To prove to someone that he was wrong/ show someone his error
He saw that he was wrong. He reasoned with the grandfather because he knew he was wrong (literally: he put him down because he was wrong. In the original: 'stood his mistake' meaning insisted on his mistake or: understood that he was wrong).
("But when they grow up, they are accustomed to giving heart and being careful in their own affairs and standing on the error" - Sefer Me'irat Eynaim Sima'n Rafet 4, and so: "And certainly this style that he wrote to you is not far-fetched. From the one who is seen to stand on his error" - Y. Burla/ Kisuphim, 34, and so: "R. Chaim Raphael stood on his error, that he might laugh "He said a little and forgot much" - Agnon/ Tamol Shilshom, 407, and so: "Blessed is the man who erred and stood on his error, for there is no man who stands on the words of the Torah, except that he stumbles in them" - Mordecai ben Yehezkel/ Sefer Hamaoshiyot Helek Rabi'i, Ha-Hanuni, 268)
Translation
Expression (Hebrew)
הֶעֱמִיד אוֹתוֹ עַל טָעוּתוֹ
Definition (Hebrew)
הֶרְאָה לוֹ שֶׁהוּא טוֹעֶה. נִמֵּק אֶת הַסִּבָּה מַדּוּעַ אֵינוֹ צוֹדֵק (מִלּוּלִית: הִנִּיחַ אוֹתוֹ הֵיכָן שֶׁהוּא טוֹעֶה. בַּמָּקוֹר:'עָמַד עַל טָעוּתוֹ' מַשְׁמָע הִתְעַקֵּשׁ עַל טָעוּתוֹ אוֹ: הֵבִין שֶׁהוּא טוֹעֶה).
("אֲבָל כְּשֶׁיִּגְדְּלוּ הֵם עַצְמָן, רְגִילִין לִתֵּן לֵב וּמְדַקְדְּקִים בְּשֶׁלָּהֶם וְעוֹמְדִים עַל הַטָּעוּת"- סֵפֶר מְאִירַת עֵינַיִם סִימָן רַפֵּט ד, וְכֵן:"וַדַּאי לֹא נָאֶה סִגְנוֹן זֶה שֶׁכָּתַב אֵלֶיךָ. מִן הָרָאוּי לְהַעֲמִידוֹ עַל טָעוּתוֹ"- י.בּוּרְלָא/ כִּסּוּפִים, 34, וְכֵן:"הָיָה ר' חַיִּים רְפָאֵל מַעֲמִידוֹ עַל טָעוּתוֹ, שֶׁיִּצְחָק לָמַד מְעַט וְשָׁכַח הַרְבֵּה"- עַגְנוֹן/ תְּמוֹל שִׁלְשׁוֹם, 407, וְכֵן:"אַשְׁרֵי הָאִישׁ שֶׁטָּעָה וְעָמַד עַל טָעוּתוֹ, שֶׁכֵּן אֵין אָדָם עוֹמֵד עַל דִּבְרֵי תוֹרָה, אֶלָּא אִם כֵּן נִכְשָׁל בָּהֶן"- מָרְדְּכַי בֶּן יְחֶזְקֵאל/ סֵפֶר הַמַּעֲשִׂיּוֹת חֵלֶק רְבִיעִי, הַחֲנוּנִי ,268)