So I was trying to read the textbook.
Chapter 0. Done.
Then Chapter 1 came along, seemingly, innocently, skipping along.
And it brought back some of my confusion (like a mudslide over my brain) about proving such simple things.
Did we not create the number system?
Is it not the way natural numbers work just the way the system was made?
Why do we have to prove it?
In what way could it possibly happen that adding one suddenly doesn't not make the next number exactly one larger?
No matter how large the number, is it not just the way of numbers to be that way?
Why, why, WHY, must it be proved?
And yet I know, without proving those simple things, we could say that everything we know in math could be wrong, because we haven't justified the most basic parts of the system.
And yet...
...
And I started working on problem set one.
I didn't understand quite why the proof worked the way it did for 3^n in class, it seems so simple and yet I don't see why it can imply that for all the numbers, why it can imply there is a pattern.
But the second question I think I actually managed to somehow workout.
Of course I have no idea what I am doing, like blindly waving about my hands, in some feeble way hoping to reach for the sun like a wilting tree.
-_-
So I am planning to go in for TA hours on Thursday to resolve my confusion and sort out the fuzziness of my head.
(And ask a lot of questions I probably should have asked in CSC165.)
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3 comments:
Good things to worry about. The most basic facts about numbers are the hardest to prove. One way past the difficulty is to just define all the difficult bits as axioms. I've seen the natural numbers defined in such a way that induction is built in to the definition, so the need to prove certain things evaporates.
So it seems that the simple is hard to prove!
I've read about the five axioms made by Euclid and how many have tried to prove the fifth using the previous four and failed...
Though I guess that isn't a good analogy. But it came to mind when I was reading about the basics of the natural numbers.
How was the definition built in?
You can just say that simple induction is a property of the natural numbers. Or well-ordering. Or complete induction. Once you build in one, you get the other two for free.
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