40
$\begingroup$

I apologize if this is a somewhat naive question, but is there any particular reason mathematicians disproportionately study the field $\mathbb{R}$ and its subsets (as opposed to any other algebraic structure)?

Is this because $\mathbb{R}$ is "objectively" more interesting in that studying it allows one to gain deep insights into mathematics, or is it sort of "arbitrary" in the sense that we are inclined to study $\mathbb{R}$ due to historical reasons, real-world applications and because human beings have a strong natural intuition of real numbers?

Edit: Note that I am not asking why $\mathbb{Q}$ is insufficient as a number system; this has been asked and answered on this site and elsewhere. Rather, why, in a more deep sense, are $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ so crucial to mathematics? Would we be able to construct a meaningful study of mathematics with absolutely no reference to these sets, or are they fundamentally imperative?

$\endgroup$
  • 15
    $\begingroup$ The first question is do mathematicians study real numbers disproportionately? $\endgroup$ – Dr Xorile Apr 28 '16 at 3:16
  • 5
    $\begingroup$ @ASKASK: If one were in the mood, one could easily play devil's advocate and argue that, in fact, every feasible measurement in the real world is only a rational number, since measuring a real number would require "infinite precision". But there are already many discussions on the internet about this, so let's not get bogged down... $\endgroup$ – Will R Apr 28 '16 at 4:14
  • 12
    $\begingroup$ The OP may be interested in spending some time reading this dialogue (or should I say trialogue?) concerning the real number system, written by Timothy Gowers of Field's medal fame. I'm not sure it entirely answers your question, but it's certainly related and it's probably worth a read. $\endgroup$ – Will R Apr 28 '16 at 4:18
  • 5
    $\begingroup$ Concerning the recent edit: you appear to have changed the nature of the question. Before, you were (and in the title, you still are) asking specifically about real numbers. But in your edit, you are talking about the whole gamut: $\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}$. This is a very different question: you are essentially now asking "why is mathematics about numbers?", to which my only answer is, "what else are you expecting?" Numbers are crucial to mathematics because mathematics, as a subject, has been built around them; they are part of the definition of the word. $\endgroup$ – Will R Apr 28 '16 at 4:38
  • 15
    $\begingroup$ You might be surprised by how few mathematicians study $\mathbb R$. There are lots of areas of mathematics, and many of them don't have much to do with $\mathbb R$. $\endgroup$ – Robert Israel Apr 28 '16 at 4:52
8
$\begingroup$

There's no way to do justice to "Why is mathematics about real numbers?" within the length constraints of a Math.SE post, but here are some relatively philosophical observations and opinions (meant to be a bit provocative, in the spirit of answering a soft question).

First, as multiple people have commented, the real numbers are not universally regarded as the be-all/end-all number system. In The Road to Reality, for example, Penrose argues that complex numbers are more fundamental for physics.

Setting that aside, why do we count, and where did natural numbers, integers, and rational numbers come from? I'm not a historian, so everything below should be regarded as a parable, biased by modern mathematical training.

Counting (both the possibility and the ability) arises from the tension between variation and uniformity in the natural world:

  • Thanks to variation, there are "different types of thing": that piece of granite, that oak tree over there, the pine tree next to it.... If we look closely at the natural world, we find it to be made of unique objects, to occur in unique, irreproduceable events. In fact, the notion of "event" is our way of cutting the solid stream of existence into temporal and spatial chunks. As Heraclitus said, you cannot step in the same river twice.

  • Thanks to uniformity, there are recognizable "classes of things": rocks, trees, snowflakes, stars, sunrises.... No two rocks (or trees, or snowflakes...) are exactly alike. At the very least, they're "in different places" or "at different times" (otherwise they'd be identical).

Once the natural world is observed to contain classes of things, "counting" is a reasonable way to measure "how many/how much". In (a paraphrase of) Kronecker's famous quotation, The integers alone were created by God. All else [in mathematics] is the work of Man. To the contrary, the natural numbers (and therefore the integers) were created by us, as well, an abstraction for enumerating distinct objects similar enough to group together for some purpose.

To make a long story short:

  • An integer is a measure of additive comparison between two natural numbers. That is, it's a thing comprising a relationship between two other things. The standard construction of the integers in set theory merely formalizes this: An integer is an equivalence class of ordered pairs $(m_{1}, n_{1})$ of natural numbers, with $(m_{1}, n_{1}) \sim (m_{2}, n_{2})$ if and only if $m_{1} + n_{2} = m_{2} + n_{1}$.

  • A rational number is a measure of multiplicative comparison between two non-zero integers. The standard construction of the rationals blah, blah, blah.

