[Bored chemist]

No you can't. Unless of course the room is infinite? But then how'd you know it's a room with walls?

Yes, I can.
I can align the table North/ South.
And there are an infinite number of angles through which I can then rotate it.
So that's an infinite number of "ways in which I can place a coffee table in a room."

Lol. Are you being serious? A room has a finite boundary condition. You might get a hell of a lot of orientations but certainly not infinite. I don't like people misleading others into such baloney!

What boundary conditions stop me choosing an arbitrary angle?
How many angles are there in the series 180, 90, 45, 22.5 ... and so on?

Which of them is forbidden?

[Eternal Student]

Hi again.

I'm aware of this thought experiment... And no. You cannot divide infinitely. There is a physical stop sign in physics .

What physical stop sign in physics?   Are you talking about an indivisible planck length?  This was mentioned in reply #53.  Otherwise lengths such as those in the tortoise and hare experiment or angles such as in Bored Chemist's example can be divided more times than any finite number of times (which is fair description of infintely many times) and it can often all be done within a finite amount of time.

Best Wishes.

[BilboGrabbins]

No you can't. Unless of course the room is infinite? But then how'd you know it's a room with walls?

Yes, I can.
I can align the table North/ South.
And there are an infinite number of angles through which I can then rotate it.
So that's an infinite number of "ways in which I can place a coffee table in a room."

Lol. Are you being serious? A room has a finite boundary condition. You might get a hell of a lot of orientations but certainly not infinite. I don't like people misleading others into such baloney!

What boundary conditions stop me choosing an arbitrary angle?
How many angles are there in the series 180, 90, 45, 22.5 ... and so on?

Which of them is forbidden?

First define how big the box is, and then you'll define the walls which would act as a boundary for all possible orientations. If we choose a simple room, which is the exact length of the table and one table stacked on the other defines the height, then how many orientations of a table can you make in this limited space?

[BilboGrabbins]

Hi again.

I'm aware of this thought experiment... And no. You cannot divide infinitely. There is a physical stop sign in physics .

What physical stop sign in physics?   Are you talking about an indivisible planck length?  This was mentioned in reply #53.  Otherwise lengths such as those in the tortoise and hare experiment or angles such as in Bored Chemist's example can be divided more times than any finite number of times (which is fair description of infintely many times) and it can often all be done within a finite amount of time.

Best Wishes.

Yes the Planck length. This is the ground rule that prevents  continuous divisibility.

[Eternal Student]

Hi.

Yes the Planck length. This is the ground rule that prevents  continuous divisibility.

   It's not a rule,  it's one possibility.   To the best of my knowledge we don't know that space is discrete rather than being continuous.

Best Wishes.

[BilboGrabbins]

Hi.

Yes the Planck length. This is the ground rule that prevents  continuous divisibility.

   It's not a rule,  it's one possibility.   To the best of my knowledge we don't know that space is discrete rather than being continuous.

Best Wishes.

It is a rule of physical length. No physicist doubts its existence. Well none I have spoken to.

[BilboGrabbins]

If you could divide infinitely, we might ask how would anything get anywhere at a fundamental length. Clearly particles do move from one place to another. We calculate such smallest possible lengths like, how long it would take light to cross the Planck length. We can calculate upper bounds of time called a Chronon. It is not equal to the Planck time however.

[BilboGrabbins]

In fact the Chronon is very similar to how long it takes light to cross a single proton 3.3×10−24s. I say similar because for an electron it corresponds to 6.27×10−24s. We measure these things by using light and from it make these possible finite lengths.

[BilboGrabbins]

The zeptosecond is still the shortest time we can measure as far as I am aware. That is the time it takes for light to cross a hydrogen atom.

[Eternal Student]

Hi again.

   It's fine for you to have your opinions.    I have my opinions and in fact I'm in the middle ground:  Space might be discrete or it might be continuous.

However, you need to be careful when you say something like this:

It is a rule of physical length. No physicist doubts its existence. Well none I have spoken to.

    It's not a universally accepted idea and you would need to cite some references.

Here's one reference that counters what you've said and there are many others.
..... space and time can be either continuous or discrete in a quantum Universe. But it means that if the Universe does have a fundamental length scale, that the CPT theorem, Lorentz invariance, and the principle of relativity must all be wrong. It could be so, but without the evidence to back it up, the idea of a fundamental length scale will remain speculative at best.....       [Taken from "This is why space needs to be continuous",   Forbes.   https://www.forbes.com/sites/startswithabang/2020/04/17/this-is-why-space-needs-to-be-continuous-not-discrete/?sh=50c668d774ea   ]

Best Wishes.

[BilboGrabbins]

Hi again.

   It's fine for you to have your opinions.    I have my opinions and in fact I'm in the middle ground:  Space might be discrete or it might be continuous.

However, you need to be careful when you say something like this:

It is a rule of physical length. No physicist doubts its existence. Well none I have spoken to.

    It's not a universally accepted idea and you would need to cite some references.

Here's one reference that counters what you've said and there are many others.
..... space and time can be either continuous or discrete in a quantum Universe. But it means that if the Universe does have a fundamental length scale, that the CPT theorem, Lorentz invariance, and the principle of relativity must all be wrong. It could be so, but without the evidence to back it up, the idea of a fundamental length scale will remain speculative at best.....       [Taken from "This is why space needs to be continuous",   Forbes.   https://www.forbes.com/sites/startswithabang/2020/04/17/this-is-why-space-needs-to-be-continuous-not-discrete/?sh=50c668d774ea   ]

Best Wishes.

I'm very up to date with these things. A more promising mathematical model has shown systems can be discrete and continuous at the same time.

