Terence Tao says AI could trigger math's biggest crisis since Gödel

Introduction

In this tutorial, we'll explore how to use a simple AI tool to help with mathematical problem-solving. As mathematician Terence Tao warns about AI's potential impact on mathematics, we'll learn how to work with AI assistance while maintaining human understanding. This hands-on approach will teach you how to use AI tools to check your math work and verify proofs, ensuring you maintain control over your mathematical reasoning.

Prerequisites

Before starting this tutorial, you should have:

• A computer with internet access
• A basic understanding of algebra (variables, equations)
• Access to an online AI tool (we'll use a free web-based tool)
• A notebook or document to write down your work

Step-by-Step Instructions

Step 1: Choose Your AI Math Tool

First, we need to select a simple AI tool that can help with mathematical problems. For this tutorial, we'll use a free online tool called Wolfram Alpha, which is excellent for checking math problems and understanding mathematical concepts.

Why: Using a tool like Wolfram Alpha allows us to verify our work and learn from AI assistance while still maintaining our own understanding of the mathematics.

1.1 Open Your Web Browser

Start by opening any web browser (Chrome, Firefox, Safari, etc.) on your computer.

1.2 Navigate to Wolfram Alpha

Type https://www.wolframalpha.com into your browser's address bar and press Enter.

1.3 Explore the Interface

Once the page loads, you'll see a large input box. This is where you'll enter your math problems. Notice how the interface is simple and focused on mathematical input.

Step 2: Learn Basic Math Input

Before we start solving problems, we need to understand how to enter math problems into the AI tool.

Why: Proper input formatting ensures the AI tool understands exactly what you're asking it to solve.

2.1 Try a Simple Equation

In the input box, type 2x + 5 = 15 and press Enter or click the equals sign.

2x + 5 = 15

This tells the AI to solve for x in the equation 2x + 5 = 15.

2.2 Observe the Results

After solving, the tool will show you the solution and often provide steps. Notice how it shows x = 5 and gives you the steps to solve it.

2.3 Try Another Example

Now try typing factor x^2 - 9 to see how the AI factors polynomials.

factor x^2 - 9

Step 3: Create Your Own Math Problems

Now that we understand how to input problems, let's create our own math challenges to solve with AI assistance.

Why: Creating your own problems helps you practice the math concepts and learn how to use AI as a learning tool rather than just a solution finder.

3.1 Write Down a Problem

Choose a basic algebra problem to solve. For example:

3x - 7 = 2x + 5

Write this problem in your notebook or document.

3.2 Solve It Yourself

Try solving the problem on paper first. This step is crucial - you need to understand the process before using AI to verify.

3.3 Enter the Problem in AI Tool

Go back to Wolfram Alpha and type your problem exactly as written:

3x - 7 = 2x + 5

3.4 Compare Results

Check if your answer matches the AI's solution. If they don't match, review your work and try again.

Step 4: Practice with Proof Verification

As Terence Tao warns, we're focusing on understanding what counts as a complete proof. Let's practice with a simple proof concept.

Why: This practice helps you understand how to verify mathematical reasoning, which is crucial as AI becomes more involved in mathematics.

4.1 Choose a Simple Proof Concept

Let's work with a basic proof: prove that the sum of two even numbers is always even.

4.2 Write Your Reasoning

First, write down your understanding:

Let's say we have two even numbers, a and b. By definition, a = 2m and b = 2n where m and n are integers.

Then a + b = 2m + 2n = 2(m + n).

Since (m + n) is an integer, 2(m + n) is even.

4.3 Use AI to Check Your Logic

Enter this reasoning into Wolfram Alpha using a query like:

prove that sum of two even numbers is even

Or just type:

even + even = even

Step 5: Learn to Question AI Responses

As Tao suggests, we must maintain critical thinking about AI assistance.

Why: AI tools can make mistakes or provide answers that seem correct but miss important details. Learning to question AI responses helps maintain mathematical rigor.

5.1 Try a Problem with Multiple Solutions

Enter a quadratic equation that has two solutions:

x^2 - 5x + 6 = 0

5.2 Compare with Your Understanding

After getting the AI's solution, verify it by substituting back into the original equation:

For x = 2: (2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0 ✓

5.3 Question the Process

Ask yourself: Does the AI show all the steps? Would a human mathematician explain it differently? What's missing from the AI's explanation?

Step 6: Document Your Learning

Finally, keep a record of your work to track how you're using AI as a learning tool.

Why: Keeping records helps you learn from each interaction and ensures you're not relying completely on AI without understanding the math.

6.1 Create a Math Journal

Keep a notebook where you write:

• Your original problem
• Your solution attempt
• AI's response
• What you learned
• Questions that arose

6.2 Reflect on the Process

After each session, ask yourself:

• Did AI help me understand the concept better?
• Could I explain this to someone else without AI?
• What parts did I still need to figure out myself?

Summary

In this tutorial, we've learned how to use AI tools like Wolfram Alpha to assist with mathematical problem-solving while maintaining our own understanding. We practiced solving equations, factoring polynomials, and working with proof concepts. Most importantly, we've learned to question AI responses and use these tools as learning aids rather than complete replacements for mathematical thinking.

As Terence Tao warns, the future of mathematics may involve AI, but human understanding and reasoning remain crucial. This hands-on approach ensures you're not just getting answers, but truly learning mathematics that AI can help verify and enhance.