timelets: (Default)

As it happened, one of Hall's contemporaries, Jacques Plante of the Montreal Canadiens, began getting the notion that being playing goal without a mask was a bad idea. Ever creative, even as a junior player, Plante began experimenting with the idea of a face protector.

https://www.nhl.com/news/jacques-plante-becomes-first-goalie-to-wear-mask/c-290431748




Now, all hockey goalies wear masks. Moreover, all college hockey players wear masks and the vast majority of professional players wear visors. How would we predict that Jacques Plante's idea would become the initial object? Also, is it a terminal object too? How should we construct a meaningful category to model that?
timelets: (Default)
It seems like Paul Romer makes a certain mistake in his approach to the theory of innovation. He assumes that characteristic function ("the recipe") is given, while it has to be discovered. In short, the set of elements is the initial object, not the terminal one.
Maybe we can think of the set of elements as the initial object, while the free group generated by it represents the terminal object.

E.g. printed book vs handwritten book (different terminal objects built using different initial objects.)
timelets: (Default)
I'm now stuck on this definition.




How can I construct a simple topos to work through the definition?

Let's start with (4) and say that B represents various kinds of sports games; S - a particular game of sports, e.g. football; 1 stands for the rules; Ω - truth table to determine the nature of the game.

Looking at (5), let PB be the score; that is, each game has a score according to the official rules. B -> PB.

Let A be a bet on the outcome of a game. g: A -> PB.

What's BxA? It looks like a matrix of games and related bets. Using f: BxA -> Ω, we can determine whether the bet was legit.

What's BxPB? It looks like a matrix of games and scores. Using epsilon: BxPB -> Ω we can determine whether the score was legit.

In this topos we can say that a unique function g maps all bets to legitimate scores.

Does this make sense?
timelets: (Default)
У человека возник хороший вопрос:
если информация истинная, то разве принятие публикой более информированных решений не даёт большее общественное благо?


С одной стороны, интуитивно очевидно, что если "истинная информация" односторонняя, то увеличение ее количества не влияет на качество решения. Поэтому, например, в суде есть принцип состязательности сторон.

С другой стороны, можно ли доказать эту интуитивную очевидность формально? Например, если мы добавляем в категорию объекты (subobjects), которые имеют тот же терминальный объект, что и все остальные объекты в категории, то это не может изменить whether the "topos square" commutes независимо от того, что находится "внутри" объекта.

upd. I think the confirmation bias has the same logical structure.
timelets: (Default)






McLane, Moerdijk. Sheaves in geometry and logic. 1999. p. 31-32

* -> [0,1]

Jul. 5th, 2017 09:38 am
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Most popular American professional sports — football, basketball, baseball, hockey — don't have a draw. Until recently, hockey ( a Canadian game) had a draw but NHL rules got changed to a combination of overtime and penalty shots, so that the public would always have their winner.

This tells you something about the culture of certainty.
timelets: (Default)
Deadline is another important terminal object of the western civilization. It simplifies decision making by imposing a hard constraint, thus creating a powerful Ω.
timelets: (Default)
I think we can model a constraint as the terminal object within a category. Once a constraint is given, rationality boils down to * -> Ω.

This is a provisional note because I don't understand initial objects yet.
timelets: (Default)
Speaking categorically of Zarathustra, he invented one of the most generic subobject classifiers - Good vs Bad.

The Greeks invented another one: mortal vs immortal.

Galileo Galilei postulated that mechanical systems with speed=0 (first derivative) as the final object are equivalent to systems with acceleration=0 (second derivative) as the final object.

Sir Isaac Newton postulated that gravitation has a final object - g (third derivative = 0).

etc. etc.
timelets: (Default)
I wonder whether the subobject classifier approach can be used to detect new developments in technology. For example, expressions like _horseless carriage_, _smarphone with no keyboard_, _driverless car_, _artificial intelligence_ indicate that the object in question doesn't belong to a boolean topos ( or is it a slice topos?).
timelets: (Default)
Thanks to bamalip I'm now learning about Subobject Classifier.




Let's play with it a bit. If we define a single issue as the whole purpose of American politics, then we can consider it to be a terminal object *. For example, when we declare "jobs for Americans" the ultimate goal for the president anybody (∀X) who is for "jobs" satisfies the Classifier Ω and is a patriotic American U. Furthermore, if we structure the debate as a Boolean Topos: "jobs vs environment", or "jobs vs self-driving cars", or "jobs vs immigration", or "jobs vs China", or "jobs vs trade", or "jobs vs Amazon", etc., everybody on the other side of "jobs" is the enemy. The simplified, "terminal" structure of the debate defines the outcome and political polarization.

upd. On a somewhat related note, the sex vs gender debate has a similar structure. Category "sex" has a terminal object, e.g. a definite chromosome characteristic, therefore it can be easily understood as a Boolean Topos. Category "gender" doesn't have a terminal object, therefore people get confused.

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