The Diagonal or Staircase Paradox
I haven't seen this in this forum, and since several members have an interest in math paradoxes I thought I would demonstrate one that interests mathematicians as well as the general public. The hypotenuse of a right triangle with each side equal to one is approximated by a "staircase" of steps having the same horizontal and vertical dimensions. Thus the total distance, adding horizontal and vertical sides, for each such staircase is 2. However, as the size of each step diminishes the staircase seems to more and more approximate a straight line, which has length the square root of two.
If it were easier to post it, I could give a similar example of a straight line between 0 and 1 that has infinite length.
Oddities that math people explain in different ways. But intuitively it sure seems like a paradox. :cool:
Comments (91)
Your example shows well that the square root of 2 cannot be a rational number, I guess.
Or not??
I thought not. I thought it's indefinite because it's multiplied by an irrational numbers. I don't see how a length can be indefinite though. Is it legit to round it to 1?
The key to dissolving the apparent paradox is to calculate the error in approximation for each tiny right triangle, then add them up. If each step has [math]\Delta x=\Delta y=\tfrac{1}{n}[/math] then the increment of error is
[math]\Delta \varepsilon =\left( 2-\sqrt{2} \right)\Delta x\text{ }\Rightarrow \text{ }Total\text{ }error=n\Delta \varepsilon =2-\sqrt{2}[/math]
Each member of the sequence of functions defined on [0,1],
[math]{{f}_{n}}(x)=\tfrac{1}{10\sqrt{n}}Sin(n\pi x),\text{ }x\in [0,1][/math],
is a smooth, wiggly curve oscillating about the line segment [0,1], getting closer and closer to that line as n increases in value. The length of the nth curve is
[math]{{L}_{n}}>\tfrac{\sqrt{n}}{5}\text{ }\to \text{ }\infty [/math],
This staircase paradox is of type 2). It seems contradictory because we expect that the length of the staircase ought to approach the length of the diagonal line that it approaches. It's not really contradictory, simply because that expectation is not right. (Mathematics is filled with many paradoxes of this kind.)
If a sequence C_1, C_2, C_3, ... of curves approaches a line L in such a way that, not only do the points of the curve approach the points of the line, but also the direction of the curve approaches the direction of the line, then the lengths of the C_n's will also approach the length of L. Otherwise it just isn't necessarily true, no matter how surprising this may be.
I know, silly games.
Yes, but in the real world... what happens when the step length approaches the Planck length?
One of the celebrated theorems in this subject is Stone-Weierstrass, which says continuous functions on an interval can be approximated by polynomials.
Approximate(ly) a straight line can be rephrased exactly as like but not identical to a straight line. In other words, the staircase never ever actually becomes a straight line. It would've been a bona fide paradox if the staircase actually became a straight line but the word "approximate" is there for a reason.
Is this really a paradox? It's like saying Roger Federer "approximates", in appearance, to Quentin Tarantino and then saying Roger Federer is Quentin Tarantino.
Maybe the staircase times infinity equals the straight line exactly
This is neat.
I just wrote the staircase after n refinements as a sum of 2n scaled indicator functions.
[math]S(n)=\sum_{i=1}^{2n} \frac{i-1}{2n} \big(\frac{i-1}{2n},\frac{i}{2n}\big)[/math].
Let the mapping of [math]S(n)[/math] to its arclength be [math]A(S(n))[/math]. Which is always 2 for any finite n.
But [math]S(n)[/math] is discontinuous.
So the limiting procedure:
[math]\lim_{n\rightarrow \infty} A(S(n))[/math]
doesn't let you take the limit inside the [math]S(n)[/math] bracket (the limit of arclength is not necessarily the arclength of the limit).
For those who don't speak math, the last bit: "letting the staircase get closer and closer to the line" doesn't entail "the length of the staircase gets closer and closer to the length of the line" since the staircase has discontinuous jumps in it.
As individual steps shrink in size, the inside corner point - the part of the step furthest from the imaginary limiting line if that line is visualized as above the steps - grows closer to that line. So, yes, there is "more and more". However, the total or accumulated error remains large.
The wiggly curve that uniformly converges to the line segment [0,1] is more entertaining, for its length becomes infinite.
Quoting Daz
Yes. These two instances rely upon point of perspective, as Mr. Wood explains. Tarski-Banach seems to depend upon the controversial Axiom of Choice, which doesn't come into play here.
