
Calculus Is Deeply Irrational - ColinWright
https://mathenchant.wordpress.com/2019/08/16/calculus-is-deeply-irrational/
======
geebee
Love it.

I remember reading (might have been here) that math and art both get difficult
at the same moment for the same reason. When you're a kid, you live in the
rational world. By this, I mean numbers that can be expressed as the ratio of
two integers. Human creations are rational. The volume of a square is a neat,
tidy equation. So is the area of a square. You can draw them easily, too,
using a ruler and clean nifty lines, and they look great. Squares are all over
human creation. You can draw a car with straight lines and squares. Wheels and
other things bring in this inconvenient number, pi, that is "irrational", but
let's just go with three point blah blah and it'll be fine. At least the
curvature is constant.

So, where are the squares in nature? Hell, where are the circles. Where is the
constant curvature. How do you draw a leaf, a tree, a face? How do you
calculate the surface volume of a leaf, or the volume of a tree?

All of a sudden, you can't measure it with the numbers you know. There is no
neat ratio of integers that will calculate the volume of that tree trunk. Or
even the volume under an easily expressed mathematical equation on a graph. In
fact once you start measuring nature, rather than the things people make,
rational numbers aren't _anywhere_. All of a sudden, you have to deal with
limits, sequences, strange numbers that can be made arbitrarily close to zero
as other numbers approach infinity. It turns out _every_ number is
"irrational", pretty much nothing is _rational_. So, instead of irrational,
let's call it Real.

Where do math and art get hard? When you start to describe things as they are,
rather than as we imagine the to be. You know, Real.

~~~
com2kid
> So is the area of a square.

Try to cut that square in half diagonally and things get irrational really
fast!

FWIW I thought calculus made a lot of sense and helped make the world make
more sense. Algebra is a fancy set of rules to manipulate rather abstract
symbols, calculus actually explains how real things work!

~~~
geebee
Good example. It's amazing how quickly you leave the rationals, even with a
square, you almost immediately need numbers that can't be expressed as the
ratio of two integers... historically, was that the first encounter with an
"irrational" number (Pythagorean theorem applied to a right isosceles
triangle)? I do remember the proof from number theory about 20 years ago,
though I could never recreate it now from memory.

------
ska
It would ruin the "joke", but probably should have be deeply transcendental...

The real numbers are much stranger than they first appear, and a lot of early
pedagogy is designed so that you don't look at this too hard.

After all, almost all reals are transcendental (but interestingly we only have
proved a small handful of them).

Most numbers people have used in "regular life" are algebraic, obviously, but
they are a countable subset so occur almost never, in the set.

~~~
hyperpallium
When do the reals start making sense (beyond hand-waving explanations)?

~~~
ska
Part of any good introduction to real analysis is going to be walking you
through a bunch of this stuff until you first understand how much you had
glossed over the details, then than the details are strange, and finally that
you have a working understanding or at least comfort with techniques to reason
about this.

Is that the same as making sense? Well to paraphrase a physicist, you don't so
much understand it as get used to it.

One of the fun things about teaching this material is that usually students
have been through years of calculus prior to it, and you get to watch the
moment when they realize all of this stuff has been "hiding in plain sight".

~~~
hyperpallium
John Von Neumann [https://math.stackexchange.com/questions/11267/what-are-
some...](https://math.stackexchange.com/questions/11267/what-are-some-
interpretations-of-von-neumanns-quote)

Does the followong seem right to you?

Intuition often comes from relating a new thing to something known. So maths,
as an abstraction of reality, initially has many sources of intuition.

Later maths never has exactly the same patterns as earlier maths (it's already
abstracted; so _same patterns_ would be the _same thing_ , though it cam
build-on). Eventually, it doesn't relate to anything known, and you have to
create that familiarity from scratch.

That's hard... but if some stuff became known in the first place, why not this
too? (One counter is that the other stuff was instinctively known, e.g. 3D
space, or at least our minds are pre-shaped to know it, e.g. language).

Assuming math is open-ended, there'll always be new stuff that doesn't relate.

------
Smaug123
Propp himself (the author) wrote a paper in 2012 called Real Analysis In
Reverse, which he links to in the post but doesn't really call attention to.
It's a lovely paper, and it talks about precisely what properties of the reals
are enough to tell you that you really are dealing with the reals - like his
example of the Constant Value Theorem from the post.

[https://arxiv.org/abs/1204.4483](https://arxiv.org/abs/1204.4483)

------
btilly
One correction. The root of rational and irrational is the word "ratio".
Rational numbers are ratios of integers. Irrational numbers are not ratios of
integers.

It is a linguistic coincidence that we wound up with words that has another
reasonable meaning.

