OK, OK, I know. There are lots and lots of proofs of the Pythagorean Theorem. (My favorite appears at the bottom of this entry, in case you want to skip to it.) I’m not trying to overthrow thousands of years of math here. I’m just wondering about something, and I’ve been unable to find any treatment of this particular problem.
Suppose you’re walking along and come to an empty lot. You have to get from the northwest corner of the lot to the southeast corner of the lot. You could just walk all the way down the west side (side A), then walk all the way along the south side (side D). You’d find the total distance equals the length of side A plus the length of side D.
We all know that to shorten your distance, you simply cut across the lot. H (the hypotenuse) does not equal A + D, but is always less than A + D.
But let’s suppose for just a minute that we don’t really know what it means to cut across. We don’t, in other words, understand diagonal. Suppose you’ve got some strange condition where you can only move north-south and east-west, but nothing in between. Now you try to take a shortcut across the lot, but you get this:
Now if you add up your distance, you’ll find that it’s (more or less) 1/2 A + 1/2 D + 1/2 A + 1/2 D = A + D ! You didn’t do any better by cutting across the lot in this way.
OK, maybe if we do a little better. What about this zigzag path?
This path looks a lot more like the hypotenuse. In fact, you can make the zigzags arbitrarily small, maybe a million turns, or a billion, or any other ridiculous number you want. Your total distance traveled is still exactly A + D ! You haven’t done any better by taking the shortcut!
And yet, if you make even a small diagonal movement, you instantly see the benefit. Consider this path, in which you travel most of A, cut across the corner, then travel most of D. Your total distance will be less than A + D. 
So what’s going on here? Clearly the hypotenuse is the best path, but I can’t approximate that hypotenuse, even a little, by using only north-south and east-west movements. What is this “diagonal” and why is it so powerful? Any thoughts?
OK, here’s the elegant proof of Pythagoras that I like, the one that Albert Einstein found on his own when as a boy he fell in love with what he called “holy geometry.”
The first thing to notice is that there are three three similar triangles in this picture. Why? Because:
1) similar triangles all have the same internal angles.
2) all triangles have the same internal angle measure (we know it’s 180 degrees, but that doesn’t matter – what matters is that all triangles have the same measure, whatever it is).
3) by 1 and 2, if two right triangles share an angle, then they have to be similar.
The three similar triangles are:
ABC, ACX, and CBX (notice that the order of the letters is crucial here, matching the appropriate angles).
In similar triangles, the ratios of the sides are equal. So the triangle CBX compared to the triangle ABC gives
a/x = c/a
Cross-multiplying gives a^2 = cx
The triangle ABC compared to the triangle ACX gives
c/b = b/(c – x)
Cross-multiply again and you get
b^2 = c^2 – cx
But we already know cx = a^2 from before, so now we have
b^2 = c^2 – a^2
Rearranging, we get the familiar
a^2 + b^2 = c^2
The Pythagorean Theorem! THAT is a beautiful bit of math.
It occurs to me that the proof depends on the very existence of triangles, and triangles only exist in a two-dimensional world. Somehow, the Pythagorean theorem emerges from the existence of two dimensions as opposed to one. I still don’t see how, but it’s interesting.
Here’s something else. If you made it through my light clock blog entry, you know that Special Relativity comes right out of the Pythagorean Theorem. The light clock is a two-dimensional argument, so it seems that even if we reduced our universe from three spacial dimensions to only two, Special Relativity would still work. Flatlander twins would still experience the paradox if one rocketed off in a two-dimensional spacecraft, then returned to find her twin aged and wrinkled (would flat creatures wrinkle?) But what isn’t clear to me is whether or not Special Relativity would apply in a one-dimensional universe, one in which everything existed on a line, since in such a universe, there are no shortcuts.
Anyway, if anyone has any thoughts on why the Pythagorean Theorem works, or if you’ve ever seen an analysis of this sort of north-south, east-west approach and why it fails, I’d love to hear about it.
34 comments
Comments feed for this article
February 5, 2009 at 6:31 pm
Nico
http://en.wikipedia.org/wiki/Manhattan_distance
February 6, 2009 at 12:40 am
stephenwhitt
Nico,
This is exactly what I needed. Thank you! Taxicab geometry, that’s fantastic!
Pretty much, wherever you go, a mathematician has been there before. Great reading.
