What makes maths beautiful?

https://newhumanist.org.uk/articles/5208/what-makes-maths-beautiful

by Hooke • 9 years ago

79 83 9 years ago

newhumanist.org.uk

paulpauper• 9 years ago

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..

dabber2• 9 years ago

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".

messe• 9 years ago

UK, Ireland, Australia, New Zealand and almost all other English speaking countries except for the US and Canada, I believe.

rhizome31• 9 years ago

Same in French, eg. J'aime les maths

certifiedloud• 9 years ago

It's common in New Zealand as well.

stablemap• 9 years ago

The original word has a plurality to it; I think it is we who did violence to "maths".

awkwarddaturtle9• 9 years ago

No. Go watch the video Houshalter linked.

https://www.youtube.com/watch?v=SbZCECvoaTA

The sensible and correct shortening is math. That's how it was originally shortened to. The brits went off-kilter in the 1900s and added the s to math.

Everyone, including brits, say mathematics IS difficult. Nobody says mathematics ARE difficult.

davnicwil• 9 years ago

Well, there's a certain logic to it: The long version of the word, Mathematics, is a regular plural. So is it so strange for the shortened version Maths to also be a regular plural?

Edit: I was imprecise with my phrasing above, I really meant to say 'takes the form of (a regular plural)' instead of 'is' - I don't know if it is or is not technically a plural - the point was only that the shortened word 'Maths' follows the same form as the long version, which seems logically consistent.

Houshalter3• 9 years ago

Is Mathematics plural? Would you say "Mathematics is cool", or "Mathematics are cool"?

awkwarddaturtle9• 9 years ago

It isn't a regular plural. What is a mathematic? Mathematic is not a noun. Mathematics is a mass noun like software or equipment.

Houshalter3• 9 years ago

Numberphile has a good video on this: https://www.youtube.com/watch?v=SbZCECvoaTA

dabber2• 9 years ago

That's a great video and exactly the depth of answer I was hoping for. Thanks!

awkwarddaturtle9• 9 years ago

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• 9 years ago

Also, maths sound awkward and silly

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

eru6• 9 years ago

Don't think there's much of an 'argument for' or 'argument against'. They are just different usages.

It's just like arguing that the German Mathematik is silly, or that the English mathematics is.

dragonwriter• 9 years ago

The argument for "maths" is that mathematics is plural.

No, mathematics is uncountable, not plural. I've always assumed it was because “math” is a distinct noun already, for a mowing of grass (were mathematics, and hence maths, plural rather than uncountable, there'd actually be more potential for confusion between “maths”, the short form of mathematics, and “maths”, the plural of “math”, which is avoided because uncountable nouns are grammatically treated as singular for most purposes, like subject/verb agreement; conversely, there is opportunity for such confusion with “math” as the short form. Admittedly, “math” for a mowing is rare enough in current usage outside of modified forms like “aftermath” that that's not big deal now.)

You wouldn't shorten software to "softe" or equipment to "equipe"

Well, sure, because the long form of latter doesn't end with an “e” and equipt is already the related adjective form. Though shortening equipment as “equip’t” is something I've encountered (the apostrophe makes unambiguous that it's a shortening and minimizes confusion with the adjective.)

praneshp2• 9 years ago

Since we are doing this, Maths is common in India as well. I'd never heard "Math' used in an Indian context.

samgranieri• 9 years ago

why not say "Mathematics"?

igravious• 9 years ago

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.

JadeNB4• 9 years ago

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.

jtc19834• 9 years ago

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."

akuma73• 9 years ago

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.

rkowalick• 9 years ago

I would argue that the first computer verified proof in some sense changed the mathematical aesthetic.

JadeNB4• 9 years ago

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.

