Quotes About Axioms

Finally there are simple ideas of which no definition can be given; there are also axioms or postulates, or in a word primary principles, which cannot be proved and have no need of proof.

OS AXIOMAS, EMBORA BÁSICOS PARA TODAS AS PROVAS, SÃO ELES MESMOS NÃO-PROVADOS OU ATÉ MESMO NÃO PROVÁVEIS... SE OS AXIOMAS SÃO NEGADOS, AS PROPOSIÇÕES DEDUZIDAS A PARTIR DOS AXIOMAS NÃO EXISTEM, VISTO QUE NÃO HÁ NADA A PARTIR DO QUE ELAS POSSAM SER DERIVADAS; A VALIDADE DO SISTEMA INTEIRO TORNA-SE SUSPEITO... OS AXIOMAS DE UMA PESSOA DETERMINAM OS SEUS TEOREMAS.

mathematics as a tree with strong roots (the Axioms), a solid trunk (Rigorous Proof) and ever growing branches blooming with wondrous flowers (the Theorems).

According to Marcia Lei Zeng and Jian Qin, an ontology describes (a)the types of things that exist (classes), (b)the relationships between them (properties), and (c)the logical ways those classes and properties can be used together (axioms).46

I am not a mathematician, but I was aware that for centuries, mathematics was considered the queen of the sciences because it claimed certainty. It was grounded on some fundamental certainties - axioms - that led to others.

Are the axioms of logic adequate to reality or are they a means and measure for us to create reality, the concept "reality", for ourselves?

The television series has respected one of the axioms of the novel: no event is allowed into it that does not have a precedent in human history.

And if you don't like that you can acquaint yourself with the axioms that neither you nor anybody else are the centre of the universe and that what you call complications are simply another name for life itself. Worry is life, and life is worry. And the absence of worry is death.

Freedom can be manifested only in the void of beliefs, in the absence of axioms, and only where the laws have no more authority than a hypothesis.

the integration of matrices is not a simple operation of adding together. It is a process of mutual interference and cross-fertilization, in the course of which both matrices are transformed in various ways and degrees. Hidden axioms, implied in the old codes, suddenly stand revealed and are subsequently dropped; the rules of the game are revised before they enter as sub-rules into the composite game. When Einstein bisociated energy and matter, both acquired a new look in the process.

There's a kind of Gödel's Theorem in human affairs: Every attempt to systemize life or to govern it by a set of axioms rich enough to encompass the totality of experience leads to a contradiction.

A dozen more questions occurred to me. Not to mention twenty-two possible solutions to each one, sixteen resulting hypotheses and counter-theorems, eight abstract speculations, a quadrilateral equation, two axioms, and a limerick. That's raw intelligence for you.

A dozen more questions occurred to me. Not to mention twenty-two possible solutions to each one, sixteen resulting hypotheses and counter-theorems, eight abstract speculations, a quadrilateral equation, two axioms, and a limerick. That's raw intelligence for you.

The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry [as did Clifford], for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought.

We can think of the difference between math and science in terms of possible worlds. Math is concerned with truths that would hold in any possible world: given these axioms, these theorems will follow. Science is all about discovering the actual world in which we live.

Whatever the degree of your knowledge, these two—existence and consciousness—are axioms you cannot escape, these two are the irreducible primaries implied in any action you undertake, in any part of your knowledge and in its sum, from the first ray of light you perceive at the start of your life to the widest erudition you might acquire at its end.

These gods are the axioms and foundational beliefs that must be accepted, a priori, rather than proven, before the belief system can be adopted, and when accepted and applied to the world allow the illusion to prevail that knowledge has been produced.

Naive, harmless people usually guide their perceptions and actions with a few simple axioms: people are basically good; no one really wants to hurt anyone else; the threat (and, certainly, the use) of force, physical or otherwise, is wrong. These axioms collapse, or worse, in the presence of individuals who are genuinely malevolent

Contrary to all prior belief, the vast continent of arithmetical truth cannot be brought into systematic order by laying down for once and for all a fixed set of axioms and rules of inference from which every true arithmetical statement can be formally derived.

The mathematician could formulate only a handful of interesting theorems without defining concepts beyond those contained in the axioms and that the concepts outside those contained in the axioms are defined with a view of permitting ingenious logical operations which appeal to our aesthetic sense both as operations and also in their results of great generality and simplicity.

the Zurich Axioms are about the world of money, and that is a world of human events. Human events absolutely cannot be predicted, by any method, by anybody. One of the traps money-world prophets fall into is that they forget they are dealing with human behavior.

1) 0 is a number. (2) The successor of any number is a number. (3) No two numbers have the same successor. (page 6) (4) 0 is not the successor of any number. (5) Any property which belongs to 0, and also to the successor of every number which has the property, belongs to all numbers.

every progression verifies Peano's five axioms.

I tell you the solemn truth, that the doctrine of the Trinity is not so difficult to accept for a working proposition as any one of the axioms of physics.