Have you ever wondered if artificial intelligence truly thinks, or does it simply parrot back what it's already seen? This question isn't just for tech nerds and philosophers anymore. It's becoming crucial for all of us as AI tools like ChatGPT become part of our daily lives—helping students with homework, assisting professionals with complex problems, and even attempting to solve ancient mathematical puzzles that stumped the brightest minds of ancient Greece.
Welcome to FreeAstroScience.com, where we break down complex scientific principles into language that actually makes sense. We're here because we believe you shouldn't need a PhD to understand the world around you. Today, we're diving into a fascinating experiment that bridges 2,400 years of human thought: what happens when you give ChatGPT the same geometry problem that Socrates used to challenge his students?
We invite you to join us on this journey. By the end, you'll understand something profound about both artificial intelligence and human learning—and you might never look at AI the same way again. Trust us, this one's worth your time.
What's This Ancient Problem That's Causing All the Fuss?
Let's start with the basics. Picture a square. Any square. Now, here's the challenge: How do you create a new square that has exactly twice the area of the original?
Sounds simple, right? Your first instinct might be to double the length of each side. If the original square has sides of 2 meters, just make them 4 meters, and you're done. Problem solved.
Except... you'd be wrong. And you'd be making the exact same mistake that a slave boy made in ancient Athens around 380 BCE .
Here's what actually happens when you double the sides:
Square Type | Side Length | Area Calculation | Result |
---|---|---|---|
Original Square | 2 meters | 2 × 2 | 4 m² |
If We Double the Sides | 4 meters | 4 × 4 | 16 m² (4× the area!) |
What We Actually Need | √8 ≈ 2.83 meters | √8 × √8 | 8 m² (2× the area) |
The mathematical formula that expresses this relationship:
If the original square has side length s, the new square needs side length s√2
New Area = (s√2)² = 2s²
The elegant geometric solution? Draw the diagonal of your original square. That diagonal becomes the side of your new square . Beautiful, isn't it? The ancient Greeks didn't have algebra, but they had geometry—and they used it brilliantly.
Why Socrates Cared About This Problem (And Why You Should Too)
Here's where it gets interesting. The philosopher Socrates wasn't just teaching geometry. He was making a radical claim about the nature of knowledge itself.
In Plato's dialogue "Meno," Socrates brings in an uneducated slave boy who's never studied mathematics. Through careful questioning—without directly teaching—Socrates guides the boy to discover the solution. The boy makes mistakes, gets frustrated, but eventually reaches the correct answer .
Socrates' stunning conclusion? We don't actually learn new things. We remember what our souls already knew from a previous existence.
This theory, called anamnesis or recollection, sparked a philosophical debate that's raged for over two millennia. Do we have innate knowledge, or do we build everything from experience?
Fast-forward to today. We're asking the same question about artificial intelligence.
So, How Did ChatGPT Handle Socrates' Test?
In early 2024, researchers from the University of Cambridge and the Hebrew University of Jerusalem decided to run a fascinating experiment. They gave ChatGPT-4 the same geometry problem that Socrates posed to the slave boy .
Their goal? Figure out whether ChatGPT's mathematical "knowledge" comes from:
- Memory (retrieving information from its training data)
- Generation (actually reasoning through problems in real-time)
The results were... complicated. And fascinating.
Round One: ChatGPT Aces the Original Problem
When first presented with the doubling the square problem, ChatGPT didn't fall into the trap. Instead of suggesting to double the sides, it immediately provided a correct algebraic solution, calculating that the new square needed sides of √8 meters .
But here's the twist: ChatGPT didn't mention Plato's elegant geometric solution at all. It went straight for algebra—solving it like a modern math student, not like an ancient Greek geometer .
Was this retrieved knowledge or generated reasoning? The researchers couldn't tell for sure. But it gets more interesting.
Round Two: When Things Got Messy
The researchers then modified the problem slightly. Instead of a square, they asked about doubling a rectangle's area while keeping its proportions .
This is where ChatGPT stumbled.
The AI suggested using the rectangle's diagonal—reasoning by analogy from the square solution. Sounds logical, right? Except it doesn't work . The diagonal of a rectangle doesn't create a proportionally similar rectangle with twice the area.
This mistake was the breakthrough moment.
Why? Because this specific error probably isn't sitting in ChatGPT's training data. No mathematics textbook teaches the wrong solution. This appeared to be ChatGPT trying to solve a problem it hadn't seen before, making an intuitive but incorrect leap .
In other words, it seemed to be generating new reasoning, not just recalling stored answers.
Round Three: Can AI Learn from Its Mistakes?
Here's where it gets really exciting. When the researchers gave ChatGPT hints and guidance—showing it the correct geometric construction—the AI managed to apply similar reasoning to yet another problem: doubling the area of an equilateral triangle .
It wasn't perfect. It needed prompting. But it showed something that looked remarkably like learning.
The "Aha" Moment: AI Has a Learning Zone Too
This brings us to one of the most profound insights from this research: the concept of ChatGPT's Zone of Proximal Development .
If you've studied education or psychology, you might recognize this term. The Zone of Proximal Development (ZPD) was introduced by Soviet psychologist Lev Vygotsky. It's the sweet spot between what you can do alone and what you can do with help from someone more knowledgeable .
For a child learning math, ZPD problems are those they can't solve independently but can solve with guidance from a teacher.
The researchers discovered that ChatGPT has something similar. There are problems it can't solve by itself, but with the right prompting—the right hints and guidance from a knowledgeable user—it can reach correct solutions .
Think about that for a moment. We're not just asking AI to retrieve information. We're potentially collaborating with it, helping it work through problems at the edge of its capabilities.
