The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.

He also gave the first purely analytic proof of the intermediate value theorem ( also known as Bolzano's theorem ).

Today he is mostly remembered for the Bolzano – Weierstrass theorem, which Karl Weierstrass developed independently and published years after Bolzano's first proof and which was initially called the Weierstrass theorem until Bolzano's earlier work was rediscovered.

Bolzano's philosophical work encouraged a more abstract reading of when a demonstration could be regarded as analytic, where a proof is analytic if it does not go beyond its subject matter ( Sebastik 2007 ).

The uniform limit of this sequence then played precisely the same role as Bolzano's " limit point ".

) It was during this period that the idea of a truly national literature and culture developed, as a rejection of Bernard Bolzano's vision of a bi-lingual and bi-cultural Czech-German state.

Bolzano's posthumously published work Paradoxien des Unendlichen ( The Paradoxes of the Infinite ) was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor, and Richard Dedekind.

Bolzano's notion of a limit was similar to the modern one: that a limit, rather than being a relation among infinitesimals, must instead be cast in terms of how the dependent variable approaches a definite quantity as the independent variable approaches some other definite quantity.

Bolzano's notion of proposition is fairly broad: " A rectangle is round " is a proposition-even though it is false by virtue of self-contradiction-because it is composed in an intelligible manner out of intelligible parts.

Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become widely known until the 1870s.

Bolzano's main claim to fame, however, is his 1837 Wissenschaftslehre ( Theory of Science ), a work in four volumes that covered not only philosophy of science in the modern sense but also logic, epistemology and scientific pedagogy.

Bolzano's original claim is that the logical realm is populated by objects of the latter kind.

Satz an Sich is a basic notion in Bolzano's Wissenschaftslehre.

( Bolzano's use of the term an sich differs greatly from that of Kant ; for Kant's use of the term see an sich.

Bolzano's primary concern is with the concrete objective meaning: with concrete objective truths or truths in themselves.

W. Schultz, Leipzig I-II 1929, III 1980, IV 1931 ; Critical edition edited by Jan Berg: Bolzano's Gesamtausgabe, voll.

Bolzano's Logic.

That is, while demonstrating the existence of such a set, it was not a constructive proof — it did not display " an object " — but rather, it was an existence proof and relied on use of the Law of Excluded Middle in an infinite extension.

His original proof relied partly on Hamilton's work on the Ricci flow.

However, his proof relied on the uniformization theorem.

While the previous proof had relied on many different methods, the new version relied almost exclusively on the behavior of cyclotomic polynomials over finite fields.

If seventeen is excluded, then Theodorus's proof may have relied merely on considering whether numbers are even or odd.

Lord justifies the resulting destruction as proof of his argument about the dangers of superhumans, pointing out the devastation that Wonder Woman and Superman could cause if they fought in a crowded area and with the fact that Superman can be brought under another's control as evidence that they cannot be relied upon.

Scarf ’ s first proof of this theorem relied on Brouwer ’ s fixed point theorem, but his hope was to provide a numerical method for computing a point in the core, making no use of fixed point theorems.

It is sometimes falsely stated that Euclid's celebrated proof of the infinitude of prime numbers relied on these numbers.

Von Reichenbach hoped to develop a scientific proof for a universal life force, however his experiments relied on perceptions reported by individuals claimed to be " sensitive ", as he himself could not observe any of the reported phenomena.

He was able to produce no objective proof of any kind to support his accusations but instead relied on testimonies from Communists and ex-Communists in presenting his case to the public.

A credulousness, a distaste for documentation, an uncritical reliance on contemporary accounts, and a proneness to assume a theory as true before adequate proof was provided were all evidences of his failure to comprehend the use of the scientific method or to evaluate the responsibilities of the historian to his reading public.

The first is the strictly scientific, which demands concrete proof and therefore may err on the conservative side by waiting for evidence in the flesh.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

Since the change to better nutrition, he feels he can report on improvements in health, though he considers the following statements observations and not scientific proof.

Most of them, the world over, operate on the same principle by which justice is administered in France and some other Latin countries: the customer is to be considered guilty of abysmal ignorance until proven otherwise, with the burden of proof on the customer himself.

At some point, he was alleged to have accompanied Swein on a pilgrimage to the Holy Land, but proof is lacking.

* Natarajan Shankar SRI International, work on decision procedures, little engines of proof, co-developer of PVS.

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

where the rule is that wherever instances of "" and "" appear on lines of a proof, "" can validly be placed on a subsequent line.

where the rule is that wherever an instance of "" appears on a line of a proof, either "" or "" can be placed on a subsequent line ;

Although there is no proof, the name " Thomas Corbett " does appear on the list of dead and missing.

The burden of proof should be on the people who make these statements, to show where they got their information from, to see if their conclusions and interpretations are valid, and if they have left anything out.

An inscription on a stone built into the wall of a summer house in Lancarffe furnishes proof of a settlement in Bodmin in the early Middle Ages.

He / she must have a valid passport and either have an invitation letter or a bank statement with enough money to survive the length of the stay in Costa Rica, plus proof of onward travel ( ticket to exit Costa Rica & legal ability to travel to the destination stated on the ticket ).

Maimonides argued that executing a defendant on anything less than absolute certainty would lead to a slippery slope of decreasing burdens of proof, until we would be convicting merely " according to the judge's caprice ".

# Every net on X has a convergent subnet ( see the article on nets for a proof ).

The first proof relies on a theorem about products of limits to show that the derivative exists.

* 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (" New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors ( i. e., polynomials ) of the first or second degree ")

In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent on the truth of this conjecture.