What happens to all the corner points in the stairs as the number of steps increases without bound? :chin:
[hide="Spoiler solution attempt"]Every point in the line becomes a "corner", so there are no corners. More precisely, [math]S(n)=\sum_{i=1}^{2n} I\big(\frac{i}{2n},\frac{i+1}{2n}\big)[/math], when conceived as a function of line position [math]S_n (x) [/math] gets little indicator function atoms [math]x I(a=x)[/math] for any [math]a \in [0,1][/math] which correspond to the corner, which is just another representation of [math]\lim_{n\rightarrow \infty} S_n (x) = \sum_{\infty} x I(a=x)=x[/math], with a in [0,1], the hypotenuse. (more precisely they get arbitrarily close to that representation)[/hide]
Quoting ssu
The difference is that in the case of polygons approximating a circle, with each successive step the error decreases (don't ask me for a proof - it's pretty messy, from what I remember), whereas in the case of the staircase the error stays the same throughout.
There is no error, hence the apparent paradox.
Here is another take I just thought of. We can define two functionals, one that gives some measure of the distance between the points of the stair functions and the diagonal (e.g. the average distance), and the other that gives the length of those functions. The shape functional steadily converges to zero, but the length functional does not.
For those that don't speak half-math, the staircase has corners where the tangent to the corner is not defined. A tangent, in 2D, is a line that is "stuck" up against the object you're interested in, intersecting one point but no other points, at least "locally", but, crucially, in only one way. Ex. if you stick a line against a circle it can only be in one way, forming a T with the radius line; a line with a different angle to the radius will intersect more points in the circle, no way to escape it.
Discontinuous above refers to points with undefined tangents, not an actual cut in the line that draws the staircase, which is a continuous line in the normal sense of being able to draw it continuously without lifting the pencil off the paper.
It is in trying to draw the derivative, all the tangent values, where the discontinuities appear. The tangent to the vertical part of the staircase is also just a vertical line, then meets a point without a undefined tangent at the corner, and then suddenly switches to the tangent being just a horizontal line following the horizontal part of the staircase; of course best to use a coordinate system at some angle to the staircase as otherwise the vertical lines are at the same x coordinate as the corners and it's not so clear what the jumping is.
A corner has no unique tangent line, which means has no derivative, which means what seems similar and doable in other calculus contexts does not work with the staircase, basically.
A better paradox, I think, is accept it just stays the same length, but we allow the staircase to get such small steps that it "occupies" the same area as the line, in the sense of not allowing the staircase steps to cross some definition of the "closest" lines adjacent to the approximating line, i.e. building the staircase to stay in a lane of area zero, but still insist it has length 2.
Wrong paradox, Tim. The wiggly curve converges uniformly to the line segment [0,1] while its length tends to infinity. Sorry I don't have the image.
Back to the Diagonal:
Imagine a straight line that rests on the outer stair corners, from the bottom most step to the topmost step. As the steps shrink that line more closely resembles the hypotenuse of the large triangle of length square root of two. Now, what can you say about all those outer corner points, which lie on this evolving line, as the number of steps becomes infinite?
What paradox is this? I tried to look it up but only found the staircase paradox itself. Is there a path that converges uniformly to the unit interval but whose length is infinite?
There must be some neat identity for elliptical functions at work here, because otherwise I wouldn't know how to calculate such a limit.
By the way, and seemed to suggest that the key to the staircase "paradox" is in some pathology of the shape, namely its corners, where the curve is not differentiable. But this is not so. Consider a similar example, where in place of straight lines there are smooth curves. I'll use half-circle arcs for simplicity:
No corners here, the curve is everywhere differentiable (although the second derivative does jump around at the intersections). As with the staircase, the amplitude of the wave tends to zero as the number of crests increases without bound. But as with the staircase, the length of the curve does not approach the length of the diagonal. We don't even have to do the calculation to see that the length of the curve does not depend on the number of crests (this is because the length of each half-circle is proportional to its diameter, and the total length of all diameters is the length of the diagonal). And so the length of the wavy curve is always [math]\frac{\pi}{\sqrt{2}}[/math].
What's more, with a simple modification we can make the length of the wavy curve increase without bound, just as in @jgill's example. Just replace half-circles with ellipses whose major axis is perpendicular to the diagonal and scaled by a factor of [math]\sqrt{n}[/math].