So, Does ChatGPT Really "Think"?
Let's be honest here. We need to be careful about anthropomorphizing AI—attributing human qualities to machines .
ChatGPT doesn't have consciousness. It doesn't "understand" mathematics the way you or we do. It's essentially a sophisticated pattern-matching system, trained on enormous amounts of text, predicting what words should come next based on statistical relationships .
But here's what makes it fascinating: From a user's perspective, ChatGPT's responses can feel remarkably thought-like. When it makes intuitive mistakes—like assuming a rectangle's diagonal would work the same way as a square's—it's exhibiting behavior that resembles human reasoning .
The researchers found evidence of both types of knowledge in ChatGPT's responses:
Retrieved knowledge: When ChatGPT eventually provided Plato's classic geometric solution, it was likely pulling from its training data
Generated knowledge: When it made creative errors on modified problems, it appeared to be reasoning through unfamiliar territory
The line between these isn't always clear. And maybe that's the point.
What This Means for Real People Using AI
We aren't just talking about abstract philosophy here. This research has practical implications for anyone using AI tools.
The Way You Ask Matters Enormously
The researchers discovered that their prompts significantly influenced ChatGPT's responses. When they asked for "elegant" solutions, the AI shifted from algebra to geometry. When they provided hints after mistakes, the AI corrected course .
This means: The quality of your interaction with AI depends heavily on how you prompt it.
If you want AI to help you explore a problem, try prompts like: "Let's work through this together" or "Can you think of another approach?"
If you want AI to retrieve established knowledge, be direct: "What's the standard solution to this problem?"
AI Makes Mistakes—And That's Actually Useful
ChatGPT's errors on the rectangle problem weren't bugs. They were features—windows into how the system processes unfamiliar challenges .
When using AI for mathematics, science, or any complex reasoning:
- Don't assume the first answer is correct
- Check the logic yourself
- Use AI's mistakes as opportunities to guide it toward better solutions
- Think of yourself as a teacher, not just a user
The Future of Human-AI Collaboration
The researchers envision systems where AI and humans work together on mathematical exploration—where chatbots combined with dynamic geometry software could help students discover mathematical principles intuitively .
We're not there yet. But we're seeing glimpses of what's possible.
The Deep Questions We Can't Ignore
This experiment forces us to confront some profound questions about knowledge itself.
Where Does Knowledge Come From?
Socrates believed knowledge was innate—already within us, waiting to be remembered. John Locke and other empiricists argued we're born as "blank slates," learning everything from experience .
ChatGPT complicates this ancient debate. Its "knowledge" comes from training data (like innate knowledge?) but it also generates novel responses to unfamiliar problems (like learned knowledge?) .
Maybe the dichotomy was always too simple. Maybe knowledge is always a dance between what we already have and what we create in the moment.
Can Machines Really Learn?
We use the word "learning" for AI, but is it the same as human learning?
When ChatGPT adjusted its approach after receiving hints, was it learning? Or just executing algorithms designed to appear learning-like?
The researchers acknowledged they don't have definitive answers. But from a user's experiential perspective, it feels like learning .
And in a practical sense, maybe that's enough.
What Makes Us Human?
If machines can solve problems, make intuitive leaps, and learn from mistakes—even if through completely different mechanisms than our brains—what does that mean for human uniqueness?
We don't think it diminishes us. If anything, this research highlights what's extraordinary about human cognition: our flexibility, our ability to teach and guide, our capacity to recognize when something is wrong and adjust.
The researchers needed deep mathematical knowledge to design their experiment, interpret ChatGPT's responses, and guide it productively . The AI couldn't do any of that on its own.
What We Learned From This 2,400-Year Journey
Let's bring this home.
We started with Socrates in ancient Athens, testing a slave boy with a geometry problem. We ended with modern researchers testing an AI with the same challenge. Across those millennia, the fundamental questions remain:
What is knowledge?
Where does it come from?
Can we truly create new understanding, or only rearrange what already exists?
The ChatGPT experiment doesn't answer these questions definitively. But it gives us new ways to think about them .
We learned that:
- AI's "knowledge" appears to be a complex mix of retrieval and generation
- The way we interact with AI dramatically shapes what it can accomplish
- Mistakes in AI systems can reveal fascinating insights about reasoning processes
- There's a "zone" where AI can work productively with human guidance
- The ancient questions about knowledge remain as relevant as ever—now applied to silicon as well as neurons
Your Mind Is Your Most Powerful Tool—Keep It Active
At FreeAstroScience.com, we have a core belief: Never turn off your mind. Keep it active, curious, questioning. As the Spanish painter Francisco Goya warned, "The sleep of reason produces monsters."
This applies doubly when interacting with AI. These tools are powerful, but they're not infallible. They need your active engagement, your critical thinking, your human judgment.
Don't accept AI outputs blindly. Question them. Test them. Guide them. Think of yourself as Socrates, drawing out better answers through skillful questioning.
We're living through a remarkable moment in history. AI is becoming more sophisticated daily. The philosophical questions that fascinated ancient Greeks are now practical concerns for anyone with a smartphone.
But here's the beautiful part: Understanding these systems better makes us more effective users. Recognizing their limitations helps us leverage their strengths. Knowing when to trust and when to question makes us better thinkers.
We hope this exploration has given you a deeper understanding of AI, knowledge, and the enduring relevance of ancient wisdom. The conversation between Socrates and the slave boy continues—now including silicon participants.
Come back to FreeAstroScience.com soon. We'll keep exploring these fascinating intersections of science, philosophy, and technology. Because your curiosity deserves clear answers, and your mind deserves to stay wonderfully, actively engaged with the incredible world we're building together.
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