> When Illustrating a mathematical idea, the first thing you need to decide is the scale.
I have spent much of my life illustrating mathematical ideas, and scale is never the first thing I decide. Most commonly it stays abstract and there is no scale; it's flexible and I can zoom in and out at will. Sometimes I will choose a scale partway through or towards the end of an explanation, if I want to use a specific analogy, but I can comfortably rescale it to something else - the scale is never fixed.
I have loved math since I was a child, and I think it depends on when you grew up and how steeped you are in reality vs. the virtual or the computer world, and how much of an abstract vs. concrete thinker you are. I was always making things in modeling clay, that greasy grey-green stuff, and so my scale was what I could make out of one brick of such stuff. I bought my first computer in 1977 (Commodore PET 2001), and the CBM ASCII set had some graphics, but nothing compared with today's graphics. My first encounter with visualization and scale was writing a program to let me know which of the four moons of Jupiter I was seeing in the sky that night. Io, Ganymede, Callisto, and Europa's orbits are almost edge-on to our view from earth, so I made Jupiter a capital O, and the moons were lowercase letters. I printed this out on a thermal printer (like a wide receipt). Cosmos was the rage on TV and I had read Einstein's Universe by Nigel Calder. I had a telescope and a microscope, so the micro and macro were very real to me. I suspect if you grew up on tablets and only built things on a 3D printer scale, you don't have that unbridled sense of the small and large except on very abstract terms. However, not a donut, not a universe-scale torus, but rather a pool donut comes to mind when I first hear torus!
I built an XYZ router table in the early 2000s out of old stepper motors. It was 8'x4', and I built stitch-and-glue wooden kayaks from the panels I cut on it. These would wind up being 16 to 22 foot long kayaks to go into the real world and have fun!
I propose a further and different "key to understanding."
I would add: the second thing to decide, besides the scale, is the Plan.
What do we mean, for example, by the "Ethical Plan." By ethical plan, I mean the purpose... "WHAT do I use mathematics for"?
Mathematics can be something immensely BIG if I use it for something important.
Or it can be miserably SMALL if I use it for something petty and trivial.
In short: even in this case, greatness depends not only on the scale, but also on the eyes of the beholder, on the Context in which it is applied, and, why not?, also on the Purpose and the ethical plan.
If mathematics were, for example, something at the service of Justice, it would be something immensely Big.
Totally agree. I really enjoyed the article, and the illustrations are really cool but scale is just something I don’t even consider. Even the very first question baffled me, when it said “Picture a torus. Is it big or small?”
I answered an unambiguous “yes”.
Also, we haven’t defined measure yet here have we? What does it even mean for something to have scale without measure?
This is one of those places where Plato really is worth reading. Plato has levels of reality that correspond to numbers. The first level, forms (also called "the monad"), is what the statement "Picture a torus" engages: contemplate an ideal torus. That torus won't have a particular color or texture or any accidental quality, just the essence of a torus, which is its shape (because torus is a shape). Size is one of those accidental qualities, and those live in the second level, which Plato calls "the bigger and smaller"—exactly what the question asks you to imagine—or "the dyad."
So, the instructions for Plato boil down to an absurdity: "contemplate the monad; what dyad do you see?" The two sentences should have nothing to do with each other in Platonic terms.
Right, I immediately saw a torus - it was light blue (that's trivial to change, but I can't have no colour if it's visual) - but it could have been the size of a bacterium or the size of a galaxy. Without any context or application, the size is undefined.
When you've mentioned that, I've noticed that by default I imagine just a shape devoid of color and texture. But I can imagine a donut, or a blue torus, but I need to explicitly think the word "blue".
A first-year physics teacher once told the class something that stuck with me (paraphrasing): "Nothing is big or small by itself. I want you to always follow these words with 'compared to ...'".
He: I think we can agree everything below the average between a Planck length and the size of the observable universe is objectively small, and everything above is objectively large. Using the geometric mean, that average is about 0.12 mm. Therefore my penis is actually large.
The geometric mean seems to be the natural mean for relative comparisons between lengths, because the mean of (Planck length, observable universe) is clearly very different from the mean of (house, observable universe).
It kinda seems like the point of the article was to talk about different mathematical illustrations, not to determine if math was big or small. Even in the article, the conclusion is that it's both. I suspect the only reason for choosing the title is to grab attention (and it worked on me).
Of course, I am extra cynical as a number theorist who can't visualize most of my field. I wrote my doctorate on Siegel modular forms, and I can honestly say I have no way to visualize them any further than numbers on a page.
Err? Peano Arithmetic is provably consistent in ZFC, but it is not in itself (if PA is consistent). Therefore if PA is consistent it is not equivalent to ZFC (regardless of whether ZFC is consistent or not)
Even before I started the video, I had a feeling it was going to lead to a kind of "introspective" mathematics that can reason about its own reasoning. I was not disappointed, thank you.
> When Illustrating a mathematical idea, the first thing you need to decide is the scale.
I have spent much of my life illustrating mathematical ideas, and scale is never the first thing I decide. Most commonly it stays abstract and there is no scale; it's flexible and I can zoom in and out at will. Sometimes I will choose a scale partway through or towards the end of an explanation, if I want to use a specific analogy, but I can comfortably rescale it to something else - the scale is never fixed.
Interesting to see such a different view.
