## I Think, Therefore I Am

René Descartes said, "I think, therefore I am." Deep stuff. I claim that I can prove it. What is wrong with this proof?

• Step 1: Descartes said "I think, therefore I am." So, we can deduce that he thought that it was true.
• Step 2: We can state that as the true statement: Descartes thought (I think, therefore I am).
• Step 3: We can use the distributive law to rephrase that: Descartes thought (I think), therefore Descartes thought (I am).
• Step 4: We can cancel out the "Descartes thought" from the previous step, leaving: (I think), therefore (I am).
• Step 5: The parentheses, in the previous step, are fairly useless. So, we deduce: I think, therefore I am.

So, what is wrong with the above proof? It actually makes some sense, except for step 4. We cannot cancel anything out of the statement in step 3, because "cancel" is something we do with an "equivalence relationship". The logical operator "therefore" is not an equivalence relationship. And Descartes' statement remains unprovable.

I invented the above proof as a joke. But, it normally requires so much explaining, that it's not a very effective joke. So, it's a logic puzzle.

There is a second flaw in my proof. Step 3 is somewhat shaky. The statement is probably true. But, I don't think that it follows logically from the previous statement.