February 5, 2009 at 6:47 pm
Casey
Whenever you subdivide, you always have the same picture, locally. Each of those little steps is going a little bit father than the hypotenuse of the little triangle, and then you are adding up all of those extra distances over many of little steps.
February 6, 2009 at 12:43 am
stephenwhitt
Casey,
Yes, I love the idea of the step-wise movement gives the same picture if you get close enough. It reminds me of the commercial about Maldelbrot and fractals from a few decades ago.
February 5, 2009 at 7:13 pm
Nathan
The approach fails because it only looks like a hypotenuse to the human observing it. Think of this – since we’re dealing with real numbers, you can scale the approximation to an infinite number of times. From the perspective of the entire parking lot, it looks to us as though the person is merely following the hypotenuse, and should take less time getting there.
However, the reality of the situation is that with each minor south-east component movement that this person takes, he could be taking the diagonal between the two instead. So, instead of moving (A / 1000000) + (B / 1000000), he could move sqrt(A^2 + B^2)/1000000 instead by taking the diagonal.
No matter how closely this zigzag line seems to approximate the diagonal hypotenuse, it only “approximates” it because it is taking a nearby path. The amount of time that this person spends actually *on* the shortest path is 0 – he touches the line on half of his turns; however, since we’re talking about a line in real numbers, the width of the line is 0 and he spends 0 time on it.
If we go the route of saying that a path nearby the hypotenuse should likewise be shorter, it is a false assumption. Would it not also be “nearby” if the person walked in a sine wave, using the hypotenuse path as an axis? However, this is obviously not the shortest path between the two points.
Finally, I think one of the problems with your logic is the switch from pure, real-number mathematics and real life. Math says that no matter how small your approximate diagonal is, you can still take a shorter path by going on the diagonal. However, real life says that if your little zigzag is approximate to the diagonal, you might as well consider them to be the same path.
Hopefully that was written in a followable method…I did like your problem and was fascinated by it…it took me a little bit to figure it out. Your conjecture on special relativity in a 1-d universe is also interesting, but I think that special relativity holds for 1 dimension as well. Why wouldn’t it? Sure, there are no “shortcuts” on a 1-dimensional line, but a spaceship traveling in three dimensions doesn’t depend on shortcuts. I guess I’m not clear on how shortcuts (as you’ve mentioned them) are paramount to the theory.
March 29, 2010 at 1:27 am
Alex
I think that Stephen’s bit about the shortcut is correct to a certain extent. The zigzags would only be the same distance as going around half the perimeter if the turns were right angles. Anything less than 90 degree turns would make it a faster route.
February 5, 2009 at 9:07 pm
n.
The step-wise movement example can be viewed as being the same a moving from point to point within the plane (x,y) with x and y being chosen from the Natural numbers (written NxN). If you start at point (1,1) and want to move to (0,0) you must first move to either (1,0) or (0,1). Hence the distance is always 2. There’s no way to construct a path that stays within NxN and is any shorter than this. You simply lack the points you would need. What you’ve observed is that it doesn’t matter how close together the points are packed if we’re still in the NxN space. By constructing a diagonal path, you are implicitly upgrading! You’re now building a path through Euclidean space, instead.
The critical difference is that NxN lacks an inner product and so the Triangle Inequality doesn’t hold there. Once you’ve “upgraded” to Euclidean Space, you get all the amazing topology that goes along with it. And the Pythagoream Theorem falls out of this.
http://en.wikipedia.org/wiki/Triangle_inequality
http://en.wikipedia.org/wiki/Inner_product_space
February 5, 2009 at 11:24 pm
Joseph Drozdik
Hi There,
I really enjoyed reading your article. Your zigzag path reminded me of the fractal problem “how long is the coastline of Britain? The more you zoom in the more details you see making the line longer.” A zigzag on a grid doesn’t approach a strait line at all as you said. The distance never shrinks from x+y axis. The best thing it does it test how good your vision is. The point where a zigzag starts looking like a line is the limit of your visions sharpness. Its not so much that that diagonals are “powerful” its just that zigzags have nothing to do with diagonal lines.
If a diagonal lines are not like zigzags. What are they like.
The answer has a lot to do with what the definition of a strait line is. My math isn’t so good so you might have to clean it up but follow me on this.
In a one dimensional world everything is layed out along a line and it is strait. (strait meaning the shortest distance between two points is along the line.) You can describe a location with a single number and you can describe a line with two numbers, one for each end. If you extend the lines past their end points all lines in this world are parallel and they are all the same.
