It seems Ashkenazi Jews of the time - including the Tosafists - lacked formal secular knowledge.
Rabbi Prof. Ephraim Kanarfogel wrote in his essay "The Tosafist Oeuvre and Torah u-Madda":
"The Ba'alei ha-Tosafot flourished in northern France and Germany (Ashkenaz) during the twelfth and thirteenth centuries. While the Jewish communities of Spain and Provence during those centuries were routinely exposed to external philosophical, scientific, and literary studies, the opportunities for exposure to the surrounding culture in Ashkenaz were severely limited...A more recent view, noting the substantive nature of medieval scholasticism, emphasized instead the role that language played in limiting cultural contact or absorption in Ashkenaz. The language of culture in Christian Europe was Latin. Ashkenazic Jewry spoke the vernacular and wrote its rabbinic corpus in Hebrew, but could not, for the most part, read or understand Latin...To be sure, Jews in northern France and Germany could understand Christian sermons that were preached publicly in the vernacular...Indeed, the Ba'alei ha-Tosafot even acquired some familiarity with Christian doctrine...But despite the evidence for contact in a number of areas, there is no indication that Jews were familiar with the bulk of Christian theological, legal, or philosophical literature. The overall orientation of the Ba'alei ha-Tosafot can be best described as Talmudocentric. They occupied themselves almost exclusively with the study and interpretation of the Talmud and other sacred texts and were not familiar, at least not formally, with philosophy, science or "the humanities". The Ba'alei ha-Tosafot do not strike us in any sense as paragons of Torah u'Madda, a point which is made even more sharply if we compare them with their contemporary, Rambam.
He refers in footnote no. 9 to a number of sources, including Prof. Avraham Grossman's "Chachmei Ashkenaz Harishonim" (The Early Sages of Ashkenaz), pg. 424, who wrote there:
"עיון פילוסופי-תיאולוגי לא מצינו כלל בשרידי יצירתם של חכמי אשכנז הראשונים, למרות העיסוק בו בקיבוצים יהודיים אחרים באותם ימים: בבל (מאז ימי רס"ג), צפון-אפריקה וספרד, ולמרות עלייתו ההדרגתית באירופה הנוצרית...על היצירה הפילוסופית-תיאולוגית של שכניהם הנוצרים ודאי ידעו חכמי אשכנז, ואולי גם הכירו חלקים ממנה, למרות שקשה היה להם להיזקק לה במקרה בשל קשיי לשון, שכן היא נכתבה בשפה הלטינית. אמנם בראשית המאה השתים-עשרה נזקק לה הרשב"ם...אך לא מצינו כן באשכנז במאה האחת-עשרה.
מכאן שההתעלמות מתחום עיוני וספרותי זה מודעת ומכוונת היא. כיון שיצירה הגותית זו לא הגיעה במאה האחת-עשרה באירופה הנוצרית להתפתחותה המלאה ולהשפעתה הרבה, אפשר שטרם נתנה את אותותיה בחברה היהודית באותה מידה שעליית המדעים והיצירה הרוחנית בח'ליפות המוסלמית...אך העובדה, שגם לאחר מכן לא ניכר רישומה בחברה היהודית באשכנז...מלמדת, כי יש לתלות זאת בראש ובראשונה במגמת ההסתגרות של יהודי אשכנז, מגמה אשר גברה והלכה במחצית השנייה של המאה האחת-עשרה..."
Translation: We have not found philosophical-theological studies within the remains of the corpus of works by the early sages of Ashkenaz, despite their usage by other collectives of Jews in those days: Babylon (since the time of Rasag), North-Africa and Spain, and despite its gradual rise in Christian Europe...the sages of Ashkenaz surely knew of the philosophical-theological corpus of their Christian neighbors, and perhaps were even familiar with parts of it, despite it being hard for them to make use of because of lingual difficulties, for it was written in Latin. Although the Rashbam made use of it in the beginning of the 12th century...but we don't find that the sages of Ashkenaz made similar use of it in the 11th century.
