
What makes maths beautiful? - Hooke
https://newhumanist.org.uk/articles/5208/what-makes-maths-beautiful
======
Animats
I had a run-in with this issue once. Many years ago, when working on a program
verification system, I had to show that two theorem provers (Boyer-Moore and
Nelson-Oppen) were working on compatible theories. If that was botched, the
system could prove false statements. So I undertook to prove, in Boyer-Moore
theory, the "axioms" of arrays used in the Nelson-Oppen prover. McCarthy
defined four classic axioms for arrays, and here are the first two.

    
    
        p=r ⇒ read(write(a, p, v), r) = v
    
        ¬(p=r) ⇒ read(write(a, p, v), r) = read(a, r)
    

This just says that if you store v at a[p], and then read it back, you get
what you put there, and you don't change any other elements.

So I set this up in Boyer-Moore notation.

    
    
        (PROVE-LEMMA SELECT-OF-STORE-1 (REWRITE) 
            (IMPLIES (AND (arrayp A) (NUMBERP I)) 
                     (EQUAL (selecta (storea A I V) I) V)))
    
        (PROVE-LEMMA SELECT-OF-STORE-2 (REWRITE) 
            (IMPLIES (AND (arrayp A) (NUMBERP I) (NUMBERP J) (NOT (EQUAL I J))) 
                     (EQUAL (selecta (storea A I V) J) (selecta A J))))
    

But Boyer-Moore theory is constructive; everything must have a definition that
can be evaluated as a function. So I had to define arrays and the "selecta"
and "storea" functions so that this would work. I could do this by defining an
array as a list of (subscript, value) tuples.

    
    
        (ADD-SHELL UNDEFINED-OBJECT UNDEFINED UNDEFINEDP ())
    
        (ADD-SHELL array-shell 
            empty-array 
            array-recognizer 
            ((array-elt-value (NONE-OF) UNDEFINED) 
             (array-elt-subscript (ONE-OF NUMBERP) ZERO) 
             (array-prev (ONE-OF array-recognizer) empty-array)))
    

This is simply a structure definition for a list. An array-shell has an array-
elt-value, an array-elt-subscript, and array-prev, which is the rest of the
elements. "array-shell" is the constructor, "empty-array" is essentially "nil"
for this type, and "array-recognizer" is a predicate which is true for objects
of type array-shell.

For array equality to work, the list of tuples had to be ordered by subscript,
so that two arrays with the same values would always have the same
representation. So selecta and storea are complicated. selecta is a recursive
lookup in a list:

    
    
        (DEFN selecta (A I) 
            (IF (EQUAL (array-elt-subscript A) I) 
                       (array-elt-value A) 
                       (IF (EQUAL (array-prev A) (empty-array)) 
                           (UNDEFINED) 
                           (selecta (array-prev A) I))))
    
    

Store into array is ugly, because it's an update of an ordered list.

    
    
        (DEFN storea (A I V)
            (IF (NOT (arrayp A))                   
                (empty-array) 
                (IF (NOT (NUMBERP I))
                    A 
                    (IF (EQUAL A (empty-array)) 
                        (IF (EQUAL V (UNDEFINED)) 
                            (empty-array) 
                            (array-shell V I (empty-array))) 
                        (IF (EQUAL (array-elt-subscript A) I) 
                            (IF (EQUAL V (UNDEFINED)) 
                                (array-prev A) 
                                (array-shell V I (array-prev A)))
                            (IF (LESSP (array-elt-subscript A) I) 
                                (IF (EQUAL V (UNDEFINED)) 
                                    A 
                                    (array-shell V I A)) 
                                (array-shell 
                                    (array-elt-value A)
                                    (array-elt-subscript A) 
                                    (storea (array-prev A) I V))))))))
    

This recursive function is complicated enough that it needs a lemma to help
with proof of termination.

