mattrolls.blogspot.com is back

[mrmorse]

Apparently I'm blogging at http://mattrolls.blogspot.com/ again. New title, new subject, new comments, new feed (the old feed is dead), lots of math nerdiness. We'll see how long it lasts.

Posted via LiveJournal app for iPad.

Comments

[jon_leonard]

It does indeed appear that comments on that site are non-functional. Maybe cross-post stuff here?

Anyway, I was trying to comment on the summation formulae: Essentially, sigma notation isn't part of the axiom set, and pretty much anything you do with it requires induction (possibly hidden in theorems). Unless you prove a theorem (about theorems!) that the process always works, you don't have any axiomatic reason that it does. Proving each individual case works, though.

[mrmorse]

I'll have to take a look at why comments aren't working. I had convinced myself that sigma notation was a formal way of saying 1+2+...+n but I guess not. Now I need to look at the definition for dot product, which I'm pretty sure was defined in sigma notation.

[jon_leonard]

Sigma notation is a formal way of wrapping up sums like that. The problem is that the informal way isn't robustly defined. Obviously your n is a power of 2, and you meant 1+2+4+8+16+...+ n, right? The "..." is asking the reader to fill in the "obvious" insides, but it sometimes turns out that it's not so obvious, and that's really hard to work with axiomatically. Sigma, on the other hand, is well defined. You start with what the rules are if the limits match (just plug in the variable), and then recursively extend the definition. But pretty much anywhere in math where you have a recursive definition, there's an invocation of induction behind it. Fortunately, this is almost never a big deal.

Dot product is usually defined in terms of sigma, yes. Sometimes it gets defined in terms of some of the useful properties instead: dot(0,0) = 0, dot(a,a) > 0 if a isn't 0, dot(a,b)=dot(b,a), dot(ca,b) = c*dot(a,b), etc. (And then we call it "an inner product", because the definition isn't unique.)

[mrmorse]

I tweaked a setting and comments should be working over there now. I posted a test comment, but more tests wouldn't hurt.