stuff

_chapters/00_about.md

@@ -3,11 +3,27 @@ layout: default
 title: About 
 ---
 
+{: style="text-align: right"}
+*"Try as you may,*
+
+{: style="text-align: right"}
+*you just can't get away,*
+
+{: style="text-align: right"}
+*from mathematics"*
+
+{: style="text-align: right" }
+Tom Lehrer 
+
+
+
 About this book
 ===
 
 This book is a product of my own endeavour of understanding category theory. It is just that as I am explaining something, I am understanding it better. It is aimed at programmers, as well as anyone else who is interested in this stuff. 
 
-The main reason I am interested in category theory is that it allows us to formalise some common concepts that we use in our daily (intellectual) lives. Much of our language is based on intuition and rightfully so: relying on intuition is a very easy way to get your point across so it is understood by other human beings. However, that is part of the problem: sometimes intuition makes it too easy to communicate with someone. So easy that he might, in fact, understand things that you haven't actually said. For example, when I say that two things are equal, it would seem obvious to you what I mean, although it isn't obvious at all (how are they equal, at what context etc). That is the place when we might want to provide a more rigorous definition of what am I saying (even if I did not have one, to begin with). But providing such definition in natural language, which is designed to use intuition as a means of communication, is no easy task. It is in these situations that people often resort to diagrams to explain their thoughts. Diagrams are ubiquitous in science and mathematics because they are an understandable way to communicate a formal concept clearly. Category theory formalises the concept of a diagram and their components - arrows and objects and creates a language for presenting all kinds of ideas.
+The main reason I am interested in category theory is that it allows us to formalise some common concepts that we use in our daily (intellectual) lives. Much of our language is based on intuition and rightfully so: relying on intuition is a very easy way to get your point across so it is understood by other human beings. However, that is part of the problem: sometimes intuition makes it too easy to communicate with someone. So easy that he might, in fact, understand things that you haven't actually said. For example, when I say that two things are equal, it would seem obvious to you what I mean, although it isn't obvious at all (how are they equal, at what context etc). That is the place when we might want to provide a more rigorous definition of what am I saying (even if I did not have one to begin with). But providing such definition in natural language, which is designed to use intuition as a means of communication, is no easy task. 
+
+It is in these situations that people often resort to diagrams to explain their thoughts. Diagrams are ubiquitous in science and mathematics because they are an understandable way to communicate a formal concept clearly. Category theory formalises the concept of a diagram and their components - arrows and objects and creates a language for presenting all kinds of ideas.
 
 In this book, we will visit those formalisms and along the way, we would see all other kinds of mathematical objects, viewed under the prism of categories.

_chapters/02_category.md

@@ -13,18 +13,18 @@ Products
 
 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?
 
-The simplest composite type, of the sets **B**, which contains **b**'s and the set**Y**, which contains **y**'s is the the *product* or **B** and **Y**.
+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**.
 
 ![Product parts](product_parts.svg)
 
-It is the set of *pairs* of **b**'s and **y**'s.
+The product is the set of *ordered pairs* of **b**'s and **y**'s. It is denoted **B x Y** and it comes equipped with two functions for retrieving the **b** and the **y** from each **(b, y)**.
 
 ![Product](product.svg)
 
-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)**.
-
 **Question**: Why is this called a product? Hint: How many elements does it have?
 
+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.
+
 Products as Objects
 ---
 
@@ -66,7 +66,6 @@ Discovered in 1921 Kazimierz Kuratowski, this one uses just the component of the
 
 ![A pair, represented by sets](pair_as_set_1.svg)
 
-
 Defining products in Terms of Functions 
 ---
 
@@ -227,6 +226,8 @@ Why is that is pretty obvious if we go back to set theory for a second (OK, mayb
 
 ![Two sets connected with multiple functions](set_arrows.svg)
 
+There are some types of categories in which only one morphism between two objects is allowed (or one in each direction), but we will talk about them later (they are also called orders).
+
 Composition
 ---
 
@@ -234,7 +235,6 @@ One of the few or maybe even the only requirement for a structure to be called a
 
 ![Composition of morphisms](composition.svg)
 
-
 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))**).

