“Suppose,” said Zeno, “that Achilles and a tortoise are planning to race”.

Such is the beginning of a famous thought experiment by an ancient philosopher. Since athletic Achilles was much faster than the slow tortoise, he let the tortoise start first. But alas, he could never catch up to it, since every time Achilles made it to where the tortoise had been, the tortoise had moved just a little further ahead. Of course Achilles was faster so he had to pass the tortoise quickly unless it had started near the finish line. So, paradox!

Most of the paradoxes of Zeno were about fractions and entireties of time and space. Can an infinite series of fractions of space add up to a finite entirety of space in a finite entirety of time? Some might say that integral calculus solves these basically mathematical problems, yet others think they point to metaphysical issues as regards to the discreteness and the continuity of time and space.

This fourfold reminds me of my previous fourfold Spacetime which dealt with succession (as parts of time), location (as parts of space), extension (as wholes of space), and duration (as wholes of time). It must have been in the back of my mind.

References:

https://equivalentexchange.wordpress.com/2011/09/23/spacetime/

[*9.122]

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# Invention and Discovery

What are the differences between invention and discovery? Ever since my post Propositions as Types I’ve been trying to determine what they are. Some say that mathematics and logic are completely human inventions and they have no correspondence to the natural world. Others say that mathematics already exists in some “Platonic” realm just waiting for our discovery. Similar to convergent evolution, the parallel invention or discovery of similar notions in mathematics lends credence to the idea that there is something “out there” just waiting for us to find it, although one could also argue that it’s merely the cultural climate along with some innate functioning of the brain. For example there is the parallel development of calculus by Newton and Leibniz. The notion of effective computability in the “Propositions as Types” paradigm also has several concurrent developments.

Modern science is based on mathematics so as one goes so goes the other. Physicist Eugene Wigner wrote a famous article on the “Unreasonable effectiveness of mathematics in the natural sciences” which has inspired a host of similarly titled articles about the “unreasonable effectiveness” of one thing for another. But the key point is that we really don’t understand the origins of mathematical thinking, or why it is so useful in helping us understand the natural world. Its value and utility seems, in fact, unreasonable.

But let’s return to the differences between invention and discovery. If something is invented, it means that it is new, freshly created. If something is discovered, it means that it already exists and it’s just waiting for us to find it. Thus the difference is between the natural and artificial, or between what exists and what didn’t exist before humans created it. Some believe the natural world itself is socially constructed, so in some sense it didn’t exist before humans saw it, or will disappear when humans stop perceiving it. This is about is arrogant as believing that the world didn’t exist before a person was born or after they die; a solipsistic view if ever there was one.

Once something is discovered, one can learn about it. Once something is invented, one can make it. Thus learning and making are tied to discovering and inventing, respectively. Inventing and discovering are required for making and learning. Of course one can also learn about an invention or how something is made, or one can learn facts about a discovery.

This fourfold of inventing, discovering, learning, and making is also related to other fourfolds. The Four Hats of Creativity seem to utilize each of these special actions for each livelihood: inventing (or creating) for the artist, discovery for the scientist, and making for the engineer (but less well learning for the designer). In addition, the Psychological Types of Jung appear to emphasize a type for each special action: intuition for invention, sensation for discovering, and cognition for learning (but less clear emotion for making).

Please compare this with a related analysis on the methods of active learning at the Tetrast (link below), where the key faculties are struggle for invention, practice for discovery, value for making, and discipline for learning.

References:

https://en.wikipedia.org/wiki/Convergent_evolution

http://tetrast2.blogspot.com/2013/04/methods-of-learning.html

[*9.86]

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# Four Bindings

Nothing in the world stands by itself. Every object is a link in an endless chain and is thus connected with all the other links. And this chain of the universe has never been broken; it unites all objects and processes in a single whole and thus has a universal character. We cannot move so much as our little finger without “disturbing” the whole universe. The life of the universe, its history lies in an infinite web of connections.

— A. Spirkin, from Dialectical Materialism

How is the universe bound up with itself, and we and everything else, as denizens of it? How are we connected to each other, and with all the other things in the universe? And how is all that stuff related to itself, and with all the other stuff that it shares space and time with?

In a previous post I mentioned chains, grids, cycles, and blocks, and associated them with the Four Causes and my fourfold Structure-Function. As I thought more about what those terms meant, I decided that they were bindings.

At first I tried to enumerate how four things could be arranged in exactly four ways. I didn’t get very far with that, but I ended up with this fourfold. It’s a gift.

References:

When I looked up “universal connections”, the first thing I saw was the following link:

https://www.marxists.org/reference/archive/spirkin/works/dialectical-materialism/ch02-s05.html

Alexander Spirkin / Dialectical Materialism

https://www.marxists.org/reference/archive/spirkin/works/dialectical-materialism/index.html

Sylvia Plath / Love Letter

http://allpoetry.com/Love-Letter

[*9.27, *9.102]

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