"The difficulty in reaching a reliable definition of 'who is a Jew' is understandably connected with the more basic problem of 'what is a Jew.'
An abstract definition of Judaism that would be both meaningful and inclusive is not at all easy.
Such an endeavor, to determine the basic principles of Judaism, has never been able to get very far, not only because the number of such basic principles is so great that they no longer serve to define anything, and also because an abstraction of Judaism cannot demonstrate its singularity.
It is true that the Maimonides’ thirteen articles of faith do have considerable positive value in this respect and do manage to distinguish the Jewish religion from others.
Nevertheless, even these do not provide a conclusive, positive depiction of a Jew.
More precisely, if we examine the more abstract principles, these define many frontiers of the Jewish religion but not its essence.
And if we insist on finding the principle of faith in the Torah that we possess, then, in fact, all the details of the Torah are brought in.
In this case, there is certainly unity, but again it is not an abstract definition.
Not in vain have sages claimed that a genuine definition of Judaism that would be both meaningful and really Jewish was not possible through any listing of a number of principles, but rather required a wide grasp of certain essentials: the acceptance of the Torah and what this involved in terms of the mitzvot, the whole structure of Halakhah, and the study of certain volumes of thought.
And yet even such very broad qualifications do not bring us closer to a precise definition.
Even the sages of the Talmud spoke of various such attempts to base the whole Torah on a limited number of principles (e.g., Shabbat 31a, Makot 24a). But ultimately these principles, no matter how splendid and satisfying they are, cannot define Judaism.
Always a significant truth had to be added: that all this referred to matters of religious practice that belonged to a specific kind of Judaism.
In short, there was no way of getting to the essence of Judaism by any abstract definition.
Neither a minimalistic religious determination nor general humanistic qualifications had the capacity of defining the specificity of the Jews.
Such minimalistic definitions may be valued for all religions or for no religion at all, but they do not really define anything.
Therefore, too, the attempt to define the Jews as a religious entity cannot serve its purpose because the more detailed and specific the qualification of the Jewish religion, the less one finds Jews who follow it, while any attempt to define the religion as that which includes all or most of the Jews would be empty of all content, and certainly of 'Jewish' content.
The more basic issue here is the question of whether it is possible to call Judaism a religion in the ordinary sense of the word."
–Rabbi Adin Steinsaltz