From here we learn that the ignoring of this literary and academic realm was deliberate. Because this corpus of thought had not yet reached its full development and influence in 11th century Christian Europe, it is possible that it had yet to take root in the Jewish society like the rise of sciences and humanities in the Muslim Caliphate...but the fact, that even after that its impression upon the Jewish society in Ashkenaz is not visible...shows, that first and foremost, this hinges on the seclusionary tendencies of Ashkenazic Jewry, a tendency that grew more and more over the second half of the 11th century..."
Prof. Efraim Urbach wrote in his book "Ba'alei Hatosafot", vol. 2, pg. 694-695:
"בהסברתן של מלים זרות לא יכלו להסתמך על ידיעתם בשפות שמהן הושאלו מלים אלו, מלבד לארמית...לפעמים גם השיגו ידיעות בעל-פה...רבו של עורך התוספות לשבת 'שמע מיהודי אחד שבא מארץ יון ואמר דבלשון יון פילוסופוס הוא דוד החכמה' (שבת קט"ז ע"א ד"ה פילוסופא). אבל בעיקר למדו את פירוש המלים מתוך השוואת המקורות...ולעיתים קרובות רק מתוך גופה של סוגיה וקשר העניינים...לאור האמצעים הדלים שהיו בידיהם, אין תימה שפה ושם נמצא פירושים תמוהים ומוזרים למלים זרות, כגון: "וקרקסאות, רוצה לומר בית הכסא בלשון ערב"...בפירוש עניינים מעין אלו ודומים להם בתחום הריאליה וההיסטוריה חסרו להם כמובן הידיעות הנחוצות; וכל הדיוק במקורות, ושפע ההקבלות לא היה בו כדי לבוא במקומה של הכרת המציאות הגיאוגרפית והריאלית..."
Translation: "When explaining foreign words they could not trust their knowledge in the languages from which those words were borrowed, besides for Aramaic...sometimes they attained knowledge passed by word-of-mouth...the rabbi of the editor of the tosfot on Shabbat "heard from a Jew who came from the land of Greece and said that in Greek philosophus means a companion of wisdom" (Shabbat 116a, D"H Philosopha). But for the most part, they learned the meaning of words from comparing with other sources...and more often only from the body of the text of the issue at hand and the relation of the matters...in light of the poor means at their hands, it should come as no surprise, therefore, that here and there we find some odd explanations to foreign words, for example: "And circuses, meaning the outhouse in Arabic"...in explanations of this sort and similar ones in the fields of reality and history they lacked, of course, the necessarily information; and all of the preciseness in the sources and the multitude of comparisons and contrasts could not replace an understanding of the geographical and realistic reality..."
Dr. Yosef Dan in his book "Torat Hasod Shel Chasidei Ashkenaz", pg. 38-39, notes the one exception:
"...לגבי הסוג הראשון, שהיא השפעה ישירה של הספרות התיאלוגית והמדעית, יש בידינו רק דוגמה מובהקת אחת, והיא גילויו של ג' ויידה, כי בכתביו של ר' אלחנן בן יקר מלונדון יש לשונות מכתביהם של סופרים נוצריים ממש. קשה להקיש מר' אלחנן על יתר חסידי אשכנז: ר' אלחנן בן יקר היה חכם נודד, שעבר ככל הנראה בצרפת הצפונית ובאנגליה, קלט מסורות ממקורות שונים, התעניין בספרות המדעית הנוכרית ובייחוד בענייני רפואה, אסטרולוגיה ותכונה, וכתביו נושאים אופי של ליקוט ידיעות תיאולוגיות ומדעיות ולא יצירה מקורית אחידה. קשה מאוד להניח, שחסידי אשכנז האחרים...היה בידם לבוא במגע ישיר עם הספרות התיאולוגית הנוצרית במקורה, בעיקר בלאטינית..."
