Your calculus textbook lied to you.
December 31, 2024 6:37 AM Subscribe
Torricelli’s trumpet is not counterintuitive – There is nothing counterintuitive about an infinite shape with finite volume, contrary to the common propaganda version of the calculus trope known as Torricelli’s trumpet. Nor was this result seen as counterintuitive at the time of its discovery in the 17th century, contrary to many commonplace historical narratives...
Actually I tried to look up the origin of this silly name but I couldn’t find it. I guess it was perhaps coined for the American market?
[mathnasium:] “It was invented by Evangelista Torricelli [wiki], a student of Galileo, who is also known for inventing the barometer!”
posted by HearHere at 6:48 AM on December 31 [1 favorite]
[mathnasium:] “It was invented by Evangelista Torricelli [wiki], a student of Galileo, who is also known for inventing the barometer!”
posted by HearHere at 6:48 AM on December 31 [1 favorite]
It's people like you who laughed at Edison for inventing the toothbrush.
posted by flabdablet at 6:56 AM on December 31 [3 favorites]
posted by flabdablet at 6:56 AM on December 31 [3 favorites]
"Lemma: Torricelli's trumpet is not counterintuitive.
Proof: First, let us define "intuitive" to mean "that which makes sense to me, a professor of mathematics with the benefit of centuries of accumulated mathematical wisdom since Torricelli came up with the idea in the first place and very strong ideas about exegetical readings of 17th century Latin texts."
Torricelli's trumpet makes sense to me, therefore Torricelli's trumpet is intuitive. QED."
People like this make terrible professors because they have lost the ability to put themselves in the place of students, particularly the students who might need the most help.
posted by jedicus at 7:00 AM on December 31 [23 favorites]
Proof: First, let us define "intuitive" to mean "that which makes sense to me, a professor of mathematics with the benefit of centuries of accumulated mathematical wisdom since Torricelli came up with the idea in the first place and very strong ideas about exegetical readings of 17th century Latin texts."
Torricelli's trumpet makes sense to me, therefore Torricelli's trumpet is intuitive. QED."
People like this make terrible professors because they have lost the ability to put themselves in the place of students, particularly the students who might need the most help.
posted by jedicus at 7:00 AM on December 31 [23 favorites]
I wanted to read his analysis of the historical evidence, but I couldn't get past the smugly insulting writing style ("idiotic," "nonsense," "comforting myth," etc.). I'm happy to be told where my intution goes wrong, but I get annoyed when someone tells me how unbelievably stupid I must be to have that intution in the first place.
posted by grimmelm at 7:01 AM on December 31 [13 favorites]
posted by grimmelm at 7:01 AM on December 31 [13 favorites]
Counterintuitive. He keeps using that word. I do not think it means what he thinks it does.
Something I find not at all counterintuitive is a mathematician revealing a touch of Engineer's Disease when opining on matters relating to language.
posted by flabdablet at 7:18 AM on December 31 [13 favorites]
Something I find not at all counterintuitive is a mathematician revealing a touch of Engineer's Disease when opining on matters relating to language.
This is mathematical paint. You can spread it as thin as you like.Likewise rigor.
posted by flabdablet at 7:18 AM on December 31 [13 favorites]
I get annoyed when someone tells me how unbelievably stupid I must be to have that intution in the first place.
Most people haven’t read Andrew Wiles’ full proof of Fermat’s Theorem, but he does use the word “tosser” at one point.
posted by Lemkin at 7:23 AM on December 31 [4 favorites]
Most people haven’t read Andrew Wiles’ full proof of Fermat’s Theorem, but he does use the word “tosser” at one point.
posted by Lemkin at 7:23 AM on December 31 [4 favorites]
(Disclaimer: not a mathematician)
I mean, it's all math on paper and 'intuitive' doesn't account for the fact that ∞ and π are 'fudging the numbers', they represent things that cannot be represented exactly by numbers.
The trumpet isn't an infinite surface with a volume of π -- π is irrational, so the better way to describe it is that as the value of the trumpet's surface area gets closer to infinity, the value of the volume gets closer to π, but it can't ever exactly equal π because π has all those pesky unending decimal places.
If you decide that π is a fixed value, which makes a lot of math easier and actually makes a lot of math more comprehensible, your equations work, but the thing with both π and ∞ is that the closer you look, the more granular you try to make the math, those numbers always have more. π is not finite either.
So their premise assumes that ∞ is infinite and π is finite is flawed based on what those symbols mean.
posted by AzraelBrown at 7:42 AM on December 31
I mean, it's all math on paper and 'intuitive' doesn't account for the fact that ∞ and π are 'fudging the numbers', they represent things that cannot be represented exactly by numbers.
The trumpet isn't an infinite surface with a volume of π -- π is irrational, so the better way to describe it is that as the value of the trumpet's surface area gets closer to infinity, the value of the volume gets closer to π, but it can't ever exactly equal π because π has all those pesky unending decimal places.
If you decide that π is a fixed value, which makes a lot of math easier and actually makes a lot of math more comprehensible, your equations work, but the thing with both π and ∞ is that the closer you look, the more granular you try to make the math, those numbers always have more. π is not finite either.
So their premise assumes that ∞ is infinite and π is finite is flawed based on what those symbols mean.
posted by AzraelBrown at 7:42 AM on December 31
Indeed, the crucial final sentence of Hobbes’s passage is difficult to parse altogether. It appears to be nonsensical on the face of it. The thing that is evident by “natural light” or intuition according to Hobbes is that: “there cannot be a solid so subtle which does not infinitely exceed every finite solid.”The points are frozen. The beast is dead. What is the difference? What, indeed, is the point?
This sounds crazy. It appears to say that any “subtle” (that is to say, small) solid must be infinite, which is obviously nonsense. I checked Hobbes’s original Latin but unfortunately that’s just as ambiguous and it doesn’t clarify anything.
posted by flabdablet at 7:42 AM on December 31
@HearHere, the "silly name" he said he couldn't find the origin of is the alternate name, Gabriel's Horn.
posted by mark k at 7:47 AM on December 31 [4 favorites]
posted by mark k at 7:47 AM on December 31 [4 favorites]
π is not finite either.
Sure it is. Your feeble numerical notation system's inability to express it precisely by some other name is not pi's problem.
posted by flabdablet at 7:48 AM on December 31 [17 favorites]
Sure it is. Your feeble numerical notation system's inability to express it precisely by some other name is not pi's problem.
posted by flabdablet at 7:48 AM on December 31 [17 favorites]
π is not finite either.
I do not understand this pronouncement. Are you saying that because it has infinite digits, there is no point on the real number line where π exists? All real numbers have infinite digits, it's just the rational ones we like to work with end with a bunch of zeroes.
FTA, I was pretty unimpressed with the whole "mathematical paint" thing. The point of infinite area and finite volume is that if the volume is finite it seems that you could fill it with REAL paint*, but that real paint would never cover the area, and that very much is non-intuitive, because how could you have it fill the shape with real paint without also covering the inside surface of it?
I did appreciate the quote on the image though, "the infinite diminution of one dimension compensated the infinite increase of the other".
*You can't though, because real paint is made of fundamental particles that take up space and eventually the trumpet will be too narrow for those fundamental particles to get any farther. But now we're back to why trying to rely on our physical experience of the world doesn't really work for infinities - ie THEY ARE NON-INTUITIVE.
posted by solotoro at 8:02 AM on December 31 [4 favorites]
I do not understand this pronouncement. Are you saying that because it has infinite digits, there is no point on the real number line where π exists? All real numbers have infinite digits, it's just the rational ones we like to work with end with a bunch of zeroes.
FTA, I was pretty unimpressed with the whole "mathematical paint" thing. The point of infinite area and finite volume is that if the volume is finite it seems that you could fill it with REAL paint*, but that real paint would never cover the area, and that very much is non-intuitive, because how could you have it fill the shape with real paint without also covering the inside surface of it?
I did appreciate the quote on the image though, "the infinite diminution of one dimension compensated the infinite increase of the other".
*You can't though, because real paint is made of fundamental particles that take up space and eventually the trumpet will be too narrow for those fundamental particles to get any farther. But now we're back to why trying to rely on our physical experience of the world doesn't really work for infinities - ie THEY ARE NON-INTUITIVE.
posted by solotoro at 8:02 AM on December 31 [4 favorites]
I learned about asymptotes in 10th grade, this is just a 3-dimensional asymptote. I find that intuitive but not for the reasons in this article.
posted by Jon_Evil at 8:10 AM on December 31 [2 favorites]
posted by Jon_Evil at 8:10 AM on December 31 [2 favorites]
how could you have it fill the shape with real paint without also covering the inside surface of it?
