ChatGPT can't multiply, but can AI do math?
TLDRThe transcript discusses the limitations of AI in performing mathematical tasks like multiplication, explaining that AI like ChatGPT relies on statistical predictions rather than understanding. It highlights the use of AI in mathematical research, particularly SAT solvers, which efficiently tackle complex problems with thousands of variables, as demonstrated in solving the Boolean Pythagorean triples problem. The summary also mentions the application of neural networks by Adam Wagner to find counterexamples in combinatorics, suggesting AI as a valuable tool for mathematicians, not a replacement.
Takeaways
- 🤖 ChatGPT struggles with multiplication because it makes predictions based on patterns in text rather than understanding the mathematical process.
- 🔢 The initial and final digits of a multiplication result can often be predicted statistically, but the middle digits require a more complex understanding that ChatGPT lacks.
- 📚 AI is currently being used by mathematicians for research, despite its limitations in understanding complex mathematical concepts.
- 🧩 SAT solvers are a type of AI used in mathematical research to solve Boolean satisfiability problems efficiently, which would otherwise take an exponential amount of time.
- 🔑 The Boolean Pythagorean triples problem was resolved using a SAT solver, demonstrating the potential of AI in tackling specific mathematical challenges.
- 🛠 SAT solvers are powerful tools but require a human touch to convert problems into Boolean sentences effectively.
- 🕊 Neural networks, another AI technique, have been used to find counterexamples in combinatorics, showcasing their potential in pure math research.
- 🎯 The cross entropy method is a technique used with neural networks to generate potential counterexamples to mathematical conjectures efficiently.
- 🔄 This method involves training a neural network to predict counterexamples, computing their validity, and retraining the network based on the results.
- 🧐 AI is unlikely to replace mathematicians but could become a valuable tool in their research, saving time and potentially uncovering insights that humans might miss.
- 🌟 The future of AI in mathematics is promising, with the potential to assist in research and solve problems that are too time-consuming for humans to tackle manually.
Q & A
Why can't ChatGPT multiply numbers accurately?
-ChatGPT fails at multiplication because it generates responses based on patterns in text it has seen before, rather than understanding the actual mathematical operations. It can predict the start and end digits of a multiplication result statistically, but the middle digits, which depend on all input digits, are more complex and often incorrect.
What is a SAT solver and how is it used in mathematics?
-A SAT solver is a software tool used to solve the Boolean satisfiability problem, which involves determining if a logical formula can be satisfied by some assignment of truth values to its variables. It applies heuristics and optimizations to efficiently handle problems with thousands of variables and has been used in mathematical research, such as solving the Boolean Pythagorean triples problem.
What was the Boolean Pythagorean triples problem and how was it solved?
-The Boolean Pythagorean triples problem asked whether it is possible to color the positive integers red and blue such that no Pythagorean triple is all one color. The answer is no, and this was proven by a SAT solver that generated a 68-gigabyte proof after two days of computation.
How do modern SAT solvers handle large problems efficiently?
-Modern SAT solvers use heuristics and optimizations to handle large problems efficiently. Instead of checking every possibility, which would take an exponential amount of time as the number of variables grows, they apply sophisticated methods to reduce the search space and find solutions more quickly.
Can AI techniques like neural networks be used for pure math research?
-Yes, AI techniques like neural networks can be used for pure math research. For example, Adam Wagner used neural networks to find counterexamples to problems in combinatorics by generating and testing graphs in a smarter, more efficient way using the cross entropy method.
What is the cross entropy method and how is it used in AI for math research?
-The cross entropy method involves training a neural network to predict how to build structures (like graphs) that are likely to be counterexamples to a conjecture. It then generates many such structures, tests them, and retrains the neural network based on the results, iteratively improving its predictions until it finds a valid counterexample.
Why do the middle digits of a multiplication result confuse ChatGPT?
-The middle digits of a multiplication result depend on the interaction of all input digits, making them complex and harder to predict with statistical methods. ChatGPT relies on patterns it has seen before, and while it can predict the start and end digits with some accuracy, the middle digits often appear random to the model.
