From: mk_thisisit
Mathematics, as a field, exists independently of human imposition; it is already present and discovered, rather than created [00:00:08] [00:01:14] [00:02:03]. It is akin to archaeology, where researchers dig to uncover results that inherently exist, even if unseen by humans [00:01:30] [00:03:54]. This means mathematical concepts, like the number three, possess their own objective existence, independent of human representation or creation [00:04:36] [00:05:00]. Mathematical truths are objective, although the process of understanding or proving them can be subjective [00:05:41].
The objective nature of mathematics leads to a significant division among mathematicians regarding how they perceive and approach these concepts.
Visual versus Abstract Approaches
Mathematicians are generally divided into those who prefer a visual image and those who favor a more abstract or algebraic way of thinking [00:02:45]. For some, like the speaker, understanding concepts is easier through visual means, making it more accessible, though they can also engage with abstract or algebraic ideas [00:03:24]. Eminent mathematicians, such as Michel Atiyah, have acknowledged the existence of both visual and non-visual (abstract) thinkers [00:03:02].
The speaker observed this division during their university studies, noting that visual thinkers were a minority among their peers [00:15:37] [00:15:40]. Despite being a visual thinker, it was not a disadvantage for them [00:15:52].
Challenges for Visual Thinkers
There is a perceived “systemic prejudice against mathematicians who are good at geometric thinking” [00:00:45] [00:19:06]. This stems from the difficulty of translating visual or geometric thinking into written proofs or algebraic manipulations required for exams [00:17:52]. While algebraic problems can be written down directly, geometric problems often necessitate an additional step of translating the visual understanding into a linear, verbal explanation [00:17:12] [00:18:06]. This translation process can make exams harder for visual thinkers compared to those who think more directly in a verbal or algebraic manner [00:19:27]. The speaker also noted that teaching visually was often not met with enthusiasm by students, who preferred formulas that could be written and manipulated [00:19:46].
Multidimensional Thinking
The human brain is not limited to three-dimensional thinking [00:11:13] [00:11:57]. Modern physics, particularly quantum mechanics, utilizes mathematical schemes that are n-dimensional or even infinite-dimensional, moving beyond simple three- or four-dimensional concepts [00:11:43]. While it might be more challenging, it is possible to conceptualize these higher dimensions in a less visual, more abstract way [00:12:04].
A prime example is Hermann Minkowski’s geometric interpretation of special relativity [00:12:37]. This framework introduces a four-dimensional geometry, where time behaves differently from spatial dimensions [00:12:42] [00:14:00]. Initially, Einstein was dismissive of Minkowski’s approach, viewing it as “mathematical sophistry” rather than “real physics” [00:22:49]. However, Einstein later realized that he would not have been able to develop the general theory of relativity without Minkowski’s visualization of a flat, multidimensional space, which could then be “curved” to describe gravity [00:23:13]. This highlights that despite personal preferences, engagement with different conceptual approaches is crucial for scientific progress.
Visual thinkers often employ “little tricks” to imagine higher dimensions, such as envisioning a three-dimensional version of a four-dimensional concept while understanding its inherent imprecision [00:13:09] [00:24:58]. The understanding of concepts like space-time in special relativity involves dimensions where time behaves distinctly from space, moving beyond simple Euclidean three-dimensionality [00:14:09].
Einstein’s Understanding of Mathematics
It is generally acknowledged that Einstein, while a brilliant physicist, had difficulties intuitively grasping certain mathematical concepts in the same way some mathematicians did [00:25:26]. This suggests that even within groundbreaking scientific work, there can be a spectrum of strengths in visual versus abstract mathematical interpretation.
The Romanticism of Mathematics
Mathematics, despite its abstract nature, can be described as “romantic” because it describes the real world in a way that feels like a fairy tale, allowing one to live in a world that is both imagined and real [00:26:12]. This is exemplified by stories like George Gamow’s “Mr. Tompkins” series, which explored physical phenomena like special relativity and quantum theory by changing parameters, inviting readers to imagine different realities [00:27:25] [00:28:26]. These stories showcase the elegance and beauty in mathematics that enables the description of the universe. The more humanity understands the physical world, the more its behavior is seen to depend on mathematics, which is subtle, sophisticated, and universal in character [00:00:24] [00:08:10].