Just by having the last little lick of paint take an awfully long time to dribble all the way down to the bottom of the can.
posted by flabdablet at 8:20 AM on December 31 [1 favorite]
Just by having the last little lick of paint take an awfully long time to dribble all the way down to the bottom of the can.
posted by flabdablet at 8:20 AM on December 31 [1 favorite]
A math professor was working her way through a theorem in class, chalking up some fearsome arcane details of functional analysis. As she neared the end of a lemma, she finished with "and from here, the conclusion is obvious".
A student raised his hand. "Professor, is it really obvious?". The professor paused, began to speak, but then sat down. She scribbled with her pencil for a bit, then began to twirl her hair. Suddenly she got up and left the room without it a word.
Twenty minutes later, she returned to the stunned class: "Yes, it is obvious".
posted by SaltySalticid at 8:31 AM on December 31 [23 favorites]
A student raised his hand. "Professor, is it really obvious?". The professor paused, began to speak, but then sat down. She scribbled with her pencil for a bit, then began to twirl her hair. Suddenly she got up and left the room without it a word.
Twenty minutes later, she returned to the stunned class: "Yes, it is obvious".
posted by SaltySalticid at 8:31 AM on December 31 [23 favorites]
Which is to say, after spending most of my life learning and teaching math, I'm not terribly interested in some "hahah that's not counterintuitive" take. Intuition in math is ultimately a cultural practice, cultivated, shared, learned and yes sometimes upset. To treat it like an objective aspect of mathematics that can be "wrong" is to do it a disservice.
posted by SaltySalticid at 8:34 AM on December 31 [9 favorites]
posted by SaltySalticid at 8:34 AM on December 31 [9 favorites]
"Inuitive", like "common sense" or "prima faci" is a stand in for " i will not and perhaps can not explain how I think and will none-the-less assert that I am right and that anyone who doesn't agree is wrong". Its neurotypical supremacy at its most pure.
As for paint, the mistake is that math is not real. Math is a fictional system of rules and consequences that often can make models that are useful at describing, wxplaining and predicting the behavior of physical, empiraclly observable systems.
Anyone who has filled an ice-cream cone with malt-bals can discover a volume filler that none-the-less can't cover the full cone and leaves voids at rhe tail..... its intuitive among us Malt-ball cone-eaters. Wait, where is everyone going?
Anyway. quantum mechanics assumes we don't have smooth continous spacial and energy dimensions, so no standard points and lines geometry object is rigorously faithful to our current quantized model.
but its a tempest in a teapot, all are just narratives with arbitrary rules used circumstantially for where/when they are useful. Reality is beyond their grasp.
posted by No Climate - No Food, No Food - No Future. at 8:35 AM on December 31 [1 favorite]
As for paint, the mistake is that math is not real. Math is a fictional system of rules and consequences that often can make models that are useful at describing, wxplaining and predicting the behavior of physical, empiraclly observable systems.
Anyone who has filled an ice-cream cone with malt-bals can discover a volume filler that none-the-less can't cover the full cone and leaves voids at rhe tail..... its intuitive among us Malt-ball cone-eaters. Wait, where is everyone going?
Anyway. quantum mechanics assumes we don't have smooth continous spacial and energy dimensions, so no standard points and lines geometry object is rigorously faithful to our current quantized model.
but its a tempest in a teapot, all are just narratives with arbitrary rules used circumstantially for where/when they are useful. Reality is beyond their grasp.
posted by No Climate - No Food, No Food - No Future. at 8:35 AM on December 31 [1 favorite]
math is not real
It’s still real to me, dammit!
[sobs]
posted by Lemkin at 8:45 AM on December 31 [1 favorite]
It’s still real to me, dammit!
[sobs]
posted by Lemkin at 8:45 AM on December 31 [1 favorite]
I understand limits and, therefore, intuit that the trumpet can have a finite volume. I'm not a mathematician, so perhaps i'm only intuiting out of ignorance?
posted by OHenryPacey at 8:46 AM on December 31
posted by OHenryPacey at 8:46 AM on December 31
Though I also disagree about the intuitiveness, I did enjoy reading this just for the sake of having a foul tempered screed to read this morning.
I also very much enjoy the term "calculus trope"
posted by Dr. Twist at 8:47 AM on December 31
I also very much enjoy the term "calculus trope"
posted by Dr. Twist at 8:47 AM on December 31
Are we going to next claim that nothing can ever equal exactly "one seventh" because it also has no finite decimal expansion?
If you measure the circumference of an actual circular object with diameter 1, it may not be correct to say that it's *exactly* equal to pi, but I'm not sure it's any more correct to say that it's *exactly* equal to any other particular number, because of the limits of your ability to measure the object precisely. Or to find a truly circular object with diameter exactly 1, for that matter.
Numbers of all types, circles, and the notion of "exact equality" are all abstractions. Very useful ones, though!
I'll admit that I always have trouble with "paradoxes" like Toricelli's trumpet, as it feels like a lose-lose whenever I'm presented with one: if the correct answer is the obvious one to me, then apparently I'm a weirdo without a normal person's intuition; if the the "intuitive" answer is the obvious one to me, then apparently I don't understand the subject well enough yet.
I'm sympathetic to the argument that the way we teach calculus, with the emphasis on limits and epsilon-delta proofs, is overly fussy and poorly motivated, but I haven't actually seen alternative introductions, I'd be curious for any recommendations....
posted by bfields at 8:47 AM on December 31 [1 favorite]
If you measure the circumference of an actual circular object with diameter 1, it may not be correct to say that it's *exactly* equal to pi, but I'm not sure it's any more correct to say that it's *exactly* equal to any other particular number, because of the limits of your ability to measure the object precisely. Or to find a truly circular object with diameter exactly 1, for that matter.
Numbers of all types, circles, and the notion of "exact equality" are all abstractions. Very useful ones, though!
I'll admit that I always have trouble with "paradoxes" like Toricelli's trumpet, as it feels like a lose-lose whenever I'm presented with one: if the correct answer is the obvious one to me, then apparently I'm a weirdo without a normal person's intuition; if the the "intuitive" answer is the obvious one to me, then apparently I don't understand the subject well enough yet.
I'm sympathetic to the argument that the way we teach calculus, with the emphasis on limits and epsilon-delta proofs, is overly fussy and poorly motivated, but I haven't actually seen alternative introductions, I'd be curious for any recommendations....
posted by bfields at 8:47 AM on December 31 [1 favorite]
I don’t quite understand why he’s so angry about it. He sounds as if somebody sent their trumpet back for a refund because they couldn’t work out how to play it.
posted by Phanx at 8:51 AM on December 31 [5 favorites]
posted by Phanx at 8:51 AM on December 31 [5 favorites]
This guy ground a menger cube up, rectally autoadministered it and got high on his own, insufferable, supply.
posted by lalochezia at 9:11 AM on December 31 [4 favorites]
posted by lalochezia at 9:11 AM on December 31 [4 favorites]
So, I am a mathematician, and it is indeed counterintuitive.
solotero is quite right, and their comment illustrates the paradox in the statement that you can fill Torricelli's Trumpet with paint (finite volume) but you can't paint it (infinite surface area and infinite length) because c'mon, it's paint and so if you fill a vessel with paint, then you've coated the inside with that paint and so ...
Viktor's blog screed about "mathematical paint" is nonsensical, because when he's talking about pouring a can of "mathematical" paint on a floor and spreading it out with a spatula, you immediately think about an actual can of paint, and if you've ever done any painting you know that no matter how long you try to stretch a can of paint, at some point you need to drive to the store and buy another can. Infinite surface area, my ass.
I actually teach about Torricelli's Trumpet (I call it Gabriel's Horn) in my calculus class, and we enjoy it because it's just fun to think about the inherent paradox. But then I remind my students that we've already seen infinitely long things add up to finite numbers: consider, for example,
0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 + 0.000003 + ...
which adds up to 0.333333.... which is just 1/3. An infinitely long list of positive numbers, with a finite sum.
