The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in his work Od Yosef Hai (Halacha 8), elaborates on the reason for Karpas, dipping celery in saltwater toward the beginning of the Seder (listen to audio recording for precise citation). He cites Rashi’s commentary to Masechet Pesahim (114) where he explains that this is done in order to arouse the curiosity of the children. Although it is not uncommon to dip vegetables in dressing during the meal, it is unusual to do so at the beginning of the meal, even before we wash our hands and eat bread. This deviation from ordinary protocol will catch the children’s attention and lead them to begin asking questions. The Ben Ish Hai explains that the Misva of telling the story of the Exodus on Pesah should be done in the form of questions and answers, as indicated in the verse, "Ki Yishalecha Bincha" – "When your son shall ask you…" (Shemot 13:14). Therefore, one is required to evoke questions for the fulfillment of this Misva, and this is the reason why we eat Karpas, to arouse curiosity so that the children will ask questions.
The Ben Ish Hai cites the comment of Rabbi Sheneur Zalman of Liadi (1745-1813), in his Halachic code, that since Karpas is eaten only to arouse the children’s curiosity, one does not have to eat a Ke’zayit of Karpas. Indeed, our custom is to eat only a small piece of celery, and not a full Ke’zayit.
The Ben Ish Hai adds that although the purpose of Karpas is to arouse the children’s curiosity, one must eat Karpas even if there are no children present, or if a person conducts the Seder alone. Hazal did not distinguish between different situations, and Karpas is thus required even if there are no children present at the Seder. By the same token, one must eat Karpas even if he knows that this will not inspire his children to ask questions.
Summary: We eat Karpas at the beginning of the Seder in order to arouse the children’s curiosity so they will ask questions. Nevertheless, it is required even if no children are present, and even if the children will not ask questions in response to the Karpas.