Often times when I told people that I was majoring in philosophy, they questioned my decision. Most of the time I didn’t take their questions seriously, but one question desired a serious answer: what is the value of philosophy? English helps us communicate better; math helps us calculate and organize important information; history teaches us about the past in order to help us better prepare for the future; but what about philosophy? Unlike other subjects that can find their value practical applications, the value of philosophy manifests itself in how it cultivation of one’s character. Philosophy opens the mind, humbles the ego, and clarifies ambiguity and misunderstanding.
Bertrand Russell, an eminent, British early twentieth century philosopher, wrote:
Philosophy, though unable to tell us with certainty what is the true answer to the doubts which it raises, is able to suggest many possibilities which enlarge our thoughts and free them from the tyranny of custom. Thus, while diminishing our feeling of certainty as to what things are, it greatly increases our knowledge as to what they may be; it removes the somewhat arrogant dogmatism of those who have never travelled into the region of liberating doubt.
- Problems of Philosophy, “The Value of Philosophy”
Typically after philosophical discussions, answers might not be found to the questions raised, but the person involved in the philosophical discussion will come to appreciate opinions other than his own. Through the questioning of his position, the person involved in philosophical discourse will learn the shortcomings of his position. His compatriots point out the problems with his position, and thus, the person comes to the humbling realization that he very well might be wrong. A person may not change his views when he sees all the flaws in it, but at least he will appreciate why others exist, and why agreement so often is hard to come by. 
A halakhically observant Jew, especially a student of Talmud and halakhah, must be open-minded, and thus understand different sides of an argument. Often times, when a person who studies the Talmud follows the flow of the Gemara, he is shocked at the conclusion of the Gemara. He finds that the opinion that contradicted his own—the opinion he was sure was wrong—was the accepted in the conclusion. Though the personal beliefs that motivated him at the beginning of his inquiry have rabbinic support, other authorities disagreed with said position—and they are sometimes the codified law. This requires the halakhist and student of the Talmud to give that view a second look, to rework the logic of the other’s argument, or even to revise his own beliefs. It takes an open-minded person to recognize this and understand why the ideas he holds dear may not fit the law. 
A judge must also be open-minded. The Gemara in Sanhedrin says that a person who wants to be a judge must be able to argue from a biblical standpoint that a sherets—an animal that according to the simple understanding of the Bible imparts impurity—is pure on a biblical level.  To do this requires a great amount of open-mindedness and creativity, given how very clearly and unambiguously the Bible states that a sherets is impure. One must be able to put aside one’s own intuitions and entertain many other possibilities, however far-fetched.
Open-minded requires one to also be humble. When a person sees the strengths in other opinions and finds the weaknesses in his opinion, it humbles him. The person faces his imperfection and learns the limits of his abilities. These are Jewish values no more than general values. Many Mussar books speak about striving for humility.  Philosophy cultivates that humility.
Furthermore, confronting the flaws in one’s positions forces one to rethink his reasoning and unearth one’s hidden assumptions and beliefs. As stated earlier, philosophy often does not give answers, but through the philosophical process a person is forced to think about his opinions and to perfect them. One might not gain the solace of answering the questions that keep him up at night, but one gains immensely from the process by refining one’s beliefs and thinking while at the same time allowing one to deeply introspect. This undoubtedly has a value in itself.
This process, though, occurs for a very basic reason: to clarify any ambiguity or misunderstanding. Judaism puts a high premium on clarification. Think about the countless commentators and super-commentators in Torah literature. Much of the commentary clarifies laws or ideas. The Mishnah in Avot says that the Sages would be careful with their words; they would make sure not to say things that have two meanings.  Philosophers slave over their papers to ensure that everything is perfectly clear; nothing in philosophy can have two meanings. If something is vague or ill-defined, other philosophers will crucify the writer.
Hence philosophy has both Jewish and general value because it cultivates open-mindedness and humility and engages one in the crucial process of clarification. Philosophy is not the only way to achieve these goals; there may be many ways to achieve these goals, but philosophy is one way of achieving these goals.
 This is an adaption of a point R. Shalom Carmy made in relation to Berkovits. Cf. Carmy, Shalom. “Review Essay: Eliezer Berkovits’s Challenge to Contemporary Orthodoxy” Torah U-Madda 13 (2005), 195-196
 Sanhedrin 17a; my explanation of the Gemara aligns perfectly with the Yad Rama on the spot, but does not contradict the other commentators.
 Example of such are Mesilat Yesharim ch. 22-23 and Hovot Ha-Levavot ch. 6
 Avot 1:11 and the commentators on the Mishnah. Focus on the Tiferet Yisrael on the Mishnah who phrases it like I have.