Question

The utopian situation in mathematics would be that the statement and the proof of every result would live "in the same world", at the same level of mathematical complexity (in a broad sense), unless there were a good conceptual reason for the contrary. The typical situation would be for a proof in finite combinatorics to be proven purely within the realm of finite combinatorics, a statement about integers to be proven using only the rationals (perhaps together with some formal symbols such as $\sqrt {2}$ and $\sqrt{-1}$), and so on. When the typical situation breaks down, the reason would be well-known and celebrated.

The prototypical field where things don't seem to work this way is Number Theory. Kronecker famously stated that "God invented the integers; all else is the work of man."; and yet, the real numbers (often in the guise of complex analysis) are ubiquitous all over Number Theory.

I am sure that this question is hopelessly naïve and standard but:

1. What is the high-concept explanation for why real numbers are useful in number theory?
2. What is the "minimal example" of a statement in number theory, for whose "best possible" proof the introduction of real numbers is obviously useful?

An alternative way of framing the question would be to ask how you would refute the following hypothetical argument:

"We know that calculus works well, so we are tempted to apply it to anything and everything. But perhaps it is in fact the wrong tool for Number Theory. Perhaps there exists a rational-number-based approach to Number Theory waiting to be discovered, whose discoverer will win a Fields Medal, which will replace all the analytic tools in Number Theory with dicrete tools."

(This question is a byproduct of a discussion we had today at Dror Bar-Natan's LazyKnots seminar.)

Update: (REWRITTEN) There has been some discussion in the comments concerning whether proofs and statements living in the same realm is "utopian". The philosophical idea underlying this question is that, in my opinion, part of mathematics is to understand proofs, including understanding which tools are optimal for a proof and why. If the proof is a formal manipulation of definitions used in the statement of the claim (e.g. proof of the snake lemma), then there is nothing to explain. If, on the other hand, the proof makes essential use of concepts from beyond the realm of the statement of the theorem (e.g. a proof of a statement about integers which uses real numbers, or proof of Poincare Duality for simplicial complexes which uses CW complexes) then we ought to understand why. Is there no other way to prove it?Why? Would another way to prove it necessarily be move clumsy? Why? Or is it just an accident of history, the first thing the prover thought of, with no claim of being an "optimally tooled proof" in any sense? For one think, if a proof of a result involving integers essentially uses properties of the real numbers (or complex numbers), such a proof would not work in a formal somehow analogous setting where there are no real numbers, such as knots as analogues for primes. For another, by understanding why the tool of the proof is optimal, we're learning something really fundamental about integers.
I'm interested not in "what would be the fastest way to find a first proof", but rather in "what would be the most intuitive way to understand a mathematical phenomenon in hindsight". So one thing that would make me happy would be a result for integers which is "obviously" a projection or restriction of some easy fact for real numbers, and is readily understood that way, but remains mysterious if real numbers/ complex analysis aren't introduced.

Answer 1

The Gödel Speedup Theorem provides some explanation why real numbers (and variants) are useful in proving statements in number theory.

Real numbers, complex numbers, and $p$-adic numbers are second-order objects over the natural numbers. Thus a proof of a number theoretic fact using such analytical devices is formally a proof of that fact in second-order arithmetic. The Gödel Speedup Theorem shows that there is a definite advantage to using second-order arithmetic to prove elementary number theoretic facts.

Gödel Speedup Theorem. Let $h$ be any computable function. There is an infinite family $\mathcal{H}$ of first-order (indeed $\Pi^0_2$) statements such that if $\phi \in \mathcal{H}$, then $\phi$ is provable in first-order arithmetic and if $k$ is the length of the shortest proof of $\phi$ in second-order arithmetic, then the shortest proof of $\phi$ in first-order arithmetic has length at least $h(k)$.

Since computable functions can grow very fast, this shows that there are true number theoretic facts that one can prove using second-order methods (e.g. complex analysis, $p$-adic numbers, etc.) but any first-order (a.k.a. elementary) proof is unfathomably long. Admittedly, the statements produced by Gödel to verify the theorem are very unnatural from a number theoretic point of view. However, it is a general fact that second-order proofs can be much much shorter and easier to understand than first-order proofs.

