What is known as the history of concepts is really the history either of our knowledge of concepts or of the meanings of words.

[..]always to seperate sharply the psychological from the logical, the subjective from the objective; never to ask for the meaning of a word in isolation, but only in the context of a proposition; never to lose sight of the distinction between concept and object.

[..]if we come only on general logical laws and on definitions, than the truth is an analytic one[..]if however it is impossible to give the proof without making use of truths which are not of a general logical nature, but belong to the sphere of some special science, then proposition is a synthetic one. For a truth to be a posteriori, it must be impossible to construct a proof of it without including an appeal to facts, i.e., to truths which cannot be proved and are not general since they contain assertions of particular objects. But ir, on the contrary, it`s truth can be derived exclusively from general laws, which themselves neither need nor admit of proof, than the truth is a priori.

[..]numerical formulae can be derived from the definitions of the individual numbers alone by means of a few general laws, and that these definitions neither assert observed facts nor pressupose them for their legitimacy.

[..]That if we pour 2 unit volumes of liquid into 5 unit volumes of liquide we shall have 7 unit volumes of liquid, is not the meaning of the proposition 5 + 2 = 7, but an application of it, which only holds good provided that no alteration of volume occurs as a result, say, of some chemical reaction.

[..]"The whole of aritmetics is innate and is in virtual fashion in us" (Leibniz)

[..]Empirical propositions hold good of what is physically or psychologically actual, the truths of geometry govern all that is spatially intuitable, whether actual or product of our fancy.[..] Axioms of geometry are independent of one another and of primitive laws of logic, and consequently are synthetic.

[..]Newton proposes to understand by number not so much a set of units as the relation in the abstract between any given magnitude and another magnitude of the same kind which is taken as unity

[..]the view of M.Cantor, when he calls mathematics an empiric science in so far as it begins with the consideration of things in the external world. On his view, number originates only by abstraction from objects.

[..]an important difference between colour and Number, that a colour such as blue belongs to a surface independently of any choice of ours. The blue colour is a power of reflecting light of certain wavelenghts; to this our way of regarding it cannot make the slightest difference. The Number 1, on the other hand, or 100 or any other Number, cannot be said to belong to the pile of playing cards in its own right, but at most to belong to it in view of the way in which we have chosen to regard it

[..]Leibniz rejects the view of the schoolmen that number is not applicable to immaterial things, and calls number a sort of immaterial figure, which results from the union of things of any sorts whatsoever, for example of God, an angel, a man and motion, which together are four. For which reason he holds that number is of supreme universality and belongs to metaphysics[..]"thus number is, as it were, a kind of metaphysical figure."

[..]While for Mill the number is something physical, for Locke and Leibniz it exists only as a notion.

[..]Berkley: "It ought to be considered that number... is nothing fixed and settled, really existing in things themselves. It is entirely the creature of the mind, or any combination of ideas to which it gives one name, and so makes it pass for a unit."

[..]I understand objective to mean what is independent of our sensation, intuition and imagination, and of all construction of mental pictures out of memories of earlier sensations, but not what is independent of reason.

[..]number is neither spatial and physical[..]nor yet subjective like ideas, but non-sensible and objective.

[..]W.S. Jevons: "It has often been said that units are units in respect of being perfectly similar to one another; but though they may be perfectly similar in some respects, they must be different in at least one point, otherwise they would be incapable of plurality. If three coins were so similar that they occupied the same place at the same time, they would not be three coins but one."

[..]not every objective object has a place.

[..]

Tags: concepts, frege, logic

"And might there not also be some reader who would say 'I accept A and B as true, but I don't accept the Hypothetical?"
"Certainly there might. He, also, had better take to football."
"And neither of these readers," the Tortoise continued, "is as yet under any logical necessity to accept Z as true?"
"Quite so," Achilles assented.
"Well, now, I want you to consider me as a reader of the second kind, and to force me, logically, to accept Z as true."
"A tortoise playing football would be -- " Achilles was beginning "-- an anomaly, of course"

Tags: football, lewis carrol, logic
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