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Claude St-Louis
Worked at Jeffery Hale-Saint Brigid's
Lives in Québec, QC Canada
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Claude St-Louis

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thank you +Refurio Anachro !
What do mathematicians want from their #notations ? Why are they so hard to learn? And why is it cool if you do? Mathematical writing is a couple of millenia(!) old. Almost all of that time it relied on hand drawn formula.

In this post I will be arguing for hand drawn notations. It has been inspired by +Xah Lee​​​​​​​​​​​​'s critique of TeX for mathematical publishing, a
very much different opinion:

I'm linking the above 2-year old post first, because it has some discussion, the latest is this one:

(1) - First and before all, the notation should reflect what is going on. To the point that everything that can be said about a structure, can also be written down. Uhm, that sounds weird... But it's not, let my try again:

What i'm demanding here is that the language has to be powerful enough to capture what you intend to say with it.

Say, you have an idea, followed by cognition, a little enlightenment so to speak, and you now want to share it with your friends. So you make up a story to convice them. Better yet, start writing it down so you can be sure it doesn't change after you fiddled it to your satisfaction.

Your story might contain nouns for all objects of interests, verbs that allow to relate them in each of the important ways, or be structured in a very different way. If your language is one of the better ones, it'll allow others to reason in interesting directions you haven't thought of before.

(2) - All notation should be concise. People working with results (to apply them, do research, or teach them) are expected to fill loads of blank space with scribbles in that notation.

I find it interesting that that often makes reading math books hard. Sure, finding answers often took one or many researchers's life-long dedication, but learning notation is no piece of cake either! Some areas easily require you to invest a year of studying to get a decent insight from scratch.

Still, it is much more economic to read those pesky books than to find out all by yourself! You know, there are lots of beginner's introductions for almost any topic, but they are fashionable, ephemeral after a while, whereas the original writings often stay popular for decades, centuries, or even longer!

Maybe mathematicians like TeX because it feels more like drawing pictures. And that one guy, Donald, felt that he should be holding the pen, because he couldn't trust people who'd typeset for him to make the best decisions. And that that'd be worth the effort!

(3) - Make it look similar to everyone else's. You do want to lean on the plenty of natural language or existing math notation to communicate with your readers. Ideally one would use the same symbols, but if there is something to describe that could be really new, that might not be the best choice:

(4) - Make it subtely differ from everyone else's. Yes, I know, I just stated the opposite opinion. It is just impossible to tell beforehand which other notation will appear next to our favorite kind of hieroglyphs:

For one, you don't know about relations to other areas that haven't been discovered yet, and in mathematical reasoning that tends to be the case more often than not!

And secondly, still, people will want to try and look if they can find such connections, even if that turns out not to be the case.

All of these reasons only make sense if we understand written mathematics as a form of communication between human beings. To make these ideas do something in the physical world, for example, tell a computer to perform a computation, is the job of engineers. Like typesetters they are better at their job than mathematicians... Oh wait, I'm contradicting myself again!

(5) - Nowadays we have another fascinating development: Proof assistants. Simple programming languages, like S R, Mathlab octave, or Mathematica Sage, that do computations (even symbolic ones, reading and printing equations) don't count here!

Well, mathematicians do compute, but their top notch results are theorems. Sometimes that means presenting the outcome of large computations, but not too many researchers end up publishing tables with numbers (or symbols) in mathematics.

So to them, a calculator isn't the right tool to write up their kind of results, and to communicate those to others. Or even to make significant progress, because calculators can only yield the easy fruits. The ones hard to get require a creative thinker to find a new way to get where no one has gone before.

But a proof assistant just might provide a significant edge over doing mathematics in the traditional way. And that would get many people writing maths in a format a computer can handle semantically.

What mathematics is, and how it is practiced are subject to change. I bet there'll always be a place for people doing geometry with a stick on a beach. But there will also be new fields born out of traditional ones.

And sometimes the medium changes, too. I'd guess that TeX's availability as open source has also been an important step in the ongoing struggle to free math papers from publishers, who lock them up, and let you have a look only if you pay them.

