ChatGPT can't multiply, but can AI do math?
TLDRThe video script discusses the limitations of AI in performing mathematical tasks, such as multiplication, due to its reliance on statistical predictions rather than true understanding. It highlights the use of AI in mathematical research, particularly through SAT solvers, which efficiently solve complex Boolean satisfiability problems, as demonstrated in the Boolean Pythagorean triples problem. Additionally, the script mentions the innovative application of neural networks by Adam Wagner to find counterexamples in combinatorics, using the cross entropy method to generate potential disproofs of mathematical conjectures. While AI is not set to replace mathematicians, it is becoming a valuable tool in their research.
Takeaways
- 🤖 ChatGPT struggles with multiplication because it makes predictions based on patterns in text rather than understanding the mathematical process.
- 🧠 ChatGPT's initial and final digits in multiplication might be correct due to statistical observations, but it fails with the middle digits as they require understanding the entire number sequence.
- 📚 AI is currently being used by mathematicians for research, indicating a collaborative relationship between humans and technology in the field of mathematics.
- 🔍 SAT solvers are a type of AI used in mathematical research to solve complex Boolean satisfiability problems efficiently.
- 🌐 The Boolean Pythagorean triples problem was resolved using a SAT solver, demonstrating the practical application of AI in solving mathematical proofs.
- 📉 SAT solvers, while powerful, require human ingenuity to convert problems into Boolean sentences and cannot replace the creative problem-solving of mathematicians.
- 🔢 Neural networks, another form of AI, have been used to find counterexamples in combinatorics, showcasing their potential in pure mathematical research.
- 💡 The cross entropy method, used in conjunction with neural networks, is a technique that can generate potential counterexamples to mathematical conjectures, saving time for mathematicians.
- 🔄 This method involves training a neural network to predict counterexamples, computing their validity, and retraining the network based on the results to improve accuracy.
- 🛠 AI is not expected to replace mathematicians but is seen as an additional tool that can aid in their research and problem-solving processes.
- 🔮 The future of AI in mathematics is promising, with the potential to uncover solutions and examples that may be beyond human capacity to find in a reasonable time.
Q & A
Why does ChatGPT sometimes fail at multiplication despite computers being able to perform the task easily?
-ChatGPT operates by making predictions based on text it has seen before, rather than truly understanding the mathematical operations. It can make statistical observations about the start and end digits of a multiplication result but struggles with the middle digits, which depend on all input digits and require more complex understanding.
What is a SAT solver and how is it used in mathematical research?
-A SAT solver is a software tool used to solve Boolean satisfiability questions. It determines if it's possible to substitute the variables in a given sentence with 'true' or 'false' to make the sentence true. SAT solvers are particularly useful for problems that can be converted into Boolean sentences and have been instrumental in solving complex mathematical problems, such as the Boolean Pythagorean triples problem.
How do modern SAT solvers manage to solve problems with thousands of variables efficiently?
-Modern SAT solvers use heuristics and optimizations to solve problems that would normally require checking an exponential number of possibilities. They have become very efficient in practice, allowing them to handle complex sentences with a large number of variables.
What is the Boolean Pythagorean triples problem and how was it resolved using a SAT solver?
-The Boolean Pythagorean triples problem asks if you can color the positive integers red and blue in such a way that no Pythagorean triple is all of one color. A SAT solver was used in 2016 to resolve this problem, with the proof generated after two days of computation, resulting in a 68 gigabyte proof file.
Can neural networks be used for pure mathematical research and how?
-Yes, neural networks can be used for pure mathematical research. For example, Adam Wagner used neural networks to find counterexamples to problems in combinatorics. He employed the cross entropy method to train a neural network to predict how to build graphs that could potentially disprove a conjecture.
How does the cross entropy method work in the context of finding counterexamples to mathematical conjectures?
-The cross entropy method involves training a neural network on certain graphs to predict how to construct graphs that are likely to be counterexamples to a given conjecture. The network generates many such graphs, and the ones that are closest to disproving the conjecture are used to retrain the network. Over time, this iterative process aims to find a graph that successfully disproves the conjecture.
Why is it unlikely that AI will replace mathematicians in the near future?
-AI lacks the deep understanding and creativity that mathematicians possess. While AI can be a powerful tool for solving specific types of problems or generating examples that would be time-consuming for humans, it still requires human insight to formulate problems, interpret results, and apply them in a broader mathematical context.
