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@ -15,8 +15,6 @@ title: About
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{: style="text-align: right" }
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Tom Lehrer
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About this book
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===
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@ -16,7 +16,7 @@ What is an Abstract Theory
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Most scientific and mathematical theories have a specific domain, to which they are related, and in which they are valid. They are created with this domain in mind and are not intended to be used outside of it. For example, Darwin's theory of evolution is created so it explains how different biological species came to be. And quantum mechanics is a description of what particles are at the low scale. Even the work of most mathematicians, although it is not bound to a specific domain, is strongly related to it, as differential equations are linked to the description of events that change over time.
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Set theory and category theory are different. They are not created for to provide a rigorous explanation of how a particular phenonium works, but to try to provide a more general framework for explaining all kinds of phenomena. Theories that are like that are called *abstract* theories. All theories *use abstraction*, else they would be pretty useless (without it Darwin would have to speak about specific animal species or even individual animals) but few are inherently abstract, so some of their core concepts are left unspecified. Or in other words, all theories are applicable outside of their domains, but set theory and category theory do not have a domain, to begin with.
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Set theory and category theory are different. They are not created to provide a rigorous explanation of how a particular phenomenon works, but to try to provide a more general framework for explaining all kinds of phenomena. Theories that are like that are called *abstract* theories. All theories *use abstraction*, else they would be pretty useless (without it Darwin would have to speak about specific animal species or even individual animals) but few are inherently abstract, so some of their core concepts are left unspecified. Or in other words, all theories are applicable outside of their domains, but set theory and category theory do not have a domain, to begin with.
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People have tried to be precise and at the same time down to Earth for centuries, and only recently discovered that "precise and down to Earth" is an oxymoron. Let's take Euclidian geometry as an example. Yes, Euclidian geometry is precise, because it is valid for all sets of objects, called ("point" "line" "angle" and "circle" etc.), which have relationships, as defined by the five famous axioms. Yes, geometry does, in many instances, describe the natural world, because there are many sets of objects which have these relations. However, its "precise" part and it's "down to Earth" part have nothing to do with each other. We can, for example, define a point as any stain on the floor of your room and that a line as a piece of duct tape, put on the same floor - that will be a completely valid application of the Euclidian laws, albeit not very useful one. Or we can try to use geometry to reason about points on the surface of the Earth, which is a very useful application, of geometry, however not of Euclidian geometry, because Euclidian geometry only describes points on a flat plane, and the Earth is not flat. You can argue that these are actually two separate theories there, which just happen to be perceived as one. You have the axioms, or the postulates on one hand, which are not useful for anything on their own, and you have applications in science and engineering which are somewhat based on them, but not quite.
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@ -27,7 +27,7 @@ Everything in set theory is defined in terms of sets. A set is a collection of t
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For example, let's construct a set, call it **G** (as grey), consisting of *all* of them. This is how we can mark it:
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For example, let's construct a set, call it **G** (as gray), consisting of *all* of them. This is how we can mark it:
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@ -41,9 +41,9 @@ The key insight about what a set it is that it enables you to reason about sever
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Subsets
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---
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Let's construct one more set. The set of *all balls that are warm colour*. I will call it **Y**, because in the diagram is coloured in **y**ellow.
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Let's construct one more set. The set of *all balls that are warm color*. I will call it **Y**, because in the diagram is colored in **y**ellow.
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Notice that **Y** contains just elements that are also present in **G**. That is, every element of the set of **Y** is also an element in the set **G**. When two sets have this relation, we may say that **Y** is a *subset* of **G** (or **Y ⊆ G**).
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@ -56,7 +56,7 @@ The set of all *red balls* contains just one ball.
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Like we said above, sets are all about for summarising *several* elements into one. Still, sets that contain just one element are perfectly valid. Simply, there are things which are one of a kind. Furthermore, if I have a function which expects a set of given items, here shouldn't be any issue if the "items" are just one item. Or to take a real-life example, the set of queens of England is a singleton set. The set of books written by the American writer Harper Lee and published during her lifetime is a singleton set - she has published just one novel.
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Like we said above, sets are all about for summarizing *several* elements into one. Still, sets that contain just one element are perfectly valid. Simply, there are things which are one of a kind. Furthermore, if I have a function which expects a set of given items, here shouldn't be any issue if the "items" are just one item. Or to take a real-life example, the set of queens of England is a singleton set. The set of books written by the American writer Harper Lee and published during her lifetime is a singleton set - she has published just one novel.
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The Empty set
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---
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@ -78,7 +78,6 @@ Functions
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> By function I mean the unity of the act of arranging various representations under one common representation.
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> Immanuel Kant, from Critique of Pure Reason
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A function is a relationship between two sets that matches each element of one set, called the *source set* of the function, with exactly one element from another set, called the **target set** of the function.
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These two sets are also called the *domain* and *codomain* of the function, or its *input* and *output*. In programming, they go by the name of *argument type* and *return type*. In logic, they correspond to the *premise* and *conclusion* (we will get there.) We might also say, depending on the situation, that a given function *goes* from this set to that other one, *connects* this set to the other. Or that it *converts* a value from this set to one of the the other one. These different terms show the many different faces of functions.
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@ -116,11 +115,11 @@ Note that the question "Who is my child?" is *NOT* a straightforward function, b
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The Identity Function
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---
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For every set **G**, no matter what it represents, we can define the function that does nothing, or in other words, a function which maps every element G to itself. It is called the *the identity function* of **G** or **idG: G → G**.
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For every set **G**, no matter what it represents, we can define the function that does nothing, or in other words, a function which maps every element G to itself. It is called the *the identity function* of **G** or **id G: G → G**.
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You can think of **idG** as a function which represents the set **G** in the realm of functions. Its existence allows us to prove many theorems, that we "know" by intuition, formally.
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You can think of **id G** as a function which represents the set **G** in the realm of functions. Its existence allows us to prove many theorems, that we "know" by intuition, formally.
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Functions and Subsets
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---
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@ -179,7 +178,7 @@ I will use the occasion to reiterate some of the more important characteristics
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Overall everything is permitted, as long as you can always provide exactly one result (also known as *The result™*) per value, and in mathematics almost always do. Actually, math is designed in a way so its operations are valid functions:
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> Every generalisation of number has first presented itself as needed for some simple problem: negative numbers were needed in order that subtraction might be always possible, since otherwise a − b would be meaningless if a were less than b; fractions were needed in order that division might be always possible; and complex numbers are needed in order that extraction of roots and solution of equations may be always possible.
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> Every generalization of number has first presented itself as needed for some simple problem: negative numbers were needed in order that subtraction might be always possible, since otherwise a − b would be meaningless if a were less than b; fractions were needed in order that division might be always possible; and complex numbers are needed in order that extraction of roots and solution of equations may be always possible.
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> Bertrand Russell, from Introduction to Mathematical Philosophy
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Note that most mathematical operations, such as addition, multiplication etc. require two numbers in order to produce a result. This does not mean that they are not functions, it means that they are just a little more fancy ones. Depending on what we need, we may present those operations as functions from the sets of *tuples* of numbers to the set of numbers, or we may say that they take a number and return a function. More on that later.
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@ -203,7 +202,7 @@ Most of the types of programming are composite types - they are a combination of
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**Question:** What is the type equivalent of subsets in programming?
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**Question:** Do you recognise some of the basic functions we defined in programming languages you know?
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**Question:** Do you recognize some of the basic functions we defined in programming languages you know?
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Some functions in programming (also called methods, subroutines, etc.) kinda resemble mathematical functions - they sometimes take one value of a given type (or in other words, an element that belongs to a given set) and always return exactly one element which belongs to another type (or set). For example here is a function which that takes an argument of type `Char` and returns a `Boolean`, depending on whether the character is a letter.
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@ -234,15 +233,15 @@ Unlike the set above, most sets that we discussed (like the empty set and single
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In order to understand Russell's paradox we will try to visualise **the set all sets that do not contain themselves**. In the original set notation we can define this set as *Let R = { x => x ∉ x }* (let R be such that it contains all sets *x* such that *x* is not a member of *x*).
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In order to understand Russell's paradox we will try to visualize **the set all sets that do not contain themselves**. In the original set notation we can define this set as *Let R = { x => x ∉ x }* (let R be such that it contains all sets *x* such that *x* is not a member of *x*).
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If we look at the definition, we recognise that the set that we just defined - *R* does not contain itself and therefore it belongs there as well.
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If we look at the definition, we recognize that the set that we just defined - *R* does not contain itself and therefore it belongs there as well.
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Hm, something is not quite right with this diagram as well - Because of the new adjustments that we made *R* **contains itself**. And removing it from the set would just bring us back to the previous situation. This is Russell's paradox. There are, of couse, multiple theories that it does not apply to.
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Hm, something is not quite right with this diagram as well - Because of the new adjustments that we made *R* **contains itself**. And removing it from the set would just bring us back to the previous situation. This is Russell's paradox. There are, of course, multiple theories that it does not apply to.
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Functional Composition
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===
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@ -305,9 +304,9 @@ Notice that the function is invertible, that is if you flip its arrows you get a
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Invertible functions are called *isomorphisms*. When there is an invertible function between two sets we can say that the sets are *isomorphic*. For example, the temperature measured in Celcius is isomorphic to the temperature, measured in Fahrenheit.
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Invertible functions are called *isomorphisms*. When there is an invertible function between two sets we can say that the sets are *isomorphic*. For example, the temperature measured in Celsius is isomorphic to the temperature, measured in Fahrenheit.
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More formally, two sets **R** and **G** are isomorphic, or **R ≅ G** if there exist functions **f: G → R** and its reverse **g R → G**, such that **f ∘ g = idR** and **g ∘ f = idG** (notice how the identity function comes in handy).
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More formally, two sets **R** and **G** are isomorphic, or **R ≅ G** if there exist functions **f: G → R** and its reverse **g: R → G**, such that **f ∘ g = id R** and **g ∘ f = id G** (notice how the identity function comes in handy).
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Isomorphism and equality
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---
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@ -329,7 +328,7 @@ Isomorphisms Between Singleton Sets
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Between any two singleton sets, we may define the only possible function.
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The function is invertible, which means that all singleton sets are isomorphic to one another.
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@ -6,14 +6,14 @@ title: Categories
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From Sets to Categories
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===
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In this chapter we will continue with set theory and at the same time we will start exploring categories, and talking about why they are important.
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In this chapter we will continue with set theory, and at the same time we will start exploring categories and talking about why they are important.
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Products
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===
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In the previous chapter, we needed a way to make a set that is a composite of other sets that we already have. For example, when we discussed mathematical functions, we couldn't define **+** and **-** functions, because we only knew about functions that take one argument. When we talked about programming, we talked about the primitive types, `Char` and `Number`, and we mentioned that most of the types are composite types. So how do we construct those?
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The main composite type, of the sets **B** (which contains **b**-s) and the set **Y** (which contains **y**-s) is called the *cartesian product* or **B** and **Y**.
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The simplest composite type, of the sets **B**, which contains **b**'s and the set **Y**, which contains **y**'s is the *product* of **B** and **Y**.
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@ -21,6 +21,8 @@ The product is the set of *ordered pairs* of **b**'s and **y**'s. It is denoted
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The product is denoted **B x Y**, and it comes equipped with two functions for retrieving the **b** and the **y** from each **(b, y)**.
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**Question**: Why is this called a product? Hint: How many elements does it have?
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The Cartesian product was first defined by the mathematician and philosopher René Descartes as a basis for the Cartesian coordinate system. But as we shall see, it has many other use-cases.
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@ -28,29 +30,29 @@ The Cartesian product was first defined by the mathematician and philosopher Ren
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Products as Objects
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---
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We established that in programming sets resemble types and functions resemble functions. Products, in this case, are like stripped-out *classes* (also called *records* or *structs*). The composite sets (the ones which form the product) are just the class's fields (also called *members*). The functions for accessing them are like what programmers call *getter methods*.
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We established that in programming, sets resemble types, and functions resemble functions. Products, in this case, are like stripped-out *classes* (also called *records* or *structs*). The composite sets (the ones which form the product) are just the class's fields (also called *members*). The functions for accessing them are like what programmers call *getter methods*.
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For example, the famous OOP example of `Person` class with `name` and `age` fields is nothing more than a product of the set of strings, and the sets of numbers (we still haven't defined strings and lists in terms of set theory but we will get to that). Objects with more than two values can be expressed as products the composites of which are themselves products.
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The famous example of object-oriented programming of a `Person` class with `name` and `age` fields is nothing more than a product of the set of strings, and the sets of numbers. (We still haven't defined strings and lists in terms of set theory but we will get to that.) Objects with more than two values can be expressed as products the composites of which are themselves products.
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Using Products to Define Numeric Operations
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---
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Products can also be used for expressing functions which take more than one argument. For example, "plus" and "minus", are functions from the set of products of two numbers to the set of numbers. (so **+: R x R → R**). Of course, we cannot draw the function itself, even partly, because it has too much arrows and it would look messy.
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Products can also be used for expressing functions which take more than one argument. For example, "plus" and "minus", are functions from the set of products of two numbers to the set of numbers. (So, **+: R x R → R**.) Of course, we cannot draw the function itself, even partly, because it has too much arrows and would look messy.
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Joking, here it is.
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Note that there are languages where the *pair* datastructure (also called a *tuple*) is a first-level construct, and multi-argument functions are really implemented in this way.
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Note that there are languages where the *pair* data structure (also called a *tuple*) is a first-level construct, and multi-argument functions are really implemented in this way.
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Defining products in Terms of Sets
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---
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Now we will define the abstract concept of a product of two sets sets in terms of sets themselves. It is not hard: the product of two sets **Y** and **B** is just the set of all possible *ordered pairs*, which contain one element of the set **Y** and one element of the set **B**. Or formally speaking **Y x B = { (y, b) }** where **y ∈ Y, b ∈ B** (**∈** means "is an element of").
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Now we will define the abstract concept of a product of two sets in terms of sets themselves. It is not hard: the product of two sets **Y** and **B** is just the set of all possible *ordered pairs*, which contain one element of the set **Y** and one element of the set **B**. Or formally speaking: **Y x B = { (y, b) }** where **y ∈ Y, b ∈ B** (**∈** means "is an element of").
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The real challenge is defining what a pair means in terms of sets. Note that the pair have to be *ordered* (formally speaking **A x B ≠ B x A for all A and B **), so it cannot be just a set of the elements - some mathematical operations (such as addition) don't care about order, but others (such as subtraction) do.
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The real challenge is defining what a pair means in terms of sets. Note that the pair have to be *ordered* (formally speaking **A x B ≠ B x A for all A and B**), so it cannot be just a set of the elements - some mathematical operations (such as addition) don't care about order, but others (such as subtraction) do.
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In programming we have the ability to assign names to each member of an object, which accomplishes the same purpose as ordering does for pairs.
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@ -69,7 +71,7 @@ Discovered in 1921 Kazimierz Kuratowski, this one uses just the component of the
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Defining products in Terms of Functions
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---
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In the previous chapter we provided a definition of a product by *zooming in* the individual elements of the sets and seeing what they can be made of. This gave us a *low-level* view of products. This time we will try to do the opposite - be as oblivious to the contents of our sets as possible. Instead of zooming in we will *zoom out*, and try define the product in terms of functions and functional composition. Effectively we will be working at a *higher level* of abstraction.
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In the previous chapter we provided a definition of a product by *zooming in* the individual elements of the sets and seeing what they can be made of. This gave us a *low-level* view of products. This time we will try to do the opposite - be as oblivious to the contents of our sets as possible. Instead of zooming in we will *zoom out*, and try to define the product in terms of functions and functional composition. Effectively we will be working at a *higher level* of abstraction.
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So let's begin with an external diagram, showing the definition of the product. Disclaimer: I know that this is a somewhat weird notation, but don't worry, we will not be using it for very long.
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@ -91,7 +93,7 @@ So how do we set apart the true product from all those "imposter" products? Simp
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1. By definition, each "impostor" can be converted to both elements of the pair.
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2. The pair is nothing more than the sum of its elements.
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More formally, in order for a set **I** to serve as an impostor for the product of **B** and **Y** and there should be two functions, which we will call **b: I → B** and **y: I → Y**. In order to prove that **I** is an impostor we need a function **I → B x Y**. That function is simply (programmers will understand this best) **(a) → b(a) x y(a)** for each **a:I**.
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More formally, in order for a set **I** to serve as an impostor for the product of **B** and **Y**, there should be two functions, which we will call **b: I → B** and **y: I → Y**. In order to prove that **I** is an impostor we need a function **I → B x Y**. That function is simply (programmers will understand this best) **(a) → b(a) x y(a)** for each **a:I**.
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@ -102,7 +104,7 @@ Notice that this definition does not rule out the sets which are isomorphic to t
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Sums
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===
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We will now study a construct that is pretty similar to the product but at the same time it is very different. Similar because, like the product, it is a relation between two sets which allows you to unite them into one without erasing their structure. Different because it encodes a different type of relation between them - a product encodes an **AND** relation between two sets, while the sum encodes an **OR** relation. For example, a parent is either a mother of a father of a child, so the set of parent's is a sum set of the sets of mothers and fathers.
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We will now study a construct that is pretty similar to the product but at the same time is very different. Similar because, like the product, it is a relation between two sets which allows you to unite them into one without erasing their structure. Different because it encodes a different type of relation between them - a product encodes an **AND** relation between two sets, while the sum encodes an **OR** relation. For example, a parent is either a mother or a father of a child, so the set of parent's is a sum set of the sets of mothers and fathers.
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@ -123,7 +125,7 @@ Like with the product, there is a low-level way to express a sum using sets alon
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But again, this distinction is only rellevant only when the two sets have common elements.
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But again, this distinction is only relevant only when the two sets have common elements.
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Defining Sums in Terms of Functions
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@ -140,11 +142,11 @@ Why can it be expressed as a function? Because it is a *many-to-one* relationshi
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What are we saying with this, if we apply it to the example, is simply that if parents are either mothers or fathers, then there surely exist a functions **mothers → parents** and **fathers → parents**.
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You might already notice that this definition is pretty similar to the previous one, and the similarities don't end here - here again we have sets that can be thought as *impostor* sums - ones for which these functions exists, but which aren't real sums (where by "real sum" we mean a set which expresses the *OR* relation and contains no additional structure).
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You might already notice that this definition is pretty similar to the previous one, and the similarities don't end here - here again we have sets that can be thought of as *impostor* sums - ones for which these functions exists, but which aren't real sums (where by "real sum" we mean a set which expresses the *OR* relation and contains no additional structure).
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All these sets are, express relationships which are more vague than the simple sum and therefore given any such set there would exist a function from the sum to it.
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All these sets express relationships which are more vague than the simple sum, and therefore given any such set, there would exist a function from the sum to it.
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For example, there exist a trivial function between the set **Y \| B** and the set **Y \| B \| R**.
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@ -156,7 +158,7 @@ This diagram captures the **OR** relation in the same way as the previous one ca
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Duality
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===
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If we have to compare the concepts of sum or and product we will find out that they are related:
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If we compare the concepts of *sum* and *product*, we will find out that they are related:
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- The *product* of two sets is related to an element of the first one *and* one element of the second one.
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- A *sum* of two sets is related to an element of the first one *or* one element of the second one.
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@ -167,14 +169,14 @@ The connection between the two has always been there, evidenced, for example, by
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Otherwise, when in category theory two concepts are captured by the same diagram, only with reversed arrows, we can say that the two concepts are **dual** to each other. That means that the concepts of *product* and *sum* are dual (this is why sum is also known as *converse product*, or *coproduct* in short).
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Otherwise, when in category theory two concepts are captured by the same diagram, only with reversed arrows, we can say that the two concepts are **dual** to each other. That means that the concepts of *product* and *sum* are dual. (This is why sum is also known as *converse product*, or *coproduct* in short.)
