There are two definitions for Judaism. One is "born to a Jewish mother" which is, of course, a recursive definition with no end condition, but you know what I mean.

The other is "performs all the righteous duties" according to the Halacha. The orthodox claim that you can't have one without the other. Certainly a Goy who performs the Mitzvoth is never considered a Jew until he is formally converted, and even then he does not have the same legal standing as a Jewish-born person.

Secular Jews usually take the first definition, extended according to their personal conviction (how does one become a Jew?), but leave off the religious stuff. We consider Judaism either a nationality or an ethnicity. The culture itself has a lot of religious overtones, but I think no culture stands still, and we should start refreshing our culture a bit.