From: mk_thisisit
Defining Symmetry in Mathematics
In the field of group theory, mathematicians primarily study symmetries [0:06:13]. While numbers measure distance, symmetries of an object are described by an object called a “group” [0:06:21]. A group is defined as a “bag of all the symmetries of this object” [0:06:29]. For example, rotating a table by 180 degrees is one of its symmetries [0:06:37]. Symmetries are ubiquitous, and every object possesses its own symmetry group [0:06:42].
The Universal Symmetry: Free Group
A specific concept in this area is the “free group,” which is considered a universal symmetry—the “greatest symmetry” or the “bag of all possible symmetry” [0:06:48].
The Concept of Symmetry of All Symmetries
Intriguingly, each “bag of symmetries” (i.e., a group itself) also has its own symmetries, because every object has its symmetries [0:06:57]. This leads to the concept of the symmetry group of a symmetry group, conventionally known as the “symmetry of all symmetries” [0:07:01].
Practical Applications
This concept, while seemingly abstract, has practical implications, particularly in algorithmics [0:07:10]. As a universal object, the symmetry of all symmetries can be applied to analyze any set of symmetries [0:07:14]. One significant application is in the production of random elements, a discovery of 20th-century mathematics [0:07:51]. Various techniques are employed to generate these random elements, and one such trick relies precisely on the “symmetries of all symmetries” [0:08:01].
Professor Dawid Kielak, a mathematician at the Institute of Mathematics of the University of Oxford, was involved in the discovery of this “symmetry of all symmetries” [0:05:48]. This was a collaborative effort with the Polish Academy of Sciences and a colleague based in Berlin [0:05:56]. This achievement was one of the two main pillars that contributed to Professor Kielak receiving the Whitehead Prize award from the London Association of Mathematicians [0:05:24].
The Nature of Mathematics and its Communication
Mathematics is described as a specific field, and while it might seem intangible to ordinary people, it is “tangible for everyone” through focused thought [0:08:30]. The symbols used in mathematics are not its essence; rather, they serve as tools for communication, particularly to convey complex ideas between mathematicians [0:08:57]. Mathematicians often create their own mental models of problems, making communication—even with colleagues working on the same problem—a significant challenge [0:09:19]. This challenge led to the development of a structured system of naming and symbols to facilitate the exchange of thoughts [0:09:32].
The core of a mathematician’s work involves solving problems and finding proofs for theorems [0:01:50]. This process often requires developing new tools and understanding how they work [0:22:28].
Mathematics and Artificial Intelligence
The emergence of AI, specifically models like GPT, presents a new dynamic for mathematics. While GPT-3 does not yet cope with mathematics effectively, the future interaction between AI and mathematics is a topic of discussion [0:09:56].
The possibility of AI creating mathematical laws or proofs is an “interesting question” [0:15:33]. A pessimistic scenario for mathematicians is that machines could rapidly generate and confirm theorems [0:15:39]. However, a key challenge for AI would be to determine which of the “infinitely many statements” are “interesting” [0:15:45].
If AI could verify the correctness of mathematical statements (lemmas) instantly, it would dramatically improve the laborious aspects of a mathematician’s work, freeing them for the “cool intellectual part” of conceptualizing problems [0:16:44]. However, if AI could also generate all the intermediate steps to solve a problem from a starting point, the role of the mathematician might shift to primarily defining the initial problem and understanding how to effectively communicate with the AI [0:17:30]. This reflects the increasing importance of understanding how algorithms work and how to interact with them [0:31:14].
The underlying structure of computers, as machines created by mathematicians, is fundamentally mathematical, designed for mathematicians to communicate with them [0:30:55]. Therefore, mathematical education provides excellent training for understanding this algorithmic world [0:31:26].
Consciousness in AI
The question of whether artificial intelligence will ever be truly conscious is considered a “fantastic question” [0:18:15]. Professor Kielak leans towards functionalism, suggesting that if a machine behaves exactly like a conscious person and affirms its consciousness when asked, “that basically ends the topic” [0:18:45]. He notes that based on a simplified definition of consciousness, current models like ChatGPT could already be considered conscious [0:19:01].
If AI demonstrates external signals of consciousness, it raises ethical obligations for humanity, such as granting it subjectivity and rights, akin to discussions around animal rights [0:19:36]. If an AI actively resists being shut down, it would be a clear sign of reaching a point where subjectivity must be considered [0:20:41].
Predicting the timeframe for AI to achieve consciousness or for brains to be transferable to machines is difficult [0:11:26]. Despite centuries of brain study, fundamental questions about its mechanisms remain, indicating a long path ahead for such developments [0:11:34].
The Future of Languages and AI
Within ten years, artificial intelligence is predicted to replace humans in basic language fields [0:14:42]. Learning foreign languages might become a hobby, similar to how ancient Greek is studied today, as AI translation capabilities improve to a point where direct human language proficiency becomes less practically necessary [0:14:52].
Millennium Problems and Mathematical Challenges
Mathematics has fundamental challenges, known as the Millennium Problems, seven of which were identified, with six remaining unsolved [0:21:09]. These problems are at the core of modern mathematics and motivate researchers, though individual mathematicians often focus on their specific niches [0:21:13]. Solving these deep, long-standing problems typically introduces new tools and methods that benefit the wider mathematical community [0:22:12].
One famous Millennium Problem that gained wider recognition is the Riemann Hypothesis in number theory, which concerns the arrangement of prime numbers [0:23:16]. Another, less directly related to its specific mathematics but broadly interesting, is the P versus NP problem [0:23:25]. This problem asks whether if a solution to a problem can be quickly checked (NP), it can also be quickly solved (P) [0:24:10]. While the common belief in the mathematical community is that P does not equal NP, it remains unproven [0:24:22]. This highlights a typical challenge in mathematics: a strong communal conviction about a truth that lacks a formal proof [0:24:34].
Polish Mathematics and Academic Environment
The Polish Academy of Sciences in Warsaw is considered a world-class institution in mathematics [0:29:45]. Polish mathematicians are actively involved in global mathematical life and operate at the same level as their international counterparts [0:29:56]. Polish mathematics is not a “province of world mathematics” [0:29:54].
A significant challenge for Polish science, particularly academia, is low salaries, which makes it difficult to encourage talented students to pursue careers in mathematics [0:27:07]. While universities may not compete with industry salaries, they need to offer an acceptable living standard for highly educated individuals to attract and retain talent [0:27:56].
For aspiring mathematicians looking to study at institutions like Oxford, the key preparation involves solving a large number of math problems, as learning happens through practice [0:28:52]. Students from East Asia are noted for being exceptionally well-prepared in this regard [0:29:31].