These jumps are what I was referring to as not everywhere differentialable.
However, I'm not saying that differentiability is the end of it, only that non-differentiability is a good indication that limits may not apply as we might intuitively expect from cases of continuously differentiable functions, and my main purpose was to explain what discontinuous meant in the context.
Non-diferentiable points can be approached in various ways that may or may not make sense given the context of the problem. Physicists, for instance, may approach these points with a important theorem called "whatever, I don't care, we'll figure it out later, or someone will eventually I'm sure".
To "solve" these problems we'd need to choose an axiomatic system. If we view these problems in Euclidean geometry, there's no axiom of infinity, so the question doesn't really make sense (you can keep making smaller staircases or smaller circles, but you can't say you have done so "infinitely many times" and each staircase or circle is indistinguishable, in some sense, from a single point along a approximate line).
Yes, that's why I said "non-differentiability is not the end to it".
But it's a good litmus test (that intuitions may not apply).
Without the non-differentiable points that are perpendicular to the axis of the line, however, it's not exactly the same problem though.
But you need to select axioms, re-define the problem as sets for instance, and so on, to then "prove" an answer relative those axioms. Without doing that my response will be "maybe".
My main purpose, as mentioned, was just to explain the definition of "discontinuous" and that normal calculus concepts may not apply. I do not have a problem with a system where the limit in these questions does not make sense, is just undefined, nor a problem with a system where the limit does make sense (and even different answers in different systems would not bother me).
You are right, it's a sufficient condition for the failure of the arc-length functional to respect the limiting procedure, not a necessary one. I believe the staircase could be approximated by some differentiable curve (replace the discontinuities with regions of sufficiently high growth, I believe polynomials would work) and cause the same issues.
Do you know a sufficient and necessary condition that characterises this sort of pathology? Other than stating "the arc-length map of the limit of the approximating series of functions is not necessarily the limit of the arc-length map of the approximating series of functions".
Er, your terminology is all over the place. A continuous function has left and right limits converging to each of its points. The staircase function is, strictly speaking, discontinuous as it is pictured in the OP, but that is just an artefact of the coordiate system. If you tilt the X-Y axes, it will become continuous.
A differentiable (or smooth) function has the first derivative at each point; the half-circle function is differentiable (again, modulo axis orientation).
An infinitely differentiable function has all derivatives; the sine function is infinitely differentiable.
There are also piecewise- versions of all these (piecewise-continuous, etc.).
Quoting fdrake
Yes, that's just what I did with the half-circle curve, and I think the sine curve (with proper scaling) would work as well.
Quoting fdrake
Interesting question, but beyond my modest pay grade, I am afraid :)
Me too, I do not have the analysis fu for that.
My terminology?
I just gave you the definition twice of continuously differentiable, the derivative function is continuous.
I've made it pretty clear I'm talking about "differentiable" in the sense of differentiable.
Quoting SophistiCat
No!
The derivative of the staircase is not continuous. I even say:
Quoting boethius
Yes, keeping the staircase with vertical steps is not continuous. But no, tilting it doesn't solve the problem, only makes the problem more obvious.
So go ahead, plot the derivative of the staircase at some angle and show how it is "continuously differentiable".
The whole point of my first comment is to clarify what when he says:
Quoting fdrake
There is only a "discontinuous jump" in the derivative, not a discontinuity in drawing the staircase. Since his comment was aimed to explain things to people that know less math, it's important to point out that "discontinuous" is often used as shorthand for "continuously differentiable" in a lot of contexts (someone who doesn't know this may wonder what is "discontinous" about the staircase).
Quoting SophistiCat
No, turn the half circles anyway you like, you will always have a vertical tangent and it will not be continuously differentiable at all points.
Proof: You can move two half opposing half circles together and get a complete circle. A complete circle can be rotated at any angle you want and remains similar. At an angle, it's even worse as now the curve will retrograde and have 2 y values for some x values; so please, tell me, what's the continuous derivative function for that? it's just circles, should be easy to derive.
Quoting SophistiCat
Yes, I recognize you can use a function that is continuously differential, like a sinewave and not have the discontinuous derivatives to deal with. But, as I mention, it's not exactly the same problem, and whatever answer you arrive at for a sine-wave, you cannot simply apply to the circles nor the staircase.
It should be clear that these are not necessarily part of the same class of problems.