    
    
        (PROVE-LEMMA storea-is-sound (REWRITE) 
            (implies (and (arrayp a) 
                (not (equal a (empty-array)))) 
                (lessp (count (array-prev a)) (count a))))
    

This says that the array-prev list component of a is always shorter than a.
The Boyer-Moore prover must prove termination for all defined functions, which
it does by examining any recursive call and proving that there is a measure
defined on the arguments which is a smaller nonnegative integer. For the easy
cases, the prover can do this automatically, but sometimes you have to define
a lemma to help it. The lemma, of course, also has to be proved.

So, given all that, the Boyer-Moore theorem prover could be turned loose to
grind out proofs of the rules for arrays. It took about 45 minutes of grinding
in 1982 on a 1 MIPS VAX 11/780\. You can rerun it now. Here's the prover,
which requires clisp.[1]. Here's the entire set of definitions and theorems we
used for program verification.[2]. Read the instructions at [1], and get the
system up to where it's done "boot-strap". Then load the file at [2]. Long,
detailed, human-readable explanations of the proof steps are displayed. The
entire proof run for all those basic theorems takes about a second, and that's
probably mostly scrolling.

So, having rigorously proved the "axioms" for arrays from first principles, I
submitted a paper to JACM. The reviewer comments I got back were mainly about
the uglyness of the approach. All that case analysis! One comment was that the
reviewer was upset about having a proof that ugly at the foundations of
computer science. The paper was rejected.

The reason it's ugly is that it doesn't get to use set theory. Informally, a
set is a collection whose order does not matter. "Order does not matter" is
not a meaningful concept in strict Boyer-Moore theory. If order matters, there
are many cases to prove, stemming from all those IF statements in the
definitions.

A generation later, fewer mathematicians would be bothered by this form of
"uglyness". But back then, machine proofs were so rare that this was
considered awful. Also, now that computers can do this in a second,
complaining that it's too complicated seems whiny.

[1] [https://github.com/John-Nagle/nqthm](https://github.com/John-Nagle/nqthm)
[2] [https://github.com/John-
Nagle/pasv/blob/master/src/work/temp...](https://github.com/John-
Nagle/pasv/blob/master/src/work/temporaryrulebase.lisp)

~~~
lomnakkus
I love these anecdotes.

Just an aside, but I was under the impression that Set Theory (these days) is
regarded as not-necessarily-all-that-well-founded or, perhaps, not ideal for
"foundations"? Is that hopelessly misguided?

If it isn't already apparent, I'm pretty much a noob in formal verification,
but I've read enough to hopefully have a very shallow understanding...?!?

EDIT: I just thought I'd add what "makes math beautiful" for me: The mere
difficulty of defining equality of 'things' and deciding what it _should_
mean.

~~~
Animats
Equality should at least have the property that

    
    
        ∀ f, x = y ⇒ f(x) = f(y)
    

In a constructive theory like Boyer-Moore, all variables have a concrete
representation, and functions can look into that representation. Thus, for
EQUAL to have the property above, the representation must be the same.

In Boyer-Moore theory, there's no "hiding", in the sense of an object with
private members. One could define a "weak equality" for sets which were lists
of values, true if the same values were in both lists. Weak equality would not
require that the lists have the same order. But you couldn't hide the
representation. Functions could peek into the list representation, and the
property above would not hold. So "weak equality" would not be very useful in
proofs.

I once proposed a sort of "object oriented" extension to Boyer-Moore theory.
The idea is that you create something like a set represented by lists, prove
some theorems about such objects, and then "hide" the private functions that
can look into the interior of the representation. Then you can define a weak
equality function p. If you can prove ∀ f, p(x,y) ⇒ f(x) = f(y) for every non-
hidden function, you have proven that p(x,y) ⇒ x = y. Then, p can be treated
as equality. Proving this requires proving it for every function already
defined. Any newly defined functions are composed from existing functions, so
the equality still holds. There's trouble with newly defined types, though,
because they come with some built-in functions such as the type predicate.

