The argument of design has several features, keying on relative complexity and tuning for use, from which one infers a designer. If we find a sand castle on a "simple" beach, which had an unusual degree of structure, and operating drawbridge and gates, its relative complexity, plus tuning for an apparent use, would lead us to infer a designer. Paley's watch, and a 747, have been prior prototypical examples of reason to infer design. But these human creations are far less complex than what we see surrounding us such as the human body or the web of life. If a far simpler object such as a robot can reasonably be inferred to not be a product of UNINFLUENCED natural phenomena, then would not something as complex and tuned as a human likewise be reasonably inferred not to be the product of natural events? Likewise, our universe appears to be remarkable in its tuning to produce life. Both of these inferences imply an intelligent creator! So what is the alternative? How do those who reject design like Richard Dawkins rebut the argument of design?
What makes the argument of intelligent design unable to stand on its own legs is that simplicity and efficiency, are hints of design, not complexity. The useless complexity of living forms, the flaws in their body are all hints at a natural, trial and error type of development.
A common example is the incredibly bad design of the human eye: photoreceptors are mounted inward, opposite to the side the light is coming, greatly reducing their efficiency. The optical nerve creates a blind spot that could have been avoided, and a significant part of the brain is dedicated to process this mess that wasn't necessary in the first place.
Some fishes, like squids, don't have this flaw, which shows that a better design was possible in the first place. Why not apply the best design to every creature ?
Compare with a watch, within which each piece has a well defined goal, is tailored precisely to accomplish this goal and nothing more with the least possible amount of material.
Anything more than a very superficial look at biology hints that life is not designed, or if there is a designer, it is not very intelligent.
Therefore a simple defence against the argument of design is to reject the premise that life appears to be designed.
The question is a little problematic because Atheism is a position on a single question. If you believe that a god exists then you are a theist, if you don't, your an atheist. Atheism has no dogma and makes no proclamations of any kind. Thus it needs no defence against arguments of design.
Also, having to defend Richard Dawkins is a little weird. I can try and explain what I think he meant though.
Now to the question itself, if complexity requires a designer, then the mind of the designer requires a designer. So all you get is an infinite regress that doesn't actually explain anything. Because if you start by saying that everything other than god requires a designer, then you are already begging what you are actually trying to prove. Which is by definition, circular.
In addition, we do not recognise design by complexity. We recognize it by contrasting it with the natural world and especially with things that arise naturally. Lastly, a lever is by no means a complex machine, but it's definitely designed and used by humans (and by a few other animals as I recently saw on the discovery channel).
You have mis-stated the Argument From Design. It is only weakly based on complexity, but instead is based on intentionality. If an object is most validly explained as a product of intentional design, then there is good reason from the existence of that object to infer a designer.
Complexity is only relevant in that it triggers a search for the understanding of the source of that complexity, because most things made by natural processes are not very complex. SOME objects we know are designed (baseball bat for example) ARE simple, and SOME natural objects are complex (see snowflakes) so simple/complex is only an indicator/trigger, not the actual condition that leads to a design inference.
The inference to design is a legitimate scientific reasoning process, and is a key feature of anthropology, criminology & forensic pathology and the SETI program.
Paley's Watch was a complex object, featuring aspects NOT found naturally at least in our experience (toothed gears, assembled movements, etc), but which ARE found in our intentionally constructed objects. Therefore, his inference that the watch was designed is entirely justified.
Life -- we don't have examples of it being designed, so the inference to a designer is far weaker. And -- there are so many oddities in life that don't fit with an intelligent design (viruses, poor organ/chemistry design elements, overpopulation/famine cycle for ecosystems etc), that design is -- suspect. When another mechanism was discovered -- natural selection -- which over sufficient time produces "designed like" features for any inherited property subject to variation and selection, then the inference to design for most features of life -- is no longer credible.
There remain three areas where an appeal to design may still be valid. The APPEARANCE of life has not been shown to be possible despite 3/4 of a century of abiogenesis research (see Freeman Dyson's The Origins of Life for a good discussion). The apparent purposefullness of evolution to create morality and reasoning implies design to Thomas Nagel (see Mind and Cosmos). And the Fine Tuning features of our universe (each of something like 30 parameters have to be in logically independent unlikely ranges for any kind of Life [not limited to organic carbon/water life] to be viable anywhere in a universe) make its intentionality for Life a plausible explanation.
The Fine Tuning argument is the strongest of these three, and has sparked a lot of speculation about natural mechanisms that could produce Apparent Fine Tuning. The current popularity of Multiverse hypotheses among cosmologists and lay people alike is because this provides a possible alternate method to arrive at Near Fine Tuning, by extending the timeline of a metaverse, generating variance, and then selecting from that variance. As an alternate explanation for Fine Tuning, it is logically possible. There has been a lot of push-back, however, from scientists and philosophers of science who point out that Multiverse plus selection is unsupported, and more importantly is UNTESTABLE (and therefore "not even wrong" as a science claim). Theist theologians can also point out that it is also an astonishingly complex hypothesis, which relies upon infinities. Complexity, untestability, and infinities used to be pointed out by non-theists as major weaknesses of theism.