What is a strait line in a two dimensional world? In a two dimensional world you need two numbers to describe a point. You need an x and y axis. A line is still the shortest distance between two points. Lets say you have a strait line drawn with a ruler on a piece of paper. You could overlay it with clear plastic with an xy grid and see your line on a graph. If you twist the plastic grid you can make your line parallel or not parallel to the x axis or y axis. Whenever it’s not parallel to your axis you can see right triangles. The line doesn’t care about your grid. It exists on its own. How your orient your grid is arbitrary and your decision. Perhaps if the universe was made of pixels like a computer monitor there would be a correct orientation but it isn’t.
If you think about things this way a right triangle can be defined as an x and y axis intersecting with a line that is not parallel to either of them.
Here’s another interesting thought about strait lines. If you imagine that you are on the line and you are traveling along it you could move forward and backward and do all the things someone in a one dimensional universe could do. If you jump off the line you suddenly break into two dimensions. Now you are floating in space and you can measure the distance between you and the closest part of the line. Those two parts (you and closest point) form another line that is perpendicular to the line you jumped off of. Any other part of the original line you look at that is not the closets part can be used to form a triangle. You can define a triangle as you and any two points you can see along the line. If one of those points is the closest you have a right triangle.
February 6, 2009 at 12:53 am
stephenwhitt
Nathan,
Thanks, that does help, particularly the comment about how in an ideal mathematical set-up, even as the zigzags approach infinity the walker still spends exactly zero time on the actual hypotenuse. That’s some very solid analysis, and it really helped me see the problem differently.
February 6, 2009 at 12:58 am
stephenwhitt
n.,
Thanks for the links. My non-Euclidean geometry is pretty weak (perhaps my Euclidean geometry is, too!) but I’ll study those later. I like your NxN space notation, and it works well with the “taxicab geometry” link from Nico. Very interesting!
February 6, 2009 at 1:00 am
stephenwhitt
Joseph,
I like the image of the straight line and the coordinate systems through clear plastic. That helps me, as does the notion that the coordinate system is an arbitrary construct. It is interesting that we live in a universe without a preferred reference frame, and that’s what makes the Pythagorean Theorem work!
February 6, 2009 at 6:38 pm
Joseph Drozdik
I thought of one more way of treating the issue that I’ll quickly share. That’s the idea of measuring distance with a string. You can take a string and connect it to one corner of a city block and pull it tight along the diagonal to the other corner and measure the distance. You can also bend the string around the corner of the block, pull it tight and measure that distance as well. To bring in your zigzag concept you can take the string that is bent around the corner and fold it in to the diagonal line. Keep folding it in from each corner to the diagonal until you have many zigzags. With a big city block and a thin string you can fold many times. Now you have a string that is tightly packed along the line. When it comes time to measure the distance however you still have to pull the string tight. Your string will uncoil and when its tight you can measure the distance.
February 6, 2009 at 10:11 pm
A.J.
Here is an interesting “tangent” thought that your entry made me think of. Although it may be stretching your metaphor a bit, another advantage of traveling the hypotenuse is accuracy. Let’s say someone needed to be at a certain point on an axis (say x=4, y dosen’t matter), and the grid is divided into 1 foot by 1 foot increments. But let’s say this person can only move in 3 foot steps. The person would not only save time and travel less distance by following the diagonal path, they would also be able to come closer to their target destination. Traveling along the hypotenuse has a higher level of discrimination than just traveling along the y axis until you get to x=4. Our imaginary traveler could not even reach x=4, he could only reach x=0, x=3, x=6, etc. However, when he travels along the hypotenuse, he can end his journey much closer to his target of x=4.
I’m sorry if this is a little confusing. I learned this fact as it relates to machining, particularly turning pieces on the lathe. Moving the cutting tool along the hypotenuse of an imaginary triangle is much more accurate than moving the cutting tool on just the x or y axis by itself. This is because the hypotenuse is longer than the other two axis, but all three movements have the same incremental movement. This little manufacturing trick is a real world application of the Pythagorean Theorem and the wonderful trigonometry that resulted from it.
February 10, 2009 at 4:11 am
stephenwhitt
AJ, this is very cool. It reminds me a little bit of a Richard Feynman story. He was listening to one girl describe to the other what he took to be a way of measuring distance on an angle. He grew interested and listened in, only to discover they were discussing knitting. It was, he said, his first realization that girls can do math! Well, that was a different, much less pc, time.