Then, someone argues that the individual numbers (0.0003, etc) in the list are getting smaller and so that's why everything adds up to a finite number. And then I remind them about this sum,
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ...,
which adds up to infinity (or better put, the sum just keeps growing and growing) even though the individual numbers in the list are getting smaller.
We all agree that the concept of infinity is just weird and paradoxical, and then we move on with our lives.
posted by math at 9:14 AM on December 31 [15 favorites]
solotero is quite right, and their comment illustrates the paradox in the statement that you can fill Torricelli's Trumpet with paint (finite volume) but you can't paint it (infinite surface area and infinite length) because c'mon, it's paint and so if you fill a vessel with paint, then you've coated the inside with that paint and so ...
Viktor's blog screed about "mathematical paint" is nonsensical, because when he's talking about pouring a can of "mathematical" paint on a floor and spreading it out with a spatula, you immediately think about an actual can of paint, and if you've ever done any painting you know that no matter how long you try to stretch a can of paint, at some point you need to drive to the store and buy another can. Infinite surface area, my ass.
I actually teach about Torricelli's Trumpet (I call it Gabriel's Horn) in my calculus class, and we enjoy it because it's just fun to think about the inherent paradox. But then I remind my students that we've already seen infinitely long things add up to finite numbers: consider, for example,
0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 + 0.000003 + ...
which adds up to 0.333333.... which is just 1/3. An infinitely long list of positive numbers, with a finite sum.
Then, someone argues that the individual numbers (0.0003, etc) in the list are getting smaller and so that's why everything adds up to a finite number. And then I remind them about this sum,
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ...,
which adds up to infinity (or better put, the sum just keeps growing and growing) even though the individual numbers in the list are getting smaller.
We all agree that the concept of infinity is just weird and paradoxical, and then we move on with our lives.
posted by math at 9:14 AM on December 31 [15 favorites]
Yeah wow, at a glance this guy's "manifesto" on math ed would make the blood boil of my (actually widely accomplished and esteemed) math ed friends. Despite (because of?) my phd in math, sometimes I think the main problem with math is the mathematicians.
posted by SaltySalticid at 9:20 AM on December 31 [3 favorites]
posted by SaltySalticid at 9:20 AM on December 31 [3 favorites]
Infinite surface area, my ass
Paging Sir Math-a-Lot.
posted by Lemkin at 9:21 AM on December 31 [8 favorites]
Paging Sir Math-a-Lot.
posted by Lemkin at 9:21 AM on December 31 [8 favorites]
I read the linked blog post and I will try to summarize it. As far as I can tell, the author, Viktor Blåsjö, is not interested in your intuition or mine, but in the intuition of seventeeth-century mathematicians.
There is a story that is told by some historians of mathematics, that Torricelli's trumpet, a solid with infinite length and finite volume, published in 1641, was considered paradoxical or counter-intuitive by Torricelli's contemporaries. Blåsjö says that this story is wrong and these historians have propagated folklore, cherry-picked their quotations, or simply failed to understand what the mathematicians they quote were actually saying. He critizes in detail Paolo Mancosu's Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (1999), suggesting that Mancosu has misunderstood or distorted some of his sources in order to make them fit the story. A substantial part of the discussion is about a single sentence from Thomas Hobbes' Principia et problemata aliquot geometrica (1674), "Solidum enim tam exile nullum esse potest, quod infinite non excedat omne solidum finitum: ut manifestum est lumine naturali." Mancosu interprets this sentence as a complaint that the finite volume of Torricelli's solid is contrary to "lumine naturali" (that is, intuition; literally "natural light"); if correct this is pretty good evidence for the "counter-intuitive" story. Blåsjö interprets this sentence as a complaint that the use of infinitesimally thin slices in the computation of the solid's volume is contrary to intuition. (In 1674, integral calculus had not yet been put on a solid footing—Leibniz started to publish his work on calculus in 1684, and Newton's Principia Mathematica appeared in 1728—so Hobbes' caution about the technique, if that's really what he was saying, was not unreasonable.)
Blåsjö's aggressive tone is a bit much, but it's directed at historians of mathematics like Mancosu and not at you or me.
posted by cyanistes at 9:40 AM on December 31 [7 favorites]
There is a story that is told by some historians of mathematics, that Torricelli's trumpet, a solid with infinite length and finite volume, published in 1641, was considered paradoxical or counter-intuitive by Torricelli's contemporaries. Blåsjö says that this story is wrong and these historians have propagated folklore, cherry-picked their quotations, or simply failed to understand what the mathematicians they quote were actually saying. He critizes in detail Paolo Mancosu's Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (1999), suggesting that Mancosu has misunderstood or distorted some of his sources in order to make them fit the story. A substantial part of the discussion is about a single sentence from Thomas Hobbes' Principia et problemata aliquot geometrica (1674), "Solidum enim tam exile nullum esse potest, quod infinite non excedat omne solidum finitum: ut manifestum est lumine naturali." Mancosu interprets this sentence as a complaint that the finite volume of Torricelli's solid is contrary to "lumine naturali" (that is, intuition; literally "natural light"); if correct this is pretty good evidence for the "counter-intuitive" story. Blåsjö interprets this sentence as a complaint that the use of infinitesimally thin slices in the computation of the solid's volume is contrary to intuition. (In 1674, integral calculus had not yet been put on a solid footing—Leibniz started to publish his work on calculus in 1684, and Newton's Principia Mathematica appeared in 1728—so Hobbes' caution about the technique, if that's really what he was saying, was not unreasonable.)
Blåsjö's aggressive tone is a bit much, but it's directed at historians of mathematics like Mancosu and not at you or me.
posted by cyanistes at 9:40 AM on December 31 [7 favorites]
We all agree that the concept of infinity is just weird and paradoxical, and then we move on with our lives.I'm sure that you don't literally say to your students "that's weird and paradoxical, let's move on with our lives," and are just summarizing pithily for this crowd, so I apologize for reading too literally—but I think this somewhat misstates what makes infinity interesting.
For example, "this statement is a lie" is paradoxical. It's kind of fun to think about, but you probably won't get any interesting math out of it. (Well, after at least 15 years as a professional mathematician, one of the few things I've learned for sure is that any statement of the form "there's nothing interesting to learn from this" is wrong. Probably the various exotic paraconsistent logics have, somewhere in their ancestry, attempts to understand this statement.)
The thing about infinity is that it is weird and counterintuitive, but (probably) not paradoxical. When a mathematically precise statement about infinity seems wrong, it's because it violates your intuition, (probably) not because math is broken. The wonder is that we have this science of mathematics that can so ably manipulate these wildly counter-intuitive things and yet not dissolve into metaphysical paradoxes.
(I feel similarly, but must speak less certainly because that's not where my training is and it's easy for the untrained to make nonsensical statements, about quantum mechanics—but that's even more dramatic, because quantum mechanics is accountable to the real world in ways that mathematics is not. If an application of mathematics to the real world fails, then we can always blame the model, whereas quantum mechanics in at least some sense is the model.)
posted by It is regrettable that at 9:42 AM on December 31 [2 favorites]
People have very different intuitions. Intuitions are a feature of people and other living beings, about real things, but also greatly influenced by and influencing culture. Objects and ideas do not have an independent "intuitiveness" factor separate from that.
An arguably modern intuition is that it is possible to have an accurate and complete model of everything, and this intuition is still alive & kicking, despite the stunning logical and scientific discoveries of a century ago.
posted by johnabbe at 9:50 AM on December 31 [1 favorite]
An arguably modern intuition is that it is possible to have an accurate and complete model of everything, and this intuition is still alive & kicking, despite the stunning logical and scientific discoveries of a century ago.
posted by johnabbe at 9:50 AM on December 31 [1 favorite]
My intuition says everything is infinite or can measure to infinite unless someone declares it is finite, often because.
Or, as my mother used to say, I was an English major, you do the math.
posted by JohnnyGunn at 9:58 AM on December 31
Or, as my mother used to say, I was an English major, you do the math.
posted by JohnnyGunn at 9:58 AM on December 31
I was with the author despite the tone until the infinitely thin paint part. Then I realized his definition of what is intuitive differs vastly from and is clearly incompatible with mine. Thanks for the post though - interesting discussion.
posted by BlackLeotardFront at 10:05 AM on December 31
posted by BlackLeotardFront at 10:05 AM on December 31
This comes across as a more advanced version of someone smugly sharing a "proof" that 1 == 0. Clearly all math is broken forever. (Narrator: It isn't.)
posted by RonButNotStupid at 10:12 AM on December 31
posted by RonButNotStupid at 10:12 AM on December 31
There’s an animated video about Hilbert’s paradox. I was OK with the fully-booked infinite-room hotel being able to fit an infinite bus full of new customers. I could even (sort of) deal with them handling an infinite number of infinite buses. But when the next bus was uncountably infinite, I decided someone must be pulling my leg.
posted by Lemkin at 10:13 AM on December 31
posted by Lemkin at 10:13 AM on December 31
For example, "this statement is a lie" is paradoxical. It's kind of fun to think about, but you probably won't get any interesting math out of it.