Answer 2

A possible explanation is that $\mathbb R$ is one of the completions of $\mathbb Q$. And why does this matter? A reason is the Hasse, or local-to-global, principle. A minimal example of this is the Hasse--Minkowski theorem, which states that if $$Q(x_1,\ldots,x_n) = \sum_{i,j=1}^n a_{ij} x_ix_j$$ is a quadratic form with $a_{ij} \in \mathbb Z$ and $\det (a_{ij}) \neq 0$, then the equation $$Q(x_1,\ldots,x_n)=0$$ has a nontrivial solution in the integers if and only if it has a solution in the reals and all the $p$-adics.

Answer 3

Consider the following

Theorem. The number of integral solutions of $x^2+y^2+z^2=4n+1$ tends to infinity as $n\to\infty$.

The only proofs (known to me) use a good deal of complex analyis. I will think of further examples as time permits.

Answer 4

The question seems to assume or at least sympathise with something along the lines of

     God created the integers and the reals are derived therefrom. 

On the contrary, it seems just as plausible that the reals are our basic intuitive data. Meaning what?

1. In the history of cognition, measuring precedes counting. For example a lion can measure the length of his leap more easily than he can distinguish between 29 and 39 wildebeests.

2. The reals are the only Dedekind complete ordered field, if "cut" is taken in a set-theoretically naive way. See e.g., "Completeness of Ordered Fields" by Hall on arxiv. The integers are the unique discretely ordered subring of the reals. The induction principle for the ring of integers (and the uniqueness of the one and only discretely ordered subring of the reals) is a consequence of the completeness of the reals: An inductive set of integers that missed some positive integer would give rise to a cut with no boundaries. The notion of continuum, in other words, can be taken as the big defining concept here, and the main underlying intuition.

3. Number Theory, or at least Diophantine Analysis, can be understood as the study of equations over the reals with solutions restricted to certain subrings. Does the question "What are the reals doing in number theory" then dissolve? Maybe.

Answer 5

Two of the most basic facts of algebraic number theory, namely the finiteness of the class number and the structure of units in rings of integers of number fields do not seem to be provable without the use of real numbers (or some use of the archimedean nature of real numbers).

Added: In fact, most finiteness results in arithmetic geometry seem to use this archimedean prime; this is the case of Mordell-Weil theorem, Mordell conjecture, finite generation of Galois cohomology groups of number fields, etc.

Afterthoughts: There are many branches of number theory and in some of them you can't even state the results without using real numbers as was pointed out elsewhere. Now, in algebraic number theory or arithmetic geometry, which does not suffer from this problem, the analogy with function fields is a quite powerful tool to try to guess what can be true, and if you look at the problem from this angle you realise that real numbers are more of a nuisance than a help : all the statements above can be proven for function fields where real numbers play no role, and many others like the Riemann hypothesis or the global Langlands correspondence still elude us in the number field setting. The fact that you have to use them to prove the above results seem to indicate that you cannot ignore this nuisance so easily... (despite the product formula that makes you believe that the information that you can extract from the archimedean prime should be readable from the others).

Answer 6

The Prime Number Theorem is a statement in Number Theory which can't even be stated, let alone proved, without real numbers.

EDIT: How about the statement that $\sum^N\phi(n)$ is asymptotic to $(3/\pi^2)N^2$ (where $\phi(n)$ is Euler's phi-function)?

Answer 7

In algebraic number theory, the class number formula, which involves real numbers, is a fundamental tool in order to understand the arithmetic of a number field. It provides a purely analytical formula for the class number, which is a fundamental invariant.

This formula is not just a nice result, it also enables to compute the class number with a computer (say for quadratic fields). I won't enter into details here, but the point is that if you know in advance that some quantity is an integer, then it suffices to approximate it within less than one half in order to determine it. Thus a computer is able to determine the class number exactly.

Of course, one can argue that a computer always works with a finite approximation, rather than with an actual real number. But it definitely seems that working with (approximations of) real numbers is useful.

More generally, it seems some number-theoretic functions can be computed more efficiently if one allows the use of real numbers (I am not an expert, but I think that the computation of Bernoulli numbers is an example of this phenomenon). It would be interesting to make this rigorous in some sense.

Answer 8

Second addition in view of the EDIT of the answer:

Some examples now focused towards understanding. How valueable/inevitable real and complex numbers are for them, I don't know.

