Principia Mathematica

The Principia Mathematica (often called PM) is a three-volume book about the foundations of mathematics. It was written by the mathematicians and philosophers Alfred North Whitehead and Bertrand Russell and was published between 1910 and 1913. This important work helped people understand how math and logic are connected.

PM had three main goals. First, it wanted to study the ideas and methods of mathematical logic as deeply as possible, using as few basic ideas and rules as it could. Second, it aimed to write math statements very clearly using special symbols. Third, it tried to solve problems that were confusing people in logic and a part of math called set theory at that time, such as Russell's paradox.

To reach its goals, PM used something called the theory of types. This theory sets rules for how math statements can be written, preventing certain mistakes from happening. The book helped make symbolic logic more popular and showed how powerful it could be. It was even listed as one of the top 100 nonfiction books in English from the 20th century by the Modern Library.

Scope of foundations laid

The Principia Mathematica focused on set theory, cardinal numbers, ordinal numbers, and real numbers. It did not include deeper ideas from real analysis, but experts knew that a lot of math could be built using the methods they chose, even if it would take a very long time.

The writers had planned a fourth volume about the foundations of geometry, but they felt too tired after finishing the third volume.

Theoretical basis

The book Principia Mathematica tries to explain the ideas and ways of mathematical logic. It aims to use as few basic ideas and rules as possible. It also wants to write math ideas in a clear way using symbols. Another goal is to solve problems in logic and grouping that were puzzling people at the time.

The book mixes ideas from logic with math in a special way. It talks about symbols like "⊢" (which means we can say something is true), "∾" (which means "not"), and "V" (which means "or"). It also talks about what is true and what is not true when using these symbols.

Ramified types and the axiom of reducibility

In the book Principia Mathematica, the authors created a special way to organize different kinds of mathematical objects called "ramified types". These types are built up step by step. For example, if you have types τ₁ through τ_m and σ₁ through σ_n, you can create a new type that deals with certain rules for combining them.

The authors found it hard to build mathematics using these rules alone, so they added something called the axiom of reducibility. This axiom says that for any complex rule, there is a simpler one that gives the same results. This made it easier to work with mathematics, even though the authors knew this axiom wasn’t perfect.

They also talked about how others tried different approaches, but these often made it hard to study certain parts of math, like numbers that go on forever without repeating. Because parts of math like these are so important, the authors believed there might be a better, more limited axiom that could work without losing too much.

Notation

Main article: Glossary of Principia Mathematica

One author notes that the notation in this work has been updated over time, making it hard for beginners to read. While much of the symbols can be changed to modern ones, some of the original notation is still debated among scholars because it represents important logical ideas.

The notation in this work is based on the work of Peano. It uses dots in a special way to act like parentheses. For example, dots can show where a group of symbols belongs together in a logical statement.

Kurt Gödel pointed out that the notation did not clearly explain the rules for how symbols should be used together. This made some proofs harder to understand.

The book uses symbols like "p", "q", and "r" with a special symbol "⊃" to build logical statements. For example, "p ⊃ q ⊃ r" needs a clear definition to understand what it means in terms of other symbols. Modern treatments would have rules to prevent such unclear strings of symbols.

The notation was developed in Chapter I, which explains the basic ideas and symbols used. It includes symbols like "=" , "⊃", "≡", "−", "Λ", "V", and "ε", along with a system of dots to organize the symbols.

The work also adopted some symbols from Frege, like the assertion sign "⊦", which is read as "it is true that". For example, to assert a proposition p, the book writes "⊦. p."

Most of the notation was created by Whitehead himself.

The dots are used like parentheses to show groups of symbols. For example, "." might stand for "and". The position of the dots shows how far they reach, which can be complex to figure out.

The book also uses special symbols for logical ideas like "if...then..." shown by "⊃", "not" shown by "∾", and "or" shown by "v". The symbol "=" with "Df" means "is defined as".

The book also introduces symbols for working with groups of things, like "ε" for "is an element of", "⊂" for "is a subset of", and "∩" for "intersection". These symbols are still used today in math.

