From: mk_thisisit
Mathematics is considered to possess its own inherent existence, independent of human creation or discovery [00:00:08]. This perspective is common among mathematicians, viewing the mathematical world as having an objective reality that is merely uncovered by humans [00:02:01].
An Objective Reality
Mathematical concepts are seen as existing independently of human thought, not as a product of human work [00:01:25]. The discovery of mathematical results is likened to archeology, where findings are already present, waiting to be unearthed, even if no one has previously observed them [00:01:30]. These results are considered objective and would exist regardless of humanity’s presence [00:04:36].
Mathematical existence is believed to be uncreated; it exists naturally and, in a sense, creates itself [00:05:00]. For example, the concept of the number “3” is considered a concept that would exist whether or not humans represent it with symbols [00:05:14].
Mathematical truth is not subjective [00:05:44]. While the understanding or perception of a truth can be subjective, the truth itself, such as “2 + 1 = 3,” exists independently [00:06:31]. Mathematics is highly abstract yet possesses permanence and an existence separate from humanity [00:07:02].
Perceiving Mathematical Truth
There is a significant division among mathematicians between those who prefer a visual image of concepts and those who favor a more abstract or algebraic approach [00:02:45]. While some find visual thinking more accessible, higher-dimensional concepts in mathematics often require non-visual or abstract understanding [00:11:46].
The human brain’s three-dimensionality does not limit the understanding of higher-dimensional mathematical schemes, which can involve n-dimensions or even infinite dimensions [00:11:49]. While visual thinking might be somewhat limited in this regard, techniques exist to think about higher dimensions in non-visual ways, or by using “tricks” to visualize them within a three-dimensional framework [00:12:07].
Systemic Prejudice in Mathematical Thinking
There may be a “systemic prejudice” against mathematicians who excel at geometric thinking [00:00:45]. Examinations in mathematics often favor a more direct, verbal, or algebraic approach over a visual one, potentially disadvantaging those who think geometrically due to the need to translate their visual understanding into a written form [00:17:52]. This can lead to visual thinkers performing less optimally in exams compared to those with non-geometric thinking styles [00:19:27].
Mathematics and the Physical World
The more humanity understands the physical world and its behavior, the more apparent its dependence on mathematics becomes [00:00:21].
The Three Realities Model
One model describes the world as comprising three interconnected realities [00:01:04]:
- Mathematical concepts: Possessing an independent existence [00:01:07].
- Physical world: The realm of physical behavior [00:01:10].
- Mental world: Where physical phenomena are “touched” or understood mentally, as mathematics itself is an idea [00:01:10], [00:02:16].
The laws of physics appear to be regulated by mathematics, specifically a type that possesses a universal character and inherent inner beauty [00:07:38]. The deeper the understanding of the physical world, the more its subtle, sophisticated, and universal dependence on mathematics is observed [00:08:10].
It is considered a dangerous viewpoint to assume that a beautiful, powerful, or universal mathematical theory must therefore be true in the physical world [00:08:47]. Instead, the approach should be to discover what works well in the physical world, which then often reveals an unexpected universality and beauty within its underlying mathematics [00:09:08]. Examples include Newton’s mechanics, which started from empirical laws (Kepler, Galileo) and led to general formalisms discovered by others (Lagrangian, Jer) [00:09:27]. However, these laws, while appearing universal, were later found to have limitations, particularly with the advent of quantum mechanics [00:10:22].
Einstein’s View and Minkowski’s Contribution
Albert Einstein, while a central figure in general relativity, initially struggled with certain mathematical concepts and intuitively grasping them [00:25:38].
Special relativity did not solely rely on Einstein for its discovery, as many individuals contributed to its development, including Lorentz [00:21:20]. Hermann Minkowski, who lived before Einstein, described special relativity in a highly geometric way using four-dimensional geometry [00:12:37]. This geometric interpretation was crucial for the later development of general relativity [00:23:16].
Einstein initially dismissed Minkowski’s geometric interpretation of special relativity as “mathematical smart talk” or “sophistry,” not “real physics” [00:22:52]. However, he later realized that this four-dimensional visualization was essential for progressing to the general theory of relativity [00:23:31]. The “relativity” in the name can be misleading, as the underlying mathematical description of spacetime ultimately describes a real, absolute geometric object, rather than merely relative points of view [00:24:07].
The “Romantic” Nature of Mathematics
Mathematics can be considered “romantic” in the sense that it allows one to inhabit a world that describes reality yet feels like a “fairy tale” [00:26:15]. This perspective highlights the imaginative and profound aspects of mathematics and its ability to model hypothetical scenarios, as seen in George Gamow’s stories, like “Mr. Tomkins,” which explored physics in an imagined world with altered parameters [00:27:46].