1. Q: What is a good way to understand that $\sum_{i=1}^n i$ is (up to small error) $n^2/2$. A: Integrate $x$ from $1$ to $n$. (Of course, this precise example is easily handled otherwise but for illustration)

2. Q: What is a good way to understand the size of $\sum_{i=1}^n i^{-1}$. A: Integrate $x^{-1}$ from $1$ to $n$.

3. Q: What is a good way to understand the size of $\sum_{i=1}^n d(i)$ where $d(i)$ is the number of divisors of $i$. A: Count the points 'under' the hyperbola $xy=n$. See here and here.

4. Q: What is a good way to understand units in rings of algebraic integers. A: Consider the points (in $R^k$) obtained by taking the logarithms of the absolute values of their $k$ (essentially) different imbeddings into the complex numbers.

5. More generally, results on linear forms in logarithms are a main tool in the study of certain Diophantine equations.

6. Q: What is a good way to understand the number of primes below $x$. A: Observe that the probabilty of a number $y$ to be prime is $y / \log y$ and integrate, getting $Li(x)$. Some details regarding a proof of the precise quality of this approach still need to be established ;).

7. In various places one will stumble somewhere in number theory over a $Q$-linear map or linear recursion. To undertand them one wants a convenient way to handle the roots of the attached characteristic polynomials, not just as formal constants.

And the list could go on and on.

I am not sure anything of this is what you are looking for.

---

There are already many answers, still I believe I have something to add.

The question talks about Number Theory (without any qualification). However, the following result is a number-theoretic one.

Let $a$ be an irrational number, then the sequence of numbers $ap$ where $p$ runs through the primes is equidistributed modulo $1$.

How to even state this without the reals, let alone prove it?

Actually, it is my impression, that many (if not most) people working on things like measures of irrationality of certain numbers, proofs of transcendence results, and so on, self-identify as Number Theorists (also look through the Number Theory MSC classification, in particular 11Jxx 11Kxx). And, results of this form are also frequently published in number theoretic journals.

Thus, without starting any philosophical discussion, but purely based on the everyday practise of Number Theory, I would say, obviously one needs reals numbers in Number Theory.

Now, returning from the defence of certain fields of Number Theory to answering your question more in its spirit:

As said by many, the reals are one completion of the rationals, sometimes it is useful to work in a complete structure.

And, since somewhat frequently in this thread the objection comes up that this or that example of an application of the reals could be avoided by doing something inconvenient, I would just like to mention that this seems to be true in many other (than Number Theory) contexts too.

---

To elaborate on the last statement a bit, and going down a slightly different line of argument:

if one wants to understand questions only involving extremely classical notions in number theory, say something like:

How does the set of divisors of a (typical) integer look like? (of course, made precise in some form)

Then, it turns out that answers to these 'discrete' questions can be obtained by 'approximating' the discrete real world by a 'continous model', think of distributions much alike as one finds them someplace else. Key word: probabilistic number theory, see for example a book by Tennenbaum or the book 'Divisors' by Hall and Tennenbaum. (And, this is not, or not mainly, the approach where one replaces the integers by some model of the integers with nice probabilistic properties, say Cramer's model; but by contrast very concrete statements about the true natural numbers that are best expressed using continous approximations.)

Also, the classical questions (mentioned by Gerry Myerson) of counting primes and other numbers defined by arithmetic properties, or derived quantities, can be consider as part of this.

So, now, could it be the case that one can replace all this by a purely discrete approach?

In some sense this seems as good a question to me as to ask, why one does, say, statistical physics rather than tracking each particle individually. Or, why all these differential equations in fluid dynamics, wouldn't it be possible one could get more insight tracking the particles constituing the liquid individually using some discrete model?

Perhaps one answer to the number theory question is that also the real world of integers (as the real real world) is simply way too complicated to allow for exact (discrete) answers, and therefore one has (at least for now, but basically I'd guess 'forever') use continous approximations to gain some insight. (Whether you encode them by reals or somehow else seems besides the point.)

And, to answer the final question, whether it would be possible that somebody comes along and sees that actually the integers are not as complicated as we think they are, well, what is truly impossible? But some of the most famous open problems in mathematics are way below such an insight; for example, what is the Riemmann Hypothesis, just an imprecise statement on a crude parameter of the set of primes.

Answer 9

I think the key intuition is that reals are easy to work with and integers are hard. Linear programming can be done in polynomial time; integer programming is NP-hard. Real closed fields are decidable, but not their integer equivalent. Diophantine equations are intractable in general, but complex analysis (which can be formulated in real terms, if desired) gives us pretty much all the information we'd want about (real) polynomials, the easiest sort of holomorphic function. Etc.