Ahem.
posted by flabdablet at 10:17 AM on December 31 [2 favorites]
Ahem.
posted by flabdablet at 10:17 AM on December 31 [2 favorites]
For example, "this statement is a lie" is paradoxical. It's kind of fun to think about, but you probably won't get any interesting math out of it.
Thinking about the liar paradox led more or less directly to Godel's proofs of the incompleteness of arithmetic.
posted by Jonathan Livengood at 10:19 AM on December 31 [1 favorite]
Thinking about the liar paradox led more or less directly to Godel's proofs of the incompleteness of arithmetic.
posted by Jonathan Livengood at 10:19 AM on December 31 [1 favorite]
Beaten by flabdablet by thiiiiiis much. :)
posted by Jonathan Livengood at 10:19 AM on December 31 [2 favorites]
posted by Jonathan Livengood at 10:19 AM on December 31 [2 favorites]
I keep coming back to that paint. It's a circular argument. "Gabriel's horn is intuitive, because it's intuitively obvious if you paint it with a special kind of paint that is also already finite in volume but able to cover infinite area."
And now I think I see where the underlying disconnect for me is...
Sorry, I just can't resist:
Metafilter: All of these people are wrong, in my opinion.
posted by solotoro at 10:22 AM on December 31 [5 favorites]
And now I think I see where the underlying disconnect for me is...
Let’s listen to Torricelli’s words again: “if one proposes to consider a solid infinitely extended, everybody immediately thinks that such a figure must be of infinite size.” That is to say, in that moment, when you are asked to picture such a shape, your “immediate” impulse is to think of one that has infinite size. That’s your “immediate” reaction. Not your considered reaction, not your mathematician’s intuition, not what you think after a bit of thought.To me, this is a bespoke distinction that isn't part of the plain-language meaning of intuition. Intuition IS what Blåsjö is calling an "immediate" reaction. Intuition is, per MW, "the power or faculty of attaining to direct knowledge or cognition without evident rational thought and inference." Once you've considered, once you've had a bit of thought, you're no longer talking about intuition.
Sorry, I just can't resist:
Metafilter: All of these people are wrong, in my opinion.
posted by solotoro at 10:22 AM on December 31 [5 favorites]
In response to flabdablet's and Jonathan Livengood's comments, I will say that I am very glad to have included the next sentence: "Well, after at least 15 years as a professional mathematician, one of the few things I've learned for sure is that any statement of the form 'there's nothing interesting to learn from this' is wrong. Probably the various exotic paraconsistent logics have, somewhere in their ancestry, attempts to understand this statement."
posted by It is regrettable that at 10:24 AM on December 31 [6 favorites]
posted by It is regrettable that at 10:24 AM on December 31 [6 favorites]
I'm really bothered by the Hobbes sentence. I wish I had some real facility with Latin because at least at first appearance, it looks like there must be a typo along the lines of adding a "not" or forgetting to add a "not" ... or maybe just another phrase that was dropped somewhere. I mean, it seems like what he should be saying is that there is no finite solid so thin that it doesn't infinitely exceed something that is infinitesimally small. And that claim does seem to be intuitively obvious, for whatever little that is worth.
posted by Jonathan Livengood at 10:32 AM on December 31 [1 favorite]
posted by Jonathan Livengood at 10:32 AM on December 31 [1 favorite]
Its neurotypical supremacy at its most pure.
Neurocentric, definitely. But "neurotypical" (besides the usual imo sloppy assumptions that term relies on) is not really the safest of bets for a math professor.
posted by trig at 10:34 AM on December 31 [4 favorites]
Neurocentric, definitely. But "neurotypical" (besides the usual imo sloppy assumptions that term relies on) is not really the safest of bets for a math professor.
posted by trig at 10:34 AM on December 31 [4 favorites]
I don't have time to read the article but I immediately assumed, exactly as cyanistes summarizes, the author is discussing what people at the time thought, and other scholars are interpreting the ancient texts wrong. It's about the philosophy and history of ideas, and if read that way the tongue-in-cheek or polemicized writing style is just deadpan humor. It's not necessarily a math teaching article.
posted by polymodus at 10:40 AM on December 31 [1 favorite]
posted by polymodus at 10:40 AM on December 31 [1 favorite]
When I was in graduate school, for physics, there was this one fellow student who couldn't get over mathematical rigor. Our instructors would use some technique, which was reasonable because the math was describing reality, and the student would be more concerned with the validity of using that technique. He did get a PhD, eventually, in math. Reading this, I had to check that it wasn't my colleague.
I currently teach physics, and I find that I have to 'crash course' math to some of my students in order to use it in some physics topic the way in which i want to cover it. I don't need the rigor, just 'rates' and 'areas' and 'very small things' -- these kids will get the rigor somewhere between the next hour and the next semester. I can understand his frustration at the apparent rigidity of the calculus curriculum -- it would be nice for students to have all of calculus before or along with physics, or even to have some harmony between topics in calculus and physics, for students taking both.
He sounds like a physicist who is trying to fix math education, because 'how hard can it be?'. The problem with that is you can't just write a manifesto and continue on; you have to work with your peers and students. He wants to challenge students and give motivations for learning the content, but his textbook meets a very specific sort of student -- my students would neither see the motivation or have a foothold on solving his problems given his wall-of-text lecture notes converted to textbook. His ideas might work if students started learning this way and maintained it, but I can't believe it would reach every student.
His whole presentation reeks of Feynmann syndrome (linked for background, this later video is more relevant i think). He knows something is insufficient in math education, and only he has the answer, and its easy because he's thinking like a physicist who can do anything!
I've shown Torricelli's trumpet to students (who were complaining about their calculus homework) and mentioned the 'painting the surface' conundrum, and the point i was trying to drive was that infinity is not a number, its a fundamentally different, weirder thing; once you start using it, your physical intuition is not always helpful. Whatever intuition he is using, he doesn't put himself in his student's seat (or anyone else's).
tl;dr: like the joke goes, i can tell he's a mathematician; he may be correct, but it will be useless.
posted by Cat_Examiner at 10:40 AM on December 31 [2 favorites]
I currently teach physics, and I find that I have to 'crash course' math to some of my students in order to use it in some physics topic the way in which i want to cover it. I don't need the rigor, just 'rates' and 'areas' and 'very small things' -- these kids will get the rigor somewhere between the next hour and the next semester. I can understand his frustration at the apparent rigidity of the calculus curriculum -- it would be nice for students to have all of calculus before or along with physics, or even to have some harmony between topics in calculus and physics, for students taking both.
He sounds like a physicist who is trying to fix math education, because 'how hard can it be?'. The problem with that is you can't just write a manifesto and continue on; you have to work with your peers and students. He wants to challenge students and give motivations for learning the content, but his textbook meets a very specific sort of student -- my students would neither see the motivation or have a foothold on solving his problems given his wall-of-text lecture notes converted to textbook. His ideas might work if students started learning this way and maintained it, but I can't believe it would reach every student.
His whole presentation reeks of Feynmann syndrome (linked for background, this later video is more relevant i think). He knows something is insufficient in math education, and only he has the answer, and its easy because he's thinking like a physicist who can do anything!
I've shown Torricelli's trumpet to students (who were complaining about their calculus homework) and mentioned the 'painting the surface' conundrum, and the point i was trying to drive was that infinity is not a number, its a fundamentally different, weirder thing; once you start using it, your physical intuition is not always helpful. Whatever intuition he is using, he doesn't put himself in his student's seat (or anyone else's).
tl;dr: like the joke goes, i can tell he's a mathematician; he may be correct, but it will be useless.
posted by Cat_Examiner at 10:40 AM on December 31 [2 favorites]
and the point i was trying to drive was that infinity is not a number, its a fundamentally different, weirder thing; once you start using it, your physical intuition is not always helpful. Whatever intuition he is using, he doesn't put himself in his student's seat (or anyone else's).
