An introduction to mathematical theorems - Scott Kennedy

Summary

This script explains the critical role of proofs in mathematics, using Euclid's axiomatic approach as a foundation. It illustrates how proofs ensure the validity of mathematical theories and their applications in various fields. An example proof demonstrates congruence in triangles, emphasizing the importance of proofs in building reliable knowledge. The script also humorously highlights proofs' relevance to winning arguments and the potential for significant rewards, like the Clay Mathematics Institute's million-dollar prize for solving 'Millennium Problems.'

Takeaways

• 📐 Proofs are essential in mathematics for establishing the validity of theorems.
• 🌟 Euclid is considered the father of geometry and was instrumental in formalizing mathematical proofs.
• 🏗 Axioms are the foundational rules upon which proofs are built.
• 🔍 Proofs ensure that mathematical theories are built on solid, verifiable foundations.
• 🏛 The concept of proof is not just limited to mathematics but extends to various fields like architecture and engineering.
• 📝 A proof involves using axioms to demonstrate that a theorem is true beyond doubt.
• 🔑 The side-side-side (SSS) congruence theorem is an example of how proofs can establish the equivalence of geometric figures.
• 🎯 The midpoint of a line segment is a key concept used in proving the congruence of triangles.
• 🏆 The Clay Mathematics Institute offers a million dollars for solving 'millenium problems', highlighting the importance of proofs.
• 🌐 Proofs are integral to advancements in technology, art, and security, among other fields.
• 🍮 The phrase 'the proof is in the pudding' metaphorically emphasizes the value of empirical evidence, akin to mathematical proofs.

Q & A

What is the significance of proofs in mathematics?

Proofs provide a solid foundation for various fields, including mathematics, to build and test theories. They ensure that mathematical theorems are true beyond doubt.

Who is Euclid and why is he important?

Euclid, who lived in Greece about 2,300 years ago, is considered the father of geometry. He revolutionized the way mathematics is written, presented, and thought about by formalizing it through axioms.

What are axioms in the context of mathematics?

Axioms are the fundamental rules or principles that are accepted as true without proof and are used as the basis for logical reasoning and argument.

How do proofs help in building mathematical theories?

Proofs use well-established axioms to demonstrate that a theorem is true. Once proven, these theorems can be used as building blocks to construct further mathematical theories.

What is the importance of proving two triangles are congruent?

Proving two triangles are congruent, meaning they have the same size and shape, is important in geometry as it allows for the application of various geometric principles and theorems.

How does the midpoint of a line segment help in proving triangle congruence?

The midpoint of a line segment helps in proving triangle congruence by establishing that two sides of the triangles are congruent, as the segments from the midpoint to the ends of the segment are equal.

What is the reflexive property in the context of congruence?

The reflexive property states that every object is congruent to itself, which is used to prove that a shared side in two triangles is congruent.

What does QED stand for and what does it signify?

QED stands for 'quod erat demonstrandum,' which is Latin for 'what was to be proven.' It is used to mark the end of a proof, signifying that the proof has been successfully completed.

Why should one study proofs?

Studying proofs can help win arguments, has potential monetary rewards, and is essential in fields like architecture, art, computer programming, and internet security.

What is the 'side-side-side' congruence theorem for triangles?

The 'side-side-side' (SSS) congruence theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

How does the concept of proofs relate to the phrase 'the proof is in the pudding'?

The phrase 'the proof is in the pudding' is a metaphor for the idea that the value of something can be judged by its results. In the context of the script, it humorously connects the importance of proofs to the enjoyment of pudding.

Outlines

00:00

📐 Introduction to Proofs in Mathematics

The paragraph introduces the concept of proof in mathematics, emphasizing its importance across various disciplines. It highlights Euclid, known as the father of geometry, who formalized mathematics through axioms. The paragraph explains that proofs are essential for establishing the validity of theorems and ideas, using the analogy of a house built on a solid foundation. An example is given to demonstrate how to prove two triangles are congruent by showing all three sides are congruent, utilizing the midpoint and reflexive property. The paragraph concludes with the historical practice of marking the end of a proof with 'QED', signifying 'quod erat demonstrandum'.