As for solving any of them. You'll need to do so relative an axiomatic system. If it's Euclidean geometry, the questions are in some sense sensible. A Euclidean procedure can create these objects under consideration as well as procedure to make half as small staircases or circles, but the limit at infinity won't be defined as there's no such tool in Euclidean geometry.
A sinewave cannot be constructed in Eucidean geometry, which maybe an interesting aside, or maybe indication that there's important difference with your circle example.
If the goal is to investigate a different but similar looking problem, that's fine, but there must be some basis to assume conclusions about something different will hold for the something else. There maybe critical differences (such as continuous differentiability), or maybe not.
Saying "ah, we can make a different object that doesn't have that problem" does not necessarily solve the problem encountered with the first object. Yes, you can make other objects and ask a similar question; no, the answer may not be the same every time you do so. There's a lot of steps to do to arrive at such a conclusion.
However, I have not said your program won't work, just pointing out some important considerations so that, er, avoid:
Quoting SophistiCat
Doesn't the "seeming straightness" of the staircase depend on the character of its visual appearance? Doesn't its visual appearance depend on facts about our visual systems and facts about the point of view we take relative to the staircase?
For instance, if you get up real close, or use a magnifying glass, you see segments of a staircase that doesn't look like a straight line anymore.
That doesn't seem so paradoxical to me. Is there another, perhaps more mathematical, paradox buried in this optical illusion?
It's called a paradox in the literature. Note the outer corner points seem to generate a line as n increases, but is the eventual line entirely composed of a countable set of points?
You can obtain the result of the other "paradox" by drawing a symmetrical sawtooth graph on [0,1] that collapses as n increases, and whose length increases without bound. I leave this as an exercise for those interested. Is the ultimate graph differentiable anywhere?
I'm beginning to sound like a math prof. Sorry! :cool:
Quoting boethius
I am not sure why you keep talking about Euclidean geometry, which, as you admit, doesn't even have the notion of a limit. You may as well be talking about group theory. Yes, I think it's pretty obvious that we are talking in the context where limits and such are defined; real analysis will do for the purpose.
Quoting Daz
No, if the arc length decreases any faster than in the examples that have been considered so far, it will converge to the length of the diagonal, as we intuitively expect. This is easy to do with any function whose distance from the diagonal can be scaled. For example, take the half-circle function and scale the peaks down by a factor of n - it will converge like a champ.
But yes, it is evident that whether the length error is constant, converging to zero or growing without bound is pretty precarious. You have to work to make sure that you get the "intuitive" result, because a lot of the times you will get something else entirely.
Quoting jgill
Yep, if your sawtooth graph doesn't have this property out of the box, you can easily make it so by multiplying it by some uniformly increasing function of n.
The reason this looks very counterintuitive to me is because if we put aside analysis and just look at what we get in the limit, every point on the staircase converges towards a matching point on the straight line - which of course has the length of the straight line. So what gives? Well, the formal answer is that the limit towards which the sequence is converging is not an element of the sequence: the limit points do not themselves lie on any staircase curve. This is not so unusual; for example, most converging rational number sequences do not converge to rational numbers.
Still, it just looks... wrong :)
Not quite, but close. The sequence actually converges uniformly to the hypotenuse, but arc length is not necessarily preserved under uniform convergence.
Note the outer corner points seem to generate a line as n increases, but is the eventual line entirely composed of a countable set of points? How can this be? :chin:
I wasn't explicit enough by what I meant by"scaled down versions".
First of all, I'm switching from the diagonal to just continuous functions defined on some interval of the real line. Same picture just viewed from another angle, but easier to talk about.
Now, what I mean by a "scaled down" is the graph of a continuous function y = f(x) that has been modified so that all linear measurements of the whole graph are shrunk by the same factor C > 1.
So instead of y = f(x) we're now looking at y = (1/C)*f(Cx).
This is essentially the same way the original stairstep pattern is scaled as it gets closer to the diagonal line. So if the graph is a bunch of semicircles of radius = 1, end to end resting on the x-axis, the arclength above a diameter like say 0 <= x <= 2 will be ?. The ratio of arclength to the length of the interval on the x-axis is ?/2.
If we shrink the whole graph down by a factor of say 100, then the semicircles now have radius = 1/100 and the arclength above a diameter, say the interval 0 <= x <= 1/50, will now be ?/100 and the ratio is again ?/100 / (1/50) = ?/2.