Except this condescends on the student even more, because it implicitly frames a disregard for "physical intuition" without any justification. Why should a student accept throwing out physical intuition (because not always helpful? a circular or self-serving logic), as if--to analogize--a student should turn off all critical thinking and throw out, say, social justice intuition if a teacher condescends a capitalist definition on economics? And so this opens a whole can of philosophical worms. Just saying. Ultimately, formal education has its limits because of time and scope. But all teachers must practice not merely empathy but actual theory of mind skills.
posted by polymodus at 10:46 AM on December 31
Except this condescends on the student even more, because it implicitly frames a disregard for "physical intuition" without any justification. Why should a student accept throwing out physical intuition (because not always helpful? a circular or self-serving logic), as if--to analogize--a student should turn off all critical thinking and throw out, say, social justice intuition if a teacher condescends a capitalist definition on economics? And so this opens a whole can of philosophical worms. Just saying. Ultimately, formal education has its limits because of time and scope. But all teachers must practice not merely empathy but actual theory of mind skills.
posted by polymodus at 10:46 AM on December 31
> “Torricelli called his geometrical object the hyperbolicum acutum (“acute hyperbolic solid”). The names “Torricelli’s Trumpet” and “Gabriel’s Horn” seem to have come into use much later, perhaps with the modern calculus textbook. The latter name refers to the archangel Gabriel and the trumpet that will announce the Day of Judgment, figuring the physical realization of Torricelli’s geometrical object as an apocalyptic incarnation of the divine. With its mind-bending combination of infinite & finite dimensions, Torricelli’s Trumpet describes an interpenetration of the ideal and the material, and invites us to imagine a music at the intersection of the impossible and the real.”[imaginaryinstruments (content note: "counterintuitive")]
posted by HearHere at 11:00 AM on December 31
Why should a student accept throwing out physical intuition
Students should not. They should use it so long as it does not contradict reality, and they should confirm intuition with experiment. As many experiments as practicable, as different as possible. There is no disregard for physical (or any other) intuition here; instead, it is noting that intuition can lead you astray, or fail to lead you at all.
By this point in the class, my students have seen many times where their intuition matches what the math shows, and what experiment shows; they have also seen many times where it does not. And, this varies between students -- they will disagree on what should happen in some situation. They haven't yet had to deal with both infinities and infinitesimals, and I find that this is a point where intuition comes up empty more often for some students
posted by Cat_Examiner at 11:29 AM on December 31 [1 favorite]
Students should not. They should use it so long as it does not contradict reality, and they should confirm intuition with experiment. As many experiments as practicable, as different as possible. There is no disregard for physical (or any other) intuition here; instead, it is noting that intuition can lead you astray, or fail to lead you at all.
By this point in the class, my students have seen many times where their intuition matches what the math shows, and what experiment shows; they have also seen many times where it does not. And, this varies between students -- they will disagree on what should happen in some situation. They haven't yet had to deal with both infinities and infinitesimals, and I find that this is a point where intuition comes up empty more often for some students
posted by Cat_Examiner at 11:29 AM on December 31 [1 favorite]
What a.... I'm not sure the word. Jerkwad??
Ya dude. It's not counterintuitive to you because you understand it. I enjoy speaking to large crowds. For many people this is counterintuitive. It doesn't make anyone wrong. Jesus what an absolute dick. Just read this extract:
Where do you think this process will stop? This is mathematical paint. You can spread it as thin as you like. How much area can you cover, if you can spread the paint thinner and thinner and thinner?
What does your “intuition” tell you? Does your “intuition” say that this spreading process with terminate after a certain number of square meters of the floor painted? Of course not. That would be idiotic.
And yet that is precisely what the standard account of Torricelli’s trumpet would have you believe. It is supposed to be “counterintuitive”, the story goes, for a finite volume of paint to cover an infinite area. Well, we have just seen that that premise is idiotic. Obviously a finite amount of paint can spread further and further, as long as you make it thinner and thinner. There is nothing “counterintuitive” about that
So ya. It's not counterintuitive if you imagine magic paint that can infitintely spread out. Except ya. Every not you guy IS NOT coming to that conclusion. I actually think I might be smart enough to say that with literally any definition of paint it does in fact have a finite coverage area. Even if you're now saying that you're comfortable calling it paint as long as quantum engagement of the literal particles making up the atoms making up the molecules making up the paint.... Welp. At some point they won't. And thing stops doing something at some point frigging equals a finite result. Big does not equal infinite.
What an absolute dick. No wonder I failed calc 2 three times and dropped out of physics as a major. Because of professors like this who absolutely had no ability to related to someone who didn't understand.
Shit.
posted by chasles at 11:36 AM on December 31
Ya dude. It's not counterintuitive to you because you understand it. I enjoy speaking to large crowds. For many people this is counterintuitive. It doesn't make anyone wrong. Jesus what an absolute dick. Just read this extract:
Where do you think this process will stop? This is mathematical paint. You can spread it as thin as you like. How much area can you cover, if you can spread the paint thinner and thinner and thinner?
What does your “intuition” tell you? Does your “intuition” say that this spreading process with terminate after a certain number of square meters of the floor painted? Of course not. That would be idiotic.
And yet that is precisely what the standard account of Torricelli’s trumpet would have you believe. It is supposed to be “counterintuitive”, the story goes, for a finite volume of paint to cover an infinite area. Well, we have just seen that that premise is idiotic. Obviously a finite amount of paint can spread further and further, as long as you make it thinner and thinner. There is nothing “counterintuitive” about that
So ya. It's not counterintuitive if you imagine magic paint that can infitintely spread out. Except ya. Every not you guy IS NOT coming to that conclusion. I actually think I might be smart enough to say that with literally any definition of paint it does in fact have a finite coverage area. Even if you're now saying that you're comfortable calling it paint as long as quantum engagement of the literal particles making up the atoms making up the molecules making up the paint.... Welp. At some point they won't. And thing stops doing something at some point frigging equals a finite result. Big does not equal infinite.
What an absolute dick. No wonder I failed calc 2 three times and dropped out of physics as a major. Because of professors like this who absolutely had no ability to related to someone who didn't understand.
Shit.
posted by chasles at 11:36 AM on December 31
He seems nice.
posted by Captaintripps at 11:47 AM on December 31
posted by Captaintripps at 11:47 AM on December 31
As a math teacher, I sometimes need to make predictions about my students' intuitions, but I certainly don't need to tell them what they should find intuitive or counter-intuitive. It's better for all parties when they tell me.
Anyway: I skimmed the blog post (which felt about 5 times as long as it needed to be), but I was interested in the point at the end about calculus versus analysis and the rhetorical uses of highlighting paradoxical results. There is a real tension between helping students to have confidence in their abilities and showing them that math never loses its ability to confound and astonish. Case in point: I have a PhD in math, and I only realized yesterday that the negative space of an arbitrarily large cannonball pile is not only connected, but contains entire lines (you can see straight through to the other side in certain directions). And, circling back to the original point, when I said "Get this..." to the nearest non-mathematician, they reported that this fact was perfectly intuitive to them.
posted by aws17576 at 11:50 AM on December 31 [7 favorites]
Anyway: I skimmed the blog post (which felt about 5 times as long as it needed to be), but I was interested in the point at the end about calculus versus analysis and the rhetorical uses of highlighting paradoxical results. There is a real tension between helping students to have confidence in their abilities and showing them that math never loses its ability to confound and astonish. Case in point: I have a PhD in math, and I only realized yesterday that the negative space of an arbitrarily large cannonball pile is not only connected, but contains entire lines (you can see straight through to the other side in certain directions). And, circling back to the original point, when I said "Get this..." to the nearest non-mathematician, they reported that this fact was perfectly intuitive to them.
posted by aws17576 at 11:50 AM on December 31 [7 favorites]
As cyanistes points out, the question the dude is considering is what would be intuitive to Torricelli's readers in 1643, not to us. To get at this, we have to remember that those readers a) did not yet have calculus; b) did not have an atom-based view of matter; and c) inherited from the Greeks a deep suspicion of anything infinite. (Thus Hobbes's view that anything claimed to be infinite was actually "indefinite".)
