# The art of mathematics.



## KaerbEmEvig (Dec 15, 2009)

Have a listen (read?):

http://www.maa.org/devlin/LockhartsLament.pdf


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## Dodecaplex (Oct 14, 2011)

Fascinating!


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## jalex (Aug 21, 2011)

As someone who is on an extracurricular maths scheme based around that author's ideal world at the same time as taking maths A-level I can say that there is a whole lot of truth in that.


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## Philip (Mar 22, 2011)

i can identify with M. Lockhart's love for math, and the heartbreaking reality that school kills math... but as far as math being an "art" like music or painting, i'm not so sure about that.

perhaps M. Lockhart doesn't realize that practicing scales for hours and hours can also be dreadful, however it doesn't necessarily kill music, because in the end music can be performed and appreciated.

on the other hand, what is the end product of math... ?  math can only be appreciated on a higher level of thought, while music speaks directly to the senses in a tangible manner.


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## jalex (Aug 21, 2011)

Philip said:


> on the other hand, what is the end product of math... ?  math can only be appreciated on a higher level of thought, while music speaks directly to the senses in a tangible manner.


He is speaking more about the creative process in maths than the end product, which is closer to an artistic process than a scientific one. And this kind of thing doesn't need a 'higher level of thought'.


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## Philip (Mar 22, 2011)

jalex said:


> He is speaking more about the creative process in maths than the end product, which is closer to an artistic process than a scientific one. And this kind of thing doesn't need a 'higher level of thought'.


i'm not sure i'd call it a creative process... it might seem creative or imaginative at first, because there are many ways to approach a problem. but the underlying process is very logical and linear: it's problem solving. in the end you get a theorem, a method, a tool, etc.

in addition, i don't see how creativity would _not_ be on a higher level?


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## jalex (Aug 21, 2011)

Philip said:


> i'm not sure i'd call it a creative process... it might seem creative or imaginative at first, because there are many ways to approach a problem. but the underlying process is very logical and linear: it's problem solving. in the end you get a theorem, a method, a tool, etc.


He's contrasting it with the constricted creative process in science in which ideas must accord with the real world. I think he's saying that when you try to solve a maths problem there are as many routes open to you as when you sit down to write a piece of music.



> in addition, i don't see how creativity would _not_ be on a higher level?


It isn't hard to derive the formula for the area of a triangle or prove that the angles in a triangle add up to 180 degrees for example provided you know the right things first. A mathematical education which put more emphasis on discovery and creativity would interest people a lot more than the 'learn question format -> plug in particular situation -> get answer' system currently in place.


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## Philip (Mar 22, 2011)

jalex said:


> He's contrasting it with the constricted creative process in science in which ideas must accord with the real world. I think he's saying that when you try to solve a maths problem there are as many routes open to you as when you sit down to write a piece of music.


yes, that's his point of view, and i disagree. basically math is logic and axioms. when you sit down to solve a problem, what you're really doing is trying out different logical paths from what you know to what you want. if you break down this process into simpler sub-processes, recombining them will never produce creativity, which can probably be modeled with the same logical sub-processes _plus_ some arbitrary variables. these arbitrary decisions lead to unique instances, while math strives to generalize.



> It isn't hard to derive the formula for the area of a triangle or prove that the angles in a triangle add up to 180 degrees for example provided you know the right things first. A mathematical education which put more emphasis on discovery and creativity would interest people a lot more than the 'learn question format -> plug in particular situation -> get answer' system currently in place.


ok, that doesn't really answer my question... although it's a good example of how math isn't a creative process. once you have a theory, you expand it or apply it (depending if you're a scientist or an engineer). physics demonstrations will intrigue most people, but start writing down equations and they lose interest. how do you bridge the gap between example and generalization? with strict, logical, hence non-creative, and possibly overly precise equations that don't exactly appeal to our low-level instincts.


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## Jeremy Marchant (Mar 11, 2010)

Speaking as someone with an honours degree in mathematics, while I agree with the general thrust of Paul Lockhart's argument, I believe that he also is guilty of trying to cram mathematics into the wrong box.

Mathematics isn't science because science is about observing, describing, explaining and predicting the natural world and mathematics is abouth the creative manipulation of patterns. But it isn't art either because - I believe - one essential component of a piece of art is that it relates to human experience, emotions and beliefs, and a triangle in a rectangle doesn't cut the mustard in that respect. Nor do quantum mechanics or hydrodynamics. 

Mathematics is mathematics. It can be used as a tool by science, as in the above examples, and it can be used as a tool by artists (golden section, perspective and so on). But, as Lockhart says, one shouldn't mistake the purpose to which the tool is put for the tool itself.


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## KaerbEmEvig (Dec 15, 2009)

Philip said:


> yes, that's his point of view, and i disagree. basically math is logic and axioms. when you sit down to solve a problem, what you're really doing is trying out different logical paths from what you know to what you want. if you break down this process into simpler sub-processes, recombining them will never produce creativity, which can probably be modeled with the same logical sub-processes _plus_ some arbitrary variables. these arbitrary decisions lead to unique instances, while math strives to generalize.
> 
> ok, that doesn't really answer my question... although it's a good example of how math isn't a creative process. once you have a theory, you expand it or apply it (depending if you're a scientist or an engineer). physics demonstrations will intrigue most people, but start writing down equations and they lose interest. how do you bridge the gap between example and generalization? with strict, logical, hence non-creative, and possibly overly precise equations that don't exactly appeal to our low-level instincts.


