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For May 2015, we will be reading Chaos: The Making of a New Science.

Please use this thread to post questions, comments, and reviews, at any time.

I was curious about that one, I will acwuire it on the upcomming month from the Library, and try to participate in any way U see fit Betsy

I wonder if one is allowed to post Qs here, or 'promote' parts of other books, smilar to content discussed here.

Sure, you can post questions or post comments that compare this book with another. I just request that you keep the discussion focused primarily on this book.

Is anybody talking about Chaos? is there another thread somewhere? I am reading it. I was hoping someone could give me an example of what is a non linear equation and how it applies to a system. So far the book is just indicating that it's hard. how can you even have an equation that pertains to a system if its chaotic? one of you geniuses please help me out here.

Hi Nancy. I haven't started the book, so I am not exactly sure what type of system you are referring to.

A linear equation is an equation when plotted will form a line. For example: y = 2x + 1



So a nonlinear equation is one that does not graph into a line (lots & lots of possibilities here-- so this is where I would need to read the book to see what type)

For example here is the graph of a 4th degree polynomial:

And I really do need to start reading -- for some reason this book has always intimidated me and I have been reluctant to pick it up. I'll try to do so before this month is over.

There are some great interactive sites for chaos theory. This is a fun one to start with:

I saw this thread on my feed from Kathy's updates and had to peek since Chaos Theory was one of my favorite math classes in college. Kathy, I hope you don't mind me butting in!

Nancy wrote: "how can you even have an equation that pertains to a system if its chaotic? one of you geniuses please help me out here. ..."

Basically, a really short explanation is that the definition of chaos in the math world is different from chaos in the real world. Mathematical chaos is basically a way of generating a series of numbers that appear to be random but have a formula behind them.

You take a given X value (horizontal axis in the graphs) then plug it into the equation to get your Y value (vertical axis). You then take that Y value and plug it back into that same equation as X and solve for Y again, and keep going.

So, as an example, given that second graph that Kathy posted (y=x^4-5x^3+5x^2+5x-6), let's start with an X of 2.
Solve for Y and you get 0.
Use 0 as X and you get -6.
Use -6 for X and you get 2520.

So, then you have the series 2, 0, -6, 2520 and so on.

You wouldn't know just by looking that 2, 0, -6, 2520 has a formula behind it, but it does.


The reason why they specify non-linear is because linear equations produce a series that doesn't look random. For example, the first equation Kathy gave was: y = 2x + 1
Starting with 0, you'd get 0, 1, 3, 7, 15, 31 etc.


There's a lot more to it, of course, but this is a really simplified explanation. I hope it makes sense!

Obviously, taking a formula and generating seemingly random numbers off of it isn't much use, but taking the seemingly random numbers and using them to derive the formula is.

Good comment, Melanti. You can always "butt in."

Thanks so much Kathy and Melanti. it is really not a hard book to read, but some of the concepts are a little fuzzy for me. Of course, seeing the graphs and concrete examples of the equations is very helpful. And the idea of starting with a certain value and then to keep plugging the result back in to generate subsequent results....I have a feeling that is going to help me.
on page 67 he talks about differential equations and then states, "Most differntial equations cannot be solved at all."
Statements like that make me itch. I see all kinds of scary characters are going to crop up. Irrational and (gasp) imaginary numbers, unsolvable equations...well it is after all a book about Chaos.

Glad I helped out, even if it was just a little bit.

Sorry I can't contribute much to the differential equation statement. My professor focused just on regular non-linear equations - he said it would take more than a one semester undergrad level class to even begin to cover the differential equation angle.

That may be one reason why I liked the class so much! It was conceptually interesting but the math never got much harder than advanced algebra/linear equations.

Well yes, solving differential equations is a trick. And yes most of them cannot be solved. I do have my degree in mathematics (years ago), but never studied chaos, so this book intrigues me.

I'll try to get going a bit more on it --

Back in the 1970s, I thought I had a fairly good educational grounding in mathematics. I got good grades in differential and intregal calculus. Chaos theory was in it's infancy, and therefore not in our cirriculum.

