Of course Jim is right to point out that there was another conception of Jewishness that developed first and foremost in Germany, that of Germans of the Mosaic persuasion, but that is a development away what was then the typical outlook (and the one that persists to this day in Russia). Of course many German Jews took the next logical step and became christians. There is yet another development in the US which was pushed by Randolph Bourne of the cultural Jew (i.e. non-believing, and not necessarily racial, but participating in Jewish culture).
There is a small controversy on these grounds in England right now, where the religious overseers of the Jewish schools (state-funded, Jewish run) here are not admitting children whose mothers are converts, because they insist that Jewishness is by matrilineal descent. To make the point the Chief Rabbi explained that those orn of Jewish mothers were entitled to attend, even if they rejected the Jewish faith, and those who were not were not welcome even if they embraced it. He meant, in effect, that Jewishness was not "a faith". Naturally (if that's the right word) parents among the Reform Jews are outraged at the decision, and it will probably cause problems with the funding authorities (which are secular).