"The second appearance of the train blocks her entirely from view and rumbles over her thoughts, as though the underlying ideas aren’t interesting enough on their own. As a mathematician, I may be biased, but I think that they are. Is the universe as infinite as we might imagine it to be?"
Writes Dan Rockmore, in "'A Trip to Infinity' and the Delicate Art of the Math Documentary/One of the most captivating concepts in mathematics is now on Netflix" (The New Yorker).
We watched that documentary last night. That is, we watched half of it. Tonight, we might watch the other half. But if we only watch half of what's left, and we keep doing that, night after night, we will never get to the end of it.
Here's the trailer — which will help you decide if you want your math enlivened/tarted up with graphics:
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God’s love for Man is infinite.
Are we talking infinity or eternity?
But if we only watch half of what's left, and we keep doing that, night after night, we will never get to the end of it.
Unless time is made up of discrete particles, in which case, like atomic decay, there will come point where it cannot be further divided. Thereafter, each night there will be a 50-50 chance that you will finish watching it.
Buzz Lightyear did it better.
Sounds like something to watch.
I also recommend Hannah frye's shows on "maths" as she calls them on prime
Documentaries these days feel like the old spaghetti test: just keep throwing strands at the wall until one sticks. There are so many images and ideas thrown at you that they are simply chaotic. It you actually want to understand something, you have to focus on it, preferably in small chunks. Documentaries like this actually increase your ignorance. I stopped watching this after 38 seconds, significantly stupider than when I started.
I really enjoyed the "Infinity Hotel" segment.
Once, years ago, I had a conversation with someone who became unreasonably angry when I said: "I don't see how there can be infinite planets. Every planet either exists or it doesn't. Put a number on each planet. It's a specific number" — and I wouldn't change my position. It's nothing to get angry about! Just say, I hear you, I see how you can't understand it, and I don't understand it either, but I believe the experts say you must be wrong.
I have a real problem with the claim "if the universe is infinite, there are infinite copies of you "out there". I understand the logic, but come on!
Should be good.
Great timing. I've been having conversations with my son who is a finitist, a mathematician who rejects the notion of infinity because it helps addressing some issues in set theory affecting the work he's doing. The subject keeps coming to my notice across a range of sources, partially because I've become interested in the Measure Problem in cosmology - how do you compute fractions of different universes of different types across the multiverse? Here's the most famous example:
"Alan Guth put it this way:[4]
'In a single universe, cows born with two heads are rarer than cows born with one head. [But in an infinitely branching multiverse] there are an infinite number of one-headed cows and an infinite number of two-headed cows. What happens to the ratio?'"
Loved the tease about the problem with multiplying by infinity in the trailer. Infinity times anything is infinity. Sort of.
BTW, there may be an infinite number of planets in the universe, but we'll never know. Because the universe is expanding, the speed of light bounds the part of it that is observable, and, being bound, there are inevitably are only a finite number of planets we could theoretically prove exist. That said, just because something is countable doesn't mean it isn't infinite.
I believe we experience infinity all around us. On a daily basis. If we only look.
Like a soccer match. Like infinity, I've never been able to make it to the end. And, it never seems to end at the end.
Or like the show "The Walking Dead". It goes on forever.
Or like the number of times Obama uses the word 'I' in a speech. It's endless.
Or the years between Detroit Lion championships. It's an incalculable number. We don't know for sure, but it could be infinite.
At what point does the time taken to fire up the app on the device, watching a swirly-gig, an ad for some irrelevant and unrelated show, outweigh any potential entertainment value of the minutes watched on the selected show?
Cantors proofs of the countably and uncountably infinite are still sound and beautiful. As G L hardy said, math is best thought of as art. But the famous article on the unreasonable effectiveness of math in science hints at meaning in math.
Countable infinity was explained by Archimedes by saying whatever number you think of, you can add one to it and find a greater number. So there is no end to numbers. Uncountable infinity mess there are the same amount of numbers between 0 and 1 as between 0 and one million. I would watch the show if I had Netflix.
She faked sincerity better than the Dems do. Take her out and add her to the list of persons to be cancelled. Thus spake the DC Mob.
Thoughts and prayer for Sienna.
Ann Althouse said...
Once, years ago, I had a conversation with someone who became unreasonably angry when I said: "I don't see how there can be infinite planets. Every planet either exists or it doesn't. Put a number on each planet. It's a specific number" — and I wouldn't change my position. It's nothing to get angry about! Just say, I hear you, I see how you can't understand it, and I don't understand it either, but I believe the experts say you must be wrong.
