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But the real interesting part here is the use of exact real arithmetic, not the category theoretic part. Category theory is just used as scaffolding. I would al

But the real interesting part here is the use of exact real arithmetic, not the category theoretic part. Category theory is just used as scaffolding. I would also say it is dangerous to see things only through the category-theoretic lens, because no doubt, as with any generalisation, there will be plenty of situations where it does not provide the right generalisation.

lidHanteyk 7y ago
Better scaffolding leads to bigger, better buildings. Don't knock the auxiliary and ancillary techniques; they're still necessary for this sort of work to be understandable. Also, from my perspective, the exact real arithmetic isn't super-interesting because it's not new; Turing was doing this sort of thing when he first got access to computers. Legend has that one of Turing's first recreational programs was computation of zeros of the Riemann zeta function, and ever since, there's been a tradition of exploring exact real arithmetic with computers and programming languages. MarshallB is one more incremental step forward in a big research programme of overcoming the inconvenient fact that we can't test real numbers for interesting properties with computers alone. At no point have I advocated only category theory. Categories are always composed of some collection of arrows (and objects, but I prefer object-free presentations!), and those arrows are always homogeneous, if not homoousios, composed of a single substance. We can study that substance on its own. Perhaps an analogy to materials science would be appropriate. We ought to study both each building material, be it sets, vector spaces, relations, graphs, or type theories; but also the architectural principles used to make buildings, like abstract algebra and category theory.
- auggierose 7y ago
  Yes, nothing against better scaffolding, as long as you recognise it for what it is. Arguably exact real arithmetic is older than inexact real arithmetic ;-) But that's not the point, I think the point is that it becomes interesting now to actually do computations with it. It helped me program solid modelling operations for triangle meshes, when I was despairing doing it with normal floats. Switching to exact real arithmetic allowed me to see things clearer, and once I had what I wanted running with exact real arithmetic, I could switch back to floats to make it faster.

lidHanteyk 7y ago

Better scaffolding leads to bigger, better buildings. Don't knock the auxiliary and ancillary techniques; they're still necessary for this sort of work to be understandable. Also, from my perspective, the exact real arithmetic isn't super-interesting because it's not new; Turing was doing this sort of thing when he first got access to computers. Legend has that one of Turing's first recreational programs was computation of zeros of the Riemann zeta function, and ever since, there's been a tradition of exploring exact real arithmetic with computers and programming languages. MarshallB is one more incremental step forward in a big research programme of overcoming the inconvenient fact that we can't test real numbers for interesting properties with computers alone. At no point have I advocated only category theory. Categories are always composed of some collection of arrows (and objects, but I prefer object-free presentations!), and those arrows are always homogeneous, if not homoousios, composed of a single substance. We can study that substance on its own. Perhaps an analogy to materials science would be appropriate. We ought to study both each building material, be it sets, vector spaces, relations, graphs, or type theories; but also the architectural principles used to make buildings, like abstract algebra and category theory.

auggierose 7y ago
Yes, nothing against better scaffolding, as long as you recognise it for what it is. Arguably exact real arithmetic is older than inexact real arithmetic ;-) But that's not the point, I think the point is that it becomes interesting now to actually do computations with it. It helped me program solid modelling operations for triangle meshes, when I was despairing doing it with normal floats. Switching to exact real arithmetic allowed me to see things clearer, and once I had what I wanted running with exact real arithmetic, I could switch back to floats to make it faster.