„The axioms which Tertium Organum embraces cannot be formulated in our language. If we attempt to formulate them in spite of this, they will produce the impression of absurdities. Taking the axioms of Aristotle as a model, we may express the principal axiom of the new logic in our poor earthly language in the following manner:
: A is both A and Not-A.
:: or
:Everything is both A and Not-A.
:: or,
:Everything is All.
But these axioms are in effect absolutely impossible. They are not the axioms of higher logic, they are merely attempts to express the axioms of this logic in concepts. In reality the ideas of higher logic are inexpressible in concepts. When we encounter such an inexpressibility it means that we have touched the world of causes.“

—  P. D. Ouspensky, Ch. XXI
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P. D. Ouspensky
1878 - 1947
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„It is remarkable that this generalization of plane geometry to surface geometry is identical with that generalization of geometry which originated from the analysis of the axiom of parallels.... the construction of non-Euclidean geometries could have been equally well based upon the elimination of other axioms. It was perhaps due to an intuitive feeling for theoretical fruitfulness that the criticism always centered around the axiom of parallels. For in this way the axiomatic basis was created for that extension of geometry in which the metric appears as an independent variable. Once the significance of the metric as the characteristic feature of the plane has been recognized from the viewpoint of Gauss' plane theory, it is easy to point out, conversely, its connection with the axiom of parallels. The property of the straight line as being the shortest connection between two points can be transferred to curved surfaces, and leads to the concept of straightest line; on the surface of the sphere the great circles play the role of the shortest line of connection... analogous to that of the straight line on the plane. Yet while the great circles as "straight lines" share the most important property with those of the plane, they are distinct from the latter with respect to the axiom of the parallels: all great circles of the sphere intersect and therefore there are no parallels among these "straight lines".... If this idea is carried through, and all axioms are formulated on the understanding that by "straight lines" are meant the great circles of the sphere and by "plane" is meant the surface of the sphere, it turns out that this system of elements satisfies the system of axioms within two dimensions which is nearly identical in all of it statements with the axiomatic system of Euclidean geometry; the only exception is the formulation of the axiom of the parallels. The geometry of the spherical surface can be viewed as the realization of a two-dimensional non-Euclidean geometry: the denial of the axiom of the parallels singles out that generalization of geometry which occurs in the transition from the plane to the curve surface.“

—  Hans Reichenbach American philosopher 1891 - 1953

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„A: "Your objection to the self-evident has no validity. There is no such thing as disagreement. People agree about everything."
B: "That’s absurd; people disagree constantly, and about all kinds of things."
A: "How can they? There’s nothing to disagree about; no subject matter. After all, nothing exists."
B: "Nonsense. All kinds of things exist, you know that as well as I do."
A: "That’s one. You must accept the existence axiom, even to utter the term “disagreement.” But to continue, I still maintain that disagreement is unreal. How can people disagree when they are unconscious beings who are unable to hold any ideas at all?"
B: "Of course people hold ideas. They are conscious beings. You know that."
A: "There’s another axiom, but even so, why is disagreement about axioms a problem? Why should it suggest that one or more of the parties is mistaken? Perhaps all of the people who disagree about the very same point are equally, objectively right."
B: "That’s impossible. If two ideas contradict each other, they can’t both be right. Contradictions can’t exist in reality. After all, A is A."
Existence, consciousness, identity are presupposed by every statement and by every concept, including that of "disagreement." … In the act of voicing his objection, therefore, the objector has conceded the case. In any act of challenging or denying the three axioms, a man reaffirms them, no matter what the particular content of this challenge. The axioms are invulnerable.
The opponents of these axioms pose as defenders of truth, but it is only a pose. Their attack on the self-evident amounts to the charge. "Your belief in an idea doesn't necessarily make it true; you must prove it, because facts are what they are independent of your beliefs." Every element of this charge relies on the very axioms that these people are questioning and supposedly setting aside.“

—  Leonard Peikoff Canadian-American philosopher 1933
Objectivism: The Philosophy of Ayn Rand (1991) ; Dialogue used to show that existence, conciousness, identity, and non-contradiction are axioms, using A as a defender of the axioms, and B as an opponent of the axioms,

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„The "poetic avant-garde" relies on fantasy and dream reality as much as the Theatre of the Absurd does; it also disregards such traditional axioms as that of the basic unity and consistency of each character or the need for a plot.“

