When I figure out or learn something, I typically find myself with an urge to share it--with family, friends or blog readers. The new knowledge just pulls me to sharing. Some of the pull is vanity. But I don't think it's all vanity. Even if I had to share the knowledge anonymously, without ever having the satisfaction of knowing if anybody ever appreciated it, I would still feel the urge to share it. This isn't a decisive argument that it's not all vanity, but it is evidence. I think I am not alone in this.
While some goods can be enjoyed alone almost as well as together with others, much of the value of knowledge seems to be the value of communal knowledge--and when I talk of "knowledge", I mean to include here "understanding", "insight" and the like. It is natural to share knowledge--that is in large part why we have language. (Self-concealment correlates with psychological and physical problems, but apparently there is insufficient evidence at present to determine what the causal relationship if any there is here--see The Psychology of Secrets by Anita Kelly. And in any case, when I talk of what is natural, I am talking of a normative naturalness, not a statistical normalcy.)
Even those who want to have an esoteric secret doctrine tend to want to have a community of cognescenti with whom they can share it. Or at the very least, and most annoyingly, they want others to know that they have secret knowledge that no one else has.
The good of knowledge is, then, incomplete when the knowledge is solitary. Likewise, the good of knowledge is incomplete when the knowledge is evanescent. While eating a chocolate can be satisfactory even though the chocolate disappears in a few seconds, and the memory fades in minutes to hours, to know something for a short bit of time and then forget it completely is to miss out something important about knowledge. Knowledge is supposed to be a stable and lasting good. We see this in Plato (though he drew from this the wrong conclusions about what can be the object of knowledge, in large part because he was an A-theorist). Suppose we were to learn the answer to some difficult scientific problem, and if we were to about to die and live never again (of course, I believe that death is not the end of life, but this is a hypothetical question), and we could pass the answer to no one. This could be quite tormenting, and it might almost have been better not to know the answer. And, no, this is not just about bragging rights.
One of the things I have wondered about is how much of the meaning of our lives would remain if death were the end of life, and all humanity were to return "again to the nebula" (to use Russell's phrase). Perhaps some valuable things might not lose that much of their savor under that hypothesis. But knowledge, I think, would be much impoverished if it were all coming to an end. One reason for that is the structure of human knowledge. Finding things out always opens more new questions, and so knowledge points to more questions which in turn point to more knowledge--Nicholas Rescher talks about this really nicely. But I think there may also be something about knowledge itself, about the connection between knowledge and eternity.
The good of knowledge, thus, seems to point to community and eternity, being incomplete without either.
For the Christian, this reflection might point to the Trinity (God's self-knowledge is essentially shared between three Persons who have one intellect), the Incarnation and beatific vision (this self-knowledge is graciously shared with us), and eschatology (our knowledge will, indeed, last--and even our bodies will rise again, so even the kind of knowledge we have as embodied beings will return). Love is greater than knowledge (in its fullness it includes knowledge but goes beyond it), but knowledge (or at least understanding, and justified true belief) is theologically significant as well. After all, Christ is Logos and Sophia.