Thanks for the story. Of the many, many things I know too little about, the mechanics of how to actually build things is at the top of the list. But I’m learning.
February 7, 2009 at 6:02 pm
Kalid
Hi Stephen, interesting article! I’ve wondered about the same question myself, and enjoy Einstein’s proof as well.
I take a different interpretation of the similarity argument (written about here: http://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/). I see it as a type of conservation law:
Area(C) = Area(A) + Area(B)
Where Area(C) is the area of a triangle with C as the hypotenuse. Since of any shape area is proportional to a side squared (the proportionality constant depends on the shape of course), it means means:
x*c^2 = x*a^2 + x*b^2
c^2 = a^2 + b^2
What I like about this viewpoint is the constant could be pi, for instance, and you get a relationship between circles (circle of radius 5 has the combined area of a circle of radius 3 and 4). Anyway, interesting thoughts! I also like to understand proofs beyond algebraic manipulation.
February 7, 2009 at 10:52 pm
stephenwhitt
Hi Kalid,
Thanks for the post. I’m loving your web site, and I look forward to spending more time there. OK if I link to it?
February 8, 2009 at 9:08 pm
Kalid
Hi Stephen, thanks for the kind words! Of course, feel free to link away
.
February 25, 2009 at 7:40 pm
marius09
Nice article.
In a way, question related to zig-zag issue is related to divergence seria:
even limit of its particular terms is 0, seria’s limit could be very well infinite.
For example, row {(1/n) | n greater than 1} has 0 as limit, but S = (1/1)+(1/2)+…(1/n)+… is divergent. “Intuition” goes wrong in both cases … All stuff involving in a way or another the infinite (as steps in the zigzag path) are almost projected to mess our evolutive instinct…:)
Marius
February 25, 2009 at 11:49 pm
stephenwhitt
Yes, the fact that that series is divergent is difficult for me to grasp. I suppose one way to think of it is if you consider that for large n, the difference between 1/n and 1/(n+1) is very small, so you see that – if you go far enough – you’re essentially adding the same amount each time.
March 2, 2009 at 9:59 pm
marius09
Hi Stephen,
I am not very sure I understood very well your point. It cannot express “same amount” because actually, more formal, if such margin would exist (more concrete: suppose it exists N so that for any n>N, |s(n+1)-s(n)|>”epsilon” – which formally express your idea; I noted s(k) sum of 1/1 + 1/2+…+1/k ) then it would involve that it exist N so that s(n+1)-s(n)>”constant epsilon”, for any n>N , which is equivalent with 1/(n+1) > “constant” for any n – and this is absurd. This shows once more how un-intuitive are issues dealing with infinite. Returning to your nice article on Pythagora “paradox”, point is that we “would expect” the zig-zag line to “fall” on the diagonal. Point is that it simply doesn’t happen, even all of us would vote it. This is the beautiful and strange power of infinite. I think.
March 3, 2009 at 12:33 am
stephenwhitt
Hi Marius,
It’s just a way for me to “see” the mathematics in my mind without going through the formal rigor. I can see why the series 1 + 1/2 + 1/4 + 1/8 converges, because each number in the series is quite a bit smaller (in fact, exactly half as much) as the one before. It turns into a zeno’s paradox situation. But the series 1 + 1/2 + 1/3 + 1/4 + 1/5 + … doesn’t converge. If you think about going way, way down the line, you end up with terms like 1/1,000,000 and 1/1,000,001. These two numbers are, of course, almost the same value, so the difference between consecutive terms is very small – unlike the converging series 1 + 1/2 + 1/4 etc.
One of my favorite responses to the Pythagorean problem was the notion that, because lines and points have no thickness, even though the zigzag touches the diagonal many, many times, the length of the diagonal covered by the zigzag is actually exactly zero. That’s a beautiful thought, and really makes the idea concrete for me.
Still, I think it’s somehow puzzling (and I can’t explain it mathematically, it just feels weird) that diagonal motion can never be decomposed as a combination of pure left-right and pure up-down motion. Instead, it is motion of a wholly different class – both left-right and up-down simultaneously.
March 26, 2009 at 5:39 pm
carisma
what it do
November 6, 2009 at 6:24 pm
John Ryskamp
Funny thing is, the Pythagorean theorem probably DOESN’T work, but we don’t yet know how. Your analysis is too “comfortable,” too mainstream, and doesn’t take account of what is going on in the history of mathematics. You’re self-satisifed–and that’s always a mistake.