Non-theists can point in each case to similar non-optimizations in these three areas as was seen for life. In Fine Tuning, our universe may be very unlikely, but it isn't anywhere close to well designed for life. In emergence of life the lateness and very local nature of that emergence suggests that life would be more of an afterthought than an original intention. And for morality and reasoning, the afterthought is even more pronounced, as reasoning took 4 more billion years to manifest, and is not present in most of life. There are therefore major weaknesses to an inference to design for each of these areas -- and the flaws in non-design explanations offered to date do not therefore lead to a justified default acceptance of a design explanation.
The possibility of our universe having been designed, to produce life, moral value, or intelligence, is therefore currently an open question.
The best way to know how Dawkins responds would be to read his book The God Delusion. As probably the most militant of the New Atheists, his argument is spelled out absolutely clearly. Part of his argument is to explain what science is and why intelligent design (ID) gets it wrong, particularly in Chapter 4, "Why There Almost Certainly Is No God".
The argument you seem to be making is the idea that because human devices, such as a Boeing 747, are less complex than the human body, that it stands to reason that someone must be responsible for its design and construction in the same way a Boeing 747 wouldn't spontaneously evolve from the side of a mountain. This is a fair question, but to understand the response, it's hard to understand if you don't understand some key tenets of physics, chemistry, and mathematics. Even the famous Fred Hoyle who was very sophisticated mathematically and scientifically got it wrong, according to Dawkins.
The phenomenon of the snowflake is familiar and well-understood. One could ask, how does a snowflake at the North Pole know how to become six-sided when it can't directly interact with a snowflake in the South Pole? And how does it go from a highly disorder mass of liquid water to a highly complex and ordered solid structure? Surely, it's more complicated than the simple art of children! We must conclude that a crystal of that complexity must have had an unseen artist. But, proper knowledge of thermodynamics and molecular geometry explains how systems move from lower complexity to higher complexity independent of intentionality.
First, under certain environmental circumstances, when a system suffers a net loss of order, parts of it can actually become more ordered in a move towards establishing dynamic equilibrium. This is called negative entropy. For instance, as water cools, the hydrogen bonds begin to play a role in forming a rigid molecule. In fact, the geometry of a snowflake is related to the tetrahedral shape of H20. This is vastly simplified but is a scientific fact.
So, clearly, if it is a fact that water and oxygen can spontaneously order itself, then other molecules follow. We arrive at Miller-Urey which shows that negative entropy exists for the organic molecules which are the basis of life. This sets the stage for abiogenesis, the origins of life from inanimate matter. From there, onto nucleic acids, likely RNA, which self-replicate and micelles which foreshadow cellular walls. It's not much of a stretch to get simple biological organisms of archaea from there.
And of course, natural selection explains nicely and neatly how random deviations in genetic material from genetic drift and other causes can continue the accumulation of biological structure resulting in very sophisticated anatomical and physiological elements. There are gaps in our knowledge, but those gaps continue to shrink and practitioners of artificial life have already engineered artificial genes de novo.
Ultimately, in skepticism, the burden of proof is put on the claimant, and so really the question is a debate of metaphysical positions. If one claims that there is supernaturalism, magical beings like gods and angels who can cast spells and defy physics, then the claimants must provide proof, and there is no scientific proof of such beings that can't better be explained by naturalism. So, those pre-Socratics who rejected the gods as myth and superfluous to the world have been validated by science, which offers better explanations for the nature of mythology by way of anthropomorphism and other aspects of evolutionary psychology.
The problem with intelligent design — and please note that I'm neither a theist nor an atheist, so I don't really have a dog in this fight — is that it has a '+1' issue. Intelligent design tries to take a middle road, allowing for all of the findings of evolution science on the assertion that some divine intelligence designed it to be so. But if some designer designed the universe to behave in the systematic, law-like manner that evolution presupposes — species evolving blindly to achieve the current world that the designer ostensibly envisioned — then what actual purpose does such a designer have? The designer is a definite '+1'; the universe would behave exactly the same way with or without a designer. Further, assuming that there is some designer, what makes us (in our temerity) assume that we are the ultimate outcome? How do we know that the design doesn't involve us wiping ourselves out so that a far more intelligent and wiser species of parakeets, octopi, or roaches — the true design the designer was reaching for — will come to ascendency. The ID argument is largely self-defeating.
I'm not averse to ID: with a little squinting it starts to look like some Eastern philosophy (karmic or daoist principles of nature). But the more 'intelligent' one presumes that designer is, the less that intelligence seems to matter in the big picture.