Consistency and criticisms

Russell once said he was surprised to find that the language of Principia Mathematica was based on Indo-European languages, after learning about the Chinese language.

Some critics felt that Principia Mathematica needed extra rules beyond basic logic. These included ideas like the axiom of infinity, the axiom of choice, and the axiom of reducibility. These extra rules made some mathematical ideas harder to express clearly.

Important questions about any system like Principia Mathematica include whether it could produce mistakes and whether there are statements it cannot prove or disprove. While basic logic was known to be reliable, the same was not certain for the rules of Principia.

In 1930, a theorem showed that basic logic could not decide every statement. Later, in 1931, another theorem explained that no powerful system can prove all true statements about numbers, nor can it prove that it itself has no mistakes.

Ludwig Wittgenstein criticized Principia Mathematica for trying to explain the basics of arithmetic, when everyday counting is actually more fundamental. He also noted that the methods in Principia work well only with small numbers and would need simpler, everyday techniques for larger ones.

In 1944, Kurt Gödel pointed out that Principia Mathematica lacked clear rules about how symbols and statements should be used, making some proofs less strong than they could be.

Contents

Part I Mathematical logic. Volume I ✱1 to ✱43

This part talks about basic ideas in logic and how they relate to groups, connections, and types.

Part II Prolegomena to cardinal arithmetic. Volume I ✱50 to ✱97

Here, the book looks at special rules for connections, especially those needed for counting.

Part III Cardinal arithmetic. Volume II ✱100 to ✱126

This part explains the idea of cardinals, which are used for counting. It shows how to add, multiply, and use powers for these numbers.

Part IV Relation-arithmetic. Volume II ✱150 to ✱186

This section talks about "relation-numbers," which are groups of matching connections. It shows how to add, multiply, and use powers for these connections.

Part V Series. Volume II ✱200 to ✱234 and volume III ✱250 to ✱276

This covers ordered groups, including those that are complete, well-ordered, and without gaps.

Part VI Quantity. Volume III ✱300 to ✱375

This part builds the groups of whole numbers, parts of numbers, and real numbers, as well as special families related to math groups.

Comparison with set theory

This section looks at how the system in Principia Mathematica (PM) compares to the usual mathematical foundations called ZFC.

One big difference is that in PM, all objects are sorted into separate types. This means that things like numbers appear in each type, which needs careful tracking. In PM, functions are treated as something that gives true or false values, and different functions can give the same results. PM also focuses more on relationships between things, while modern math often focuses on functions. In PM, numbers called cardinals and ordinals are grouped differently for each type, unlike in ZFC where they are all grouped together in one way. The way PM handles certain math operations with these numbers also differs from the usual methods used today.

Differences between editions

The main text of Principia Mathematica stayed the same between its first and second editions, but some changes were made. In the second edition, Volumes 1 and 2 were reset to use fewer pages, while Volume 3 was reprinted as it was.

Volume 1 of the second edition included five new parts. There was a 54-page introduction by Bertrand Russell talking about changes they would have made with more time. Appendix A, 15 pages, discussed something called the Sheffer stroke. Appendix B, numbered as *89, looked at a math idea called induction. Appendix C, 8 pages, talked about propositional functions. There was also an 8-page list of definitions that served as an index for the many symbols used in the book.

In 1962, Cambridge University Press put out a shorter paperback version with parts of the second edition of Volume 1. This included the new and old introductions, the main text up to *56, and Appendices A and C.

Legacy

The book Principia Mathematica helped people learn more about symbolic logic and showed how important it is. It also showed that studying math and logic together can lead to great discoveries. Even though it has some problems, the book influenced later work in logic, such as important ideas by Gödel's incompleteness theorems.

Even though the special symbols used in the book weren’t used by many, people still study Principia Mathematica a lot. They look at it to understand its history or to learn more about how math and logic work together. The book was also chosen as one of the top 100 nonfiction books of the 20th century by the Modern Library.