Sunday, March 11, 2018

An Introduction to Math, by chris the cynic, Part 1: One to one correspondence, sheep, natural numbers, and addition thereof.

[This is also what happens when I have no internet.  Written at the same time as the volumes on exponentiation, but given much less attention.]

We don't know what the first math was.

I don't have the internet right now, so I can't check if crows or gorillas or such can do math.  Whether or not other extant animals can, you know that earlier hominids would have had it in them, and modern humans were around for a very long time before they started writing things down.  The result is that extensive, if informal, mathematics had been worked out before writing was a thing.

It's worth distinguishing “math” from “numbers”.  It's really easy to work out what the first number was.  It was one.
I have one cat.
I have one pair of pears.
I have one set of dishes.
I have one deck of cards.
I have one flock of sheep.
I have one pile of Legos
I have one . . . empty fucking space where my stuff was supposed to be!
Anything, no matter how numerous or vacuous, can be described as one collection.

You can always have a group, a portion, an amount, or a lack.*

That, however, isn't math.  It's just a number.  It doesn't tell us what the first math was.

One theory is that the beginning was 1 to 1 correspondence.  That can easily be used to keep track of things without any formal understanding of anything.

It goes like this:

I tally my sheep by putting rocks in my pouch:
I have one sheep, I put a rock in my pouch.
I have another sheep, I put another rock in my pouch.
I have one last sheep, I put another rock in my pouch.
I check on my sheep using the rocks in my pouch:
One rock.  One sheep.
Another rock.  Another sheep.
The only number used is “one”, which we've already discussed is basically a gimme, yet it can be scaled to any number of sheep (or other discrete things) it is possible to have.  (It is not possible to have infinite sheep or one over pi sheep.)

You do need to be aware that if a sheep dies you must take a rock out, and if a sheep is born you must put a rock in.  With those considerations taken care of, it allows you to make sure you've got your whole flock even if there are too many for you to keep track of in your head and you cannot count.

One to one correspondence isn't just the math of sheep and stones.  It's also the math of tally marks.  This lends itself nicely to the creation of the natural numbers.

You give names to certain collections of tally marks.  One tally mark is called “one”, for that only makes sense.  You look upon “││” and call it “two”.  You look upon “│││” and call it “three”.

You can keep on going forever, but without a system it would be hard to keep track of all those names.  We'll get back to that later.  For now, regardless of names, the key point is that we've got all of the natural numbers (we have the set {1, 2, 3, 4, 5, . . .}).

We haven't defined any operations yet.  The numbers just sit there.

The basic operations are pretty easy to figure out.  Addition, in particular, comes very naturally.

You take two and put it next to three.  Then count up the tally marks or make a new, combined, tally with one mark for each mark in the two individual tallys.

││+│││ = │││││

Congratulations you just learned that 2 + 3 = 5.

It's also plainly clear that x + y = y + x.  .  You don't even need to redraw the marks to change one to the other, just scratch out the plus after the first x marks and stick one in after the first y marks.  So points for discovering commutative property as well.

The associative property is likewise easy to demonstrate.  It doesn't matter where you insert a “+” or parentheses, the number of tally marks doesn't change.

It's worth noting that we don't get any new numbers here.  Addition, on its own, is fully capable of giving us more numbers.  Infinity plus one of them, in fact.  But to do that we need to have a negative number and we have not yet reached that point.

Thus far everything is simple and intuitive, and honestly I'm just stringing words together now because I kind of stopped writing this after the first sentence of the previous paragraph so the content sort of ran out.

Still, it seems like if I'm going to post this I should have some sort of ending.  Not sure what that should be though.  See you next time, if such a thing exists.

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* The last part, “a lack”, does get resistance.  People don't generally dispute that 0 =  1 × 0, so they don't dispute that zero is one zero and their objection isn't mathematical in nature, but the idea that nothing, in the abstract, can be said to be one [thing] does get resistance even when that [thing] is a collection with nothing in it.

That's . . . not a major problem for us right now, but it is worth noting that (modern) mathematics is built upon a foundation of set theory and the set with nothing in it is precisely one set.  It's also monumentally important.  It's usually written as “{}”, “Ø”, or “ø”.  It is called “the null set” or “the empty set”.

1 comment:

  1. The perspective you offer here puts me in mind of Kronecker's famous saying: "God made the integers, and the rest is the work of man." The just-so story you give is pedagogically very good, but I wonder if it's really the sort of way numbers came to be in our minds. Of course, the way I do math is so ahistorical I have no grounds to complain either way.