"The boiling frog story is generally offered as a metaphor cautioning people to be aware of even gradual change lest they suffer eventual undesirable consequences. It may be invoked in support of a slippery slope argument as a caution against creeping normality. It is also used in business to reinforce that change needs to be gradual to be accepted. Oppositely, the expression "boiling frog syndrome" is sometimes used as shorthand to invoke the pitfalls of standing pat.
The story goes like this:
"Place a frog in a pot of boiling water and it will jump out to try to escape. However, if you place the frog in a pot of room temperature water, it will sit there contently. You can then gradually increase the temperature and the frog will stay in the water and boil to death."
What's interesting about this story is that it isn't true. If you drop a frog in boiling water, it will not jump out, it will die. If you drop a frog in a room temperature pot, it will jump out because frogs don't hold still. If it is comfortably sitting in the water and you raise the temperature, it will try to escape.
Why is this myth so prevalent? It comes from an experiment done by Friedrich Goltz while searching for the location of the soul. He found that the frogs would indeed stay in the water and gradually boil, IF THEIR BRAINS HAD BEEN REMOVED, but intact frogs would try to escape at 25 C/77 F.
So what really killed the frogs?
Not the boiling water. Not their own indifference to their condition. It was the person above them who limited their capacity to think for themselves.
So what's the takeaway?
The only frogs who were in danger of boiling were those who were not permitted to use their brains. Those who were able to exercise their mental capacity to observe their condition escaped, and prevented their own demise. The hot water was indeed dangerous to the frogs, but the way they were able to escape the danger was by thinking and acting appropriately.
There indeed may be dangers in the ever-changing world around us, but the greater danger we face is when we stop thinking about our own condition. We need to examine things critically, and if anyone in an authority position tries to limit our capacity to do so, we need to jump out of the pot."
In this post I will be arguing for hand drawn notations. It has been inspired by '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
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 for posting about it here:
The picture I found because used it here:
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!
Here's my layman's understanding.
Fill a tube with glass frit, run salty water through it between membranes and induce a little electric current... this causes clumping of salty sections and less salty sections throughout the running water as it goes through the tube. not all that useful as it's kinda dispersed. and hard to get at.
Then they figured out something awesome. If you hit that tube with a shockwave, all the heavy clumps (the salty ones) wind up on one side, and the less heavy (clean water) winds up on the other, easy to split off into two new pipes.
Much cheaper than boiling all the water, and collecting condensation. Much easier than pushing the water through membranes that tend to clog a lot, and need constant repair/replacement. Much faster as you do it with running water, not batches in vats, and it looks like it can probably scale.
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 (http://arxiv.org/abs/1509.00403) 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.
The picture comes from the 2010 paper New States of Matter Suggested by New Topological Structures (http://arxiv.org/abs/1012.2698) 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: http://www.technologyreview.com/view/422055/topologist-predicts-new-form-of-matter/
Brunnian links are named after the German mathematician Hermann Brunn (1862–1939). The Borromean rings (https://en.wikipedia.org/wiki/Borromean_rings) are special case of Brunnian links (https://en.wikipedia.org/wiki/Brunnian_link)
Efimov states were predicted by Russian physicist V.N. Efimov in 1970: https://en.wikipedia.org/wiki/Efimov_state
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: http://arxiv.org/abs/1509.01554
(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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