We’ve made considerable progress understanding the Pythagorean Theorem as a piece of constructivist mathematics. Constructivism, as you know, stands for the proposition that argumentation inherently ends in paradox.
So some arbitrary insertion in the argument must be made, in order to “head off” this “paradox.” It is called the “constructivist intervention.” And it is usually an intervention which shares a deeply held prejudice, so it is hard to spot.
As it happens, there are no paradoxes: their compulsion is not their logical content, it is their rhetorical artifice.
It is funny you should say that special relativity “comes out of” the Pythagorean Theorem. This is because we have now discovered the constructivist intervention in Einstein’s formulation of the relativity of simultaneity. If you look at the train experiment in RELATIVITY, you will see it explicitly (it is implicit in the 1905 paper): it is the notion of a “natural” coincidence, which Einstein inserts with respect to M and M’.
That took a hundred years to locate. Your comment suggests that the constructivist intervention in the Pythagorean Theorem is this “natural” coincidence of points.
That is a pretty well known assumption now, but it has not yet been precisely located in the Pythagorean Theorem. If you’re so smart, where is it?
Since the relativity of simultaneity has now been “disproved” (although, according to Einstein’s “Geometry and Experience,” it was never supposed to have logical content–due to the constructivist intervention, constructivist “arguments” don’t have logical content), we can no longer get to General Relativity.
So we can no longer say that the Pythagorean Theorem is invalid under the Standard Model.
However, this only means that the Pythagorean Theorem is once again at issue, with “natural” coincidence being a candidate for the constructivist intervention in that Theorem, as you imply.
So the question for you and your readers is:
where is the constructivist intervention in the Pythagorean Theorem?
For more on the influence of constructivism in twentieth-century ideas, see
Ryskamp, John Henry, Paradox, Natural Mathematics, Relativity and Twentieth-Century Ideas (June 17, 2008). Available at SSRN: http://ssrn.com/abstract=897085
For more on the revolution in set theory which has renewed interest in the logical status of the “paradoxes,” see A. Garciadiego, BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC ‘PARADOXES.’
November 7, 2009 at 12:15 pm
stephenwhitt
Wow. Thanks for the comment. You sound like someone I could learn from. I hope you’ll indulge me.
“You’re self-satisifed–and that’s always a mistake.”
Actually I don’t think I am. That’s why I asked the question. Some of the responses pointed to some interesting ideas, and now I feel better that at least some have asked the same questions. But I agree that being self-satisfied is a mistake – particularly for me, since I don’t know and really want to.
“Constructivism, as you know, stands for the proposition that argumentation inherently ends in paradox.”
Nope, didn’t know that.
“And it is usually an intervention which shares a deeply held prejudice, so it is hard to spot.”
This is an interesting idea. I think I recognize it often in the history of science. Advances happen when someone asks the question that everyone thought they knew the answer to already.
“As it happens, there are no paradoxes: their compulsion is not their logical content, it is their rhetorical artifice.”
Can you say this again? I don’t understand anything after the colon.
“It is funny you should say that special relativity “comes out of” the Pythagorean Theorem.”
The reason I said this is that many of the explanations I’ve read of Special Relativity (including one of my favorites, the “light clock”) have the Pythagorean Theorem embedded within. And of course the form of the equations has a term in the denominator that comes directly from the A^2 + B^2 = C^2 solution. I always thought that was quite beautiful. Perhaps it’s not fundamental, but it helped me reach whatever level of understanding I have (perhaps not much).
“If you’re so smart, where is it?”
Beats the poo out of me.
“For more on the influence of constructivism in twentieth-century ideas, see
Ryskamp, John Henry, Paradox, Natural Mathematics, Relativity and Twentieth-Century Ideas (June 17, 2008). Available at SSRN: http://ssrn.com/abstract=897085”
I will read this and try to learn as much as I can. I hope to hear from you again.
December 10, 2009 at 8:15 pm
John Ryskamp
There’s a clue in your presentation of the Theorem. The argument in it sounds like Zeno’s paradox. Zeno was a great critic of Pythagoras, and I wonder if his “paradoxes” are implied criticisms of the Theorem. Of course, Zeno’s “paradoxes” are not paradoxes.
By compulsion I mean, there’s something which makes us feel there is “something to” the argument. But it turns out not to be logical content, only a clever rhetorical approach which hides a flaw. A lot of “paradoxes” refer to an infinite domain, so we cannot define crucial terms, although the references are thrown in “casually” and seem to be ideas on which we all “agree,” so there seems nothing we need to examine.