If you pursue this, you probably get to some theory of types, but I didn't
want to go there, so I gave this up.

~~~
lomnakkus
> Equality should at least have the property that ∀ f, x = y ⇒ f(x) = f(y)

My impression was that even that might be a bit contentious[1]. Well, maybe
not contentious, but maybe... disputable? It also requires a theory of
functions, or at least a syntax to derivation rules.

That was just to be absurdly nitpicky, I _think_ I understand what you're
saying here, and it's much appreciated.

[1]
[https://pdfs.semanticscholar.org/806a/92153c4b8d344edec45eec...](https://pdfs.semanticscholar.org/806a/92153c4b8d344edec45eece4ba105afd3dda.pdf)

------
Houshalter
I'm having trouble finding the source of this quote, but here:

>Many mathematicians have the opposite opinion; they do not or cannot
distinguish the beauty or importance of a theorem from its proof. A theorem
that is first published with a long and difficult proof is highly regarded.
Someone who, preferably many years later, finds a short proof is "brilliant."
But if the short proof had been obtained in the beginning, the theorem might
have been disparaged as an "easy result." Erdős was a genius at finding
brilliantly simple proofs of deep results, but, until recently, much of his
work was ignored by the mathematical establishment.

And here is an interesting study on what mathematicians consider beautiful:
[https://www.gwern.net/docs/math/1990-wells.pdf](https://www.gwern.net/docs/math/1990-wells.pdf)
They found that mathematicians varied a great deal in what they considered
beautiful. And that there isn't any consensus on what is considered beautiful.
It also changes over time:

>There was a notable number of low scores for the high rank theorems 9 Le
Lionnais has one explanation [7]: "Euler's formula e^pi*i = - 1 establishes
what appeared in its time to be a fantastic connection between the most
important numbers in mathematics... It was generally considered 'the most
beautiful formula of mathematics' . . . Today the intrinsic reason for this
compatibility has become so obvious that the same formula now seems, if not
insipid, at least entirely natural."

I think the interestingness or surprisingness of a mathematical idea is a
temporary thing. Once you really deeply understand something, it's no longer
surprising or interesting. It should just feel obvious and simple.

This goes along with Jurgen Schmidhuber's theory of art
([http://people.idsia.ch/~juergen/creativity.html](http://people.idsia.ch/~juergen/creativity.html)).
That humans crave novelty and unexpectedness in all areas, and that is what we
consider beautiful. But once you get very familiar with something, it loses
it's novelty.

~~~
jacobolus
Euler’s formula says (in one interpretation) that a half-turn rotation in the
plane is the same as a reflection through a point, a statement which is
obvious to children who think about it for a bit. It’s phrased in a cute way,
but “most beautiful formula of all time” etc. is really a stretch.

The tricky part is thinking of rotations as a type of “number” with a
multiplicative structure, in which case the [additive] angle measure becomes
the logarithm of the rotation, letting us write our half-turn rotation using
the fancy-looking exp( _πi_ ) instead of ½⥀ or rot(180°) or whatever.

If you asked someone whether ⤺² = ⥀ (or similar) was the most beautiful
formula in mathematics they would probably laugh or look at you funny.

~~~
ouid
is there anything interesting to you in the notion that cos(x)+isin(x) is the
unique analytic continuation of e^x in the complex plane?

It's not just a rotation by definition, I don't think.

~~~
jacobolus
If you think about it for a bit, it’s clear that this must be so.

A circle’s circumference is proportional to its radius, so if you want to have
a conformal map between a rectangular grid in one space and a polar-coordinate
grid in another space, the spacing of the polar-coordinate grid lines in the
radial direction must necessarily be proportional to the radius so that they
can match the radius-dependent spacing in the tangential direction.

Anyhow, there are many interesting features of the structures here. Both
rotation and logarithms are very interesting and important both in mathematics
and in science/engineering. I just don’t think Euler’s identity in particular
is worthy of so much attention.

~~~
ouid
Sure, but I think most people believe that cos(x)+isin(x)=e^ix because that's
how it is defined, in which case e^(i\pi)=-1 isn't even a result. It is at
least the tiniest bit more interesting than that.

------
thearn4
I'm a mathematician, and never quite got the emphasis on "beauty" that some
obsess over. Maybe I'm just too deep into the applied side. But mathematical
beauty to me is unexpected terseness or conciseness in a result. Which is
similar to how I would define beautiful source code I think.