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Category Theory - brief definition
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===
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|
||||
Maybe it is time to see what a category is. Well, a category consists of objects (an example of which are sets) and morphisms which go from one object to another (which can be viewed as functions) and which should be composable. We can say a lot more about categories, and even present a formal definition, but for now it is suffice for you to remember that sets are one example of a category and that categorical objects are like sets, except that we don't *see* their elements. This is why category-theoretic notions being captured by the external diagrams and strictly set-theoretic notions being captured by internal ones.
|
||||
Maybe it is time to see what a category is. Well, a category consists of objects (an example of which are sets) and morphisms which go from one object to another (which can be viewed as functions) and which should be composable. We can say a lot more about categories, and even present a formal definition, but for now it is sufficient for you to remember that sets are one example of a category and that categorical objects are like sets, except that we don't *see* their elements. This is why category-theoretic notions are captured by the external diagrams, and strictly set-theoretic notions are captured by internal ones.
|
||||
|
||||
What other categories, or applications of category theory are there, other than sets? We already discussed one - types in programming languages. Remember that we said that programming types (classes) are somewhat similar to sets and programming functions are somewhat similar to functions between sets? This is another example of a connection between two things that we can be defined using category theory:
|
||||
What other categories, or applications of category theory are there, other than sets? We already discussed one - types in programming languages. Remember that we said that programming types (classes) are somewhat similar to sets, and programming functions are somewhat similar to functions between sets? This is another example of a connection between two things that can be defined using category theory:
|
||||
|
||||
|
||||
| Category Theory | Set theory | Programming Languages |
|
||||
@ -183,11 +185,11 @@ What other categories, or applications of category theory are there, other than
|
||||
| Objects and Morphisms | Sets and Functions | Classes and functions |
|
||||
| N/A | Element | Object |
|
||||
|
||||
This diagram illustrates how category theory allows us to see the big picture when it comes to sets and similar structures - when we are at the realm of sets we can view the set as a collection of individual elements. In category theory we don't have such notion, but we saw how taking this notion away allows us to define concepts such as the sum and product sets in a whole different and more general way.
|
||||
This table illustrates how category theory allows us to see the big picture when it comes to sets and similar structures - when we are at the realm of sets we can view the set as a collection of individual elements. In category theory we don't have such notion, but we saw how taking this notion away allows us to define concepts such as the sum and product sets in a whole different and more general way.
|
||||
|
||||
**NB: Do note how the word "Object" is used in both programming languages and in category theory, but for completely different things. The equivalent a categorical object is equivalent to a class in programming language.**
|
||||
|
||||
Looking at the table I cannot help but notice the somehow weird, but otherwise completely logical symmetry (or perhaps "reverse symetry") between the the world as viewed through the lense of set theory, and the way it is viewed through the (inverted) lens of cathegory theory:
|
||||
Looking at the table I cannot help but notice the somehow weird, but otherwise completely logical, symmetry (or perhaps "reverse symmetry") between the world as viewed through the lenses of set theory, and the way it is viewed through the (inverted) lens of category theory:
|
||||
|
||||
| Category Theory | Set theory |
|
||||
| --- | --- |
|
||||
@ -195,11 +197,11 @@ Looking at the table I cannot help but notice the somehow weird, but otherwise c
|
||||
| Objects and Morphisms | Sets and functions |
|
||||
| **N/A** | Element |
|
||||
|
||||
By switching to external diagrams, we lose sight of the particular (the elements of our sets), but we have gained the ability to see the whole universe that we have been previously trapped in. The whole realm of sets, can be thought as one category, a programming language can also be thought as a category. The concept of a category allows us to find and analyse similarities between these and other structures.
|
||||
By switching to external diagrams, we lose sight of the particular (the elements of our sets), but we have gained the ability to see the whole universe that we have been previously trapped in. Just as the whole realm of sets can be thought as one category, a programming language can also be thought as a category. The concept of a category allows us to find and analyze similarities between these and other structures.
|
||||
|
||||
|
||||
|
||||
One remark before we go - the last paragraphs I sound as if I'm *comparing* categories and sets and rooting for categories. I don't want you to get the wrong impression that the two concepts are somehow competing with one another.
|
||||
One remark before we go: in the last paragraphs I sound as if I'm *comparing* categories and sets and rooting for categories. I don't want you to get the wrong impression that the two concepts are somehow competing with one another.
|
||||
|
||||
Perhaps that notion would be somewhat correct if category and set theory were meant to describe *concrete* phenomena, in the way that the theory of relativity and the theory of quantum mechanics in physics. Concrete theories are conceived mainly as *descriptions* of the world, and as such it makes sense for them to be connected to one another in some sort of hierarchy. Abstract theories, like category theory and set theory, on the other hand, are more like languages for expressing such descriptions. They still can be connected, and are connected in more than one way, but there is no inherent hierarchy between the two and therefore arguing over which of the two is more basic, or more general, is just a chicken-and-egg problem, as you would see in the next chapter.
|
||||
|
||||
@ -208,23 +210,23 @@ Defining Categories (again)
|
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|
||||
Every category theory guide (including mine) starts by talking about set theory, however looking back I really don't know why that is the case - most books that focus around a given subject don't start by introducing an entirely different subject. Perhaps the set-first approach is the best way to introduce people to categories. Or perhaps using sets to introduce categories is just one of the things that people just do because everyone else does it. But one thing is for sure - we don't need to study sets in order to understand categories. So now I would like to start over and talk about categories as a first concept. So pretend like it's a new book.
|
||||
|
||||
So, a category is a collection of objects (things) where the "things" can be anything you want. Consider, for example, these ~~colourful~~ grey balls:
|
||||
So, a category is a collection of objects (things) where the "things" can be anything you want. Consider, for example, these ~~colorful~~ gray balls:
|
||||
|
||||
|
||||
|
||||
A category consists of a collection of objects as well as some arrows connecting some of them to one another, that we call (the arrows) morphisms.
|
||||
A category consists of a collection of objects as well as some arrows connecting some of them to one another. We call the arrows, *morphisms*.
|
||||
|
||||
|
||||
|
||||
Wait a minute - we said that all sets form a category, but at the same time any one set can be seen as a category on its own right (just one which has no morphisms). This is true and an example of a phenomenon that is very characteristic of category theory - one structure can be examined from many different angles and may play many different roles, often in a recursive fashion.
|
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|
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This particular analogy (a set as a category with no morphisms) is, however, not very useful. Not because it's in any way incorrect, but because category theory is *all about the morphisms*. If in set theory arrows are nothing but a connection between a source and a destination, in category theory it's the *objects* that are nothing but a source and destination for the arrows that connect them to other objects. This is why, in the diagram above, the arrows, and not the objects, are coloured: the category of sets should really be called the category of set functions.
|
||||
This particular analogy (a set as a category with no morphisms) is, however, not very useful. Not because it's in any way incorrect, but because category theory is *all about the morphisms*. If in set theory arrows are nothing but a connection between a source and a destination, in category theory it's the *objects* that are nothing but a source and destination for the arrows that connect them to other objects. This is why, in the diagram above, the arrows, and not the objects, are colored: the category of sets should really be called the category of set functions.
|
||||
|
||||
Speaking of which, note that objects in a category can be connected by multiple arrows and that arrows having the same source and target sets does not in any way make them equivalent (it does not actually mean that they would produce the same value).
|
||||
|
||||
|
||||
|
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Why is that is pretty obvious if we go back to set theory for a second (OK, maybe we really *have* to do this from time to time) - there are, for example, an infinite number of functions that go from number to boolean and the fact that they have the same source type and the same target type (or the same *type signature*, as we like to say) does not in any way make them equivalent to one another.
|
||||
Why that is true is pretty obvious if we go back to set theory for a second. (OK, maybe we really *have* to do this from time to time.) There are, for example, an infinite number of functions that go from number to boolean, and the fact that they have the same input type and the same output type (or the same *type signature*, as we like to say) does not in any way make them equivalent to one another.
|
||||
|
||||
|
||||
|
||||
@ -239,9 +241,9 @@ One of the few or maybe even the only requirement for a structure to be called a
|
||||
|
||||
Formally, this requirement says that there should exist an operation (denoted with the symbol **•**) such that for each two functions **g: A → B** and **f: B → C**, there exists exactly one function **(f • g): A → C**.
|
||||
|
||||
**NB:** Note that functional composition is written from right to left. e.g. applying **g** and then applying **f** is written **f • g** and not the other way around (you can think of it as a shortcut to **f(g(a))**).
|
||||
**NB:** Note that functional composition is written from right to left. e.g. applying **g** and then applying **f** is written **f • g** and not the other way around. (You can think of it as a shortcut to **f(g(a))**.)
|
||||
|
||||
Again, note that this criteria is not met by just *any* morphism with this type signature. There is *exactly one* morphism that fits these criteria and there may be some which don't.
|
||||
Again, note that this criteria is not met by just *any* morphism with this type signature. There is *exactly one* morphism that fits these criteria, and there may be some which don't.
|
||||
|
||||
|
||||
|
||||
@ -263,11 +265,11 @@ Functional composition is special not only because you can take any two morphism
|
||||
|
||||
|
||||
|
||||
This approach for building stuff is often used in programming. To see some examples, you don't need to look further than the way the pipe operator in bash (`|`), that feeds the standard output of a program with the standard input of another program, is (ab)used (if you *want* to look further, note that there is a whole programming paradigm based on functional composition, called "concatenative programming").
|
||||
This approach for building stuff is often used in programming. To see some examples, you don't need to look further than the way the pipe operator in bash (`|`), that feeds the standard output of a program with the standard input of another program, is (ab)used. (If you *want* to look further, note that there is a whole programming paradigm based on functional composition, called "concatenative programming".)
|
||||
|
||||
But let's get back to the math. If we carefully review the definition above can see that it can be reduced to multiple applications of the following formula: given 4 objects and 3 morphisms between them **f** **g** **h**, combining **h** and **g** and then combining the end result with **f** should be the same as combining **h** to the result of **g** and **f** (or simply **(h • g) • f = h • (g • f)**.
|
||||
But let's get back to the math. If we carefully review the definition above, we can see that it can be reduced to multiple applications of the following formula: given 4 objects and 3 morphisms between them **f** **g** **h**, combining **h** and **g** and then combining the end result with **f** should be the same as combining **h** to the result of **g** and **f** (or simply **(h • g) • f = h • (g • f)**).
|
||||
|
||||
**Task:** show how the definition can be reduced to the formula (the approach resembles mathematical induction).
|
||||
**Task:** show how the definition can be reduced to the formula. (The approach resembles mathematical induction.)
|
||||
|
||||
This formula is true if and only if this diagram commutes.
|
||||
|
||||
@ -275,9 +277,9 @@ This formula is true if and only if this diagram commutes.
|
||||
|
||||
Given that all our category-theoretic diagrams commute, we can say, in such cases, that the formula and the diagram are equivalent.
|
||||
|
||||
This formula (and the diagram also) is the definition of a property called **associativity**. Being associative is required for functional composition to really be called functional composition (and for a category to really be called category). It is also required for our diagrams to work, as diagrams can only represent associative structures (imagine if the diagram above does not commute - it would be super weird).
|
||||
This formula (and the diagram) is the definition of a property called **associativity**. Being associative is required for functional composition to really be called functional composition (and for a category to really be called category). It is also required for our diagrams to work, as diagrams can only represent associative structures. (Imagine if the diagram above does not commute - it would be super weird.)
|
||||
|
||||
Associativity is not just about diagrams, for example when we express relations using formulas, associativity law means that *the brackets don't matter*. The formulaic definition of associativity is **(h • g) • f = h • (g • f)**).