What is the role of AI in assisting mathematicians in their research?
-AI serves as a tool that can help mathematicians by automating certain tasks, generating examples, and solving specific types of problems more efficiently than humans. It can save time and provide new insights, but it is not a substitute for the intuition and expertise of mathematicians.
How does the accuracy of ChatGPT's predictions depend on the text it has been trained on?
-ChatGPT's accuracy in making predictions is heavily dependent on the quality and diversity of the text it has been trained on. The more comprehensive and varied the training data, the better ChatGPT can make accurate and relevant predictions.
What are some limitations of using AI for mathematical research?
-AI has limitations in understanding the underlying principles of mathematics and may struggle with problems that require deep conceptual understanding. Additionally, converting mathematical problems into a format that AI can process can be challenging and may not always be possible.
Can AI techniques other than SAT solvers and neural networks be used for mathematical research?
-Yes, there are various AI techniques beyond SAT solvers and neural networks that can be applied to mathematical research. These may include machine learning algorithms, data mining, and other forms of computational intelligence that can assist in pattern recognition, data analysis, and problem-solving.
What is the significance of the 68 gigabyte proof generated by the SAT solver in the context of the Boolean Pythagorean triples problem?
-The 68 gigabyte proof is significant because it demonstrates the capability of SAT solvers to handle and solve complex mathematical problems that are beyond human capacity to solve manually. It also highlights the potential of AI to contribute to mathematical research by providing solutions that would be impractical to obtain through traditional methods.
Outlines
🤖 AI's Limitations in Mathematical Computation
The paragraph discusses the limitations of AI, specifically ChatGPT, in performing mathematical operations like multiplication. It explains that while AI can predict the beginning and end digits of a product based on statistical observations from text, it struggles with the middle digits due to the complexity involved. The AI's predictions are based on patterns it has seen before, not an actual understanding of mathematics. This highlights the current gap between AI's capabilities and the expertise of mathematicians.
🔍 AI as a Tool in Mathematical Research
This section explores how AI is being utilized by mathematicians for research, focusing on SAT solvers. SAT solvers are used to determine if a Boolean satisfiability problem can be solved by substituting 'true' and 'false' for variables and applying reduction rules. Modern SAT solvers are efficient and can handle problems with thousands of variables. The paragraph provides an example of how a SAT solver was used to solve the Boolean Pythagorean triples problem in 2016, emphasizing that while powerful, these tools require human ingenuity to convert problems into a format they can solve.
🧠 The Potential of Neural Networks in Pure Mathematics
The final paragraph discusses the use of neural networks in pure mathematical research, citing a paper by Adam Wagner. Wagner used neural networks and the cross entropy method to find counterexamples to combinatorial conjectures. The method involves training a neural network to predict how to build graphs that could disprove a conjecture, generating many such graphs, and then retraining the network based on which ones were closest to disproving the conjecture. This technique is highlighted as a potentially time-saving tool for mathematicians and a novel way to apply AI in mathematical research.
Mindmap
Keywords
💡Multiplication
💡Predictions
💡Statistical Observations
💡Language Models
💡SAT Solver
💡Boolean Satisfiability
💡Heuristics
💡Combinatorics
💡Cross Entropy Method
💡Neural Networks
💡Counterexamples
Highlights
ChatGPT struggles with multiplication due to its prediction-based approach rather than true understanding.
ChatGPT's initial and final digits in multiplication are often correct, but it fails in the middle digits.
Large language models like ChatGPT rely on statistical observations rather than deep understanding of math.
AI is currently being used by mathematicians for research, despite not being able to replace them.
SAT solvers are a type of AI used in mathematical research to solve Boolean satisfiability problems.
SAT solvers can handle sentences with thousands of variables with the help of heuristics and optimizations.
The Boolean Pythagorean triples problem was resolved using a SAT solver in 2016.
A 68 gigabyte proof was generated by a SAT solver, showcasing the power of this AI tool.
Converting math problems into Boolean sentences is a clever task that requires human input.
Neural networks are being explored for pure math research, as demonstrated by Adam Wagner's work.
The cross entropy method is used to find counterexamples to combinatorics problems using neural networks.
Neural networks can be trained to generate graphs that are likely counterexamples to conjectures.
The process of retraining neural networks with closer disproofs can save mathematicians time.
AI techniques are expected to become another tool for mathematicians, not replace them.
The potential of AI in finding examples that no human would have time for is intriguing.