Thinking about emergence and math this morning, and how we can deal with emergence if we systematically discard pieces of the math toolkit.

The fascinating thing, though, is that we don't end up with the empty set (no pun intended).

Emergence is when the whole is different/greater than the sum of its parts.

Important question: when doesn't that happen?

In the real world, I'd say never. It is only when we abstract that we can have a concept of quantity where, say, two apples can be summed together and make an aggregate with the same qualities.

In reality, apples are distinct and every apple is special. Each grouping of two apples has qualities that are distinct from other groups of two. This doesn't even get (yet) into "you can't compare apples to oranges" territory.

In fairness, some things are more alike than others and we can use use quantitative math to sum them, but we have to discard the differences... not just the differences, but also the emergent properties that invariably come about when we group things.

All of this comes into sharp relief when we group heterogenous things. We can add red paint and yellow paint, but the sum is not "two paints" except very abstractly. It is orange.

Or, if we don't have word for 'orange', 'red-yellowness.'

This is emergence. The properties of the grouping are different than the properties of the constituents.

Again, where doesn't this happen?

So, if you were to describe this, you'd have a very weird sort of math. You have objects with mappings to qualities. The objects can be placed into sets, and the sets themselves can have different qualities.

The power-set (set of all subsets) of any set of objects can each have a distinct set of qualities, and there's nothing that allows us to predict them completely.

This is the essence of art, discovery, and (if we are honest with ourselves) a lot of engineering.

So, is there math here? Yes, but not much. It's sets and subsets, and the recognition that aggregates are distinct and not amenable to summation.

Sometimes, when I'm designing, I make little sketches of nested boxes. Each box can contain several boxes, and each can have one or more lines (rays) emanating from it. These are the names of capabilities or qualities. I call these properties.

If a box contains other boxes, its lines (rays) are 'emergent' (derived) properties.

The sketches are a thinking tool. Most of the time I go "forward." I start with a many boxes that are the "things I have" and then make boxes groups of them to think about what the emergent properties might be.

Other times, I go "backward." I make a big box and I put lines on it to show the properties I want, and then I fill the box with other boxes (with their own properties) that might produce the emergent effect I need.

Sketching is just one way of applying this frame.

To end this thread, I'll just say that I like to see design as the discovery and accommodation of emergent effects. Seeing what's there, and what you can do with it.