_chapters/04_order.md

@@ -247,10 +247,19 @@ The poset example with the number dividers is also isomorphic to an inclusion or
 
 ![Divides poset](divides_poset_inclusion.svg)
 
-Let's elaborate a bit about the isomorphism between the number poset and the prime inclusion order. It is comprised of 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) and one function from the number poset to the prime inclusion order (which is an operation called *prime factorization* of a number, consisting (unsurprisingly) of finding the set of prime numbers that result in that number when multiplied with one another (as we said, this set is unique, and hence these functions really do comprise an isomorphism)).
+Order isomorphisms
+---
+
+This might be a good time to elaborate a bit about what an order isomorphism is. 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: 
+
+- 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) 
+- 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). 
+
 
 ![Divides poset](divides_poset_isomorphism.svg)
 
+For sets, an isomorphism means just that the functions are inverse of each other. But as orders, besides being sets, have the arrows that connect them there is an additional requirement for a pair of functions to form an isomorphisme - to be an isomorphism, a function has *to respect those arrows, in other words should be *order preserving*. More specifically, applying the function (let's call it F) to any two elements in one set (**a** and **b**) should result in two elements that have the same corresponding order in the other set (so **a ≤ b** iff **F(a) ≤ F(b)**. 
+
 Birkhoff's representation theorem
 ---
 
@@ -265,6 +274,8 @@ I won't go into details about this result, I would only mention that the "prime"
 
 By the way, the partial orders that are *NOT* distributive lattices are also isomorphic to inclusion orders, it is just that they are isomorphic to inclusion orders that do not contain all possible combinations of elements.
 
+<!-- TODO in a Power set P(X), the meet of a collection of subsets is their intersection, and the join is their union -->
+
 Lattices
 ===
 
@@ -347,7 +358,7 @@ What about that other law that was required in order to be a category - the iden
 
 ![Reflexivity](reflexivity.svg)
 
-So it's official - preorders are categories (sounds kinda obvious, especially after we also saw that orders can be reduced to sets and functions using (through the inclusion order) and sets and functions form a category in their own right). 
+So it's official - preorders are categories (sounds kinda obvious, especially after we also saw that orders can be reduced to sets and functions (the inclusion order) and sets and functions form a category in their own right). 
 
 And since partial orders and total orders are preorders too (as they obey those two laws), they are categories as well.
 
@@ -362,7 +373,7 @@ That is in the contrast with the category of sets where there are potentially in
 Products and sums
 ---
 
-While we are rehashing diagrams from the previous chapters, let's look at the diagram defining the *sum* of two objects in a category. 
+While we are rehashing diagrams from the previous chapters, let's look at the diagram defining the *coproduct* of two objects in a category. 
 
 ![Joins as coproduct](coproduct_join.svg)
 

_chapters/05_logic.md

@@ -11,12 +11,12 @@ Now let's talk about one more *seemingly* unrelated topic, just so we can surpri
 What is logic
 ===
 
-Logic, has a very peculiar role, so in order to understand it well it's important to understand where it stands in the grand schema of things: 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. of that which really exists. For example, if the laws of physics show how particles behave in our universe (or multiverse), we might use logic to deduce how would they behave in any universe that is possible to exist (under a given set of postulates, real or made up). The key is that everything that is actual is also possible, and so all sciences are (or should be) be based on 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. of that which really exists. For example, if the laws of physics show how particles behave in our universe (or multiverse), we might use logic to deduce how would they behave in any universe that is possible to exist (under a given set of postulates, real or made up). The key is that everything that is actual is also possible, and so all sciences are (or should be) be based on logic. But at the same time (and that's sometimes overlooked) nothing real is purely logical.
 
-Logical "proofs"
+Logical proofs
 ---
 
-OK, let's be more specific. Logic aims to study the *rules* by which knowing one thing leads you to conclude or (*prove*) that some other thing is also true, regardless of the things's domain (or scientific discipline) i.e. by only refering to their form.
+OK, let's be more specific. Logic aims to study the *rules* by which knowing one thing leads you to conclude or (*prove*) that some other thing is also true, regardless of the things's domain (e.g. scientific discipline) i.e. by only referring to their form.
 
 On top of that, it (logic) tries to organize those rules in what are called *logical systems* (or formal systems as they are also called) - these are collections of rules for manipulating proposition that have give you the maximum expressive ability without adding anything extra.
 
