George, you're right, I could have said discontinuity, but I think the term non-linearity expresses a slightly different sort of phenomenon. Not complete discontinuity, but a significant inflection in the curve that is not predictable from the previous model but can be induced once the new model is properly understood and integrated.

It seems to me to be an important epistemiological phenomenon that isn't properly understood as yet. Chaos theory and dynamical complexity are trying hard, I think, but I don't know enough to be able to comment meaningfully on those subjects.

http://en.wikipedia.org/wiki/Complex_system

The other issue is the role of rationalism. I have long been an empiricist but of late I've been moved to contemplate a more rationalist approach. Modern scientific practice has little room for rationalism that is not empirically based, but the whole foundation of science is such rationalism. Abduction in the service of rationalism rather than empiricism is what creates new paradigms, while grounded theory demands that any such rationalist flights of fancy must be able to be fitted to empirical observations. If they are not they are labelled "philosophy" or perhaps "mysticism" depending on how the labeller feels about them. Emotion is important, even in empiricists...

Anyway, I'm still grasping at this. It's not an easy topic.

Antiseptic,

>>Not complete discontinuity, but a significant inflection in the curve that is not predictable from the previous model but can be induced once the new model is properly understood and integrated.<<

Mathematically, what you are describing is a function (of time) that at a given point is not discontinuous, only its derivative, the tangent direction, is; the graph is broken there, forming something like a cusp, so that one cannot predict the “future behavior” of the function from its “past”, not even approximately, as it is the case at points where there is no discontinuity of the tangent. In mathematics this has nothing to do with non-linearity.

This is first year calculus, in distinction to chaos theory and dynamical systems (dealing with non-linear differential equations) which are very abstract but clearly defined and understood parts of pure mathematics. They can be, and are, used to model many situations in physics, and perhaps in some sense can also be used to represent some epistemological procedures, as you rightly indicate.

However, I am suspicious of explanations that mix mathematics with the field where it is to be applied (as they do in the wikipedia link you provided), although this is how scientists, including physicists, interested in representations of physical reality (and not in mathematical structures as such ) work. A trivial example of what I have in mind: a child has to learn about numbers through counting oranges and bunnies, but once it got the idea of a number, he/she does not have to refer to oranges to learn about arithmetics; actually the mixing in of oranges when learning about e.g. prime numbers would be irritating.

In my dictionary, empiricism is the theory that all knowledge is derived from sense-experience, and rationalism is the theory that reason rather than experience is the foundation of certainty in knowledge. In this sense rationalism with its “rather than” seems more embracing, open to qualifications, than empiricism with its rigid “all”. Of course, it all depends on what one understands by “certainty in knowledge”, “experience”, “reason”.

Thanks George, I'm constantly reminded that my maths is not what it used to was (and that was pretty average, at best). However, I was simplifying a little, because it seems to me that chaos and complexity are critical to a proper explanation of the sort of abductive cognitive leaps that I'm describing. Perhaps from one chaotic attractor to another? As I said, I'm still grasping at this, I can't claim to have any kind of developed hypothesis, let alone understanding.

Just remembered this article I read a while back:

http://www.australiancatholics.com.au/content/view/243/

Brother Guy Consolmagno, astronomer at the Vatican Observatory

thanks fort the link poiroi..<<He adds that both..[science/relig]..respond to ‘hidden-ness’ or mystery in the same way:

‘The nature of faith is to keep growing.
The nature of science is to not be complete.’

According to Brother Guy,..the opposite of both science and faith is actually certainty...‘No scientist is certain’, he remarks.

‘If we were certain,..there would be no more reason to do science.’..>>

IT Also reconfirms..the close link to study of god
via study of his creation

i couldnt assist with input..
re antiseptic/george conversation..

so back tracked..
to antiseptics quote..<<..Creation theology is a good example..of the non-linearity I referred to:.."In the beginning,..God created the heavens and the Earth"...

<<There's no linear progression ..rom nothing to something,..just a sudden emergence..of something unpredictable from chaos.>>

first thought
chaos..not applicable...in theological sense
as we can presume..'the face of the deep;'..darkness..of the heavens was in direct linear progression..from its creation..;to..'let there be light..

that..*when created/revealed ..'was waste and void'..
which we can presume god was able to see..*sans the light..[logical deduction/linear progression]....

the deep/liquid earth began rotating/[night day]
centrifugal force raised the Gondwanaland plate/..*
firmament

-divided the atmospheric gaseous/waters...
from the liquefied..[no.ice at that stage]

micro flora/fauna...man..etc
all reasonable deductions..by lineal deduction?

anyhow
love the fresh inputs
and enjoying the helpfull/kindness all are showing

Poirot,

I am joining OUG in thanking you for the link. I did not know of the article (of course, I knew of Guy Cosolmagno), not even of the website.