Because the OP does not specify an axiomatic system but describes the problem essentially in Euclidean geometry. The OP doesn't say "what does real analysis say about this?".
This is also not a highly technical mathematical forum, so people unfamiliar with real analysis may wonder how the problem presents in Euclidean terms.
Furthermore, it is a fairly usual question of interest "what is the simplest system in which the problem can be described?", and the problem of making the staircase or circles smaller and smaller can be described in Euclidean geometry.
The OP doesn't even talk about limits, just this process of making things smaller.
As other posters, have noted, the process of making things smaller doesn't change anything, "zoom in" and the shape is similar. This could be a satisfactory answer for some.
The "problem" arises when we want to consider, to take your circle example, the situation where "all points" on the line are the center of a circle; that we cannot zoom in and see circles.
I don't have time to fully investigate the characteristics of such an object, cause there's a pandemic, but considering the lack of intellectual rigour is essentially the cause of building a system so fragile to the pandemic in the first place as well as failing to contain it "because stocks might go down", I feel I need to be even more vigilant in these troubling times on all fronts!
Yes, you can smooth out corners to make a curve continuously differentiable; no, whatever conclusions are drawn from doing that don't automatically apply to the corner case, maybe we're interested in investigating the corners and want to deal with what happens when, trying to take the limit of shrinkifying the stair lengths, essentially every point becomes non-differentiable (that the object is "only corners", or at least all the rational points are defined as corners or some kind of scheme like this; may or may not be of interest to people here).
My intuition for these kinds of problems tells me my intuition may or may not be correct for these kinds of problems. Making the circles or the corners or the sinewave periods "dense" could have surprising results, which may not be the same for each case, or maybe even different ways to construct these kinds of object within the same system, not to mention different systems, with different results.
This isn't a criticism of your approach. It's totally valid to consider the case of the circle as a closely related problem to the stairs, and the case of the sinewave as closely related to the circles. It's not valid to assume conclusions automatically propagate backwards to all these cases, an argument is needed for why this would be so (lot's of theorems have "only necessarily valid for continuously differentiable functions" warning attached); nor is it automatic that those previous harder problems are not interesting because a simpler problem has been found. It happens all the time that very subtle differences make an approach work for one problem and not another and that harder problems are chosen over simpler ones. It's all I'm pointing out, I'm sure you agree now that my position is totally clear.
Looked at this some more. A general explanation seems to be "the arc length functional is not a continuous functional in the infinite norm (uniform convergence norm) on the space of continuous functions". We can treat the staircase as an example of this fact (after rotating it to make it continuous).
What does that mean, to say that the notion of vertical and horizontal lengths is incommensurate with the notion of diagonal lengths?
It is somewhat subtle to define for which subsets of the plane the concept of length makes sense, and to define exactly what the length is of such a subset. But in the standard definition, there is no distinction between diagonal lengths and horizontal or vertical ones.
Putting more symbols on it.
Let [math]C[/math] be the set of continuous functions over [math]\mathbb{R}[/math], then define the function [math]d : C \times C \rightarrow \mathbb{R}^+[/math] by [math]d(f_1,f_2) = \sup_{x \in \mathbb{R}} |f_1 (x) - f_2 (x)|[/math]. [math]d[/math] is a metric, so [math](C,d)[/math] can be thought of as a topological space.
We want to consider functionals over [math]C[/math], which are mappings [math]C \rightarrow \mathbb{R}[/math]. Consider some subset [math]M[/math] of [math]C[/math] and a function [math]f_0 \in M[/math]. Call a functional [math]L[/math] continuous at a function [math]f_0 \in M[/math] when and only when for all [math]\epsilon > 0[/math] there exists an open [math]U \subset M[/math] with [math]f_0 \in U[/math] such that [math]| L(f) - L(f_0) | < \epsilon[/math].
(Edit: an open ball with radius R centred at a function f (the collection of which generates this topology) seem to be the collection of functions whose supnorm is less than R; so can be thought of as the set of continuous functions which converge uniformly to f with R as the "minimal" least upper bound which can be used in a uniform convergence proof to f for all the ball elements.)
Stipulate that continuous functions have an arc length, and let the approximating staircase with [math]2n[/math] steps be [math]S(n)[/math]. If we consider [math]f_0[/math] as the straight line, [math]S(n)[/math] has arclength 2 for arbitrarily small [math]d[/math]-balls around [math]f_0[/math], which contain an [math]S(n)[/math] by uniform convergence.
