--- /dev/null
+---
+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:
+
+<div class="imgcenter"><img class="thumb" src="/images/2014/03/diagram.png"
+/></div>
+
+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/
--- /dev/null
+---
+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.
+
+<p style="text-align:center; font-weight:bold;">***</p>
+
+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.
projects / thetarpit.git / commitdiff