Admittedly the "mathematical paint" business is probably a wrong move— that's intuitive to topologists, counter-intuitive to probably everyone else. (But we still can't assume that it was counter-intuitive in the same way to Torricelli's original readers.)
posted by zompist at 11:59 AM on December 31
Admittedly the "mathematical paint" business is probably a wrong move— that's intuitive to topologists, counter-intuitive to probably everyone else. (But we still can't assume that it was counter-intuitive in the same way to Torricelli's original readers.)
posted by zompist at 11:59 AM on December 31
A mathematician and their mathematician friends walk into a bar. The first mathematician orders a beer. The second mathematician orders half a beer. The third mathematician orders a quarter of a beer. The fourth mathematician orders an eighth of a beer. The bar tender looks out, sees an infinitely long line of mathematicians waiting to order, then pours two beers and tells them “Take these and get the fuck out of my bar.”
If you can understand this joke of how a finite amount of beer can serve an infinite number of mathematicians, then it’s not counterintuitive or a paradox.
Some people are annoyed when people claim simple things are complicated because it seems like mystification and aggrandizing and gatekeeping. Other people get annoyed when those people claim that the thing is simple because it makes them feel “stupid” or like a “failure” because they couldn’t understand this purportedly simple thing.
Text is notoriously ambiguous, but I’m giving the author here the benefit of the doubt in that they meant “This thing people say is supposedly impossible is easy to understand once you read this explanation” without an added “so you must be an idiot if you didn’t understand it before.”
posted by AlSweigart at 12:05 PM on December 31 [1 favorite]
If you can understand this joke of how a finite amount of beer can serve an infinite number of mathematicians, then it’s not counterintuitive or a paradox.
Some people are annoyed when people claim simple things are complicated because it seems like mystification and aggrandizing and gatekeeping. Other people get annoyed when those people claim that the thing is simple because it makes them feel “stupid” or like a “failure” because they couldn’t understand this purportedly simple thing.
Text is notoriously ambiguous, but I’m giving the author here the benefit of the doubt in that they meant “This thing people say is supposedly impossible is easy to understand once you read this explanation” without an added “so you must be an idiot if you didn’t understand it before.”
posted by AlSweigart at 12:05 PM on December 31 [1 favorite]
Also, and I say this with all sincerity: He’s not an asshole, he’s just Dutch.
posted by AlSweigart at 12:33 PM on December 31
posted by AlSweigart at 12:33 PM on December 31
Technically, from his staff page, he looks to be a Swede, working in the Netherlands for some time.
posted by cardboard at 12:44 PM on December 31
posted by cardboard at 12:44 PM on December 31
Calculus used to mean "counting pebbles" and the abstraction in what Maths calls Real Analysis turned the pebbles into "however small your pebble, we can find a smaller pebble that's a better approximation" -- which gives you a limited volume of paint that doesn't care about minimal thin-ness to cover area, we can always make it thinner so it will cover the area we need.
(I have, ofc, fed the troll.)
salotoro: real numbers have infinite digits, it's just the rational ones we like to work with end with a bunch of zeroes.
bfields (and maybe b-rings also): Are we going to next claim that nothing can ever equal exactly "one seventh" because it also has no finite decimal expansion?
There's uh (I guess) countably infinite rational numbers with non-terminating decimal expansion. The short sequence for 1/7 in base 10 (our conventional counting numbers) was going to be my example; 0.142857142857... which also carries the convention that those ever-smaller pebbles let 1/9=0.111.. and 9/9 = 1 = 0.9999... = 7/7 = 0.7+0.28+0.014+0.0056+0.00035+0.000049+...
This involves Gödel-type "the rules are broken" which does take rigour to think through.
posted by k3ninho at 1:23 PM on December 31
(I have, ofc, fed the troll.)
salotoro: real numbers have infinite digits, it's just the rational ones we like to work with end with a bunch of zeroes.
bfields (and maybe b-rings also): Are we going to next claim that nothing can ever equal exactly "one seventh" because it also has no finite decimal expansion?
There's uh (I guess) countably infinite rational numbers with non-terminating decimal expansion. The short sequence for 1/7 in base 10 (our conventional counting numbers) was going to be my example; 0.142857142857... which also carries the convention that those ever-smaller pebbles let 1/9=0.111.. and 9/9 = 1 = 0.9999... = 7/7 = 0.7+0.28+0.014+0.0056+0.00035+0.000049+...
This involves Gödel-type "the rules are broken" which does take rigour to think through.
posted by k3ninho at 1:23 PM on December 31
>I'm really bothered by the Hobbes sentence. I wish I had some real facility with Latin because at least at first appearance, it looks like there must be a typo along the lines of adding a "not" or forgetting to add a "not" ... or maybe just another phrase that was dropped somewhere.
It bothers me too! But there's a contemporary English translation (not by Hobbes) which reads as follows:
Also, when Hobbes talks about 'the light of nature' he doesn't mean intuition, but something more like synderesis, which we might translate as 'natural reason'. So this whole argument about whether Hobbes did or didn't find it counter-intuitive is slightly beside the point.
posted by verstegan at 1:59 PM on December 31 [2 favorites]
It bothers me too! But there's a contemporary English translation (not by Hobbes) which reads as follows:
So that so absurd a Proposition as this, an Infinite is equal to a Finite, ought not to be ascribed to Torricellius; for there can be no Solid so small, which, being Infinite, doth not exceed every finite Solid, as is manifest by the Light of Nature.That's still not very clear. But 'if it's infinite, it can't be finite' seems to be the gist of it, and Mancusu's paraphrase ('Hobbes believed that natural light teaches us that an infinitely long solid, however subtle, must exceed in volume any finite solid') doesn't seem obviously untrue to Hobbes's meaning.
Also, when Hobbes talks about 'the light of nature' he doesn't mean intuition, but something more like synderesis, which we might translate as 'natural reason'. So this whole argument about whether Hobbes did or didn't find it counter-intuitive is slightly beside the point.
posted by verstegan at 1:59 PM on December 31 [2 favorites]
Still thinking about Hobbes' take on Torricelli. I think it helps to look at what he says in the whole passage at the end of Chapter 12 (on the infinite) in his 1674 book on principles and problems of geometry. Here is the last paragraph, with a translation slightly cleaned up from what you get from Google Translate:
Google rendered that first (taking the whole passage at once) as: Therefore, such an absurd proposition as this, that the Infinite is equal to the Finite, should not be attributed to Torricelli, for there can be no solid so thin that the Infinite does not exceed every finite solid, as is evident from natural light.
But then if you give Google just that sentence, it comes back with: "Therefore, a proposition as absurd as this, that the Infinite is equal to the Finite, should not be attributed to Torricelli, for there can be no solid so thin that it does not infinitely exceed every finite solid, as is evident from natural light."
Blåsjö's rendering of the sentence is roughly, "Therefore, such an absurd proposition as this, that the Infinite is equal to the Finite, should not be attributed to Torricelli, for there can be no solid so thin that an infinite stack of such solids does not exceed every finite solid, as is evident from natural light."
And on reflection -- and taking the whole passage into account -- that rendering strikes me as very plausible. It's basically reading Hobbes as suspicious of both infinities and infinitesimals.
But I'm not sure that agreeing as to how to interpret Hobbes here helps with Blåsjö's claim that Torricelli's trumpet wasn't counterintuitive to 17th century people (or thinkers or mathematicians more specifically).
posted by Jonathan Livengood at 2:04 PM on December 31 [3 favorites]
Infinitum autem a Mathematicis (saepissime dicitur pro Indefinito. Indefinitum autem est idem quod quantumvis magnum, & aliquando pro Infinite parvo, modo non fit Nihil. Dividi enim in Infinitum, id est in Nihila quantitas nulla potest, Dicitur etiam Infinitum aliquando pro Quantum est possibile. Infinitum autem proprie dictum nihil est nifi superet mensurarum datarum numerum omnem assignabilem. Sed demonstratum esse aiunt a Torricellio Solidum quoddam acutum Hyperbolicum, etiam hoc sensu Infiniti, aequale esse Cylindro cuidam, cujus quidem basis habeat diametrum aequalem dimidiae basi Hyperbolae, altitudinem vero aequalem ejusdem Hyperbolae axi transverso. Demonstrationem hujus Problematis faepius & attente legeram; neque quicquam inveni Paralogismi. Inveni tamen distantiam quam Torricellius supponit Infinitam, intelligi de distantia indefinita; nec potuisse ab ipso aliter intelligi, qui Principio utitur Cavalleriano de Indivisibilibus in valde multis demonstrationibus; quae Indivisibilia Cavallerii talia sunt ut eorum aggregatum aequale possit esse cuicunque datae magnitudini aequale. Itaque propositio tam absurda quam haec est, Infinitum Finito esse aequale, Torricellio ascribi non debet, Solidum enim tam exile nullum esse potest, quod Infinite non excedat omne Solidum finitum, ut manifestum est lumine naturali. Absurditas illa Arithmeticorum est disputantium de Infinito, & superficiem & solida mensurantium per lineas sine latitudine, qui inter Arithmeticam & Geometriam nullam animadvertentes differentiam, radicem numeri (quae numeri sui quadrati pars est) pro eadem re habuere cum Figurae quadratae latere, quanquam latus sui quadrati partem non esse consiteantur. De Mathematica dixi. Quid Algebristae in contrarium dicturi sunt expecto.
