A Quote by Gottfried Leibniz

When a truth is necessary, the reason for it can be found by analysis, that is, by resolving it into simpler ideas and truths until the primary ones are reached. It is this way that in mathematics speculative theorems and practical canons are reduced by analysis to definitions, axioms and postulates.

Gottfried Leibniz

Quote Author

Gottfried Leibniz
German - Philosopher
July 1, 1646 - November 14, 1716

When a truth is necessary, the reason for it can be found by analysis, that is, by resolving it into simpler ideas and truths until the primary ones are reached.

Gottfried Leibniz

There are also two kinds of truths, those of reasoning and those of fact. Truths of reasoning are necessary and their opposite is impossible, and those of fact are contingent and their opposite is possible. When a truth is necessary its reason can be found by analysis, resolving it into more simple ideas and truths until we reach those which are primitive.

Gottfried Leibniz

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.

Gottfried Leibniz

Any reductionist program has to be based on an analysis of what is to be reduced. If the analysis leaves something out, the problem will be falsely posed.

Thomas Nagel

There is a certain way of searching for the truth in mathematics that Plato is said first to have discovered. Theon called this analysis.

Francois Viete

[Mathematics] is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing - one great, glorious thing. Whether it is differential topology, or functional analysis, or homological algebra, it is all one thing. ... They are intimately interconnected, they are all facets of the same thing. That interconnection, that architecture, is secure truth and is beauty. That's what mathematics is to me.

Paul Halmos

There are several kinds of truths, and it is customary to place in the first order mathematical truths, which are, however, only truths of definition. These definitions rest upon simple, but abstract, suppositions, and all truths in this category are only constructed, but abstract, consequences of these definitions ... Physical truths, to the contrary, are in no way arbitrary, and do not depend on us.

Georges-Louis Leclerc, Comte de Buffon

For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.

Michael Frayn

There is synthesis when, in combining therein judgments that are made known to us from simpler relations, one deduces judgments from them relative to more complicated relations. There is analysis when from a complicated truth one deduces more simple truths.

Andre-Marie Ampere

What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. Roughly speaking, people know that it deals with numbers, figures, with relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes.

Stanislaw Ulam

Buddhist epistemologists do argue that rational analysis leads to the conclusion that rational analysis cannot give us infallible access to truth, including that one. That's not self-defeating, though; it only induces an important kind of epistemic humility and a clearer view of what we do when we reason. We engage in one more fallible human activity among many.

Jay L. Garfield

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.

Francis Bacon

Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.

Ivars Peterson

When I was an institutional broker in a former life, I was a believer in the merits of using technical analysis. I found that it was a very useful tool that complemented the much more mainstream tools generically referred to as fundamental analysis.

Rick Santelli

You cannot get involved in debate on 'MOTD'. You can do it on Sky because they've got hours and hours. We've got a couple of minutes. It's a very disciplined show. Our primary purpose is to show the action, and the analysis is very secondary. We have lots of people who would prefer no analysis. We have lots of people who would prefer more analysis.