~~~
smallnamespace
> linguistic coincidence

It's _not_ a coincidence. The root of both ratio and rational come from latin
_ratio_ , gerund of _reri_ , 'to calculate, to reckon, to think'.

To the ancients, just like us, computation, thinking, and reasoning are
linked. Irrational numbers were literally numbers that 'could not be reckoned'
in the normal sense.

You can even go further with this analogy. The halting function is not
computable because there is no finite procedure that tells whether any program
halts. But this objection to infinity is also what the ancient Greeks had to
continuity (Zeno's paradoxes) and irrational numbers, as there is no finite
procedure that results in an exact answer. In fact, there is no _bounded_
algorithm that can distinguish sqrt(2) on all possible numeric inputs.

~~~
xamuel
Nonsense, the digits of sqrt(2) most definitely _are_ computable and there are
bounded algorithms that can check, given inputs n and k, whether or not n is
the kth digit of sqrt(2).

~~~
smallnamespace
I said 'all inputs' for a reason. If I give you an infinite sequence that is
sqrt(2) you cannot in finite time say it actually is.

Now in practice this isn't a problem, but it reveals the problem with
finitism.

~~~
moefh
That doesn't seem to be a fair argument, you can also do that with rational
numbers: if I give you an infinite sequence that is exactly 1/3, can you in
finite time say it actually is?

If you _choose_ to represent numbers as infinite streams of digits, then
obviously you can't compare them for equality in a finite amount of time.

The issue is: is it possible to use real numbers in a way that sidesteps this
problem? In general, it's impossible -- most real numbers are uncomputable! --
but for some useful subsets (beyond the rational numbers) it's possible.

~~~
smallnamespace
There's a fair point there, which is that some computations may look hard
simply because we picked a difficult representation, but I'm not sure that
applies here.

There's finite, exact representation for 1/3 in whole numbers, namely itself,
but as far as I know (?) there isn't one for sqrt(2) unless, say, your choice
of representation is the root of some polynomial. Is there a bounded procedure
that shows whether any two polynomials represent the same set of roots?

Getting back to the original etymological question, my point is that
_irrational_ in the sense of 'can't be reckoned, computed' is very close to
how we would see it. The Greeks simply had a different conception of
computation, one grounded in finitism and constructing things geometrically.

~~~
messe
> Is there a bounded procedure that shows whether any two polynomials
> represent the same set of roots?

Yes. Two polynomials share a root if their resultant vanishes.

------
wodenokoto
Irrational has two meanings:

“Not logical”, which by definition does not apply to math.

“Not a ratio”, e.g., a number that cannot be written as the division (ratio)
of two whole numbers.

It should be no surprise to anyone, that calculus deals with the latter, but
the title would allude to the article dealing with the former.

~~~
coldtea
Yes, and in fact the ancient Greek term, of those who invented the thing, was
neither ("illogical" or "not a ratio").

It was "alogos" (later "arritos"), with the meaning "inexpressible", "which
cannot be spelt out" \- in the sense that you can never fully write out e.g.
the square root of two the way you can a natural or a rational number like 2/3
or 1/4.

Unfortunately "alogos" also meant "illogical", so that's where the confusion
stems...

~~~
perl4ever
Well, maybe the point is that "illogical" _should_ mean inexpressible and the
whole issue is the wrong connotations of it.

------
hprotagonist
_Why does calculus involve so many irrational numbers?_

Because there are uncountably many more irrational numbers than there are
rational numbers.

~~~
roywiggins
There's uncountably many more noncomputables than computables too, but the
noncomputables hardly crop up, even though they come along with the reals
also.

~~~
jerf
If you did want to salvage a "calculus without real numbers" the computables
are probably more promising. All of the criticisms of the article still
entirely apply to them, but there's a better chance you could come up with
substitute definitions that may yield some useful concepts; they're closed on
a more useful operations. I believe this is an actual studied area of math.
But it's certainly more complicated than using real numbers. We may not teach,
ahem, "real" real numbers to high school students, but you'll get to it fairly
early in dedicated college math courses at the undergrad level.

~~~
inflatableDodo
I've wondered before if there could be a well defined set of numbers between
the integers and reals (perhaps with its own class of infinite set) that
nicely excludes the uncomputables. If there is, I suspect that it might be
what the universe is actually using and if I was going to go looking for it,
I'd follow the breadcrumbs from umbral moonshine and go sniffing around the
monster group.

~~~
roywiggins
Other than the rationals? Those are all computable. Or the algebraics.

Whether there's an infinity between the naturals and the reals depends on the
continuum hypothesis, which is independent of the usual axioms of set theory,
so you can pick.

------
tempsolution
I don't like clickbait... The article starts to suggest that calculus is hard
to reason about and "irrational". In the end all it is really saying is that
calculus requires irrational numbers, well doh! And guess what, you even need
complex numbers. And the question is not about "shopping" numbers, there is a
history to all of that. If you want something universal, use complex numbers.
Try to teach that to a pre-schooler and you will find that you might wanna
start with something simpler.