|
||||
Associativity is not just about diagrams. For example when we express relations using formulas, associativity just means that brackets don't matter in our formulas (as evidenced by the definition **(h • g) • f = h • (g • f)**).
|
||||
|
||||
And it is not only about categories either, it is a property of many other operations on other types of objects as well e.g. if we look at numbers, we can see that the multiplication operation is associative e.g. **(1 . 2) . 3 = 1 . (2 . 3)**. While division is not **(1 / 2) / 3 = 1 / (2 / 3)**.
|
||||
|
||||
@ -295,17 +297,18 @@ It's important to mark this morphism, because there can be (let's add the very i
|
||||
Isomorphisms
|
||||
---
|
||||
|
||||
Why do we need to define a morphism that does nothing? It's because morphisms are the basic building blocks of our language and we need this one to be able to speak properly. For example, once we have the concept of identity morphism defined we can have a category-theoretic definition of an *isomoprhism* (which is important, because the concept of an isomorphism is very important for cathegory theory): An isomorphism between two objects (**A** and **B**) consists of two morphisms - (**A → B**. and **B → A**) such that their compositions are equivalent to the identity functions of the respective objects.
|
||||
Why do we need to define a morphism that does nothing? It's because morphisms are the basic building blocks of our language, and we need this one to be able to speak properly. For example, once we have the concept of identity morphism defined, we can have a category-theoretic definition of an *isomorphism* (which is important, because the concept of an isomorphism is very important for category theory): An isomorphism between two objects (**A** and **B**) consists of two morphisms - (**A → B**. and **B → A**) such that their compositions are equivalent to the identity functions of the respective objects.
|
||||
|
||||
Here is how this looks when expressed using a formulas:
|
||||
|
||||
Objects **A** and **B** are isomorphic
|
||||
iff there exist mophisms
|
||||
iff there exist morphisms
|
||||
**f: A → B**
|
||||
**g: B → A**
|
||||
such that
|
||||
**f • g = idB**
|
||||
**g • f = idA**
|
||||
**f • g = id B**
|
||||
and
|
||||
**g • f = id A**
|
||||
|
||||
And here is the same thing expressed with a commuting diagram.
|
||||
|
||||
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@ -12,7 +12,7 @@
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@ -140,7 +140,7 @@ The above set is not linearly-ordered - although the connection establishes the
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Greatest and least
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---
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Although posets don't give us definitive answer to who is better than who, some of them still can give us an answer to the more important question (in sports, as well as in other domains), namely *who is number one*, who is the champion, the player who is better than anyone else, or more generally the element that is bigger than any other element.
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Although partial orders don't give us definitive answer to who is better than who, some of them still can give us an answer to the more important question (in sports, as well as in other domains), namely *who is number one*, who is the champion, the player who is better than anyone else, or more generally the element that is bigger than any other element.
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We call such elements the *greatest element* and some (not all) partial orders do have such element - in our last diagram **M** is the greatest element, in this diagram, the green element is the biggest one.
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@ -237,21 +237,21 @@ We can define what is called the *inclusion order* of those sets, in which **A**
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Note that the *join* operation in an inclusion order is the set union, and the *meet* operation as set intersection.
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This diagram might remind you of something, because if we take the colors that are in each of its set and mix it into one color, we get the color-blending poset that we saw earlier.
|
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This diagram might remind you of something, because if we take the colors that are in each of its set and mix it into one color, we get the color-blending partial order that we saw earlier.
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The poset example with the number dividers is also isomorphic to an inclusion order - the inclusion order of all possible sets of *prime* numbers, including repeating ones (or alternatively the set of all *prime powers*). This is confirmed by the fundamental theory of arithmetic, which states that every number can be written as a product of primes in exactly one way.
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The order example with the number dividers is also isomorphic to an inclusion order - the inclusion order of all possible sets of *prime* numbers, including repeating ones (or alternatively the set of all *prime powers*). This is confirmed by the fundamental theory of arithmetic, which states that every number can be written as a product of primes in exactly one way.
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Order isomorphisms
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---
|
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We mentioned order isomorphisms several times already so this is about time to elaborate a bit about what they are. Take the isomorphism between the number poset and the prime inclusion order as an example. Like an isomorphism between any two sets, it is comprised of two functions:
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We mentioned order isomorphisms several times already so this is about time to elaborate a bit about what they are. Take the isomorphism between the number partial order and the prime inclusion order as an example. Like an isomorphism between any two sets, it is comprised of two functions:
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- One function from the prime inclusion order, to the number poset (which in this case is just the *multiplication* of all the elements in the set)
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- One function from the number poset to the prime inclusion order (which is an operation called *prime factorization* of a number, consisting of finding the set of prime numbers that result in that number when multiplied with one another).
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- One function from the prime inclusion order, to the number order (which in this case is just the *multiplication* of all the elements in the set)
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- One function from the number order to the prime inclusion order (which is an operation called *prime factorization* of a number, consisting of finding the set of prime numbers that result in that number when multiplied with one another).
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@ -276,7 +276,7 @@ By the way, the partial orders that are *not* distributive lattices are also iso
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Lattices
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===
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In the previous section we mentioned what *lattices* are - they are posets, in which every two elements have a *join* and a *meet*. So every lattice is also partial order, but not every partial order is a lattice (we will see even more members of this hierarchy). Most partial orders that are created based on some sort of rule, like the ones from the previous section, are also lattices when they are drawn in full, for example the color-mixing poset.
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In the previous section we mentioned what *lattices* are - they are partial orders, in which every two elements have a *join* and a *meet*. So every lattice is also partial order, but not every partial order is a lattice (we will see even more members of this hierarchy). Most partial orders that are created based on some sort of rule, like the ones from the previous section, are also lattices when they are drawn in full, for example the color-mixing order.
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@ -292,7 +292,7 @@ Our color-mixing lattice, has a *greatest element* (the black ball) and a *least
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Semilattices and trees
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---
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Lattices are posets that have both *join* *and* *meet* for each pair of elements. Posets that just have *join* (and no *meet*), or just have *meet* and no *join* are called *semilattices*. More specifically, posets that have *meet* for every pair of elements are called *meet-semilattices*.
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Lattices are partial orders that have both *join* *and* *meet* for each pair of elements. Partial orders that just have *join* (and no *meet*), or just have *meet* and no *join* are called *semilattices*. More specifically, partial orders that have *meet* for every pair of elements are called *meet-semilattices*.
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Before Width: | Height: | Size: 12 KiB After Width: | Height: | Size: 24 KiB |
@ -6,26 +6,28 @@ title: Logic
|
||||
Logic
|
||||
===
|
||||
|
||||
Now let's talk about one more *seemingly* unrelated topic just so we can "surprise" ourselves when we realize it's category theory (this time there will be another surprise in addition to that, so don't fall asleep. Also, in this chapter I will not merely transport you to a different branch of mathematics, but an entirely different discipline - *logic*.
|
||||
Now let's talk about one more *seemingly* unrelated topic just so we can "surprise" ourselves when we realize it's category theory. By the way, in this chapter there will be another surprise in addition to that, so don't fall asleep.
|
||||
|
||||
Also, I will not merely transport you to a different branch of mathematics, but to an entirely different discipline - *logic*.
|
||||
|
||||
What is logic
|
||||
===
|
||||
|
||||
Logic is the science of the *possible*. As such, it is at the root of all other sciences, all of which are sciences of the *actual*, i.e. that which really exists. For example, if science explains how our universe works then logic is the part of the description which is also applicable to any other universe that is *possible to exist*.
|
||||
Logic is the science of the *possible*. As such, it is at the root of all other sciences, all of which are sciences of the *actual*, i.e. that which really exists. For example, if science explains how our universe works then logic is the part of the description which is also applicable to any other universe that is *possible to exist*. A scientific theory aims to be consistent with both itself and observations, while a logical theory only needs to be consistent with itself.
|
||||
|
||||
Logic studies the *rules* by which knowing one thing leads you to conclude or (*prove*) that some other thing is also true, regardless of the things' domain (e.g. scientific discipline) and by only referring to their form.
|
||||
Logic studies the *rules* by which knowing one thing leads you to conclude (or *prove*) that some other thing is also true, regardless of the things' domain (e.g. scientific discipline) and by only referring to their form.
|
||||
|
||||
On top of that, it (logic) tries to organize those rules in *logical systems* (or *formal systems* as they are also called).
|
||||
|
||||
Logic and mathematics
|
||||
---
|
||||
|
||||
Seeing this description, we might think that the subject of logic is quite similar to the subject of set theory and category theory, as we described it in the first chapter. Only, there instead of the word "formal" we used another similar word, namely "abstract" and instead of "logical system" we said "theory". This observation would be quite correct - today most people agree that every mathematical theory is actually logic plus some additional definitions added to it. For example, part of the reason why *set theory* is so popular as a theory for the foundations of mathematics is that it adds just a single primitive to the standard axioms of logic which we will see shortly - the binary relation that indicates *set membership*. Category theory is close to logic too, but in a quite different way.
|
||||
Seeing this description, we might think that the subject of logic is quite similar to the subject of set theory and category theory, as we described it in the first chapter - instead of the word "formal" we used another similar word, namely "abstract", and instead of "logical system" we said "theory". This observation would be quite correct - today most people agree that every mathematical theory is actually logic plus some additional definitions added to it. For example, part of the reason why *set theory* is so popular as a theory for the foundations of mathematics is that it adds just one single primitive to the standard axioms of logic which we will see shortly - the binary relation that indicates *set membership*. Category theory is close to logic too, but in a quite different way.
|
||||
|
||||
Primary propositions
|
||||
---
|
||||
|
||||
A consequence of logic being the science of the possible is that in order to do anything at all in it, we should have an initial set of propositions that we accept as true, or false. These are also called "premises", "primary propositions" or "atomic propositions" as Wittgenstein dubbed them.
|
||||
A consequence of logic being the science of the possible is that in order to do anything at all in it, we should have an initial set of propositions that we accept as true or false. These are also called "premises", "primary propositions" or "atomic propositions" as Wittgenstein dubbed them.
|
||||
|
||||
|
||||
|
||||
@ -34,7 +36,7 @@ In the context of logic itself, these propositions are abstracted away (i.e. we
|
||||
Composing propositions
|
||||
---
|
||||
|
||||
At the heart of logic, as in category theory, is the concept of *composition* - if we have two or more propositions that are somehow related to one another, we can combine them into one using a logical operator, like "and", "or" "follows" etc. The result would be a new proposition, not unlike the way in which two monoid objects are combined into one using the monoid operation and actually some logical operations do form monoids, like for example the operation **and**, with the proposition **true** serving as the identity element.