Considerable work is under way on demolishing the “paradoxes.” A very important book is Garciadiego’s BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC ‘PARADOXES.’
But a shorter route to finding the constructivist intervention in the Pythagorean theorem, may be to evaluate it in the context of the mistake slipped into the relativity of simultaneity, the “natural” coincidence Einstein slips into the train experiment (see his book RELATIVITY).
Nobody ever took Einstein seriously in his presentation of constructivism in “Geometry and Experience,” although Einstein felt that that address was the best formulation of his ideas he had ever achieved. But he actually practiced what he preached. There is no logical content in the idea of the relativity of simultaneity, Einstein did not believe there was any such thing as logical content, and he carried out that idea by making a constructivist intervention in the relativity of simultaneity. “Natural” is that intervention (“fallt zwar…zusamman”). It’s the “zwar” which is the intervention.
Is there a “zwar” in the Pythagorean theorem? I think, yes, although it’s probably tucked away in a place and we have a prejudice against looking in that place.
In order to see whether you understand the logical role of “zwar,” you ought to look at the 1905 paper, because Einstein does NOT use it there. Then your task is to show that the formulation of the relativity of simultaneity in the 1905 paper is the same formulation Einstein used in the train experiment in RELATIVITY, but expressed in a different way.
So then, what, precisely, is the relation of the Pythagorean theorem to the relativity of simultaneity as Einstein expresses it in RELATIVITY.
For example,
where is Einstein’s “M” in the Pythagorean theorem?
where is Einstein’s “M’” in the Pythagorean theorem?
Making these identifications will take us a long way toward identifying the constructivist intervention in the Pythagorean theorem.
I maintain that the Pythagorean theorem is constructivism. Am I right?
December 13, 2009 at 9:30 pm
stephenwhitt
I think we’re speaking two different languages. You mentioned earlier that the Pythagorean Theorem doesn’t work. Why is it, then, that whenever I take a piece of plywood and cut it at a right angle on one corner and a diagonal on the other, I get two sides whose length squared add up to the square of the diagonal? Seems to me that is a pretty good indication that the theorem works.
What interests me is why the universe works in such a way that such a shortcut (the diagonal path) is possible. I’m also interested in why in the light clock the diagonal path is longer than the straight path. In other words, why does 1 + 1 = 1.414….. instead of 1 +1 = 1 or 1 + 1 = 2.
Sorry I’m not answering your questions. I think you’ve assumed a lot of knowledge (for instance, what is “zwar?”) that I lack.
December 13, 2009 at 11:46 pm
John Ryskamp
stephenwhitt
I think we’re speaking two different languages. You mentioned earlier that the Pythagorean Theorem doesn’t work. Why is it, then, that whenever I take a piece of plywood and cut it at a right angle on one corner and a diagonal on the other, I get two sides whose length squared add up to the square of the diagonal? Seems to me that is a pretty good indication that the theorem works.
NOT REALLY. YOU’RE NOT LOOKING AT THE LOGIC, YOU’RE HIDING BEHIND THE SUPPOSED ‘THINGS WE ALL KNOW AND GRANT.’ THE ‘PLYWOOD,’ THE ‘RIGHT ANGLE,’ AND ‘LENGTH SQUARED,’ ALL ARE TERMS OF ART WITHIN THE DISCOURSE OF THE PYTHAGOREAN THEOREM.
WHICH IS FINE. BUT YOU’RE ‘GEE GOSH AND GOLLY I’M JUST PLAIN FOLKS’ DOESN’T CUT ANY ICE WITH ME. YOU’RE STILL WITHIN THE ISSUES RAISED BY THE PYTHAGOREAN THEOREM. CALLING IT ‘PLYWOOD’ AND SAYING IT MAKES SENSE TO YOU, DOESN’T ADDRESS THE ISSUE. STICK TO THE ISSUE, INSTEAD OF REACHING FOR THE AD HOC. BAD RHETORICAL STRATEGY, BECAUSE IT’S EASY TO SEE THROUGH.
SO NOW WE’RE BACK TO THE LOGIC, THE PLYWOOD NOTWITHSTANDING.
THE FACT IS THAT EINSTEIN WAS A CONSTRUCTIVIST AND AN ANOMALY HAS BEEN IDENTIFIED WITHIN THE RELATIVITY OF SIMULTANEITY, ONE WHICH IS CONSISTENT WITH HIS CONSTRUCTIVIST ORIENTATION.
