Quotes About Axioms

A science is something which is constructed from truth on workable axioms. There are 55 axioms in scientology which are very demonstrably true, and on these can be constructed a great deal.

Paradoxes are less paradoxical in their reference to truth than most of the most plausible axioms.

Intuitive minds, on the contrary, being thus accustomed to judge at a single glance, are so astonished when they are presented with propositions of which they understand nothing, and the way to which is through definitions and axioms so sterile, and which they are not accustomed to see thus in detail, that they are repelled and disheartened.

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… 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 productions of the mind and hand seem very numerous in books and manufactures. But all this variety lies in an exquisite subtlety and derivations from a few things already known, not in the number of axioms. VIII

Proverbs are the literature of reason.

Boarding-House Geometry DEFINITIONS AND AXIOMS All boarding-houses are the same boarding-house. Boarders in the same boarding-house and on the same flat are equal to one another. A single room is that which has no parts and no magnitude. The landlady of a boarding-house is a

The results show how unreliable peer consensus can be, even when it is a peer consensus of highly intellectual people, if those people share a very similar vision of the world and treat its conclusions as axioms, rather than as hypotheses that need to be checked against facts.

people share a very similar vision of the world and treat its conclusions as axioms rather than as hypotheses that need to be checked against facts.

Undoubtedly his greatest contributions were in the field of psychohistory. Seldon found the field little more than a set of vague axioms; he left it a profound statistical science.… … The best existing authority we have for the details of his life is the biography

In the Principia Mathematica, Bertrand Russell and Alfred Whitehead attempted to give a rigorous foundation to mathematics using formal logic as their basis. They began with what they considered to be axioms, and used those to derive theorems of increasing complexity. By page 362, they had established enough to prove "1 + 1 = 2.

The result of our previous discussion is that our axioms, if interpreted as meaningful statements, necessarily presuppose a kind of Platonism, which cannot satisfy any critical mind and which does not even produce the conviction that they are consistent.

But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.

Only someone who (like the Intuitionist) denies that the concepts and axioms of classical set theory have any meaning could be satisfied with such a solution, not someone who believes them to describe some well-determined reality. For in reality Cantor's conjecture must be either true or false, and its undecidability from the axioms as known today can only mean that these axioms do not contain a complete description of reality.

...we find what he says true, as far as we have experienced, and we can judge no further but by larger experience — for axioms in philosophy are not axioms till they have been proved upon our pulses. We read fine things, but never feel them to the full until we have gone the same steps as the author.

What are your Axioms, and Categories, and Systems, and Aphorisms? Words, words.... Be not the slave of Words...

You know, in formal logic, an inconsistent set of axioms can be used to prove anything at all. Once you have a single contradiction, A and not A, there's nothing you can't derive from it.

Philosophy begins when one learns to doubt—particularly to doubt one's cherished beliefs, one's dogmas and one's axioms.

Philosophy begins when one learns to doubt—particularly to doubt one's cherished beliefs, one's dogmas and one's axioms. Who knows how these cherished beliefs became certainties with us, and whether some secret wish did not furtively beget them, clothing desire in the dress of thought? There is no real philosophy until the mind turns round and examines itself. Gnothi seauton, said Socrates: Know thyself.

One thing only I know, and that is that I know nothing, Philosophy begins when one learns to doubt - particularly to doubt one's cherished beliefs, one's dogmas and one's axioms. Who knows how these cherished beliefs became certainties with us, and whether some secret wish did not furtively beget them, clothing desire in the dress of thought? There is no philosophy until the mind turns round and examines itself. Gnothi seauton, said Socrates: Know thyself.

GOOD DESIGN IS SIMPLE. You hear this from math to painting. In math it means that a shorter proof tends to be a better one. Where axioms are concerned, especially, less is more. It means much the same thing in programming. For architects and designers, it means that beauty should depend on a few carefully chosen structural elements rather than a profusion of superficial ornament.

No very good sense can be given to the idea that the elements of Euclidean geometry may be found in nature because either everything is found in nature or nothing is. Euclidean geometry is a theory, and the elements of a theory may be interpreted only in terms demanded by the theory itself. Euclid's axioms are satisfied in the Euclidean plane. Nature has nothing to do with it.

By refraining from reducing multiplication to addition we are enabled through these axioms to banish continuity, which is so difficult to fix precisely, from the logical structure of geometry.

Axioms are often cherished long after reason would counsel their abandonment.