|
||||
At the heart of logic, as in category theory, is the concept of *composition* - if we have two or more propositions that are somehow related to one another, we can combine them into one using a logical operators, like "and", "or" "follows" etc. The result would be a new proposition, not unlike the way in which two monoid objects are combined into one using the monoid operation. And actually some logical operations do form monoids, like for example the operation **and**, with the proposition **true** serving as the identity element.
|
||||
|
||||
|
||||
|
||||
@ -47,29 +49,29 @@ Important to note that **∧** is the symbol for **and** and **∨** is the symb
|
||||
The equivalence of primary and composite propositions
|
||||
---
|
||||
|
||||
It is important to stress that although in the leftmost proposition the green ball is wrapped in a gray ball to make the diagram prettier propositions that are composed of several premises (symbolized by gray balls, containing some other balls) are not in any way different from "primary" propositions (single-color balls) and compose in the same way
|
||||
When looking at the last diagram, it is important to stress that, although in the leftmost proposition the green ball is wrapped in a gray ball to make the diagram prettier, propositions that are composed of several premises (symbolized by gray balls, containing some other balls) are not in any way different from "primary" propositions (single-color balls) and that they compose in the same way.
|
||||
|
||||
|
||||
|
||||
Modus ponens
|
||||
---
|
||||
|
||||
As an example of a proposition that contains multiple levels of nesting (and also a great introduction of the subject of logic in its own right), consider one of the oldest (it was already known by Stoics at 3rd century B.C.) and most famous propositions ever, namely the *modus ponens*.
|
||||
As an example of a proposition that contains multiple levels of nesting (and also as a great introduction of the subject of logic in its own right), consider one of the oldest (it was already known by Stoics at 3rd century B.C.) and most famous propositions ever, namely the *modus ponens*.
|
||||
|
||||
Modus ponens is a proposition that states that if proposition **A** is true and also if proposition **(A → B)** is true (that is, if **A** implies **B**), then **B** is true as well. For example, if we know that "Socrates is a human" and that "humans are mortal" (or "being human implies being mortal"), we also know that "Socrates is mortal".
|
||||
Modus ponens is a proposition that states that if proposition **A** is true and also if proposition **(A → B)** is true (that is if **A** implies **B**), then **B** is true as well. For example, if we know that "Socrates is a human" and that "humans are mortal" (or "being human implies being mortal"), we also know that "Socrates is mortal."
|
||||
|
||||
|
||||
|
||||
Let's dive into it. The proposition is composed of two other propositions in a **follows** relation where the proposition that follows (**B**) is primary , but the proposition from which **B** follows is not (let's call that one **C** - so the whole proposition becomes **C → B**).
|
||||
Let's dive into it. The proposition is composed of two other propositions in a **follows** relation, where the proposition that follows (**B**) is primary, but the proposition from which **B** follows is not primary (let's call that one **C** - so the whole proposition becomes **C → B**.)
|
||||
|
||||
Going one more level down, we notice that the **C** propositions is itself composed of two propositions in an **and**, relationship - **A** and let's call the other one **D** (so **A ∧ D**), where **D** is itself composed of two propositions, this time in a **follows** relationship - **A → B**. But all of this is better visualized in the diagram.
|
||||
|
||||
Tautologies
|
||||
---
|
||||
|
||||
Because the content of propositions in logic is abstracted away, we often cannot tell whether a given proposition is true or false without knowing their values. However, with propositions such as *modus ponens* we can: modus ponens is *always true*, regardless of which are the propositions which form it and whether they are true or false. If we want to be fancy, we can also say that it is *true in all models of the logical system*, a model being a set of real-world premises are taken to be signified by our propositions.
|
||||
We often cannot tell whether a given composite proposition is true or false without knowing the values of the propositions that compose it. However, with propositions such as *modus ponens* we can: modus ponens is *always true*, regardless of whether the propositions that form it are true or false. If we want to be fancy, we can also say that it is *true in all models of the logical system*, a model being a set of real-world premises are taken to be signified by our propositions.
|
||||
|
||||
For example, our previous example would not stop being true if we substitute "Socrates" with any other name, nor if we substitute "mortal" for any other quality that humans possess.
|
||||
For example, our previous example will not stop being true if we substitute "Socrates" with any other name, nor if we substitute "mortal" for any other quality that humans possess.
|
||||
|
||||
|
||||
|
||||
@ -83,17 +85,12 @@ Here are some more complex (less boring) tautologies (the symbol **¬** means "n
|
||||
|
||||
|
||||
|
||||
We will learn how to determine which propositions are a tautologies, but first let's see why is this important at all i.e. what are tautologies good for.
|
||||
We will learn how to determine which propositions are a tautologies shortly, but first let's see why is this important at all i.e. what are tautologies good for: tautologies are useful because they are the basis of *axiom schemas*/*rules of inference*. And *axiom schemas* or *rules of inference* serve as starting point from which we can generate other true logical statements by means of substitution.
|
||||
|
||||
Logical systems
|
||||
===
|
||||
|
||||
Tautologies are useful because they are the basis of *axiom schemas*/*rules of inference*. And *axiom schemas* or *rules of inference* serve as starting point from which we can generate other true logical statements by means of substitution.
|
||||
|
||||
Axiom schemas
|
||||
Axiom schemas/Rules of inference
|
||||
---
|
||||
|
||||
Realizing that the colors of the balls in modus ponens are superficial, we may want to represent the general structure of modus ponens that all of its variations have.
|
||||
Realizing that the colors of the balls in modus ponens are superficial, we may want to represent the general structure of modus ponens that all of its variations share.
|
||||
|
||||
|
||||
|
||||
@ -103,48 +100,47 @@ Note that the propositions that we plug into the schema don't have to be primary
|
||||
|
||||
|
||||
|
||||
Rules of inference
|
||||
This is it about *axiom schemas* and *rules of inference are almost the same thing except they allow us to actually distill the conclusion from the premises. For example in the case above, we can use modus ponens as a rule of inference to proves that **a or b** is true.
|
||||
|
||||
All axiom schemas can be easily applied as rules of inference and the other way around.
|
||||
|
||||
Logical systems
|
||||
---
|
||||
|
||||
Rules of inference specify how axiom schemas are applied. Axiom schemas and rules of inference are almost the same thing. All axiom schemas can be easily applied as rules of inference and the other way around e.g. in the case above, we can use modus ponens as a rule of inference to proves that **a or b** is true.
|
||||
|
||||
Completeness of logical systems
|
||||
---
|
||||
|
||||
Knowing that we can use axiom schemas/rules of inference to generate new propositions, we might ask whether it is possible to create a small collection of such schemas/rules that is curated in such a way that it enables us to generate *all* other propositions. You would be happy (although a little annoyed, I imagine) to learn that there exist not only one, many such collections. And yes, collections such as the one above are what we call *logical systems*.
|
||||
Knowing that we can use axiom schemas/rules of inference to generate new propositions, we might ask whether it is possible to create a small collection of such schemas/rules that is curated in such a way that it enables us to generate *all* other propositions. You would be happy (although a little annoyed, I imagine) to learn that there exist not only one, but many such collections. And yes, collections of this sort are what we call *logical systems*.
|
||||
|
||||
Here is one such collection which consists of the following five axiom schemes **in addition to the inference rule modus ponens** (These are axiom schemes, even though we use colors).
|
||||
|
||||
|
||||
|
||||
Proving that this and other similar logical systems are complete (can really generate all other propositions) is due to Godel and is known as "Godel's completeness theorem".
|
||||
Proving that this and other similar logical systems are complete (can really generate all other propositions) is due to Gödel and is known as "Gödel's completeness theorem" (Gödel is so important that I specifically searched for the **ö** letter so I can spell his hame right.)
|
||||
|
||||
Conclusion
|
||||
---
|
||||
|
||||
We now have an idea about how do some of the main logical constructs (axioms, rules of inference) work. But in order to prove that they are true, and to understand *what they are*, we need to do so through a specific *interpretation* of those constructs. We will now look into two interpretations - one very old and the other, relatively recent. This would be a slight detour from our usual subject matter of points and arrows, but I assure you that it would be worth it. So let's start.
|
||||
We now have an idea about how do some of the main logical constructs (axioms, rules of inference) work. But in order to prove that they are true, and to understand *what they are*, we need to do so through a specific *interpretation* of those constructs.
|
||||
|
||||
We will look into two interpretations - one very old and the other, relatively recent. This would be a slight detour from our usual subject matter of points and arrows, but I assure you that it would be worth it. So let's start.
|
||||
|
||||
Classical logic. The truth-functional interpretation
|
||||
===
|
||||
|
||||
> Beyond the world that we inhabit and perceive every day there exist the *world of forms* where all ideas and concepts that manifest themselves in the objects that we perceive reside e.g. beyond all the people that have ever lived, there lies the prototypical person, and we are people only insofar as we resemble that person, beyond all the things in the world that are strong, lies the ultimate concepts of strength, from which all of them borrow etc. And although, as mere mortals, we live in the world of appearances and cannot perceive the world of forms, we can, through philosophy, "recollect" with it and know some of its features.
|
||||
> Beyond the world that we inhabit and perceive every day, there exist the *world of forms* where reside all ideas and concepts that manifest themselves in the objects that we perceive e.g. beyond all the people that have ever lived, there lies the prototypical person, and we are people only insofar as we resemble that person, beyond all the things in the world that are strong, lies the ultimate concepts of strength, from which all of them borrow etc. And although, as mere mortals, we live in the world of appearances and cannot perceive the world of forms, we can, through philosophy, "recollect" with it and know some of its features.
|
||||
|
||||
The above is a summary of a worldview that is due to the Greek philosopher Plato and is sometimes called Plato's *theory of forms*. Originally, the discipline of logic represents an effort to think and structure our thoughts in a way that they apply to this world of forms i.e. in a "formal" way. Today, this original paradigm of logic is known as "classical logic". Although it all started with Plato, most of it is due to the 20th century mathematician David Hilbert.
|
||||
|
||||
The existence of the world of forms implies that even if there are many things that we, people, don't know, at least *somewhere out there* there exists answer to every question. In logic, this translates to *the principle of bivalence* that states that *each proposition is either true or false*.
|
||||
|
||||
Due to this principle, propositions in classical logic can be aptly expressed with set theory using boolean set (so either true or false).
