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. — Raymond

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?

By your logic, when I walk a mile on a flat surface, I should say I walked more than a mile. — T Clark

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.

What is it? Steps, itty-bitty ones Or a Straight line? Make up your mind, will ya! The ol' switcheroo!

It's a fun paradox! Although the staircase becomes very close to a straight line, we should remember that none of its outward corners ever fall on the diagonal line that goes through the staircase. Therefore, the lengths are never equal.

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.

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.

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).

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). — sime

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.

This is the stupidest discussion I have ever seen on the forum... Well, that's not true. Pretty stupid though. Here's my favorite:

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.

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. — sime

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.

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. — 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...

This is the stupidest discussion I have ever seen on the forum... Well, that's not true. Pretty stupid though. Here's my favorite:

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. — T Clark

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.

Someone here even thinks they float straight up a stair... — Raymond

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?

Is that all it takes to get rid of this dumb-ass paradox - a motor and some gears? — T Clark

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...

it's impossible not to hop or wobble when climbing the stairs... — Raymond

Not if it's an escalator. Step on and move in a straight diagonal line to the top.

Not if it's an escalator. Step on and move in a straight diagonal line to the top. — T Clark

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.

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. — sime

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:

The paradox: 2=sqrt2, while 1/2=1/2.

Please die, thread. Poor thing, I should never have started you. — jgill

This, as so many others, is The Thread That Doesn't End.

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 [repeat]

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 [repeat] — T Clark

Sounds an awful lot like life. Except for the repeat maybe.

the paradox is due to intuitions that aren't compatible with the definition of the classical Euclidean topology. — sime

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 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? — Raymond

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 (ε, δ)-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 ε and player one then chooses a value for δ in response. If δ is such that |f(x) - L| < ε whenever |x - b| < δ , 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(δ) - 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.

Scale issue! As we zoom in and zoom out, things that matter don't matter. I am the center of the universe. :lol:

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.

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. — Raymond

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.