Performing Mathematical Operations on Nonsense

Let us define the number i as equal to the square root of -1.  So i cannot be positive or negative, but all real numbers are positive or negative—so i is imaginary.  I am pretty much the farthest thing from a mathematician, but i strikes me as being something that we think we have some understanding of, but we really don’t, similar to saying “There is either a red square-circle or there is not a red square-circle.”

But the funny thing is, we can perform operations with i:

(2i)(4i) = (2 · 4)(ii), which equals (8)( i2), which equals (8)(-1), which equals -8.

So from something that doesn’t really make sense, namely “i = the square root of -1,” we get something that makes perfect sense.  How much of philosophy is like this?

Possibility and Nonsense

Before talking about the nature of arguments in my Intro to Logic class, I start off talking about inferential relationships between statements more generally.  So I ask them to consider what else must be true , e.g., if “Todd is dead” is true and if “Bob loves Jill” is true.

Two of the claims that people said followed from “Todd is dead” were:

1) There is at least one dead person.

2) There is a reason for Todd’s death.

I used this opportunity to talk about the difference between logical and causal possibility.  I take it that 1) is logically necessary in relation to “Todd is dead” and that 2) is causally necessary.  We can imagine a world in which people die for no reason, or something like that.

This led to the students’ asking about whether claims following from “Bob loves Jill” were causally or logically implied.  Someone asked whether it could be possible for someone not  to be able to love someone else and if so whether it would be causal or logical.  I said we could imagine a person having some kind of chemical imbalance or the like such that it would be causally impossible for him to love anyone.  But this led me to ask the class whether my water bottle’s not being able to love anyone is a causal or logical impossibility.  It is not so clear, is it?

This reminds me of an interesting but difficult passage in “Part II” of Wittgenstein’s Philosophical Investigations, where he writes:

“A new-born child has no teeth.”—”A goose has no teeth.”—”A rose has no teeth.”—This last at any rate—one would like to say—is obviously true!  It is even surer than that a goose has none.—And yet it is none so clear. For where should a rose’s teeth have been? The goose has none in its jaw. And neither, of course, has it any in its wings; but no one means that when he says it has no teeth.—Why, suppose one were to say: the cow chews its food and then dungs the rose with it, so the rose has teeth in the mouth of a beast. This would not be absurd, because one has no notion in advance where to look for teeth in a rose. ((Connexion with ‘pain in someone else’s body’.))

So, we might say that it is obviously true that my water bottle cannot love anyone, but is that not more than just odd sounding?  Is it a causal impossibility that makes us say this?  We might imagine the water bottle imbued with a spirit by a magician or god mightn’t we?

What about these three statements:

A) Either it is raining or it is not raining.
B) Either there is a black unicorn or there is not a black unicorn.
C) Either there is a red square-circle or there is not a red square-circle.

In the context of asking about immediate inferences, we might say that you can’t infer anything about the world, so to speak, by the truth of A) and B).  But should we say the same about C)?  If the idea of a square-circle is incoherent, then what could C) possibly mean?  Is C) true?  If it is false, is it necessarily false?  Is it nonsense?