AI Can Do Maths Now, and it's Wild

Another Roof
17 Mar 202431:18

TLDRThe video discusses AlphaGeometry, an AI by DeepMind that tackles Olympiad-level geometry problems, surpassing human averages. While it marks a leap in AI reasoning, the speaker questions if it truly advances mathematical reasoning and its impact on mathematics. They dissect AlphaGeometry's solution process, comparing it to human problem-solving, and contemplate AI's role in future mathematical discovery, emphasizing the importance of beauty in mathematical proofs.


  • 💡 In 1997, Deep Blue defeated the highest-rated chess player, in 2011, Watson beat Jeopardy champions, and in 2016, AlphaGo bested a Go champion.
  • 🤖 AlphaGeometry, developed by DeepMind, can solve Olympiad-level geometry problems, outperforming average human participants.
  • 📈 AlphaGeometry combines a language model, similar to a specialized version of ChatGPT, and a symbolic deduction engine to solve problems.
  • 🔍 The language model suggests steps like plotting midpoints or drawing diameters, while the deduction engine applies geometrical theorems.
  • 🏆 AlphaGeometry solved a 2008 International Mathematical Olympiad problem in 40 steps, close to its average of 55 steps.
  • 🔄 AlphaGeometry's approach involves iterating suggestions and deductions until the problem is solved, sometimes resulting in redundant steps.
  • 📚 The video explains the process of AlphaGeometry solving an Olympiad problem, highlighting its logical but sometimes convoluted reasoning.
  • 💬 Commentators compare AlphaGeometry's dual-system approach to a right brain generating ideas and a left brain analyzing them.
  • 🚀 While AlphaGeometry's achievements are significant, the video argues that its problem-solving method resembles brute force rather than true reasoning.
  • 🔮 The potential of AI in mathematics is explored, with the presenter suggesting that AI-assisted proofs might miss out on creative and elegant solutions.

Q & A

  • What significant milestones in AI development were mentioned in the video?

    -The video mentions several significant milestones in AI development: Deep Blue defeating the highest-rated chess player in 1997, Watson winning against Jeopardy champions in 2011, and AlphaGo beating a human champion at the game Go in 2016.

  • What is Alpha Geometry, and why is it significant?

    -Alpha Geometry is an AI developed by DeepMind that can solve Olympiad-level geometry problems, often outperforming the average human participant. It represents a major advancement in AI reasoning and a potential step towards artificial general intelligence.

  • How does Alpha Geometry solve geometry problems?

    -Alpha Geometry uses two systems: a language model that suggests ideas like plotting points or drawing lines, and a symbolic deduction engine that applies geometric theorems and makes logical deductions based on the suggestions.

  • What approach does Alpha Geometry take when solving problems?

    -Alpha Geometry combines creativity and logical deduction, often trying multiple steps and making inferences until it solves the problem. Its process can be somewhat meandering and includes many redundant steps.

  • What are some properties of triangles discussed in the video?

    -The video covers the circumcircle (a unique circle passing through all three vertices of a triangle) and its circumcenter, the altitudes of a triangle (lines from each vertex perpendicular to the opposite side), and the orthocenter (the point where all three altitudes intersect).

  • What was the Olympiad problem that Alpha Geometry solved in the video?

    -Alpha Geometry solved a problem from the 2008 International Mathematical Olympiad, which involved proving that certain points on a triangle all lie on a circle.

  • How did Alpha Geometry's solution differ from traditional proofs?

    -Alpha Geometry's solution was longer and more convoluted, with many redundant steps. Traditional proofs often aim to be more concise and illuminating, clearly showing why the result holds.

  • What are the implications of AI like Alpha Geometry for the future of mathematics?

    -AI like Alpha Geometry could assist in solving problems, but there are concerns about whether it can produce proofs that are as insightful or beautiful as those created by humans. There is also a question of whether AI-assisted proofs will generate new theoretical insights or merely solve problems using existing knowledge.

  • What criticisms does the video offer about AI-generated proofs?

    -The video criticizes AI-generated proofs for often lacking elegance and insight. The concern is that AI might solve problems without contributing to the beauty and deeper understanding that human-generated proofs provide.

  • What are the potential benefits and drawbacks of AI in solving mathematical problems?

