From: Lucian Mogosanu Date: Sat, 12 Apr 2014 09:12:57 +0000 (+0300) Subject: posts: 01d, 01e X-Git-Tag: v0.4~25 X-Git-Url: https://git.mogosanu.ro/?a=commitdiff_plain;h=47e9c04f4652c2b6bbcf2f5f6f31f248bb292ec1;p=thetarpit.git posts: 01d, 01e --- diff --git a/images/2014/03/diagram.png b/images/2014/03/diagram.png new file mode 100644 index 0000000..33ab45b Binary files /dev/null and b/images/2014/03/diagram.png differ diff --git a/images/2014/03/diagram.tex b/images/2014/03/diagram.tex new file mode 100644 index 0000000..46bd6ee --- /dev/null +++ b/images/2014/03/diagram.tex @@ -0,0 +1,17 @@ +\documentclass{article} +\usepackage{tikz} +\usetikzlibrary{positioning} +\begin{document} +\begin{tikzpicture} + % Tell it where the nodes are + \node (A) {$a_p$ $b_p$}; + \node (B) [below=of A] {$c_p$}; + \node (C) [right=of A] {$a_n$ $b_n$}; + \node (D) [right=of B] { $c_n$}; + % Tell it what arrows to draw + \draw[-stealth] (A)-- node[left] {\small $\mbox{add}$} (B); + \draw[-stealth] (B)-- node [below] {\small $l$} (D); + \draw[-stealth] (A)-- node [above] {\small $l$} (C); + \draw[-stealth] (C)-- node [right] {\small $+$} (D); +\end{tikzpicture} +\end{document} diff --git a/posts/y00/01d-on-numbers-structure-and-induction.markdown b/posts/y00/01d-on-numbers-structure-and-induction.markdown new file mode 100644 index 0000000..9212053 --- /dev/null +++ b/posts/y00/01d-on-numbers-structure-and-induction.markdown @@ -0,0 +1,123 @@ +--- +postid: 01d +title: On numbers, structure and induction +excerpt: Peano arithmetic for the dumb +date: March 29, 2014 +author: Lucian Mogoșanu +tags: math +--- + +In "[The miracles that matter][1]", Mircea Popescu gives a beautiful +description of numbers, starting from set-theoretical constructs and building +the sets of numbers we are so used to: naturals and integers, fractional +numbers, and finally, real and complex numbers, from which rise the +mathematical wonders that make the world go round. This dispels the myth that +such simple things are also simplistic; on the contrary, they are quite +profound, one being able to argue that they lie at the very core of human +thought. + +But what if we defined numbers, and for the sake we will limit ourselves to +natural numbers in this article; what if we defined numbers in a slightly +different way? We could name it "constructivistic", as in building the concept +and/or structure of numbers in an axiomatic way, starting from almost nothing +at all[^1]. + +Let's start from a set called $\mathbb{P}$[^2], consisting of objects which we +will define in the following way: + +* $Z$ is the Peano "zero": the smallest natural number there can be. We could + theoretically choose any other number to be our "smallest number", but then + that wouldn't make much of a difference, would it? Therefore $Z \in + \mathbb{P}$. +* For any given element in $n \in \mathbb{P}$, there exists + $n' \in \mathbb{P}$, defined as $n' = S(n)$, where $S$ is an endomorphism + over $\mathbb{P}$. In other words, $S : \mathbb{P} \rightarrow \mathbb{P}$ is + a morphism which generates (unique) elements of $\mathbb{P}$. It also has the + effect of imposing an ordering on $\mathbb{P}$, which is of course very + important, but we'll leave this detail aside for now. + +Now let's define another morphism, starting from the binary relation +$\mathbb{P} \times \mathbb{P}$, to $\mathbb{P}$. We will name it $\text{add}$: + +$\text{add} : \mathbb{P} \times \mathbb{P} \rightarrow \mathbb{P}$, such that +$\begin{array}{ll}\text{add}(x,Z) &= x \\ +\text{add}(x,S(y)) &= S(\text{add}(x,y))\end{array}$ + +We can also define a predecessor function $P$, as follows: + +$P : \mathbb{P} \rightarrow \mathbb{P}$, +$P(S(x)) = x$. + +Notice that $P$ is a partial morphism: it's actually only defined on +$\mathbb{P} \setminus \{Z\}$, as there is no predecessor for our "zero" object. +To make $P$ a total function, we'd have to take $\mathbb{P}$, add the notion of +signedness and "double" it with the same elements having the minus sign. The +same problems would arise for products and fractions, exponentiations and +roots, and so on and so forth. We'll keep things as simple as possible (and no +simpler) by remaining in the context of our little monoid over addition. + +We can easily show that $\mathbb{P}$ and $\mathbb{N}$ are equivalent sets and +that the two addition operations are also equivalent[^3]: let's define a +morphism $l$ which "lifts" objects in $\mathbb{P}$ to numbers in $\mathbb{N}$: + +$l : \mathbb{P} \rightarrow \mathbb{N}$, +$\begin{array}{ll}l(Z) &= 0 \\ +l(S(x)) &= 1 + l(x) \end{array}$ + +To demonstrate that addition works the same for both sets, we'll start from two +arbitrary objects $a_p, b_p \in \mathbb{P}$ and we'll "lift" them to +$a_n = l(a_p)$ and $b_n = l(b_p)$ respectively, so that +$a_n, b_n \in \mathbb{N}$. We now have to show that the sum operations over the +two objects, and the two numbers respectively, are equivalent. Thus we define +$c_p = \text{add}(a_p, b_p)$ and $c_n = a_n + b_n$. We have to show that + +$c_n = l(c_p)$. + +To do that, we'll bother using a proof mechanism called a *commutative +diagram*. For simplicity, I will represent $a$'s and $b$'s as pairs and abuse +notation a bit, by which I mean that we are applying the lifting function $l$ +on each element of the pair. The final result looks like this: + +
