(1) First, we need to know what we're talking about, and that's less easy than it sounds. What is Cynicism? We face a number of problems in answering that question; they are not sufficient to prevent us from answering it, but we need to keep in mind the obstacles to confidence here. The first is that we have relatively little information about the Cynics; the second is that most of what we do have is fragmentary and late; and the third is that the Cynic became a common archetypal figure.
We do not even know for sure why they are called Cynics. The Greek word Kynikos means doggish. There are conflicting stories about why they were given this name. On one story, it is because they began with the school of Antisthenes -- Antisthenes was one of the students of Socrates -- who is said to have taught in the Cynosarges. We don't know for sure what 'Cynosarges' meant, but obviously it looks like it has the 'dog' root. According to another story, Antisthenes was nicknamed Haplokuon, which means something like 'plain dog'. Again, for reasons we don't know. According to another, and much more famous, idea, it became attached to Diogenes of Sinope, supposedly a student of Antisthenes, because of his dog-like (i.e., shameless) behavior. The reason it's worth pointing this divergence out is that unlike other movements we don't know how it became a movement. In addition, you'll notice that all three stories end up linking the Cynics to Antisthenes -- about whom we mostly know only very indirect things, outside of some stories told by Xenophon -- and thus to Socrates. And one of the things that Stoics will later do is use the Cynics as one of their links to Socrates, through Antisthenes; any of the three might have arisen to support this link, so we can't even lay foundations with regard to the Cynics without worrying about Stoic mediation long after the fact.
Most of what we know about the Cynics comes through very late sources, the most famous of which is Diogenes Laertius (c. AD 300). Diogenes Laertius himself is difficult to assess; sometimes he is extremely reliable and sometimes extremely unreliable, depending on his sources. There still seem to have been Cynics around, in the sense of people calling themselves Cynics; but it's difficult to assess exactly how they relate to their forebears. We should be cautious about this. Some people calling themselves Cynics may well have just called themselves such on the basis of common stories about Cynic philosophers, in the way that people calling themselves Socratic today are not part of any ongoing Socratic movement, but just people who, finding ideas associated with Socrates in some medium or other, just start calling themselves Socratics. There's a perfectly legitimate sense in which they are what they say, but it's a very different kind of historical association.
Likewise, people who are called Cynics may or may not always have been Cynics; just as, if I were to call a friend 'John the Stoic', that could mean anything from his having fully imbibed Stoic ideas to his reminding me vaguely of what I've heard about Stoics. This is related to the fact that the Cynic was often used as a figure of example in philosophical, especially Stoic, arguments and discussions. Epictetus's Discourses, Book 3, for instance, which discusses Cynicism, is a philosophical argument in which the Cynic functions more as a type than as any historical description. A number of our other sources for Cynicism are Stoic, like Musonius Rufus, and in each case they aren't just doing history but arguing for a philosophical position. In a similar way, Julian the Apostate has two Orations (the Sixth and the Seventh), from which bits and pieces can be gleaned, but he argues that those in his day calling themselves Cynics do so without proper regard for their predecessors in the name, and that the basic principles of the early Cynics are in fact nothing other than universal philosophical principles in a particularly obvious form. Beyond that we have some anonymous works -- the Cynicus that used to be attributed to Lucian of Samosata and the Cynic Epistles. All of these together, including, interestingly, the Cynic Epistles just on their own, also give us a sort of schizophrenic Cynicism: Cynics are characterized in entirely inconsistent ways. At one point they'll be characterized as ascetical, at another as hedonistic; at one point their primary characteristic is said to be harshness, and at another their primary characteristic is said to be gentleness; at one point they'll be characterized as pure philosophers, at another as ignorant people trying to cover their stupidity with a name. Sometimes (as in Julian) the inconsistency seems to be historical, in that one form of Cynicism is treated as untrue to what Cynicism was (the historicity of which is not always clear); sometimes (as in the Cynic Epistles) the inconsistency seems to arise from traces of an ongoing debate about what Cynicism was in the first place.
None of this is to say that we can say nothing about the Cynics; we have plenty we can say, but when we do so we must keep in mind that we are always doing a fair amount of selection and reconstruction from the evidence. And on this regard, it does not seem actually to matter much how carefully we date things -- even if we're dealing with contemporary sources, which we can manage to an extent when talking about the first century, we can't always take them at complete face value -- and if we took them all straight we'd have an incoherent mess. By all means, let's talk about the Cynics; but let us do so with caution.
(2) What are some reasons why one might link the early Christians to Cynic movements? There seems to have been a revival of Cynic groups in the period, and there were people calling themselves Cynics all over the Empire, so there's the right kind of temporal and geographical contiguity to make influence possible. It's difficult to pin down what Cynics were doing in Galilee, but this is not in itself a serious problem. Geographical mobility in the Empire, while not always easy, was fairly common -- perhaps the most famous example is the missionary journeys of Paul. Groups, movements, and ideas diffused fairly easily through the Empire. So the remote conditions for influence are actually fairly promising.
