The story of mathematical proof – with John Stillwell

The Royal Institution

16 Mar 202344:03

EducationalLearning

32 Likes 10 Comments

TLDRThe speaker emphasizes the importance of proof in mathematics, arguing that it is as fundamental as calculation. They explain the concept using the Pythagorean theorem and highlight the evolution of mathematical proofs from visual to algebraic methods. The history of algebra and calculus is discussed, showcasing the contributions of al-Khwarizmi, Descartes, and Newton. The talk delves into the development of logic and computation, touching on the works of Leibniz, Boole, and Frege. It concludes with insights from Cantor, Dedekind, and Gödel, illustrating the inherent limitations of proof and computation in mathematics, underscoring the perpetual need for new mathematical ideas.

Takeaways

📚 The speaker emphasizes the importance of proof in mathematics, which is often overlooked in favor of calculation.
🧩 The Pythagorean theorem is highlighted as a fundamental concept in geometry, with a visual proof involving rearranging triangles to form squares.
🤔 The speaker raises a critical point about the certainty of visual proofs, suggesting that more rigorous methods like Euclid's are necessary for true validation.
📖 Thomas Hobbes' initial disbelief and subsequent appreciation of Euclid's proof illustrates the power of logical deduction from axioms in geometry.
🔍 The Greeks' rigorous approach to proof was partly due to the discovery of the irrationality of the square root of two, which challenged their numerical theories.
📐 The axiomatic method of Euclid, starting from self-evident statements and using logic to derive theorems, has become a model for mathematical development.
🌐 Algebra, which developed separately from Greek mathematics, was justified geometrically, especially in the early methods for solving quadratic equations by 'completing the square'.
📈 The evolution of algebra into a more arithmetic-like discipline allowed for the abstraction from geometric representations to symbolic manipulation.
⭕️ Descartes' analytical geometry and Newton's 'universal arithmetic' show the increasing dominance of algebraic methods in describing geometric figures and solving problems.
🔢 The introduction of calculus in the 17th and 18th centuries provided a powerful tool for calculating areas, volumes, slopes, and lengths, often with simpler methods than previous geometric approaches.
💡 Leibniz's idea that logic could be performed by calculation foreshadowed the development of algorithms and the recognition that computation has its inherent limitations.

Q & A

What is the main theme of the speaker's discussion in the script?
-The main theme of the speaker's discussion is the relationship between proof and calculation in mathematics, emphasizing the importance of proof and how it has evolved through history.
Why does the speaker believe that proof doesn't get enough recognition in mathematics teaching?
-The speaker believes that proof doesn't get enough recognition because while everyone knows mathematics is about calculation, it is more fundamentally about proof, which is a concept that is often overlooked.
What is the Pythagorean theorem and why is it significant in the script?
-The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It is significant in the script as it serves as an example to illustrate the concept of proof in geometry.

Outlines

00:00

📘 Introduction to Proof in Mathematics

The speaker thanks Lisa, viewers, Princeton University Press, and the Royal Institution. They emphasize the importance of proof in mathematics, stating it is often under-recognized in teaching. Proof and calculation are described as two sides of the same coin, with a focus on the Pythagorean theorem's visual proof.

05:01

🔍 The Pythagorean Theorem and Euclid's Proof

A detailed explanation of the Pythagorean theorem, including a visual proof using triangles and squares. The speaker highlights the difference between visual proofs and Euclid's more elaborate, logical proof, which impressed philosopher Thomas Hobbes. This illustrates the process of breaking down complex proofs into simpler, self-evident propositions.

10:06

📏 The Pythagorean Discovery of Irrational Numbers

The speaker discusses the Pythagoreans' discovery of the irrationality of root two, which revealed a divide between geometry and numbers. This led to the development of the axiomatic method in Euclidean geometry, where axioms and logic derive theorems, forming the foundation of modern mathematics.

15:06

🔢 The Evolution of Algebra and Geometry

The development of algebra, starting from geometric solutions to quadratic equations by al-Khwarizmi, is explained. The transition from geometric to algebraic methods led to the efficient calculation and the merging of algebra and geometry, exemplified by Descartes' analytic geometry.

20:08

🔄 Calculus and Infinite Processes

The speaker describes the origins of calculus and its ability to calculate areas, volumes, and lengths through infinitesimal processes. Key figures such as Newton and Leibniz are mentioned, highlighting their contributions to making complex calculations easier, despite initial logical contradictions.

25:09

🧮 Boolean Algebra and Predicate Logic

The history of logic and its reduction to computation is traced from Leibniz's ideas to Boole's algebra. Boolean algebra simplifies propositions into true or false values, leading to the development of a logical system that can express mathematical truths. Frege's predicate logic provided an axiomatic system for all mathematics, paving the way for modern logic.

30:10

🔗 The Limitations of Computation and Proof

The diagonal argument and its implications for computation and logic are discussed. The speaker explains how this argument reveals limitations in algorithms and axiom systems, showing that certain mathematical problems cannot be solved by computation alone. This highlights the inherent need for new mathematical ideas.

35:11

📈 The Infinite and Real Numbers

The differentiation between discrete and continuous infinities, illustrated by counting points in a plane versus points on a line. The speaker uses a thought experiment involving removing points to show the complexities of counting real numbers, emphasizing the unique properties of the continuous infinity.

40:15

🔄 Diagonal Argument and Computable Numbers

The diagonal argument's application to real numbers demonstrates the impossibility of listing all computable numbers. This leads to the conclusion that no axiom system can generate all mathematical truths, as shown by the work of Post, Turing, and Gödel. This underlines the limitations of both computation and logical systems.