Houshalter3• 9 years ago

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 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). 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.

jacobolus6• 9 years ago

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.

eru6• 9 years ago

It is beautiful that this interpretation maps (complex) numbers and geometry so well---even if the formula itself doesn't look so beautiful and surprising anymore.

gizmo686• 9 years ago

Euler's formula is "e^ix = cos(x) + i sin(x)", which arguably encodes the "trick" of thinking of e^ix being a rotation. Once you understand this relationship, the special case of x=pi becomes obvious and boring. If you arrived at that special case without first hitting the more general understanding, than I can see calling it "beautiful" because it is so suggestive of something deeper going on.

Of course, when we present this identity as the most "beautiful formula", it is often to people who don't have any understanding of what it means to take something to a complex power, so the correct response should be to look at you funny.

ouid4• 9 years ago

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.

alexvoda• 9 years ago

I love the way you wrote that? Why do mathematicians have a predisposition to overload symbols? I hope in the future, mathematical notation will evolve to make full use of Unicode and of names with more than 1 letter.

JadeNB4• 9 years ago

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.

I guess by "reflection through a point" you mean "the composition of two reflections through orthogonal axes through the point", right? Otherwise this cannot possibly be true, since a rotation is orientation-preserving but a reflection is orientation-reversing.

theoh• 9 years ago

Saw a lecture by Zeki (mentioned in the article) at an art college summer school last year. He showed "e to the i pi" and also a Ramanujan equation. Of course his results are that people see the elegance of e to the i pi but don't see beauty in the apparent arbitrariness of the Ramanujan equation. I have to admit I don't remember what his sample was -- definitely mathematicians (but of what level?). This is an issue that was raised by art colleagues -- would you determine the ultimate aesthetic value of an artist's work by using an unqualified audience?

Zeki is a bit of a buffoon with a tendency to assume that he understands other people's domains well enough to talk about them. It may not be arrogance on his part but there's a definite sense that he doesn't let complexity and nuance get in the way of a good story.

thearn4• 9 years ago

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.

marcuslager2• 9 years ago

What you said made me want to write surpisingly terse and concise code. It's what I shall try to do from hereon.

dwaltrip• 9 years ago

Terseness and conciseness are great, but should be balanced with readability and ease of editing.

ice109• 9 years ago

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• 9 years ago

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

Someone3• 9 years ago

In pure math, the journey is the reward; in applied math, the destination is, and how you get there is unimportant.

That, IMO, is why applied math rarely is beautiful. Pure math often is because no sane person would spend time doing ugly work that doesn't even lead to useful results.

Computational proofs such as that of the four color theorem have changed that a bit. That may be the reason some mathematicians don't accept them as proofs.

Similarly, "publish or perish" may have led to more proofs in pure mathematics that are more hard work than inspiration, and of course, zillions of hours have been spent looking for counterexamples of theorems in pure mathematics.

dkarapetyan• 9 years ago

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

Someone3• 9 years ago

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.

ouid4• 9 years ago

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

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

BigChiefSmokem• 9 years ago

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• 9 years ago

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.

Animats3• 9 years ago

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 [2] https://github.com/John-Nagle/pasv/blob/master/src/work/temp...

Animats3• 9 years ago

Just to look at, here are the actual machine proofs from running our verifier theory through the Boyer-Moore prover.[1] AI, the early years, at work. Here's one proof, for commutativity of TIMES for the natural numbers.

    (PROVE-LEMMA COMMUTATIVITY-OF-TIMES (REWRITE) (EQUAL (TIMES X Y) (TIMES Y X)))
    
         Give the conjecture the name *1.
    
         We will appeal to induction.  Two inductions are suggested by terms in
    the conjecture, both of which are flawed.  We limit our consideration to the
    two suggested by the largest number of nonprimitive recursive functions in the
    conjecture.  Since both of these are equally likely, we will choose
    arbitrarily.  We will induct according to the following scheme:
          (AND (IMPLIES (ZEROP X) (p X Y))
               (IMPLIES (AND (NOT (ZEROP X)) (p (SUB1 X) Y))
                        (p X Y))).
    Linear arithmetic, the lemma COUNT-NUMBERP, and the definition of ZEROP inform
    us that the measure (COUNT X) decreases according to the well-founded relation
    LESSP in each induction step of the scheme.  The above induction scheme
    produces the following two new conjectures:
    
    Case 2. (IMPLIES (ZEROP X)
                     (EQUAL (TIMES X Y) (TIMES Y X))).
    