I have to wonder, though, whether formal mathematics can be a true description of the universe we live in without any experimental data.

Here's an obvious example. Albert Einstein used "thought experiments" to arrive at what he thought was a true description of reality. After he arrive at conclusions regarding Relativity Theory, he strived to express his results in terms of mathematics. He needed the help of a couple of professional mathematicians to accomplish that goal.

Today, however, theoretical physicists generally use mathematics as their sole approach to describing physical reality, WITHOUT REGARD TO EXPERIMENT! String theorists are the most obvious practicioners of this approach, but many cosmologists accept the working principal that mathematical findings must necessarily describe reality.

This approach seems to me much like that of religious thought. I assume certain given ideas about reality (god, satan, etc.) and reason from there. My conclusions can't be challenged, if I apply strict logic.

But no experiment has given any reason to believe in gods or satans. Likewise, no experiment has given any reason to believe that mathematic reasoning necessarily describes reality.

It is great that we can describe what we know of the universe mathematically. But, it doesn't necessarily follow that the mathematics that we generate describes the universe.

Just when I was thinking that if I only learned enough math, the workings of the universe would be crystal clear :-(
ok Kathy, can you give me an example of a differential equation that can be solved, and one that can not? where do these equations come from?
I am not too far along in the book because I have a horrible habit of reading more than one book at a time. As soon as I finish with "The Third Chimpanzee" I can commit more to Chaos. End of month is approaching and I don't want to get thrown out of Å·±¦ÓéÀÖ!!

I can't figure out how to do the math symbols in a Å·±¦ÓéÀÖ box, but here is a link to a paper I found of examples of some solvable differential equations for a university math class.

Nancy,
I don’t know where you stand in mathematics, but if you are not familiar with the concepts of Derivative and Integral, to read the article that Kathy provided would be very impractical.

The crucial difference between algebraic equations and differential equations is rather simple though. When we solve algebraic equations (we are familiar with them since middle school), the unknown that we are looking for is a NUMBER(s). The unknown for differential equations is a FUNCTION(s).

I hope you are familiar with the concept of a function. At any rate, it’s easy to get the idea just by googling it up.
Roughly, a function is a law how two different qualities depend on each other. If some things are connected in this world, the way they are connected we express in terms of functions.

And you were right in your earlier sentiment; it is hard to argue against the statement that the Language of Nature is Mathematics. That is, those laws how things connected to each other � the functions � are best presented when written mathematically.

It’s another story that we start with observations and experiments anyway, and only then, are trying to build corresponding mathematical models based on the former. That’s how mathematical science has been working in the past, anyway.

People,

There seems to be some uncertainty about what constitutes a differential equation. Most science uses differential equations: the reason being that for example forces, accelerations etc. are related by F = ma (Newton!) and the acceleration, a, is the second derivative of position.

A simple (first order) differential equation would then be dx/dt = v (constant velocity in a straight line). This would be integrated (or solved) to give simply x = vt + A. A is the unknown constant that can only be found with more information, i.e. where the object starts.

Physics is full of these equations and more complex ones as well.

When I was in high school in the UK I learnt how to solve DEs for my maths A level. There are all sorts of interesting things that you come across such as
ad2y/dx2 + bdy/dx + cy = 0

Such equations can be solved using some special techniques similar to the solution of the quadratic equation ax2 + bx + c = 0. Being nonlinear (it has a quadratic term) there is the prospect of chaotic solutions.

Chaos is well known to atmospheric sciences (my expertise) which prevents detailed weather forecasts much beyond 7 days. After this time small perturbations to the initial state magnify and make the forecast unusable. It is many years since I read James Gleick's book on chaos. I don't recall it containing any mathematics of any consequence, so those of you without maths or science degrees won't have any problems.

[sorry about the notation: this word processor doesn't seem to accept HTML]

I think parts of this discussion are jumping around being too high-level and, perhaps, too low. I think there is also a bit of being "too generous" when it comes to the power of mathematics as it intersects with the real world. First let's do some mathematics. I assume Nancy has a basic high school education covering algebra and functions.

