- What is the Pilpul methodology? (Examples demonstrating application of this methodology would be appreciated.)
- Who utilizes this methodology? (Which yeshivot learn this way?)
- Where did this methodology originate? (Who started it?)
- Anybody can ask a question
- Anybody can answer
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Pilpul is not so much a method of learning as it is a method of applying one's learning. It is thought to have started with R' Yaakov Pollak in the 15-16th century, and involves the linking together of disparate texts on the basis of lexical and thematic similarities. It has its detractors, but chiefly because it has been taken to illogical extremes by those who would use it to demonstrate their learning, rather than to apply it.
Its most frequent application is in the realm of drash. Rabbonim who have employed it include R' Yonasan Eybeschütz and the Beis haLevi, among others. The extent to which it can be employed in the development of halakha lema'aseh has long been the subject of some controversy. In fact, R' Mordekhai Yoffe (the Levush) writes in his introduction that he had once subscribed to this method but that he now rejects it wholeheartedly. Other famous opponents to pilpul include the Maharal of Prague, the Vilna Gaon and the Netziv.
Sources for the above:
Menachem Elon, Jewish Law: History, Sources, Principles (4 vols; trans. Bernard Auerbach and Melvin Sykes; JPS, 1994), III:1345ff;
Gil S. Perl, The Pillar of Volozhin: Rabbi Naftali Zvi Yehuda Berlin and the World of Nineteenth-Century Lithuanian Torah Scholarship (Academic Studies Press, 2012);
Allan Nadler, The Faith of the Mithnagdim: Rabbinic Responses to Hasidic Rapture (Johns Hopkins University Press, 1997), 127ff;
Shaul Stampfer, Lithuanian Yeshivas of the Nineteenth Century: Creating a Tradition of Learning (trans. Lindsey Taylor-Guthartz; Littman Library of Jewish Learning, 2012), 44-45 and 106-107.
For further reading, see also:
Erica S. Brown, "Orthodoxy and the Search for Spirituality in Jewish Adult Education", Jewish Spirituality and Divine Law (ed. Adam Mintz and Lawrence Schiffman; Yeshiva University Press, 2005), 271-295 (288 n32).
To add to Shimon's answer, a prime example of Pilpul is any sefer written by R' Yosef Engel. His entire way of learning is collecting various sugyos and basically smashing them together to prove his point. Often, he starts off with a חקירה, a theoretical/conceptual inquisition, which has two possible answers. He then collects all the sugyos that he can find (which are usually very numerous) to show that the Tanaim/Amoraim hold one way or another, or that his חקירה is really a מחלוקת between them.
R' Yosef Engel does not avoid the problem of bizarre outcomes. He will sometimes take his חקירה to it's logical extreme, with strange results. (For example, in סימן כד of ספר אתוון דאורייתא, he concludes that a זר should be חייב מיתה בידי שמים for destroying תרומה. This is a basically unheard-of חומרה with no proof whatsoever, other than the fact that it is the product of R' Yosef Engel's assumptions.)
This derech halimud is basically out of style. Those who are not learning for in-depth analyzation will not be learning anything at all like this. For those who are looking for in-depth analyzation, the derech of R' Chaim Soloveichik has basically taken over many Yeshivos, and he had a totally different way of learning, usually not involving smashing several sugyos together but taking one sugya/concept and breaking it down into its theoretical components.