no local and deterministic model for a single electron’s behavior [00:09:29].
Furthermore, some classical mechanics problems exhibit unpredictable behavior. For instance, a ball placed on certain hill potentials can start moving in an unpredictable direction (right or left) without any mathematical ability to predict its fate [00:25:10]. This occurs because the differential equations describing such potentials do not meet conditions (e.g., Lipszyc condition) for a unique solution [00:26:01].
Interplay Between Physics and Mathematics
There is a close relationship between mathematics and physics. Mathematics is essential for creating better models in physics [00:01:03]. Historically, calculus was invented by a physicist (Newton) [00:06:50]. Notable figures like Isaac Newton and Roger Penrose exemplify individuals who were both natural philosophers and mathematicians, blurring the lines between the fields [00:07:30].
Mathematicians are inspired by physics [00:17:47] and computer science, which is sometimes called “the new physics” [00:17:52]. There are many examples of mathematics proving useful in physics, even in abstract concepts like string theory [00:18:35]. The Fields Medal has even been awarded to physicists like Edward Witten for their mathematical contributions [00:18:49].
Despite this, some argue that today, physics needs empirical observations more than new mathematical discoveries for further breakthroughs, especially in quantum gravity [00:19:23]. While quantum gravity theories exist, there’s no empirical way to determine which, if any, is true [00:19:31].
Philosophical Perspectives
Roger Penrose, a Nobel laureate, proposes three existences:
- The independent existence of mathematics, rooted in Platonic ideas [00:13:48].
- The existence of physical laws, found in elementary particles and interactions [00:13:55].
- Mental existence, referring to human consciousness [00:14:04].
Mathematicians, according to one view, are humble scientists who cultivate their “own little plot,” without the fundamental goal of describing the universe [00:14:17]. They claim to have no ambitions of describing specific reality, merely offering tools that physicists use [00:14:46].
Philosophy’s role in the development of physics is viewed skeptically by some physicists, who consider it primarily intellectual entertainment [00:37:26]. However, the development of logic, formally happening in philosophy departments, is recognized as a “very interesting” and dynamic area, although not always directly applicable to physics [00:38:40].
The debate over the nature of Artificial Intelligence (AI) highlights a philosophical concept that AI might be a “new zone of human existence,” rather than merely a tool or an inhuman being, providing a framework for discussion that transcends simple definitions [00:40:33].