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I would consider this more an artifact of the modelling than a fundamental point about reality. After all, how do you determine when a new product adds a dimension, or changes existing coefficients? As long as a product is perfect replacement of some existing product, only better along an existing axis, that's easy.

But in reality, new products/inventions, even improvements on existing ones, are usually not that simple. They add an extra dimension, more freedom to find better solutions to problems. But in a high-level, low dimensional description, this freedom can be collapsed into a change in parameters, or really added as extra dimension, if the effects are important enough.

Funny thing is, I am currently working on shape optimization, where it is completely natural to change the number of parameters used to describe the shape, and thus the dimension of the problem.

A related field is order reduction, where you try to (locally) approximate a physical phenomenon by its most important modes. If there is a change in the physics, you can either modify the modes, but keep the same number of them, or you might find that for the new situation more modes are required to describe it well enough.

I would suggest this is a good analogy for your innovation/improvement distinction

by GreatZamfir on Fri Feb 22nd, 2008 at 08:07:51 AM EST
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Well, a new dimension corresponds to a new manufacturing process, with different inputs. As long as there is substitutability you don't have "true" innovation.

I am familiar with dimension reduction (proper orthogonal modes, principal componets, factor analysis...) and you're right, at some level the number of variables is a matter of choice. But you still have to be able to close the system of equations. You can always ascribe the effect of all the neglected modes to "noise", though.

We have met the enemy, and he is us — Pogo

by Carrie (migeru at eurotrib dot com) on Fri Feb 22nd, 2008 at 03:10:26 PM EST
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