jestingrabbit wrote:@mr-mitch: firstly, I think its important to note that there are two quite separate things going on here. There is the rational geometry aspect of what wildberger is saying, and there is the deprecation of real numbers aspect. I'm happy to argue that both make a certain amount of sense, but I want to make it clear that there are two distinct things going on.
I recognize this, but they're not very separate topics. In order to abandon the real numbers you'll need to use the rational numbers, and rational geometry. If the reals offer much more than the rationals, and they do, and the geometry in real numbers offers more than the geometry of the rationals, and it does, then I do not see any reason to abandon the real numbers. They work. If someone isn't comfortable with the formality of the definition of the real numbers, and subsequent properties and applications of the real numbers, then this is what should be studied.
jestingrabbit wrote:The simplicity of the formulae is not just a matter of aesthetics, they are computationally much quicker to work with for things like solving triangles.
Even for finding the spread of an angle between two lines there is a simple algebraic expression. So, computationally and aesthetically, the formulae associated with rational trigonometry are superior.
But it's not superior, it's the same. Let's remember that any function you do use to describe an angle must be one-to-one over a reasonable domain. If you're constantly going back and forth between angles and sines then of course you're going to have computational issues. I don't think you can claim rational geometry is the reason for such advances, it exists in the real geometry (intolerant pun not intended) and I do believe such ideas have been around for a long time. In secondary school, inverting the sine would be the last thing I'd do when asked for an angle. Let's also not forget that angles are much more important when you consider proportions. They ratio of the square of sines is not equal to the ratio of the corresponding angles.
To abandon angles all together is a little shortsighted; a lot of problems can be solved considering the angles, but not so with the spread (eg circles).
jestingrabbit wrote: it clarifies that Euclid's postulates are deficient in specifically delineating the kind of geometry that we typically mean when we talk about Euclidean geometry ie they don't guarantee that all lines through the centre of a circle intersect the circle.
I would say this is more of a limitation of the rationals compared to the reals. The reason is because such non-intersections are irrational.
jestingrabbit wrote: When we talk about approximate solutions, we tend to just be talking about truncating our precise solution (or more precisely, calculating to some precision and stopping) and presenting that as an approximate solution. That is a very vague way of going about things.
I don't think that's very vague at all. The decimal system is only meant to approximate, it's not ideal for representing any number, even most rational numbers. You could choose a different base but you'll always run into the same problems when you reach the negative powers of the base. The vagueness doesn't come from the real numbers, rather the decimal system. They're not entirely linked.
One of the problems I had with the youtube link above is he states about computation and checking two programs produce the same output. While this particular thought is quite irrelevant to the real numbers (any real number is not necessarily a decimal number, in fact most of them aren't, something which is often overlooked) it's the same for any program that computes a rational number in the decimal system, too.
The problem that real numbers aren't decimal numbers cannot be solved, but that's not a good reason for abandoning the real numbers. You don't need infinite precision, and indescribable numbers are useless.
jestingrabbit wrote:Does working only in the reals allow us to determine if there is a point in Q2 that is precisely on the unit circle with the difference of the coordinates less than any epsilon?
Assuming the appropriate (set theory) axioms, definitely, as Q2 [imath]\subset[/imath] R2.
Algorithmically, perhaps. Any rational number in the decimal system is of two classes, finite or repetitive. Any algorithm that solves a problem in R2 should would output an integer and a decimal string. During computation of that string, in polynomial time, you can check to see if there are cycles. As soon as a cycle is detected, you would then assume this is the solution and use the 999 trick to form a rational number. Then, perform substitution which can easily be checked. If it isn't a valid then you continue computation. If it isn't valid and you run into the same cycle you run into approximations; is it worth continuing the computation or is it sufficient to return the currently computed rational number?
The repeated digits in 1/7 in order are 142857
If it turns out the solution cycles 1428571428571428571428572, then this number is incredibly close to 1/7.
Such precision (based on detecting these cycles) is well defined and is directly related to the length of the cycles. But why, in the practical sense, you would want to detect a rational number with a large amount of decimals (perhaps about 35), I've no idea. Instruments aren't accurate to such a small scale.
jestingrabbit wrote:So saying that, for instance, "Any solution you have in the reals can be converted to the rationals, as an approximation" is I think demonstrative of a really vague way of thinking that working only with the reals tends to inculcate.
I agree that this does occur with various lies to children. Secondary students are taught to desire decimals instead of working with fractions. I hate working with decimals, but rational numbers aren't the solution.
jestingrabbit wrote:However, even if we have a perfectly logical exposition of real numbers, we don't get around the fact that the arithmetic isn't defined by algorithms, but by supertasks or limits (which are, in general, another kind of supertask), and that there is no meaningful way to test the equality of real numbers, nor, indeed, to precisely describe an arbitrary real without a supertask.
Definitely, but the question, in a practical scenario, is only limited to precision again. You won't ever need to know if two real numbers are the same, only up to a certain point which is simple.
You cannot describe arbitrary reals without supertasks and limits because most reals are indescribable. The ones that you can describe are either rational or irrational. You can then describe the irrationals by their minimal polynomials, or if they're non algebraic then, I'm not sure. Wouldn't this be more interesting to research?
The thing that I find overlooked the most when we're talking about solutions and the real numbers is that mathematics is at most a model. Why would you settle for "appears to touch" when you can often solve for an algebraic irrational number; in the model. Whether or not this solution applies to the real world is irrelevant. All you really need is an approximation.
Our current system provides all this, much more than the rational system.