|
||||
The existence of the world of forms implies that even if there are many things that we people don't know, at least *somewhere out there* there exists an answer to every question. In logic, this translates to *the principle of bivalence* that states that *each proposition is either true or false*. Due to this principle, propositions in classical logic can be aptly represented using set theory by boolean set, which contains those two values.
|
||||
|
||||
|
||||
|
||||
*Primary propositions* are just a bunch of boolean values, *logical operators* as functions that take a one or several boolean values and return another boolean value. *Composite propositions* are, in this case, just the results of the invocation of these functions.
|
||||
According to the classical interpretation, you can think of *primary propositions* as just a bunch of boolean values, *logical operators* are functions that take a one or several boolean values and return another boolean value (and *composite propositions* are, just the results of the application of these functions).
|
||||
|
||||
Let's review all logical operators in this context.
|
||||
|
||||
The *negation* operation
|
||||
---
|
||||
|
||||
Let's begin with the negation operation. Negation is a unary operation, which means that it is a function that takes just *one* argument of type boolean and (like all other logical operators) returns a value of type boolean.
|
||||
Let's begin with the negation operation. Negation is a unary operation, which means that it is a function that takes just *one* argument and (like all other logical operators) returns one value, where both the arguments and the return type are boolean values.
|
||||
|
||||
|
||||
|
||||
@ -155,18 +151,18 @@ The same function can also be expressed in a slightly less-fancy way by this tab
|
||||
| True | False |
|
||||
| False | True |
|
||||
|
||||
Such tables are called *truth tables* and they are ubiquitous in classical logic . They can be used not only for defining operators but for proving results as well.
|
||||
Tables like this one are called *truth tables* and they are ubiquitous in classical logic. They can be used not only for defining operators but for proving results as well.
|
||||
|
||||
Interlude: Proving results by truth tables
|
||||
---
|
||||
|
||||
Having defined the negation operator, as we did above, we are in position to prove the first of the axioms of the logical system we saw, namely the *double negation elimination*. In natural language, this axiom is equivalent to the observation that saying "I am *not unable* to do X" is the same as saying "I am *able* to do it".
|
||||
Having defined the negation operator, we are in position to prove the first of the axioms of the logical system we saw, namely the *double negation elimination*. In natural language, this axiom is equivalent to the observation that saying "I am *not unable* to do X" is the same as saying "I am *able* to do it".
|
||||
|
||||
|
||||
|
||||
(despite its triviality, the double negation axiom is probably the most controversial result in logic, we will see why later.)
|
||||
|
||||
If we view logical operators as functions, from and to the set of boolean values, than proving axioms involves composing several of those functions into one function and observing its output. More specifically, the proof of the formula above involves just composing the negation function with itself and verifying that it leaves us in the same place from which we started.
|
||||
If we view logical operators as functions from and to the set of boolean values, than proving axioms involves composing several of those functions into one function and observing its output. More specifically, the proof of the formula above involves just composing the negation function with itself and verifying that it leaves us in the same place from which we started.
|
||||
|
||||
|
||||
|
||||
@ -186,7 +182,9 @@ Each proposition in classical logic can be proved with such diagrams/tables.
|
||||
The **and** and **or** operations
|
||||
---
|
||||
|
||||
OK, you know what **and** means and I know what it means, but what about those annoying people that want everything to be formally specified (nudge, nudge). Well we already know how we can satisfy them - we just have to construct the boolean function that represents **and**. Because **and** is a *binary* operator, instead of a single value the function would accept a *pair* of boolean values.
|
||||
OK, *you* know what **and** means and *I* know what it means, but what about those annoying people that want everything to be formally specified (nudge, nudge). Well we already know how we can satisfy them - we just have to construct the boolean function that represents **and**.
|
||||
|
||||
Because **and** is a *binary* operator, instead of a single value the function would accept a *pair* of boolean values.
|
||||
|
||||
|
||||
|
||||
@ -210,7 +208,7 @@ We can do the same for **or**, here is the table.
|
||||
|
||||
**Task:** Draw the diagram for **or**.
|
||||
|
||||
Using those tables, we can also prove some axiom schemas we can later use:
|
||||
Using those tables, we can also prove some axiom schemas we can use later:
|
||||
|
||||
- For **and**: **p ∧ q → p** and **p ∧ q → q** "If I am tired and hungry, this means that I am hungry".
|
||||
- For **or**: **p → p ∨ q** and **p → p ∨ q** "If I have a pen this means that I am either have a pen or a ruler".
|
||||
@ -218,9 +216,9 @@ Using those tables, we can also prove some axiom schemas we can later use:
|
||||
The *implies* operation
|
||||
---
|
||||
|
||||
Let's now look into something less trivial: the *implies* operation, (also known as *entailment*). This operation binds two propositions in a way that the truth of the first one implies the truth of the second one. You can read **p → q** as "if **p** is true, then **q** must also be true. The entailment is a binary one, it is represented by a function from an ordered pair of boolean values, to a boolean value.
|
||||
Let's now look into something less trivial: the *implies* operation, (also known as *entailment*). This operation binds two propositions in a way that the truth of the first one implies the truth of the second one. You can read **p → q** as "if **p** is true, then **q** must also be true.
|
||||
|
||||
Here is the truth table.
|
||||
Entailment is also a binary function - it is represented by a function from an ordered pair of boolean values, to a boolean value.
|
||||
|
||||
| p | q | p → q |
|
||||
|---| --- | --- |
|
||||
@ -229,19 +227,19 @@ Here is the truth table.
|
||||
| False | True | True |
|
||||
| False | False | True |
|
||||
|
||||
Now there are some aspects of this which are non obvious so let's go through every case.
|
||||
Now there are some aspects of this which are non-obvious so let's go through every case.
|
||||
|
||||
1. If *p* is true and *q* is also true, then **p** does imply **q** - obviously.
|
||||
2. If *p* is true but *q* is false then **q** does not follow from **p** - cause *q* would have been true if it did.
|
||||
3. If *p* is false but *q* is true, then **p** still does imply **q**. What the hell? Consider that by saying that *p* implies *q* we don't say that the two are 100% interdependent e.g. the claim that "drinking alcohol causes headache" does not mean that drinking is the only source of headaches.
|
||||
4. And finally if **p** is false but **q** is false too, then **p** still does imply **q** (just some other day).
|
||||
|
||||
It might help you to remember that **p → q** (**p** implies **q**) is true when **-p ∨ q** (either **p** is false or **q** is true.)
|
||||
It might help you to remember that in classical logic **p → q** (**p** implies **q**) is true when **-p ∨ q** (either **p** is false or **q** is true.)
|
||||
|
||||
Proving results by axioms/rules of inference
|
||||
---
|
||||
|
||||
Let's examine the above formula, stating that **p → q** is the same as **-p ∨ q**.
|
||||
Let's examine the above formula, stating that **p → q** is the same as **¬p ∨ q**.
|
||||
|
||||
|
||||
|
||||
@ -254,33 +252,36 @@ We can easily prove this by using truth tables.
|
||||
| False | True | **True** | True | True | **True** |
|
||||
| False | False | **True** | True | False | **True** |
|
||||
|
||||
But It would be much more intuitive if we do it using axioms and rules of inference. To do so, we have to start with the formula we have (***p → q**) plus the axiom schemas and arrive at the formula we want to prove (**-p ∨ q**). The formulas that are used at each step are specified at the right-hand side, the rule of inference is modus ponens.
|
||||
But it would be much more intuitive if we do it using axioms and rules of inference. To do so, we have to start with the formula we have (**p → q**) plus the axiom schemas, and arrive at the formula we want to prove (**¬p ∨ q**).
|
||||
|
||||
Here is one way to do it. The formulas that are used at each step are specified at the right-hand side, the rule of inference is modus ponens.
|
||||
|
||||
|
||||
|
||||
To really prove that the two formulas are equivalen we have to also do it the other way around (start with (**-p ∨ q**) and (***p → q**)).
|
||||
Note that to really prove that the two formulas are equivalen we have to also do it the other way around (start with (**¬p ∨ q**) and (**p → q**)).
|
||||
|
||||
Intuinistic logic. The BHK interpretation
|
||||
Intuitionistic logic. The BHK interpretation
|
||||
===
|
||||
|
||||
Although the classical truth-functional interpretation of logic works and is correct in its own right, it doesn't fit well the categorical framework that we are using here: It is too "low-level", it relies on manipulating the values of the propositions, it doesn't in any way show the connection between **and** and **or** that we already saw. For these and other reasons (mostly other, probably), in the 20th century a whole new school of logic was founded, called *intuinistic logic*.
|
||||
If classical logic is based on *set theory*, intuinistic logic is based on *category theory* and related theories. If *classical logic* is based on Plato's theory of forms, then intuinism began with a philosophical idea originating from Kant and Schopenhauer: the idea that the world as we experience it is largely predetermined of out perceptions of it. As the mathematician L.E.J. Brouwer puts it.
|
||||
Although the classical truth-functional interpretation of logic works and is correct in its own right, it doesn't fit well the categorical framework that we are using here: It is too "low-level", it relies on manipulating the values of the propositions. According to it, the operations **and**, **or** and **implies** are just 3 of the 16 possible binary logical operations and they are not really connected to each other.
|
||||
|
||||
For these and other reasons (mostly other, probably), in the 20th century a whole new school of logic was founded, called *intuitionistic logic*. If we view classical logic as based on *set theory*, then intuitionistic logic would be based on *category theory* and its related theories. If *classical logic* is based on Plato's theory of forms, then intuinism began with a philosophical idea originating from Kant and Schopenhauer: the idea that the world as we experience it is largely predetermined of out perceptions of it. As the mathematician L.E.J. Brouwer puts it.
|
||||
|
||||
> [...] logic is life in the human brain; it may accompany life outside the brain but it can never guide it by virtue of its own power.
|
||||
|
||||
Classical and intuinistic logic diverge from one another right from the start - because intuinistic logic deals with *constructing* a proof rather than *discovering* or *unveiling* a universal truth, it is *off with the principle of bivalence*, that is, we have no basis to claim that each statements is necessarily *true or false*. For example, there might be a statements that might not be provable not because they are false, but simply because they fall outside of the domain of a given logical system (the twin-prime conjecture is often given as an example for this.)
|
||||
Classical and intuitionistic logic diverge from one another right from the start: because according to intuitionistic logic we are *constructing* proofs rather than *discovering* them or *unveiling* a universal truth, we are *off with the principle of bivalence*, that is, we have no basis to claim that each statements is necessarily *true or false*. For example, there might be a statements that might not be provable not because they are false, but simply because they fall outside of the domain of a given logical system (the twin-prime conjecture is often given as an example for this.)
