From: mk_thisisit
Mathematics is fundamentally a tool, and like any tool, it can be applied in various ways, including purposes deemed morally ambiguous [00:01:21], [00:02:51]. The argument that mathematics itself is to blame for societal division is dismissed, as human division is considered a natural and early development, evident even in ancient Egypt [00:01:32], [00:01:37], [00:01:51].
Mathematics as a Versatile Tool
Today, mathematics is employed for a wide range of activities, from building and creation to control and resource distribution [00:02:00]. As a language of precision, it possesses the potential to describe an ever-increasing scope of phenomena [00:02:35], [00:02:45]. This expanding utility of mathematics is viewed positively by mathematicians [00:02:23].
The ability of mathematics to describe reality versus creating new concepts is a deep philosophical question [00:09:31]. Most professional mathematicians feel that mathematics primarily discovers [00:09:39]. Often, mathematical discoveries initially appear to have no natural application, but later physicists find real-world phenomena that behave exactly like the mathematical models [00:10:14], [00:10:26]. This suggests that mathematical concepts are discovered, and subsequently found to exist in nature, even if they require laboratory creation [00:11:05], [00:11:11]. For example, two-dimensional objects are often studied in mathematics due to their simplicity, and physicists can create interfaces between magnetic fields that behave like these theoretical two-dimensional objects [00:10:49], [00:10:55]. This interplay highlights the strong connection between physics and mathematics.
Fundamental Concepts in Mathematics
Division and Inversion
In mathematics, formal “division” as commonly understood does not exist [00:00:28], [00:03:56]. Instead, there is an “inversion operation” [00:00:30], [00:04:01]. To divide by a number, one actually multiplies by its inverse [00:04:04]. The only number that cannot be inverted is zero, making division by zero forbidden [00:03:09], [00:03:30], [00:04:18]. This concept, like Plato’s Allegory of the Cave, suggests that everyday division is merely a “shadow on the wall” of the more fundamental operation of inversion [00:06:16], [00:06:21].
Similarly, formal “subtraction” doesn’t exist; it is seen as “addition” with an inverse number [00:04:33], [00:04:35]. Different mathematical systems allow for different types of inversions (e.g., with respect to addition or multiplication), which is the focus of algebra [00:04:50].
Fractions
Historically, even in the 15th century, figures like Leonardo da Vinci did not fully grasp the concept of fractions as we understand them today [00:01:15], [00:06:55]. They might have thought of “fractions” as a sequence of numbers, like 3, 9, 27 representing 1/3 [00:07:07]. A deep understanding of fractions requires knowledge of higher mathematics, specifically “equivalence relations” and “equivalence classes” [00:07:17], [00:07:38].
Euclid’s Axioms and Modern Proofs
Euclid’s axioms, though revolutionary in their time for establishing a system of axioms and proofs to build mathematics, were not entirely correct in their derivations [00:07:47], [00:08:13]. Modern mathematics, influenced by Hilbert’s formalization program from over a century ago, has moved beyond Euclid’s specific approach, focusing on rigorous proofs verifiable by computers [00:08:24], [00:08:50]. Modern axioms are more abstract, with concepts like number reversal being axiomatic, and division derived from them [00:09:01].
The Nature of Mathematical Truth
Gödel’s Incompleteness Theorems
Kurt Gödel’s theorem, mentioned in relation to Roger Penrose’s theory of consciousness, demonstrates that within any consistent axiomatic system, there will always be true statements that cannot be proven from those axioms [00:17:53], [00:18:28]. This challenged Hilbert’s dream of building all of mathematics from a finite set of axioms [00:18:19], [00:18:51]. Gödel’s theorem implies that it’s impossible to prove that mathematics is consistent [00:21:12]. This introduces an element of “faith” in mathematics, as one must believe in its consistency to make certain arguments [00:19:13], [00:21:16].
Consciousness and Quantum Phenomena
Roger Penrose’s theory suggests that consciousness arises from quantum phenomena at the level of elementary particles or microtubules in the brain [00:20:02], [00:20:13], [00:20:16]. His argument, based on Gödel’s theorem, claims that because the human brain can understand true statements that are unprovable by algorithms, consciousness cannot be purely algorithmic and must involve a quantum algorithm [00:32:00], [00:20:31], [00:20:40], [00:20:51]. However, this reasoning is flawed because it relies on the unprovable assumption that mathematics is consistent [00:20:55], [00:21:00].
Mathematics and the Universe
The role and perception of mathematics in understanding the universe is a constant theme.
Dimensions
The world is commonly perceived as three-dimensional, or four-dimensional if time is included [00:10:40], [00:10:42], [00:11:36]. However, theories like string theory propose higher dimensions, potentially up to 11 or even 27 [00:10:44], [00:11:22]. A holographic model in physics suggests that our perceived four-dimensional reality might be an illusion, with true existence in one less dimension, where quantum phenomena in a three-dimensional space-time give rise to what we see [00:11:49].
Research into higher dimensions, particularly odd-numbered ones (e.g., 130s dimensions), involves understanding their “hyperbolic” nature, where “directions” multiply very quickly [00:13:08], [00:13:10], [00:13:38], [00:13:51]. The goal is to show how such spaces can be derived from time-evolving lower-dimensional spaces [00:14:04].
Examples of four-dimensional spaces include space-time itself, or product spaces created by combining two lower-dimensional objects, such as a “donut with three holes” and a “donut with 11 holes” to form a four-dimensional object [00:14:35], [00:14:57]. Mathematics also considers spaces with infinitely many dimensions, which are distinct but equally significant [00:15:24], [00:15:29].
