Shaking the Foundations of Mathematics

So, I was reading The Nothing That Is: A Natural History of Zero by Robert Kaplan this morning. It can be a bit pretentious (incessant quote-dropping, referring to his main concept "the Great Paradigm"), but it also has brilliant descriptions. Descriptions of historical developments in mathematics, but also descriptions of the properties of numbers that are so fundamental that one forgets how fascinating they are. For example, he points out the somewhat magical evolution of numbers from adjectives ("your four sheep and my eight sheep make a flock of twelve sheep") to nouns ("four and eight is twelve").

As a result, I was thinking that mathematics was born from the idea that you could abstract numbers from things. That is, four sheep and four sheep makes 8 sheep, AND 4 stones and 4 stones makes 8 stones, and so you can use stones to count your sheep and all the adding and subtracting and whatever works out fine.

(Sarah: Why on Earth were you thinking that?
Bob: Oh, contemplating the foundations of math and what it relates to comes with the territory. )

But we don't know that adding sheep and adding stones always works out. We just know that it almost always works out, and when it doesn't work out, usually it's because we lost a stone or a sheep or something, and eventually they turn up.
The only way the stones and sheep (and therefore, ALL numbers) wouldn't really match up is if the amount of things changed while we weren't looking. If sheep were flickering in and out of existence. Ridiculous!
...
But of course, we don't know where the sheep are when we're a sheep short, we just usually find them later. But sometimes, we lose stuff and it just seems to disappear. And sometimes we find it again, but somewhere we definitely checked before. So maybe this flickering in and out of existence does happen.

(Sarah: Oh, it TOTALLY does.
It is an especially common phenomenon among socks.
It is an especially common phenomenon within dryers.)

What's more, I am a big fan of Dad's observation that "There are things that are true, and things that are false, and also things whose truth value is not so fixed. In particular, there are statements that are true if you believe them." (Don't believe me? How about "I can't do math?" I'm not saying disbelieving it will make it false, but you can't simultaneously believe you're incapable of it and do it, regardless of what "it" is.)

So what if the sheep and stones and keys and socks would flicker MORE, except that we BELIEVE in mathematics?

(Sarah: That's entirely true. If you don't believe in socks getting lost in the dryer, they won't be lost in the dryer. But misplace one sock and BELIEVE it got lost in the dryer and you will never pull a matching pair of socks from your dryer again.)

If so, the extent of things flickering in and out of existence is, like an inverse Tinkerbell thing, derived from people's innumeracy, specifically their failure to believe in the consistency of numbers. After all, when we say "hmm...I've got two more stones here than sheep...oh well, there must be two more sheep out there I haven't seen", then we go look for the sheep. If we just said "Oh well, more stones than sheep", we may never see those sheep again. And they might not even BE anymore.

(Sarah: You must post this discussion to the blog. I mean what is a blog with an INNUMERACY tag if no one posts about innumeracy?
It would be as though innumeracy doesn't exist.
It went the way of the sheep for which there are stones.)

In conclusion:

• First, it's a good thing we are teaching kids times tables and stuff, because if we weren't, we could lose whole people (Amelia Earhart), whole ships (in the Bermuda Triangle), whole cities (Atlantis)...OMG, maybe we have!
  
  (Sarah: We totally did.
  No one believed a woman could fly around the world . . . and Amelia Earhart ceased to exist.)
  
• And Second, this is a dangerously self-fulfilling thing. All the wrong answers from all the students aren't wrong in the sense that there's an objective truth they're failing to fulfill, they're wrong because we desperately cling to what we need to be right.
  

(Sarah: I had this dream last night where the I said, "I live in a constant state of the contingency of if-ness."
I'm not making that up.
It's totally true.
And then you started this conversation.
And my dream came true.
God, I have to start having more interesting dreams.
Bob: That is wild. And doesn't "contingency" mean "if-ness"?
Sarah: Which is exactly what I thought when I woke up.
Like word for freakin' word.
Bob: We're uncanny. Like X-Men.
Sarah: LOL
If only we were self-regenerating and our bones covered in Adamantium also like X-Men.
Gotta run.
Bob: ttyl)

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