[BilboGrabbins]

For instance: https://iopscience.iop.org/article/10.1088/1367-2630/12/11/115001

[hamdani yusuf (OP)]

Indeterminate: the hidden power of 0 divided by 0

You've all been indoctrinated into accepting that you cannot divide by zero. Find out about the beautiful mathematics that results when you do it anyway in calculus.  Featuring some of the most notorious "forbidden" expressions like 0/0 and 1^∞ as well as Apple's Siri and Sir Isaac Newton.

In his book “Yearning for the impossible” one my favourite authors John Stillwell says “…mathematics is a story of close encounters with the impossible and all its great discoveries are close encounters with the impossible.”  What we talk about in this video and quite a few other Mathologer videos are great examples of these sort of close encounters.

[Bored chemist]

If we choose a simple room, which is the exact length of the table and one table stacked on the other defines the height, then how many orientations of a table can you make in this limited space?

No, lets not pick a stupid example, carefully chosen so that you seem to be right. It's particularly stupid choice given that I already specified an important parameter; I said I can rotate the table in the room. So the grownups know that your scenario is irrelevant.
Let's pick a real one. The coffee table in the room I'm in is about 1.5 m long and 0.7 m wide and the room is roughly square and  about 4 metres on each side.
In reality there's an infinite number of possible angles I can align it on.

[BilboGrabbins]

If we choose a simple room, which is the exact length of the table and one table stacked on the other defines the height, then how many orientations of a table can you make in this limited space?

No, lets not pick a stupid example, carefully chosen so that you seem to be right.

I beg your pardon? Chosing simple constructions in physics is actually EXACTLY how we should approach your stupid thought experiment.

[Eternal Student]

Hi.

Indeterminate: the hidden power of 0 divided by 0 .........  (and video provided)...

I can review the video, if that helps:
    It was a 12 minute video.  You can safely watch it in 6 minutes if you choose double-speed.  Calculus was discussed and L'Hopitals rule.   Something controversial was said about 5 seconds before the end, so you may like to slow it down for that.  If you're new to calculus, this video might provide some useful graphics and explanation.  If you're interested in L'Hopitals rule the video doesn't provide enough detail to help much.

Best Wishes.

[hamdani yusuf (OP)]

Transcendental numbers powered by Cantor's infinities

In today's video the Mathologer sets out to give an introduction to the notoriously hard topic of transcendental numbers that is both in depth and accessible to anybody with a bit of common sense. Find out how Georg Cantor's infinities can be used in a very simple and off the beaten track way to pinpoint a transcendental number and to show that it is really transcendental. Also find out why there are a lot more transcendental numbers than numbers that we usually think of as numbers, and this despite the fact that it is super tough to show the transcendence of any number of interest such as pi or e. Also featuring an animated introduction to countable and uncountable infinities, Joseph Liouville's ocean of zeros constant, and much more.

[Eternal Student]

Hi again.

I've got to ask:   Do you know this Mathlogger person?  Why are you promoting these videos?   They're good - but what is this for?  It takes time to watch these and you don't seem to want to discuss anything about them.

Best Wishes.

[hamdani yusuf (OP)]

Hi again.

I've got to ask:   Do you know this Mathlogger person?  Why are you promoting these videos?   They're good - but what is this for?  It takes time to watch these and you don't seem to want to discuss anything about them.

Best Wishes.

No. I posted here because they're related to the topic in this thread. They give us insight on how professional mathematicians are thinking about the problem. Or what they think is the best way to teach the problems to non-mathematicians.
I have posted my criticism on Cantor's diagonal argument which is used in the videos. So I hope I'm not criticizing the wrong argument, which would make it a straw man fallacy.

[Eternal Student]

Hi.

First of all I've just finished watching the second video about transcendetal numbers.  That was 17 minutes long.

They give us insight on how professional mathematicians are thinking about the problem.

   Not really.   There's nothing especially new that was presented in those videos.  Most of this was known before 1900.  Textbooks and lecture courses present the material in a way that is more useful to professional mathematicians.   The videos, especially the Mathologger videos make it clear that they were trying to appeal to the general public and required only what he described as "common sense" to undertstand them.

...Or what they think is the best way to teach the problems to non-mathematicians.

   This is more likely.   However, Mathologger wasn't really even trying to teach these topics in the usual sense of teaching.  The videos were just about getting people interested and getting some views.  They were what people describe as  "edu-tainment"  and not specifically as education.  To phrase this another way, they were more for entertainment than anything else.
    On the positive side, getting people interested is an important part of teaching and these videos would seem to be doing a good job of that.

I have posted my criticism on Cantor's diagonal argument which is used in the videos.

    Perhaps you should summarise your criticism on Cantor's diagonal argument again.   I can only find  references like this in thread:

I think that diagonal argument produces more problems and inconsistencies, rather than being useful to solve other problems. So, getting rid of it could help mathematics to move forward and restore its consistency.

   So, if I'm right then your criticism of the diagonal agrument is that it produces unusual consequences.
Sadly (actually it's not necessarily "sad", it's just really ineteresting), it does  but  "getting rid of it"  isn't a realistic option.  It's a valid procedure following from the axioms of the ZFC axiom systsem and so the consequences it reveals are properties that structures we construct from the ZFC axioms, like Real numbers and subsets of these, would have.
    Do you think there is some fundamental flaw in Cantor's diagonalisation method?   This would certainly be worth identifying.
OR ELSE,   are you proposing that the ZFC axioms are not the most appropriate axiom system for the Real Numbers?  I mean, you could well be right there,   there is no universal agreement about what the fundamental axioms of Mathematics should be,   it's just that   ZF (with or without the Axiom of Choice) seems to work well.

Best Wishes.