I have loved math since I was a child, and I think it depends on when you grew up and how steeped you are in reality vs. the virtual or the computer world, and how much of an abstract vs. concrete thinker you are. I was always making things in modeling clay, that greasy grey-green stuff, and so my scale was what I could make out of one brick of such stuff. I bought my first computer in 1977 (Commodore PET 2001), and the CBM ASCII set had some graphics, but nothing compared with today's graphics. My first encounter with visualization and scale was writing a program to let me know which of the four moons of Jupiter I was seeing in the sky that night. Io, Ganymede, Callisto, and Europa's orbits are almost edge-on to our view from earth, so I made Jupiter a capital O, and the moons were lowercase letters. I printed this out on a thermal printer (like a wide receipt). Cosmos was the rage on TV and I had read Einstein's Universe by Nigel Calder. I had a telescope and a microscope, so the micro and macro were very real to me. I suspect if you grew up on tablets and only built things on a 3D printer scale, you don't have that unbridled sense of the small and large except on very abstract terms. However, not a donut, not a universe-scale torus, but rather a pool donut comes to mind when I first hear torus! I built an XYZ router table in the early 2000s out of old stepper motors. It was 8'x4', and I built stitch-and-glue wooden kayaks from the panels I cut on it. These would wind up being 16 to 22 foot long kayaks to go into the real world and have fun!
I propose a further and different "key to understanding."
I would add: the second thing to decide, besides the scale, is the Plan.
What do we mean, for example, by the "Ethical Plan." By ethical plan, I mean the purpose... "WHAT do I use mathematics for"?
Mathematics can be something immensely BIG if I use it for something important. Or it can be miserably SMALL if I use it for something petty and trivial.
In short: even in this case, greatness depends not only on the scale, but also on the eyes of the beholder, on the Context in which it is applied, and, why not?, also on the Purpose and the ethical plan.
If mathematics were, for example, something at the service of Justice, it would be something immensely Big.
It sounds like you ain't a fan of recreational mathematics?
Totally agree. I really enjoyed the article, and the illustrations are really cool but scale is just something I don’t even consider. Even the very first question baffled me, when it said “Picture a torus. Is it big or small?”
I answered an unambiguous “yes”.
Also, we haven’t defined measure yet here have we? What does it even mean for something to have scale without measure?
This is one of those places where Plato really is worth reading. Plato has levels of reality that correspond to numbers. The first level, forms (also called "the monad"), is what the statement "Picture a torus" engages: contemplate an ideal torus. That torus won't have a particular color or texture or any accidental quality, just the essence of a torus, which is its shape (because torus is a shape). Size is one of those accidental qualities, and those live in the second level, which Plato calls "the bigger and smaller"—exactly what the question asks you to imagine—or "the dyad."
So, the instructions for Plato boil down to an absurdity: "contemplate the monad; what dyad do you see?" The two sentences should have nothing to do with each other in Platonic terms.
Right, I immediately saw a torus - it was light blue (that's trivial to change, but I can't have no colour if it's visual) - but it could have been the size of a bacterium or the size of a galaxy. Without any context or application, the size is undefined.
When you've mentioned that, I've noticed that by default I imagine just a shape devoid of color and texture. But I can imagine a donut, or a blue torus, but I need to explicitly think the word "blue".
> Also, we haven’t defined measure yet here have we?
Kilograms, obviously.
A first-year physics teacher once told the class something that stuck with me (paraphrasing): "Nothing is big or small by itself. I want you to always follow these words with 'compared to ...'".
Sewing machine.
https://xkcd.com/2754/
She: It's not that big.
He: I think we can agree everything below the average between a Planck length and the size of the observable universe is objectively small, and everything above is objectively large. Using the geometric mean, that average is about 0.12 mm. Therefore my penis is actually large.
She: I shouldn't have married a physicist.
> Using the geometric mean, that average is about 0.12 mm
That's... actually kinda cool to know.
It's true!
https://www.wolframalpha.com/input?i=Geometric+mean+of+%28Pl...
Average? So around half the size of the observable universe?
They specified the geometric mean.
The arithmetic mean (what you're thinking of) of 1 and 100 is 50.5.
The geometric mean of 1 and 100 is 10. It gives a sense of the average magnitude.
They edited the comment, previously it did not mention geometric mean.
The geometric mean seems to be the natural mean for relative comparisons between lengths, because the mean of (Planck length, observable universe) is clearly very different from the mean of (house, observable universe).
I've always loved this recording of Thurston talking about branched coverings and knot complements using big knots: https://www.youtube.com/watch?v=IKSrBt2kFD4
It kinda seems like the point of the article was to talk about different mathematical illustrations, not to determine if math was big or small. Even in the article, the conclusion is that it's both. I suspect the only reason for choosing the title is to grab attention (and it worked on me).
Of course, I am extra cynical as a number theorist who can't visualize most of my field. I wrote my doctorate on Siegel modular forms, and I can honestly say I have no way to visualize them any further than numbers on a page.
If you run a statistical sample on modern maths books, it becomes clear that maths is usually about 6cm x 5cm.
Good article.
Math is smaller than the smallest and bigger than the biggest.
It's also deep, it goes all the way to the bottom.
> The world of mathematics is both broad and deep, and we need birds and frogs working together to explore it. -- Freeman Dyson
Weird Things Happen When Math Gets Too Expressive
https://www.youtube.com/watch?v=EVwQsvof7Hw
Peano arithmetic is sufficiently expressive enough to be equivalent to any possible future theory of mathematics.
Err? Peano Arithmetic is provably consistent in ZFC, but it is not in itself (if PA is consistent). Therefore if PA is consistent it is not equivalent to ZFC (regardless of whether ZFC is consistent or not)
I am referring to this slide : https://youtu.be/EVwQsvof7Hw?t=1646
Even before I started the video, I had a feeling it was going to lead to a kind of "introspective" mathematics that can reason about its own reasoning. I was not disappointed, thank you.
Physics, Topology, Logic and Computation: A Rosetta Stone - https://arxiv.org/abs/0903.0340
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Obviously a torus is the size of a doughnut.
Doesn’t math come down to =
Yes.
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