But the Infinite is very often said by Mathematicians for the Indefinite. The Indefinite is the same as however great, and sometimes for the Infinitely small, as long as it does not become Nothing. For no quantity can be divided into the Infinite, that is, into Nothing, and the Infinite is sometimes said for whatever is possible. But the Infinite, properly speaking, is nothing, if it exceeds all assignable numbers of given measurements. But they say that it was demonstrated by Torricelli that a certain acute Hyperbolic Solid, also in this sense of Infinity, is equal to a certain Cylinder, the diameter of whose base has indeed been equal to half the base of the Hyperbola, and the height is equal to the transverse axis of the same Hyperbola. I had read the demonstration of this Problem many times and attentively; and I did not find any Paralogism. However, I found that the distance which Torricelli supposes to be Infinite, is understood of an indefinite distance; nor could it have been understood otherwise by him, who uses Cavalleri's Principle of Indivisibles in very many demonstrations; which Indivisibles of Cavalleri are such that their aggregate can be equal to any given magnitude. Therefore, such an absurd proposition as this, that the Infinite is equal to the Finite, should not be attributed to Torricelli, for there can be no solid so thin that the Infinite does not exceed every finite solid, as is evident from natural light. That absurdity belongs to the Arithmeticians who argue about the Infinite, and measure surface and solids by lines without width, who, noticing no difference between Arithmetic and Geometry, have considered the root of a number (which is part of its square number) to be the same thing as the side of a square figure, although the side is not a part of its square. I have spoken of Mathematics. I await what the Algebraists will say to the contrary.The crucial sentence is this one: Itaque propositio tam absurda quam haec est, Infinitum Finito esse aequale, Torricellio ascribi non debet, Solidum enim tam exile nullum esse potest, quod Infinite non excedat omne Solidum finitum, ut manifestum est lumine naturali.
Google rendered that first (taking the whole passage at once) as: Therefore, such an absurd proposition as this, that the Infinite is equal to the Finite, should not be attributed to Torricelli, for there can be no solid so thin that the Infinite does not exceed every finite solid, as is evident from natural light.
But then if you give Google just that sentence, it comes back with: "Therefore, a proposition as absurd as this, that the Infinite is equal to the Finite, should not be attributed to Torricelli, for there can be no solid so thin that it does not infinitely exceed every finite solid, as is evident from natural light."
Blåsjö's rendering of the sentence is roughly, "Therefore, such an absurd proposition as this, that the Infinite is equal to the Finite, should not be attributed to Torricelli, for there can be no solid so thin that an infinite stack of such solids does not exceed every finite solid, as is evident from natural light."
And on reflection -- and taking the whole passage into account -- that rendering strikes me as very plausible. It's basically reading Hobbes as suspicious of both infinities and infinitesimals.
But I'm not sure that agreeing as to how to interpret Hobbes here helps with Blåsjö's claim that Torricelli's trumpet wasn't counterintuitive to 17th century people (or thinkers or mathematicians more specifically).
posted by Jonathan Livengood at 2:04 PM on December 31 [3 favorites]
Also, when Hobbes talks about 'the light of nature' he doesn't mean intuition, but something more like synderesis, which we might translate as 'natural reason'. So this whole argument about whether Hobbes did or didn't find it counter-intuitive is slightly beside the point.
This is a really good point!
posted by Jonathan Livengood at 2:08 PM on December 31
This is a really good point!
posted by Jonathan Livengood at 2:08 PM on December 31
The rendering as "there can be no Solid so small, which, being Infinite, doth not exceed every finite Solid, as is manifest by the Light of Nature" is also really interesting, since it suggests something like, "the smallest infinitely big solid is still infinitely bigger than any finite solid," which I agree is right in line with Mancosu's reading.
posted by Jonathan Livengood at 2:12 PM on December 31 [1 favorite]
posted by Jonathan Livengood at 2:12 PM on December 31 [1 favorite]
So, did anyone here ever study calculus using a book that made more than a cursory mention of Toricelli ? "Calculus was first defined and studied by Newton and Liebnitz. If you argue in my class about who was first you get an F. BTW, they were building on earlier work by Toricelli. Now on to differentiation." That's about the sum total I got.
posted by ocschwar at 2:43 PM on December 31
posted by ocschwar at 2:43 PM on December 31
We all agree that the concept of infinity is just weird and paradoxical, and then we move on with our lives.
posted by math
Our very finite lives.
posted by Pouteria at 5:02 PM on December 31 [1 favorite]
posted by math
Our very finite lives.
posted by Pouteria at 5:02 PM on December 31 [1 favorite]
I actually really appreciated their example of splitting a cube in two to provide an intuitive example of area vs volume
Start with a one inch cube. Split it in half and move the top half to the side. Split that top half in half and do the same. Continue.
So, basically, you get a sequence of slices that are half as thick, but each have a top area of 1x1. The volume remains the same but the area increases by 1 at each slice.
Although if I've done the math right we hit a physical limit at about 25 slices, or about the thickness of graphine.
Which just illustrates that infinity is weird, and that up and down one's intuition can go wrong.
posted by indexy at 6:02 PM on December 31
Start with a one inch cube. Split it in half and move the top half to the side. Split that top half in half and do the same. Continue.
So, basically, you get a sequence of slices that are half as thick, but each have a top area of 1x1. The volume remains the same but the area increases by 1 at each slice.
Although if I've done the math right we hit a physical limit at about 25 slices, or about the thickness of graphine.
Which just illustrates that infinity is weird, and that up and down one's intuition can go wrong.
posted by indexy at 6:02 PM on December 31
For folks interested in the current math/calculus education issues that have been raised in this thread, I highly recommend looking into the research on students’ concept images for limits (Part 2, Part 3) and other key calculus concepts (links give a reasonable introductory overview with lots of citations to get you started).
posted by eviemath at 6:12 PM on December 31 [3 favorites]
posted by eviemath at 6:12 PM on December 31 [3 favorites]
I only realized yesterday that the negative space of an arbitrarily large cannonball pile is not only connected, but contains entire lines
That's perfectly in line with my own intuition, probably because as a small child I played with a box of grapefruit and recall being able to see the bottom of the box before taking any out.
It's an observation that also fits in nicely with the way, in an actual honeycomb, the wax is thicker wherever three cells meet than wherever two do.
And now that you've raised it again, the connected thing makes sense to me too. Any given pair of adjacent spheres in the pile touches only at one point, so there's no way to form a seal that would stop water finding a way past them. Which is of course why sandy soils drain so well.
Intuition is basically the raw, unfiltered output of the brain's pattern recognizer. It's therefore a property of people, not of the things that the people are working with. To claim that some or other mathematical conclusion is or is not counterintuitive, then, is not even wrong; it's a straight-up category error, akin to arguing at length about whether red paint is or isn't Wednesday.
posted by flabdablet at 7:14 PM on December 31 [3 favorites]
That's perfectly in line with my own intuition, probably because as a small child I played with a box of grapefruit and recall being able to see the bottom of the box before taking any out.
It's an observation that also fits in nicely with the way, in an actual honeycomb, the wax is thicker wherever three cells meet than wherever two do.
And now that you've raised it again, the connected thing makes sense to me too. Any given pair of adjacent spheres in the pile touches only at one point, so there's no way to form a seal that would stop water finding a way past them. Which is of course why sandy soils drain so well.