For a piece about change in mathematics, have a look at John Von Neumann's short "The Mathematician":

Thanks to +Stefan Huber​​​​​​​​​​​​ for posting about it here:

The picture I found because +Richard Green​​​​​​​​​​​​ used it here:

"Continued fractions"

It's by Lucas Vieira Barbosa, read Richard's post to find out more about him.

Still reading? Are you ready for more on complex numbers? Look out for my upcoming post!

4 comments on original post
John Gambini's profile photoRefurio Anachro's profile photo
+John Gambini​​, you can read off equations for phi, since all subtowers are equal to phi:

phi = 1 + 1/phi

By the way, what about representing continued fractions recursively?
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Look. Think. Share.
Thank you for caring !
Got this as a reshare from +George Rodenbaugh and originally from +Thomas Power. The only reason I made an original post is because it wasn't a public share. This is something that should definitely be shared publicly. Thanks +Thomas Power and +George Rodenbaugh.
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Got this from avaaz and I want to share it with you:

Dear friends,

Quietly, globally, billions of bees are dying, threatening our crops and food. But if Bayer stops selling one group of pesticides, we could save bees from extinction.

Four European countries have begun banning these poisons, and some bee populations are already recovering. But Bayer, the largest producer of neonicotinoids, has lobbied hard to keep them on the market. Now, massive global pressure from Avaaz and others has forced them to consider the facts -- and in 24 hours, Bayer shareholders will vote on a motion that could stop these toxic chemicals. Let’s all act now and shame the shareholders to stop killing bees.

The pressure is working, and this is our best chance to save the bees. Sign the urgent petition and send this to everyone -- let's reach half a million signers and deliver it directly to shareholders tomorrow in Germany!

Bees don't just make honey, they are vital to life on earth, every year pollinating 90% of plants and crops -- with an estimated $40bn value, over one-third of the food supply in many countries. Without immediate action to save bees, many of our favourite fruits, vegetables, and nuts could vanish from our shelves.

Recent years have seen a steep and disturbing global decline in bee populations -- some bee species are already extinct and some US species are at just 4% of their previous numbers. Scientists have been scrambling for answers. Some studies claim the decline may be due to a combination of factors including disease, habitat loss and toxic chemicals. But increasingly, independent research has produced strong evidence blaming neonicotinoid pesticides. France, Italy, Slovenia and even Germany, where the main manufacturer Bayer is based, have banned one of these bee killers. But, Bayer continues to export its poison across the world.

This issue is now coming to the boil as major new studies have confirmed the scale of this problem. If we can get Bayer shareholders to act, we could shut down once and for all Bayer’s influence on policy-makers and scientists. The real experts -- the beekeepers and farmers -- want these deadly pesticides prohibited until and unless we have solid, independent studies that show they are safe. Let's support them now. Sign the urgent petition to Bayer shareholders now, then forward this email:

We can no longer leave our delicate food chain in the hands of research run by the chemical companies and the regulators that are in their pockets. Banning this pesticide will move us closer to a world safe for ourselves and the other species we care about and depend on.

With hope,

Alice, Antonia, Mia, Luis, Ricken, Stephanie, Pascal, Iain, Ari and the whole Avaaz team


Studies fault Bayer in bee die-off (Christian Science Monitor)

2 Studies Point to Common Pesticide as a Culprit in Declining Bee Colonies (New York Times)

Leaked document shows EPA allowed bee-toxic pesticide despite own scientists’ red flags (Grist)

Bayer’s Annual Stockholder’s Meeting (ASM website)
Pesticides are killing bees and threatening our food supply. In 24 hours, shareholders at the biggest chemical producer, Bayer, could vote to stop their toxic production. Massive public pressure has f...
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I'm not a blues amateur but this got my attention.
This man has utterly captivated me. You play some mean blues white boy.
Adam “Godlypus” Boenig's profile photo
This cover is amazing, no matter what your music of preference.
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Claude St-Louis

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Fascinating! I need to learn more about this...
Links of links, and higher structures

A Brunnian link is a collection of linked loops with the property that cutting any one of the loops frees all the others. This picture, which comes from a paper by Nils A. Baas, shows a second order Brunnian link. This consists of six linked loops in a circle, but each of the linked loops is itself a Brunnian link of four linked loops, coloured purple, orange, beige and cyan.