    -The potential benefits include solving problems more quickly and efficiently, especially those with immediate practical applications. The drawbacks include the risk of missing out on new theoretical developments and the cultural value of beautiful, human-created proofs.



🤖 AI's Advancement in Solving Olympiad-Level Geometry Problems

The video introduces Alpha Geometry, an AI developed by Deep Mind that can solve Olympiad-level geometry problems, outperforming the average human participant. The script discusses the significance of this AI in the context of AI reasoning and its potential impact on mathematics. It also critiques the lack of detailed coverage on Alpha Geometry's solutions, emphasizing the need to understand its problem-solving process. The video promises to delve into the AI's methodology and its implications in the field of mathematics.


📚 Dissecting Alpha Geometry's Problem-Solving Approach

This paragraph provides an in-depth look at how Alpha Geometry tackles geometry problems, using a combination of a language model and a symbolic deduction engine. It describes the AI's step-by-step process in solving a problem from the 2008 International Mathematical Olympiad, highlighting the systematic yet meandering approach it takes. The explanation covers the AI's method of generating ideas and making logical deductions, comparing it to human problem-solving strategies. The paragraph also touches on the potential of AI to revolutionize mathematics education and problem-solving.


🔍 A Closer Examination of Alpha Geometry's Solution to an Olympiad Problem

The script presents a detailed analysis of Alpha Geometry's solution to a specific Olympiad problem involving cyclic quadrilaterals and angles. It notes the AI's redundancy in restating deduced facts and compresses some steps for clarity. The summary walks through the AI's logical progression, from defining points and using geometric theorems to deducing angles and proving the cyclic nature of certain quadrilaterals. The paragraph also reflects on the AI's proof methodology, questioning whether it truly demonstrates reasoning or is simply following a complex algorithm.


🤔 Reflections on the Nature and Implications of AI in Mathematics

The script reflects on the broader implications of AI's role in mathematics, particularly in problem-solving. It questions whether Alpha Geometry's approach to solving geometry problems can be generalized to other fields and whether it represents a true leap in AI reasoning. The video maker, identifying as a mathematician, shares personal opinions on the potential of AI to discover new knowledge and verify solutions, while also expressing reservations about the lack of elegance and creativity in AI-generated proofs.


🎨 The Aesthetics of Mathematical Proofs and AI's Limitations

This paragraph delves into the concept of beauty in mathematics, contrasting the creative and aesthetic aspects of human-generated proofs with the more mechanical approach of AI like Alpha Geometry. It discusses the cultural and inspirational value of mathematical proofs and the potential loss if AI were to replace human mathematicians in discovering and proving new theorems. The script emphasizes the importance of the 'aha' moment in problem-solving and the unique joy that comes from resonating with logical poetry in mathematics.


🚀 The Future of AI in Mathematics and Its Cultural Impact

The final paragraph contemplates the future role of AI in mathematics, particularly in solving unsolved problems and the potential cultural impact of AI-generated proofs. It raises concerns about AI's ability to capture the beauty and creativity inherent in mathematical problem-solving and the risk of AI replacing human mathematicians, leading to a loss of cultural richness. The video concludes by acknowledging the achievements of Alpha Geometry while tempering excitement with a critical perspective on the current state and future of AI in mathematics.



💡Deep Blue

Deep Blue is an IBM supercomputer that gained fame for being the first computer system to defeat a reigning world chess champion, Garry Kasparov, in a match in 1997. In the context of this video, Deep Blue represents a significant milestone in artificial intelligence's ability to perform complex tasks, such as playing strategic games, which was a precursor to the advancements discussed in the video.


Watson refers to IBM's AI platform that competes with human intelligence. It is known for winning the quiz show 'Jeopardy!' in 2011, showcasing AI's capability in understanding natural language and complex information retrieval. The video uses Watson as an example of AI's progress in different domains, including game playing and problem-solving.


AlphaGo is a computer program developed by DeepMind that plays the board game Go. It made history by defeating a world champion Go player in 2016, demonstrating AI's advanced capabilities in pattern recognition and strategic thinking. The video script mentions AlphaGo to highlight the continuous evolution of AI in mastering complex games beyond chess.

💡Alpha Geometry

Alpha Geometry is an AI developed by DeepMind, designed to solve Olympiad-level geometry problems. The video discusses this AI's ability to outperform the average human participant in geometry problem-solving, which is seen as a leap in AI reasoning towards artificial general intelligence. It is a central topic of the video, illustrating the potential impact of AI on mathematics.