~~~
ice109
I got pretty far in math (ms in cs with half of the classes being legit math
classes) I never got the obsession either. in general it always felt to me
like a bunch of awkward people trying to ascribe aesthetic value to something
that had none. it also always felt self-aggrandizing: here look at how
beautiful is this game I play where I made up the rules.

edit: down votes on hn without comments? boo

~~~
joe_the_user
Well, most of your post involves insulting a fairly broad group of people.

------
veli_joza
"What makes math beautiful?" Having it rendered correctly would be a good
start.

Maybe it doesn't work just for me, but displaying "a2 + b2 = h2" in article
about math is really ungraceful degradation.

------
dkarapetyan
Much of complex analysis and linear algebra I think is beautiful and can be
easily made accessible to the general layman.

~~~
Someone
I think you overestimate "the general layman". For example, complex analysis
requires one to accept the existence of the square root of minus one and that
multiplying by it is equivalent to a rotation.

In my experience, "the general layman" simply doesn't possess enough
suspension of disbelief to get past such hurdles.

~~~
ouid
linear algebra, though, I feel is an excellent example.

Everyone has solved a system of linear equations in school. The concept is
pretty accessible.

~~~
Someone
In 2D, I'll give you that, but in general, I would say "has struggled
solving", rather than "has solved", and not even 'everyone'.

Few can, for example, reliably compute the intersection point of a line l'
parallel to l through a point p with a plane P, and many would panic when
given a problem in N>3-dimensional space.

I know people who _memorized_ formulas for solving linear systems as used in
macro-economics (on the order of 10 dimensions, but lots of zeroes in their
matrix), and, because of that, couldn't work when given a slightly different
model to solve (for example, if taxation became be a constant plus a fraction
of income instead of just a fraction of income, or if capital gains tax were
introduced).

~~~
ouid
What on earth are you talking about?

The first step in a linear algebra course is to codify the method that you use
in middle/high school to solve systems of equations of the form
a00x0+a01x1+...=b0....

That algorithm sets the stage for everything. The pace is natural, the math is
accessible, no geometric interpretation is ever required, and it's built for
problems that people of even average intelligence can understand. Frankly,
it's astonishing we don't teach it in high school.

------
igravious
Is a beautiful poem still beautiful if it is written in a language that you
don't understand?

I conjecture that the inherent beauty of mathematics is overstated. The beauty
of Euler’s identity is like the sublime beauty of a Himalayan peak shrouded in
thick fog. A lot of the pleasure and reward stems as much from the effort to
climb above the fog to view the peak in all its majesty as from any structural
formal beauty. There are only so many ways formal structures can relate to
each other aesthetically.

At least that's what I think today.

~~~
JadeNB
> There are only so many ways formal structures can relate to each other
> aesthetically.

I think that a lot of the inherent beauty of mathematics (there is also the
beauty of the climb, as you very nicely described it) comes precisely from the
realisation that, sensible as this claim is, it doesn't seem to be true: there
seem always to be more relationships out there to discover, and their hidden
and surprising natures can make them all the more beautiful once uncovered.

~~~
jtc1983
Yes, this ^^

There is even a _mathematical_ case to be made that there are always "more
relationships out there to discover." One of the more plausible, sober
interpretations of the various limitative phenomena in the foundations of
mathematics discovered throughout the 20th century is that mathematics is not
reducible to fintary symbol manipulation. Perhaps this is a bit loose and
controversial, but the point is just that we don't necessarily need to solely
lean on the observed fact that there have always been "more relationships out
there to discover."

------
paulpauper
parsimoniousness and specificity. Each number and symbol serves a specific
function/role that if rearranged or changed, the whole system/equation would
cease to work, but also that the symbols themselves encode so much information
and meaning within them..

------
akuma73
Not all math is beautiful IMHO.

The proof of the 4 color theorem "reduced the infinitude of possible maps to
1,936 reducible configurations (later reduced to 1,476) which had to be
checked one by one by computer and took over a thousand hours."

Of course beauty is subjective and not well defined. Some might consider that
proof beautiful.

