Last week, at lunch time I sat down in front of a woman who after few seconds spent looking at me, decided that I looked clever, and started asking me questions about the four Maxwell equations of Electromagnetism that are displayed in the cafeteria as Kings College (she eventually explained that the display had been a mystery for her for a long time). After a while she asked me "... by the way, since you are a mathematician, why cannot we divide by zero in mathematics ?"
I explained and unfortunately she didn't get it, and said that my answer was "academic and not logical" (and wondered why a simple question could have such a complicated answer...). This left me wondering about the episode for many hours and even talking about it with my students the following days, until something occurred to me. She simply only didn't get the structure of the proof I was using: a proof by contradiction.
I got her email-address today and below is the part of the email I sent to her in which I clarify the entire thing. Turns out that explaining what is a proof by contradiction to a non mathematician (and why/how they work) is not absolutely trivial (but I guess that reasoning about thinking patterns, only sounds funny to mathematicians). No wonder why it took us so long to isolate the rules of Natural Deduction :-)
Now that I have access to a writing medium (which is like superpowers for mathematicians ^_^), let me explain again what I said...
1. The question of "why is it that zero doesn't have an inverse", is simple only in appearance. Humans have only recently acquired knowledge, technology and wide spread high school education (and even this latter is not as wide spread as we all would like to believe). For most of our recorded history the every day life of humans did not involve asking one self questions about arithmetic subtleties. In fact, the very fact that you can ask this question is the proof that you are yourself the result of a long social evolution and in quite an advance state of knowledge.
2. The question not being simple, is likely to require an answer which is itself not totally trivial. The answer, to be understood, require about as much, if not more, mental sophistication as what was required to ask the question in the first place.
3. The answer that I gave to you last week, which you said was "academic" was the simple application (to your question) of a thinking pattern that all humans are familiar with and which goes as follows: To justify that something is not true, assume that it is true, draw an impossible conclusion and then conclude that the assumption simply could not be true.
You have used this pattern 10000 times since you were born, without realizing that you did. Let me give you one example.
- Friend: You, I know you cannot see the sky right now, but do you think it's raining ?
- You: No, it's not raining.
- Friend: How can you be so sure ?
- You: Well..., if it was raining the ground would be wet, and I can see that it is not, so it's not raining.
- Friend: Fair enough.
- - When you said "if it was raining", you assumed the opposite of what you know is true (or more exactly the opposite of what you want to show).
- - When you said "the ground would be wet", you draw a logical consequence of your assumption.
- - When you said "and I can see that it is not", you point out the fact that the conclusion is impossible, or contradictory, or does not match reality.
- - When you said "so it's not raining", you are basically saying, that from all you have said so far, you are now fully entitled to say that really it's not raining.
So now, let me do it with the problem of the inverse of zero (or dividing by zero, which is the same thing)
- Friend: Pascal, why cannot we divide by zero ?
- Pascal: Because zero doesn't have an inverse (dividing by zero, and finding the inverse of zero are the same thing by the way).
- Friend: How can you be so sure ?
- Pascal: Well, if zero had an inverse, then the product of zero and this inverse would have two different values 1 and 0 simultaneously (1 because the product of any number by its own inverse is 1, and 0 because the product of zero times any number is zero). In fact we would have that 0 equal 1 as a necessary logical conclusion. Since zero and one are clearly not equal, then zero doesn't have an inverse.
- Friend: Fair enough.
The mechanism is the very same
- - "if zero had an inverse" : assumed the opposite of what I know is true
- - "then the product of zero and this inverse would have two different values 1 and 0 simultaneously (1 because the product of any number by its own inverse is 1, and 0 because the product of zero times any number is zero). In fact we would have that 0 equal 1 as a necessary logical conclusion" : drawing the conclusion.
- - "Since zero and one are clearly not equal" : pointing out the fact that the conclusion is impossible, or contradictory, or does not match reality.
- - "then zero doesn't have an inverse" : end of the proof.
Now, a last note for this entry. A question nobody really ever asks, which is in fact the most interesting question anybody could ask after having been told about proofs by contradiction the first time...
The question is: why is it that reaching a contradiction allows us to state the opposite of the assumption, in other words, why can we then claim what we wanted to prove as being true ? Answer is relatively simple: we, humans, a technological biological species, have an untold faith in the fact that the universe itself is not illogical and not a place where contradictions happen. This faith has been built-in inside us as part of our mental evolution over hundred of thousand, in not million of years. It actually underlies most of our thinking patterns.
Of course, there is nothing mysterious or mystical here. The fact that we have that (this "faith") built-in, is the mere psychological acknowledgment of the fact that physical laws apply uniformly regardless of location and time.