Concept
Theorem
Overview
in mathematics, a statement that has been proved
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
In the original texts
Passages from public-domain books where Theorem is discussed.
The theorem is evident for _a_ = 1. We verify analytically that if it is true for _a_ = [alpha], it will be true for _a_ = [alpha] + 1. 2º I say that _a_ × _b_ = _b_ × _a_. The theorem has just been proven for _b_ = 1. We could verify analytically that if it is true for _b_ = [beta], it will be true for _b_ = [beta] + 1.
Henri Poincaré, The Foundations of Science
states what it is; names the idea; opens on it
Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
states what it is
All the other theorems are demonstrated: that is to say, a reason of knowing is given, the truth of which everyone is bound to acknowledge. The logical truth of the theorem is thus shown, but not its transcendental truth (v. §§ 30 and 32), which, as it lies in the reason of _being_ and not in the reason of _knowing_, never can become evident excepting by means of intuition. This explains _why_ this sort of…
Arthur Schopenhauer, On the Fourfold Root of the Principle of Sufficient Reason, and On the Will in Nature
states what it is; names the idea
The four angles being thus ascertained, and two sides of the quadrilateral being radii of the earth; the two remaining sides and the diagonal, or in other words, the moon's distance from the two places of observation and from the centre of the earth, could be ascertained, at least in terms of the earth's radius, from elementary theorems of geometry. At each step in this demonstration we take in a new induction…
John Stuart Mill, A System of Logic
names the idea
The fact is not improbable: but a mathematician like Newton, who seemed to arrive _per saltum_ at principles and conclusions that ordinary mathematicians only reached by a succession of steps, certainly could not have performed the comparison in question without being led by it to the _a priori_ ground of the law; since any one who understands sufficiently the nature of multiplication to venture upon multiplying…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
states what it is; names the idea; opens on it
Philosophers who discussed Theorem
Related ideas
- Belief
- Fractal
- Formal language
- Four color theorem
- Falsifiability
- Self-evidence
- Gödel's completeness theorem
- Argument
- Interpretation (logic)
- Zermelo–Fraenkel set theory
- Necessity and sufficiency
- Foundations of mathematics
- Propositional logic
- Proof theory
- Archimedes
- Mathematics
- Mathematical object
- Mathematical logic
- Model theory
- Validity (logic)
- Natural number
- Number theory
- Probability theory
- Symbol (formal)
Opposing perspectives
Place in the tradition
Part of Dialetheism.