The arclength functional is probably nowhere continuous in that sense? There will always be a series of approximating curves you can construct that have any desired arclength that uniformly converge to the desired function.
Well, no, it doesn't, because there isn't any problem so long as we stay with Euclidean geometry (as rightly noted). The apparent problem only arises when we introduce the notion of a limit, and perhaps other implicit assumptions.
Quoting jgill
Quoting boethius
"Almost none" of the limit points on the diagonal (let's just call it that for brevity) is a corner point, for the simple reason that there is only a countable number of them. Also, keep in mind that the diagonal (which we interpret as the limit point of the sequence of curves) is not itself part of the sequence and does not have the same properties. Every member of the sequence is piecewise-differentiable, while the diagonal is, of course, everywhere differential.
Do you think it's the case that in the limit the corner points become dense in the straight line, despite remaining countable? Edit: it seems either that is true, or the notion of corner breaks down in the limit (and so the corner points are dense in the line by virtue of every line point being a corner point).
Aren't the corner points just the dyadic rationals of the form [math]\frac{a}{2^n}[/math]? In which case they're dense.
Ah yes! You can see them as the end points of the interval expressions here, should've seen that sooner.
The corner points correspond indeed to to rational dyadics, as seen on rulers. The denominators being powers of two. The paradox here is that the stepped diagonal has length two, while appearing to have a length of the square-root of two.
It's indeed somehow similar to the use of Feynman diagrams in quantum field theory, each higher order diagram giving increasingly smaller and increasingly more contributions to an interaction process. The first stair represents the first order process. One right angle on the line. (inside the square). The second order process, introducing two extra vertices (which in a Feynman diagram can be put in in a variety of ways though), is corresponding to the second stair. Again two vertices are added for a third order diagram, corresponding to virtual particles. These four extra vertices can already be added in a lot of ways, each diagram contributing to the scattering process.On the staircase though, four new angles are added. Not two. The two more vertices are added again, giving rise to a new spectrum of diagrams. Ad infinitum! All contributing with a fastly decreasing weight. Normally second order contributions will do. I have done these calculations, but it's boring! The similarity with this staircase is enlightening! In any book on diagrammatics, this enormity can be seen.See, for a small example, here:
https://i.pinimg.com/736x/27/ad/3b/27ad3b1f4776c5d591a4b84e889433aa--feynman-diagram-science-art.jpg
To make a line infinite without adding a second dense dimension is impossible. A Peano curve (Giuseppe Peano already made this in the end of the 19th century) is a one dimensional infinite line that's fit in a square (like a 2-meter DNA string is fit into a small nuclei, though the two ends of DNA don't identify with infinity). This curve is a predecessor of fractals, which would occupy a fraction of the square.
Except the sum of the [s]chord[/s] side lengths on the polygons you've shown do approach the circumference of the circle. If I remember correctly, this is how pi was first estimated.
It's not a paradox. The diagonal line represents the distance I travel going from the bottom to the top - ?2. It doesn't matter whether I use stairs or a smooth diagonal, that's the distance I'm travelling.
Isn't the distance traveled on the stairs always 2? You not go in a straight line on the stairs. Only on a flat slope, if not slippery. The paradox is that the length of the stairs seems sqrt2 but is 2.
Isn't that just calculus?
How is an infinite line between 0 and 1 constructed?
If I walk a mile on a flat surface, how far did I walk? A mile. I don't count the distance my feet moved when they went up and down as I walked. Why would I do that on the stairs? My feet don't follow the discontinuous path of the stair.
I feel myself always hopping on a stair. My center of mass seems not to go in a straight line. Maybe you walk the stairs while your CoM floats linea recta.
A Short Note: Extending the Diagonal Paradox
Quoting T Clark :sad:
Quoting T Clark
The path is continuous. Sounds like you just float up. :gasp:
When you're walking up the stairs, your feet go up and down, just like they do when you're walking on a flat surface. I guess you're center of mass doesn't go in a straight line in either case. By your logic, when I walk a mile on a flat surface, I should say I walked more than a mile.
On a straight surface, your CoM can stay on one height while walking, as on a diagonal upward (wrt to the diagonal). On a stair you push yourself up each step.
So you go from 0 to 1 on a sine with zero wavelength? Does the line become a rectangle? With area 1xamplitude of sine? Great article! Is an infinite complex path projected on 0-1?
