Before any theory of evolution is worth stating, it is worth being precise about what the theory has to explain, because the usual summary leaves out half of it.
This course assumes no biology beyond school level. It does assume you are willing to treat a two-hundred-year-old argument as an argument rather than as a mistake, because the case for design was made carefully, was believed by careful people, and was not defeated by ridicule.
The watch on the heath
In 1802 William Paley opened Natural Theology with a thought experiment that everybody remembers and few people state fairly. Crossing a heath, you strike your foot against a stone. Asked how the stone came to be there, you might reasonably answer that for all you know it had lain there forever. Now suppose you find a watch. The same answer will not do, and Paley is careful about why.
His reason is not that the watch is complicated. It is that the watch's parts are put together for a purpose, and that the purpose fails if the arrangement is disturbed. The spring is coiled to store force, the train of wheels is cut to transmit it at a rate, the escapement releases it in equal beats, the hands are geared to the beats. Change the tooth count on one wheel and the watch does not tell slightly worse time; it tells no time at all. Paley's inference is from the co-adaptation of parts to an end, and the strength of the argument lies there.
He then argues that eyes are worse for his opponent than watches. The eye has a transparent cornea in front and a light-sensitive layer behind, at the distance that brings a distant object into focus. It has a lens whose refractive index changes from the outside in, correcting an aberration that a uniform lens would show. It has an iris that opens and closes with the light, a duct that keeps the front surface wet, and a lid that wipes it. Fish, which look through water, have a nearly spherical lens with a much higher power, because water and cornea refract almost alike and the cornea does no work. Paley noticed that too, and treated it as a designer adjusting an instrument to a different medium.
Do not be tempted to answer that the eye is imperfect. Paley knew it was, and his argument does not need perfection: an old watch that gains five minutes a day is still obviously a watch. The argument needs only that the parts are arranged for a function they would not perform if arranged differently.
Example. State the logical form of Paley's argument in three lines, and identify which line a critic must attack.
Premise one: objects whose parts are co-adapted to an end are produced by an intelligence that intended the end. Premise two: organisms have parts co-adapted to ends. Conclusion: organisms are produced by such an intelligence. Premise two is a straightforward observation and is true; anyone denying it has misunderstood the target. So the whole weight sits on premise one, which is not an observation at all but a claim that intelligent design is the only process that produces co-adaptation. It is a claim about the space of possible causes, and it can only be defeated by exhibiting another cause that does the job. That is exactly what the rest of this course does, and it is why "the eye is badly built" is a bad reply and "here is a process that builds eyes" is a good one.
Now you. Paley writes that the argument is not weakened if the watch sometimes goes wrong, nor if there are parts whose use you cannot make out. Why does he add the second point, and what would genuinely weaken his case?
Answer
He adds it because the obvious counterattack is to point at some organ nobody can explain and claim it shows there was no plan. Paley's reply is that ignorance of a purpose is not evidence of the absence of one, which is fair. What would genuinely weaken the case is a part that is co-adapted to an end and whose particular form is what an unguided history would leave behind rather than what a designer would choose: not a useless part, but a working part built the awkward way a modified inheritance would build it. Later lessons produce those, and it matters that the challenge is this specific.
How good is the fit, in numbers
"Looks designed" is not a measurement, so take one. The human retina holds roughly 1.2 × 10⁸ rods and 6 × 10⁶ cones, and the optic nerve leaving it carries about 10⁶ fibres, so the eye compresses its input by more than a hundredfold before sending it anywhere. At the centre of the fovea, cones are packed at about 2.4 μm between centres, which corresponds to about half an arcminute of visual angle. Measured acuity for a person with good sight is one arcminute, which is what that spacing allows and no better. The optics and the sampling are matched to each other.
That matching is what needs explaining, and it recurs everywhere you measure. The bones of a bird's wing are hollow with internal struts, which is what an engineer does when stiffness matters and mass is expensive. Haemoglobin binds oxygen cooperatively, so that its loading curve is steep exactly across the range of partial pressures between lung and tissue rather than across some other range. None of this is vague. Each is a quantitative fit between a structure and a job.
How much change, and how slowly
Since Paley's premise one is a claim about what unguided processes can do, the honest question is how much accumulated change an eye actually requires. In 1994 Dan-Erik Nilsson and Susanne Pelger built a deliberately pessimistic model. They started with a flat patch of light-sensitive cells backed by pigment, and allowed only small changes: deepen the pit, narrow the aperture, let the jelly filling it acquire a gradient of refractive index. At every stage they required the structure to be an improvement in spatial resolution over the one before, and they took each step to be a 1 per cent change in whatever dimension was changing.
Example. Nilsson and Pelger found their sequence needed 1,829 steps of 1 per cent. By what total factor does the structure change, and if the whole sequence takes 364,000 generations, what is the average change per generation?
Each step multiplies by 1.01, so the total factor is . Taking logarithms, , and , a change of about eighty million-fold. Spread over 364,000 generations, the per-generation factor is , which is : five thousandths of one per cent per generation. For an animal breeding once a year that is 364,000 years, which is a geological instant.
