# Equivalence

## Difference that makes no difference.

It is said that information is a difference that makes a difference. In a way this is a bit poetic and obscure but it captures an abstract aspect of information. Physically, information can stuff like pixels on a computer screen, but the physical changes that happen because of the information can be huge.

Equivalence, in that spirit, might be seen as a difference that makes no difference. For instance, 1+3 is one statement and 3+1 is another statement that is physically different but means exactly the same - a difference that makes no difference.

When I go out about my errands each day, there are many paths I could take to the store.

Each path is different but is equivalent because they all end up at the same place.

Of course some paths are longer and if time spent walking is significant then they are not equivalent in that sense. But many of the paths have the same time spent walking and I can actually feel the equivalence.

As I walk along the crowded sidewalk it doesn't matter if I go left or right around a person coming towards me. It's actually interesting how people unconsciously negotiate with each other how to move so they don't collide.

I'm working my way through a math book (Advanced Algebra and Calculus Made Simple) and in math equivalence has a surprising role. What happens is that there can be many ways of calculating a result; that is, they all give the same result. But some ways of doing it involve much less work.

For instance, say I plot the curve traced by an equation. (That is for each value of x in a range, calculate a y position and plot that on the screen.) There is an easy to comprehend way of doing that called substitution which involves choosing a value of x and calculating the pixel position and plotting it. This can be very slow.

There is a method of doing that that involves an idea called the Remainder Theorem that does the same thing very much faster. It produces a result that is equivalent to the substitution method.

It's like "You take the high road and I'll take the low road, and I'll be to Loch Lomond 'afore ye".

I studied this stuff long ago when I was a physics student at university. Then I went to live in the woods and got out of practice and couldn't do stuff like calculus anymore but I retained some core ideas about it. Years later when I was writing computer programs that simulate physical processes I found I could simulate calculus within my simulation.

But that method is very slow. Calculus would be faster but equivalent and I'm trying to get so I can use it.

I like thinking about the physics that underlies the physics we know. A key concept in the physics we know is that everything is made of something else. Water is made of atoms and atoms are made of protons, electrons and neutrons. The protons and neutrons are made of quarks - but then, what are the electrons and quarks made of?

Nobody knows. One idea is called string theory - that is that everything is made of tiny strings that vibrate in 11 dimensions. The theory is a tad complex and nobody has been able to show that it actually corresponds to reality because so far there is no way to test its conclusions.

An interesting thing about string theory is that seemingly different systems produce the same results.

I don't know more about this than handwaving but Wikipedia tells us that "Type I string theory turns out to be equivalent by S-duality to the SO(32) heterotic string theory. Similarly, type IIB string theory is related to itself in a nontrivial way by S-duality."

Equivalence seems to be an artifact of the way we perceive reality. As Kant said, we work with descriptions of reality and not the "thing in itself". So equivalence seems to be a matter of perspective.

I look out my window at a grove of trees and see a very complex scene. I can go downstairs and outside and look at that grove and see a very different complex scene.

I suspect that equivalence is no more than that.

What do you think?

I open the floor.