@@ -25,12 +25,7 @@ What does "prove" mean in this context? Simple - when we are able, using the rul
 Logic and mathematics 
 ---
 
-All of the concepts that we studied here are formal concepts, so we can say that we have been doing logic throughout this book. And we would be quite correct - every mathematical theory is 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 set theory (in particular the Zermelo–Fraenkel flavour of it) adds just one single primitive to the standard axioms of logic which we will see shortly - the binary relation that indicates *set membership*. So set theory is very very close to logic exacly how they relate is probably outside the scope of that book. Category theory is close to logic too, but in a quite different way (this is not outside our scope, so we will examine the connection later.)
-
-Here I will quote Bertrand Russell, and the way he explains logic and maths, using geometry as an example:
-
-> But since the growth of non-Euclidean Geometry, it has appeared that pure mathematics has no concern with the question whether the axioms and propositions of Euclid hold of actual space or not: this is a question for applied mathematics, to be decided, so far as any decision is possible, by experiment and observation. What pure mathematics asserts is merely that the Euclidean propositions follow from the Euclidean axioms--i.e. it asserts an implication: any space which has such and such properties has also such and such other properties. Thus, as dealt with in pure mathematics, the Euclidean and non-Euclidean Geometries are equally true: in each nothing is affirmed except implications. All propositions as to what actually exists, like the space we live in, belong to experimental or empirical science, not to mathematics; when they belong to applied mathematics, they arise from giving to one or more variables in a proposition of pure mathematics some constant value satisfying the hypothesis, and thus enabling us, for that value of the variable, actually to assert both hypothesis and consequent instead of asserting merely the implication.
-> Bertrand Russell, from The Principles of Mathematics (1903)
+All of the concepts that we studied here are formal concepts, so we can say that we have been doing logic throughout this book. And we would be quite correct - every mathematical theory is 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 set theory (the Zermelo–Fraenkel flavor, for example) adds just one single primitive to the standard axioms of logic which we will see shortly - the binary relation that indicates *set membership*. So set theory is very close to logic. Exactly how they relate is probably outside the scope of that book. Category theory is close to logic too, but in a quite different way (this is not outside our scope, so we will examine the connection later.)
 
 The elements of logic
 ===
@@ -38,7 +33,7 @@ The elements of logic
 Primary propositions
 ---
 
-A consequence of the above (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 (or "values" as Russell calls them) that we accept as true. These are also called called "premises", "primary propositions" or "atomic propositions" as Wittgenstein dubbed them. 
+A consequence of the above (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 (or "values" as Russell calls them) that we accept as true, or false. These are also called "premises", "primary propositions" or "atomic propositions" as Wittgenstein dubbed them. 
 
 ![Balls](balls.svg)
 
@@ -49,11 +44,11 @@ Composing propositions
 
 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 monoid objects are combined into a new monoid object using the monoid operation. 
 
-Moreover, some logical operations do form monoids, like the operation **and** with the proposition **true** serving as the identity element.
+Actually, some logical operations do form monoids, like the operation **and** with the proposition **true** serving as the identity element.
 
 ![Logical operations that form monoids](logic_monoid.svg)
 
-However logic is not *just* a monoid, as logic studies not one but many logical operations and they ways in which they relate, for example, in logic we might be interested in the distributivity of the **and** and **or** operations. 
+However, unlike group theory, logic has not one but *many* logical operations and studies *the ways in which they relate*, for example, in logic we might be interested in the law of distributivity of the **and** and **or** operations and what it entails.
 
 ![The distributivity operation of "and" and "or"](logic_distributivity.svg)
 
@@ -62,7 +57,7 @@ Important to note that **∧** is the symbol for **and** and **∨** is the symb
 The equivalence of primary and composite propositions
 ---
 
-Do note that in the leftmost proposition, the green ball is wrapped in a grey ball just to make the diagram prettier - propositions that are composed of several premises (symbolized by grey balls, containing some other balls) are not in any way different from "primary" propositions (single-color balls). 
+Do note that in the leftmost proposition, the green ball is wrapped in a gray ball just 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). 
 
 As a result of this is that we can compose propositions with multiple levels of nesting (*recursively* as the computer science people say). 
 
@@ -71,70 +66,72 @@ As a result of this is that we can compose propositions with multiple levels of
 Modus ponens
 ---
 
-As an example of a proposition that contains multiple levels of nesting, consider one of the most famous propositions ever, namely *modus ponens*. 
+As an example of a proposition that contains multiple levels of nesting (and a great introduction of the subject in its own right), consider one of the most famous propositions ever, namely *modus ponens*. 
 
-Modus ponens states that if **A** is true and if also **A → B** is true (if **A** implies **B**), then **B** is true as well. 
+Modus ponens is a proposition that states that if **A** is true and if also **A → B** is true (if **A** implies **B**), then **B** is true as well. For example, if we know that "Socrates is a human" and that "Being human implies being mortal", we also know that "Socrates is mortal". 
 