      This simplifies, applying TIMES-ZERO and TIMES-NON-NUMBERP, and opening up
      the functions ZEROP, EQUAL, and TIMES, to:
    
            T.
    
    Case 1. (IMPLIES (AND (NOT (ZEROP X))
                          (EQUAL (TIMES (SUB1 X) Y)
                                 (TIMES Y (SUB1 X))))
                     (EQUAL (TIMES X Y) (TIMES Y X))),
    
      which simplifies, unfolding the functions ZEROP and TIMES, to:
    
            (IMPLIES (AND (NOT (EQUAL X 0))
                          (NUMBERP X)
                          (EQUAL (TIMES (SUB1 X) Y)
                                 (TIMES Y (SUB1 X))))
                     (EQUAL (PLUS Y (TIMES Y (SUB1 X)))
                            (TIMES Y X))).
    
      Appealing to the lemma SUB1-ELIM, we now replace X by (ADD1 Z) to eliminate
      (SUB1 X).  We employ the type restriction lemma noted when SUB1 was
      introduced to constrain the new variable.  We must thus prove:
    
            (IMPLIES (AND (NUMBERP Z)
                          (NOT (EQUAL (ADD1 Z) 0))
                          (EQUAL (TIMES Z Y) (TIMES Y Z)))
                     (EQUAL (PLUS Y (TIMES Y Z))
                            (TIMES Y (ADD1 Z)))).
    
      However this further simplifies, applying the lemma TIMES-ADD1, to:
    
            T.
    
    
    That finishes the proof of *1. Q.E.D.

Boyer and Moore were trying really hard to make the results understandable to traditional mathematicians.

[1] https://github.com/John-Nagle/pasv/blob/master/src/work/temp...

lomnakkus4• 9 years ago

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.

jtc19834• 9 years ago

Use of the phrase "not-necessarily-all-that-well-founded" is unfortunate since standard set theory studies the properties of extensional, well-founded objects in general. Though clearly you didn't have the technical understanding of well-foundedness in mind. Just might confuse ppl because there is a branch of set theory called "non-well-founded set theory" that is perfectly respectable -- I.e., and "well-founded" in the sense you intend -- but not about the same objects studied in the standard context.

I believe what you have in mind regarding set theory as "not ideal for 'foundations'" (why is the word foundations in quote, btw?) is that there is a recent push to reexamine the prospects more intensional foundations based on type theory. The most promising and currently fashionable version of this push is Homotopy Type Theory (HoTT), which has been given press and prestige from the IAS at Princeton and interest from Fields Medalist Vladimir Vovodsky (sp?). There are plausible reasons given, though often polemical, for preferring some sort of type theoretical or category theoretical foundations over set theoretical ones. Perhaps most importantly, for readers of HN, the comparatively easy formalization into computer-checkable proof and computer-assisted proof (look up, for instance, Coq).

That said, there is a long way to go before HoTT (or something like it) displaces traditional foundations and set theory. For one thing, there is no obvious axiomatization or way to confirm that HoTT is a suitable foundation for extant non-foundational mathematics. For one thing, standard type theory (MLTT) plus Vovodsky's univalence axiom is proof-theoretically equivalent to a subsystem of analysis. I would guess that, for instance, you can't prove that all Borel sets are measurable using MLTT plus univalence (but this is totally speculative and I would appreciate being corrected).

Last -- I suspect the only mathematicians who find set theory proofs beautiful are set theorists. Most other mathematicians don't really give af about set theory.

Animats3• 9 years ago

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.

yarg• 9 years ago

Math isn't beautiful, beauty is mathematical.

noiv• 9 years ago

formula, please.

veli_joza• 9 years ago

"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.

agumonkey• 9 years ago

- wholemeal view

- knowing out of ignorance (probabilities)

- compression of knowledge

What makes maths beautiful?

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