~~~
JackFr
> And guess what, you even need complex numbers.

That's not true, and that is the point. To prove the main theorems of
calculus, you don't need complex numbers, but you do need irrationals.

When you first learn calculus, you do one or two epsilon-delta proofs, and
then your teacher gets a little hand wavy about limits and you move on to the
real work of derivatives and integrals, cause the limits stuff intuitively
makes sense. When you continue on in Real Analysis or Topology of the Real
Line, you discover that your intuition lied to you, and concepts like open and
closed sets and intersections and accumulation points are important and are in
general non-obvious.

Counterexamples In Analysis

[https://www.amazon.com/Counterexamples-Analysis-Dover-
Books-...](https://www.amazon.com/Counterexamples-Analysis-Dover-Books-
Mathematics/dp/0486428753/ref=asc_df_0486428753/?tag=hyprod-20&linkCode=df0&hvadid=312168166316&hvpos=1o1&hvnetw=g&hvrand=1608085467130596869&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9067609&hvtargid=pla-514661979117&psc=1)

(Pdf version)
[https://pdfs.semanticscholar.org/a4e7/eb352e4c44bf75d8fabaf7...](https://pdfs.semanticscholar.org/a4e7/eb352e4c44bf75d8fabaf7594494151cb322.pdf)

------
ncmncm
"Thank you very much, but I will buy the Computable numbers, instead. They do
not require abandoning my sanity (although I am aware I must still abandon
rationality)."

In particular, the reals mean you accept measure theory, which in one of its
key results says you can build two spheres of radius x from one sphere of
radius x, and no holes. The computables don't go there.

There are infinitely many computable numbers, but only _countably_ many -- no
more than of natural or rational numbers. That turns out to be enough for
everything sane you want to do.

Of course you need altered versions of the key theorems of calculus, because
the computables are not continuous in the "real" sense. The numbers blur a
bit, instead, to cover the gaps, much as water manages to behave like a fluid
despite being made of nothing but discrete particles. The differences are a
PITA but keep you on the straight and narrow. You get the same answers for
everything that makes sense, and no answer for things that don't.

All the supposedly real numbers you will ever encounter are computable (too).
Roots, pi, e, anything representable with a Taylor series. So you don't really
give anything up.

Reals make a good enough approximation, which means you don't need to go
hungry. Pretend you're using reals. Nobody needs to know. Everybody else is,
too. The sane, anyway.

~~~
joppy
Measure theory also tells you that if you managed to make two solid balls of
radius 1 from cutting up a single solid ball of radius 1, then some of the
intermediate pieces you cut the original ball into must have been non-
measurable sets. Non-measurable sets are known to behave counterintuitively,
and are avoided when using measure theory (since they cannot be assigned a
volume).

Is “garbage in, garbage out” so unreasonable?

~~~
aeneasmackenzie
Why use a system where you have to check if you put garbage in?

~~~
joppy
You have to check if a number is zero before you divide by it, you need to
check if a number is positive before you take its log, you need to check it a
set is measurable before you measure it or integrate over it. I don’t really
see how any of these are significantly different.

------
hyperpallium
I always thought calculus should work equally well with rationals, because
they are continuous (in that you can always get closer...)

~~~
nerdponx
_rationals, because they are continuous (in that you can always get
closer...)_

You sort of _can 't_ get arbitrarily close. And that's the whole reason
calculus doesn't work for rationals.

Rationals fundamentally have "gaps", and correspond 1:1 with integers.

~~~
jpmattia
> _You sort of can 't get arbitrarily close._

Is there an obvious concrete example? I've never spent proper time studying
Real Analysis, and I confess I have the same intuition as OP: That you could
use rationals to approximate irrationals to arbitrary precision.

~~~
gianduja
You're right, you can.

What's special about real numbers is that, if you have a sequence of reals for
which the distance between two consecutive elements approaches zero, then
there exists a real number that's the limit of the starting sequence.

This isn't the case with rational numbers, eg. the sequence 1.4, 1.41, 1.414
... (EDIT: these are increasing approximations of sqrt(2)) satisfies the
hypothesis but there's no single rational number this sequence approaches.

This property is called completeness. The real numbers are a complete
topological space, whereas the rationals aren't.

~~~
Koshkin
To be fair, your sequence seems to approach 1.414141414... which _is_ a
rational number.

~~~
gianduja
Thanks, I was producing approximations of sqrt(2), I should have said so.