THIS MEANS THAT THE PYTHAGOREAN THEOREM IS ONCE AGAIN ‘AT ISSUE.’ THAT IS, IT IS INVALID UNDER THE STANDARD MODEL, BUT IF WE DON’T REACH THE STANDARD MODEL, BECAUSE OF ‘NATURAL’ COINCIDENCE, THEN DOES THE PYTHAGOREAN THEOREM HAVE LOGICAL CONTENT.
I’M NOT SAYING THAT IT DOESN’T. WHAT I AM SAYING IS THAT GIVEN THE VAST INFLUENCE OF CONSTRUCTIVIST THINKING, IT IS LIKELY THAT THE PYTHAGOREAN THEOREM IS A PIECE OF CONSTRUCTIVISM, AND THAT THEREFORE IT CONTAINS A CONSTRUCTIVIST INTERVENTION WHICH DEPRIVES IT OF LOGICAL CONTENT. STICK TO THIS ISSUE.
What interests me is why the universe works in such a way that such a shortcut (the diagonal path) is possible. I’m also interested in why in the light clock the diagonal path is longer than the straight path. In other words, why does 1 + 1 = 1.414….. instead of 1 +1 = 1 or 1 + 1 = 2.
THE NOTION OF A ‘DIAGONAL’ IS PROBABLY MERELY AN ARTIFACT OF CONSTRUCTIVISM. DID THAT EVER OCCUR TO YOU? OF COURSE NOT, BECAUSE YOU NEVER GAVE A THOUGHT TO CONSTRUCTIVIST THINKING. BUT NOW YOU HAVE A REASON TO GIVE IT A THOUGHT, BECAUSE THE PYTHAGOREAN THEOREM IS ONCE AGAIN ‘AT ISSUE’ SINCE ITS LOGICAL CONTENT IS NOT UNDERMINED BY THE STANDARD MODEL (SINCE WE DON’T LOGICALLY GET TO THE STANDARD MODEL).
Sorry I’m not answering your questions. I think you’ve assumed a lot of knowledge (for instance, what is “zwar?”) that I lack.
YOU REALLY ARE NOT PAYING ATTENTION, ARE YOU? THE QUESTION IS NOT, WHAT IS ‘ZWAR.’ THE QUESTION IS, WHAT ROLE DOES IT PLAY IN THE FORMULATION OF THE RELATIVITY OF SIMULTANEITY? iT PLAYS THE ROLE OF A CONSTRUCTIVIST INTERVENTION. LOGICALLY, HAVING SAID THAT, WE DON’T REACH THE QUESTION, ‘WHAT IS ‘ZWAR?’ I THINK THE LOGIC SUGGESTS THAT THE NEXT STEP IS,
IS THIS CONSTRUCTIVIST INTERVENTION, ALSO THE CONSTRUCTIVIST INTERVENTION IN THE PYTHAGOREAN THEOREM?
SINCE ‘ZWAR’ INTERESTS YOU, ASK: WHAT IS THE PYTHAGOREAN IDEA OF ‘ZWAR,’ AND WHERE DO WE FIND IT EXPRESSED WITHIN THE PYTHAGOREAN THEOREM ITSELF? THAT IS, DON’T REACH OUTSIDE THE TERMS OF THE ARGUMENT (THAT ‘PLYWOOD’). SPEAK EXCLUSIVELY IN THE TERMS OF THE ARGUMENT, AS IF YOU ARE THE ARGUMENT SPEAKING TO YOUR OWN QUESTION.
IN SHORT, AVOID THE AD HOC.
December 14, 2009 at 2:11 am
stephenwhitt
Didya ever see the episode of Gilligan’s Island where the Professor meets some guy who knows a lot more than he does? The Professor feels all inferior, so he has to write cheat notes on his hands and such. I’ll have to go write some cheat notes and then I’ll get back with you after I’ve learned certain key things, like for one what constructivism actually is. Because I don’t know.
Also I’m moving, so I won’t have Internet access until at least Thursday. So thanks for the things to think about, and I’ll come back with my cheat notes and then maybe I can start learning from you.
December 14, 2009 at 2:58 am
John Ryskamp
Start by reading A. Garciadiego, BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC ‘PARADOXES.’
Constructivism is the idea that argumentation necessarily ends in paradox. Therefore, if you DON’T want your argument to end in paradox, you must make an arbitrary intervention in argument, to “deflect” it from “paradox.”
