Part of the branch of mathematics called "Foundations of Mathematics" is the decomposition of things such as numbers, arithmetical operations and mathematical functions into more basic things.
Mathematical concepts can be defined in terms of more basic concepts, which in turn can be defined in terms of even more basic concepts, until all mathematical concepts have been defined in terms of a small number of basic concepts.
I think this is pretty cool.
Before I learned about Foundations, I thought that numbers just are what they are, never that they might be made up of something simpler.
It was like finding out for the first time that every physical object is made of protons, electrons and neutrons.
What happens if we take something—the number one, for example—and decompose it all the way down to the most basic concepts?
What would it look like?
I created this web page to answer this question.
Here, I've taken definitions from the textbook Mathematical Logic, which is Willard Quine's account of Foundations, and scripted the page to expand these definitions interactively.
In Quine's system, everything is composed of just four basic concepts: class membership (), universal generalization (), joint denial (logical NOR, or ), and class abstraction ().
This page shows the 48 mathamatical entities defined in the "List of Definitions" at the end of Mathematical Logic.
All are expandable except D1 (negation), whose expansion I have disabled.
Unlike most of the other definitions, when D1 is expanded, it is doubled in size, rather than just having a fixed number of symbols added to it.
The definitions on this page contain many negations of sub-expressions, and those sub-expressions contain negations of their own sub-expressions, and those sub-expressions contain negations, and so on, so as negations are expanded, the doubling becomes a quadrupling, the quadrupling becomes an octupling, and the size of a definition grows exponentially.
Expanding the more advanced definitions requires more memory than an Internet browser allows.
The "Expansions" view is where these expansions are shown. To view a symbol on the "Expansions" view, click on its item in the list of definitions and it will be displayed at the bottom of the page. To expand a symbol, check the checkbox next to it in the list of definitions. When a definition's checkbox is checked, the symbol will be expanded wherever it appears in a definition.
The mathematical notation on this page is the notation of Mathematical Logic.
I've attempted to create a variant of this page written with modern notation; to view it, click here.
The mathematical notation on this page is modern, and not Quine's original notation. Since there is no one standard system of notation in modern mathematics, my choices of symbols are in some cases arbitrary. For some entities, I could not find a customary modern symbol, so I kept Quine's notation. To view a version of this page that uses Quine's original notation, click here.
These are the definitions in the "List of Definitions" included at the end of Mathematical Logic. They are reproduced here with permission.