12/11/22, 8:02 AM
This is the science today. You cannot go against the science. Your argument in this case demonstrates the fallacy of the 'science is settled' because we do not know one war or the other how many planets there actually are.
I had a discussion on relativity, positing that 'time' is actually linear. I was told I was wrong based on various examples of time differences and demonstration of relativity. My counter was that each of our physical dimensions are linear in each direction, but if you move from one point to another in one direction, the total distance 'traveled' is different depending on your distance traveled in each of the other dimensions. Maybe a bad argument, but the ability to put forward an argument and counter with reasoning (no matter *how* bad) is (one of the) points of science.
All these shows try to be so clever but usually do a terrible job at explaining what they are talking about. "A black hole is like an endless garbage disposal...." Ugh.
I wonder if our hostess would approve of Zenos Paradox as a legal strategy
https://xkcd.com/1153/
Slightly related:
Flatland: A Romance of Many Dimensions ...
,,, is a satirical novella by the English schoolmaster Edwin Abbott Abbott, first published in 1884 by Seeley & Co. of London. Written pseudonymously by "A Square", the book used the fictional two-dimensional world of Flatland to comment on the hierarchy of Victorian culture, but the novella's more enduring contribution is its examination of dimensions.
Wikipedia
It's one thing to understand the idea of infinity, but quite another to understand how it can exist in reality.
"It's one thing to understand the idea of infinity, but quite another to understand how it can exist in reality."
How can it not?
There are days I convince myself that the universe must be closed; how can it go on forever? And then you pile the "therefore there are infinite copies of yourself argument" on top of it and that seems to seal the deal.
But then, how can the universe be so close to flat (i.e. it is not closed) without actually being flat? That coincidence is too hard to believe as well.
In order to express “infinity” we have to use the inverse concept closer to our perception of reality. In the end, inFINity, is literally not the end.
Insondable is the Spanish word for un-fathomable. Again, there’s a need to approach the mind of God indirectly, as it were.
Infinity is in the Justice Potter realm.
We can only say what infinity is not.
I'll be watching so long as neither Neil deGrasse Tyson nor Bill Nye are involved.
"But if we only watch half of what's left, and we keep doing that, night after night, we will never get to the end of it."
Eventually, you'll reach the Planck limit and there won't be a half left to watch.
One formula for Pi is = 1+1/4+1/9+1/(16)+1/(25)+...
Pi starts with 3.14 but it never ends. So, if you do the math for each and every circle, you'd never finish either. Math helps humans understand the relationships between things and how they move, but those things have no problem moving, doing, and rolling without us around.
https://mathworld.wolfram.com/PiFormulas.html
Countable infinity was explained by Archimedes by saying whatever number you think of, you can add one to it and find a greater number. So there is no end to numbers. Uncountable infinity mess there are the same amount of numbers between 0 and 1 as between 0 and one million.
Valid or not, the way I think about it is this- one can, given enough time, count all of the integers between any two given end points no matter how far apart. Whereas one can never count all of the number values between any two consecutive integers. And, yes, Cantor's proof is one of the most beautiful and understandable ones I have ever encountered.
"My counter was that each of our physical dimensions are linear in each direction,"
Pretty sure general relativity says that's not right.
There are a finite number of planets created from the big bang—if you gave each one a number, at some point, they would all be numbered. But if space is infinite, there may be an infinite number of big bangs out there, in which case the number of planets is infinite. But there is only an infinite number of planets if there are/were an infinite number of planet-creating events.
And if you want struggle with the concept of infinity, think about this classic paradox.
The mathematical object has a finite volume, but an infinite surface area.
Infinity is interesting, but I have only a finite time to ponder it, and I don't aim or claim to "understand" Infinity. I don't even see how a human CAN "understand" Infinity.
You may grasp (or think you grasp) certain human concepts about Infinity but that's different.
"It's one thing to understand the idea of infinity, but quite another to understand how it can exist in reality."
I have the same thought when it comes to Sam Britton.
"But if we only watch half of what's left, and we keep doing that, night after night, we will never get to the end of it."
This reminds me of an off-color joke about the never-ending war of the engineers versus the mathematicians. Wanna read it?
....
Oh, yeah? Try 'n stop me!
Two sophomores were conducted into a testing room, Jones was a math major, and Smith a budding engineer. The proctor seated them in two chairs arranged along the north wall. Then a nude and nubile maiden entered and took a seat next to the south wall.