 ![Modus ponens](modus_ponens.svg)
 
 Let's dive a little deeper. The proposition is composed of two other propositions in a **follows** relation where the proposition that follows (**B**) is primary (or at least could be, again, pretty diagrams before all), but the proposition from which **B** follows is not. Let's call that one **C** - so the whole proposition becomes **C → B**. 
 
-Going deeper, we have to mention that of course 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**.
+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**.
 
 Tautologies
 ---
 
-Because the content of our propositions is abstracted away, we often cannot tell whether a given proposition is true or false. However, with propositions such as *modus ponens* we can: modus ponens is *always true*, regardless of whether the propositions it is composed of are true, or we can also say that it is *true in all models of the system*, which is the same thing (a model being a set of real-world premises are taken to be signified by our propositions). 
+Because the content of our propositions is abstracted away, we often cannot tell whether a given proposition is true or false. However, with propositions such as *modus ponens* we can: modus ponens is *always true*, regardless of whether the propositions which form it are true. If we want to be fancy, we can also say that it is *true in all models of the 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.
 
 ![Variation of modus ponens](modus_ponens_variations.svg)
 
-Such always-true propositions are called *tautologies* (their more famous counterparts are the *contradictions*). The simplest tautology (which might remind you of something), is the one which states that a proposition implies itself (e.g. "All bachelors are unmarried").
+Propositions that are always true are called *tautologies*. And their more-famous false counterparts are the *contradictions* (you can turn each tautology into contradiction by adding a "not"). 
 
+The simplest tautology, is the one which states that a proposition implies itself (e.g. "All bachelors are unmarried"). It may remind you of something.
 
 ![Identity tautology](tautology_identity.svg)
 
-Here are some more complex (less boring) tautologies, where the connection between the left and the right part not so apparent (the symbol **¬** means "not"/negation.
+Here are some more complex (less boring) tautologies (the symbol **¬** means "not"/negation.
 
 ![Tautologies](tautology_list.svg)
 
-We will learn how to prove that a given proposition is a tautology, but first let's see what are tautologies good for.
+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.
 
 Logical systems
 ===
 
-
-Tautologies are the basis of *axiom schemas* and of *rules of inference* which is almost the same thing. That is, they can serve as starting point from which we can generate all true logical statements by means of substitution. And axiom schemas/rules of inference form logical systems, as we mentioned before. But let's not get ahead of ourselves.
+Tautologies are useful because they are the basis of *axiom schemas* or *rules of inference* (which is almost the same thing): they can serve as starting point from which we can generate other true logical statements by means of substitution. And axiom schemas/rules of inference form logical systems, but let's not get ahead of ourselves.
 
 Axiom schemas
 ---
 
-Realizing that the colors of the balls in modus ponens are superficial, we may want to represent the general structure that all of its versions share.
+Realizing that the colors of the balls in modus ponens are superficial, we may want to represent the general structure that all of its variations have.
 
 ![Modus ponens](modus_ponens_schema.svg)
 
-This structure, the one that looks like a coloring book in our example, is called *axiom schema*. And the propositions produced by it are *axioms*.
+This structure (the one that looks like a coloring book in our example) is called *axiom schema*. And the propositions that are produced by it are *axioms*. 
+
+Note that the propositions that we plug into the schema don't have to be primary. For example, having the proposition **a** (that is symbolized below by the orange ball) and the proposition stating that **a** implies **a or b** (which is one of the tautologies that we saw above), we can plug those propositions into the *modus ponens* and realize that **a or b** is true. 
+
+![Using modus ponens for rule of inference](modus_ponens_composite.svg)
+
 
 Rules of inference
 ---
 
-All axiom schemas can be easily used as procedures for creating new propositions out of existing ones (*rule of inference*). For example, having the proposition **a** (that is symbolized below by the orange ball) and the proposition stating that **a** implies **a or b** (which is one of the tautologies that we saw above), we can plug those propositions into the *modus ponens* axiom schema and use it as a rule of inference to proves that **a or b** is true.
-
-![Using modus ponens for rule of inference](modus_ponens_composite.svg)
-
-Every *axiom schema* can also be potentially a rule of inference, although in practice, having one rule of inferences is enough to construct a logical system.
+Most axiom schemas can be easily applied as rules of inference i.e. as procedures for declaring propositions that follow from true propositions as also true 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
 ---
 
-OK, we started talking about logical systems again, without first explaining what they are. Apologies. Anyways, knowing that we can use axiom schemas/rules of inference to generate new propositions, we might ask whether it is possible to have a small collection of such schemas/rules that is curated in such a way that it enables us to generating *all* other tautologies, as well as all propositions that are true given a set of premises. You would be happy (although a little annoyed, I imagine) to learn that there exist not only one, many such collections. 
+OK, we started talking about logical systems again, so let's explain what they are: Knowing that we can use axiom schemas/rules of inference to generate new propositions, we might ask whether it is possible to have a small collection of such schemas/rules that is curated in such a way that it enables us to generating *all* other tautologies. 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*. 
 
