timelets: (Default)
У bamalip недавно был пост про математику, об отношении логики и абстракции. Поскольку я не математик, а пользователь, то отношение к предмету у меня чисто потребительское. То есть мне скорее интересно, почему математика, включая логику и абстракцию, полезна в человеческом сообществе (а в сообществе дельфинов - нет).

По-моему, дело в том, что в отличии от естественных наук (natural sciences), разнообразные искусственные науки (artificial sciences - computer science, engineering, finance, architecture, etc.) занимаются конструированием новой реальности. Использование математики позволяет быстро создавать сложные непротиворечивые (consistent) штуки, причем делать это с применением принципа разделения труда. Здесь логика работает на непротиворечивость, а абстракция на специализацию.

Для сравнения, в естественных биологических системах непротиворечивая сложность достигается путем эволюции в течение многих миллионов лет.

timelets: (Default)
11. ...Our language can be regarded as an ancient city: a maze of little streets and squares, of old and new houses, of houses with extensions from various periods, and all this surrounded by a multitude of new suburbs with straight and regular streets and uniform houses.

L.W. Phil. Inv. 4ed. p. 11


A street is an expression made of houses and navigation spaces.
timelets: (Default)
Probably it's not new, but the idea that the reality (whatever it means) can be described completely in words, music, formulas, protein codes, etc. implies that the reality is a free monoid "implemented by nature" and we have a functor that maps the reality to words, music, formulas, etc.

On the other hand, a subset of the reality can be described in such terms. For example, human breathing is a regular process that lasts between one's birth and death. For all intents and purposes, we can not only describe, but also synchronize it with texts and music, i.e. sing songs.



Most economists consider the reality to be even simpler than that because in their models it evenutally maps to R (benefits - costs) and linear time T.
timelets: (Default)
Going back to the conversation about forgetful functors.

Stanford Encyclopedia of Philosophy on Model Theory:
Sometimes we write or speak a sentence S that expresses nothing either true or false, because some crucial information is missing about what the words mean. If we go on to add this information, so that S comes to express a true or false statement, we are said to interpret S, and the added information is called an interpretation of S. If the interpretation I happens to make S state something true, we say that I is a model of S, or that I satisfies S, in symbols ‘I ⊨ S’. Another way of saying that I is a model of S is to say that S is true in I, and so we have the notion of model-theoretic truth, which is truth in a particular interpretation.


Whenever we see a statement S we can assume that it is missing the I portion. In short, the agent created a mapping from SxI to S, using a forgetful functor F: SxI -> S.

Right?

When I say "любой переход к модели - это forgetful functor" I mean that, to figure out the model I for S we have to figure out F. Most likely, my original statement is incorrect in mathematical sense.

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