> 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.
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 ...'".
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.
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
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)
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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Yes.
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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?
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.
> Also, we haven’t defined measure yet here have we?
Kilograms, obviously.