Only if it pleases you to do so.
I can imagine though that I only push backwardly on the stairs, but I have to push myself upwardly too. I think you can direct your CoM linea recta only on flat surfaces. Nice problem! Let's solve... I think you have to use your body and muscles to let your body move straight up, but usually taking the stairs is taking the long way home.
The staircase method of approximation is used a lot in mathematics. But mathematicians don't look at the length of the path: they look at the area occupied by the shape. Luckily, the area is much more straightforward—nothing paradoxical there!
The area below the triangles (between the triangles and the diagonal) goes to zero, while the length of all triangle sides stays the same, i.e, two. The paradox lies in the fact that both shapes look the same but aren't. The diagonal and fine stairs both have the same area beneath them (1/2) but they are different in nature.
Consider for example, that it is impossible to visualise or perceive an extensionally infinite staircase, or a perfectly straight path, or vanishingly small point, or a precise angle. The instability, ambiguity and uncertainty that characterises mental imagery and perception complements the realities of mathematical undecidability and finitistic reasoning that intuitionistic geometry recognises and which classical geometry ignores, while Brouwer's theory of choice sequences parallels how one visualises or recognises "infinity" (i.e. as a finite random truncation of a vaguely sized process).
Whatever this means, the paradox is not that big a deal. Aristotle may have been aware of the fact that arc length is not preserved in this kind of process.
In my opinion, the philosophical paradox is only solvable having gained an intuitionistic understanding of the continuum and of point-free topology, due to the fact that intuitionism is better fitted to the phenomenology of mathematical judgement.
You can solve this paradox intuitively. Infinitesimals are not points. You can envision the straight diagonal as a countable and sommable sequence of infinitesimals dl (the length being the integral, sqrt2). At the same time, the infinitesimal stair is a row of infinitesimals in the horizontal direction, dx (integral 1, total horizontal length), and a row of infinitesimals in the vertical direction, dy (total length 1). You can even view the first staircase a differential: one dx in the x direction and one dy vertically. There are no points involved here, as the continuum can't be constructed with points. It can with differentials. Even when they are infinitely small. A dx is larger than x. You can even attach a measure on it. The continuum is not a collection of points.
Quoting sime
I can visualize both the infinite line as well as the point, perfectly straight line or infinitely small or long staircase.
The paradox lies in the fact that the stair length seems to be that of a straight line. Which it isn't. Someone here even thinks they float straight up a stair...
hehe You're welcome. But stupid or not, the paradox is due to intuitions that aren't compatible with the definition of the classical Euclidean topology. Rather than insist that our intuitions are wrong and that the mathematics is right, we can instead insist that our intuitions are right by switching to an arguably more realistic axiomatization of geometry in which the paradox is dissolved or doesn't arise in the first place, such as computational geometry or intuitionism.
Can you honestly intuit an extensionally infinite staircase that is arbitrarily close to a diagonal line yet remains different in length? The concept of differentiation is similarly philosophically problematic, due to the ghost of departed quantities.
Ha! Float straight up a stair. That guy must be really stup...Hey... wait a minute. What if it's an escalator. Is there still a paradox?
Is that all it takes to get rid of this dumb-ass paradox - a motor and some gears?
The ghost of departed quantities. Sounds great. And this sounds great too:
"In my opinion, the philosophical paradox is only solvable having gained an intuitionistic understanding of the continuum and of point-free topology, due to the fact that intuitionism is better fitted to the phenomenology of mathematical judgement"
But it's not true. I don't know what you prefer: "pretty stupid", which it is not, but the member claiming this is just sticking his too big nose in this debate he clearly doesn't understand (that's why he intervenes like an angry child, calling it stupid), or that it's not true...
Differentials are funny things. They are not points, but infinitely small pieces of a continuum. The small stairs has the same length as the big one. The smooth diagonal has a different structure as the infinitely small stair. You could put the differentials in a variety of ways together around the diagonal. Mutually orthogonal, like a stairs, or in a general zig-zag pattern, which will lead to a total length bigger than sqrt2. Maybe even an infinite length. Can one project all parallel differentials placed together to form an infinite line, squeeze together on the diagonal? If you rotate all dx on the infinite line 90 degrees, can the be layed side by side on the diagonal?
