There are no bearded men in the world - What goes wrong in this proof?

Solution 1:

The (lack of a) definition of what constitutes a beard is the flaw.

Solution 2:

This is the so-called "Sorites paradox", or "heap problem", which is usually expressed in terms of a pile of sand and the same inductive problem. The Wikipedia article I've linked has a summary of the philosophical objections, but basically Eckhard is correct. Personally I've always thought of this sort of argument as having a hidden step in which the arguer carefully moves the definition of "pile" away from whatever semantic space the might-be-a-pile is about to be moved to.

Solution 3:

I don't agree that the lack of definition of what is a beard is the flaw. It's a flaw, sure, but I don't think it's the central flaw here.

The problem is more fundamental than that: this is the misapplication of sharply mathematical concepts to real world concepts that have what we might (no pun intended) call fuzzy definitions. The reality is that there is no definition of beard based on "number of whiskers" nor any sharp line that clearly divides "beard" from "not beard". We might even vary our idea of what constitutes a beard based on context. Among our widely clean shaven, and neatly trimmed, society we might consider even a feeble growth a beard whilst the same facial hair displayed among Edwardian gentleman would be mocked as barely worthy of a teenage boy.

Solution 4:

The base case isn't problematic, as I doubt anyone would say that a man with a single whisker was bearded. The induction step, though, rests on the assumption that if $k$ hairs isn't enough to be called a beard, then neither is $k+1$ hairs. This is an extremely problematic claim, as (together with the base step) it is equivalent to stating that no finite number of hairs is enough to constitute a beard. Since a given person has only finitely many hairs on his face, then the induction step takes for granted that no person has a beard in order to prove that no person has a beard. Circular logic is bad, m'kay?

Ultimately, this fake proof amounts to trying to prove a claim about something that is not defined (or only vaguely defined). We can't logically discuss such objects, so such a pursuit will ultimately be fruitless.

Solution 5:

There are three ways to disagree with an argument: 1 - Identify an ambiguous or incorrectly-used term 2 - Disagree with the premise 3 - Show that the conclusion does not actually follow from the premises

In this case, the simple reason to argue it's wrong is that disagree with the premise you stated: "Assume as induction hypothesis that the statement holds true for n = k hair"

The slightly more complex reason is the implied assumption that the number of individual hairs is what determines the presence or absence of a beard.