It seems to me that the set of all numbers really used by mathematics and physics is countable, because they are defined by means of a finite set of symbols and, eventually, by computable functions.
Since almost all real numbers are not computable, it seems that real numbers are a set too big and most of them are unnecessary.
So why do mathematicians love this plethoric set of numbers so much?
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From a physics point of view, all you ever produce as the outcome of a measurement is a rational number. 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. Each measurement is an approximation to the 'actual outcome', whatever it may be. Now, if all goes well, the entire countable sequence of rationals is a Cauchy sequence which is the actual outcome.
Now, mathematically it is convenient to be able to speak of this 'actual outcome' as a single object in a nice system of measurements. One way to do so is define the real numbers. Then the outcome of an experiment really is a real number. 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. 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. The fact that you only use a fraction of those numbers is not particularly relevant.
Another reason for introducing the real numbers is that they exhibit pleasant computational properties. Perhaps most importantly is that they form a complete metric space. 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. Again, this is a small price to pay for having a fantastic accompanying mathematical theory.
Finally, don't forget that it is not always a single number that is sought as the result of a computation. 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. Then again it is crucial to have a system where you can use lots of tools and the reals provide that.
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Because the completeness of real numbers, which make sure limit has closure, then one could use calculus.
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Real numbers include all the numbers you might want to use - including limits of convergent sequences of rational numbers. It is true that only countably many of these can ever be defined, but we don't know which we might want in advance.
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 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.
Admitting 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. Without this, demonstrating the existence of a point relevant to the problem in hand may require a more complex case-by-case analysis.
A similar enriching of possibilities occurs in measure theory, where countably additive measures make a huge difference.
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Almost all physical models use PDEs - partial differential equations (and ODEs). Without real numbers, how do you do PDEs? It is one thing to measure quantities in the laboratory, but without PDEs, our mathematical descriptions of nature would be very poor indeed.
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Do not be compelled into thinking uncountable sets are necessary for doing mathematics. Even though the traditional set-theoretic foundation is as valid as anything from a technical standpoint, I think people have come up with more onthologically satisfactory answers by working in computable universes.
You might like to look into:
Ron Maimon's answer to "Was mathematics invented or discovered?" (for a classic but computable approach)
Type Theory this is the foundational theory I find most satisfactory
Even the main theory used by mathematicians (first-order logic + ZFC) has countable models, which means that there is a set which is countable (in meta-theory) but which satisfies the axioms of real numbers. The diagonal argument then becomes the statement that it is not possible to construct a bijection (inside ZFC) between $\omega$ and $\mathbb{R}$. 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.
Regarding the impossibility of working with differential equations, this is just not the case. To solve a differential equation is to find an algorithm which approximates the solution arbitrarily. This could be a closed form, an integral, or some numerical method. 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. Every single function and procedure treated in classical calculus is computable.
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. 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.
As 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). Their 'uncountability' can be seen as just a useful abbreviation.
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Suppose you have a square of area 1 cm.You want to calculate length of the diagonal.If you don't have notion of real number then you can not find length of the diagonal.Another way to say this that in a countable system you can always find some equation that has no solution in that system.For example in rational number system there exist no solution of the equation $x^2-2=0$. Note:
Mathematician always concern with mathematical structure in which every equation has solution.In natural number system there exist no solution of the equation $X+2=1$ .So mathematician extended system of natural number to set of integers.Again there exist no solution of the equation $x^2-2=0$ in rational number system.So mathematician extended the rational number system to real number system.Again the equation$X^2+1=0$ has no solution in real number.So mathematician extended real number system to complex number system.Indeed in complex number system every equation has solution.So they stopped...:)
Hope this helps.
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Mathematicians don't compute things. Computers compute things.
Mathematicians are not limited by computation, much like Turing machines are not bounded by $2^{1024}$ gigabits of tape.
The real numbers are a tool, and it turns out to be useful. Since mathematicians are utilitarians, they like useful things. So they use the real numbers. If you want to ask why do physicists need the real numbers, you should ask that on Physics.SE or some other community centered around physics.
Why do mathematicians like the real numbers? Because they are closed under taking limits, which makes them ideal for talking about approximations by rational numbers, something we can easily comprehend (this is true at least in principle the notion of a rational number).
Since these are closed we don't have to worry and keep track whether or not our sequence is convergent, is the sequence itself is computable, do these properties hold or not. We have an arbitrary sequence, if we know it's a Cauchy sequence, then we know that it has a limit.
This is why mathematicians tend to embrace the axiom of choice when they work outside "the countable domain". When you want to make a statement about "all commutative rings with a unit", you don't want to start adding conditions on and on and on, that will help your proof go through. You want to say "Take any such and such object, then we can do this and that". Period. Utilitarianism galore.
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If you want to take a serious look at 'Math without Reals', you could do worse than to read Errett Bishop's book, 'Foundations of Constructive Analysis'. Over the history of modern math, there have been several people who have set out to extirpate the reals, as it were: Kronecker, Brouwer, etc. Bishop is known for delivering the maximum of useful results with the minimum of philosophical decoration.
