Are All Horses White?

Here’s a homework assignment I had a few weeks ago in Elementary Number Theory:

Find the flaw in the following ‘proof by induction.’

CLAIM: Alexander the Great has been always riding a white horse

PROOF: It suffices to prove that all horses are white, since then, in particular, the horses ridden by Alexander the Great were always white.  To prove the latter, let us employ mathematical induction.

One can certainly find a white horse.  This establishes the base for induction. 

Suppose that any k horses are white, k≥1.  Let us show that any (k+1) horses are white.  Take k+1 horses.  Choose any k of them.  By assumption, these are all white.  Take one of the chosen k horses and exchange it with the (k+1)-st horse.  The new group of k horses, by assumption consists of white horses only and the new (k+1)-st horst is now white, because it was chosen from among white horses.  Thus all of the k+1 horses are white and the validity of the inductive step has been established.

By the principle of mathematical induction, all horses are white.

Basically, the idea is to prove that every horse that has ever existed is white, so therefore every horse that Alexander the Great has owned has been white.  The proof says that there is a group of k horses, where k is any number.  Since this proof starts where k=1, let’s take one horse.  Because any k horses has to be white, the horse has to be white.  When you add a second horse, you have a group of k+1 horses.  Because k=1, the added second horse is also a group of k.  Because any group of k horses must be white, both horses in the set of k+1 have to be white.

Since we showed that the proof is solid for any group of k=1 horses, let’s look at a larger sample.  For inductive proofs, we have to show that, for any number k that is a natural number (any number greater than 0: 1,2,3,…) the statement must be true for k+1.  

For example, let k=3.  There are three white horses in the group.  If you add one horse to the group, you will have k+1 horses, or 4.  Based on the statement that three (k) horses in the group HAVE to be white, you can take any random grouping of three horses and they must all be white horses.  This being said, if you take a group of 3 (k) horses and it includes the horse that was added last (the 4^th horse), all of the horses in that group must be white.  That cycle would continue until it is proven that every horse in the group of k+1 (4) horses HAS to be white.

That’s OBVIOUSLY not true.  But it seems airtight at first glance, right?

Here’s the problem: We started with a group of k=1.  Of COURSE every horse in a group with only one horse in it is going to have only ONE color of horse.  Duh.  If we started with a bigger group, such as k=2, you would significantly cut down on this error as you can’t make the same assumption that both horses in a group where k>1 are the EXACT same color.

Well, there’s the mathematical version!  Let’s look at it from a few other points of view.

PHYSICS:

Think about the definition of color.  Color is the refraction of specific wavelengths of light from different objects.  Black is the absence of light or when all light is absorbed into an object and no color is reflected (by the way, black is not actually a color as it is a LACK of color/light).  Color is, therefore, dependent on the presence of light.  It can therefore be argued that all things are black when light is absent.  Also, it can be argued that everything is black… plus light and color.  Speaking from a physicist’s point of view, everything is black and color is a side-effect of the presence of light.

BIOLOGY:

Consider an albino horse.  PURE white. 

Albino horses are the result of a genetic mutation where melanin is not being made.  Melanin is the pigmentation of skin, hair, and eyes.  Without melanin, it is impossible for there to be any color present in the fur, skin, or eyes of a horse.  Therefore, speaking from a biologist’s standpoint, EVERY horse could be called white. 

If albinism is the absence of melanin, then every horse is white… plus color!  Every horse contains the potential to be albino as they have the same properties of an albino horse… plus melanin.  Horses are all white, plus color! Think of it like an equation:

Albino=Horse-Melanin
Horse=Albino+Melanin

Conclusion?  There are always multiple ways to look at one problem.  Don't get tunnel vision, you might miss out on some really amazing facts!