Cyclic Universe Theory
Roger Penrose, in his book Cycles of Time, proposes that the universe is cyclical, with each Big Bang being the final phase of a previous universe [00:24:46]. He uses a mathematical concept of scale change where a vastly expanded, “boring era” universe (almost empty of particles) can be mathematically “squeezed” to become the Big Bang of the next universe [00:24:51], [00:25:03], [00:25:10], [00:25:17].
The Question of Existence
A profound philosophical question in mathematics is “why is there something rather than nothing?” [00:26:02]. From a mathematical perspective, the “empty set” is the simplest and most elegant, where mathematics “works best” [00:26:12]. The existence of anything beyond the empty set is therefore a puzzle [00:26:30], [00:26:33]. This leads to the idea that if anything exists, then all possible universes must exist due to mathematics’ inherent elegance [00:26:57], [00:27:01], [00:27:04]. This perspective offers solace to mathematicians, ensuring that their work, no matter how abstract, corresponds to a “real problem in some universe” [00:27:17]. This belief touches on the independent existence of mathematics.
Achievements in Mathematics
Award-Winning Research: Symmetries of all Symmetries
A recent significant achievement is an article recognized as the best in algebraic topology in the last five years, though its core subject is group theory [00:29:18], [00:29:24]. The research concerns the “t_zd property” for “automorphisms of free groups,” which can be colloquially described as the “symmetries of all symmetries” [00:29:41], [00:29:52], [00:30:04]. These groups are universal symmetries, and their automorphisms are symmetries of these universal symmetries [00:29:58]. The proof demonstrated that these groups share properties with classical linear groups, specifically that they cannot be symmetries of Euclidean spaces [00:30:12], [00:30:43], [00:30:54].
This theorem allows for the creation of “expanders,” which are special types of graphs that can quickly process vast amounts of information, used in algorithms like those for sorting on Twitter [00:31:17], [00:32:13], [00:32:22]. The discovery of this theorem involved a computer program that output logs of changing constants, with the final verification coming from the computer [00:35:10], [00:35:29].
Fibration Theorem
Another significant theorem focuses on “fiberization,” a phenomenon where a complex space can be understood as a smaller object changing over time [00:48:57], [00:49:01]. This concept helps to understand higher-dimensional spaces by decomposing them into lower-dimensional “fibers” evolving through time [00:49:03], [00:49:57]. The theorem generalizes a previous result about three-dimensional manifolds with hyperbolic geometry, showing that any object with a specific algebraic property will fiberize [00:50:18], [00:50:21], [00:50:31].
Coherence of Groups with One Relator
Recent work in mathematics, by Marco Linton and Andrej Hakin, proved the “coherence of groups with one relator” [00:33:04], [00:33:11]. Groups, representing the symmetries of objects, can be infinitely numerous, requiring methods like “presentation” to encode them [00:33:16], [00:33:26]. A “relator” is a rule that describes how different sequences of operations within the group are equivalent [00:33:52], [00:33:58]. Groups with only one such rule have been studied for over a century due to their frequent appearance and comparative ease of study [00:34:11], [00:34:16].
Mathematics and Humanity
Infinity
In mathematics, “infinity” means that a finite list cannot contain all elements of a set [00:36:36]. While there are different kinds of infinities, group theory often deals with the “smallest infinity” [00:36:46]. The concept of infinity can be counterintuitive, as illustrated by Zeno’s paradox, where an infinite number of steps can be completed in a finite time [00:37:54], [00:38:28]. Modern mathematics, particularly topology, provides answers to these paradoxes by understanding how finite sections can be “extended to an infinite line” through concepts like homeomorphism [00:38:40], [00:38:50].
The Beauty of Connections: Euler’s Formula
Euler’s formula () is considered the most elegant in mathematics [00:40:51], [00:40:56]. Its beauty lies in combining fundamental constants from seemingly disparate areas of mathematics:
- e: The base of the natural logarithm, related to derivatives [00:42:00].
- π (pi): The ratio of a circle’s circumference to its diameter [00:42:04].
- i: The imaginary unit, where i² = -1 [00:42:06].
- 1 and 0: The additive and multiplicative identities [00:41:11].
Geometrically, the formula means that rotating a segment from 0 to 1 on a complex plane by 180 degrees (pi radians) results in -1 [00:41:17]. The natural fit of this formula into nature is fascinating and indicates a deep, underlying connection [00:41:51], [00:42:24]. Understanding it fully requires knowledge of complex numbers and Taylor series [00:42:38], [00:42:41], [00:42:46].
This leads to the question of whether the universe itself might be “complex” rather than “real” [00:43:57]. If humans evolved thinking in complex numbers, the connection between classical and quantum physics might be immediately obvious [00:44:02].
Communication Challenges
Mathematicians face communication difficulties, even among themselves, especially when dealing with highly abstract or idiosyncratic ways of thinking [00:39:06], [00:39:27], [00:40:03]. However, long-term collaborations, such as those with doctoral students, foster better communication over time [00:39:33].
Mathematics and Technology
Mathematics is increasingly integrated into daily life through technology, particularly computers. Computers are fundamentally mathematical machines, designed to perform mathematical operations [00:54:35]. The development of computer science is considered a significant achievement of mathematics [00:53:15].
The use of AI like ChatGPT is seen as another powerful tool, useful for linguistic services and proofreading [00:51:29], [00:51:33]. While it’s conceivable that generative AI could eventually generate mathematical proofs, this isn’t yet a reality [00:51:50]. The role of humans in a world with such advanced tools would be to master their use, much like electricity in modern life [00:52:26], [00:52:36].
Unsolved Problems
One of the most anticipated breakthroughs in mathematics is the solution to the Riemann Hypothesis [00:54:53]. Fermat’s Last Theorem, which remained unproven for 300 years, was finally solved by Andrew Wiles, ushering in the new field of arithmetic geometry [00:55:37], [00:55:53]. This highlights the ongoing importance of mathematical research.