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What makes maths beautiful?
- paulpauper 9y agoparsimoniousness 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..
- dabber 9y agoIs 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 9y agoUK, Ireland, Australia, New Zealand and almost all other English speaking countries except for the US and Canada, I believe.
- certifiedloud 9y agoIt's common in New Zealand as well.
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- stablemap 9y agoThe original word has a plurality to it; I think it is we who did violence to "maths".
- awkwarddaturtle 9y agoNo. Go watch the video Houshalter linked. https://www.youtube.com/watch?v=SbZCECvoaTA 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.
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- davnicwil 9y agoWell, 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.
- Houshalter 9y agoIs Mathematics plural? Would you say "Mathematics is cool", or "Mathematics are cool"?
- TheOtherHobbes 9y agoIt's a special case like physics, optics, dynamics, economics, and so on. The simple (linguistically obscure, but let's pretend) way to understand it is that there's an implied "the study of" before the word. So although you are studying multiple elements of a topic, the study itself is singular.
- awkwarddaturtle 9y agoIt isn't a regular plural. What is a mathematic? Mathematic is not a noun. Mathematics is a mass noun like software or equipment.
- Houshalter 9y agoNumberphile has a good video on this: https://www.youtube.com/watch?v=SbZCECvoaTA https://www.youtube.com/watch?v=SbZCECvoaTA
- dabber 9y agoThat's a great video and exactly the depth of answer I was hoping for. Thanks!
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- awkwarddaturtle 9y agoI 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 9y ago> Also, maths sound awkward and silly Which, when in the UK, becomes: "Also, 'math' sounds awkward and silly"
- awkwarddaturtle 9y ago> 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 9y agoIt 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 9y ago> 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.
- praneshp 9y agoSince we are doing this, Maths is common in India as well. I'd never heard "Math' used in an Indian context.
- samgranieri 9y agowhy not say "Mathematics"?
- praneshp 9y agoActually school time tables, college transcripts, etc all say Mathematics. Only everyday communication uses the shortened form.
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- igravious 9y agoIs 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 9y 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.
- jtc1983 9y agoYes, 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 9y agoNot 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 9y agoI would argue that the first computer verified proof in some sense changed the mathematical aesthetic.
- JadeNB 9y 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.
- Houshalter 9y agoI'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 9y agoEuler’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.
- thearn4 9y agoI'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.
- marcuslager 9y agoWhat you said made me want to write surpisingly terse and concise code. It's what I shall try to do from hereon.
- dwaltrip 9y agoTerseness and conciseness are great, but should be balanced with readability and ease of editing.
- marcuslager 9y agoNow there are too many rules. I'll just go back to writing crap code.
- ice109 9y agoI 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
- computerex 9y 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. Relative to the average Joe on the street you may have gotten "pretty far" in mathematics, but not relative to math majors. I'd expect undergrads to destroy you. And certainly nowhere near the prodigies of mathematics like Gauss or Euler. Since you got pretty far in mathematics, you must know of https://en.wikipedia.org/wiki/Euler%27s_identity https://en.wikipedia.org/wiki/Euler%27s_identity e^(ipi)+1=0 The fact that this one equation relates some of the fundamental constants is pretty profound, and some would say, beautiful. The fact that a human like you or me came up with this true assertion is also, beautiful. Have you heard of Srinivasa Ramanujan?
- dkarapetyan 9y agoMuch of complex analysis and linear algebra I think is beautiful and can be easily made accessible to the general layman.
- Someone 9y agoI 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.
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- ouid 9y agolinear algebra, though, I feel is an excellent example. Everyone has solved a system of linear equations in school. The concept is pretty accessible.
- Someone 9y agoIn 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 9y agoWhat 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.
- BigChiefSmokem 9y agoIn 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 9y 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.
- Animats 9y agoI 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/temporaryrulebase.lisp https://github.com/John-Nagle/pasv/blob/master/src/work/temp...
- yarg 9y agoMath isn't beautiful, beauty is mathematical.
- noiv 9y agoformula, please.
- deleted 9y ago[deleted]
- adam_seeliger 9y agoLeaving the "s" off maths. That is what makes "maths" beautiful.
- veli_joza 9y 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 9y ago- wholemeal view - knowing out of ignorance (probabilities) - compression of knowledge