-I think that the first one is due to Gottlob Frege and it is composed by the *modus ponens* rule of inference, that we already saw and five axiom schemes that all follow the structure **A → B** (these are axiom schemes, not axioms, eventhough we use colors).
+For example, a complete logical system is the collection of the following five axiom schemes **in addition to the inference rule modus ponens** (These are axiom schemes, even though we use colors).
 
 ![A minimal collection of Hilbert axioms](min_hilbert.svg)
 
-And yes, collections such as the one above are what we call logical systems. Technically they would be *complete* logical systems and a collection that is not capable of generating all other propositions would be *incomplete logical systems*, but who has time for incomplete logical systems?)
+Technically they should be called *complete* logical systems and a collections that are not capable of generating all other propositions would be *incomplete logical systems*, but who has time for incomplete logical systems? 
+
+Proving that this and other similar logical systems are complete is due to Godel and is known as "Godel's completeness theorem".
 
-The proving that this and other similar logical systems are complete is due to Godel and is known as "Godel's completeness theorem".
 
 Classical logic
 ===
@@ -143,45 +140,42 @@ We now have an idea about how do some of the main logical constructs (axioms, ru
 
 > 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.
 
-The above is my 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. And this is what we will be looking at in this chapter.
+The above is my 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.
 
-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 answer to every question, whether it would be affirmitive or negative i.e. that ultimately *each proposition is either true or false*. For this reason, propositions in classical logic can be aptly expressed as functions which output boolean values.
+Truth-functional interpretation
+===
+
+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 i.e. that ultimately *each proposition is either true or false* (this is known as *the principle of bivalence*). Due to it, propositions in classical logic can be aptly expressed as functions which output boolean values.
 
 ![The set of boolean values](boolean_set.svg)
 
-- We can view primary propositions as simple functions that return a boolean value and don't take any arguments.
+We can view *primary propositions* as simple functions that return a boolean value and don't take any arguments.
 
-- We can view logical operators as functions that take a bunch of boolean values and return another boolean value.
+We can view *logical operators* as functions that take a bunch of boolean values and return another boolean value.
 
-- Composite propositions are, in this case, just the results of the invocation of these functions.
-
-We will see why this simple and *seemingly* self-evident observation (that propositions can be either true or false) is so important and how it vastly simplifies both the definition of both propositions and logical operators and the solving of some problems - all you need to do is to ennumerate all (two) possibilities.
+*Composite propositions* are, in this case, just the results of the invocation of these functions.
 
 The *negation* operation
 ---
 
-Let's begin with the the negation operation, because it is the simplest one, as it is an unary operation, which means it takes just one argument. It can be represented by this function.
+Let's begin with the negation operation. Negation is a unary operation, which means that it is a function that takes just *one* argument.
 
 ![negation](negation.svg)
 
-It can also be expressed in a sligtly less-fancy way by this table.
+The same function can also be expressed in a slightly less-fancy way by this table.
 
 | p | ¬p |
 |---| --- | 
 | True  | True  | 
 | False  | False |
 
-Such tables are called *truth tables* and they are ubiquitos in classical logic, not only for defining operators, but for proving results as well.
+Such tables are called *truth tables* and they are ubiquitous in classical logic, not only for defining operators, but for proving results as well.
 
-Proving results by truth tables
+Interlude: Proving results by truth tables
 ---
 
-All axioms and rules of inference aside, each proposition in classical logic can be proved by means of truth tables alone. Actually, proving a proposition by means of axioms and rules means nothing other than the fact that it can be also proven by truth tables. This is so, because axioms are themselves are proven by means of 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 equivalen 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, 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".
 
