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katastrofa
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 25th, 2018, 12:05 pm

Who's Vico? Giambattista Vico - the philosopher and social scientist?
 
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Cuchulainn
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 25th, 2018, 12:21 pm

Who's Vico? Giambattista Vico - the philosopher and social scientist?
“Giambattista Vico was a practical roundheaded Neapolitan. It pleases Croce to consider him as a mystic, essentially speculative, “disdegnoso dell’ empirismo.” It is a surprising interpretation, seeing that more than three-fifths of his Scienza Nuova is concerned with empirical investigation.
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katastrofa
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 25th, 2018, 1:00 pm

The scholars of Age of Enlightenment sound more rigorous to me than the most of science nowadays.
 
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Cuchulainn
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 30th, 2018, 5:29 am

A great example of how to motivate Cauchy convergence is the following:

Consider using Differential Evolution (DE) to minimize a function in n space, Choose an initial population of NP individuals (vectors) all of which ultimately converge to the unknown optimum. Convergence is reached when they all converge to each other.
The distance between the population members determines convergence. There are several ways to measure, e.g. maximum distance between trial vectors and best vector or difference between best and worst objective values.
Each generation is one iteration in the evolution scheme.
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Traden4Alpha
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 30th, 2018, 11:01 am

A great example of how to motivate Cauchy convergence is the following:

Consider using Differential Evolution (DE) to minimize a function in n space, Choose an initial population of NP individuals (vectors) all of which ultimately converge to the unknown optimum. Convergence is reached when they all converge to each other.
The distance between the population members determines convergence. There are several ways to measure, e.g. maximum distance between trial vectors and best vector or difference between best and worst objective values.
Each generation is one iteration in the evolution scheme.
And what if convergence is suboptimal? What if the superior solution is to randomly flip between the multiple optima?

For example, some tribes of native hunters of caribou used a randomizer to determine where to hunt so the prey could never learn which places to avoid.
 
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Cuchulainn
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 30th, 2018, 12:00 pm

And what if convergence is suboptimal? What if the superior solution is to randomly flip between the multiple optima?

This is a mathematical issue (necessary and sufficient conditions) to be addressed. This has been addressed in the literature. In general, DE converges to the global optimum in contrast to gradient methods.

Think of a population as a vector of 'haploid' bitsets, essentially.

I'm not interested in biological evolution here. Don't want 2 simultaneous discussions. It adds no precision.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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katastrofa
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 30th, 2018, 5:55 pm

A great example of how to motivate Cauchy convergence is the following:

Consider using Differential Evolution (DE) to minimize a function in n space, Choose an initial population of NP individuals (vectors) all of which ultimately converge to the unknown optimum. Convergence is reached when they all converge to each other.
The distance between the population members determines convergence. There are several ways to measure, e.g. maximum distance between trial vectors and best vector or difference between best and worst objective values.
Each generation is one iteration in the evolution scheme.
In what way does this Gedankenexperiment motivate Cauchy convergence?
 
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Traden4Alpha
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 30th, 2018, 6:50 pm

And what if convergence is suboptimal? What if the superior solution is to randomly flip between the multiple optima?

This is a mathematical issue (necessary and sufficient conditions) to be addressed. This has been addressed in the literature. In general, DE converges to the global optimum in contrast to gradient methods.

Think of a population as a vector of 'haploid' bitsets, essentially.

I'm not interested in biological evolution here. Don't want 2 simultaneous discussions. It adds no precision.
Are there physical systems that embody DE? Or is DE another thing from the minds of mathematicians?
 
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katastrofa
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 30th, 2018, 8:17 pm

A harmonic oscillator? I'm not sure why you think the Cauchy convergence is not present in physical system - I gave you several examples, and if you read some more on mathematical physics, you'll see see that it's ubiquitous.
 
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Traden4Alpha
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 30th, 2018, 10:03 pm

A harmonic oscillator? I'm not sure why you think the Cauchy convergence is not present in physical system - I gave you several examples, and if you read some more on mathematical physics, you'll see see that it's ubiquitous.
But a harmonic oscillator never actually converges below some threshold epsilon that is a function of some noise floor (thermal, quantum fluctuations, etc.) and system design.

Such systems may be approximated by Cauchy convergence but they are not really ever convergent. It's only a model and that is a product of the mind.
 
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outrun
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 31st, 2018, 7:46 am

It's only a model and that is a product of the mind.
That's the main point of this topic.

Image
 
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Cuchulainn
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 31st, 2018, 9:42 am

The distance from reality to concept >> distance from concept to realty. 
"Compatibility means deliberately repeating other people's mistakes."
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katastrofa
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 31st, 2018, 11:25 am

A harmonic oscillator? I'm not sure why you think the Cauchy convergence is not present in physical system - I gave you several examples, and if you read some more on mathematical physics, you'll see see that it's ubiquitous.
But a harmonic oscillator never actually converges below some threshold epsilon that is a function of some noise floor (thermal, quantum fluctuations, etc.) and system design.
Do you mean damping? BTW, everything in physics in a harmonic oscillator! :-)

Going back to DE, they are the base of variational principles and as such the foundation of the whole known Universe workings: mechanics, conservation laws, thermodynamics, general relativity, etc. Of course, there are examples of systems which go beyond that picture, e.g. open systems will asymptotically approach, but never reach the stationary state (dictated by the least* action principle).

*) If someone corrects me that the "stationary action principle" is a more correct name I will locate them with my quantum annealer probe and annihilate with a reversed polarity neutron gun.
 
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outrun
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 31st, 2018, 11:29 am

The distance from reality to concept >> distance from concept to realty. 
This is "theory-laden" too. What distance metric, and what space are we going to position both "the concept" and "reality"?
Do we have a final model of reality as well as how concepts are represented in our brain? Does "a brain" even exist? Can we generalize about them: are all brains the same?

Our brain might not even be able to represent the ultimate model of reality?

My cat sometimes watches TV (the VPRO has a show where all the main actors are kittens, ..and then there is a tortoise who's the butler,.. he loves that show). I often think: what is my cat thinking when it watches the TV? What does it think of all the cuts?  I'm sure it has no good model of what it is looking at: how does the TV work, how are TV productions made and paid for? Cats know nothing about money, all they know is how to efficiently torture you into getting them food, ..and they know that a laptop is nice and warm when you're busy working on it.

My cat lives in a different reality.
 
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Cuchulainn
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 31st, 2018, 11:49 am

And what if convergence is suboptimal? What if the superior solution is to randomly flip between the multiple optima?

This is a mathematical issue (necessary and sufficient conditions) to be addressed. This has been addressed in the literature. In general, DE converges to the global optimum in contrast to gradient methods.

Think of a population as a vector of 'haploid' bitsets, essentially.

I'm not interested in biological evolution here. Don't want 2 simultaneous discussions. It adds no precision.
Are there physical systems that embody DE?  Or is DE another thing from the minds of mathematicians?
Instead of always asking questions, maybe try to answer the question yourself.
This is about the 6th time you have asked this question without having given an answer. Or a counterexample.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl