The Ontario Science Centre once had a "proof" that was made using wooden strips that were about an inch wide and perhaps a quarter of an inch deep. There was a right angled triangle with squares on each side. It looked just like typical diagrams except that (a) it was mounted on the wall, (b) the triangle and the squares were covered with glass, (c) there was exactly enough coloured liquid inside to fill the square on the hypotenuse. Every minute or so, the triangle rotated in such a way as to allow all the water from the square on the hypotenuse to flow into the squares on the other two sides. Of course, the fit was exact. I remember watching the exhibit, thinking about it, and suddenly feeling a greater depth of understanding, not of geometry, but of trigonometry! Of course, not a proof, but a brilliant exhibit. (Here is a similar demo on youtube: https://www.youtube.com/watch?v=CAkMUdeB06o)
It only shows that the Pythagorean theorem holds for that one particular set of side lengths. A mathematical proof requires that you show it is true for all possible side lengths.
An analogy would be that I take two equal-length sticks and say "given any two sticks they will be the same length". I have an example (the two sticks I'm holding) but this does not amount to a proof of my statement (and the statement is obviously incorrect).
It’s interesting to wonder if a contraption might be built whereby some slider adjusts the sizes of the two smaller squares, while maintaining the right angle (i.e. keeps the corner on a circle with the hypotenuse as diameter). It would be much more mechanically complicated and harder to build, but pretty awesome.
That's easy enough, but it changes the total area, which means some kind of drainage is required, which complicates the demonstration of equal-area squares.
As long as you don’t change the size of the large square, the total area of the two smaller squares must be the same. That’s just a restatement of the Pythagorean Theorem.
Demonstrations based on physical objects aren't mathematically precise. Who's to say that the wood doesn't flex, or that the fluid doesn't compress? You can see that the areas are equal to within measurement tolerance, but not that they're exactly equal.
Here's a classic 'physical' pictorial proof that 31.5 = 32.5:
Besides the issue that a physical demonstration only shows that the theorem is true for one particular triangle, reality is sometimes more tricky than one thinks. See for example this https://en.wikipedia.org/wiki/Missing_square_puzzle
Further, it doesn't prove exact mathematical equality. Depending on the scale of the demonstration and the accuracy of the observations, it demonstrates approximate equality up to a certain precision. (You could also demonstrate π=3.14.)
the thing that bugs me about that is that its using volume to illustrate a fact about area. Now of course it's legitimate assuming all the rectangular prisms have the same depth but it still feels a bit like some sleight of hand is taking place.
> Now of course it's legitimate assuming all the rectangular prisms have the same depth
But I still consider it to add an extra element to the demonstration that may potentially confuse people who look at it (or have them wondering 'are they all the same depth or not?'), which is something you don't want in an educational demo.
Since we are on HN and talking of the Pythagorean theorem, I must mention E. W. Dijkstra's proof. https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/E... he proves the generalization sgn(α + β - γ) = sgn(a² + b² - c²) also by using similar triangles. Incredibly neat.
Whoa, that IS neat. Thanks for sharing! A key part of his reformulation is π = α + β + γ (the sum of the internal angles of a triangle equal 2 right angles). That statement is, funny enough, equivalent to the parallel postulate.
Going on a tangent now: lately I've been thinking that the "dead horse" of the Pythagorean theorem is actually trying to tell us that flat (as opposed to fractal!) dimensions come from composable self-similarity (squares).
Yeah if the parallel postulate is out then this obviously doesn't stand. Starting from the North Pole, walking down the prime meridian to the equator then turning right and walking along the equator finally turning right to the north again at the right point so you walk through New Orleans creates a triangle on a sphere with three right angles...
It's not labelled this way but notice that the BIG squares are identical, having sides of a+b. Only the four triangles are rearranged. Just subtract the four identical triangles.
In the second arrangement, a c^2 area is present in addition to the same four triangles. In the first arrangement this had been rearranged to an a^2 and a b^2 area.
It's important to assure yourself the triangles are the same and the big square is the same - there is no tiny hidden sliver or something, and right angles are preserved, there are no shenanigans.
To be honest the second image with the tilted c² is enough. From that picture alone you can figure the outer square has the same area as the inner square + 4 times the triangle area:
Yours does use some elementary algebra[1], which wasn't used in the Chinese geometric proof. I wonder if ancient Chinese mathematicians could simplify (a+b)^2 symbolically, or even did symbolic algebra this way?
I'm sure they knew that the area of triangles, which you also use, is (ab/2) - but the purely visual proof needs nothing more than the knowledge that the area of a square is the square of the length of its sides. (And I guess some obvious facts like that no matter how you divide an area the sum of the areas of its parts will be same - the reason I mention tiny slivers is in some fake geometric proofs this intuitive knowledge is abused.
> Einstein’s predictions, during a solar eclipse in 1919—was asked if it was really true that only three people in the world understood the theory, he said nothing. “Don’t be so modest, Eddington!” his questioner said. “On the contrary,” Eddington replied. “I’m just wondering who the third might be.”
> When Arthur Eddington—the British astrophysicist who led the team that confirmed Einstein’s predictions, during a solar eclipse in 1919—was asked if it was really true that only three people in the world understood the theory, he said nothing. “Don’t be so modest, Eddington!” his questioner said. “On the contrary,” Eddington replied. “I’m just wondering who the third might be.”
In the first quote it is implied that Einstein is being asked the question. Then it isn't clear who Eddington was. The questioner, I thought at first. Then no, a phone typo of Einstein? I found it confusing and the larger quote cleared it up.
Every time Pythagoras is brought up, I can't help but think of the cult of Pythagoras as well.
One of the tenants is the refutation of beans. For what reason, I have no idea. I've always found it strange that a mind so capable was also equally capable of such folly.
The same cult found a proof that sqrt(2) is irrational, and kept it hidden because they believed natural numbers and their ratio's to be everything.
Then some dude Hippasus found the same proof and talked about it openly. The pythagorean cult murdered him for it.
(I checked wikipedia [1] and apparently, historians disagree on whether this is true)
Trying to figure out the proof of a² + b² = c² by myself, without looking up the solution, was somehow exiting. Being exposed to a riddle and trying to find the solution is kinda cool.
However, not solving it after 10 minutes left me feeling a bit dumb... :)
Real (non-trivial, non-obvious) problems that someone hasn’t seen before can take hours, days, weeks, years, whole careers, or sometimes centuries to solve. Some of them later turn out to be impossible (and for many we still just don’t know).
Real math education would have students grappling with relatively open-ended problems that take significant amounts of rumination and some cleverness to solve. It would explicitly encourage/reward close critical reading, creative brainstorming, planning, strategic thinking, generalization and specialization, executive control (e.g. time management), error checking, and clarity of exposition (including when asking for help after being stuck). There would be no shame in throwing out incorrect hypotheses, asking for clarification, getting stuck on a problem, making subtle mistakes which could serve as good examples for future improvement, etc. But skill and stamina at such work must be trained slowly, starting from an early age.
The problem is that current (US) math education instead pre-chews everything, assigns students lists of exercises almost identical to what they saw someone solve before, and mostly tests memorization/recall and willingness to do the same trivial task over and over for hours despite being terribly bored, under purely extrinsic motivation.
For people used to such math homework, the standard response to a single problem which takes >5 minutes to work through is to give up.
Depends on what the goals are. I don't think most counties want a large fraction of their population becoming mathematicians. They basically want most of their population able to use math not discover new areas.
What do you think it means to “use math”? If you want to train human spreadsheets you’re wasting everyone’s time: the spreadsheet already exists as a product for electronic computers, and is much faster and more reliable than a human will be at doing huge piles of arithmetic. (But anyway, even if you wanted to train expert performance at arithmetic, the way schools currently go about this is horribly inefficient and doesn’t take into consideration research on neuroscience / psychology.)
If you just want to train people to be unthinking drones who can follow narrowly specified procedural rules without understanding their context or meaning, then I guess the current system is a relative success.
In general, the point of mathematics education in primary/secondary school is not to train future mathematicians, but to teach people important problem-solving skills. The same skills are (to some extent anyway) useful in essentially any field you might name, from childcare to plumbing to legal analysis to fine art.
In particular: self-confidence that hard problems can be tackled and that anything one person can do another typically can also with training and effort, time management, lateral thinking, learning when to keep trying a strategy vs. when to switch and try something else, salvaging useful partial results from failed efforts, drawing diagrams, careful record-keeping of works in progress, more generally externalizing problem state so that it can be worked with outside your head, converting fuzzy problems into precise formal terms to they are amenable to careful logical analysis (including making explicit all of the assumptions involved in the model chosen), exploring the relationships between different problems, investigating specific concrete examples of general rules and generalizing from particular cases to abstract theorems, searching/skimming published literature for solutions to problems that are too much to handle or finding relative experts to ask for help and knowing how to do so productively, clearly explaining an original problem and its context and any simplifying assumptions and then clearly explaining a solution step by step, checking solutions by solving a problem multiple ways or doing quick sanity checks, .....
I could probably keep listing more here, but you get the idea. Anyone who plans to do any kind of real-world technical work will be at a huge advantage if they have significant amounts of problem-solving practice going back to childhood.
You might similarly protest that we should not bother reading and analyzing novels in school because few careers explicitly require reading/writing fiction, or that we should not bother with physical education courses because few careers require playing dodgeball, or that we should not bother with music courses because few careers require skill at playing the recorder, etc. etc.
There is a wold of difference between solving for X and proving P =/!= NP. Collage level math like DiffEq is still the kiddy pool with well known approaches that work.
Writing a new graphics engine, or baking a cake takes applying existing techniques, but not the fluid exploration of unsolved frontiers. So, when I say use math I mean take advantage of what exists not nessisarily add anything new.
So, yea we want problem solvers, but not thinkers.
Why do you think (paraphrased) “I think secondary school mathematics should focus on solving non-obvious problems” has anything to do with “proving P ≠ NP” per se? I’m obviously not suggesting that we should assign famous unsolved research problems to secondary students. I would instead hope we could assign students a variety of problems taking them between 10 minutes and a few weeks to solve (at their current level of skill), with emphasis placed on smart problem-solving efforts rather than on sorting students based on who gets the most right answers.
Most American undergraduate differential equations courses are taught as a list of recipes with little room for thought. Rather comparable to elementary school arithmetic drills frankly, though obviously involving more built up preparation. https://web.williams.edu/Mathematics/lg5/Rota.pdf
However, it is possible to assign difficult problems to students at any level from age 3 onward (see http://www.msri.org/people/staff/levy/files/MCL/Zvonkin.pdf for an example of real mathematics instruction for preschool students; for primary students look up the work of Dienes; at the middle school level I think some Russian programs are pretty good https://bookstore.ams.org/MAWRLD-7/ etc.). It just takes more work for teachers to provide feedback about student solutions to such problems, it’s less amenable to grading by rote (and therefore not easy to check via standardized tests), and it takes more significant focus/attention/decisionmaking by teachers from moment to moment (and ideally more teacher background preparation). The students learn the subject more deeply, enjoy the process more, and learn significantly more transferrable skills.
Porting a graphics engine from one platform to another very similar platform (after having ported many other software projects between the same pair of platforms) or baking a cake just like all the others you have baked before might take nothing more than skillful application of well-established procedures, but making a new graphics engine in the first place (assuming it does something novel) or inventing a recipe for a new type of cake definitely takes problem solving skills.
I am saying there are different classes of what you are lumping under problem solving skills. Coming up with a new cake recipe does not involve building a new oven or measuring system. It's a well constrained problem. Use known techniques and apply some time and money gets a new recipe.
P vs NP on the other hand might not be possible to solve.
Yes, but inventing a new cake recipe (especially one fairly different than what you have baked before) uses a completely different set of skills than baking a cake, is the point.
You need to develop hypotheses about cake baking, test them empirically (e.g. by baking many cakes while varying one ingredient systematically), cross-apply knowledge from other cooking/baking experience, figure out workarounds to any problems that come up, at some point develop a high-level goal (e.g. mix a particular pair of flavors), and then check that your result matched your previsualized goal, tweaking the recipe in response to feedback until it comes out the way you want, keeping detailed notes matching recipes to results, etc. You need to have a much deeper understanding of cake ingredients and baking chemistry, and you need to work a lot harder at a higher level to invent recipes than to follow them.
If you are a cookbook author designing your recipe to be implemented by unskilled homemakers using unstandardized ingredients and equipment, or if you are a food chemist for an pre-packaged cake mix company, you might have an even larger set of concerns and required skills to invent a new cake recipe.
The kind of skills you learn while inventing new cake recipes might also be useful for solving other kinds of engineering problems. The kind of skills you need to follow someone else’s recipe to bake a cake are much more limited and domain-specific.
I must admit I still don’t understand why P vs. NP has anything to do with primary/secondary math education.
The problem is that current (US) math education...
Worth noting that the US tried the kind of math education you are suggesting (called the New Math initiative) and it failed miserably. The math education we are seeing today is largely born out of a counter reaction to that failure.
The New Math was something fairly different than what I am suggesting. It was an attempt at an alternate curriculum for primary/secondary school based on higher-level / more abstract mathematical topics, partially displacing study of arithmetic.
The New Math curriculum per se wasn’t so terrible, though it certainly had flaws (like anything invented from scratch out of context and not slowly developed and tweaked over time in response to feedback in a real-world setting). The bigger problem was that the proponents of the New Math didn’t have much buy-in from students, parents, teachers, school administrators, or the broader society, didn’t really do any outreach or teacher training, didn’t really produce enough supporting materials, and just dumped the curriculum on schools without support.
Parents and teachers didn’t know what to make of the curriculum (were unqualified to teach with or assess it), and didn’t feel involved in the process, and as a result there was a lot of opposition.
But what I’m talking about is not teaching different subjects per se, but teaching whatever subject in a different way, focused more on solving problems and thinking than on precisely mimicking teacher’s demonstrations or memorizing formulas. The current typical math pedagogy is patronizing, emphasizes memorization/recall and very careful attention to details (sometimes irrelevant details about formatting), teaches students that they shouldn’t try to think for themselves and teaches them to conflate getting the right answer with being “smart” or “good at math” and that anyone who makes a mistake or doesn’t know how to get the answer is “stupid” or inherently incapable.
I'm not in the US, but both of those seem to be greatly focused on the curriculum. OP is instead talking about the style of teaching and learning, which could be applied to practically any curriculum.
The intuitive geometric description I sometimes use with people is to have them imagine two duplicate rectangles and cut a diagonal in each to form four equivalent triangles. Mentally label the hypotenuse c, the long side b, and the short side a. Then arrange the four triangles into a square with the perimeter consisting of sides of length a+b. Now from the outside area is simply (a+b)^2. But computed from the inside area will consist of 4 * (1/2 a*b) + the area of the central tilted square with sides c or c^2. Simple algebra yields a^2 + 2ab + b^2 = 2ab + c^2 or a^2 + b^2 = c^2. A more fun problem is imagining why there are 180 degrees in any triangle. And for that matter where did that 360 degrees in a circle come from (last one hint: imagine 6 duplicate equilateral triangles and arrange them into a hexagon, assign the Babylonian magic number 60 to each vertex and sum the interior central angles, after using one side to draw a circle around the hexagon).
Had to stop reading long enough to see if I could prove it myself using similar triangles. This is definitely cleaner than what I came up with though, which involved creating a similar triangle with side b1 set equal to side c of the original.
Can anyone point to a proof that for similar non-isosceles right triangles, the area is proportional to the square of a side? Can that be generalized to other shapes?
When you scale a figure, say by doubling all distances, the area quadruples. Scale by factor k and the area goes up by a factor k^2.
So if the area proportion between the triangle and the square is T:S, and we scale both figures by the same factor k, we get areas k^2* T and k^2* S respectively.
