Question: What ideas did this Italian philosopher, poet, essayist, and philologist, who wrote the poem L'infinito, come in touch with while living in a secluded town in the conservative Papal States?

Context paragraphs:
Mazzatello: Mazzatello (abbreviated mazza) was a method of capital punishment used by the Papal States from the late 18th century to 1870 involving the infliction of head trauma.  The method was named after the implement used in the execution: a large, long-handled mallet or pollaxe.  According to Abbott, "mazzatello" constituted "one of the most brutal methods of execution ever devised, requiring minimal skill on the part of the executioner and superhuman acquiescence by the victim".  Megivern cites "mazzatello" as one example of an execution method devised by the Papal States that "competed with and in some instances surpassed those of other regimes for cruelty".

Anconine Republic: The Anconine Republic (Italian: Repubblica Anconitana ) was a revolutionary municipality formed on 19 November 1797.  It came about after a French victory at Ancona in February 1797, and the consequent occupation of the city.  It existed in the region of Marche, with Ancona serving as its capital.  Despite the Treaty of Campo Formio stating that Ancona and the surrounding region had to be returned to the Papal States, the municipality proclaimed the decadence of papal rule, under French protection.  The subsequent tension led to general conflict with the Pope and the French occupation of the whole of the Papal States.  Ancona was incorporated into the Roman Republic on 7 March 1798.  It had a consul as its head.

List of people executed in the Papal States: This is a list of people executed in the Papal States under the government of the Popes or during the 1810–1819 decade of French rule.  Although capital punishment in Vatican City was legal from 1929 to 1969, no executions took place in that time.  This list does not include people executed by other authorities of the Roman Catholic Church or those executed by Inquisitions other than the Roman Inquisition, or those killed in wars involving the Papal States, or those killed extrajudicially.

Caeca et Obdurata: Caeca et Obdurata Hebraeorum perfidia (named for its Latin incipit, meaning "the blind and obdurate perfidy of the Hebrews") was a papal bull, promulgated by Pope Clement VIII on February 25, 1593, which expelled the Jews from the Papal States, effectively revoking the bull "Christiana pietas" (1586) of his predecessor Pope Sixtus V.  Prior to 1586, Pope Pius V's bull "Hebraeorum gens sola" (1569) had restricted Jews in the Papal States to Rome and Ancona.

List of Prime Ministers of the Papal States: This is a list of the prime ministers of the Papal States.  Officially the prime minister was recognized as Minister of Interior (Italian: "Ministro dell'Interno" ), like the title of "chief of the government" was reserved to the Cardinal Secretary of State, the office was created from the granting of the Statute of the Papal States in 1848 to 1850, when Pope Pius IX disclaim the Statute after the Roman Republic's fall in 1849.

Armistice of Bologna: The Armistice of Bologna was a treaty signed between the Papal States and the French First Republic on 23 June 1796.  It resulted in a ceasefire between the two parties that was intended to last until a permanent peace treaty could be signed (the 1797 Treaty of Tolentino).  The terms of the armistice included the payment of 15 million livres in cash in addition to goods and works of art to the French and significant territorial reductions.  A temporary Austrian victory against the French at the Siege of Mantua persuaded Pope Pius VI to renounce the armistice in September 1796.  The French subsequently invaded the Papal States and enforced the terms of the armistice.

Paul Zacchias: Paul Zacchias or Paolo Zacchia (1584-1659) was an Italian physician, teacher of medical science, forensic medicine, medio-legal jurist, philosopher, and poet.  He is said to have been personal physician to Pope Innocentius X and Pope Alexander VII.  Zacchias was also legal adviser to the Rota Romana, the highest Papal court of appeals, and head of the medical system in the Papal States.  His most well known book in three volumes, "Quaestiones medico-legales" (1621-1651) established legal medicine as a topic of study.

Giacomo Leopardi: Giacomo Taldegardo Francesco di Sales Saverio Pietro Leopardi (] ; 29 June 1798 – 14 June 1837) was an Italian philosopher, poet, essayist, and philologist.  He is widely seen as one of the most radical and challenging thinkers of the 19th century.  Although he lived in a secluded town in the conservative Papal States, he came in touch with the main ideas of the Enlightenment, and through his own literary evolution, created a remarkable and renowned poetic work, related to the Romantic era.  The strongly lyrical quality of his poetry made him a central figure on the European and international literary and cultural landscape.

L'infinito: "L'infinito" (] ; English: The Infinite ) is a poem written by Giacomo Leopardi probably in the autumn of 1819.  The poem is a product of Leopardi's yearning to travel beyond his restrictive home town of Recanati and experience more of the world which he had studied.  It is widely known within Italy.

Roman Republic (18th century): The Roman Republic (Italian: "Repubblica Romana" ) was proclaimed on 15 February 1798 after Louis Alexandre Berthier, a general of Napoleon, had invaded the city of Rome on 10 February.  The Roman Republic was a client republic under the French Directory composed of territory conquered from the Papal States.  Pope Pius VI was exiled to France and died there in 1799.  It immediately took the control of the other two former-papal revolutionary administrations, the Tiberina Republic and the Anconine Republic.  The Roman Republic was short-lived, as the Papal States were restored in October 1799.

Supporting paragraphs: L'infinito Giacomo Leopardi
Answer: Enlightenment