Answer 10

A possible candidate for a "minimal" result about integers that is a "projection" of a result about reals: the group structure of the solutions of the Pell equation $x^2-dy^2=1$ for $d$ a nonsquare positive integer.

The (positive branch of the) curve $x^2-dy^2=1$ has parametric equations $x=\cosh t$, $y=\frac{1}{\sqrt{d}}\sinh t$, and hence it is isomorphic to the real numbers $t$ under addition. If we let $x_t=\cosh t$, $y_t=\frac{1}{\sqrt{d}}\sinh t$ then the sum of points $(x_s,y_s)$ and $(x_t,y_t)$ is $(x_s x_t+dy_s y_t,x_s y_t+y_s x_t)$.

It follows that the sum of integer points is an integer point (and likewise the difference). Hence the integer points form a subgroup of the points on the curve. Moreover, we can see that this abelian group is infinite cyclic (assuming a nontrivial integer point), generated by the integer point with smallest nonzero $t$ value.

Answer 11

Someone once suggested on MO that this is because on the one hand Matiyasevich's theorem shows that no algorithm can solve Diophantine equations over $\mathbb{Z}$ (and the corresponding result is not known over $\mathbb{Q}$), and on the other hand this is possible over $\mathbb{R}$ because of quantifier elimination. It is also possible over $\mathbb{Q}_p$ for every $p$, and this suggests that one would really like to have some sort of local-to-global principle so that special types of Diophantine equations can be solved, and if not, to understand how it fails...

Anyway, I am still not sure I totally agree with the sentiment of the first sentence. Think of it this way: if I want to understand a category $C$ and I understand a category $D$ very well, and moreover I have a functor $F : C \to D$, then it stands to reason that I can learn something about a problem in $C$ by translating it to a problem in $D$, where I understand what is going on very well. Nothing is mysterious about this process other than possibly the construction of the functor $F$, and in the case of number theory $F$ is something like the analytification functor from varieties over $\mathbb{Z}$ or $\mathbb{Q}$ to varieties over $\mathbb{R}$ or $\mathbb{C}$ and the existence of this functor is not so hard to understand.

Answer 12

To continue on a theme of a couple answers thus far:

The real numbers, along with the $p$-adics for each rational prime $p$ are the completions of the rational numbers. Regarding arithmetic objects (such as quadratic forms or algebraic varieties in general, among others) defined over $\mathbb{Q}$ as being defined over these completions endows them with additional structure that one can exploit.

This "exploitation" comes in various forms, including the Hasse principle mentioned in other posts (points over $\mathbb{Q}$ beget points over completions and the latter are some kind of evidence for the former).

Then there is also the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves and its various generalizations. These purport to relate global ("over $\mathbb{Q}$") things like values of a global $L$-function to a host of data that is germane to the various local completions of $\mathbb{Q}$. The period that turns up in the BSD conjecture, for example, is really a facet of the elliptic curve regarded over the reals where one can integrate forms in the usual sense. The Tamagawa numbers, on the other hand, come from the various $p$-adic completions of $\mathbb{Q}$.

I guess the theme in all of this is that global fields lack structure that their various completions possess, and embedding them in these completions gives extra insight and "handles" to grab onto in order to study the global field and the objects defined over it.

Answer 13

I would like to say that at least to some extent we need the real numbers and not just rational approximations.

Suppose we were being completely formal and we asked the question, ``Is there some algebraic number $\beta>0$ such that for all $n$ there exist $p_n , q_n \in \mathbf{Z}_{> 0}$ with $\beta \ne \frac{p_n}{q_n}$ and there exists a fixed $\delta >1$ such that $|\beta - \frac{p_n}{q_n}| < \frac{1}{q_n^{1+\delta}}$'' ?

The answer of course is no by the Thue-Siegel-Roth Theorem. But moreover, we know that we can only discover a measure zero subset of the transcendental numbers in this way. So while we can approximate real numbers by rationals all day long, the amount of information we get out of a rational approximation can vary wildly.

Answer 14

I'll admit I'm not number theorist but here is my take on why the field has to embrace the completions of $\mathbb{Q}$. It is a simple reason: $e$. Other branches of math have run across interesting numbers that aren't algebraic which deserve studying in their own right. So not only should number theory as a whole (not necessarily every practitioner) should take the reals as valid objects of study.