By reflexivity, we must have 4<=4 in the preorder. Because 2*2 equals 4, 4<2 must hold in the preorder. By a similar argument, 8<4 should hold in the preorder as 4*2 equals 8, however as we already know that 4<=2 and 4<=4 must hold, it follows that 4 is a lower bound of 2 and 4, and as 8<4 holds by one of our assumption, 8 cannot be the glb of 2 and 4. It thus follows that there cannot be a preorder on R (or on N, Q, Z for that matter) s.t. cartesian product/glb is multiplication.

]]>It looks like you were working with the case where h is not 0; as you noted, the single-step difference gives you the type of contexts with any number of holes, e.g. for f(x) = x³, Δf(x)/Δx = (f(x+1) – f(x))/(x+1 – x) = 3x² + 3x + 1. That is, there are three contexts with one hole, three with two holes, and one with three holes.

The difference operator delta can actually be cast in terms of a q-derivative by taking q to be an operator on x rather than a ring element, e.g. q(x) = x+1. I asked John Baez about this yesterday and he told me that if F is any automorphism of a commutative algebra A over some field K, then for any k in K and a in A, we can define q(a) = k(F(a) – a). The result is something called a “twisted derivation”.

]]>http://blog.sigfpe.com/2010/08/divided-differences-and-tomography-of.html

I think it’s pretty amazing that the same definition has these two entirely different interpretations.

You can go further. Both derivatives and divided differences of types can be seen as special cases of constraining types with regular expressions: http://blog.sigfpe.com/2010/08/constraining-types-with-regular.html

]]>The light bouncing off the atmosphere and into our eyes is polarized just like light bouncing off of a glossy magazine or the surface of a pool. The angle of polarization depends on the position of the sun. Facing north, as the sun rises and then sets, the angle of polarization goes through a 180 degree turn. I chose the contours of the template in this post such that when the sun is at an angle A, the ray from the center of the sundial at angle B filters cos(A-B) of the light. The result is that the darkest part of the polarized film will behave like an hour hand and will go through a 360-degree turn from sunrise to sunset, roughly 6AM to 6PM.

]]>I found this while looking for polarization sundials. I wonder if you would like to explain a little more how it is supposed to work.

best,

Gert

Some time ago the universal thing was called “object”. “Everything is an object.” Now it is “design pattern”. “Everything is a design pattern”. I see progress, guys. That’s fantastic. ]]>

Or could we define a category where zero and one were separate objects?

]]>The unit is property number 3 in each case.

> It’s also important to mention that “flatten” is associative.

I didn’t mention any of the axioms that a monad has to satisfy, since I didn’t want to get into the category theory in the main text. But yeah, flatten should be “associative” in the sense that if you have a list of lists of lists, it doesn’t matter in which order you apply flatten to get rid of the parentheses. It should also satisfy the left and right unit laws, in the sense that if you wrap each element in the list in brackets and then flatten, you get back the original list, and if you wrap the entire list in brackets and then flatten, you get back the original.

It all depends on what rules you’re playing by. If you use Zermelo–Fraenkel set theory with the axiom of choice, then Cantor’s proof holds. If you use constructive Zermelo–Fraenkel set theory, then you can’t construct the table, and the proof above suffices to show there’s only one infinity. And the ultrafinitists deny that even the natural numbers are infinite!

]]>If we take away the … at the end of each number and consider all 2-bit binary values:

0: 00

1: 01

2: 10

3: 11

inverse of the diagonal: 10.

But 10 is in the list! In order for Cantor’s construction to work on our 4 numbers, each one would have to have 4 bits.

But if we’re considering 4-bit numbers, we can list 16 of them.

So for any list of numbers in base b that each have k digits, Cantor’s construction only works if we have no more than k elements in the list. But there are b^k possible elements, and it’s easy to enumerate them all.

For example, in our example of base 2 values with 2 bits, Cantor only produces an outsider if your list has 2 or fewer elements, even though it’s obvious that our list should have 4 elements.

So Cantor basically keeps moving the goal posts exponentially — “wait, I need one more digit to make sure my (decimal) number isn’t in your list.” “But if we add one more digit, I can fit 10x as many numbers in my list!”

So whether you accept Cantor’s argument depends on what you think happens out in those “…”s.

I’m only comfortable thinking in terms of things I could construct, so I call myself a Constructivist. But it’s kind of like being a skeptic at a revival; it limits the range of conversation topics.

http://fdiv.net/2012/04/01/objectivist-c ]]>

I liked the book, especially the parts about the video game. But 90 percent of the book is a terrorist/mafia thriller. Not that I have anything against that, I just usually prefer science fiction. ]]>