|
||||
|
||||
So, intuinistic logic is not bivalent, i.e. we cannot have all propositions reduced to true and false.
|
||||
So, intuitionistic logic is not bivalent, i.e. we cannot have all propositions reduced to true and false.
|
||||
|
||||
|
||||
|
||||
One thing that we still do have are propositions that are "true" in the sense that a proof for them is given - the primary propositions. So with some caveats (which we will see later) the bivalence between true and false proposition might be thought out as similar to the bivalence between the existence or non-existence of a proof for a given proposition - there either is a proof of it or there isn't.
|
||||
One thing that we still do have there are propositions that are "true" in the sense that a proof for them is given - the primary propositions. So with some caveats (which we will see later) the bivalence between true and false proposition might be thought out as similar to the bivalence between the existence or absense of a proof for a given proposition - there either is a proof of it or there isn't.
|
||||
|
||||
|
||||
|
||||
This bivalence is at the heart of what is called the Brouwer–Heyting–Kolmogorov (BHK) interpretation of logic, something that we will look into next.
|
||||
|
||||
The original formulation of the BHK interpretation is not based on any particular mathematical theory. Here we will first illustrate it using the language of set theory (just so we can abandon it a little later).
|
||||
The original formulation of the BHK interpretation is not based on any particular mathematical theory. Here, we will first illustrate it using the language of set theory (just so we can abandon it a little later).
|
||||
|
||||
The **and** and **or** operations
|
||||
---
|
||||
@ -298,26 +299,27 @@ Saying that **A** implies **B** (**A → B**) would just mean that there exist a
|
||||
|
||||
|
||||
|
||||
And the *modus ponens* rule of inference is just the fact that if we have a proof of **A** we can call this function (**A → B**) to obtain a proof of **B**.
|
||||
And the *modus ponens* rule of inference is expressed by the fact that if we have a proof of **A** we can call this function (**A → B**) to obtain a proof of **B**.
|
||||
|
||||
(Note that in order for this to work, we also need to define the functions in terms of sets i.e. we need to have a set representing **A → B** for each **A** and **B**. We will come back to this later.)
|
||||
|
||||
The *negation* operation
|
||||
---
|
||||
|
||||
So according to BHK interpretation saying that **A** is true, means that that we possess a proof of **A** - simple enough. But it's a bit harder to express the fact that **A** is false: it is not enough to say that we *don't have a proof* of **A** (the fact that don't have it, doesn't mean it doesn't exist). Instead, we must show that claiming that it is true *leads to contradiction*.
|
||||
So according to BHK interpretation saying that **A** is true, means that that we possess a proof of **A** - simple enough. But it's a bit harder to express the fact that **A** is false: it is not enough to say that we *don't have a proof* of **A** (the fact that don't have it, doesn't mean it doesn't exist). Instead, we must show that claiming that **A** is true leads to a *contradiction*.
|
||||
|
||||
To express this, intuinistic logic defines the constant **⊥** which plays the role of *False* (and is also known as "absurdity" or "bottom value"). **⊥** is defined as the proof of a formula that does not have any proofs. And the equivalent of false propositions are the ones that imply that the bottom value is provable (which is a contradiction). So **¬A** is **A → ⊥**.
|
||||
|
||||
To express this, intuitionistic logic defines the constant **⊥** which plays the role of *False* (and is also known as "absurdity" or "bottom value"). **⊥** is defined as the proof of a formula that does not have any proofs. And the equivalent of false propositions are the ones that imply that the bottom value is provable (which is a contradiction). So **¬A** is **A → ⊥**.
|
||||
|
||||
In set theory, the **⊥** constant is expressed by the empty set.
|
||||
|
||||
|
||||
|
||||
And the observation that propositions that are connected to the bottom value are false is expressed by the fact that if a proposition is true, i.e. there exists a proof of it, there can be no function from it to the empty set.
|
||||
And the observation that propositions that are connected to the bottom value are false is expressed by the fact that if a proposition is true, i.e. there exists a proof of it, then there can be no function from it to the empty set.
|
||||
|
||||
|
||||
|
||||
The only way for there to be such function is if the set of proof of the proposition is empty as well.
|
||||
The only way for there to be such function is if the set of proofs of the proposition is empty as well.
|
||||
|
||||
|
||||
|
||||
@ -325,64 +327,60 @@ The only way for there to be such function is if the set of proof of the proposi
|
||||
|
||||
**Task** Look up the definition of function and verify that there does exist a function *from the empty set* to itself (in fact there exist a function from the empty set to any other set.
|
||||
|
||||
Classical VS Intuinistic logic
|
||||
Classical VS intuitionistic logic
|
||||
---
|
||||
|
||||
Although intuinistic logic seems to differ a lot from classical logic, it actually doesn't - if we try to deduce the axiom schemas/rules of inference that correspond to the definitions of the structures outlined above, we would see that they are virtually the same as the ones that define classical logic. With one exception concerning the *double negation elimination axiom* that we saw earlier, a version of which is known as *the law of excluded middle*.
|
||||
Although from first glance intuitionistic logic seems to differ a lot from classical logic, it actually doesn't - if we try to deduce the axiom schemas/rules of inference that correspond to the definitions of the structures outlined above, we would see that they are virtually the same as the ones that define classical logic. With one exception concerning the *double negation elimination axiom* that we saw earlier, a version of which is known as *the law of excluded middle*.
|
||||
|
||||
|
||||
|
||||
This law is valid in classical logic and is true when we look at it in terms of truth tables, but there is no justification for it in the BHK interpretation - a fact that spawned a heated debate between the inventor of classical logic David Hilbert and the inventor of intuinistic logic L.E.J. Brouwer, known as *the Brouwer–Hilbert controversy*.
|
||||
This law is valid in classical logic and is true when we look at it in terms of truth tables, but there is no justification for it in the BHK interpretation - a fact that spawned a heated debate between the inventor of classical logic David Hilbert and the inventor of intuitionistic logic L.E.J. Brouwer, known as *the Brouwer–Hilbert controversy*.
|
||||
|
||||
Logics as categories
|
||||
===
|
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|
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Leaving the differences between intuinistic and classical logics aside, the BHK interpretation is interesting because it provides that higher-level view of logic, that we need in order to construct a interpretation of it based on category theory.
|
||||
Leaving the differences between intuitionistic and classical logics aside, the BHK interpretation is interesting because it provides that higher-level view of logic, that we need in order to construct a interpretation of it based on category theory.
|
||||
|
||||
Such higher-level interpretations of logic are sometimes called an *algebraic* interpretations, *algebraic* being an umbrella term describing all structures that can be represented using category theory like groups and orders.
|
||||
Such higher-level interpretations of logic are sometimes called *algebraic* interpretations, *algebraic* being an umbrella term describing all structures that can be represented using category theory, like groups and orders.
|
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|
||||
The Curry-Howard isomorphism
|
||||
---
|
||||
|
||||
Programmers might find the definition of the BHK interpretation interesting for other reason - it is very similar to a definition of a programming language: propositions are *types*, the **implies** operations are *functions*, **and** operations are composite types (objects) and **or** operations are *sum types* (which are currently not supported in most programming languages, but that's a separate topic.) Finally a proof of a given proposition is represented by a value of the corresponding type.
|
||||
Programmers might find the definition of the BHK interpretation interesting for other reason - it is very similar to a definition of a programming language: propositions are *types*, the **implies** operations are *functions*, **and** operations are composite types (objects), and **or** operations are *sum types* (which are currently not supported in most programming languages, but that's a separate topic.) Finally a proof of a given proposition is represented by a value of the corresponding type.
|
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|
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|
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This similarity is known as the *Curry-Howard isomorphism*.
|
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|
||||
**Task:** The Curry-Howard isomorphism is also the basis of special types of programming languages called "proof assistants" which help you verify logical proofs. Install a proof assistant and try to see how it works (I recommend Mike Nahas's Coq Tutorial).
|
||||
**Task:** The Curry-Howard isomorphism is also the basis of special types of programming languages called "proof assistants" which help you verify logical proofs. Install a proof assistant and try to see how it works (I recommend the Coq Tutorial by Mike Nahas).
|
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|
||||
Cartesian closed categories
|
||||
---
|
||||
|
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Knowing about the Curry-Howard isomorphism and knowing also that programming languages can be described by category theory may lead us to think that *category theory is part of this isomorphism as well*. And we would be quite correct, this is why it's sometimes known as the Curry-Howard-*Lambek* isomorphism, Lambek being the person who discovered the categorical side.
|
||||
Knowing about the Curry-Howard isomorphism and knowing also that programming languages can be described by category theory may lead us to think that *category theory is part of this isomorphism as well*. And we would be quite correct, this is why it is sometimes known as the Curry-Howard-*Lambek* isomorphism, Lambek being the person who discovered the categorical side.
|
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|
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Let's examine this isomorphism (without being too formal about it). As all other isomorphisms, it comes in two parts.
|
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|
||||
The first part is finding a way to convert a logical system into a category - this would not be hard for us, as sets form a category and the flavor of the BHK interpretation that we saw is based on sets.
|
||||
Let's examine this isomorphism (without being too formal about it). As all other isomorphisms, it comes in two parts. The first part is finding a way to convert a logical system into a category - this would not be hard for us, as sets form a category and the flavor of the BHK interpretation that we saw is based on sets.
|
||||
|
||||
|
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|
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**Task:** See whether you can prove that logic propositions and entailments forms a category. What is missing?
|
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|
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The second part involves converting a category into a logical system - this is much harder, as in order to do it, we have to enumerate the criteria that a given category has to adhere to in order for it to be "logical".
|
||||
The second part involves converting a category into a logical system - this is much harder. To do it, we have to enumerate the criteria that a given category has to adhere to, in order for it to be "logical". These criteria have to guarantee that the category has objects that correspond to all valid logical propositions and no objects that correspond to invalid ones.
|
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|
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|
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|
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These criteria have to guarantee that a category has objects that correspond to all valid logical propositions and no objects that correspond to invalid ones.
|
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|
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Categories that adhere to these criteria are called *cartesian closed categories*, but we won't look into them directly. Instead, we would present logic using a similar but simpler structures that we already examined - orders.
|
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Categories that adhere to these criteria are called *cartesian closed categories*. We won't describe them here directly, but instead we would start with a similar but simpler structures that are instance of them and that we already examined - orders.
|
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|
||||
Logics as orders
|
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---
|
||||
|
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We will now do something that is quite characteristic of category theory - examining a concept in a more limited version of the theory, in order to make things simpler for ourselves.
