From: mk_thisisit
Mathematics is understood to possess an inherent existence, independent of human creation or imposition [00:00:08]. This perspective views mathematics as something that is discovered, akin to archaeology, where results are unearthed rather than invented [00:01:30]. Mathematical theorems and concepts hold their own reality, persisting even if never explicitly discovered by humans [00:01:41].
The Independent Existence of Mathematics
Many mathematicians share the view that the mathematical world exists independently of humanity [00:02:01]. Humans do not create it; they uncover or discover parts of it [00:02:06]. While physical phenomena can be tangibly touched, mathematics is purely conceptual and is “touched mentally” [00:02:16].
Mathematical results are considered objective and permanent, existing regardless of human presence [00:04:36]. For instance, the concept of the number 3 is independent of human existence and would be there whether or not humans ever existed [00:05:32]. While the way we represent it (e.g., with symbols) is human-dependent, the concept itself is not [00:05:27]. Mathematical truth is generally not considered subjective [00:05:44]; proofs provide objective sequences of steps that lead to truth, even if the individual’s understanding of that truth can be subjective [00:05:59].
Visual vs. Abstract Thinking in Mathematics
There is a significant division among mathematicians regarding their preferred way of perceiving mathematical concepts: some prefer a visual image, while others prefer a more abstract or algebraic approach [00:02:45]. Sir Roger Penrose, for example, finds it easier to understand things visually [00:03:27]. This difference in thinking styles can even impact academic performance, as verbal or direct methods may lead to better exam results compared to visual thinking that requires translation into written solutions [00:19:35]. There is a perceived “systemic prejudice” against mathematicians who excel at geometric thinking in academia [00:19:10].
Mathematics as the Language of Physics
The more humanity comprehends the physical world, its mechanisms, and behaviors, the more evident it becomes that it fundamentally relies on mathematics [00:00:24].
The “Three Worlds” Model
A common model for understanding reality divides it into three interconnected realms:
- Mathematical concepts [00:01:07]: Possessing an independent existence [00:07:30].
- Physical world [00:01:07]: Describing physical behavior [00:07:35].
- Mental world [00:01:10]: The human mind’s interaction with these concepts.
The laws of physics appear to be regulated by a special kind of mathematics—one that is universal, possesses intrinsic beauty, and is subtle and sophisticated [00:07:41].
The Interplay between Mathematics and Physics
It is a common belief that if a mathematical theory is elegant, powerful, or beautiful, then the physical world must operate according to its principles [00:08:53]. However, this is considered a “dangerous point of view” [00:08:47]. Instead, the preferred approach is to observe what works effectively in the physical world and then discover that, for some reason, it possesses inherent universality and beauty [00:09:11].
This role of mathematics in understanding the universe can be seen in historical developments:
- Newtonian Mechanics: Isaac Newton formulated a general theory based on the laws of gravity from Johannes Kepler and fundamental laws from Galileo Galilei [00:09:31]. This led to general formalisms, later developed by figures like Joseph-Louis Lagrange, that describe physical behavior in a highly generalized manner [00:10:04].
- Quantum Mechanics: Newtonian laws were found to be less universal than initially thought in the realm of quantum mechanics [00:10:29]. Quantum mechanics operates on very different, though still highly mathematical, principles than those useful in Newtonian mechanics [00:10:59].
Influence and Limitations of Mathematics in Scientific Theories
A crucial example of the physics and mathematics their interplay and implications is the development of relativity:
- Special Relativity: This theory did not solely depend on Albert Einstein for its discovery [00:21:20]. Key ideas existed before him, notably from Hendrik Lorentz and Hermann Minkowski [00:21:32]. Minkowski described special relativity in a four-dimensional geometric way (space-time), which is now considered a revelation and simplifies understanding the laws of special relativity [00:12:47].
- General Relativity: Einstein initially dismissed Minkowski’s geometric interpretation as “mathematical smart talk” or “sophistry,” believing it wasn’t “real physics” [00:22:54]. However, he later realized that he could not have developed the General Theory of Relativity without incorporating Minkowski’s four-dimensional, flat space visualization, which then becomes curved to describe general relativity [00:23:13]. The name “relativity” itself can be misleading, as the underlying reality is a single, absolute geometric object rather than a relative set of viewpoints [00:24:07].
Einstein’s approach to physics was primarily intuitive; he reportedly had difficulty grasping certain mathematical concepts in the same intuitive way mathematicians did, though he was undoubtedly a brilliant physicist [00:25:32].
The Human Brain and Mathematical Understanding
A question arises whether our understanding of mathematical truth is limited by the three-dimensionality of the human brain [00:11:19]. However, mathematics applied in quantum mechanics often deals with schemes that are not three- or four-dimensional, but can be n-dimensional or even infinite-dimensional [00:11:43]. While visual thinking is often based on three dimensions, it can be considered somewhat limited for these higher dimensions, where a more abstract approach is beneficial [00:12:10].
Even with four-dimensional space-time, visualization is possible using “little tricks,” such as thinking about it in terms of a two-dimensional space and one-dimensional time, which yields non-Euclidean geometric concepts due to the different behavior of the time dimension [00:14:05].
The “Romantic” Aspect of Mathematics
Mathematics can be considered “romantic” in the sense that it allows one to live in a world that describes reality, yet feels almost like a fairy tale [00:26:15]. Authors like George Gamow, with his “Mr. Tomkins” stories, effectively conveyed complex physics concepts (like special relativity or quantum theory) through imaginative narratives, making them more accessible and perhaps instilling a sense of wonder [00:28:26].