First: Calculus.

Part 1: Differential Calculus
Do you remember slope? That's the m-piece of y = mx + b. It had that formula with distance of y over distance of x?

y = mx + b, is a linear equation and that slope formula works for the linear case. Slope tells us something specific about the equation, namely the rate of change (or, as I like to say, how fast the function is moving). Figuring out the slope of a non-linear equation is far more complex, and this is what "differential calculus" is for.

When you see people above using notation like dy/dx, this is a fancy way of saying "m". There's this whole process developed by Newton and Leibniz on doing this. The "dy" and "dx" are called differentials. A very low level way to see this... take our linear equation above.

y = mx + b

If I "differentiate" this I get:

dy/dx = m

Which you expect, since I said above dy/dx was the slope. (don't worry about why the other terms go away, I'm trying to get to differential equations).

Part 2: Integral Calculus

Let's say, all you know is dy/dx = m, but we want to, somehow, get back to the original function that gave us this slope. Well the great minds in the late 1600's figured out something called "integral calculus", which is, essentially, the inverse operation of differential calculus... because what's the use of only going one way? So, when I'm faced with:

dy/dx = m

Then we integrate it:

y = mx + b

We are back where we started. Integral calculus does all kinds of special stuff, like give us areas if we have bounded regions. But for our purposes, we just need to know that it gives us a variable plus a constant.

Differential Equations:

Differential equations mix all this stuff up. Another way of saying dy/dx is to use y' instead (read y-prime). So, we put all this together like:

y' + y = 0

This would read "the rate of change of a function PLUS the function itself!" Essentially we want to find the function where you add these two things together so that when added they equal zero.

A lot of the solutions to basic differential equations revolve around the equation y = e^x, so in the case above the solution function is y = ce^(-x). I realize, that's not going to be clear, but I feel it would take to long to explain in this format.

Differential Equations give us strange situations to think about, they're like strange puzzles. Chaotic systems are even harder to figure out.

I hope all that made sense...

Philosophy of Mathematics:

I have huge issues with this. I don't want to get into a giant debate about this, since it doesn't have much to do with the book. But I've noticed people (not just in this discussion) seem to think mathematics is far more powerful than it actually is. Often times there is confusion about "how real" mathematics is and what that implies in the real world.

Here's the first thing... math isn't "real", like a desk or chair is real. Math is a total abstraction. Mathematics is never bound by our reality. Mathematics isn't a science. There are no "experiments" or "testing labs" or anything like that. Mathematics, for lack of a better term, is a head game in the purest sense.

However, it turns out that some things in mathematics are quite useful for studying reality. For example, if you want to look at cars, they are on wheels. Wheels are circular. Well, the geometry of circles is very well explored in mathematics, so we can use a lot of that geometry to leverage against how we want to study wheels in the real world. The real major thing to walk away with here is that there are no true circles in the real world, not in the mathematical sense. Reality will always have an atom slightly out of place etc, so that you never have as perfect a circle as we can think about in mathematics. Mathematics is an idealized approximation for what we may encounter in the real world. Hardy referred to this difference as "mathematical reality".

Sure, mathematics is used in Physics, but when it comes to the sciences, experiments trump EVERYTHING. You can develop the most beautiful, elegant theory about the universe, but if it doesn't hold up to experiment, you're done, it's false. I don't care if your math made great leaps and bounds in making more elegant mathematics, it's not a good model for reality. This is because math is not bound by our reality. However, science is bound to it.

Now, when you get into theoretical work you use math to try and predict things. These are merely guidelines, if you will, to help point us in a direction that might give us answers. Take the Higgs Boson, for example. Higgs did all kinds of math years ago about this thing the mathematics said could exist. They had no idea if this thing existed until they developed an experiment to find it! Things like String Theory have all kinds of problems, because there have been, essentially, no experiments to validate their truth. Those people are just doing math. It's beautiful math... but is it real, in the sense your chair is real? It might not be... and I think people get a little lost when trying to figure out the two.

Sorry this is so long... kudos to anyone who bothered to read it!