Constructivism is at least as old as Aristotle, who felt the need to “avoid” “paradox.” The question is, is Pythagoras a constructivist, and if so, where is the constructivist intervention in the theorem.
Perhaps, if the Pythagorean theorem is not an argument, the key lies in showing WHY it is not an argument.
February 3, 2010 at 5:32 am
Guillermo Bautista
hi. I have also created a blog about Pythagorean Theorem here:
http://math4allages.wordpress.com/2010/02/03/pythagorean-theorem/
you may also want to check the other articles here:
http://math4allages.wordpress.com/view-posts-by-topics/
February 4, 2010 at 12:14 am
stephenwhitt
Great stuff. I like your 0! = 1 entry. Very non-intuitive idea, but well done.
January 6, 2011 at 1:59 pm
Robert
Is this reply/feed still in use? i have an issue with the Pythagorean Theorem and wonder if anyone is still reading here?
thanks,
Robert
January 7, 2011 at 12:09 am
stephenwhitt
I’d love for you to post your thought. Don’t know if I can help, but I certainly learned a lot from many of the other posts on this entry.
April 26, 2013 at 1:19 pm
Peter Zelchenko
Well, I’m not going to play the “I’m dumber than anyone else here” game, but I’m perplexed by a lot of this. Lately I’ve been working on developing “urban orienteering” for Boy Scouts. We’ll be starting in Jackson Park, the site of the 1893 Columbian Exposition and of the then-largest building on Earth. Also the site of the first automobile race, which catapulted the car into world prominence (talk about a paradox), and of the University of Chicago, the Ivy League school where I’ve had my lantern out for years searching in vain for an honest man.
Inside the park, it’s just like orienteering. Take your compass, point it in a prescribed direction, go a prescribed distance, do the next leg. (We’ll call it “s’morienteering” because we plan on splitting the team into three groups, one for each ingredient. That will give them an attainable and compelling goal.) Once you get out of the park, you have buildings blocking your way. In geometric Chicago especially, you will have some geometry and maybe even trigonometry and the laws of sines and cosines (which ratios, as an engineer, seem more direct to me than the proofs of geometry — but then that’s constructivism according to Mr. Ryskamp).
To teach this honestly, I had better be able to explain these steps. And for three months I’ve been wracking my brain on it, because it’s not been good enough for me simply to know that c^2 = a^2 + b^2, that the squares work that way. It’s a thing of beauty in its way, but it does not teach triangulation. I want to know WHY these squares are put where they are to solve this problem, and WHY we square rather than do something else. Physics and math are mere convenience, coincidence, religion, magic, until I can see it demonstrated, preferably visually. I like a lot of the demonstrations above (particularly the discussion of minutely subdivided stairsteps), but it too often seems to enter the magical rather than the practical.
Euclid I.47 attempts to explain by showing similar parts against the squares and angles, and then in effect showing that half-parallelograms are the same size as others “in the same parallels.” I thought Pappus’ generalization helped simplify that. Pappus keeps us in the realm of parallelograms, and gives us a better visual explanation (see below). But then both of these keep taking us back to the question of the squares. All explanations I’ve found (“hinged dissection”?) are simply transforming geometric shapes to account for the squares.
I thought Jon Ryskamp was starting us toward an honest path, but soon his increasingly grating and sphinxlike tone and stance, spouting Russell and German, seemed to me to exemplify better than anything here the very magic he claims to want to dispel. He’s done nothing even to convince me he has the first clue. As a young girl, my mother used to spend most of her afternoons with Kenneth Burke’s family in New York City. Burke was a kind and generous expositor, and he could have explained Russell very well. But simply being close to his mind didn’t make her (or me) any better off. If Ryskamp himself is so smart, he’ll be explaining rather than simply trying to sound smart, assuming his part is even on the path. I think that all of us together can answer this question.
http://aleph0.clarku.edu/~djoyce/java/elements/bookII/propII12.html
http://www.cut-the-knot.org/pythagoras/PappusPyth.shtml
http://www.math.ufl.edu/~rcrew/texts/pythagoras.html
In a series of two 2010 University of Chicago papers, I complained about the teaching of chaos theory from a similar stance:
http://pete.zelchenko.com/2010/Vandervoort%20Paper%201%20d3%2006-09-10.doc
http://pete.zelchenko.com/2010/Vandervoort%20Paper%202%20d4%2006-18-10.doc