One common line of argument is based on reconstructing Q. Matthew, Mark, and Luke, the Synoptic Gospels, have a significant amount of overlap. One of the major candidate explanations of this is the Four Source Hypothesis, that Matthew and Luke shared two sources: Mark and another document, no longer extant, that is usually given the label Q. If you operate on this hypothesis and take Q to be what Matthew and Luke share that is not in Mark, you get a lot of sayings in loose narrative frameworks. Assuming (as the hypothesis would make probable) that this was, more or less, what Q would have actually been like, you can ask, were there any other kinds of works that exhibit a similar format or structure? And one possible example is the chreia. A chreia is essentially a pithy saying set in its occasioning circumstances. There are lots and lots of these chreiai devoted to Cynics; indeed most of what we might know about Cynics themselves, as opposed to about ideas attributed to them, comes from exactly such stories. Chreia-style anecdotes are fairly simple, whereas Q seems to have more elaborated stories, but it isn't unheard-of for people to take a chreia and expand it or develop it, and certainly at some point people started putting Cynic chreiae together in order to get Cynic Lives.
None of this on its own is very strong. You could, thus far, compare Q with the Analects of Confucius, if only China were just as close geographically: story-forms of this kind can spontaneously arrive. In the Roman Empire they were specifically cultivated, to be sure, but this is as much a problem as anything. Writing or developing chreia was a standard rhetorical and literary exercise; virtually anyone capable of writing a long work in Greek would have deliberately written or developed chreiai at some point as part of the usual practice in polishing their Greek. It would be extraordinarily stupid just to find chreia-like things and conclude immediately that they were signs of Cynic influence. So we'd need more, and it would have to do with content. Many of the stories in Q exhibit denunciation; denunciation is a common means by which Cynic parrhesia or frankness was expressed. One thing that might possibly be used to distinguish a Cynic idea from a Stoic one is the radicalness of the former (Stoics tend to think in terms of orderly, systematic change rather than sudden conversion or radical transformation); and certainly Q seems to insist on radical transformation. One position associated with the Cynics is the idea that poverty or simplicity is happy or blessed, makarios, which we certainly find in Q. Indeed, if you take the Beatitudes, you could give a plausibly Cynic reading of each one. One of the loose parallels with the Golden Rule is found in a chreia about Diogenes preserved by Stobaeus; asked how to master himself, Diogenes replies that one should reproach oneself with the same things with which one reproaches others. The list could be extended.
The argument through all of this is not that Q is just a Cynic Life, or that all the elements of Q are found in Cynic Lives; there are themes that clearly are not. But if you were to ask what other ancient works Q is most like, in format and content, an obvious answer would be the various works devoted to the Lives of the Cynics.
Of course, one problem arises immediately here, namely that the argument depends on comparing a reconstructed first-century Cynicism with a reconstructed Q document. I've already mentioned the cautions required by the first. Downing is actually quite good at navigating the potential dangers on this front. The other problem, of course, is Q itself. While the Four Source Hypothesis is still quite popular, if we were to switch to another major candidate solution for the Synoptic Problem, the increasingly popular Farrer Hypothesis, Q vanishes, because on that hypothesis Matthew used Mark and Luke used Mark and Matthew -- there's no need for a Q at all, since all the commonality attributed to Q is just in Matthew. And the Gospel of Matthew is not so obviously like a Cynic Life. This is not fatal for the argument; one of the things Downing is careful to do is to show that there is still quite a bit of resonance between Cynicism and themes and ideas found in Matthew, Mark, and James. After all, it is logically necessary that some of the possible content echoes in Q be found in Matthew! Nonetheless, it is true that the case is weakened if we remove Q from the equation.
So far, then, we have temporal contiguity, geographical contiguity, similarity of some content, and (possible) similarity of genre. One possible worry is that all this similarity may have more to do with how the stories were written up than with Jesus and his disciples being influenced by Cynics -- the similarity must be more than could be attributed just to the influence of Cynic genres on Greek literature and culture as a whole. Thus it's important for Downing to build up the number of parallels, which he attempts to do in Christ and the Cynics and Cynics and Christian Origins.
