[HN Gopher] Is math big or small?
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Is math big or small?
Author : robinhouston
Score : 77 points
Date : 2026-04-12 08:25 UTC (1 days ago)
(HTM) web link (chessapig.github.io)
(TXT) w3m dump (chessapig.github.io)
| N_Lens wrote:
| Good article.
|
| Math is smaller than the smallest and bigger than the biggest.
| lioeters wrote:
| 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
| contraposit wrote:
| 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.
| lioeters wrote:
| 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
| gottheUIblues wrote:
| 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)
| contraposit wrote:
| I am referring to this slide :
| https://youtu.be/EVwQsvof7Hw?t=1646
| mkl wrote:
| > 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.
| seanhunter wrote:
| 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?
| mkl wrote:
| 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.
| red75prime wrote:
| 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".
| Pay08 wrote:
| > Also, we haven't defined measure yet here have we?
|
| Kilograms, obviously.
| Telemakhos wrote:
| 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.
| aledevv wrote:
| 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.
| eru wrote:
| It sounds like you ain't a fan of recreational mathematics?
| eggy wrote:
| 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!
| volemo wrote:
| _Yes._
| fuglede_ wrote:
| 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
| lefra wrote:
| 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 ...'".
| cubefox wrote:
| 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.
| leni536 wrote:
| Average? So around half the size of the observable universe?
| quietbritishjim wrote:
| 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.
| leni536 wrote:
| They edited the comment, previously it did not mention
| geometric mean.
| cubefox wrote:
| 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).
| zahlman wrote:
| > Using the geometric mean, that average is about 0.12 mm
|
| That's... actually kinda cool to know.
| bo1024 wrote:
| Sewing machine.
|
| https://xkcd.com/2754/
| cammasmith wrote:
| 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.
| krm01 wrote:
| Doesn't math come down to =
| gweinberg wrote:
| Obviously a torus is the size of a doughnut.
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