One of the reasons that Baas is interested in such linked structures is because of their connections with physics. The Efimov effect in quantum mechanics refers to a bound state of three bosons in which the attraction between any two bosons is too weak to form a pair. In other words, removing any one of the particles results in the other two falling apart, like the links in the Borromean rings, a famous example of a Brunnian link. The hope is that the study of more general Brunnian links will predict generalized versions of the Efimov effect.

Baas expands on these themes in the recent paper On Higher Structures ( which discusses the concept of hyperstructures in mathematics and their possible applications. The paper gives an intuitive outline of the notion of “hyperstructure”; the rigorous definition, which appears in some of Baas's earlier works, is given in terms of category theory. 

The basic idea is that a hyperstructure consists of a set of bonds at various levels: 0, 1, 2 and so on. The bonds at level 0 are objects with properties. A 1-bond binds together some of the properties of the level 0 objects. In turn, the 1-bonds have properties of their own, which are bound together by 2-bonds, and so on.

The paper elaborates the underlying philosophy as follows: 
Much of the intuition around hyperstructures comes from thinking of them as evolutionary structures. They are designed and defined in the same way as evolution works: collections interact forming new bonds of collections with new properties, these being selected for further interactions forming the next level of bonds, etc. In a sense, nature or the environment acts as a kind of observer (or “observation sheaf”). The success of evolutionary structures makes their theoretical counterparts — hyperstructures — a useful design model.

The paper explains in detail how hyperstructures can capture the essence of organized structures, and discusses how the theory might be applied to understanding biological systems (such as in genomics) and democratic structures, as well as to designing and synthesising new molecules and materials.

Relevant links 

The picture comes from the 2010 paper New States of Matter Suggested by New Topological Structures ( by Nils A. Baas. In the recent paper, Baas acknowledges A. Stacey and M. Thaule for helping with the diagrams.

Here's a 2010 popular article about Baas's paper of that year:

Brunnian links are named after the German mathematician Hermann Brunn (1862–1939). The Borromean rings ( are special case of Brunnian links (

Efimov states were predicted by Russian physicist V.N. Efimov in 1970:

Baas's recent paper appears in the General Mathematics section of the arXiv. This is remarkable because this section is typically used as a dumping ground for papers that the arXiv moderators don't like, such as this recent paper about the Riemann hypothesis whose authors don't understand what they're doing:

(Maybe I should have had a Relevant links of links section, with a link to a list of links?)

#mathematics #scienceeveryday #spnetwork arXiv:1509.00403
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I'm discovering this extraordinary poet and this is exactly what I needed right now. Hope you will like it too.
Thanks to +Michael Baker for the post.
Acquainted with the Night
by Robert Frost

I have been one acquainted with the night.
I have walked out in rain -- and back in rain.
I have outwalked the furthest city light.

I have looked down the saddest city lane.
I have passed by the watchman on his beat
And dropped my eyes, unwilling to explain.

I have stood still and stopped the sound of feet
When far away an interrupted cry
Came over houses from another street,

But not to call me back or say good-bye;
And further still at an unearthly height,
O luminary clock against the sky

Proclaimed the time was neither wrong nor right.
I have been one acquainted with the night.
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Good to know. Thank you +Olivia Faith Ladene Esenbock !
This is rather enlightening. The illusion of choice. Via +reddit
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One more proof that Winston Churchill was right: sport is bad for your health.
When this photograph was first published in Africa Geographic, BBC Wildlife, later in Paris Match and the Daily Mail, it resulted in a flurry of e-mails, phone calls and letters from around the world asking if the image was a fake. The image became the most talked about of shark photograph ever. The photograph is real, no photoshop, no digital manipulation, no nothing, in fact it was shot on slide film Fuji Provia 100 using a Nikon F5 Camera and 17-35 mm lens
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Claude St-Louis

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Québec, QC Canada
  • Jeffery Hale-Saint Brigid's
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Claude St-Louis's +1's are the things they like, agree with, or want to recommend.

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