💡Artificial General Intelligence (AGI)

Artificial General Intelligence refers to the hypothetical ability of an AI to understand, learn, and apply knowledge across a wide range of tasks at a level equal to or beyond that of a human. The video mentions AGI in the context of Alpha Geometry's achievements, suggesting that AI's ability to solve complex geometry problems could be a step towards achieving this broader intelligence.

💡Language Model

A language model in the context of AI is a system that is trained to predict and generate human-like text or speech. In the video, the language model is part of Alpha Geometry's system, suggesting geometrical ideas and working alongside the symbolic deduction engine to solve problems. It exemplifies the collaborative approach between different AI components.

💡Symbolic Deduction Engine

A symbolic deduction engine is a component of AI that applies logical reasoning to deduce facts or conclusions from a set of premises. In Alpha Geometry, this engine works in tandem with the language model to validate and build upon the suggestions made, ultimately solving geometry problems. The video describes this dual-system approach to demonstrate AI's problem-solving methodology.

💡International Mathematical Olympiad (IMO)

The International Mathematical Olympiad is a prestigious annual competition for high school students, testing their skills in solving complex mathematical problems. The video script uses a problem from the 2008 IMO paper to illustrate Alpha Geometry's capabilities, emphasizing the difficulty level and the AI's performance in solving such problems.


A circumcircle is a circle that passes through all the vertices of a triangle. The video explains the concept of the circumcircle and its center, the circumcenter, as part of the geometry problem-solving process. It is used to demonstrate the application of geometrical properties in Olympiad-level problems.


The orthocenter of a triangle is the point where its three altitudes intersect. An altitude in a triangle is a line segment from a vertex to the opposite side (or its extension), and it is perpendicular to that side. The video discusses the orthocenter in the context of the IMO problem, showing how it relates to the problem's solution.

💡Cyclic Quadrilateral

A cyclic quadrilateral is a four-sided figure (quadrilateral) where all four vertices lie on the circumference of a circle. The video mentions the property of cyclic quadrilaterals, where opposite angles sum to 180 degrees, as a key concept used by Alpha Geometry in its proof, highlighting the importance of this property in geometrical reasoning.


In mathematics, a proof is a logical argument that establishes the truth of a statement. The video discusses Alpha Geometry's approach to providing proofs for geometry problems, comparing it to human problem-solving methods. It also raises questions about the quality and elegance of AI-generated proofs, emphasizing the value of mathematical proofs beyond their correctness.

💡Beauty in Mathematics

The concept of beauty in mathematics refers to the aesthetic appreciation of mathematical concepts, proofs, and theories. The video script reflects on the importance of beauty in mathematics, suggesting that AI-generated proofs may lack the elegance and creativity found in human-generated proofs, which are often valued for their insight and inspiration.


AI has made significant strides in solving complex problems, as demonstrated by Alpha Geometry, an AI developed by DeepMind.

Alpha Geometry can solve Olympiad level geometry problems, outperforming the average human participant.

The AI uses a combination of a language model and a symbolic deduction engine to generate solutions.

The system suggests ideas and makes deductions based on geometrical theorems, working in tandem to solve problems.

Alpha Geometry's approach to problem-solving has been likened to the human brain's right and left brain functions.

The AI successfully solved a problem from the 2008 International Mathematical Olympiad in 40 steps.

The proof process involves a deep understanding of triangle properties, such as circumcircles and orthocenters.

Alpha Geometry's solution to the Olympiad problem is complex and involves numerous steps of logical deduction.

The AI's proof methodology is not always straightforward and can be seen as meandering.

Despite its success, the proof provided by Alpha Geometry may not always be the most elegant or illuminating.

The potential of AI in mathematics extends beyond problem-solving to the discovery of new synthetic theorems.

AI's role in mathematics raises questions about the future of problem-solving and the value of beauty in proofs.

The cultural impact of mathematics is tied to the human experience of discovering and proving results.

AI-generated proofs may lack the creative and aesthetic elements that are valued in mathematical proofs.

The development of AI in mathematics could lead to a reevaluation of what constitutes a 'beautiful' proof.

The video concludes with a call for a balanced view of AI's role in mathematics, acknowledging its potential alongside its limitations.