Among the things that cannot be nullified when they exist in a mixture are things that are considered a biryah, or "whole entities". The Gemara in Hullin (99b) states this in reference to the gid hanashe, the forbidden sinew, in just two words - biryah shani - an entity is different (and not batel, nullified). The logic for this seems clear - some things are so significant in themselves that, as long as they remain whole, they never lose their identity, even when they exist in a much larger mixture. [The possible hashkafic implications of this should be clear, and Rabbi Blanchard spoke last Wednesday about one of them - that as long as we retain our integrity, and do not let ourselves be pulled apart in every which way, we will never lose who we are, regardless of what situations or circumstances we find ourselves.]
The key question is what are the criteria that determine if something is a biryah? This is debated by the Rishonim, with almost all of them turning to the discussion of biryah in another context -the rule that if on eats a whole bug, for example, one is liable for lashes even though he has not eaten the minimum quantity, a kizayit, the size of an olive (Makkot 17a). All Rishonim (except Rambam) assume that the definition of biryah is the same in both areas - lashes and resistance to bitul - but they debate what this definition is. Tosafot (Zevachim (72a), s.v. Vi'Livtal and Hullin (96a) s.v. Mey) states that the criterion for biryah is a function of how the Torah articulated the prohibition of the object. If the Torah said "Do not eat the gid hanashe" or "Do not eat a bug," then if one eats a whole gid or a whole bug, she violates, regardless of the size, because this is exactly what the Torah prohibited. If, however, the Torah said, "Do not eat neveilah, meat not slaughtered correctly," then since this does not describe a specific object, one only violates if one eats a minimum quantity.
Ramban, Ran, and other Rishonim disagree. They emphasize that an object must have an independent identity, which it would lose when divided, and that it can be numbered. So, a bug is a biryah, because half of a bug is no longer a bug, and 5 bugs is 5 bugs. This is different from neveilah, where half of a quantity or twice a given quantity, is just more or less neveilah, not half, or fewer, or many. (This criterion is also true for Tosafot, who focuses on the Biblical prohibition, since when the Torah forbids a "thing" qua the whole thing, it can only be talking about whole entities. However, for these Rishonim it is not a function of how the Torah has formulated the prohibition, but just of the thing as an entity). These Rishonim demand, moreover, that the thing still be whole - to retain its significance. They also add additional criteria for something to be have the special status of biryah - it must also: (1) have come from a live animal - fully inanimate entities are not as significant and (2) must have started its existence with its forbidden status - if the status came later it is considered an accidental quality, and not definitional.
The debate of these positions - is it based on how the Torah prohibited it, or on criteria that define the importance of the item - is really a debate about the relationship between the two halakhic areas where biryah is significant - lashes and resistance to bitul. Why does one definition apply equally to both fields? How do we understand when field A is linked to field B? On way to understand this is that A defines B - because one gets lashes for this bug, it is important in itself, and therefore cannot become batel. This is Tosafot's position - the starting point is the Torah's prohibition, and understanding when one gets lashes. The other possibility - in general - is that B defines A - because it is not batel, therefore one gets lashes. Not only does this not make sense, but it cannot be the case here since the resistance to bitul is only a rabbinic law, and the lashes is a Biblical law. This approach is understandably not adopted by any Rishon. The third possibility, however, is not that A defines B or B defines A, but rather that a third factor - C - determines both A and B. That is the approach of Ramban and Ran - the concept of chashivut, significance ("C") is determined by independent criteria, and then has an impact on two separate laws - (1) something qualitatively significant warrants lashes, although it is a small quantity, and (2) something qualitatively significant is resistant to bitul.
Shulkhan Arukh and Rema rule like Ramban and Ran, and require all the criteria mentioned above. Now, cases of biryah are rare, but there is one case which is very common and causes a lot of halakhic issues, and that is the case of bugs. If a bug is in my lettuce or broccoli, although there is more than 60 times the amount of broccoli to the size of the bug, it is not batel because a bug is a biryah - it is defined as such in the Gemara, and it satisfies all the criteria mentioned above. How, then, can we ever eat vegetables, if even with pesticides and the like, there is always a chance that there might be some bug hiding in the leaves?
