Education and Word and Number Hidden Vagueness

These mini-essays help students of any age to re-understand education in a deeper and more connected way.

They look for “circum-spective” intelligence. (Not in the sense of prudential or cautious but in the sense of “around-looking.”)

One of the things to begin to see is that explaining things in schools is misleading “ab initio” (i.e., from the beginning).

Let’s do an example:

In basic algebra, you’re asked: what happens to (x2 – 1)/(x – 1) as x “goes to” (i.e., becomes) 1.

If you look at the numerator (thing on top), x2 is also 1 (since 1 times 1 is 1) and (1 – 1) is zero. The denominator is also (1 – 1) and zero.

Thus you get 0 divided by 0.

You’re then told that’s a no-no and that’s because zeros and infinities lead to all kinds of arithmetic “bad behavior” or singularities.

You’re then supposed to see that x2 – 1 can be re-written as (x – 1)(x + 1) and since “like cancels like,” you cancel the x – 1 is the numerator and denominator and “get rid” of it.

This leaves simply x + 1. So, as x goes to 1, x + 1 goes to 2 and you have a “legitimate” answer and have bypassed the impasse of 0 acting badly (i.e., zero divided by zero).

If you re-understand all this more slowly you’ll see that there are endless potential confusions:

For example: you cannot say that (x2 – 1)/(x – 1) = x + 1 since looking at the two sides of the equal sign shows different expressions which are not equal.

They’re also not really equivalent.

You could say that coming up with x + 1 is a workaround or a “reduced form” or a “downstream rewrite” of (x2 – 1)/(x – 1).

This reminds us of the endless confusions in high school science: if you combine hydrogen gas (H2) with oxygen gas (O2) you don’t get water (H2O). Water is the result of a chemical reaction giving you a compound.

A mixture is not a compound. Chemistry is based on this distinction.

Math and science for that matter, are based on taking a formula or expression (like the one we saw above) and “de-cluttering” it or “shaking loose” a variant form which is not identical and not the same but functionally equivalent in a restricted way.

A lot of students who fail to follow high school or college science sense these and other “language and number” problems of hidden vagueness.
School courses punish students who “muse” to themselves about hidden vagueness. This behavior is pre-defined as “bad woolgathering” but we turn this upside down and claim it is potentially “good woolgathering” and might lead to enchantment which then underlies progress in getting past one’s fear of something like math or science or anything else.

One is surrounded by this layer of reality on all sides, what Wittgenstein calls “philosophy problems which are really language games.”

Think of daily life: you say to someone: “you can count one me.” You mean trust, rely on, depend on, where count on is a “set phrase.” (The origin of the phrase and how it became a set phrase is probably unknowable and lost in the mists of time.)

“You can count on me” does not mean you can stand on me and then count something…one, two, three.

In other words in all kinds of language (English, say, or math as a language) one is constantly “skating over” such logic-and-nuance-and-meaning issues.

The genius Kurt Gödel (Einstein’s walk around buddy at Princeton) saw this in a deep way and said that it’s deeply surprising that languages work at all (spoken, written or mathematical) since the bilateral sharing of these ambiguities would seem deadly to any clarity at all and communication itself would seem a rather unlikely outcome.

You could also say that drama giants of the twentieth century like Pinter, Ionesco and Beckett, intuit these difficulties which then underlie their plays.

All of this together gives you a more “composite” “circum-spective” view of what is really happening in knowledge acquisition.

Essay 89: Physics AI Predicts That Earth Goes Around the Sun

from Nature Briefing:

Hello Nature readers,

Today we learn that a computer Copernicus has rediscovered that Earth orbits the Sun, ponder the size of the proton and see a scientific glassblower at work.

Physicists have designed artificial intelligence that thinks like the astronomer Nicolaus Copernicus by realizing the Sun must be at the center of the Solar System. (NASA/JPL/SPL)

AI ‘Discovers’ That Earth Orbits the Sun [PDF]

A neural network that teaches itself the laws of physics could help to solve some of physics’ deepest questions. But first it has to start with the basics, just like the rest of us. The algorithm has worked out that it should place the Sun at the centre of the Solar System, based on how movements of the Sun and Mars appear from Earth.

The machine-learning system differs from others because it’s not a black that spits out a result based on reasoning that’s almost impossible to unpick. Instead, researchers designed a kind of ‘lobotomizedneural network that is split into two halves and joined by just a handful of connections. That forces the learning half to simplify its findings before handing them over to the half that makes and tests new predictions.

