Infinite regress



Big fleas have little fleas, Upon their backs to bite 'em, And little fleas have lesser fleas, and so, ad infinitum. An infinite regress or homunculus fallacy is when an argument relies on a series of never-ending propositions, where the validity of one proposition depends on the validity of the one which follows and/or precedes it.

The infinite regress is a close sibling of circularity, wherein the premises provide support for the conclusion, which in turn provides support for said premises to begin with, which in turn…

Why it's logically fallacious
Neither can there be a separated infinite number: for number, or what has number, is countable, and so, if it is possible to count what is countable, it would be possible to traverse the infinite. Aristotle posits an argument that shows an infinite regress to result in a contradiction. Formed using predicate logic, the proof reads like this:


 * 1) Nx⊃Cx
 * 2) Cx⊃Tx
 * 3) ∴Nx⊃Tx (1, 2 hypothetical syllogism)
 * 4) Ix⊃~Tx
 * ∃x(Nx·Ix)
 * Na·Ia
 * Na (6, simplification)
 * Ta (3, 7 modus ponens)
 * Ia (6, simplification)
 * ~Ta (4, 9 modus ponens)
 * Ta·~Ta (8, 10 conjunction)
 * 1) QED ∃x(Nx·Ix) results in a contradiction
 * 2) ∴There does not exist a number that is infinite

Let's define the terms. Nx⊃Cx reads "if x is a number, then x is countable." Cx⊃Tx reads "if x is countable, then x is traversable." Ix⊃~Tx reads "if x is infinite, then x is not traversable." ∃x(Nx·Ix) reads "there exists an x such that x is a number and x is infinite," and is a supposition for the sake of argument. Now, 'countable' and 'traversable' need to be defined. Aristotle regarded numbers as made up of composite parts. If Aristotle had thought of the number 42, he would have thought that it was composed of 42 individual parts. This is what he means by 'countable'. 'Traversing' is the act of counting. So, if a number is countable, then counting the individual parts and finally reaching the number is traversing, which means the number is traversable. Aristotle says that if a number is truly infinite, it can't be traversed because the end of the number can't ever be reached. Given the definitions of the terms and the logical validity of the argument, Aristotle concluded that there exist no infinite numbers.

Intelligent design
One example of a viciously infinite regression arises in intelligent design creationism, which states that there are problems in the theory of Darwinian evolution by natural selection which can only be resolved by invoking a designer or first cause without proposing a solution to the immediate question, "Who designed the designer?" Despite that, the response to this is an example of special pleading: creationists assert that every being needs a cause, but God is an eternal presence which did not need a cause. No evidence for this has ever been presented for peer review, or critical analysis of any kind.

Turtles
The anecdote illustrates a popular example of infinite regress:

A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: "What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise." The scientist gave a superior smile before replying, "What is the tortoise standing on?" "You're very clever, young man, very clever", said the old lady. "But it's turtles all the way down!

Homunculi
The term "homunculus" first appeared in writing on alchemy, De Natura Rerum (1537), referring to what later became known as sperm after the invention of the microscope. In folklore and in literature, the term "homunculus" often refers to a miniature fully-formed human.

In the Eastern Bloc, homunculus has referred to attempts to remold people to be "without sexual, high intellectual or high emotional 'centres'". More recently, Daniel Kalder has used homunculus to refer primarily to the heads of puppet states who felt compelled to follow the party line while at the same time not showing any innovation from the party canon. Stalinist examples include of Mongolia,  of Bulgaria,  of Czechoslovakia, Enver Hoxha of Albania, Kim Il Sung of North Korea, and  of the Soviet Union.

Motor homunculus
The motor homunculus or somatotopic homunculus, is a visualization tool used in neuroanatomy to illustrate which parts of the brain control which parts of the body.