Франция, 15 августа. Орлеан

Луара

или я чего не понимаю в геометрии

"Поскольку пятый постулат определяет метрические свойства однородного пространства, отсутствие его в абсолютной геометрии означает, что метрика пространства не определена, и большинство теорем, связанных с измерениями (например, теорема Пифагора) не могут быть доказаны в абсолютной геометрии." 

милейшая книга, Геометрия Бураго

линк 

Париж, 16 августа

Вид из нашего окошка. Пляс Пигаль с Мулен Ружем за углом (да что в них...)


Музей романтизма. 150 лет назад.


Музей романтизма. месяц назад. Жалюзи перекрасили.


Уличная еда на Монмартре.


Ну это вы знаете.


Вид с Sacre Coeur


Вид с Sacre Coeur


Вид с Sacre Coeur


Граффити на Sacre Coeur. Некоторые очень любопытны.

It's complicated

progress

 https://github.com/yishn/tikzcd-editor

"what is a physical theory?"

src 

What is a physical theory? We might be able to agree that whatever it is, we demand it to have at least the following ingredients:

a) it provides us with a thing called the space of states of a physical system;

b) it provides us with a thing called the collection of sensible propositions about the states of the physical system;

c) it provides us with a way to evaluate any proposition on any element of the space of states such as to obtain something like a truth value which is a measure for the degree to which the proposition holds for that state.

What is a proposition?

I’ll essentially review my review here, which did receive a bit of positive feedback.

A little (maybe a little more) reflection shows that a good way to characterize the nature of propositions about some collection, V, is to realize that propositions P about V should be equivalently described by two properties:

• every proposition P maps every element in V to the truth value in the collection Ω of truth values;

and

• every proposition corresponds precisely to the sub-collections

SP⊂V$$$$

for which the proposition P “holds”.

 

So in order to be able to talk about propositions we need to work internally to a context T$T$ which has the at least the necessary properties for this to make sense. Such a context is, by definition, called a topos.

What is a topos.

So a topos is a category T with the property that it contains a an object Ω, such that morphisms from any other object V into Ω correspond precisely to subobjects of V:

 

(P:V→Ω)↔(SP⊂Ω),$$$$

 

and has on top of that the right properties for this statement to make good sense in the first place.

Given a topos T, we can nicely satisfy our requirements a), b) and c) by picking a state space object in T.

What is a state space object?

A state space object is a pointed topos; a topos T together with a fixed chosen object

V∈Obj(T).$$$$

 

Given any state space object V∈Obj(T), we define

•$\bullet$ the collection of states to be the elements of V, i.e. the morphisms

ψ:1→V$$$$

in T (here 1$1$ denotes the terminal object in T);

 

• the collection of propositions to be the morphisms

P:V→Ω.$$$$

 

This is nice, because there is a beautifully obvious evaluation of proposition on states now, taking values in truth values, namely the very composition of these two kinds of morphisms

1−→−P(ψ)Ω:=1→ψV→PΩ.$$$$

 

Notice that this is not really specific to physical theories. It is rather just the mere minimum of structure to reason about anything at all: to make propositions.

Now, at last, things are getting clearer to me. Halleluja. Also, I feel like I have to start a course of Topos Theory at our seminar.