J. Andrew, On 12/30/08, J. Andrew Rogers <[email protected]> wrote: > > > On Dec 30, 2008, at 12:51 AM, Steve Richfield wrote: > >> On a side note, there is the "clean" math that people learn on their way >> to a math PhD, and then there is the "dirty" math that governs physical >> systems. Dirty math is fraught with all sorts of multi-valued functions, >> fundamental uncertainties, etc. To work in the world of "dirty" math, you >> must escape the notation and figure out what the equation is all about, and >> find some way of representing THAT, which may well not involve simple >> numbers on the real-number line, or even on the complex number plane. >> > > > What does "dirty math" really mean? There are engineering disciplines > essentially *built* on solving equations with gross internal inconsistencies > and unsolvable systems of differential equations. The modern world gets > along pretty admirably suffering the very profitable and ubiquitous > consequences of their quasi-solutions to those problems. But it is still a > lot of hairy notational math and equations, just applied in a different > context that has function uncertainty as an assumption. The unsolvability > does not lead them to pull numbers out of a hat, they have sound methods for > brute-forcing fine approximations across a surprisingly wide range of > situations. When the "clean" mathematical methods do not apply, there are > other different (not "dirty") mathematical methods that you can use.
The "dirty" line is rather fuzzy, but you know you've crossed it when instead of locations, things have "probability spaces", when you are trying to numerically solve systems of simultaneous equations and it always seems that at least one of them produces NANs, etc. Algebra was designed for the "real world" as we experience it, and works for most engineering problems, but often runs aground in theoretical physics, at least until you abandon the idea of a 1:1 correspondence between states and variables. Indeed, I have sometimes said the only real education I ever got in AI was > spending years studying an engineering discipline that is nothing but > reducing very complex systems of pervasively polluted data and nonsense > equations to precise predictive models where squeezing out an extra 1% > accuracy meant huge profit. None of it is directly applicable, the value > was internalizing that kind of systems perspective and thinking about every > complex systems problem in those terms, with a lot of experience > algorithmically producing predictive models from them. It was different but > it was still ordinary math, just math appropriate for the particular > problem. Bingo! You have to "tailor" the techniques to the problem - more than just "solving the equations", but often the representation of quantities needs to be in some sort of multivalued form. The only thing you could really say about it was that it produced a lot of > great computer scientists and no mathematicians to speak of (an odd bias, > that). Yea, but I'd bet that you got pretty good at numerical analysis ;-) With this as background, as I see it, hypercomputation is just another >> attempt to evade dealing with some hard mathematical problems. >> > > > The definition of "hypercomputation" captures some very specific > mathematical concepts that are not captured in other conceptual terms. I do > not see what is being evaded, ... which is where the break probably is. If someone is going to claim that Turing machines are incapable of doing something, then it seems important to state just what that "something" is. since it is more like pointing out the obvious with respect to certain > limits implied by the conventional Turing model. I wonder if we aren't really talking about analog computation (i.e. computing with analogues, e.g. molecules) here? Analog computers have been handily out-computing digital computers for a long time. One analog computer that produced tide tables, now in a glass case at the NOAA headquarters, performed well for ~100 years until it was finally replaced by a large CDC computer - and probably now with a PC. Some magnetic systems engineers still resort to fish tank analogs rather than deal with software. Steve Richfield ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=123753653-47f84b Powered by Listbox: http://www.listbox.com