Very good, Adam. Thank you. I took calculus, but it was 30+ years ago. Great tutorial.

I am not a mathematician, although I have a mathematical background from my remote past; yet my personal philosophical attitude towards math and reality is identical to that of Adam’s (with the precision to the gist he provided above; of course, this blog is not the right place to go deeper on the subject).

I only wanted to mention here that that philosophical attitude is just an opinion, a fashion of the age so to speak, an opinion on Reality, Mathematics, Science, and relationship between them. (If the word ‘opinion� sounds too low, the word worldview stands for the same here). This opinion is beyond Mathematics and Science themselves and some will sure disagree with it. Platonists for one will. (Platonism is still popular among some mathematicians even today, btw). For some, this ‘Phenomenalogical� Reality (what Adam, me and the kind call The Reality) is only a superficial part of Reality, and for them, a number, as for instance, may be more real than a chair.

What I’ve said may sound trivial, but I thought it would not hurt to mention it. Because, we all here have so diverse backgrounds that I guess it’s easy for many to lose sight which part is science, which is pure math, and what is just somebody’s point of view. I think Adam was driving at something similar. (Btw, Adam did a good job of refreshing the concepts of derivative, integral, and differential equation for those who studied them say 30+ years ago. As for those who never studied those things... I don’t know� Maybe it was helpful for some naïve but sharp brains).

We still appreciate and study the science and the math ancient Greeks developed. As for their worldview (in particular, the relationship between math and reality, gods?)� I guess it may seem bizarre to many of us. I believe, say in a thousand years, our achievements in math and science will still be valued. But will our posterity appreciate our current fashionable philosophy? And how about the worldview of them descendants if presented to us? I think a worldview is a very historical thing and we may not be able to comprehend it if even somebody dropped it on us with a time machine. Can we tell which part of our current fashionable worldview (including the relationship between math and science) is not a function of historicity?

Adam wrote: "I think parts of this discussion are jumping around being too high-level and, perhaps, too low. I think there is also a bit of being "too generous" when it comes to the power of mathematics as it i..."

Thanks Adam ... it is helpful, and I understood it up to the point about ce. Which I assume you are talking about Euler's number and I'm pretty fuzzy on that.
I realize the book is really not all that mathematically challenging in the sense that it does not get into complicated equations. But it does allude to it. The part about fractals, which are so beautiful, does make one want to wrap one's mind around the math, which I believe involves imaginary numbers as well as fractional dimensions.
And that is as far as I've gotten so far.

Kathy wrote: "I can't figure out how to do the math symbols in a Å·±¦ÓéÀÖ box, but here is a link to a paper I found of examples of some solvable differential equations for a university math class. ..."

It won't come up!

Bigollo wrote: "Nancy,
I don’t know where you stand in mathematics, but if you are not familiar with the concepts of Derivative and Integral, to read the article that Kathy provided would be very impractical.

The..."

I am vaguely recalling my old calculus lessons. Which I mostly forgot since I never have occasion to use them!

Nancy wrote: "Kathy wrote: "I can't figure out how to do the math symbols in a Å·±¦ÓéÀÖ box, but here is a link to a paper I found of examples of some solvable differential equations for a university math class..."

Hmm. The link works for me.

The link works for me too. It pulls up a pdf document. Maybe you need to wait a little bit for it to download the document.

just says Not Found. U RL not found on this server. tried on the android and the computer.

Very cool stuff. The Koch Curve: infinitely long perimeter contained within a finite area. The Mandelbrot sets are gorgeous.

Very nice zoom into Mandelbrot set:

(makes me dizzy)

I finished the book, but not sure what to think of it. While the subject matter is very interesting, my motivation for reading declined: the science part too was hand-wavy, the short biographical glimpses into the scientists' motivations was not enough to get a feel why they were doing it.

I wondered what kind of progress or break-throughs in this field were made in the 30 years since this book came out. Anyone has a good resource on that?

Bigollo wrote: "Nancy,
I don’t know where you stand in mathematics, but if you are not familiar with the concepts of Derivative and Integral, to read the article that Kathy provided would be very impractical.