—  Martin Esslin Playwright, theatre critic, scholar 1918 - 2002
Context: The "poetic avant-garde" relies on fantasy and dream reality as much as the Theatre of the Absurd does; it also disregards such traditional axioms as that of the basic unity and consistency of each character or the need for a plot. Yet basically the "poetic avant-garde" represents a different mood; it is more lyrical, and far less violent and grotesque. Even more important is its different attitude toward language: the "poetic avant-garde" relies to a far greater extent on consciously "poetic" speech; it aspires to plays that are in effect poems, images composed of a rich web of verbal associations. The Theatre of the Absurd, on the other hand, tends toward a radical devaluation of language, toward a poetry that is to emerge from the concrete and objectified images of the stage itself. The element of language still plays an important part in this conception, but what happens on the stage transcends, and often contradicts, the words spoken by the characters. In Ionesco's The Chairs, for example, the poetic content of a powerfully poetic play does not lie in the banal words that are uttered but in the fact that they are spoken to an ever-growing number of empty chairs. Introduction : The absurdity of the Absurd

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„It is impossible by means of inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects. [Footnote: ] If this axiom be denied for all temperatures, it would have to be admitted that a self-acting machine might be set to work and produce mechanical effect by cooling the sea or earth, with no limit but the total loss of heat from the earth and sea, or in reality, from the whole material world.“

—  William Thomson British physicist and engineer 1824 - 1907
Mathematical and Physical Papers, Vol.1 http://books.google.com/books?id=nWMSAAAAIAAJ p. 179 (1882) "On the Dynamical Theory of Heat with Numerical Results Deduced from Mr Joule's Equivalent of a Thermal Unit and M. Regnault's Observations on Steam" originally from Transactions of the Royal Society of Edinburgh, March, 1851 and Philosophical Magazine iv, 1852

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„What needs to be done is that fundamental, ontological conceptions of reality need to be redone. We need a new language, and to have a new language we must have a new reality...“

—  Terence McKenna American ethnobotanist 1946 - 2000
Context: Our ability to destroy ourselves is the mirror image of our ability to save ourselves, and what is lacking is the clear vision of what should be done... What needs to be done is that fundamental, ontological conceptions of reality need to be redone. We need a new language, and to have a new language we must have a new reality... A new reality will generate a new language, a new language will fix a new reality, and make it part of this reality.

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„The Protestant churches generally hold that the elements of the sacrament are flesh and blood only in a tropical sense; they nourish our souls as meat and the juice of it would our bodies. But the Catholics maintain that they are literally just that; although they possess all the sensible qualities of wafer-cakes and diluted wine. But we can have no conception of wine except what may enter into a belief, either —
# That this, that, or the other, is wine; or,
# That wine possesses certain properties.
Such beliefs are nothing but self-notifications that we should, upon occasion, act in regard to such things as we believe to be wine according to the qualities which we believe wine to possess. The occasion of such action would be some sensible perception, the motive of it to produce some sensible result. Thus our action has exclusive reference to what affects the senses, our habit has the same bearing as our action, our belief the same as our habit, our conception the same as our belief; and we can consequently mean nothing by wine but what has certain effects, direct or indirect, upon our senses; and to talk of something as having all the sensible characters of wine, yet being in reality blood, is senseless jargon. Now, it is not my object to pursue the theological question; and having used it as a logical example I drop it, without caring to anticipate the theologian's reply. I only desire to point out how impossible it is that we should have an idea in our minds which relates to anything but conceived sensible effects of things. Our idea of anything is our idea of its sensible effects; and if we fancy that we have any other we deceive ourselves, and mistake a mere sensation accompanying the thought for a part of the thought itself. It is absurd to say that thought has any meaning unrelated to its only function. It is foolish for Catholics and Protestants to fancy themselves in disagreement about the elements of the sacrament, if they agree in regard to all their sensible effects, here or hereafter.
It appears, then, that the rule for attaining the third grade of clearness of apprehension is as follows: Consider what effects, which might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.“

—  Charles Sanders Peirce American philosopher, logician, mathematician, and scientist 1839 - 1914
The final sentence here is an expression of what became known as the Pragmatic maxim, first published in "Illustrations of the Logic of Science" in Popular Science Monthly, Vol. 12 (January 1878), p. 286 <!-- ; also in Collected Papers of Charles Sanders Peirce, Vol. 5, p. 388 -->

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„If scientific reasoning were limited to the logical processes of arithmetic, we should not get far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.“

—  Vannevar Bush American electrical engineer and science administrator 1890 - 1974
Context: If scientific reasoning were limited to the logical processes of arithmetic, we should not get far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability. The abacus, with its beads strung on parallel wires, led the Arabs to positional numeration and the concept of zero many centuries before the rest of the world; and it was a useful tool — so useful that it still exists.

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