Translation: "...about the first type, which is the direct influence of the theological and scientific literature, we have only one clear example, and that is the discovery of G. Vajda, that in the writings of Rabbi Elchanan ben Yakar of London1 there are terms clearly gleaned directly from the writings of Christian writers2. It is difficult to infer from Rabbi Elchanan to the other Ashkenaz Pietists: Rabbi Elchanan was a nomadic sage, who it seems most likely passed through northern France and England, absorbed traditions from different sources, was interested in the non-Jewish scientific literature and in particular, that of the fields of medicine, astrology and geometry, and his writings bear characteristics of anthologies of theological and scientific ideas, rather than original, homogenous works. It is difficult to assume that the other Ashkenaz Pietists...had it within them to come in direct contact with the Christian theological literature in its original form, mostly in Latin..."3
Dr. Reimund Leicht wrote in "The Reception of Astrology in Medieval Ashkenazi Culture" that though medieval Ashkenazi Jews were not open to or simply disinterested in the study of science as disciplines separate from the Talmud, it seems that they did study the astrology necessary to understand astrological discourses in the Talmud and other sources. He brings, for example, Rashi on Eruvin 56b, where Rashi shows proficient understanding in the field, and even quotes from Rabbi Shabbtai Donnolo's book on astrology (!). Another example is Rabbi Yosef Bechor-Shor on Beresheet 2:2 and Shemot 1:214. He also points to examples from the circle of Chasidei Ashkenaz, where much astrological material can be found, such as in the works of the Roke'ach and Rabbi Avraham ben Azriel in Arugot Habosem. Leicht points to these examples and others as evidence that astrology was formally studied in Ashkenazi yeshivot - although it was always in a Torah-based context, never as an independent subject, and it seems that only Jewish texts were used for study material, such Rabbi Shabbtai's Sefer Chackmoni, the Baraita d'Shmuel and the Baraita d'Mazalot.
1 Not much is known about Rabbi Elchanan. Cecil Roth wrote a bit about him in "The intellectual activities of medieval English Jewry", but I haven't managed to access this book. Georges Vajda in his essay "Perush R' Elchanan ben Yakar L'Sefer Yetzirah" (Al-Yad, Vol. 16a) compiled the short colophons written by Rabbi Elchanan in different versions of his commentary on Sefer Yetzirah: א. אני אלחנן יצחק בר יקר למדתי זה הספר לפני אדם שלמדו לפני רבינו יצחק הזקן זלה"ה ופירשתיו בעיר לונדריש (I, Elchanan Yitzchak bar Yakar, learned this book from a man who learned it from Rabbeinu Yitzchak, and I wrote a commentary on it in the city of London). ב. ספר זה שעשיתי וחברתי אלחנן בן יקר מגזע רבינו שמעון הגדול וכו' (This book which I made and authored, Elchanan ben Yakar from the stock of Rabbi Shimon the Great). ג. אני אלחנן עשיתי זה הספר ולקחתי מדברי רב סעדיה ומדברי ר' אברהם אבן עזרא החכם ומדברי ר' שבתאי הרופא ( I, Elchanan, made this book and I took from the teachings of Rav Saadya and from the teachings of Rabbi Avraham Ibn Ezra the wise and from the teachings of Rabbi Shabbtai the Healer).
2 One example, mentioned by Vajda in the third World Congress of Jewish Studies, is that Rabbi Elchanan apparently got from Honorius Augustodunensis the idea of the three social classes - priests, knights and farmers - originating from Noach's three sons: Shem, Yefet and Cham (Canaan).
3 It occurs to me that Rabbi Elchanan might have chosen to settle in England because there's some evidence that the England community, though being a spiritual branch of the Ashkenzic community, was a little more open in terms of intellectual studies: According to Cecil Roth in "The Jews in the English Universities", some of England's most prominent rabbis at the time - such as: Rabbi Berachiah Hanakdan and Rabbi Moshe ben Hanessiah - were connected to England's universities. They weren't active students, but it may be surmised that the local students were interested in discussing and debating with them on various matters.
4 According to Leicht, the book by "חכם צח" is Avraham bar Chiya's Sefer Megillat Hamegalleh.