I remember watching a documentary once, and it kind of made the case that Einstein wasn't an isolated genius, but more like Mark Zuckerberg. Basically organizing and putting together other scientists work,then unfairly getting all the credit. Anyone seen this documentary before? Its been years since I watched it, so memory is vague.
For what it's worth, organizing other scientists work from a new perspective and having a unified framework is the highest form of genius a scientist can achieve. That is why for example Einstein is credited with relativity rather than Poincare.
I find this style a bit frustrating to read. I just want to know what his proof is, but there's so much fluff in the article that the actual meat is kind of hidden. So anyway, "his" proof is to drop a height to the hypotenuse and use similar triangles. Ah, right, I know this proof and I remember it as the canonical proof my grade 9 geometry textbook presented.
this is the New Yorker not an maa article. it's supposed to be entertaining for a typical New Yorker reader. I swear man some people are so tone deaf it's frustrating: the whole world doesn't cater to people like us. at least try to make at attempt to empathize.
edit:
besides it's literally labeled steps 1-5 (i had no trouble skipping the "fluff" and finding the proof).
I'm sure other people like this style of article. I'm just telling you I don't.
Even the steps 1-5 are a bit fluffy for me. If you had just told me "drop a height to the the hypotenuse and use similar triangles" I would have known what was meant, more succinctly.
Do you have a suggestion on where to get unfluffed math news? I sometimes look at the AMS notices or Terry Tao's blog, but they don't cover quite the same breadth as things like Quanta magazine. I don't know where to get the big news for major events from an unfluffed source. Usually by the time the news has hit Quanta magazine or the NYT, it has already been circulating somewhere else long enough to be old news. I just don't know where else to look.
I guess, but that's a bit of a firehose. Is there really nothing intermediate between the NYT and just scouring all the journals? Something like the AMS notices or Terry Tao's blog, but slightly more frequently updated?
I don't see why it has to be pretension or conceit. Personally I am often frustrated by mainstream maths/science coverage, but that's not because I think I'm special or I'm pretending to be smart: in fact I'm very ignorant, only moderately intelligent, and couldn't follow the journal articles even if I had the time and inclination. My preference is for layman's explanations that guide me through a topic and help me to achieve some level of genuine understanding, and it's perfectly sincere. I'm sure there are people like me but smarter or more knowledgeable, whose preferences are similar but pitched a bit higher.
But this isn't the whole world, or the New Yorker for that matter -- it's the Hacker News comment section. jordigh's frustration is one that could easily be shared by many HN readers. (And in any case it was not presented as anything more than a personal reaction.) I genuinely don't understand why you responded so sharply, and for whatever it's worth I found jordigh's comment useful: it provided a tl;dr of the proof, and helped me decide whether the article would be enjoyable to read.
Eh, if I don't get what I want, so be it. I'm not demanding it. I thought you might know of something like it. If I can get what I was looking for, all the better, though.
Maybe I'm just a very specialised audience: I know enough mathematics to find some popularisations boring but I'm not a research mathematician where I can get gossip via word-of-mouth like I presume must happen. Maybe that's why there just isn't a publication like what I want. There aren't that many maths school dropouts like me who would want to read it.
Sympathetic, but noscript compatibility has been out the window for a decade at least. You can't reliably book travel, interact socially, read email, buy basic retail items, etc, without being extremely selective about providers. I'm with you philosophically, but that ship sailed long ago.
If the New Yorker would use the proper format of SVG here (or even PNG — the diagrams are all JPEGs), they would likely compress even better; most of them also include a ton of whitespace either side of the actual image. Testing one of them, the size falls by ~28% if you correct for all this.
On mobile, where my connection is less reliable, I also have a strong distaste for lazy loading: it squanders all the time available from when I started reading until now. Then, when the image is finally in view, then we start the long haul of fetching it, and risk that I've again lost good connectivity.
Conversely, I like that it doesn't waste my data if I only get 1/2 way through an article and have to close it because I'm out and about and the world around me may interrupt me reading something on my phone.
why are you making a comment that's completely irrelevant to the source text to attack a perception of the new yorker's writing style that isn't even present?
