# Explain to a Six-year-old

Cryptographic Hash Functions

Carathéodory’s Extension Theorem

Monoids in the Category of Endofunctors

Dirac Notation

Gödel's Incompleteness Theorems

GPT Nailed It, Episode 4.
ChatGPT version: Mar 14, 2023; model: gpt-3.5-turbo

## Cryptographic Hash Functions

Chat transcript (translated from Chinese with the help of ChatGPT; substantially revised both manually and by ChatGPT)
• How to explain cryptographic hash functions to a six-year-old child?

• Explaining cryptographic hash functions to a six-year-old child might be challenging because it is a relatively complex concept. Here is a simple analogy that you can use to help explain:

"Did you know that when we play treasure hunt games, we usually use a map to find the treasure's location? Now imagine a special way of hiding treasure that is unique because the treasure map only tells us where the treasure is, but it doesn't show us the path to get there. Therefore, only those who already know the way can get to the treasure. Cryptographic hash functions are just like this special way of hiding treasure. They can hide our treasure (such as online accounts) and only those who know the correct path (the correct password) can find the treasure. Also, cryptographic hash functions can help us determine if someone is the real treasure owner because only the owner knows the correct path. So, cryptographic hash functions can help us protect our important information and prevent others from easily finding and stealing it."

## Carathéodory’s Extension Theorem

Chat transcript
• How to explain Carathéodory’s Extension Theorem to a six-year-old child?

• Carathéodory's Extension Theorem is a mathematical concept that is quite complex and may be difficult for a six-year-old to understand. However, if you'd like to introduce the concept to a child, you could try using an analogy or a simplified explanation.

One possible way to explain it is by using the idea of building blocks. Imagine that you have a set of blocks of different shapes and sizes. You can use these blocks to build many different structures, but sometimes you might need a bigger block or a different shape to complete your design. Carathéodory's Extension Theorem is like adding more blocks to your set so that you can build even more complex structures.

In mathematical terms, the theorem states that if you have a set of simple functions (like building blocks) that can be combined in different ways, you can extend them to a larger set of more complex functions. This is useful because it allows mathematicians to work with more complicated functions that can describe more complex phenomena, like waves or fluid dynamics.

Of course, this is a simplified explanation and the actual theorem is much more complex. However, it might be a good starting point to introduce the idea to a child and encourage their curiosity about mathematics.

## Monoids in the Category of Endofunctors

Chat transcript
• How to explain monoids in the category of endofunctors to a six-year-old child?

• Hi! I think I can help you explain monoids in the category of endofunctors to a six-year-old child.

So, imagine you have a big toy box filled with different toys, like dolls, cars, and building blocks. You love playing with these toys and making all sorts of fun things with them.

Now, let's imagine that you have a special toy that's really cool. It's not just any toy - it's a toy that you can use to make other toys! This special toy is like a magic wand that can turn a car into a boat or a doll into a robot.

In math, we call this special toy an "endofunctor." An endofunctor is a way of transforming one thing into another thing. But not just any thing - endofunctors only work on one particular type of thing.

Now, let's say you have two of these special toys - one that turns cars into boats and another that turns boats into airplanes. If you use the car-to-boat toy first and then the boat-to-airplane toy, you end up with an airplane.

In math, we call this a "composition" of endofunctors. You're using one endofunctor and then using another one on the result. When you do this, you get a new endofunctor that's like the combination of the two toys.

Finally, let's say you have a special rule that says whenever you use a toy, you have to put it back in the toy box. You can't keep it out to play with later.

In math, we call this a "monoid." A monoid is like a special set of toys that you can play with, but you have to follow certain rules when you use them.

So, when we talk about monoids in the category of endofunctors, we're talking about special combinations of these toy-like tools that can be used to turn one thing into another thing, and we have to follow certain rules when we use them.

I hope that helps explain monoids in the category of endofunctors in a way that a six-year-old can understand! Let me know if you have any other questions.

## Dirac Notation

Chat transcript (translated from Chinese with the help of ChatGPT)
• How to explain the Dirac notation in quantum mechanics to a six-year-old child?

• Explaining the Dirac notation in quantum mechanics to a six-year-old child may be challenging as it requires some basic knowledge of mathematics and physics concepts. However, we can try to explain it through metaphors.

Firstly, you can tell the child that the Dirac notation is like a magical symbol that helps us understand some of the laws in the microscopic world. We can imagine it as a mystical symbol, just like the ones that wizards draw in books. However, this symbol is not used for magic, but to help us study the movement of particles in the microscopic world.

You can tell the child that the Dirac notation was invented by a British physicist named Paul Dirac. This symbol can help us predict the movement of electrons and explain why electrons can only exist in specific energy levels.

You can use some concrete examples to help the child understand. For instance, you can tell the child to imagine a glass ball rolling in a bowl. This glass ball can only roll in the bowl and cannot jump out of it. In the microscopic world, electrons are also like glass balls, they can only exist in specific energy levels and cannot jump between energy levels arbitrarily.

Lastly, you can tell the child that although the Dirac notation may seem magical, it is just a symbol that helps us understand the laws of nature. We still need to continue learning and exploring to better understand the mysteries of the microscopic world.

## Gödel's Incompleteness Theorems

Chat transcript (translated from Chinese with the help of ChatGPT)
• How to explain Gödel's incompleteness theorems to a six-year-old child?

• Explaining Gödel's incompleteness theorems to a six-year-old child may be challenging, as it is a relatively complex mathematical theory. However, you can try to introduce the concept in a simple way.

Firstly, you can tell the children that some problems cannot be solved using mathematical formulas or methods. Just like some riddles or problems, we may need some other tricks or methods to find the answer.

Then, you can try to explain the core idea of Gödel's incompleteness theorems, which is that some statements within a mathematical system cannot be proven or disproven within that system. This means that even if a system seems to have many rules and formulas, there are still some propositions that cannot be proven or disproven.

You can use analogies to help children understand this concept. For example, you can ask them to imagine a maze with many doors and passages. However, some doors and passages cannot be accessed because the rooms they lead to are locked, and we do not have the keys. These locked doors and passages are like propositions within a mathematical system that cannot be proven or disproven; we cannot enter these rooms, nor can we solve these problems.

Finally, you can tell children that although Gödel's incompleteness theorems may sound abstract and mysterious, they are actually an interesting theory in mathematics. They make us realize that even if we think we know a lot, there are always some things that cannot be fully understood or solved. This also reminds us to remain humble and curious, to keep learning and thinking.