![]() That doesn't mean that everything we ever want to speak of is rational or even algebraic, of course - for example, if we're talking about harmonic oscillations, then the ratio between the maximal instantaneous speed and the amplitude is the phase velocity, and the ratio between the phase velocity and the frequency of the oscillation is $2\pi$, which is transcendental. We don't need real numbers, in the sense that we could do without them if we really had to. Their 'uncountability' can be seen as just a useful abbreviation. The book Classical Topology and Combinatorial Group Theory champions this point of view, giving a proof of the Jordan Curve Theorem which is much easier than the standard one.Īs you can see, most interesting mathematics takes place in computable sets, and the topics that don't are also dealt with with finistic reasoning (the proofs in functional analysis and set theory fit inside books after all). Topology is precisely the work of taming uncountable spaces into chunks with invariants in such a way we can use finitistic arguments to prove theorems. Every single function and procedure treated in classical calculus is computable. ![]() The $\epsilon-\delta$ definiton of limit is algorithmic: if you give me an $\epsilon$, I must have a procedure for finding a $\delta$ that approximates the limit well enough. This could be a closed form, an integral, or some numerical method. To solve a differential equation is to find an algorithm which approximates the solution arbitrarily. Regarding the impossibility of working with differential equations, this is just not the case. To me there couldn't be a stronger sign that uncountable sets are inadequate when our system of reasoning simply can't filter them unambiguously. Mathematicians are not limited by computation, much like Turing machines are not bounded by $2^$. Without this, demonstrating the existence of a point relevant to the problem in hand may require a more complex case-by-case analysis.Ī similar enriching of possibilities occurs in measure theory, where countably additive measures make a huge difference. It is particularly useful to know that the real numbers are essentially a unique model for the key defining properties, so when we prove theorems we know we are all talking about the same thing.Īdmitting countable limits of rationals, rather than simply finite limits, turns out to have far-reaching consequences - for example, we can prove the intermediate value theorem. These numbers are packaged in a convenient form which enables results to be proved for all the numbers we might encounter on our mathematical travels. It is true that only countably many of these can ever be defined, but we don't know which we might want in advance. Real numbers include all the numbers you might want to use - including limits of convergent sequences of rational numbers. Then again it is crucial to have a system where you can use lots of tools and the reals provide that. Sometimes what one needs to know is whether or not an integral converges, but the actual value where it converges to is immaterial, and whether or not it is a computable number in any sense is irrelevant. Again, this is a small price to pay for having a fantastic accompanying mathematical theory.įinally, don't forget that it is not always a single number that is sought as the result of a computation. Now, without that property most of analysis fails, so we really need the reals to be complete, and that necessarily means lots of numbers out of our reach. Perhaps most importantly is that they form a complete metric space. The fact that you only use a fraction of those numbers is not particularly relevant.Īnother reason for introducing the real numbers is that they exhibit pleasant computational properties. That's not a big deal, and its a rather small price to pay for having a really convenient system of numbers of great relevance to physics to work with. Now, when you do that you find that are actually created lots and lots of new real numbers, most of which can't ever be obtained as the outcome of anything. Then the outcome of an experiment really is a real number. One way to do so is define the real numbers. Now, mathematically it is convenient to be able to speak of this 'actual outcome' as a single object in a nice system of measurements. Now, if all goes well, the entire countable sequence of rationals is a Cauchy sequence which is the actual outcome. Each measurement is an approximation to the 'actual outcome', whatever it may be. More accurately, any well-defined physics experiment is reproducible, and thus one can always produce a finite sequence of rationals out of a potentially infinite sequence rationals of outcome measurements per experiment. From a physics point of view, all you ever produce as the outcome of a measurement is a rational number.
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