We often hear people speak of something as being beautiful. For example, many believe that mathematics is a beautiful subject. What they mean by being beautiful varies by context, and is often regarded as a sort of subjective sentiments. That is, by a common belief, if someone does not "feel it", then it is impossible to explain "it" to them.
But is it really? For example, many abhor mathematics and do not believe that it is beautiful. Is it truly impossible for me to communicate my appreciation of the subject's aesthetic qualities? I'd like to believe it is not the case given that as human beings we share more or less equivalent mental faculties. If I were to advocate beauty of mathematics, what would be appropriate way of doing it so that the communication is philosophically sound? To provide a non-example, some mathematicians argue that mathematics is beautiful because it describes order and symmetry. But why is description of order and symmetry necessarily beautiful? So I don't think such arguments are philosophically sound or of much value as far as to convince a "non-believer" is concerned.