“The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel axiom in a rather complicated way. He realized that, so worded, this axiom lacked the self-sufficiency of the other nine axioms, and there is good reason to believe that he avoided using it until he had to. Many Greeks tried to find substitute axioms for the parallel axiom or to prove it on the basis of the other nine. …Simplicius”

— Morris Kline

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 177
Context: The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel axiom in a rather complicated way. He realized that, so worded, this axiom lacked the self-sufficiency of the other nine axioms, and there is good reason to believe that he avoided using it until he had to. Many Greeks tried to find substitute axioms for the parallel axiom or to prove it on the basis of the other nine.... Simplicius cites others who worked on the problem and says further that people "in ancient times" objected to the use of the parallel postulate.

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "The attempt to avoid a direct affirmation about infinite parallel straight lines caused Euclid to phrase the parallel a…" by Morris Kline?

Morris Kline 42

American mathematician 1908–1992

Related quotes

“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.”

Hans Reichenbach (1891–1953) American philosopher

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.
The Philosophy of Space and Time (1928, tr. 1957)

“…the mathematician uses an indirect definition of congruence, making use of the fact that the axiom of parallels together with an additional condition can replace the definition of congruence.”

Hans Reichenbach (1891–1953) American philosopher

The Philosophy of Space and Time (1928, tr. 1957)

“In the field of non-Euclidean geometry, Riemann… began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length.
…he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom… In brief, there are no parallel lines. This … had been tried… in conjunction with the infiniteness of the straight line and had led to contradictions. However… Riemann found that he could construct another consistent non-Euclidean geometry.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 454

“To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.”

Bertrand Russell (1872–1970) logician, one of the first analytic philosophers and political activist

As quoted in Williams' Weighing the Odds: A Course in Probability and Statistics (2001), p. 498
Attributed from posthumous publications

“The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried.”

Francis Bacon book Novum Organum

Aphorism 19
Novum Organum (1620), Book I
Context: There are and can be only two ways of searching into and discovering truth. The one flies from the senses and particulars to the most general axioms, and from these principles, the truth of which it takes for settled and immovable, proceeds to judgment and to the discovery of middle axioms. And this way is now in fashion. The other derives axioms from the senses and particulars, rising by a gradual and unbroken ascent, so that it arrives at the most general axioms last of all. This is the true way, but as yet untried.

“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 book Tertium Organum

Source: Tertium Organum (1912; 1922), Ch. XXI

“[Relativist] Rel. There is a well-known proposition of Euclid which states that "Any two sides of a triangle are together greater than the third side." Can either of you tell me whether nowadays there is good reason to believe that this proposition is true?
[Pure Mathematician] Math. For my part, I am quite unable to say whether the proposition is true or not. I can deduce it by trustworthy reasoning from certain other propositions or axioms, which are supposed to be still more elementary. If these axioms are true, the proposition is true; if the axioms are not true, the proposition is not true universally. Whether the axioms are true or not I cannot say, and it is outside my province to consider.”

Arthur Stanley Eddington (1882–1944) British astrophysicist

Space, Time and Gravitation (1920)

“One states as axioms several properties that it would seem natural for the solution to have and then one discovers that the axioms actually determine the solution uniquely. The two approaches to the problem, via the negotiation model or via the axioms, are complementary; each helps to justify and clarify the other.”

John Nash (1928–2015) American mathematician and Nobel Prize laureate

"Non-cooperative Games" in Annals of Mathematics, Vol. 54, No. 2 (September 1951)
1950s
Context: We give two independent derivations of our solution of the two-person cooperative game. In the first, the cooperative game is reduced to a non-cooperative game. To do this, one makes the players’ steps of negotiation in the cooperative game become moves in the noncooperative model. Of course, one cannot represent all possible bargaining devices as moves in the non-cooperative game. The negotiation process must be formalized and restricted, but in such a way that each participant is still able to utilize all the essential strengths of his position. The second approach is by the axiomatic method. One states as axioms several properties that it would seem natural for the solution to have and then one discovers that the axioms actually determine the solution uniquely. The two approaches to the problem, via the negotiation model or via the axioms, are complementary; each helps to justify and clarify the other.

“[M]y axiom runs as follows: "The whole is not identical with a part." This axiom leads us at once to a problem. What relation has the part to the whole?”

Karl Pearson (1857–1936) English mathematician and biometrician

The Ethic of Freethought (Mar 6, 1883)

“It has long been an axiom of mine that the little things are infinitely the most important.”

Arthur Conan Doyle book The Memoirs of Sherlock Holmes

Source: The Memoirs of Sherlock Holmes

Help us to complete the source, original and additional information

Morris Kline 42

Related quotes

Related topics