Mindmap

Keywords

💡Proof

Proof in mathematics refers to a logical argument or a chain of reasoning that demonstrates the truth of a statement with absolute certainty. In the video, the speaker emphasizes the importance of proof, suggesting that while many people associate math with calculation, it is equally, if not more, about establishing proof. The Pythagorean theorem is used as an example, where both a visual proof and a more elaborate Euclidean proof are discussed to illustrate different approaches to mathematical validation.

💡Calculation

Calculation is the process of performing arithmetic operations to obtain a numerical result. The speaker mentions that while calculation is well-recognized in mathematics, it is often seen as separate from proof. However, the video argues that calculation and proof are two sides of the same coin, with the development of algebra and calculus showing how calculation can lead to new proofs and theorems.

💡Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The video uses this theorem to discuss both a visual proof involving geometric shapes and a more complex proof by Euclid, highlighting the different ways in which mathematical truths can be established.

💡Axiomatic Method

The axiomatic method is a framework for mathematical knowledge that starts with self-evident statements, known as axioms, from which other statements are derived through logic. The speaker explains that the Greeks, exemplified by Euclid's 'Elements', used this method to ensure the certainty of geometric truths, which contrasts with the more visual proofs that could be more easily understood but less rigorous.

💡Algebra

Algebra is a branch of mathematics that uses symbols and the rules of arithmetic to manipulate these symbols. The video discusses the development of algebra, starting from geometric interpretations of equations by al-Khwarizmi to the symbolic manipulations that became more prevalent around 1600. The transition from geometric to algebraic representations is highlighted as a shift from visual to calculation-based proofs.

💡Geometry

Geometry is a branch of mathematics concerned with questions of shape, size, relative position, and the properties of space. The video contrasts geometry's reliance on visual proofs with the more abstract, calculation-based proofs that emerged with the development of algebra and calculus. Geometry is also shown to be susceptible to the limitations imposed by the uncountability of real numbers.

💡Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal expansions. The video discusses the Pythagoreans' discovery of the irrationality of the square root of two, which challenged their belief that all numbers could be expressed as ratios of natural numbers, leading to a 'divorce' between geometry and number theory.

💡Diagonal Argument

The diagonal argument, as presented in the video, is a method used to demonstrate the uncountability of the real numbers. It involves creating a new number by altering each digit in an infinite list of numbers, ensuring that the new number is different from all those listed. This argument shows that no algorithm can list all computable numbers, highlighting a fundamental limitation in the power of computation and proof in mathematics.

💡Limitations of Computation

The video discusses the inherent limitations of computation, particularly in relation to the real numbers and the concept of infinity. It uses the diagonal argument to show that there are problems in mathematics for which no algorithm can provide all correct answers, indicating that there are unsolvable problems within the realm of computable numbers.

💡Non-Euclidean Geometries

Non-Euclidean geometries are geometric systems that do not subscribe to Euclid's parallel postulate. The video mentions the discovery of these geometries in the 19th century, which necessitated a common foundation for all geometries, leading to a deeper exploration of the real numbers and the concept of infinity.

💡Predicate Logic

Predicate logic is a more complex form of logic that includes quantifiers and variables that can represent a wide range of mathematical entities. The video explains that predicate logic is strong enough to express all of mathematics and includes logic ingredients necessary for proving theorems. However, it also points out that the completeness of predicate logic, like Frege's system, does not provide an algorithm for determining the validity of all propositions.

Highlights

The speaker emphasizes the importance of proof in mathematics, which is often overlooked in favor of calculation.

The concept of proof and calculation are two sides of the same coin, highlighting the interconnectedness of these mathematical approaches.

A visual proof of the Pythagorean theorem is presented, showcasing the geometrical understanding of mathematical concepts.

The need for rigorous proof in geometry led to the development of the axiomatic method, exemplified by Euclid's work.

Thomas Hobbes' initial disbelief and subsequent fascination with Euclid's proof illustrates the power of mathematical reasoning.

The discovery of the irrationality of the square root of two challenged the Pythagorean worldview and led to a separation between geometry and number theory.

Al-Khwarizmi's method for solving quadratic equations through geometry laid the foundation for modern algebra.

The transition from geometric to algebraic methods in solving equations increased efficiency and abstracted away from visual representations.

Descartes' work in 1637 marked a significant shift towards using algebraic equations to describe geometric shapes.

The development of calculus in the 17th and 18th centuries revolutionized the way mathematicians calculate volumes, areas, slopes, and arc lengths.

The first calculus textbook by l'Hôpital introduced the concept of curves being made of infinitesimal straight segments, a foundational idea in calculus.

The limitations of early calculus were acknowledged, with mathematicians recognizing the need for a more rigorous understanding of infinitesimals.

Leibniz's vision of reducing logic to computation foreshadowed the development of algorithms and computational methods in mathematics.

Boole's work in 1847 introduced Boolean algebra, which simplified logical operations into an algebraic form.

Frege's system of predicate logic provided a comprehensive framework for understanding the validity of logical statements.

The concept of infinity was explored in the 19th century, leading to a deeper understanding of the real numbers and the distinction between discrete and continuous infinities.

Cantor's diagonal argument demonstrated the uncountability of real numbers, revealing fundamental limitations in computation and axiomatic systems.

The limitations discovered through the diagonal argument imply that there are mathematical problems without solutions that can be found by an algorithm.

Gödel's incompleteness theorems are highlighted, showing that no axiomatic system can capture all mathematical truths.

The conclusion emphasizes the enduring need for creativity in mathematics, as it cannot be fully mechanized due to its inherent limitations.

Transcripts

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Related Tags

Math Proof Calculation Pythagorean Geometry Algebra Euclid Irrational Numbers Logic Algorithms Computation Limits