Mystic Math Land
Mystic Math LandReals are abstractions. Of course all mathematical concepts are abstractions, but why are these particular abstractions necessary? Could we not define our mathematics economically in terms of the range of variation of our empirical devices - as fractions and intervals of this range - and do away with the reals? What can we do with the reals that we can't do without them?''
Look at the classical problems they addressed; the commensuration of the hypotenuse with the sides of a triangle, or squaring the circle. We don't use them to do these things in our engineering; we use simple rational approximations and nothing but. If we never use them, why have them at all?
Reals are very useful abstractions: they give you lots of good results directly applicable all over the place. Humanity wouldn't be the same if it wasn't for IsaacNewton, GottfriedWilhelmLeibniz, GeorgCantor,... and all the mathematicians that developed these abstractions.
Newton and Leibniz used no limits, of course. Leibniz spent a lot of time on infinitesimals, but only in response to metaphysical critics of his calculus. His original paper makes no reference to infinitesimals, infinities, nor limits, nor seem to require same. I'm also unaware of any empirical uses of Cantor's results.
See the EwDijkstra quote on philosophers (I put it there on purpose) for the typical view of mathematicians with regards to intuition, philosophies, "real things" and such. You seemed to be complaining about other abstraction being somehow "better" than reals. That's a baseless assertion as long as you don't point to such other theory(-ies). That this "other theory" could be developed and it "could" turn out more "intuitive" or somehow better than reals, is an assertion that even if true has absolutely no value.
I'm happy that the reals make a metaphysical context within which you can frame the rationals. The question is simply whether this context is necessary. If not, it's fair to inquire whether it may not be subject to a FrameProblem, and whether some alternative framework might be possible or preferable. As for a particular alternative, you could start with some of the suggestions on ThereIsNoInfinity - quantize the distinguishable regions of your empirical device, to within whatever resolution is necessary to cover its range of description. Then deal with rationals and forget the rest.
Your other assertion was that nothing in the physical universe has a measure that is a real number. Your assertion is interesting in itself, and has been raised by others, but from a mathematical perspective, it is a) philosophical, b) heavily context dependent, and c) outside of mathematics. So even if true, it has no impact whatsoever on mathematics as long as you can still use reals as an "abstraction" and not the "real thing". I hope this clears up the subject a bit.
No, I'm afraid not. Math can create seven different species of boojum on a Tuesday, prove them incomplete and inconsistent on a Wednesday, and use them to make pretty graphic pictures all the rest of the week. Physics adopts these math representations because they're economical and require no reinvention work for the physicist. But in this it takes on beaucoup baggage and it's not clear that this is harmless to physics.
That's exactly what I'm asking. Why are these abstractions necessary? Could we not define our mathematics economically in terms of the range of variation of our empirical devices - as fractions and intervals of this range - and do away with the reals? What can we do with the reals that we can't do without them?
The problem is determining the range of our output. The number 10^10^10^10 is ridiculously large, and will never come up in any real world applications. So one might consider excluding it from the, I don't know what you would call them, let's say relevant numbers. But if relevant numbers can't be made arbitrarily large, and can't be found arbitrarily close together, then there must be a largest one. But pretty much every relevant number has a relevant successor, so unless we make an extremely unpleasant exception, we are stuck with 10^10^10^10. It doesn't matter if we expect it to apply to anything, it's necessary for the system to be logically coherent.
Don't start with 10^10^10^10. Start with 1, meaning the entire possible range. Then chop the range up into tiles and label them consistently. For example, 1/2 would mean the whole first half of the range, and 2/2 the whole second half of the range. Map the labels onto bits in a bit string but note that these are not points - ThereAreNoPoints. Continue chopping up until you have bit strings with sufficient resolution to describe your measurements. If you need to chop some more, chop away to whatever extent you need. If you discover you need to renormalize to a larger range, add a bit on the (BigEndian) left.
Other number systems are admittedly more subtle, and largely because a priori they don't need to be applicable to anything. It just so happens that physicists find them immensely useful for describing effects they observe, and I can assure you, there are effects which can't be described without them. For instance, if you restrict yourself to rational slopes and intercepts for lines, and rational centers and radii for circles, then you will find lines and circles that come arbitrarily close together but do not actually have any points of intersection. This is an annoyance, and one that is easily remedied by simply defining the points to be there. After all, the points and their coordinates are an abstraction to begin with, so there shouldn't be any harm in extending their use accordingly. Doing this for lines and circles gives you all the square roots. The only coherent way to do it for intersections of curves in general, with sufficient conditions on what you expect intersections to be like, or equivalently with an intuitive definition for what counts as a point, is to use the real numbers.
Accepting for a moment ThereAreNoPoints, there remain regions of intersection. Our physical measurements contain an ambiguity; shouldn't our theoretical results do likewise?
Now you may argue in response that curves don't really exist, and in a certain sense you'd be right. But we need to introduce some kind of abstractions to make sense of reality. These things are chosen from the various "mystical" mathematical systems, and when it comes down to it, the natural and rational numbers are inadequate for describing various properties we want to hold. Things like the intermediate value theorem, essentially equivalent to the above, or the rotational invariance of space (this was discussed on ThereIsNoInfinity) just don't work with the natural numbers. To say more, I would need to know more about what it is you are trying to describe and what it is about the natural numbers that you think make them any less of an abstraction. What do you want your numbers and points to do?
I've sketched something here but it's not "mine". I'm asking a question, not giving an answer.
Btw, IsaacNewton and GottfriedWilhelmLeibniz got by without limits because their work was not established on a rigorous basis. That's fine if you watch where you step, but freely manipulating quantities the way they did can in fact get you to contradictory results. That's why Weierstrass and others introduced limits, to put the work on a firm axiomatic basis. You can get by them by introducing infinitesimals or some other equivalent system, but you need something or else the results of the model will not be coherent.
Agreed here.
Pardon me for butting in, and not really re-reading the entire page (although I've read it several times in the past):
BuckminsterFuller said, "Nature isn't using Pi". Here's why: http://www.angelfire.com/mt/marksomers/65.html
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