~~~
JadeNB
> Not all math is beautiful IMHO.

Surely no-one claims that it is? Speaking of 'beautiful code' does not invite
the rebuttal that there is also ugly code, just encourages the appreciation
that ugly code isn't all there is, and that formal systems can be æsthetically
pleasing.

------
BigChiefSmokem
In code I define beauty as the ability to do a lot with just a little.
Simplicity while at the same time providing robustness is what gives me this
feeling.

E=mc^2 is considered the greatest equation in history because of this reason.
It makes it simple to explain to someone not well-versed in math what is
actually behind the symbology.

~~~
sidusknight
>E=mc^2 is considered the greatest equation in history because of this reason.

Only people with a "pop-culture" idea of math&physics think that.

------
agumonkey
\- wholemeal view

\- knowing out of ignorance (probabilities)

\- compression of knowledge

------
yarg
Math isn't beautiful, beauty is mathematical.

~~~
noiv
formula, please.

------
dabber
Is it a UK thing to add the "s" to math? I see it pretty often but it never
seems right. I've always used the word as an unchanging irregular plural
similar to "sheep".

~~~
awkwarddaturtle
I may be biased since I'm american but I always felt "maths" to be silly. The
argument for "maths" is that mathematics is plural. But it is plural in the
sense it is a "mass noun" like software or equipment. You wouldn't shorten
software to "softe" or equipment to "equipe". Also, maths sound awkward and
silly and went spoken pretty much similar to math.

~~~
timthorn
> Also, maths sound awkward and silly

Which, when in the UK, becomes: "Also, 'math' sounds awkward and silly"

~~~
awkwarddaturtle
> Which, when in the UK, becomes: "Also, 'math' sounds awkward and silly"

But math came from england via the oxford dictionary.

[https://youtu.be/SbZCECvoaTA?t=50](https://youtu.be/SbZCECvoaTA?t=50)

Try this. Do you say "maths are fun" or "maths is fun"? What about physics. Do
you say physics is difficult? Or physics are difficult.

Is it singular or plural?

~~~
sbmassey
It is uncountable. Nobody serious justifies 'maths' based on it being plural,
it is just a different contraction of the word 'mathematics' than what you are
used to.

~~~
awkwarddaturtle
> Nobody serious justifies 'maths' based on it being plural

That's the only justification. What other justification is there?

> it is just a different contraction of the word 'mathematics' than what you
> are used to.

Yes. Because the british mistakenly assumed mathematics was a "regular
plural". For what reason would you keep the s at the end?

Do you shorten economics to econS or to econ? Do you takes ECONS 101?

The explanation was given here.

[https://youtu.be/SbZCECvoaTA?t=50](https://youtu.be/SbZCECvoaTA?t=50)

Why are you so resistant to FACTS and HISTORY? I don't get people who argue
even after given evidence.

~~~
consz
>That's the only justification. What other justification is there?

That those who hear it can understand what the speaker is talking about -- I
would say _that_ is the only justification necessary (indeed, for any word),
and is more than justified for the term "maths".

~~~
awkwarddaturtle
That is a justification to continue using it. Sure. But that isn't the
justification for shortening mathematics to maths.

All I'm saying is that it was shortened to maths due to a misunderstanding of
what mathematics is. And all I'm saying is that when british people saying the
"correct" way to shorten mathematics is to shorten it to "maths", I'm saying
their JUSTIFICATION is wrong. Because mathematics is not a regular plural.

Okay? I'm explaining it as simply as possible. Use maths if you want. If that
is what is used in britain, fine. I'm just saying the JUSTIFICATION for why
british people say it is "correct" is wrong. Okay?