 ![Double negation elimination formula](double_negation_formula.svg)
 
@@ -193,7 +187,7 @@ If we want to be formal about it, we might say that applying negation two times
 
 ![The identity function for boolean values](boolean_identity.svg)
 
-If we are tired of diagrams, we can represent the composition diagram above as table. This kinds of tables are called *truth-tables* and they are the usual more of representation used.
+If we are tired of diagrams, we can represent the composition diagram above as table. This kinds of tables are called *truth-tables* and they are the usual more of representation used. Each proposition in classical logic can be proved by means of truth tables alone.
 
 | p | ¬p | ¬¬p |
 |---| --- | --- |
@@ -202,42 +196,6 @@ If we are tired of diagrams, we can represent the composition diagram above as t
 
 Despite its triviality, the double negation axiom is probably the most controversial result in logic, we will see why later.
 
-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.
-
-![Entailment](entailment.svg)
-
-Here is the truth table.
-
-| p |	q | p → q |
-|---| --- | ---   |
-| True  | True  | True  |
-| True  | False | False |
-| False | True  | True  |
-| False | False | True  |
-
-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 it 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 two factors:
-    - By saying that *p* implies *q* we don't say that the two are 100% interdependent e.g. the claim that "drinking causes headache" does not mean that drinking is the only source of headaches, and it won't be refuted the fact that you can get headache without drinking.
-    - The conclusion that *p* implies *q* is reached only if all four cases are satisfied, so this events means nothing by itself.
-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.) Because the arguments that truth functions take have just two possible values (**true** and **false**), it is not hard to prove that by enumerating all possible combinations of the arguments.
-
-| p |	q | p → q | ¬p  | q   |  ¬p ∨ q |
-|---| --- | ---   | --- | --- | --- |
-| True  | True  | **True**  | False | True | **True** |
-| True  | False | **False** | False | False | **False** |
-| False | True  | **True**  | True  | True | **True** |
-| False | False | **True**  | True  | False | **True** |
-
-You can see that truth tables don't scale well for longer problems. In a sec we will see how we can prove this result of this using some predefined set of axioms. But let's go through the **and** and **or** relations real quick.
-
 The **and** and **or** operations
 ---
 
@@ -264,54 +222,178 @@ And we can use as axioms a pair of pretty obvious propositions that we can prove
 - 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".
 
-Logical systems redux
+The *implies* operation
 ---
 
-After we defined the main logical operators and saw and proved some propositions (which we called tautologies) are universally valid, using truth tables, we are now in position to finish our talk about logical systems which prompted this chapter. If you recall, a logical system is a collection of tautologies and rules that allow us to deduce all other tautologies.
+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.
 
-There are two main classes of such logical systems, the first one are the so-called *Hilbert-style* proof systems (also called *Frege-style*) and the other one are the systems of *natural deduction*, the main difference between the two being whether they rely on *axioms* or on *rules of inference* as a methods for proving propositions.
+Here is the truth table.
 
-- *Natural deduction* systems consist only of *rules of inference* (they don't have axiom schemas).
-- *Hilbert-style* systems, on the other hand rely mainly on *axiom schemas* as means of proving propositions. They do, however, contain one rule of inference, just because otherwise there would be no way to combine the generated propositions - usually that rule is the *modus ponens* rule, that we already saw.
+| p |	q | p → q |
+|---| --- | ---   |
+| True  | True  | True  |
+| True  | False | False |
+| False | True  | True  |
+| False | False | True  |
 
-With some variations Hilbert-style system consist of the following axioms.
+Now there are some aspects of this which are non obvious so let's go through every case.
 
-![A minimal collection of Hilbert axioms](min_hilbert.svg)
+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 it 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 two factors:
+    - By saying that *p* implies *q* we don't say that the two are 100% interdependent e.g. the claim that "drinking causes headache" does not mean that drinking is the only source of headaches, and it won't be refuted the fact that you can get headache without drinking.
+    - The conclusion that *p* implies *q* is reached only if all four cases are satisfied, so this events means nothing by itself.
+4. And finally if **p** is false but **q** is false too, then **p** still does imply **q** (just some other day).
 
-And, as an example, we can use those axioms to prove the And let's see an example of a proof using these axioms
+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.) Because the arguments that truth functions take have just two possible values (**true** and **false**).
+
+| p |	q | p → q | ¬p  | q   |  ¬p ∨ q |
+|---| --- | ---   | --- | --- | --- |
+| True  | True  | **True**  | False | True | **True** |
+| True  | False | **False** | False | False | **False** |
+| False | True  | **True**  | True  | True | **True** |
+| False | False | **True**  | True  | False | **True** |
+
+You can see that truth tables don't scale well for longer problems. 
+
+p → q
+
+if p is true
+
+   p ∨ -q
+
+if p is false
+
+  -p ∨ q
 
-But wait, how can we be *sure* that a given system is really capable of deducing all tautologies that exist in the world? We can be sure, because it was proven by a guy called Godel, who is like the greated logician ever, a result known as Godel's completenes theorem. However we won't get into that result right now.
 
