Notices by Jorge Stolfi (jorgestolfi@mas.to)

Summary of the Starship program so far:

Full stack launches: 6
Second stage landings: 0
Fully reusable recoveries: 0
Insertions into orbit: 0
Tons of payload in orbit: 0
Crewed missions: 0
Fuel transfers in orbit: 0
Insertions in Lunar orbit: 0
Lunar landings: 0
Lunar liftoffs: 0
Billions of gov $ spent: 4
Promised Moon landing: 2021

@futurebird @dymaxion @whknott

🧵> In three dimensions it is more complicated: besides the obvious Euclid-like axioms, it seems one need also an axiom that says "for any eight lines A1,A2,A2,A4 and B1,B2,B3,B4, if 15 of the pairs Ai,Bj intersect, then the 16th will intersect too".

And it only got worse in higher dimensions.

So in the end I gave up on the axiomatic approach. I defined a model with real coordinates, and said "n-dim projective geometry is is any geometry isomorphic to this"

@futurebird @dymaxion @whknott

My PhD thesis was about a variant of projective geometry in n dimensions. As a wannabe mathematician, I spent months trying to define that geometry by a set of axioms. In two dimensions, it is basically Euclidean geometry without circles -- no compass, just ruler, so its axioms are straightforward.🧵>

@futurebird @dymaxion @whknott

It is not just that the axioms are arbitrary, but that they cannot be specified precisely without contradiction. If you can't define what a "set" is, how can you expect everybody to agree on whether something is a set of not?

Euclid himself assumed a couple of things that should have been explicitly stated as axioms, such as "If a line, not passing through any vertex of a triangle, meets one side of the triangle then it meets another side."

@futurebird @dymaxion @whknott

🧵> Some math theorems depend on the axiom of choice. If I got it right, it says that, given any (finite or infinite) set of non-empty and disjoint sets, there is a set that consists of one element from each of those sets. Seems pretty obvious and lame, but it leads to several counterintuitive results...

@futurebird @dymaxion @whknott

"Let X be the set of all elements that don't belong to X" That definition of the set X is *obviously* invalid; but how do you define rigorously what is a valid definition?

The complement C of a set X is supposed to be all the elements that do not belong to X. But since sets can be elements of other sets, the set C is itself an element of C. Is that okay? if not, how do we fix that?🧵>

@futurebird @dymaxion @whknott

🧵> And, finally, since Gödel we know that, no matter which axioms we choose, there will be infinitely many math statement that are true but cannot be proved starting from those axioms. Indeed, *practically all* true statements have no proofs -- because proofs are countable, but true statements are uncountable.

@futurebird @dymaxion @whknott

🧵> Furthermore, over the last couple of centuries we have realized that those "fundamental axioms" are not as "true" as Euclid assumed. General relativity and the Heisenberg principle make Euclid statements wholly unreal -- like axioms about gods, ghosts or dragons. The foundations of set theory are a mess. 🧵>

@futurebird @dymaxion @whknott

🧵> I don't see THAT -- going all the way to "fundamental axioms" -- as a good thing. On one hand, the choice of axioms is arbitrary: one could take any sufficiently large set of theorems from Euclid, declare them to be the axioms, and then derive his axioms from these.🧵>

@futurebird @dymaxion @whknott

🧵> But, as you say, math teaching at school indeed tries more: it tries to derive every math statement all the way from a few "fundamental axioms". That is a leftover from the days when the only serious math taught at school was geometry, and geometry meant Euclid's book.

🧵>

@futurebird @dymaxion @whknott

Indeed the emphasis of math teaching, in high school and early college, is in the sort of logic that enables one to gain trust in a mathematical statement C from facts A and B that one trusts are true.

And that is a good thing, because most problems the student will require algorithms or formulas that he did not learn at school.

🧵>

@sofiav

Thanks!

But does that term imply that the *whole* government, *including the law makers*, follows the laws? I think I saw it used also for very specific situations, like a single case. Somewhat in the sense of "throwing the book" at the defendant.

saw someone describe LLMs as "an encyclopedia on steroids" and i think that's basically incorrect in every way, isn't it? i would have gone for something more like "a thesaurus on acid"

What is the English translation of the Portuguese term "estado de direito"?

That would be a regime where (1) most people -- especially government officials -- respect the laws, and most of those who don't get punished by the state, and (2) the laws are made and changed only according to procedures specified by the law.

It is not the same as "democracy", although in practice it is a requisite thereof. Its opposite includes regimes where those who make the laws do not have to follow them.

Big AI companies are considering building their own nuclear power plants.

I say 'fine' -- provided that the plants are built far from any inhabited place, and both the datacenters and corporate headquarters are built on top of them.

Then, when a plant chooses to go Fukushima, it will be pure gain, no loss.

@alter_kaker @Patricia

District voting may have made sense back when the main issues that people cared about were connected to their land. Thus the Rep from Distric X was supposed to defend the interests of the people from District X.

Today, however, most people care about issues that are not tied to their land or their district.

@alter_kaker @Patricia

The US two-party system is largely a consequence of district-based voting, with one representative per district, voting and single-round elections.

The current district-based system means that a political or social class that is "only" 30% of the population, but spread over the whole country,will probably end up with 0% reps in Congress. 🧵>

@BrianJopek

Is that a /The Onion/ article, perchance?

🧵> But the truth is weirder! According to the paper below (pag 20) the probability for a 3x3 0-1 matrix having zero determinant is only a bit larger (~66%), but then it starts to decrease with n, being only ~45% for a 8x8 matrix...

https://arxiv.org/pdf/math/0511636

Let A be an nxn matrix with random independent 0 or1 entries. What is the probability that A has zero determinant, as a function of n?

Without thinking, I would have guessed that the probability quickly decreases with n. But for n = 1 the probability is 50%, and for n = 2, surprise, it is 10/16 = 61.5%.

So I revised my guess to be "the probability of the determinant being zero tends to 1 as n goes to oo". 🧵>