How might AI save mathematicians time in their research?
-AI can save mathematicians time by automating the generation and testing of potential counterexamples to conjectures, as well as solving specific types of problems (like Boolean satisfiability) more efficiently than traditional methods, allowing mathematicians to focus on more complex aspects of their work.
Will AI replace mathematicians in the future?
-It is unlikely that AI will replace mathematicians in the near future. While AI can be a powerful tool to aid in mathematical research, many problems require human creativity and insight to convert into forms that AI can handle or to interpret the results. AI is more likely to complement mathematicians rather than replace them.
What are the limitations of using SAT solvers in mathematics?
-SAT solvers are limited to problems that can be converted into Boolean satisfiability questions. Many mathematical problems do not fit this form and thus cannot be solved using SAT solvers. Additionally, the process of converting a problem into a suitable Boolean sentence often requires significant human effort and expertise.
Outlines
🤖 AI's Limitations in Mathematical Precision
This paragraph discusses the limitations of AI, specifically ChatGPT, in performing precise mathematical operations like multiplication. It explains that while AI can make accurate predictions based on statistical observations of the beginning and end digits of numbers, it struggles with the middle digits due to their complexity. The paragraph also touches on the use of AI in mathematical research, introducing the concept of SAT solvers and their application in solving Boolean satisfiability problems, with a notable example being the Boolean Pythagorean triples problem.
🔍 SAT Solvers and Their Role in Mathematical Research
The second paragraph delves deeper into the use of SAT solvers in mathematical research. It explains how SAT solvers work by substituting 'true' and 'false' for variables in sentences and applying reduction rules to determine if a sentence can be made true. The paragraph highlights the efficiency of modern SAT solvers in handling thousands of variables and their significance in solving complex mathematical problems, such as the Boolean Pythagorean triples problem, which required a 68-gigabyte proof generated over two days of computation.
🧠 The Potential of Neural Networks in Pure Mathematics
The final paragraph explores the application of neural networks in pure mathematical research, referencing a paper by Adam Wagner that used neural networks to find counterexamples to problems in combinatorics. It describes the cross entropy method, a technique that trains a neural network to generate graphs likely to disprove a conjecture, and then refines the network based on the closest attempts. The paragraph concludes by acknowledging the potential of AI as a tool for mathematicians, without expecting it to replace them, and expresses excitement for future applications of AI in mathematical research.
Mindmap
Keywords
💡Multiplication
💡Statistical Observations
💡Language Models
💡SAT Solver
💡Boolean Satisfiability Problem
💡Heuristics
💡Neural Networks
💡Cross Entropy Method
💡Combinatorics
💡AI in Math Research
Highlights
ChatGPT struggles with multiplication due to its predictive nature rather than true understanding.
AI's middle digits in multiplication are often incorrect because it relies on statistical patterns rather than computation.
Large language models like ChatGPT are not yet capable of outperforming mathematicians in complex tasks.
AI is being utilized by mathematicians for research, particularly with SAT solvers.
SAT solvers are used to determine if Boolean sentences can be satisfied with true or false values.
Modern SAT solvers can efficiently solve problems with thousands of variables using heuristics and optimizations.
The Boolean Pythagorean triples problem was resolved using a SAT solver, producing a 68 gigabyte proof.
SAT solvers are powerful but require the skill to convert problems into Boolean sentences.
Neural networks have been used in pure math research to find counterexamples to conjectures.
Adam Wagner used neural networks and the cross entropy method to disprove combinatorics conjectures.
The cross entropy method trains a neural net to generate graphs that may serve as counterexamples.
This technique retrains the neural net based on the graphs closest to disproving a conjecture.
AI techniques like neural networks could save mathematicians time by disproving false conjectures.
AI is unlikely to replace mathematicians but can be a valuable tool in their research.
The potential of AI in pure math research is vast, with the possibility of finding examples that no human could.