It's unsurprising that both abstractions were invented: If two people have flocks of sheep (say), it's natural to ask "who has more sheep, and by how many?". It's natural to represent debts as negative integers. If a flock of sheep (well...) must be divided among several people, it's natural to ask "What is each person's portion?", and to use rational numbers to represent the answer.

The real numbers obviously arose many centuries later, under pressures of Archimedes' method of exhaustion (for which one needs numbers representing "limits of rational sequences"), and were formalized two millennia after that in order to put calculus on a solid logical footing.

I won't even touch the complex numbers, partly for last of time and space (heh), but mostly because Penrose (and many others) do so far more competently, with the depth the subject deserves.

$\endgroup$
  • 3
    $\begingroup$ There is some evidence that natural numbers are not a convention "created by us", by actually an a priori function of the brains of humans and some other animals. I've seen a video of Noam Chomsky hypothesizing that an evolutionary mutation was required to permit the brain to entertain limitless counting, and that language would be an extrapolated development after that. $\endgroup$ – Daniel R. Collins Apr 28 '16 at 17:34
  • $\begingroup$ See: youtube.com/… $\endgroup$ – Daniel R. Collins Apr 28 '16 at 17:38
  • 3
    $\begingroup$ You're not really addressing why, specifically, we use real numbers instead of some other extension of the rationals, which is, I think, the most important aspect of the question. $\endgroup$ – Kyle Strand Apr 29 '16 at 21:36
  • 1
    $\begingroup$ @Kyle: Fair enough. I was musing on why the natural, integer, and rational numbers arise naturally for humans, given the OP's clarification, "Note that I am not asking why $\mathbb{Q}$ is insufficient as a number system; this has been asked and answered on this site and elsewhere. Rather, why, in a more deep sense, are $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ so crucial to mathematics? Would we be able to construct a meaningful study of mathematics with absolutely no reference to these sets, or are they fundamentally imperative?" $\endgroup$ – Andrew D. Hwang Apr 29 '16 at 23:17
  • 2
    $\begingroup$ @Daniel: Interesting; thank you for the video link. Again, I was deliberately being a bit provocative in saying humans invented the natural numbers, but although Peano's axioms for the natural numbers are simple and close to everyday experience, I don't see that the natural numbers exist in a manner independent of human minds. Practically speaking (e.g., in terms of classical computation), I see no way to distinguish the infinite set of naturals from a sufficiently long interval $\{0, 1, 2, ..., N\}$. (Maybe this is a semantic issue with the meaning of "exist".) $\endgroup$ – Andrew D. Hwang Apr 29 '16 at 23:46
49
$\begingroup$

I've often asked myself the same thing, and this is what I tell myself. $\mathbb R$ is (up to order-preserving field isomorphism) the only totally ordered, complete field. This is pretty big news, because these two nice structures lead to so many others we find useful to study in math. $\mathbb R$ (and more generally $\mathbb R^n$) is so great because a plethora of these fundamental "structures" studied in math are present in (at least some subset of) $\mathbb R$. When we learn of new concepts, it's natural (crucial) to seek examples, and we often find solace in the usual first stop -- $\mathbb R^n$.

Here's a poor-at-best survery of some of the aforementioned structures that $\mathbb R$ has.

Algebra

  • Group -- we can combine elements, i.e., $a + b$, invert them, i.e., $a^{-1}$.
  • Field -- we get more ways to combine elements, $+, -, \times, \div$.
  • Ordered field -- we get to do things like transitivity, i.e., $a < b \wedge b < c \implies a < c$, and "add inequalities", i.e., $a \leq b \wedge c \leq d \implies a + c \leq b + d$.
  • Vector space -- linear algebra's pretty important. Arrow-like addition is very physical.

Analysis

  • Completeness -- analysts love sequences... to converge. This allows for a lot of "take a sequence..." arguments which start with a probably-desired sequence that ends up being Cauchy.
  • Compactness -- we always want to exploit compactness in analysis, and $\mathbb R^n$ has a particularly nice characerization of it.
  • Hilbert Space -- we all love Hilbert space. Orthogonality is a useful tool. So is the spectral theorem.
  • Measure space -- measuring is very physical, and crucial for integrating! $\mathbb R$ is the natural setting for the famous Lebesgue measure, and all measures map into the "subset" $[0,\infty]$ of $\mathbb R$. For Riemann integration, the (Darboux) definitions hinge on the least upper bound property of $\mathbb R$.