There is no paradox. The stair is just no diagonal. So you can't go up the stairs in a straight line. Guess you have to buy some gear for that. A lift or a constraining device to place your body in while walking the stairs. Though I doubt that even that is possible. The stairs paradox: it's impossible not to hop or wobble when climbing the stairs...
Not if it's an escalator. Step on and move in a straight diagonal line to the top.
If the stair is an escalator it's an escalator. Or a stair in a smooth disguise. Yes of course if the stair is flat, you can go straight up. I wrote that already. The elevator doesn't walk the stairs and makes you float upwards.
Try the Taxi Cab metric.
Please die, thread. Poor thing, I should never have started you. This is wandering in the direction of threads having hundreds of posts devoted to the meanings of "2+2=4" :worry:
This, as so many others, is The Thread That Doesn't End.
[i]This is the thread that doesn't end.
It just goes on and on my friend.
Some people started writing it, not knowing what it was.
Now they'll continue writing it forever just because,
This is the thread that doesn't end[/i] [repeat]
Sounds an awful lot like life. Except for the repeat maybe.
Why? Can't we imagine small pieces of lines put together non parallel? Even orthogonal? The pieces are no points, however small you choose them. If you tried to construct the diagonal with points, laid side by side, you always need more points which can be fit in between. A line can be cut in pieces but not in points.
Differentials, i.e. infinitesimals cannot denote regions of Euclidean space, due to the fact the reals are an Archimedean field, which prohibits the definition of infinitely small intervals. Yet infinitesimals are indispensable to analysis, due to the mathematical importance of potential infinity, of which they are the reciprocal concept.
According to Cauchy
"When the successive numerical values of a variable decrease indefinitely so as to be smaller than any number, this variable becomes what is called infinitesimal , or infinitely small quantity... One says that a variable quantity becomes infinitely small when it's value decreases numerically so as to converge to the limit zero"
In other words, an infinitesimal should not be understood as being a quantity, but understood as referring to a variable that refers to a non-infinitesimal value chosen at random from a monotonically decreasing process whose limit is zero. In practice, the use of an "infinitesimal" is analogous to running an algorithm that generates it's respective process, then stopping the algorithm after a finite random amount of time and using the last value obtained as the value of the infinitesimal variable (which is necessarily a non-infinitesimal quantity)
More generally, the (?, ?)-definition of a limit of a function f(x) at some point b has a similar interpretation, namely as a process denoting a winning strategy in a sequential game played between two players. Player one first fixes a value for L, then in every round of the game player two chooses a positive value for ? and player one then chooses a value for ? in response. If ? is such that |f(x) - L| < ? whenever |x - b| < ? , then player one wins the round. If player one has a strategy for winning every round, then the limit is L. But all meaningfully defined games must eventually terminate, which in this case is when player two decides to quit, making the eventual value of |f(?) - L| a random positive quantity determined by player one's last move.
So on reflection, the philosophical paradox raised by the OP is resolved purely through careful inspection of the limit concept; for to say that a sequence of finite staircases comes "arbitrarily close" to a diagonal line, is only to assert that a staircase randomly drawn from the respective process comes boundedly close to the diagonal line, where the looseness of the bound is a monotonically decreasing function of the staircase's position in the sequence.
It's all too easy to accidentally commit the fallacy of absolute infinity.
This paradox shows that intervals dx are not the same as x. All dx tògether have length 2, because you lay them together mutually orthogonal. Points can't be laid aside mutually orthogonal. The continuum can't be constructed from points x. But it can from dx's.
Only finite intervals exist in the standard euclidean space, but this doesn't matter because infinitesimals aren't even quantities, meaning that limits and their approximations never meet in the plane, which resolves the paradox.
The paradox is essentially the paradox of the equality between x and dx. The dx contains an infinity of intermediate points x.
You might understand the paradox differently to me, but for me the paradox concerns only the concept of length, and since points are volumeless they cannot contribute to the paradox. Not to mention the fact that the OP's visual demonstration of the paradox only made reference to line lengths and a limiting argument.
From this perspective, the paradox is reproducible by using a point-free topology consisting of the lattice of right-angled triangles with rational-valued endpoints, with an analogous dissolution to what i mentioned above.
I tend to agree. The continuum can't be broken up in points, but still points are used in defining it, by pointing at the fact that you always forget to mention a point. Differentials include the missing elements as they are different from lengths, which gives the paradox 2=sqrt2, a seemingly contra-intuitive opinion, but true.