The enormous usefulness of mathematics is something bordering on the mysterious … There is no rational explanation for it … The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve…

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{ Better Explained }:

How do you draw an elephant?

• Pencil the structure using ovals, rectangles, and so on
• Ink the final result, taking the lines you want
• Erase the underlying pencil structure, revealing the elephant

…

Is tracing different from drawing? You bet. Tracing is mimicry — we don’t know why a line is there. We just start in one corner and work our way around. Sure, we might make a pretty elephant — but can we draw one with a different trunk? Standing on two hind legs? Probably not.

Math is similar: we “teach” by tracing a student through the steps of a proof. But there’s an underlying pencil structure that was in the mind’s eye of the proof’s author that we’re not seeing. We’re walking the student along the drawing (“Here is the head, here is the trunk, here is the leg”) without show the mindset that created the proof (“The head is an oval, connected to a larger oval for the body; the legs are cylinders, which we smooth out.”).

If we’re lucky, the student generalizes the steps and creates their own pencil structure.

…

Sometimes we create “nice-looking elephants” through trial and error. Later on, we realize there’s a common structure that can simplify future efforts. True learning is about discovering and exploring these structures, not simply generating the pretty elephants.

••••••

OS:

I love Better Explained. I wish they covered more topics, but I feel like I’ve been looking for this my whole life — the intuitive explanation.

Funny; it reminds me of another { 3rd grade story }: besides that it was a group project wherein we had to write something long-ish, the context is irrelevant. The two girls I was sitting with were editing what I’d written, and were trying their hardest to convince me that my paragraph was too long (or too short?) because “paragraphs are 5 sentences long”. Someone had taught them that, without explaining the point that a paragraph depends on how long it takes to convey a thought. A creative writer knows that a paragraph can be a single word or three pages, as necessary.

For the longest time, I inwardly face-palmed thinking about that. Who was to blame? A bad teacher, or students who couldn’t grasp subtleties and abstractions? But… how can I laugh at them now? Truly, my entire mathematical experience (up until recently) has been exactly this way. I’ve been tracing mathematical elephants.

Einstein … was mightily influenced by what the ex–patent clerk called Mach’s Principle[:] “the inertia of every element of matter is due to its interaction with all the other elements of matter in the universe.”

We haven’t fully followed that investigative road, Mead said. “Instead what we’ve done is we’ve treated isolated objects as if all their attributes were just given us, and [we] haven’t asked where they came from,” he said. “Things like the inertia of an object, the rest energy of an object, the velocity of light — all those things. We have a list of fundamental constants that we’re not allowed to ask where they come from.”

If we want to get that stalled 100-year-old revolution unstuck, Mead said, we’ve got to ask – and discover – where those constants come from, and not just believe in them as handed down by academics and buried in mountains of math. We need to discover their basis in the interactions and interrelationships of all matter in the universe.

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Finally completed Intro to Physics at Udacity!

In the scheme of things, a basic physics course isn’t a big deal. But, I’m excited about this! This pursuit is a huge part of something I want to dedicate my life to, and here’s one more step in the right direction.

The course was an amazing educational experience, thanks to the instructors Andy & Jonathan, the students who participated in the forums, and the Udacity team. I learned a lot and it was the most fun I’ve ever had with math, which is { saying something }.

Hoping to take a more advanced course soon. Until then I’ll be busy with { these } and Khan Academy.

…when women were reminded — even subtly — of the stereotype that men were better than women at math, the performance of women in math tests measurably declined. Since the reduction in performance came about because women were threatened by the stereotype, the psychologists called the phenomenon “stereotype threat.”
…
“Everyone experiences stereotype threat. We are all members of some group about which negative stereotypes exist … And in a situation where one of those stereotypes applies … we know that we may be judged by it.”
…
“For a female scientist, particularly talking to a male colleague, if she thinks it’s possible he might hold this stereotype, a piece of her mind is spent monitoring the conversation and monitoring what it is she is saying, and wondering whether or not she is saying the right thing, and wondering whether or not she is sounding competent, and wondering whether or not she is confirming the stereotype,

realcleverscience asked: Just a comment: I think many people - including myself - know that the Greeks and others had primitive notions of evolution, but they never discussed probable causes for it. The Islamic quote is interesting precisely bc it seems to touch on natural selection. That said, yeah, the educational system is woefully inadequate.