+ +The diagram commutes, which means that $l$ can be seen as a functor mapping +$\text{add}$ to number addition[^4]. ▪ + +A few aspects are worth noting. Firstly, we have shown that natural numbers in +$\mathbb{P}$ are a higher-level interpretation of the natural numbers described +as set cardinalities. That is, in addition to describing something very similar +to counting using fingers, they also have a sort of structure established by +the two constructors which define them. In other words, they also present a +deeper algebraic and axiomatic interpretation. + +Secondly, both $S$ and $\text{add}$ denote, through the presence of recurrence, +a kind of inductive reasoning which stands at the basis of the numbers in +$\mathbb{P}$. This leads us to the concept of "catamorphism", or "fold", used +to represent these operations over more generic structures such as lists, +monoids etc. Numbers are definitely a given, i.e. Gödel's incompleteness +theorems show that we possess limited reasoning in regard to them, but they can +be used to describe other structures! + +Finally, this approach provides an equivalent, yet different framework for the +construction of mathematical proofs. While this might seem unimportant for +small proofs such as the one above, let's think of the impact for large proofs +such as those involving millions of lines of hieroglyphs. + +Thus it's not only that miracles happen right before our eyes, but also that +these patterns appear all throughout the vast landscape of mathematics, their +discovery and the understanding and interpretation of their depth being left to +us, the intelligent, and yet so very dumb ones. + +[^1]: In fact such a theory could only start from the same set-theoretical +constructs, no matter what other conventions we'd make along the way. + +[^2]: From the italian mathematician Giuseppe Peano, who postulated these +axioms. + +[^3]: I'll leave the demonstration for $P$ as an exercise for the reader. + +[^4]: The proof for the reverse mapping will also be left as an exercise for +the reader. + +[1]: http://trilema.com/2014/the-miracles-that-matter/ diff --git a/posts/y00/01e-the-mirror.markdown b/posts/y00/01e-the-mirror.markdown new file mode 100644 index 0000000..924c2aa --- /dev/null +++ b/posts/y00/01e-the-mirror.markdown @@ -0,0 +1,73 @@ +--- +postid: 01e +title: I die when I look in the mirror. +date: April 12, 2014 +author: Lucian Mogoșanu +tags: storytime +--- + +I fell asleep. I fell asleep $n$ times composed, in the algebraic sense. + +I woke up. I was a black woman in a white room, in a white bed with white +sheets. I got up and looked in the only mirror in the room. I died. +Instantaneously. + +I woke up again; still a black woman in a white room, in a white bed with white +sheets. I got up, looked in the same mirror and died again. Instantaneously. + +I woke up yet again, a third time. This time I was myself. Not my real self, +but a self with a more squarely face and a hair too well arranged to be my real +self, but it felt like myself. Not like my real self, but the self it felt like +was the self I was expecting it to be. + +This time I told myself I wouldn't die anymore. + +

***

+ +I'm in Jerry's apartment; he has a room full of gadgets: mobile phones, +consoles, wearables, from the 20th century to now. I don't know Jerry too well, +but I don't know any other guys with so much knowledge on console gaming. He +hands me a Gameboy and tells me it's great; he helps me switch the batteries, +since I can't figure out how to do it, the blasted cover has a really weird +configuration. I tell him I've never owned a gaming console, but I've played +some console games in my life: from the SNES to Terminator clones sold in +Romania, I've tried most of them over at friends in the neighbourhood. Nowadays +I get to play all the oldies on emulators. + +Then Jerry put a cartridge in and told me to turn it on. Suddenly the whole +perspective changes. + +The intro shows a real-life photo of a blonde woman and a spaceship drawn in a +3D setting. The spaceship flies through space and lands (or is cast away, I +can't tell) on a foreign planet not shown yet. Instead, the ship's crew comes +into focus in the same photorealistic detail. + +... then suddenly, the faces degenerate into pixeled 2D sprites and they keep +oscillating between the photo and the sprites for a couple of times for each +crew member. I can't remember any of the names or the faces, but I know that I +(the player) am one of them. + +The game changes to a Final Fantasy-esque map, a top-down view with topographic +details, representing a desert planet with some kind of dunes. Now, the really +peculiar thing is that the dunes aren't really dunes, but more like volcanoes +with craters in the top. So I climb one of the "dunes" and reach the crater; +the crater seems to "open", becoming something similar to a black hole. The +image zooms in. + +The game perspective turns first-person, but with very vague details. I find +myself in an almost empty room, without any doors. I try to pass through one of +the walls and thus I get to enter in another, completely different room, like +from another world. I do this again and I suddenly find myself on a basketball +field, playing basketball. The image becomes clearer and clearer as I pass +through walls. + +After a while I reach an empty (detailed, photorealistic) room with a mirror. I +look in the mirror. I don't die. + +I realise, however, that the game learns my self as I play it, it adapts to my +way of playing and in the process it creates a model of Me, transposing Me +into the game. I wonder how this is possible with nowadays' technology and I +stop on the clear thought that mapping an entire person into a game is indeed +theoretically possible. + +All images of myself are only mirrors, after all, and the mirrors are tar pits.