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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.

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Embedding from books doesn't work on dreamwidth, so here's a snapshot:

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It is different in the world of spirit...
the bread, only he who was in anguish finds repose, only he who descends into the under world rescues the beloved, only he who draws the knife gets Isaac.

--- Fear and Trembling, by Soren Kierkegaard.
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297. Of course, if water boils in a pot, steam comes out of the pot, and also a picture of steam comes out of a picture of the pot. But what if one insisted on saying that there must also be something boiling in the picture of the pot?

Ludwig Wittgenstein. Philosophical Investigations.
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89. For logic seemed to have a peculiar depth — a universal significance. Logic lay, it seemed, at the foundation of all the sciences. For logical investigation explores the essence of all things. It seeks to see to the foundation of things, and shouldn’t concern itself whether things actually happen in this or that way. —– It arises neither from an interest in the facts of nature, nor from a need to grasp causal connections, but from an urge to understand the foundations, or essence, of everything empirical.

Not, however, as if to this end we had to hunt out new facts; it is, rather, essential to our investigation that we do not seek to learn anything new by it. We want to understand something that is already in plain view. For this is what we seem in some sense not to understand.

- L.W. Phil. Inv. 4th ed. p. 46-47.
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There are various problems as regards language. First, there is the problem what actually occurs in our minds when we use language with the intention of meaning something by it; this problem belongs to psychology. Secondly, there is the problem as to what is the relation subsisting between thoughts, words, or sentences, and that which they refer to or mean; this problem belongs to epistemology. Thirdly, there is the problem of using sentences so as to convey truth rather than falsehood; this belongs to the special sciences dealing with the subject-matter of the sentences in question. Fourthly, there is the question: what relation must one fact (such as a sentence) have to another in order to be capable of being a symbol for that other? This last is a logical question, and is the one with which Mr Wittgenstein is concerned.
--- Bertrand Russel. Introduction to Tractatus Logico-Philosophicus, by Ludwig Witgentstein. 1922.
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...nothing in the universe
Is born unique and grows unique, alone,
But all belong to a species, very many
Of the same kind.
Therefore one must accept that sky
And earth and sun, moon, sea and all else that exists
Are not unique, but in number numberless.

- Lucretius, Trans. Melville, On the Nature of the Universe. Book 2, line 1077.

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В короткой истории с запретным плодом (Gen 3:1) есть основной набор современных моральных принципов (кроме Liberty) с упором на Authority & Sacrilege. Причина всех несчастий - нарушение запрета Евой и змеем. В соответствии с порядком установленным в самом начале, Адам выше всех животных и не может принимать от них советы (Gen 1:26). Но место Евы в иерархии не было прописано в явном виде - явная ошибка Создателя (no associativity!), которая всем вышла боком. С другой стороны, признание этой ошибки, видимо, и привело к смягчению приговора: изгнание из рая вместо смерти (для Адама и Евы) и постоянная вражда со всеми живущими для змея (explicit associativity rules).

Наверное, каждый элемент системы моральных принципов надо моделировать отдельной категорией.

f:(action, context) -> judgement
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It's interesting to watch (from afar) how the process of self-selection is working out

Before the Haidt's book, I'd be tempted to comment and show logical and factual mistakes in the original post. But now I know that it would be a waste of my time because "you can’t change people’s minds by utterly refuting their arguments." Kinda sad.
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Как-то так. Надо почистить картинки и разжевать каждый шаг, но логика, кажется, правильная.

The Achiles Dilemma


The Achilles Dilemma OLOG




Odysseus OLOG

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Going over Treatise of Intuitionist Logic, by JG Granstorm for the second time (see fig and text below):
Screen Shot 2016-03-17 at 11.00.37 AM
Screen Shot 2016-03-17 at 11.01.57 AM

This points to an possible interesting connection between imagination that creates concepts and the process of "learning by doing" that converts the concepts into things. The more complex the thing that embodies the concept, the more communications we need to have with other people who participate in doing the thing. As a result, the expression side of the triangle begins to grow. (upd. In a dynamic model we need expressions to communicate between the same person at diffent time periods.)

Based on this model, to represent a system in terms of the Category Theory, we would have to have at least three types of Objects that play Sources and Targets, and  at least three (or likely six?) types of Arrows that go between them. For example, an individual's artistic sci-fi imagination can be formalized as: ImaginationA: Concept -> Expression,
while Edison-style craftsman imagination as: ImaginationE: Concept -> Thing

Furthermore, if we want to model an evolution of the system, we would have to introduce all these objects at at least two times periods, e.g. today and tomorrow. This will cause us add three more arrows, e.g.
EvolveConcept: ConceptAtTime1 -> ConcepAtTime2

* I'm using "thing" instead of "object" to avoid confusion with objects in the Category Theory.
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Does God (or any other deity of your choice) have a solution to the Entscheidungsproblem? If yes, how are we going to find out whether the answer is not a trick of the Devil?

(1) The most optimistic position is that there is a systematic method to establish either A true or ∼A true, for any proposition A. I take this position to imply a positive solution to Hilbert’s Entscheidungsproblem, in direct contradiction with the result gained by Church and Turing.30 Thus, this position is self-contradictory. J.G. Granstrom. Treatise on Intuitionist Logic.
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I was always suspicious of analogies because I knew that any two objects could be grouped together for whatever reason. But I never bothered to find a logical proof of that. Well, here's how Peirce does it in just one paragraph for any group of objects*,

This principle is that any plurality or lot of objects whatever have some character in common (no matter how insignificant) which is peculiar to them and not shared by anything else. The word "character" here is taken in such a sense as to include negative characters, such as incivility, inequality, etc., as well as their positives, civility, equality, etc. To prove the theorem, I will show what character any two things, A and B, have in common, not shared by anything else. The things, A and B, are each distinguished from all other things by the possession of certain characters which may be named A-ness and B-ness. Corresponding to these positive characters, are the negative characters un-A-ness, which is possessed by everything except A, and un-B-ness, which is possessed by everything except B. These two characters are united in everything except A and B; and this union of the characters un-A-ness and un-B-ness makes a compound character which may be termed A-B-lessness. This is not possessed by either A or B, but it is possessed by everything else. This character, like every other, has its corresponding negative un-A-B-lessness, and this last is the character possessed by both A and B, and by nothing else. It is obvious that what has thus been shown true of two things is, mutatis mutandis, true of any number of things. Q. E. D.
In any world whatever, then, there must be a character peculiar to each possible group of objects.

He gives his proof in a free text form because the modern notation for propositional logic has not been invented yet. In any case, in a true Pragmatist fashion he doesn't stop with the proof, but goes further,

so long as we regard characters abstractly, without regard to their relative importance, etc., there is no possibility of a more or less degree of orderliness in the world, the whole system of relationship between the different characters being given by mere logic; that is, being implied in those facts which are tacitly admitted as soon as we admit that there is any such thing as reasoning.

In other words, unless we have a system of value judgments that enables us to make decisions about relative importance, the logic itself can't settle issues.



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