Keywords

💡Proof

A proof in mathematics is a logical argument that establishes the truth or validity of a theorem or proposition. It is a rigorous demonstration that relies on previously established theorems, axioms, or definitions. In the script, proofs are highlighted as the foundation for various disciplines, including mathematics, logic, and engineering. The importance of proofs is emphasized through the example of proving that two triangles are congruent, which is a fundamental concept in geometry.

💡Axioms

Axioms are fundamental, self-evident principles or assumptions that are accepted without proof. They serve as the starting point for logical reasoning and derivation of theorems. In the context of the video, Euclid is credited with establishing axioms as the 'rules of the game' in mathematics, which must be used to prove theorems. Axioms are the basis upon which the structure of mathematical proofs is built.

💡Euclid

Euclid of Alexandria, referenced in the script, is a seminal figure in the history of mathematics, often referred to as the 'father of geometry'. His work, 'Elements', laid the foundation for geometrical proofs and formalized the approach to mathematics that is still used today. Euclid's influence is evident in the script's discussion of proofs and the historical context in which he revolutionized mathematical thought.

💡Theorem

A theorem is a statement that has been proven on the basis of previously established statements, such as axioms or other theorems. The script uses the example of proving the congruence of two triangles to illustrate how a theorem is proven. The importance of theorems is underscored by the fact that they are the building blocks of mathematical knowledge.

💡Congruent

In geometry, two figures are said to be congruent if they have the same shape and size. The script provides an example of proving that two triangles are congruent by showing that all corresponding sides and angles are equal. This concept is central to the discussion of proofs in geometry and serves as a practical application of the proof concept.

💡Midpoint

The midpoint of a line segment is the point that divides it into two equal parts. In the script, the concept of midpoint is used to establish that two sides of the triangles are congruent, which is a step in proving that the triangles are congruent overall. This term is integral to the script's proof example and demonstrates the application of geometric definitions in proofs.

💡Reflexive Property

The reflexive property states that any object is identical to itself. In the script, this property is used to assert that the shared side of the two triangles is congruent to itself, which is a step in the proof. The reflexive property is a fundamental concept in logic and mathematics that is often used in proofs to establish self-evidence.

💡QED

QED is an abbreviation for 'quod erat demonstrandum', a Latin phrase meaning 'what was to be demonstrated'. It is traditionally used at the end of a mathematical proof to indicate that the proof is complete. The script mentions Euclid's use of QED, highlighting the historical continuity and the sense of accomplishment associated with completing a proof.

💡Millenium Problems

The Clay Mathematics Institute's Millenium Problems are a set of seven unsolved problems in mathematics, with a $1,000,000 prize offered for the solution to each. The script mentions these problems to illustrate the real-world significance and potential impact of proofs in mathematics. Solving these problems would represent a monumental achievement in the field.

💡Pudding

The phrase 'the proof is in the pudding' is an idiom used in the script to humorously emphasize the practical value and necessity of proofs. It suggests that the effectiveness or value of something can only be judged by its results. In the context of the video, it light-heartedly reinforces the importance of proofs in confirming the validity of mathematical theories.

Highlights

Proofs provide a solid foundation for various disciplines.

Euclid is considered the father of geometry and a pioneer of proofs.

Euclid revolutionized the way mathematics is written, presented, and thought about.

Axioms are the fundamental rules of mathematics established by Euclid.

Proofs are essential to validate theorems and avoid building on false premises.

Proofs ensure the integrity of mathematical structures, like a house built with correct beams.

A proof demonstrates that a theorem is true beyond a doubt using established rules.

The example of proving two triangles are congruent illustrates the process of a proof.

