We care only about a very small and narrow subset of possible programs, not any arbitrary one. It's possible to solve this kind of problem in large enough classes of program to be useful.
> 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!
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".
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.
Fatalism is widespread, but not nearly universal enough that we can say it was the norm 15000 years ago.
For that matter, people who were pretty fatalist were still capable of using chance for purposes of fairness. The democrats in ancient Athens come to mind. I'm also pretty sure the (Christian) apostles' use of chance was also more about avoiding a human making the decision, than about divination.
I'm not saying divination isn't a thing, I'm saying there are examples of use of chance where it doesn't seem like divination.
Athenians selected through sortition didn't seem to act much like they believed they were chosen by the gods, and they defended their institutions mainly as wisdom, not as revelation.
And the apostles, being Jews, had a big taboo about using chance to determine God's will, but apparently not against using chance to fill vacancies.
There are bible passages suggesting the outcome of lots is God's will, and there are passages condemning divination. You can find them from the same links you posted above. But at the time of the apostles, it was a no-no to use chance to figure out God's will.
Please don't just shake links out of your sleeve, and talk to me instead. Do you think the Athenians acted like they were chosen by the gods when their number came up?
Don't you see a difference between the situations where chance could clearly have been used simply as a mechanism for fairness / avoiding a biased choice, and things like reading the movement of the birds or interpreting the shape of molten lead thrown into water?
Even in things like the goat choice in the bible you link above, I think it may be more about fairness than divination. Because as far as I know, the priests actually got to eat the sacrificial goat, but not the scapegoat they chased into the wild. So was it really about divining which goat God hated more, or was it maybe about "don't cheat by keeping the juicy goat for yourselves and chasing away the mangy one!"?
Yes, but so too is a modern western framing of these “dice” as “gambling” objects.
And also, the esteem in recognizing them as prefiguring a skill or system of thought that fund managers and FDA panels use today. In a roundabout way, it praises our own society’s systems by recognizing an ancient civilization for potentially having discovered some of their mathematical preliminaries.
They found 239 unique sets of dice from 130 tribes across 30 linguistic stocks. Although many of them are "binary lots" there is clear evidence that games of chance are extremely widespread in ancient North America
> His final report includes illustrations and descriptions of 293 unique sets of Native American dice from “130 tribes belonging to 30 linguistic stocks,” and it notes that “from no tribe [do dice] appear to have been absent”. In addition, Culin cites and quotes at length 149 ethnographic accounts of how these dice were used to power games of chance and for gambling. Based on this record, Culin suggested that “the wide distribution and range of variations in the dice games points to their high antiquity”.
> No prehistoric dice have ever been discovered in the eastern part of North America.
Come on, you don’t really think modern statistics might’ve come about from Europeans taking inspiration in the gambling practices of nomadic peoples in remote southwestern parts of North America. No need to pay lip service to every scold.
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