Next FDA Chief Will Face Ongoing Challenges

U.S. President Donald Trump has nominated radiation oncologist Stephen Hahn to lead the Food and Drug Administration (FDA). If the Senate confirms Hahn, who is the chief medical executive of the University of Texas MD Anderson Cancer Center, he’ll be leading the agency at the centre of a national debate over e-cigarettes, prompted by a mysterious vaping-related illness [archived PDF] that has made more than 2,000 people sick. A former FDA chief says Hahn’s biggest challenge will be navigating a regulatory agency under the Trump administration, which has pledged to roll back regulations.

Do We Know How Big a Proton Is?

A long-awaited experimental result has found the proton to be about 5% smaller than the previously accepted value. The finding seems to spell the end of the ‘proton radius puzzle’: the measurements disagreed if you probed the proton with ordinary hydrogen, or with exotic hydrogen built out of muons instead of electrons. But solving the mystery will be bittersweet: some scientists had hoped the difference might have indicated exciting new physics behind how electrons and muons behave.

Contingency Plans for Research After Brexit

The United Kingdom should boost funding for basic research and create an equivalent of the prestigious European Research Council (ERC) if it doesn’t remain part of the European Union’s flagship Horizon Europe research-funding program [archived PDF]. That’s the conclusion of an independent review of how UK science could adapt and collaborate internationally after Brexit — now scheduled for January 31, 2020.

Nature’s 150th anniversary

A Century and a Half of Research and Discovery

This week is a special one for all of us at Nature: it’s 150 years since our first issue, published in November 1869. We’ve been working for well over a year on the delights of our anniversary issue, which you can explore in full online.

10 Extraordinary Nature Papers

A series of in-depth articles from specialists in the relevant fields assesses the importance and lasting impact of 10 key papers from Nature’s archive. Among them, the structure of DNA, the discovery of the hole in the ozone layer above Antarctica, our first meeting with Australopithecus and this year’s Nobel-winning work detecting an exoplanet around a Sun-like star.

A Network of Science

The multidisciplinary scope of Nature is revealed by an analysis of more than 88,000 papers Nature has published since 1900, and their co-citations in other articles. Take a journey through a 3D network of Nature’s archive in an interactive graphic. Or, let us fly you through it in this spectacular 5-minute video.

Then dig deeper into what scientists learnt from analyzing tens of millions of scientific articles for this project.

150 Years of Nature, in Graphics

An analysis of the Nature archive reveals the rise of multi-author papers, the boom in biochemistry and cell biology, and the ebb and flow of physical chemistry since the journal’s first issue in 1869. The evolution in science is mirrored in the top keywords used in titles and abstracts: they were ‘aurora’, ‘Sun’, ‘meteor’, ‘water’ and ‘Earth’ in the 1870s, and ‘cell’, ‘quantum’, ‘DNA’, ‘protein’ and ‘receptor’ in the 2010s.

Evidence in Pursuit of Truth

A century and a half has seen momentous changes in science, and Nature has changed along with it in many ways, says an Editorial in the anniversary edition. But in other respects, Nature now is just the same as it was at the start: it will continue in its mission to stand up for research, serve the global research community and communicate the results of science around the world.

Features & Opinion

Nature covers: from paste-up to Photoshop

Nature creative director Kelly Krause takes you on a tour of the archive to enjoy some of the journal’s most iconic covers, each of which speaks to how science itself has evolved. Plus, she touches on those that didn’t quite hit the mark, such as an occasion of “Photoshop malfeasance” that led to Dolly the sheep sporting the wrong leg.

Podcast: Nature bigwigs spill the tea

In this anniversary edition of BackchatNature editor-in-chief Magdalena Skipper, chief magazine editor Helen Pearson and editorial vice president Ritu Dhand take a look back at how the journal has evolved over 150 years, and discuss the part that Nature can play in today’s society. The panel also pick a few of their favorite research papers that Nature has published, and think about where science might be headed in the next 150 years.

Where I Work

Scientific glassblower Terri Adams uses fire and heavy machinery to hand-craft delicate scientific glass apparatus. “My workbench hosts an array of tools for working with glass, many of which were custom-made for specific jobs,” says Adams. “Each tool reminds me of what I first used it for and makes me consider how I might use it again.” (Leonora Saunders for Nature)

Quote of the Day

“At the very least … we should probably consider no longer naming *new* species after awful humans.”

Scientists should stop naming animals after terrible people — and consider renaming the ones that already are, argues marine conservation biologist and science writer David Shiffman. (Scientific American)

Yesterday was Marie Skłodowska Curie’s birthday, and for the occasion, digital colorist Marina Amaral breathed new life into a photo of Curie in her laboratory

(If you have recommended people before and you want them to count, please ask them to email me with your details and I will make it happen!) Your feedback, as always, is very welcome at

Flora Graham, senior editor, Nature Briefing