"Here's the game, gentlemen," said the proctor. "When I clap my hands you may move your chair halfway between your current position and the girl. Then I will clap a second time and you may move halfway between your new position and the girl, and so on. Understand?"
Both responded in the affirmative.
"Good. You have one hour."
**CLAP!**
Smith lifted his chair and moved halfway across the room. Jones, however, moved not an inch.
"See here, Mr. Jones," said the proctor, "did you not understand my instructions, or are you just not interested?"
"I'm quite interested, but your conditions are an analog of Xeno's Paradox -- the impossibility of completing an infinite number of tasks in a finite amount of time", said Jones.
"Quite right, Mr. Jones. Care to explain yourself, Mr. Smith?"
"Philosophy ain't in my syllabus, sir," replied Smith, "but I figured I'd get close enough sooner or later."
"There are a finite number of planets created from the big bang—"
Don't think you can say that. If "our" big bang produced an open universe, space goes on forever and there are likely an infinite number of planets in it.
Can't be sure, but it _sounds_ like those obnoxious self-important fools who take an actual mathematical or physical concept and turn it into a buzzword. "All of our understanding has to be modified to include the quantum concept of probability..." "Einstein said everything is relative, even time and space, and therefore..."
Mathematicians have been dealing with infinity as something you can actually study and discuss coherently - rigorously - since Georg Cantor in the 19th century. You don't need buzzwords; it's a well-developed part of modern mathematics.
Enigma writes, "One formula for Pi is = 1+1/4+1/9+1/(16)+1/(25)+..."
Are you sure about that? What you have there is the sum of the reciprocals of the squares of the positive integers. That summation converges on about 1.64 which isn't even a good approximation of pi/2.
Moon Duchin
Nomen est omen, as they used to say.
There was chatter online last month about NASA spotlighting its female scientists (allegedly because they didn't have enough scientists of color to fill a television program).
Moon Duchin, who combines a Harvard Math/Women's Studies background with LGBTQIA+ activism and drawing reapportionment maps for Democrats, looks like just the sort of person the government and media want to spotlight.
"If "our" big bang produced an open universe, space goes on forever and there are likely an infinite number of planets in it."
Wouldn't that big bang still be banging?
If the universe is infinite and the universe is expanding, is it really the universe that is infinite, or is it the "non-universe"?
It's one thing to understand the idea of infinity, but quite another to understand how it can exist in reality.
Immanuel Kant explained all that.
Unfortunately, no one has been able to understand Kant.
How to make mathematics boring. As to "does infinity exist", what we count as things are mental abstractions built from of sensations. Are there such things as lines, edges, and surfaces? Infinities of various sorts live in the imagination and are quite real there, whether or not they are perceived as existing depends on the observer.
"I don't see how there can be infinite planets. Every planet either exists or it doesn't. Put a number on each planet. It's a specific number" — and I wouldn't change my position.“
That’s like saying there can’t be an infinite number of numbers because each one has a number.
PBS Nova had an episode recently discussing zero to infinity. It was very good, as Nova tends to be. Here’s a link.
https://www.pbs.org/wgbh/nova/video/zero-to-infinity/
Take the over on number of hoos and wows in the AP class. Take the under on time when all kids
in the other classes fall asleep.
An infinite ladder is not a ladder. - Wittgenstein
"Wouldn't that big bang still be banging?"
"Big bang" isn't really a precise concept. If by it we mean the period of inflation, and then the dumping of all that "energy" into particle formation, that's over.
Though it is true that planets are still forming, like the infinity + 1 concept.
"... what it would mean for a mathematical object like infinity to 'exist'."
That's a tough one. First figure out what it means for a mathematical object like seven to 'exist'.
"Don't think you can say that. If "our" big bang produced an open universe, space goes on forever and there are likely an infinite number of planets in it."
I understand the idea of the matter in the universe continually expanding and making new space as it goes and new planets coming into being, but I don't understand how at any given moment in time the planets can be infinite in number. We're talking about an immense orb. Either it's there or it isn't. If it's there, you can put an number on it. Do you contend that there are mysterious extra there-and-not-there planets — the +1 that is sometimes used to say why there must be infinite numbers? Yes, there COULD be one more, but if there IS one more, it exists and could take its own distinct number.
"That’s like saying there can’t be an infinite number of numbers because each one has a number."
No it's not, because numbers are in the mind. The planets are actual things — very large ones too. It's crazy to say that an idea is the same thing as an object. I can think of a unicorn, but there are no unicorns.
