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Is it still worth studying discrete mathematics in the era of large language models?
Is it still worth studying discrete mathematics in the era of large language models?
Indeed, large language models are becoming increasingly good at solving many standard mathematical problems. However, they still make mistakes, both in calculations and in reasoning.
Thus, responsibility for verifying the result still lies with the user, and the ability to distinguish a correct solution from an incorrect one becomes even more important. To evaluate a solution, one must understand the mathematical language in which it is written; studying this language is one of the main goals of this textbook.
For example, a model might think that \(8.8-8.11=-0.31\). If you don't know how addition and subtraction work, you won't be able to recognize that the model has made a mistake. As another example, suppose you asked a model to prove that the \(n\)-th Fibonacci number is not less than \(2^{n/2}\) for all \(n \ge 6\). The model might produce the following solution:
We prove by induction on \(n\). Base case \(n=6\): \(F_{6}=8 \ge 8=2^{6/2}\). Induction step \(n-1 \to n\): \[F_{n} =F_{n-1}+F_{n-2}\ge 2^{\frac{n-1}{2}}+2^{\frac{n-2}{2}}=2^{\frac{n}{2} -1}(\sqrt{2}+1)>2^{\frac{n}{2}-1}\cdot 2 = 2^{\frac{n}{2}}.\]If you don't know what mathematical induction is, or if you have little experience working with it, you might not notice the mistake in this proof. A similar incorrect argument could even be used to “prove” the obviously false statement that all Fibonacci numbers are even.
At the same time, large language models often solve problems correctly, and discussing ideas and solutions with them can be useful. We recommend the following approach. Whenever you need to solve a problem, don't rush to read our solution or ask a large language model for a solution:
- Don't deprive yourself of the unique satisfaction of solving a problem!
- If you solve a problem yourself, both the solution and its methods will stick in your memory much better.
- Even if you fail to solve the problem, the time will not be wasted: you will try different approaches and identify exactly what makes the problem difficult. This, in turn, will help you study the given solution with greater interest and appreciation.