 Intuinistic logic
 ===
 
-<!--
-> Now this is the story all about how
-> Classical logic got flipped, turned upside down
-> And I'd like to take a minute, just sit right there
-> I'll tell you bout the ideas of a guy called Brouwer
--->
+In the 20th century some people tried to really put the postulates of classical logic and set theory to the test by trying to base all mathematics on them. Doing so exposed some issues with classical logic, such as Godel's incompleteness theorem, which lead some of them to the conclusion that logic deals just with subjective ideas that exist only in our minds. This lead to the development of a new kind of logic, known as *intuinistic logic*.
 
-The foundations of classical logic (which was the topic of the last chapter) remained undisputed from its inception untill the 20th century when people tried to really put them to the test by using it as a basis of all mathematics. It was then when came some disturbing results came out. Resultss like Godel's *other* famous theorem - his *in*completeness theorem was published, stating that systems that are able to represent arithmetics, contain some statements that are undecidable, so they cannot be proven to be true, nor false (note that this does not contradict Godel's completeness theorem, as it speaks about arithmetics in particular, while the completeness theorem is about general logic).
-
-In this times, a new philosophical mathematical and logical school of thought, known as *intuitionism* (also called *constructivism*) was born. If *classical logic* is based on Plato's theory of forms, then intuinism beginned with a philosophical idea originating from Kant and Schopenhauer, that is more or less the opposite - it is the notion that forms are not a thing in itself, but are just subjective ideas that exist only in our minds and the idea that the world as we experience it is largely predetermined of out perceptions of it. Or in the words of the mathematician who founded this school, L.E.J. Brouwer:
+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. Or in the words of the mathematician who founded this school, L.E.J. Brouwer:
 
 > [...] 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.
 
+But if logic deals with *constructing* rather than *discovering* then we have no basis to claim that each statements is necessarily *either true or false*. For example, there might be a statements that might not be provable (the twin-prime conjecture is often given as an example), simply because they fall outside of the domain of the system that we are working with, but that does not make them false. This wouldn't be such a big deal if the fact that each statement is either true or false weren't the *basis of truth-tables*, which is how all logical operators are defined. 
 
-The BHK Interpretation.
----
+The BHK Interpretation 
+===
 
-Aside from bening much humbler regarding the validity of its results, the intuinistic school differed from the classical logic over one very important point - they thought that since mathematics is just an activity and truth is only in our heads, then it is necessarily subjective and incomplete, therefore we have no basis to claim that each statements is necessarily *either true or false*, (which if you remember is the basis of classical logic and truth-tables).
+So, due to the reasons outlined above, intuinistic logic is not bivalent, we cannot have all propositions reduced to a value of the boolean set (to true and false). 
 
-The *negation* operation
----
+![The True/False dichotomy](true_false.svg)
 
-The *implies* operation
----
+But one thing that we still do have 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 dichotomy between the existence or non-existence of a proof for a given proposition may be taken as similar to the proposition being true or false - there either is a proof of a given proposition or there isn't.
+
+![The proved/unproved dichotomy](proved_unproved.svg)
+
+This is known as the as the Brouwer–Heyting–Kolmogorov (BHK) interpretations of intuinistic logic 
 