(3) All of this is good as far as it goes. But the points at which we are dealing with less than certainty are adding up quite quickly. One can worry that the Cynics are amorphous enough that it is a mistake to think of them as a unified movement capable of serving as a causal mediator in our explanation. We are picking and choosing what counts as Cynic. To be sure, what we are picking and choosing is plausible and fits the evidence well, but we could choose rather different things, as well. We run into the worry here that the parallels are due in great measure to our selecting out the aspects of what we know about Cynicism that fit with Christianity. We have to face the worry that a lot of the parallels on the board are fairly generic, and can be found even among non-Cynics. The chreia-like format of Q is perfectly explicable without appealing to Cynic influence, assuming there was even a Q. The obvious context of Jesus and his disciples is Jewish, and while this doesn't rule out Cynic influence and Downing is careful to argue that genre-wise Q and the Gospels are quite a divergence from common Jewish writing, one immediately has to face the worry that this difference has as much to do with the fact that our Gospels are being written in Greek, in a broader Empire culture, as with any actual influence. And there is a difference between Cynic genre and genre by which we know something about the Cynics -- it is true that chreiai, for instance, are often about Cynics, but we have only limited information on how these anecdotes would have functioned for Cynics, and we know that many of the philosophers transmitting such information about the Cynics are not actually Cynics. Barring a few works, most of what we know about the Cynics comes from Stoics or people obviously influenced by them. And the Stoics raise another worry. We have independent reason to think that Stoicism, a far more widely influential philosophical school, had a definite influence at least on early Christianity (e.g., Paul adapting vocabulary and forms that were common among the Stoics), so some of this supposed Cynic influence may just be Stoic influence on how the writers portrayed their subject. Other worries could be, and have been raised.
The problem is not any one of these individually. The problem is that there are so many points at which one can raise this kind of worry. This has led to a fairly common accusation against the Cynic hypothesis, namely, that it is irrefutable, not because it is solidly established, but because there are so many little parts that can be tweaked that you can force it to fit any evidence short of a text by Jesus being asked his views of Cynicism and replying, "What's Cynicism?" Actually, you could even make it fit that, since obviously Jesus could be influenced by Cynicism without either him or his disciples knowing that it was Cynicism he was being influenced by. Wax noses are hard to break; that's probably good with real wax noses but it is a flaw with the figurative ones.
A good example of this problem arises when we consider one argument that has been discussed at some length. Here is Mark 6:7-11 (NIV):
Calling the Twelve to him, he began to send them out two by two and gave them authority over impure spirits.
These were his instructions: “Take nothing for the journey except a staff—no bread, no bag, no money in your belts. Wear sandals but not an extra shirt. Whenever you enter a house, stay there until you leave that town. 11 And if any place will not welcome you or listen to you, leave that place and shake the dust off your feet as a testimony against them.”
This has parallels in Luke 9:1-5 and Matthew 10:1,5-14; some have argued that the story is both in Mark and Q. Regardless, one of the things critics have pointed out is that the begging bag is commonly associated with the image of the Cynic. Downing replies to this that it was by no means universal. And that is true. But the form of argument here is actually the same form Downing relies on; he's constantly saying that this or that is such that the reader would have naturally taken it to be Cynic. It is based on the image of the Cynic. And what the critics are pointing out is that you can find examples such that it would have clashed with any such expectations of what a Cynic would be. Finding parallels, Downing ignores the diversity of the Cynics and the fact that occasionally contradictory things are said of them; faced with a counter, he relies on the diversity. But we need to consider not only the Cynic-like features, but also the things in the text that are very un-Cynic-like. Downing is not afraid to admit such elements -- but he simply doesn't do what is required to take them into account in the argument itself.
(4) These are only a sample of the arguments on the table. It would be interesting if we could put Cynicism alongside Stoicism and Middle Platonism as philosophical influences on the early Christian community as manifested in the New Testament. I would say that Downing has done the service of bringing to the fore quite a few interesting parallels; this is enough to raise the idea of Cynic influence here from a mildly interesting idea to a genuinely intriguing one. But it seems quite clear that it falls short of doing anything more.
We can see this if we look at it the opposite way, as a question in the history of philosophy. Given the parallels, contiguity, and the like, would this suffice to treat the relevant New Testament texts as sources of information about first-century Cynicism? And the answer is quite clearly that it is not. What is more, the hypothesis doesn't actually seem to explain much that is not explicable on the basis of independently known causal factors -- Judaism, broader Greek culture, and the like. From a HoP perspective, the genre argument is weak by its very nature, and content parallels in the history of philosophy are often results of convergences in environmental factors rather than direct causal link, and there are enough other possible causal routes involved here that even if there is some kind of influence, it may be too farflung or tenuous to trace. This is especially true given how diverse the Cynics seem to be and how secondhand and contradictory a lot of information about them is. The hypothesis merely posits a fact of influence and not a clear means thereof; there's nothing wrong with this, but it has the implication that even if true it couldn't tell us much about the diffusion of Cynic ideas. The argument largely exhausts itself in establishing its point and we don't get much real illumination on either Cynics or early Christians from it. The parallels remains rough parallels and there's not much more to say. Moreover, the Cynics are so diverse a group, and our information about them so stylized, that while it's worth further inquiry, there is a real possibility that this is as far as we can get on the point. In HoP, since we aren't historians but philosophers using historical methods, we might posit the influence as an abstract exercise just to see the results, without worrying too much about whether we were dealing with history or reasoned counterfactual speculation, and the possibility of historicity would certainly be worth a footnote or comment on occasion. But this is a long way from the sort of historical claims proponents of the Cynic hypothesis are making, which don't seem to handle the Cynics as a philosophical movement very well, and seem to keep our understanding of the Cynics and the early Christians largely where they were before.