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Math and Life

Unicorns Exist, because 1 ≠ 1

  1. All chipmunks are communists.

  2. I am a chipmunk.

  3. Therefore, I am a communist.

This argument is comically absurd. Chipmunks are not predisposed to have an opinion on complex political theory,1 and I am, in fact, not a chipmunk.

But it’s not all bad. For one thing, it led to my making this photo:

chipmunk.png

For another, it sort of makes sense. If you pretend for a moment that all chipmunks are indeed communists, and that I am, as a matter of fact, a chipmunk, it would follow that I am communist.

When I say “it makes sense” I mean that it follows a certain logical order (specifically, a syllogism). If I were to replace “chipmunk” with “human,” and “communist” with “mammal,” I’d get a perfectly reasonably argument:

  1. All humans are mammals.
  2. I am a human.
  3. Therefore, I am a mammal.

Notice that the structure of the argument didn’t change, only the words. This creates a game of mad libs with logical argument:

  1. All dragons are sweet grandmas.
  2. I am a dragon.
  3. Therefore, I am a sweet grandma.

or

  1. All toasters are liquid oxygen.
  2. I am a toaster.
  3. Therefore, I am liquid oxygen.

The point here is that we can logically reason with wacky, out of the way, or even downright wrong premises. That’s what I’m going to be doing shortly. We’ll begin with something weird, and end up with unicorns.


But before we get to the unicorns, we need to talk about some logic.

When we say “[claim 1] OR [claim 2],” we mean that at least one of the claims is true. If both claims are false, then the entire statement is false. If one claim is true, then the entire statement is true. Let’s try some examples:

“The ukulele is a stringed instrument OR basketball is a sport” (True)

“Bears are reptiles OR my pencil is a cat” (False)

“I am human, OR today is Friday” (True)

That last one is a little interesting. This blog will be posted on a Friday, but I have no idea what day you are reading this. Still, I confidently tell you that the statement is true. Why? Going back to the definition of “OR,” if one claim is true then the entire statement is true. The first claim, “I am human” is certainly true, so it doesn’t matter what the second claim is.

This means I can substitute any proposition in for the second claim, and the statement is still holds:

“I am human, OR I am a cat” (True)

“I am human, OR the Earth is flat” (True)

This process of starting with a true statement then adding “...OR [any other statement]” is called a “Disjunction introduction,” which is just a fancy of saying “add an OR.”

Wikipedia’s example of this trick illustrates it quite nicely:

“Socrates is a man. Therefore, Socrates is a man or pigs are flying in formation over the English Channel.”


One more piece of logic. Consider this statement: “I am either at home or at the movies.”

If I’m not at home, where am I?

Because an OR statement guarantees at least one of the claims is true, and we know that I’m not at home, I have to be at the movies. This is called a disjunctive syllogism.


Premise 1: 1 = 1

Premise 2: 1 ≠ 1

“but—”

Hold your buts.2 There are unicorns at the end of this rainbow.

  1. From premise 1, 1 = 1.
  2. Because 1 = 1, we can use a disjunction introduction to say “1 = 1 OR unicorns exist”
  3. From premise 2, 1 ≠ 1.
  4. If “1 = 1 OR unicorns exist,” and "1 ≠ 1" (premise 2), then unicorns must exist by disjunctive syllogism.

It may seem like I’ve pulled some trick, but every step is valid logical inference. For the formal logic fans, I pulled this from wikipedia:

principle of explosion.PNG

So did I do something wrong? Do unicorns really exist?3

unicorn1.jpg

As you’ve probably guessed, the problems arise when I assumed that 1 ≠ 1 AND 1 = 1.

Contradictions are explicitly forbidden under most4 systems of logic. And you can see why: replace “Unicorns exist” with literally anything you want (like your mom's phone number), and it becomes true.

That concept, that any statement can be proven from a contradiction, is called the Principle of Explosion.

How interested you are in this is a good measure of whether you would enjoy majoring in philosophy.


You may shrug your shoulders indifference. It’s not like there are any real contradictions you could use this on. Even if there were, you wouldn’t be able to conjure a pizza in a puff of logic.

However, if you work in a field whose entire foundation is built upon rigorous logic, say math or philosophy, you might care a bit more.

Contradictions are a big deal in math. Such a big deal that it caused a complete upheaval of set theory in the early 1900s over one simple set:

  (from wikipedia)

(from wikipedia)

I won’t get into the nitty-gritty, but this basically says that you can define a set (the first part) that implies a logical contradiction (last part), which, as we have seen, isn’t good.

This spelled the end for naive set theory, and led to a new set5 of axioms for the budding branch of mathematics.


Of course, all this has little to no practical consequence, but it is interesting to think of what happens when you break the rules. Who knows, out of nothing, you may create a strange new universe.


1And even if they did, whos to say they're commies?

2I am continually disapppointed that this usage of this phrase hasn't caught on.

3Even this unicorn seems perplexed

4There are actually some that don't. You can read up here.

5Ahem.

Image sources can be found here.