------
squirrelicus
This has nothing to do with calculus and everything to do with the fact that
there are an infinite number of irrational numbers between any two rational
numbers. Indeed, if you randomly select a real number, you can be _almost
certain_ that you will select an irrational number. Of course, _almost
certain_ has a specific rigorous meaning: the probability of not selecting (in
this case, an irrational number) is greater than zero, but less than all real
numbers, i.e. Infinitesimal.

Edit: spelling

Edit: the infinitesimal belongs not in the real numbers, but in the hyperreal
numbers, if you want to learn more

------
Nasrudith
I am reminded of the ironic irrationality of ancient thinking insisting that
rational numbers numbers must exist with things like Pythagoreans murdering
over it - apocryphal admittedly. But the same pattern appears for other
numbers and systems like the complex and imaginary and non-euclidian geometry
which caused "rational" people to be filled with such rage.

It is an interesting pattern that anyone who insists upon their "rational" or
"sane" systems in the face of reality tends to actually deeply unhinged.

------
scarmig
Naive question:

Can you build something like calculus on a number system weaker than the
reals, but stronger than the rationals? What's the weakest you could go?

~~~
sverona
It's unclear what precisely you mean by 'weaker', but if you mean with regard
to completeness - the reals are the so-called _completion_ of (that is, the
smallest complete metric space containing) the rationals; so you're kind of
stuck.

~~~
liuyao
You can weaken "completeness" though. In each of its many manifestations,
completeness asks for the existence of something for _all_ subsets (sequences,
functions, ...) of reals. We can instead ask for the ones that are definable
by finite means (aka computable, constructive).

To find (=construct) the extreme value, for example, you probably won't be
able to find a general procedure; it has to be treated case by case. Now
you'll appreciate how easy real numbers are. It gives the assurance that the
extreme values exist.

I would not have thought it would interest Hacker News readers, but here's
more about computable numbers:
[https://observablehq.com/@neobourbaki/computable-
numbers](https://observablehq.com/@neobourbaki/computable-numbers)

~~~
sverona
This is really cool. My advisor is a strict constructivist and will tell
anyone with ears about finitism after a few beers, and I've always found it a
useful philosophical position.

------
slaymaker1907
I liked his comment that irrationals really aren’t that unreasonable in the
common sense of the term. There are so many questions on Quora which
demonstrate people are really uncomfortable with transcendental/irrational
constants like pi and e. While you can approximate those as much as you like,
the true monsters like Chaitin’s constant cannot be approximated at all.

------
carapace
Sort of a tangent but uh...

[https://en.wikipedia.org/wiki/Surreal_number](https://en.wikipedia.org/wiki/Surreal_number)

> ...the surreal number system is a totally ordered proper class containing
> the real numbers as well as infinite and infinitesimal numbers...

Invented by Conway while studying Go.

------
your-nanny
My first inkling that I had no clue about the reals was when I first thought
hard about the fact that you can have a bijection between intervals of
different length, eg (0, 1) and (0, 10).

------
bumbledraven
> To paraphrase Obi-Wan Kenobe: The completeness property of the reals is what
> gives calculus its power. It surrounds the set of real numbers and
> penetrates it. It binds the number line together.

:)

------
kavalec
Calculus: the art of dividing by zero without getting caught.

~~~
perl4ever
That phrase makes me think of this:

[https://en.wikipedia.org/wiki/Renormalization](https://en.wikipedia.org/wiki/Renormalization)

------
dr_dshiv
Eudoxus! A Pythagorean mathematician who studied under the philosopher-king
Archytas and Plato.

He also put forward a compelling argument for hedonism, preserved by
Aristotle.

------
commandlinefan
> try to measure both a side and a diagonal of a square using the same
> measuring-stick, you’ll run into trouble

Because you never end up with whole numbers both ways?

~~~
mjcohen
Correct. Incommensurable - lacking a common measure. The side and diagonal of
an isosceles right triangle are incommensurable - there is no length that goes
into both an integer number of times. There is a proof of this in Euclid.

------
dnprock
I think the reason is infinitesimal. Calculus touches infinities. And
infinities are related. So we end up using infinities to describe each other.

------
Koshkin
So, TL;DR:, one cannot build a meaningful calculus using the rational numbers
alone. Not unexpected, judging by its history... A more interesting (to me,
anyway) fact is that one can build (or "model") any number system - _all_
integers, rationals, reals, etc. - using just the set of natural numbers (i.e.
positive integers).

~~~
Jaxan
Well building the reals from rationals is not easy at all. It took a few
decades to get a reasonable definiton. And in constructive mathematics, there
are several competing (non equivalent) definitions of the reals. It's a very
deep topic, yet we seem to understand (at least in calculus) what real numbers
are.

------
rq1
For completeness’ sake, yes.

------
eugf_
Nice bait