|
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|
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So we already saw that a logical system, along with a set of its propositions, forms a category.
|
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So we already saw that a logical system along with a set of primary propositions forms a category.
|
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|
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|
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|
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If we assume that there is only one way to go from proposition **A**, to proposition **B** (or that there are many ways, but we are not interested in the difference between them), then logic is not only a category, but a *preorder* in which the relationship "bigger than" is taken to mean "implies".
|
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If we assume that there is only one way to go from proposition **A**, to proposition **B** (or there are many ways, but we are not interested in the difference between them), then logic is not only a category, but a *preorder* in which the relationship "bigger than" is taken to mean "implies".
|
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|
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|
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|
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@ -405,7 +403,7 @@ By now you probably realized that the **and** and **or** operations are the brea
|
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|
||||
Here comes the first criteria for an order to represent a logical system accurately - *it has to have **meet** and **join** operations for all elements*. Having two elements without a meet would mean that you would have a logical system where there are propositions for which you cannot say that one or the other is true. And this not how logic works, so our order has to have meets and joins for all elements. Incidentally we already know how such orders are called - they are called *lattices*.
|
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|
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One more important law concerning the **and** and **or** operations that is not always present in the **meet**-s and **join**-s concerns the connection between the two, i.e. way that the **and** and **or** operations distribute, over one another.
|
||||
One important law of the **and** and **or** operations, that is not always present in the **meet**-s and **join**-s concerns the connection between the two, i.e. way that they distribute, over one another.
|
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|
||||
|
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|
||||
@ -420,7 +418,7 @@ Wait, where have we heard about distributive lattices before? In the previous ch
|
||||
The *negation* operation
|
||||
---
|
||||
|
||||
In order for a distributive lattice to represent a logical system, it has to also have objects that correspond to the values **True** and **False**. But in order to mandate that these objects exist, we must first find a way to specify what they are in order/category-theoretic terms.
|
||||
In order for a distributive lattice to represent a logical system, it has to also have objects that correspond to the values **True** and **False**. But to mandate that these objects exist, we must first find a way to specify what they are in order/category-theoretic terms.
|
||||
|
||||
A well-known result in logic, called *the principle of explosion*, states that if we have a proof of **False** (or if "**False** is true" if we use the terminology of classical logic), then any and every other statement can be proven. And we also know that no true statement implies **False** (in fact in intuinistic logic this is the definition of a true statement). Based on these criteria we know that the **False** object would look like this when compared to other objects:
|
||||
|
||||
@ -438,15 +436,16 @@ So **True** and **False** are just the *greatest* and *least* objects of our ord
|
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|
||||
|
||||
|
||||
This is another example of the categorical concept of duality - * **True** and **False** are dual to each other*, (which makes a lot of sense if you think about it).
|
||||
This is another example of the categorical concept of duality - ***True** and **False** are dual to each other*, (which makes a lot of sense if you think about it).
|
||||
|
||||
So in order to represent logic, our distributive lattice has to also be *bounded* i.e. it has to have greatest and least elements.
|
||||
So in order to represent logic, our distributive lattice has to also be *bounded* i.e. it has to have greatest and least elements (which play the roles of **True** and **False**).
|
||||
|
||||
The *implies* operation
|
||||
---
|
||||
|
||||
Finally, if a lattice really represents a logical system (that is, it is isomorphic to a set of propositions) it also has to have *function objects* i.e. there needs to be a rule that identifies a unique object **A → B** for each pair of objects **A** and **B**, such that all axioms of logic are followed.
|
||||
How would this object be described? You guessed it, using categorical language i.e. by recognizing a structure that consisting of set of relations between objects, in which (**A → B**) plays a part.
|
||||
|
||||
How would this object be described? You guessed it, using categorical language i.e. by recognizing a structure that consists of set of relations between objects in which (A → B) plays a part.
|
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|
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|
||||
|
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@ -21,7 +21,7 @@
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Before Width: | Height: | Size: 14 KiB After Width: | Height: | Size: 17 KiB |
@ -1,4 +1,7 @@
|
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Invertible
|
||||
invertible
|
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coproduct
|
||||
Coproduct
|
||||
morphism
|
||||
Morphism
|
||||
coproducts
|
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@ -31,6 +34,7 @@ Monoid
|
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Monoids
|
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isomorphism
|
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isomorphisms
|
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Isomorphisms
|
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Hasse
|
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Antisymmetry
|
||||
connexity
|
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@ -50,10 +54,21 @@ non-abelian
|
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Dih3
|
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composable
|
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forall
|
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<<<<<<< HEAD
|
||||
codomain
|
||||
Euclidian
|
||||
subtypes
|
||||
Immanuel
|
||||
iff
|
||||
concatenative
|
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structs
|
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DeMorgan's
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getter
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OOP
|
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=======
|
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BHK
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yey
|
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intuinistic
|
||||
Intuinistic
|
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intuitionistic
|
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intuinism
|
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Brouwer
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schemas
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@ -75,6 +90,9 @@ Curry-Howard-Lambek
|
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Curry-Howard
|
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Lambek
|
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cartesian
|
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<<<<<<< HEAD
|
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Functor
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functor
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functors
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Functors
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isomorphism
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@ -87,4 +105,3 @@ Monoidal
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morphisms
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codomain
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iff
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invertible
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|
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2
index.md
2
index.md
@ -3,4 +3,4 @@ layout: default
|
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title: index
|
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---
|
||||
|
||||
[](00)
|
||||
[](00_about)
|
||||
|
||||
33
proposal.md
33
proposal.md
@ -29,14 +29,14 @@ I am writing a primer in category theory and various related concepts in "higher
|
||||
|
||||
My book serves as chapter 0 going through the gist of the material covered by other similar introductory books, but doing so in a way that would enable non-mathematicians to swift through with ease.
|
||||
|
||||
Reading it would enable my readers to effortlessly go through any academic introduction to category theory, as well as to start tacking resources that use category theory as a tool to threat other subjects.
|
||||
Reading it would enable my readers to effortlessly go through any academic introduction to category theory, as well as to start tackling resources that use category theory as a tool to treat other subjects.
|
||||
|
||||
About the author
|
||||
---
|
||||
|
||||
I am a nerd who works in web programming with background in design and technical communication. I first discovered category theory through programming, but then started pursuing it in its own right. Finding math more and more fascinating.
|
||||
I am a Bulgarian nerd who works in web programming with background in design and technical communication. I am into functional programming, from where I first learned about category theory but then started finding math more and more fascinating.
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Currently living in Sofia with my wife and kid.
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Born and living in Sofia with my wife and kid.
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|
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Similar resources
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===
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@ -164,7 +164,7 @@ Category theory is very *visual* in its nature. Category-theoretic diagrams are
|
||||
|
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However, books on category theory typically have as many diagrams as other math textbooks. I consider this a huge missed opportunity for making the subject more approachable. This was the original motivation for the creation of CTI.
|
||||
|
||||
Besides being more in quantity, the diagrams of CTI use many techniques from traditional communication design (such as the use of color for example) in order to illuminate the different subjects and abstractions that I am covering.
|
||||
Besides being more of them, the diagrams of CTI use many techniques from traditional communication design (such as the use of color for example) in order to illuminate the different subjects and abstractions that I am covering.
|
||||
|
||||
Both books that I am reviewing have less diagrams than CTI. From the resources that I have examined, the only author who stresses on diagrams as much as I do is Tai-Danae Bradley (her blog is [https://www.math3ma.com/](https://www.math3ma.com/).)
|
||||
|
||||
@ -226,6 +226,22 @@ Feedback
|
||||
|
||||
My book received a lot of positive feedback and a lot of people shared it on social media.
|
||||
|
||||
Quotes
|
||||
---
|
||||
|
||||
> "The range of applications for category theory is immense, and visually conveying meaning through illustration is an indispensable skill for organizational and technical work. Unfortunately, the foundations of category theory, despite much of their utility and simplicity being on par with Venn Diagrams, are locked behind resources that assume far too much academic background.
|
||||
>
|
||||
> As Gilbert Strang offered in his critique of the typical pre-calc courses in ["Too Much Calculus"](http://siags.siam.org/siagla/articles/Strang2001.pdf) in favor of linear algebra, similar arguments can be made for promoting statistics, discrete mathematics, or category theory.
|
||||
>
|
||||
>Should category theory be considered for this academic purpose or any work wherein clear thinking and explanations are valued, beginner-appropriate resources are essential. There is no book on category theory that makes its abstractions so tangible as "Category Theory Illustrated" does. I recommend it for programmers, managers, organizers, designers, or anyone else who values the structure and clarity of information, processes, and relationships."
|
||||
|
||||
[Evan Burchard](https://www.oreilly.com/pub/au/7124), Author of "The Web Game Developer's Cookbook" and "Refactoring JavaScript"
|
||||
|
||||
> "The clarity, consistency and elegance of diagrams in 'Category Theory Illustrated' has helped us demystify and explain in simple terms a topic often feared."
|
||||
|
||||
[Gonzalo Casas](https://gnz.io/), Software developer, leading courses at digital/robotic fabrication for PHD students at ETH Zurich
|
||||
|
||||
|
||||
Comments
|
||||
---
|
||||
|
||||
@ -243,6 +259,12 @@ Comments
|
||||
|
||||
> Thank you for creating Category Theory Illustrated, for me the book makes the concepts easier to understand and build an intuition. (received by email)
|
||||
|
||||
> It's pretty cool that you can show the bridge between two different logic systems and it actually is grokkable. In other words I had no chance of understanding this concept from the Wiki article alone which is where I found it. (received by email)
|
||||
|
||||
> What an amazing write-up!
|
||||
> I've picked up those things peace by peace form Wikipedia to be able to understand the slang in Haskell land. But it was a long and puzzling process. This great summary offered here will hopefully help other people in the future get a coherent picture more quickly. (I hope the SEO is good so people will find it. I'm at least going to recommend it form now on whenever someone asks related questions).[Source](https://news.ycombinator.com/reply?id=28661220&goto=item%3Fid%3D28660157%2328661220)
|
||||
|
||||
|
||||
Discussions
|
||||
---
|
||||
|
||||
@ -250,5 +272,8 @@ Discussions
|
||||
|
||||
[HackerNews](https://news.ycombinator.com/item?id=26658111)
|
||||
|
||||
[HackerNews again](https://news.ycombinator.com/item?id=28660157)
|
||||
|
||||
|
||||
[Twitter](https://twitter.com/search?q=category-theory-illustrated&src=typed_query&f=live)
|
||||
|
||||
|
||||
Loading…
Reference in New Issue
Block a user