Certainly, the intellectual prowess of the Greeks hasn’t gone unnoticed…
sort of:

My comment about the educational system… mostly a reaction to the canon of what is passed on — all mostly as singular stories sans connections. At least, that’s been my experience in schools.

This post reminded me that I didn’t even know about Eratosthenes (!) until watching Carl Sagan’s Cosmos a few of years ago (well into college), and because Eastern contributions to science & maths, like Al Jahiz’s, are often (largely) ignored. Again, my experience, but also basing that off of similar complaints heard ‘round the web.

As for the Anaximander addition, it was mostly for my own archival/research purposes — I’m interested in abiogenesis and had no idea about him, prior.

re: { Struggle for Existence }, 800’s CE

Evolution of Regularites?

{ Rule 30 }

&

Stephen Wolfram on
{Computation and the Future of Mathematics }

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A rabbit probably requires a f*ck ton of energy to conjure… it’d be interesting to see if someone could do the math.

I started by finding the E of a 1kg rabbit (using e=mc^2), but I don’t know if that’s the right place to start… & needs more information: Internal energy U, Pressure P and volume V (of rabbit’s container?) — the latter is probably arbitrary.

(via freshphotons)

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Art by { dvdp }

THE PHYSICS OF SPIRALS?

staceythinx:

Digits is a poster series by James Adame designed for a campaign to promote classroom visits by professionals that use math and science in their jobs. 

About the project:

This campaign was created for an initiative of the State of Mass. school board to show kids the importance of studying Math and Science…We wanted to show students that Math and Science isn’t scary- it makes dreams come true and surrounds us in daily life in everything we do.

••••••

I disagree with this idea.

This isn’t any different from anything they’re been doing in school for years, except it looks a little prettier.

I know this much: it wouldn’t have worked for me — I hated math in school. HATED it. Did well enough, but knew I’d almost never have to use it in my job (and I was right — I don’t). Now, years later, I’m actually doing Trig review for fun.

What happened is that I realized the inherent magic of it. By magic, I mean the math of physics, of Alan Turing, of the Golden Ratio, of the ancient Greeks! Whereas, unfortunately, the stuff above just brings the whole process down to the “kid’s level”. Kids, who love magic and superheros and pirates and fantasy and crazy shit… and they’re telling them about dull, commonplace things like bullies and… what’s up there? Wedgies? Ok, the invisibility one is pretty cool. But it isn’t real, unlike the aforementioned examples.

Let’s not be afraid to bring the wonder of the very real, mysterious world we live in into the classroom — Hell, into our daily lives.

(via freshphotons)

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{ Simple mathematical pattern describes shape of neuron ‘jungle’ }

A 2/3 power law: L = (3/4π)^1/3 × V^1/3n^2/3
where n is the number of dendritic sections to make up the tree, L is the total length of these sections, and V is the total volume

{ A scaling law derived from optimal dendritic wiring }

Authors:
    Hermann Cuntza,b,c,1,
    Alexandre Mathya, and
    Michael Häussera

Abstract:
The wide diversity of dendritic trees is one of the most striking features of neural circuits. Here we develop a general quantitative theory relating the total length of dendritic wiring to the number of branch points and synapses. We show that optimal wiring predicts a 2/3 power law between these measures. We demonstrate that the theory is consistent with data from a wide variety of neurons across many different species and helps define the computational compartments in dendritic trees. Our results imply fundamentally distinct design principles for dendritic arbors compared with vascular, bronchial, and botanical trees.

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nightsinneon:

Diagrams from Geometrical psychology, or, The science of representation: an abstract of the theories and diagrams of B. W. Betts (1887) by Louisa S. Cook, which details New Zealander Benjamin Bett’s remarkable attempts to mathematically model the evolution of human consciousness through geometric forms.