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Several years ago in a textbook I read this example as a faulty use of proof by induction. I never really realized why it fails. Here it goes:

Theorem. There are no bearded men in the world.

  • Proof by induction

Base case: Suppose a person has n=1 facial hair. That's not enough to be called a beard.

Induction step: Assume as induction hypothesis that the statement holds true for n = k hair, meaning the person has n = k facial hair that are not enough to constitute a beard. Adding one hair to the set would not matter and the statement would still hold true.

Therefore no bearded man exists in the world.

What's the flaw here?

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The base case looks pretty suspicious to me. You could pick yourself, your aunt or something and remark that person has no beard, but "a person" is way too general to begin with. The following step (the inductive one) is even worse: there is no inductive step at all! – DonAntonio Apr 5 '13 at 16:51
This is the Sorites Paradox: – Alvaro Maggiar Apr 5 '13 at 16:52
@DonAntonio: This "proof" is that since no finite number of facial hairs constitute a beard, and every person has a finite number of facial hairs, then no person has a beard. The induction is "showing" that no finite numbers of facial hairs constitute a beard. (As Joe Turner has mentioned, this is a version of the sorites paradox.) – inactive... for now Apr 5 '13 at 16:58
This is similar to the induction proof that all natural numbers are a whole lot less than a million. – Thomas Andrews Apr 5 '13 at 16:59
you might also be interested, while we're on the topic of false induction proofs, in this classic example – Coffee_Table Apr 5 '13 at 17:10
up vote 11 down vote accepted

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.

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+1 for noting a central problem not yet mentioned (and for "fuzzy definitions," even if you didn't intend it). – Cameron Buie Apr 5 '13 at 23:53
The fuzzier the beard, the less fuzzy the statement, "it is a beard becomes". – Baby Dragon Apr 6 '13 at 6:01
+1. The accepted answer from Eckhard explains that a mathematician would be uncomfortable reasoning about 'beards' without definition, but it doesn't actually address at all why the chain of reasoning fails - what is different between the argument as applied to the word 'beard' as conventionally (loosely) defined and to a mathematically idealized concept of 'beard' (e.g. a collection upon a face of 50 or more hairs). – Mark Amery Apr 6 '13 at 12:29
@MarkAmery I agree. Eckhard's answer is reasonable and good enough, that's why I accepted it. But this one tackles the issue at a more fundamental level. Thereby I now set this one as as the best answer. I'm new to SO, I hope changing the accepted answer is not frowned upon. – Omid Rohanian Apr 6 '13 at 13:00
A relevant quote describing your problem: You ask whether something "is" or "is not" a category member but can't name the question you really want answered. - from In this case, the fuzzy question is 'does this particular mass of fuzz belong to the category "beards"?'; assuming that question is cleanly answerable 'yes' or 'no' is the key mistake. Every time the argument says "X cannot be rightly called a beard", they're making a sort-of-maybe-true statement, not a TRUE one. Induction on sort-of-true statements doesn't work. – Mark Amery Apr 6 '13 at 13:32

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

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A good example as to why the first step in mathematics is always to construct a precise definition for what you want to study. – Jack M Apr 5 '13 at 22:59
My dictionary defines beard as "a growth of hair on the chin and lower cheeks of a man's face". This doesn't exclude 1 hair being a beard. So the base case is false. Then again, this definition also implies that there's no such thing as a bearded lady. – Barmar Apr 6 '13 at 10:48

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.

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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.

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Then only Chuck Norris has a beard. – Jonathan Rich Apr 5 '13 at 16:52
Well, I certainly won't be the one to tell him otherwise. – Cameron Buie Apr 5 '13 at 17:00

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.

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How is this getting debated! It is nonsense to try and prove something is beard when you don't even know even know what a beard is!