The midpoint of a line segment is a key concept used in proving triangle congruence.

Congruence of sides is a method to prove that two triangles are the same size and shape.

The reflexive property states that everything is congruent to itself.

The side-side-side congruence theorem is used to prove triangle congruence.

QED is used to mark the end of a proof, signifying what was to be proven.

Studying proofs can sharpen the mind and help win arguments.

The Clay Mathematics Institute offers a million dollars for solving 'the millenium problems'.

Proofs are integral to architecture, art, computer programming, and internet security.

Proofs are a universal concept, as illustrated by the saying 'the proof is in the pudding'.

Transcripts

What is proof?

And why is it so important in mathematics?

Proofs provide a solid foundation for mathematicians

logicians, statisticians, economists, architects, engineers,

and many others to build and test their theories on.

And they're just plain awesome!

Let me start at the beginning.

I'll introduce you to a fellow named Euclid.

As in, "here's looking at you, Clid."

He lived in Greece about 2,300 years ago,

and he's considered by many to be the father of geometry.

So if you've been wondering where to send your geometry fan mail,

Euclid of Alexandria is the guy to thank for proofs.

Euclid is not really known for inventing or discovering a lot of mathematics

but he revolutionized the way in which it is written,

presented, and thought about.

Euclid set out to formalize mathematics by establishing the rules of the game.

These rules of the game are called axioms.

Once you have the rules,

Euclid says you have to use them to prove what you think is true.

If you can't, then your theorem or idea

might be false.

And if your theorem is false, then any theorems that come after it and use it

might be false too.

Like how one misplaced beam can bring down the whole house.

So that's all that proofs are:

using well-established rules to prove beyond a doubt that some theorem is true.

Then you use those theorems like blocks

to build mathematics.

Let's check out an example.

Say I want to prove that these two triangles

are the same size and shape.

In other words, they are congruent.

Well, one way to do that is to write a proof

that shows that all three sides of one triangle

are congruent to all three sides of the other triangle.

So how do we prove it?

First, I'll write down what we know.

We know that point M is the midpoint of AB.

We also know that sides AC and BC are already congruent.

Now let's see. What does the midpoint tell us?

Luckily, I know the definition of midpoint.

It is basically the point in the middle.

What this means is that AM and BM are the same length,

since M is the exact middle of AB.

In other words, the bottom side of each of our triangles are congruent.

I'll put that as step two.

Great! So far I have two pairs of sides that are congruent.

The last one is easy.

The third side of the left triangle

is CM, and the third side of the right triangle is -

well, also CM.

They share the same side.

Of course it's congruent to itself!

This is called the reflexive property.

Everything is congruent to itself.

I'll put this as step three.

Ta dah! You've just proven that all three sides of the left triangle

are congruent to all three sides of the right triangle.

Plus, the two triangles are congruent

because of the side-side-side congruence theorem for triangles.

When finished with a proof, I like to do what Euclid did.

He marked the end of a proof with the letters QED.

It's Latin for "quod erat demonstrandum,"

which translates literally to

"what was to be proven."

But I just think of it as "look what I just did!"

I can hear what you're thinking:

why should I study proofs?

One reason is that they could allow you to win any argument.

Abraham Lincoln, one of our nation's greatest leaders of all time

used to keep a copy of Euclid's Elements on his bedside table

to keep his mind in shape.

Another reason is you can make a million dollars.

You heard me.

One million dollars.

That's the price that the Clay Mathematics Institute in Massachusetts

is willing to pay anyone who proves one of the many unproven theories

that it calls "the millenium problems."

A couple of these have been solved in the 90s and 2000s.

But beyond money and arguments,

proofs are everywhere.

They underly architecture, art, computer programming, and internet security.

If no one understood or could generate a proof,

we could not advance these essential parts of our world.

Finally, we all know that the proof is in the pudding.

And pudding is delicious. QED.