 The **and** and **or** operations
 ---
 
-Formalizing logic with *orders*
+If the existence of a proof of a proposition is taken to mean that the proposition is true, then the definitions of **and** is rather simple - the proof of **A ∧ B** is just *a pair* containing a proof of **A**, and a proof of **B** e.g. *a set-theoretic product* of the two (see chapter 2). The principle is similar - if you can construct the pair of **A** and **B** i.e. if both proofs do exists, then **A ∧ B** is provable too.
+
+![And in the BHK interpretation](bhk_and.svg)
+
+**Question:** what would be the **or** operation in this case?
+
+The *implies* operation
+---
+
+In this case, saying that **A** implies **B** (**A → B**) would just mean that there exist a function which can convert a proof of **A** to a proof of **B**. 
+
+![Implies in the BHK interpretation](bhk_implies.svg)
+
+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**.
+
+**Task:** In order for this to work, you need to define the function in terms of sets. Like with *and* you can do this by using the concept of a pair, work out the details.
+
+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*.
+
+To express this, intuinistic logic defines the constant "false" (**⊥**) defined as the proof of a formula that does not have any proofs (also known as "absurdity" or "bottom value"). And this allows for us to label false propositions as ones that imply the bottom value. So in intuinistic logic **¬A** as just a shorthand for **A → ⊥**. In set theory, this constant can be expressed by the empty set.
+
+![False in the BHK interpretation](bhk_false.svg)
+
+And the observation that false propositions are connected to the bottom value 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.
+
+![False in the BHK interpretation](bhk_false_function.svg)
+
+The only way for there to be such function is if the set of proof of the proposition is empty as well.
+
+![False in the BHK interpretation](bhk_false_function_2.svg)
+
+**Task:** Look up the definition of function and verify that there cannot exist a function from any set *to the empty set* 
+
+**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
+---
+
+Although intuinistic logic seems to differ a lot from classical logic, it actually isn't - if we try to outline the schemas/rules of inference that correspond to the definition of the structures outlined above, we would see that they are virtually the same as the ones that define classical logic. With one exception - the of *double negation elimination* that we saw earlier, which is also known (in a slightly different form as the *law of excluded middle*. 
+
+![The formula of the principle of the excluded middle](excluded_middle_formula.svg)
+
+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 intuinistic logic - 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*.
+
+Logics as categories
 ===
+
+Leaving the differences between intuinistic and classical logics aside, the BHK interpretation is interesting because it provides a bit of the "higher-order" view of logic, that we need in order to  define it in terms of category theory.
+
+The Curry-Howard correspondence
+---
+
+Programmers might find the definition of the BHK interpretation very similar to a definition of a programming language, and it indeed is - this similarity is known as the *Curry-Howard correspondence* - propositions are *types*, the **implies** operations are *functions*, **and** operations are composite types (objects) and **or** operations are *sum types* (which are not supported in some programming language). And a proof of a given proposition is represented by a value of the corresponding type. 
+
+![Logic as a programming language](logic_curry.svg)
+
+**Task:** The Curry-Howard correspondence 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).
+
+Logic as a category
+---
+
+When seeing set-theoretic description of the BHK interpretation and the Curry-Howard correspondence, we might start suspecting that *logical propositions form a category* and they indeed do. 
+
+![Logic as a category](logic_category.svg)
+
+We already mostly know how to define this category, so I won't get into detail.
+
+**Task:** See whether you can prove that logic propositions and entailments forms a category. What is missing?
+
+Cartesian closed category
+
+
+Once you see that logic is a category, you might think you've seen it all. You might think that can already skip the rest of the chapter, or throw the whole book away, while putting on your sunglasses and jumping in your convertible car. But you would be missing out, as there is an even simple structure that captures all of the concepts that we saw, while providing some interesting insights on what logic is.
+
+If we assume that there is only one way to go from proposition **A**, to proposition **B** (or there are many ways, but they are equivalent), then logic is not only a category, but a *preorder* (a category that has just one morphism between any two objects).
+
+![Logic as a preorder](logic_preorder.svg)
+
+Furthermore, if we count propositions that follow from each other (or sets of propositions that have the same truth value and can be proven by the same proof) as equivalent, then logic is a *partial order*.
+
+![Logic as an order](logic_order.svg)
+
+And so it can be represented by a Hasse diagram, yey. 
+
+![Logic as an order](logic_hasse.svg)
+
+This representation of logic is know as *algebraic* representation - it means that 
+
+Which orders represent logic and what laws does an order have to obey so it is isomorphic to a logic? We will see if we examine 
+
+### The **and** and **or** operations
+
+If you remember 
+
+### The *implies* operation
+
+
+### The *negation* operation

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_chapters/06_functors.md

@@ -0,0 +1,6 @@
+
+Hom set
+===
+
+Terminal objects
+===

dictionary.txt

@@ -1,6 +1,7 @@
 coproduct
 morphism
 coproducts
+morphism
 morphisms
 preorders
 Preorders
@@ -47,3 +48,20 @@ non-abelian
 Dih3
 composable
 forall
+BHK
+yey
+intuinistic
+Intuinistic
+intuinism
+Brouwer
+schemas
+Nahas's
+Coq
+Kolmogorov
+entailments
+Logics
+logics
+bivalence
+Zermelo
+Fraenkel
+distributivity