Now you. Suppose the same 1,829 steps were spread over 36,400 generations instead. What is the change per generation, and does the answer change the force of the argument?
Answer
Ten times fewer generations means ten times the change in each: , or five hundredths of one per cent per generation, still far below anything a breeder would struggle to achieve. The force of the argument does not come from the exact number. It comes from the fact that the required per-generation change stays negligible across two orders of magnitude of assumed timescale, so the conclusion is not sensitive to the guess. What the model does not show is that this is how eyes actually evolved. It shows that time is not the obstacle, which is a narrower and more defensible claim than it is usually reported as.
The second fact, which design does not predict
Now the half that is usually left out. Alongside the fit of organisms to their circumstances there is a second, quite different regularity, and it is the one that actually decided the argument.
Pick up the forelimb of a human, a bat, a whale and a mole. The jobs could hardly be more different: manipulating, flying, swimming, digging. Yet all four contain one bone in the upper segment, two in the lower, a cluster of small bones at the wrist, and five rays beyond, in that order, connected the same way, developing in the same sequence in the embryo, and supplied by nerves that leave the spinal cord at the same levels. Richard Owen, who was no evolutionist, named this in 1843: homology, sameness of structure regardless of function, as against analogy, sameness of function regardless of structure. A bat's wing and a bird's wing are analogous. A bat's wing and a whale's flipper are homologous.
Homology is not what a designer optimising each animal for its job would produce. A dolphin has no use for five separate digits welded inside a flipper; a mole's hand would be better as a single blade. The constraint is not functional, which means it is historical, and it is the sort of constraint you get from copying an existing design rather than from choosing a good one.
The pattern is nested, which is the strong claim
The deeper point is not that similarity exists but that it is arranged in a very particular way. Species share characters in groups within groups, never in overlapping sets. Every animal with mammary glands also has three middle-ear bones, a single lower-jaw bone, a four-chambered heart and hair. Every one of those with a placenta also has the mammary glands, and so on inward. You do not find a group that shares feathers and mammary glands and nothing else, or one that shares the mammalian ear with the reptilian jaw and picks its lung type from a third group.
Linnaeus imposed this arrangement in 1735 because it worked, and he had no theory of why it should. It is worth seeing how strong the claim is. Nothing forces a set of characters to be nestable. Twenty objects can be arranged into 2.2 × 10²⁰ distinct branching patterns, so an arbitrary set of characters spread over twenty species would almost certainly disagree about which pattern to draw. Living characters agree, over and over, drawn from anatomy, embryology, biochemistry and, later, sequence. That agreement is a fact that any theory has to explain, and design does not predict it: a designer reusing good solutions would be free to give the whale's flipper to the fish and the fish's fin to the whale.
Example. Three characters are scored across four species. Species A, B and C have hair; A and B have a placenta; A, B, C and D all have a backbone. Is this set nested, and what would a non-nested set look like?
It is nested. Backbone covers {A, B, C, D}, hair covers {A, B, C}, placenta covers {A, B}, and each set sits wholly inside the one before, so the three characters draw a single consistent hierarchy. A non-nested set would be one where hair covered {A, B, C} and some fourth character covered {C, D} while a fifth covered {A, D}: those overlap without containment, and no single branching diagram accounts for all three. Real biological characters overwhelmingly behave like the first case, and that is the observation, not the theory.
Now you. Wings occur in bats, birds and insects; the character "has wings" therefore covers a set that is not nested inside the mammal set or the bird set. Does that break the pattern?
Answer
No, and seeing why is the whole skill. "Has wings" is a functional description, not a structure. Looked at as structures the three are different objects: a bat's wing is a hand with skin stretched between elongated fingers, a bird's wing is a forelimb with feathers on a fused hand, an insect's wing has no bones at all and is not a limb. Coded as what they are made of rather than what they do, each falls inside its own group and the hierarchy is intact. Conflicts of this kind are the normal difficulty in reading the pattern, they are called convergence, and later lessons meet cases where distinguishing homology from convergence is genuinely hard rather than easy.
What a rival theory has to supply
So there are two explananda, and they pull in different directions. The first is adaptation: the quantitative fit of structure to function, which really does look like the product of a designer and which Paley stated better than anyone. The second is the nested pattern of homology, which looks nothing like a designer's work and everything like a genealogy.
A rival to Paley therefore has a specific job list. It must produce co-adaptation of parts without foresight. It must produce it in a way that leaves a nested pattern rather than an arbitrary one. It must account for organisms carrying structures that are useless or awkward for their present job but make sense as inherited equipment. And, since about 2.1 million species have been described and the total is estimated near 8.7 million, it must generate that number of distinct kinds from however many it started with.
It must also be cheap in one specific currency. Any process working by small accumulated changes needs time on a scale nobody in 1802 had any reason to grant. Paley's world was a few thousand years old. Whether the earth could afford the 364,000 generations that even a pessimistic model of an eye demands is not a biological question at all, and the next lesson goes to the rocks to settle it.