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Simply put, because they are the most powerful natural descriptor of quantity that we have.
To elaborate only briefly, we can easily describe certain real numbers that we encounter regularly ($\sqrt{2}$ or $\pi$). Throw in the fact that most of our theorems are more powerful and general when proven for the Reals (and often simpler) and you have reasons to include them and none to remove them.
Indeed, Reals are a very natural thing to conceive of - that alone would suffice. Go outside and tell me that the world is merely made up of Rationals.
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We don't need real numbers, in the sense that we could do without them if we really had to.
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. Or, in exponential decay, the ratio between the instantaneous relative rate of decay and the half-life is $\log 2$, another transcendental.
Still one can have a purely rational mathematics, where instead of talking about transcendental ratios as numbers in themselves, we speak about processes that can produce ever finer rational bounds for the ratios, to any desired precision. So instead of having a single number $\pi$ we would have a succession of rational intervals, plus a theorem you can get the circumference of a circle with diameter 1 to fit into any of these intervals if you compute it precisely enough. We would then go on to prove a lot of theorems about how such sequences-of-ever-tighter-rational-bounds relate to each other and behave in different situations.
However, during this proof program we would find that there is a lot of standard boilerplate arguments about such sequences that we find ourselves repeating again and again. Being mathematicians, our instinct is to generalize, so we'd want to combine those boilerplate arguments into a set of clean reusable lemmas with assumptions that are easy to work with.
Once we completed this work, and after polishing by a few generations of students and textbook authors, we would find that we had reinvented the reals, in the guise of a general theory of sequences-of-ever-tighter-rational-bounds. Even though our eventual interest would still be the rational bounds, the general theory is easier to apply in arguments.
We find that our general theory applies to more sequences-of-ever-tigher-rational-bounds than the computable ones. That doesn't really bother us, because it still produces true results when applied to sequences that we can compute, and explicitly excluding the noncomputable ones would be more work for no clear benefit. In the few cases where it is of particular interest whether the resulting sequence is computable or not, we can go beyond the theory and do more careful work closer to the purely-rational substratum. But in most cases that is not important enough to us to make the increased complexity of doing so seem worthwhile.
It's not as if the multitude of reals that we don't need cost anything as long as we're not using them -- the theorems about reals that we need wouldn't be any easier to prove if we excluded the unnecessary ones.
In a few cases it would be technically convenient to have an enumeration of the reals to work with -- but it is usually routine to replace that with an enumeration of a dense subset of reals plus an assumption of continuity. Certainly that bit of footwork is much less work than what it would take to add in explicit restrictions to "interesting reals" in every theorem about reals we prove.
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Your argument, which can be rephrased as:
There are only countably many real numbers which could ever be defined, so all the rest are useless
is fallacious. There are plenty of times in mathematics when we want to deal with large sets, without caring whether we can explicitly define all of them.
As an example: compact Hausdorff spaces have properties which make them interesting to mathematicians. But any countable example must have an isolated point, and we'd like to know if there are examples without isolated points.
An obvious example of a compact Hausdorff space with no isolated points is given by closed interval of real numbers, such as $[0,1]$. We can't define every element of $[0,1]$, but we can still say interesting things about it - for example, it is compact.
On the other hand, let $A$ be the set of all definable elements of $[0,1]$. Then $A$ and $[0,1]$ have precisely the same sets of definable elements, but $A$ is not compact (by the Baire category theorem). So even though we can't define any elements of $[0,1]\setminus A$ explicitly, these undefinable numbers do have an effect which we can define.
Why is this? One answer comes from talking about induction principles. You're probably familiar with the principle of mathematical induction - if $T$ is some set of natural numbers such that $1\in T$ and $n\in T\Rightarrow n+1\in T$, then $T=\mathbb N$ - but there are versions which apply to much larger sets than the natural numbers, and even to objects like the class of all sets which are too large to be sets. The $\in$-induction scheme is the following:
Let $\phi$ be a property of sets such that whenever $\phi$ holds for all elements of a set $x$, it must hold for $x$ as well. Then $\phi$ holds for all sets.
If you're used to thinking about induction in the following way:
We have $1\in T$, so we must have $2\in T$, and then we must have $3\in T$, and so on...
then it might seem confusing to induct over all sets. There are certainly examples of sets that can't be defined; indeed, any real number has a unique representation as a set. Yet we can still induct over all sets; it seems as if mathematics is somehow allowing us to make infinitely many calculations at once.
The reason we can do this is that induction principles are proved using arguments by contradiction, which turns out to be an extremely powerful tool. I believe that in constructivist logic (where statements can be neither true nor false) then there really is no extra benefit that you can get by including numbers that can't be defined, so your question does get to the heart of some pretty deep mathematical logic.
For further reading, have a look at this fictional dialogue written by Timothy Gowers.
It is much harder to avoid "unnecessary" real numbers than to accept them.