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This is essentially what the other answers say. Can you add something? – robjohn Apr 5 '13 at 18:00
Yes I realise now the answer by @Eckhard essentially says that same as me, however I missed this at first glance. The only other thing I would have to add is that it is quite interesting that everyone thinks they know what a beard is. But in reality, we don't! When has anyone defined to us what a beard actually is? – user58514 Apr 5 '13 at 18:44
Isn't that true for most words we use? No one learns a language by precise definitions, it's done by inference and extrapolation. Even dictionaries don't have definitions precise enough to solve the sorites paradox. By your reasoning, we don't know what anything is! – Barmar Apr 6 '13 at 10:44
But that is my technical point! We don't! – user58514 Apr 6 '13 at 11:01

This might repeat what others have said, but the induction principle requires that the predicate applies to the natural numbers. Clearly, the predicate 'x is a person with n facial hairs' doesn't apply to the naturals. Even though it's not a proof my mathematical induction, it still is an intuitively good argument, making use of repeated applications of modus ponens, together with the principle that for any n, if a person is n facials hairs is unbearded then a person with n+1 is unbearded. There are a number of solutions to the sorities argument in the philosophical literature. See, for example, Timothy Williamson's book Vagueness.

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Induction step: Assume as induction hypothesis that the statement holds true for n = k hair, meaning the person has n = k facial hair that are not enough to constitute a beard. Adding one hair to the set would not matter and the statement would still hold true.

Precisely define "beard" first. Once you do that, the above statement will be false, since you'll have some $k$ where adding one hair makes it a beard.

Mathematics only deals with precise definitions. In almost any mathematics course, they will spend a lot more time that you think necessary to define things. Now you know why.

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The implicit "definition" of a beard as an arbitrary number of facial hairs is the primary issue. The proof starts with a very easy to grasp fact; 1 facial hair does not a beard make. A variable $k$ is then given the value 1, and the number of facial hairs an individual has, $n=k=1$, is not a beard.

By adding one hair to $k$, we don't change the answer; two hairs is still no beard. Three hairs, same thing. The proof then concludes that, for a given number of hairs n, because $n=k=1$ is no beard and $n=k+1$ doesn't change the answer, there is no $k$, produced by incrementing from the base case, that constitutes a beard. The "proof" is essentially stating that an arbitrary and undefined number of hairs $x$ is needed to make a beard, and $n=k<x \therefore n\neq x$ for all $n$.

It's a "boiling a frog" argument; drop a frog in a pan of boiling water and he'll hop out. But, put him in a pan of cold water, and heat it up by degree, the frog will just sit there, because each degree feels the same as the last. There is, however, a qualitative difference between water that is 211*F, and water that is 212*F, that does not exist between any two adjacent degrees of water temperature < 212 (until you get to 32). By the same token, somewhere between one hair and 8 million hairs, you'd call it a beard; perhaps it's scraggly, perhaps it's just a goatee, but a goatee is a beard and a scraggly beard is a beard.

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There is? I would like to know it so I could spend my morning counting my hairs and making sure whether or not I am bearded or just hairy. – Asaf Karagila Apr 10 '13 at 16:50
That's the whole point; because there is no consensus on how many hairs constitutes a beard, the "proof" tries to get away with saying that no number of hairs constitutes a beard. – KeithS Apr 10 '13 at 18:36
Well, from my view point anything I can't count is infinite anyway; so there are infinitely many hairs on my face. Huzzah! I am a bearded man. – Asaf Karagila Apr 10 '13 at 18:37
Not quite. About 615 hairs grow on each square centimeter of skin. Most are microscopic, but still hair. The full human body has approximately 14 to 15 square feet of skin. Doing a little arithmetic, that works out to approximately 8.5 million hairs growing on your body. Your face and head is at most 10% of that surface area, so if every inch of your head were covered in hair you'd still only have maybe one million hairs on your head. Now here's the kicker; every square inch of your head is covered in hair, including your jaw, so every human being (male or female of any age) is bearded. – KeithS Apr 10 '13 at 18:47
You can be "breaded" to your heart's content; what you and your GF do in the bedroom is not my concern. My point stands; the initial argument made no definition of a facial hair as needing to be visible. In law, ambiguity in a document benefits the party that did not draft it. I didn't draft it, so hair's hair no matter how visible, so your girlfriend and Billy Gibbons are equally bearded (and Billy Gibbons most definitely has a beard... Just sayin.). :P – KeithS Apr 10 '13 at 19:06

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