Intuition is basically the raw, unfiltered output of the brain's pattern recognizer. It's therefore a property of people, not of the things that the people are working with. To claim that some or other mathematical conclusion is or is not counterintuitive, then, is not even wrong; it's a straight-up category error, akin to arguing at length about whether red paint is or isn't Wednesday.
posted by flabdablet at 7:14 PM on December 31 [3 favorites]
Speaking as a language nerd, this is mostly frustrating because it is a missed opportunity to trot out dozens of contemporary examples of the expressions he claims to have better translations of, as evidence that his translations are in fact better.
posted by No-sword at 11:30 PM on December 31 [1 favorite]
posted by No-sword at 11:30 PM on December 31 [1 favorite]
I am sorry cyanistes, did you mean seventeeth-century methematicians?
I have not read the article, fell in a rabbit hole trying to figure out a base Pi numbering system so that Pi is completely normal and it is numbers like “2” that are weird.
posted by Dr. Curare at 7:16 AM on January 1
I have not read the article, fell in a rabbit hole trying to figure out a base Pi numbering system so that Pi is completely normal and it is numbers like “2” that are weird.
posted by Dr. Curare at 7:16 AM on January 1
I don’t quite understand why he’s so angry about it. He sounds as if somebody sent their trumpet back for a refund because they couldn’t work out how to play it.
This is just sort of his whole style. Grumpy close-reading in service of, as the podcast title promises, vociferous statements of opinion. He spent, not the first episode, but the entire first season of the podcast detailing, with receipts, his opinion that Galileo was a shitty scientist who doesn't live up his latter-day veneration in pop science communication.
I feel like he manages to make a reasonable argument in all this about the poor foundation of the central trope—that the presentation of Torricelli's trumpet as a historical landmine of a paradox is more a too-good-to-check story than something justified by the historical record and that it's not clear that practicing mathematicians of the time were melting down about it—but his style is as always very grudging and pedantic and hyperfocused on a specific mathematical/historical context in a way that is unhelpful unless grudging historical pedantry is precisely what you're in the mood for. And then he tosses on declarations of what is and isn't counterintuitive without really allowing for commonplace use of the word, because he wants to argue about the historical text rather than talk about the degree to which Torricelli's trumpet is, in fact, weird in a lay sense.
Which I feel like is the middle ground in this: whether as a mathematical object it is counterintuitive or not is more a subjective statement about any given person's intuition and mathematical background and ways of thinking than it is any objectively resolvable point. But we can all pretty much agree that it's a weird thing! A lot of cool math stuff is weird. Paradoxes are weird. Infinity is weird. High-dimensional geometry is weird. Basically anything that contradicts at some surface level the tangible, finite, three-dimensional analog experiences of the mammalian brain tends to feel weird!
This guy ground a menger cube up
First of all, how dare you,
posted by cortex at 9:40 AM on January 1 [2 favorites]
This is just sort of his whole style. Grumpy close-reading in service of, as the podcast title promises, vociferous statements of opinion. He spent, not the first episode, but the entire first season of the podcast detailing, with receipts, his opinion that Galileo was a shitty scientist who doesn't live up his latter-day veneration in pop science communication.
I feel like he manages to make a reasonable argument in all this about the poor foundation of the central trope—that the presentation of Torricelli's trumpet as a historical landmine of a paradox is more a too-good-to-check story than something justified by the historical record and that it's not clear that practicing mathematicians of the time were melting down about it—but his style is as always very grudging and pedantic and hyperfocused on a specific mathematical/historical context in a way that is unhelpful unless grudging historical pedantry is precisely what you're in the mood for. And then he tosses on declarations of what is and isn't counterintuitive without really allowing for commonplace use of the word, because he wants to argue about the historical text rather than talk about the degree to which Torricelli's trumpet is, in fact, weird in a lay sense.
Which I feel like is the middle ground in this: whether as a mathematical object it is counterintuitive or not is more a subjective statement about any given person's intuition and mathematical background and ways of thinking than it is any objectively resolvable point. But we can all pretty much agree that it's a weird thing! A lot of cool math stuff is weird. Paradoxes are weird. Infinity is weird. High-dimensional geometry is weird. Basically anything that contradicts at some surface level the tangible, finite, three-dimensional analog experiences of the mammalian brain tends to feel weird!
This guy ground a menger cube up
First of all, how dare you,
posted by cortex at 9:40 AM on January 1 [2 favorites]
Are we just doing math jokes now?
Three logicians walk into a bar. The bartender says, "do you all want a beer?"
"I don't know" says the first
"I don't know" says the second
"Yes" says the third.
Also base pi is fine as a system of numeration, it's just that everything has a non-terminating expansion except for (a subset of) the ring Z[\pi, \pi^{-1}].
(Did I see somewhere that you can do Mathjax or otherwise get nice math into MeFi? That would be nice for some threads)
posted by SaltySalticid at 10:04 AM on January 1 [1 favorite]
Three logicians walk into a bar. The bartender says, "do you all want a beer?"
"I don't know" says the first
"I don't know" says the second
"Yes" says the third.
Also base pi is fine as a system of numeration, it's just that everything has a non-terminating expansion except for (a subset of) the ring Z[\pi, \pi^{-1}].
(Did I see somewhere that you can do Mathjax or otherwise get nice math into MeFi? That would be nice for some threads)
posted by SaltySalticid at 10:04 AM on January 1 [1 favorite]
Twenty minutes later, she returned to the stunned class: "Yes, it is obvious".
SaltySalticid, that is both a hilarious joke in its own right as well as a perfectly precise parody of what this guy is doing. Standing ovation.
posted by straight at 12:09 PM on January 1
SaltySalticid, that is both a hilarious joke in its own right as well as a perfectly precise parody of what this guy is doing. Standing ovation.
posted by straight at 12:09 PM on January 1
I'd like to think the author would have a good-natured chuckle at his own expense if someone responded, "So, it's intuitive if we give you 5000 words to explain why it's intuitive?"
posted by straight at 12:13 PM on January 1
posted by straight at 12:13 PM on January 1
Part of the problem is that he's using "intuitive" in a very idiosyncratic way. In its original technical sense, an intuition is an immediate cognition of its object, and "intuition" contrasts with "speculation" and "discursive thought." At least, that's more or less the way European philosophers used the term from Anselm (I think) down through at least Kant, Mill, and Peirce. (This is one reason why I thought 's verstegan's comment on synderesis and the light of nature was very nice.)
Today, as a term of art in philosophy thing's are more complicated, since the older sense is mostly lost and there is a lot of debate about what exactly we have in mind with "intuition" -- some saying it's something like an intellectual perception or seeming (with a distinctive phenomenology), some saying it's (default) inclination or disposition to believe, and some saying it's just another word for opinion or for some sub-type of belief. See the SEP entry on intuition for more. But I don't think anyone who is serious about treating intuition as some kind of special cognitive phenomenon, ability, capacity, or faculty will agree with Blåsjö's usage.
And if we're thinking about ordinary usage, then it seems to me at least that intuition has essentially the sense of "unsupported, initial gut reaction," which you can see is close to the old technical sense and at least retains the idea that an intuition is not the result of deliberate, reflective thought.
posted by Jonathan Livengood at 12:40 PM on January 1 [2 favorites]
Today, as a term of art in philosophy thing's are more complicated, since the older sense is mostly lost and there is a lot of debate about what exactly we have in mind with "intuition" -- some saying it's something like an intellectual perception or seeming (with a distinctive phenomenology), some saying it's (default) inclination or disposition to believe, and some saying it's just another word for opinion or for some sub-type of belief. See the SEP entry on intuition for more. But I don't think anyone who is serious about treating intuition as some kind of special cognitive phenomenon, ability, capacity, or faculty will agree with Blåsjö's usage.
And if we're thinking about ordinary usage, then it seems to me at least that intuition has essentially the sense of "unsupported, initial gut reaction," which you can see is close to the old technical sense and at least retains the idea that an intuition is not the result of deliberate, reflective thought.
posted by Jonathan Livengood at 12:40 PM on January 1 [2 favorites]
The historical mathematicians avoiding statements involving infinity made me think of the "proof" that π is 4. Count me among those that think intuition cannot be trusted. It's useful sure, and with experience one can learn to avoid some of the gotchas that can lead to proofs of things that just ain't so. But taking the limit of sequences does require some care.
posted by mscibing at 7:16 PM on January 1
posted by mscibing at 7:16 PM on January 1
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posted by dis_integration at 6:48 AM on December 31 [7 favorites]