That's what the essay is talking about. You're not doing mathematics, you're doing calculations. You're just another example that school deludes people into misconception that mathematics = using given algorithms to determine the answer of a known problem. No! Mathematics is about discovering the answer to a mystery! You got it all wrong, I'm afraid.

Actually, you're so wrong it's hurting my eyes to read this. You know how many ways to find out the area of a circle there are? Dozens! Mathematics isn't about remembering that A=pi*R^2 (I actually had to look it up on Google to be sure I remember it correctly) - it's about finding it out yourself. I did this myself in the first class of high school, I thought: why not:

Draw a circle around a regular triangle. Cut the triangle into three identical equilateral triangles (with the base equal to L and the sides equal to R). Do the same with a square. Find out the relationship between the number of sides of my regular polygon and the angle between the adjacent sides. Why? Because R=const. Which means that if I know the angle, I also know the angle at the base of the equilateral triangle (it's exactly two times smaller!) - this means I can calculate the area of my equilateral triangle. My regular triangle is made of three identical equilateral triangles, my square - of four equilateral triangles (just a different angle at the base). Now, if I can just make the number of sides of my regular polygon go to infinity (sides of my polygon will go to zero), I can find out the area of the circle because my circle is composed of an infinity of equilateral triangles with A=0. What's beautiful that in this case limes of infinity*0=area of the circle. That's mathematics! If this isn't creativity, I don't know what is!

Nobody told me the algorithm. Nobody gave me any tips. The teacher was busy cranking out those polynominal problems to solve. How boring. Now, finding out the area of a circle from scratch is awesome. Later I found out that Galileo was the first person to publish this 'method'. How cool is that!


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## Philip (Mar 22, 2011)

well congrats! obviously there's a little bit of Galileo in you, very cool.



KaerbEmEvig said:


> That's what the essay is talking about. You're not doing mathematics, you're doing calculations. You're just another example that school deludes people into misconception that mathematics = using given algorithms to determine the answer of a known problem. No! Mathematics is about *discovering* the answer to a mystery! You got it all wrong, I'm afraid.


yes, that's how i view it: you *discover* math, you *create* art.


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## kv466 (May 18, 2011)

Math is everywhere and in everything. Music and art would be nothing without it. By itself, an art? Sure, if you want it to be...not sure about having to think at a higher level...just have a higher level of appreciation for all things around you.


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## Philip (Mar 22, 2011)

is a baby going to appreciate math? can a baby enjoy (or at least react to) music? that's what i meant by "higher level". it's hierarchical. the knee jerk reflex is low-level, playing chess is high-level; physics is low-level, sociology is high-level.


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## KaerbEmEvig (Dec 15, 2009)

Philip said:


> is a baby going to appreciate math? can a baby enjoy (or at least react to) music? that's what i meant by "higher level". it's hierarchical. the knee jerk reflex is low-level, playing chess is high-level; physics is low-level, sociology is high-level.


Is it going to appreciate Dostoyevsky? I doubt it.

I discovered the answer to the mustery, but I also devised the method, created the bridge to get to the other side.


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## Igneous01 (Jan 27, 2011)

fascinating article, I share my sentiments with the author.

I remember 6 months ago studying matrices, binomial theorem, set theory, permutations and combinations, network theory, and cryptography. It was incredible and interesting learning about these forms of calculation. It opened up a whole new way of thinking about problems.

However, the textbook was completely non-intuitive. One thing I did not like was that some topics had no historical background to them, no foundation as to why this works, no simple way of explaining where I can use this. Lots of exercises and problems to make sure you remember, but look at me now - I forgot most of it because of the way it was presented. 

Some problems were very good though, like how many possible combinations exist in a lock that has 60 numbers and uses 4 of them as the key? The chances of winning the lottery if there are 60 numbers to choose from and 9 numbers have to be taken, that cannot be repeated? These questions are what made the math fun, in fact Its the only thing I still remember how to calculate.

But Binomial theorem had no such questions, and I could never fully memorize it or work with it, because there was no engaging problem to work with. Solve for n? how is that the least bit engaging? And of course, no application of it being used on a real world problem, it felt as if i was doing pseudo-math, calculating something that had no reason for existing in the first place.

This is why physics was so engaging at my school, I had an amazing physics teacher, always talking about questions and problems in a imaginary context, and how to solve it. for example:

If you were lifting weights because you wanted to be stronger, would you work out more with those same weights, if it was done underground in the mantle of the earth?

To me theres nothing more fascinating then a question like this - here because you are deep beneath the earth, there is a stronger gravitation pull on your body, because you are closer to the dense mass of the earth, with the entire crust above pushing down on you! Lifting weights that were maybe 10kg would feel like hundreds of kg at this depth!

You could even calculate the net force pushing down on you at this depth if you really wanted to know. That is mathematics, that is physics, that is beauty right there.

But unfortunately, its not taught like this in math, in science I always had gifted teachers who understood teaching for what it is, and made me appreciate math and physics in an imaginary and artistic context. 

Alas, if I only knew where Matrices could be applied in a problem, then I would have more use for it.


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