Finish the basics of logic
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ep2.tex
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ep2.tex
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@ -21,105 +21,103 @@
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\date{\today}
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\title{Mathematics of Programming Languages}
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\setep{02}{What is Mathematics?}
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\setep{02}{Basics of Logic}
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\begin{document}
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\maketitle
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\begin{frame}{Why did I choose to make this?}{}
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\begin{frame}{Why logic?}{}
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\begin{itemize}
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\item Originally, I just wanted to start from logic
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\item I read a tweet which made me think not everyone knows what math really is
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\item I started to ask around online and even on the street
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\item Logic serves as the backbone of computer science
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\item it provides the fundamental principles and tools necessary for understanding,
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designing, and building complex systems
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\item It forms the basis of logical reasoning, proofs, problem solving, and program correctness.
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\end{itemize}
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\end{frame}
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\begin{frame}{What is logic?}{}
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\begin{itemize}
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\item One of the key aspects of Mathematics is unambiguity
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\item The study of the principles of reasoning
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\item Logic is tool to remove ambiguity from our arguments and thoughts
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\item Most of the definitions of formal logic have been developed so that they agree with the
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natural or intuitive logic used by people.
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\item The difference between formal logic and intuitive logic exists to avoid ambiguity and obtain consistency.
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\item To put it simply logic is all about what counts as a good argument. A good argument is a valid argument. One that, preserves truth from premises to conclusion.
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\end{itemize}
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\end{frame}
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\begin{frame}{Arguments}{Propositional logic}
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\begin{itemize}
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\item An argument is a sequence of statements aimed at demonstrating the truth of an assertion.
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\item The assertion at the end of the sequence is called the \textbf{conclusion}.
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\item The preceding statements are called \textbf{premises}.
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\item As an example:\\
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\begin{flalign*}
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&\text{If} \overbrace{\text{Earth is a planet}}^{\textit{premise}} \text{then} \overbrace{\text{Earth is round}}^{\textit{conclusion}}\\
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&\text{Earth is a planet}\\
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\therefore \hspace{0.2em} &\text{Earth is round}.
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\end{flalign*}
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\end{itemize}
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\end{frame}
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\begin{frame}{Arguments}{Symbolic logic}
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\begin{itemize}
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\item In logic, the form of an argument is distinguished from its content.
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\item Logic, won't help you determine the intrinsic merit of an argument’s content.
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\item It will help you analyze an argument’s form to determine whether the truth of the conclusion follows \textit{necessarily} from the truth of the premises.
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\begin{flalign*}
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&\text{If} \overbrace{\text{Earth is a planet}}^{\textit{p}} \text{then} \overbrace{\text{Earth is round}}^{\textit{q}}\\
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&\overbrace{\text{Earth is a planet}}^{\textit{p}}\\
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\therefore \hspace{0.2em} &\overbrace{\text{Earth is round}}^{\textit{q}}.
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\end{flalign*}
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\item Convension: We will use letters \textit{p}, \textit{q}, and \textit{r} to represent
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component sentences.
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\end{itemize}
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\end{frame}
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\begin{frame}{Proposition}{}
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\begin{itemize}
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\item In any mathematical theory, new terms are defined by using those that have been
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previously defined. (E.g. A mathematical system)
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\item This process has to start somewhere
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\item In logic, the words \textbf{\textit{sentence}}, \textbf{\textit{true}}, \textbf{\textit{false}} are the initial
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undefined terms
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\item A \textbf{proposition} (or \textbf{statement}) is a sentence that is true or false but not both.
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\item For example:
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\begin{itemize}
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\item To my surprise, the majority of people got it wrong too
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\item Even other engineers and scientists
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\item ``Earth is a planet'' (true)
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\item ``Earth is flat'' (false)
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\item $x + 5 > 0$ (Depends on the value of $x$)
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\item ``How are you?'' (not a proposition)
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{What was the tweet all about?}{}
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\begin{figure}
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\centering
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\pic{ep2/tweet.png}{0.4}
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\end{figure}
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\begin{frame}{Compound propositions}{}
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\begin{itemize}
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\item We can use \textbf{logical connectives} to connect propositions together to form compound propositions.