Geometry

  • Metric Space -- we can measure distances $d(p,q)$ between points. This is very physical. The triangle inequality is here too, which is even more useful in normed spaces, where it reads $\|u + v\| \leq \|u\| + \|v\|$, because it leads to many useful estimates in analysis.
  • Manifolds -- things that by definition locally look like $\mathbb R^n$. Many "objects" that we deal with early in math are manifolds (we just didn't know it at the time).
  • All of the separation axioms (Hausdorff, regular, normal, ...).
  • All of the countability axioms (separable, Lindelof, ...).

Not shown (for the sake of space and the inevitable lack of completeness) is the interrelatedness between many of these properties for $\mathbb R$, which is another indispensable virtue of $\mathbb R$.

$\endgroup$
  • 1
    $\begingroup$ This similar question has a very good answer. $\endgroup$ – Jon Warneke Apr 28 '16 at 5:38
  • 5
    $\begingroup$ Absolutely. Manifolds are basically defined as "let's keep all the nice properties of $\mathbb{R}^n$". $\endgroup$ – Turion Apr 28 '16 at 11:19
  • 1
    $\begingroup$ Isn't $\mathbb R$ the only totally-ordered complete Archimedean field? $\endgroup$ – Ben Millwood Apr 28 '16 at 21:56
  • 2
    $\begingroup$ Given a totally-ordered field, completeness is equivalent to the least-upper-bound property, and the Archimedean property follows from the least-upper-bound property. $\endgroup$ – Jon Warneke Apr 28 '16 at 22:57
  • 1
    $\begingroup$ Note that it's possible to define a somewhat-analogous concept of "computable completeness" that does apply to the computable reals, even though the traditional Dedekind (or equivalent) definition clearly does not apply: math.stackexchange.com/a/418011/52057 $\endgroup$ – Kyle Strand Apr 29 '16 at 21:45
7
$\begingroup$

A nice property of real numbers is that they are complete: every Cauchy sequence converges.

In analysis, mathematicians like to study spaces that are complete. People study Banach spaces rather than ordinary normed spaces; study Hilbert spaces rather than ordinary inner product spaces. A space that is not complete does not have as nice properties as complete spaces.

Ever since Pythagoras' disciple discovered that $\sqrt 2$ is irrational, that was the beginning that signals that the rational numbers $\mathbb{Q}$ is not enough to represent all quantities.

However, there are some "problems" with the real numbers, and I know at least one professor (not to mention his name here) who does not believe in the real numbers. His reason, if I remember correctly, is that once you go deeper, a real number is not just a string of decimals: it is a equivalence class of Cauchy sequences. Not only is each Cauchy sequence infinite, each equivalence class is infinite (uncountable I think). This is the price to pay for dealing with real numbers.

$\endgroup$
3
$\begingroup$

It's also worth noting that this representative of is a strong bias towards analysis. An algebraist generally cares minimally about $\mathbb{R}$, generally field extensions of $\mathbb{Q}$ and $\mathbb{F}_p$ are far more interesting. The badly named so-called Real Numbers are very much unimportant in Combinatorics and Number Theory compared to other sets, but mathematicians spend much more time talking about the Real Numbers because there is a strong bias, both in collegiate education, and younger education, towards analytic topics over algebraic or topological ones.

$\endgroup$
  • 1
    $\begingroup$ I'm not sure I agree with this. Even if the thing I'm really interested in is algebraic extensions of $\mathbb{Q}$, I will end up studying completions like $\mathbb{C}$ and $\mathbb{Q}_p$. If I think I'm a combinatorialist, I might still end up using generating functions and doing countour integrals for asymptotics (e.g. to prove Stirling's approximation). Perhaps the reals are over-emphasized in the standard curriculum, but they don't cleanly belong to "analysis", and I would think that the vast majority of mathematicians care about them in some form or other. $\endgroup$ – Slade Apr 28 '16 at 19:42
  • 2
    $\begingroup$ I mean, yes it comes up (I'm not saying it doesn't), but the idea of R as foundational is basically an analysis thing. $\endgroup$ – Stella Biderman Apr 28 '16 at 19:49
0
$\begingroup$

An amazingly large part of Riemann-Stieltjes integration theory, as given in Apostol's Mathematical Analysis, can be developed in the setting of rational number system. Completeness property of real number system is needed, only when we try to prove that every continuous function is integrable.

It's the existential proofs of various theorems in Mathematics, that are the things of real challenge. They take 90% energy of a mathematician. Applied mathematicians and theoretical physicists don't take bother in proving existential theorems, and thus, they save their 90% energy.

The whole edifice of Dedekind's construction of real number system is meant to prove: There exists a complete ordered field.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.