This question is not new, and is addressed in Shulkhan Arukh (siman 84), and by many Achronim. Of course, what should be noted, is that many Achronim state that it is impossible to totally be free of bugs (this was before the age of pesticides, so the situation was even worse), and that at most only the most righteous of people refrained from eating vegetables. Rather than saying that everyone was sinning, they tried, when possible, to find ways to defend the practice.
The first question to ask is what are the chances that there are bugs present, and does one really have to check if most of the time there are not bugs? Halakha distinguishes between three possible scenarios - one where bugs are most likely present (huchzak bi'tolayim) - in this case it is considered like they are definitely there, and one cannot eat the vegetable without rigorous inspection. The other extreme is where it is only a distant concern that bugs might be present - the likelihood is miyut she'eino matzuy - a infrequent minority of cases. In such cases, one can assume they are not present, and does not even have to check - this is what we do with, say, cucumbers and tomatoes. Then there is the middle category, miyut hamazoy, a frequent minority of cases. Although there is a principle of following the rov, the majority, the Rishonim state that when there is a miyut hamazoy of problematic cases, we have an obligation to check if possible. This creates an obligation to check fresh vegetables that often have bugs to ensure that no bugs are present. While the exact line between a common and uncommon minority is not clear, most poskim use the definition of 10%, and require that vegetables be checked whenever there is a 10% or greater of the vegetable shows a presence of bugs.
Now, there are actually two ways a bug can be present in a vegetable. It can be readily discernable to the naked eye - as such, it is nikkar, discernable, and not batel even Biblically. Bitul only works when the prohibited item is lost and cannot be identified, not when it can readily be seen. The other scenario is that it is not obvious, it is hidden in the broccoli, say, but it exists there as a whole entity - as a biryah. In this case, it is batel Biblically, but not rabbinically. Thus, for leafy vegetables, such as Romaine lettuce and spinach, etc., where there is a miyut hamazoy of occurrence of bugs, and, when inspected, bugs can be seen on the leaf, one must check to make sure there are no bugs. To do otherwise would risk violating the Biblical prohibition since such bugs would not be nullified even according to the Torah. However, for cluster vegetables, such as broccoli and asparagus, the bugs cannot be seen without pulling them apart, soaking them, thrashing them in water, and the like. Under such concerns the bugs are not nikkar and we only have the Rabbinic a problem of biryah.
Is there a way to limit this rabbinic problem of biryah? Arukh HaShulkhan suggests a few, all based on the principle that biryah, at the end of the day, is based on something's chashivut, significance, and that that can be undermined by various factors. So, says Arukh HaShulkhan, there are opinions that when biryah is only the tiniest part of the mixture - less than 1 in 960 or 1 in 1,000 - that it is totally insignificant and it is batel. There are also opinions that if the entity never appears by itself, but always on the host vegetable, it loses its independent significance. And, finally, his own opinion, that when the thing has a terrible taste, although it is a natural taste, it loses significance and can become batel. R. Shlomo Zalman Aurbach adds in a similar vein, that in a case where the bug cannot be seen by natural light, it is not significant, and can be batel. Both Arukh HaShulkhan and R. Shlomo Zalman Aurbach state that while such a bug satisfies the criteria of biryah for lashes, nevertheless, the principle of resistance to bitul is not directly based on lashes, but based on significance, and that can be undermined by the factors mentioned above. This is completely in line with the Shulkhan Arukh 's ruling that biryah is based on independent criteria of chashivut, and not on the criteria of lashes per se.
There is an easier way to be lenient here. The fact that this resistance to bitul is only rabbinic allows for leniencies in the case of doubt. Since there is a doubt if there are bugs there, and since biryah is a rabbinic concern, we can be lenient and allow these cluster vegetables without checking (assuming there is no reason to be concerned for infestation, which would make it muchzak bi'tolayim). This is the position that the Star-K, under the authority of Rav Heinemann, took for many years, and only changed recently to adopt the same standards as other kosher industries. This is also the position that I personally follow and rule for others, which allows people to eat vegetables without onerous demands and with full halakhic observance.