The..."
Thank you Bigollo. I got a new computer and finally was able to view the link. You're right ... I didn't get very far. It turns out that I've forgotten how to integrate. It looks like a great link but it's a bit too advanced for me.
It turns out in the book he kind of glosses over the math parts anyway. Which in my case is probably good, except that it leaves me with more questions than answers.

amalia wrote: "Very nice zoom into Mandelbrot set:

(makes me dizzy)

I finished the book, but not sure what to think of it. While the subject matter is very interesting, my..."
That is the coolest video I have ever seen. I want to see it on IMAX. Very trippy.
I think I agree with you about the book.

Having a problem understanding Feigenbaum's calculations ... he keeps coming up with the number 4.669. But what is this number and where is he getting it? It evidently has to do with period doublings in populations. And it's a ratio. But I just don't comprehend it.

Hello? Is anyone actually reading this book?

Nancy wrote: "Hello? Is anyone actually reading this book?"

Hi there. I've been out of town and my copy is a hardback so I didn't take it with me. Hope to get to reading it more & finish it now that I am back. The next two months group reads don't interest me, so I will have no distractions from those discussions.

Nancy wrote: "Hello? Is anyone actually reading this book?"

I had all intentions to read the book, checked it out from the library, started to read, but bogged down due to several other reads at the same time. Somehow this book did not win the first priority with me. I'm still hoping to read it in June, switching to July group read in July.

Good! well, I am looking forward to your impressions. Kathy, you're forgiven...every extra ounce on a trip becomes a burden :-)

So far we seem to have a history of the people and science involved in the discovery of Chaos. I can see where if you don't have a background in math or physics that the author seems to be tempting you with the mention of the different types of equations and theories.

Yes, and I find this frustrating with many science books. I think most people, in the US at least, sort of get gypped on education in math. Or maybe we learn it but don't use it for anything, then when it comes time to apply it to something, we are lost.
Anyway, after reading this book, I had to redefine "chaos" in my mind. I had always equated it with entrophy and randomness (think a puppy left alone in dorm room for a few hours). This is not what the term means at all, apparently.

I found the same thing Nancy - chaos and randomness being different things that is. As far as the math goes, I was surprised that many of the equations were simple, with the seemingly straightforward issue of feedback bringing about the chaotic behaviour...

It took some faith for those on the edge of Chaos study to go out on a limb and ignore the naysayers at first. We meet some extraordinary people & scientists in these chapters.

True. I read an article today about why science needs silos in The Conversation, asserting that isolated perseverance is the key to discovering something new, since joining the herd would mean missing these results. (It also indicates collaboration is needed - both/and situation overall).

Interesting part I read today on how the Soviet Union where the scientists/mathematics worked together had actually come upon Chaos theory earlier than the Western World where science is broken into different disciplines that rarely talk to each other, let alone work with each other.

Yes, I hadn't realized that there is such a gulf between math people and physics people. From the outside, they seem so related.
I too am fascinated by the feedback loops that generate chaotic results. Not what I would expect at all.

I'm still reading -- slowly. Give me another month and I should finish. Interesting, but not fascinating, can't put it down.

I hear you.
"Bonk" should go faster!

I'd like to read this book...I'm a biology student and I've just taken an exam about Math Models for demography and evolution. It was also about deterministic chaos models: Robert May's logistic map is amazing!

Yes Kikyosan, do read it. it's not really hard to read, I just had a lot of questions regarding the math, which was not gone into in detail and a bit beyond me, anyway. With what you've been studying it will probably be clear to you.

Yes, I can manage math language if it is about derivative, integrative and differential calculus, systems and stuff like these. And I've just discovered this book will be sold by a local newspaper in some weeks. What a coincidence! So happy. I will be able to post some comments after summer.

Great! I look forward to reading your impression of it. I was enchanted by the pictures of fractals and Mandelbrot sets!

Nancy wrote: "Great! I look forward to reading your impression of it. I was enchanted by the pictures of fractals and Mandelbrot sets!"

They are beautiful!