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\item For example:
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\begin{itemize}
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\item $\overbrace{\text{\say{Earth is a planet}}}^{\textit{p}}$ \textbf{or} $\overbrace{\text{\say{Sun is a planet}}}^{\textit{q}}$
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\item $\overbrace{\text{\say{Alice is at work}}}^{\textit{p}}$ \textbf{and} $\overbrace{\text{\say{Bob is at work}}}^{\textit{q}}$
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\item $\overbrace{\text{\say{Earth is \textbf{not} flat}}}^{\textit{p}}$
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{Why do we have to start from here?}{}
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\begin{frame}{More symbols}{}
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Let's rewrite the propositions from before using symbolic variables:
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\begin{itemize}
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\item Understanding Mathematics helps us to think better
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\item To asking the right question
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\item There's no consensus on the definition of mathematics, but I'll give you mine
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\item My goal is to get you to think
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\item $\overbrace{\text{\say{Earth is a planet}}}^{\textit{p}}$ \textbf{or} $\overbrace{\text{\say{Sun is a planet}}}^{\textit{q}}$: ($p \lor q$)
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\item $\overbrace{\text{\say{Alice is at work}}}^{\textit{p}}$ \textbf{and} $\overbrace{\text{\say{Bob is at work}}}^{\textit{q}}$: ($p \land q$)
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\item $\overbrace{\text{\say{Earth is \textbf{not} flat}}}^{\textit{p}}$: ($\neg p$)
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\end{itemize}
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\end{frame}
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\begin{frame}{So, What is mathematics?}{}
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\begin{itemize}
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\item ``Mathematics is the language in which God has written the universe''
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\item To put it simply, Mathematics is a language with a specific set of properties
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\item Usually, a language enables us to express something
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\item For example, musical notation is the language to express music
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\end{itemize}
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\end{frame}
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\begin{frame}{Language of Music}{}
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\centering
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Anyone who speaks the language of musical notations understands this sheet
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\begin{figure}
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\centering
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\pic{ep2/music_score.png}{0.65}
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Maths is the same, I can express my thoughts about something in terms of mathematics
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\end{figure}
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\end{frame}
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\begin{frame}{Maths the Framework}{}
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\begin{itemize}
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\item One can produce nonsense in Maths as well
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\item But maths provides a framework that eliminates nonsense, errors and ambiguity
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\item Mathematics is a precise language
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\end{itemize}
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\end{frame}
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\begin{frame}{Not just a language}{}
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\begin{itemize}
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\item It's not just a language, it's a language + reasoning, it's a tool for reasoning
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\item It's a big collection of some people's careful thoughts
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\item In form of provable and precise statements
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\end{itemize}
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\end{frame}
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\begin{frame}{So, What is mathematics?}{}
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\begin{itemize}
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\item By Mathematics, it is possible to connect one statement to others.
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\item Mathematics is a way of going from one set of statements to another via reason.
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\item It's an interconnected web, how an idea in one field drives you to others
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\end{itemize}
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\end{frame}
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\begin{frame}{What are those statements all about?}{}
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\begin{itemize}
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\item Humans are pattern recognizing machines. (We will talk about it more in the episode)
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\item Mathematics is all about generalizing those patterns via abstractions
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\item And study the relations between those abstractions
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\item Usually, there are more than one way to describe the same thing
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\item different between doing math and using math
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\begin{itemize}
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\item Mathematicians want to make their reasoning as general as possible
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\item But math users usually want special cases
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\end{itemize}
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\end{itemize}
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\end{frame}
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\begin{frame}{}
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\end{frame}
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\begin{frame}{Contact}{}
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Please, share your thoughts and ideas or researches and papers that you want me to have a look at via:
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\begin{itemize}
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@ -0,0 +1,10 @@
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\usepackage[sfdefault]{roboto} %% Option 'sfdefault' only if the base font of the document is to be sans serif
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\usepackage[T1]{fontenc}
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\mdseries
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\documentclass{article}
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\begin{document}
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\begin{math}
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\psi(x_b, t_b) = \int_{-\infty}^{+\infty} K(b, a)\psi(x_a, t_a)dx_a
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\end{math}
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%% K(b, a) = (\frac{m}{ih(t_b - t_a)})^\frac{1}{2} exp \frac{im(x_b - x_a)^2}{2\hbar(t_b - t_a)}
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\end{document}
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@ -30,6 +30,8 @@
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\RequirePackage[normalem]{ulem}
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\RequirePackage{amsmath}
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\RequirePackage{amssymb}
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\RequirePackage{mathtools}
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\RequirePackage{dirtytalk}
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\RequirePackage{capt-of}
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\newcommand*{\setep}[2]{\def\@ep{#1}\def\@eptitle{#2}}
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