"I understand the idea of the matter in the universe continually expanding and making new space as it goes and new planets coming into being, but I don't understand how at any given moment in time the planets can be infinite in number. "
I think you're thinking space is finite in expanse. It might very well be infinite. Infinite now. Also infinite in the past.
"No it's not, because numbers are in the mind. The planets are actual things —"
There's a respectable school of thought that mathematics doesn't just describe reality, it IS reality.
Thanks. I may watch it just to test the hypothesis "Netflix Originals without "fuck" is not a null set."
Let me put it this way. Say that the mean planet density in the universe is 1 planet per cubic light-year (although any non-zero number works). Say also that "space extends infinitely far - a proposition that is consistent with all observations and that is part of the cosmological model favored by many physicists and astronomers -" (Brian Greene, The Hidden Reality, p. 6). Then there are clearly an infinite number of planets in our universe.
What it does mean for logical domains and associated constructs.
"That’s like saying there can’t be an infinite number of numbers because each one has a number."
“No it's not, because numbers are in the mind. The planets are actual things — very large ones too. It's crazy to say that an idea is the same thing as an object..”
That’s exactly my point. You think numbering planets in your mind limits their number. That’s wrong. The infinitude or not of planets is like the infinitude or not of the universe—it’s an empirical question that cannot be answered by pure logic. If it could we would know the answer.
"If by it we mean the period of inflation, and then the dumping of all that "energy" into particle formation, that's over."
How can there be an infinity of planets without an infinity of particles? If particle formation is "over", then there must be a finite number of particles.
"There's a respectable school of thought that mathematics doesn't just describe reality, it IS reality."
Your description sounds too much like the Scholastic notion that if you can imagine it, it must exist.
In mathematics, like any language, it is possible to create valid statements (sc. equations) that are nevertheless total crock as far as testable reality is concerned. The history of physics is littered with beautiful maths that did not pan out experimentally.
"How can there be an infinity of planets without an infinity of particles? If particle formation is "over", then there must be a finite number of particles."
If space is infinite, there are an infinite number of particles. They were all created at the big bang (saying "particles" is glossing over that particles are created from energy, and vice versa, all the time, but there has been no net increase).
"Your description …"
It's not my description, and I don't necessarily ascribe to it.
We watched that documentary last night. That is, we watched half of it. Tonight, we might watch the other half. But if we only watch half of what's left, and we keep doing that, night after night, we will never get to the end of it.
Reminds me my Xeno's Restaurant joke, in which you eat half your dinner tonight, then half of what's left tomorrow night, and so on. (There are long-time patrons that are looking at their remaining meals with an electron microscope.)
We watched that documentary last night. That is, we watched half of it. Tonight, we might watch the other half. But if we only watch half of what's left, and we keep doing that, night after night, we will never get to the end of it.
Reminds me my Zeno's Restaurant joke, in which you eat half your dinner tonight, then half of what's left tomorrow night, and so on. (There are long-time patrons that are looking at their remaining meals with an electron microscope.)
If space is infinite, there are an infinite number of particles. They were all created at the big bang (saying "particles" is glossing over that particles are created from energy, and vice versa, all the time, but there has been no net increase).
That's verging on nonsense.
Explanations that posit fewer entities, or fewer kinds of entities, are to be preferred to explanations that posit more; does that ring a bell? You're positing infinite entities to support a speculation without evidence.
This is my problem with so much of math theory and science theory.
OK, so lets say you're right. What changes? And The answer is usually - nothing.
For example, if the theory of relativity is wrong, what happens?
A couple of people beat me to Zeno's paradox. There's also the alternative Achilles and the tortoise version (discussed in Godel, Escher, and Bach).
>>"Philosophy ain't in my syllabus, sir," replied Smith, "but I figured I'd get close enough sooner or later."
The version of the joke that I remember from high school or so had to do with getting close enough "for all practical purposes."
>>Cantors proofs of the countably and uncountably infinite are still sound and beautiful.
>>If it's there, you can put a[] number on it.
Doesn't address Althouse's point, but that's the heart of Cantor's proof: you *can't* put a number on the (irrational) numbers between zero and one. So, aleph nul (the integers continuing on to infinity) isn't the end of infinity, so to speak. AFAIK, it still hasn't been determined whether there's anything between aleph nul and the set of numbers between zero and one. Though you *can* put a number on all of the rational numbers between zero and one.
--gpm
"That's verging on nonsense."
No, it's actually vanilla cosmology.
I'm not sure what you are objecting to. Is it the possibility that space may be of infinite extent? Is so, please see Brian Greene's quote I provided at 3:24pm. You can google Brian Greene if necessary.
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