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

Yes, physical systems have minimum epsilons for various physical reasons. Digital computers have a complex minimum epsilon linked to data type and the exponent of floating point. That said, some numerical algorithms can converge to true zero epsilon.

Yet the bigger mathematical issue remains that a seemingly converging sequence that NEVER actually converges below some epsilon is not a Cauchy sequence. Thus, mathematical truths about Cauchy sequences may not apply to such systems.

(Note: these "resolution limit" issues affect more that just Cauchy sequences. That leads to the ironic and uncomfortable realization that "real" numbers are not real. Useful approximations, yes. Real in the black-and-white mathematical sense, no.)

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

(Note: these "resolution limit" issues affect more that just Cauchy sequences. That leads to the ironic and uncomfortable realization that "real" numbers are not real. Useful approximations, yes. Real in the black-and-white mathematical sense, no.)

The nice thing about models of reality is that there are so many to choose from. Numerical analysts seem to be quite comfortable with these (interrelated) models.

[font=-apple-system, Helvetica Neue, Helvetica, sans-serif]Yet the bigger mathematical issue remains that a seemingly converging sequence that NEVER actually converges below some epsilon is not a Cauchy sequence. Thus, mathematical truths about Cauchy sequences may not apply to such systems. [/font]

Why is it bigger in your opinion and what are the consequence of so-called non-convergence?

Can you give a concrete example?

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

uncomfortable realization that "real" numbers are not real. Useful approximations, yes. Real in the black-and-white mathematical sense, no

There are models in the problem domain and in the solution domain and they operate on views of similar (bit not quite the same) objects. But people tend to slur their words.

In FORTRAN 77 there are no reals, just REAL*4 and REAL*.8.

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

The real issue here concerns infinity vs. finiteness.

Indeed. And how to get infinity (both countable and uncountable) into a computer, step-by-step.

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

lovenatalya wrote:
The real issue here concerns infinity vs. finiteness.

In any kind of "real world", there is a minimal positive $\epsilon$. In numerical analysis, it is the computer floating point limit. In quantum mechanics, it is the Plank length. In optics, it is the wavelength of the light you are using. There is also the Heisenberg uncertainty principle lower bound when you consider two non-commutative operators. Correspondingly there is also the upper bound. With a given minimal $\epsilon$, a Cauchy sequence can be defined, and its length is finite, since there is a lower bound on the difference amongst the sequence after some index.

The mathematical version of the Cauchy sequence is taking the aforementioned Cauchy sequence to infinity. The prototype of  Infinity is the natural number. We can understand the mathematical version or infinite version of the Cauchy sequence as the (countably) infinite intersection of all the physical versions or all the finite versions of Cauchy sequence with all kinds resolution/precisions.

Another good example is the case of  stochastic process $X(\omega, t)$that satisfies an SDE in an uncountable space $\Omega X (0,T)$. This can't be put into a computer so we need to project these spaces into finite dimensions. For example,classic MC take M draws and NT mesh points. For a random ODE on the other hand with white noise we can replace $\Omega$ by a countable Karhunen-Loeve Fourier Expansion (which we then truncate). The resulting ODE is than solve using an ODE solver.

The Method of Lines (MOL) and FEM mappings are similar.

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

The notion of Cauchy sequences is deeplly rooted in natural sciences, especially in the measurement theory and all disciplines routinely using statistics. It's more explicit in modern physics, and the least obvious in the most obvious facts of epistemically observable (or non-observable) world, I would say.

Imagine that you measure the temperature in the room. You measure it 10 times and average the result. It's not a precise result - you'd need to measure it ad infinitum with an infinitely precise thermometer to achieve an absolute precision, but it's technically impossible (because of the beast from the east, climate change, etc.). However, you know that the result should converge to some value (modulo the measurement errors and disturbances) even if you'll never learn it, and your averaged result is in its proximity (which can be estimated).

Yyou need Cauchy sequence to describe the measurement in the micro scale. When you study classical objects, e.g. a ball that you can see and throw in a chosen direction, the Euclidean space suffices to describe the situation. However, if the ball was microscopically small - an electron, its properties wouldn't be that obvious. You'd have to measure them somehow. The measurement amounts to scanning through all possible configurations/states of the microscopic world hoping to find the one with the "ball": it can be anywhere and fly at any speed in any direction, but you can check a single value of each at once. You need a mathematical construct which can accommodate all possible results. The construct is one of the Euclidean space generalisations - the Hilbert space. The fact that it contains all such possible states of the system stems from its completeness, and the completeness relies on the Cauchy convergence.

Another random example, there was a popular game when I was a child: you had a limited time to guess what the other person had in mind based on their answers to a series of your questions. The Cauchy sequence makes it possible to win this game.

Choose what's Cauchyer for ya.

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

The construct is one of the Euclidean space generalisations - the Hilbert space. The fact that it contains all such possible states of the system stems from its completeness, and the completeness relies on the Cauchy convergence.

Good remark. In Hilbert spaces H you can define an inner product and then a norm induced by that inner products. All kinds of norms can be defined. So, a Cauchy sequence will converge to an element _in_ H,

There are spaces that are not complete.

Some spaces only have a metric and then induced  norm. Banach spaces are complete metric spaces. These with Sobolev spaces) are the heavy-duty spaces for applied functional analysis and standard fodder for PDE/FEM. This was my favourite stuff as undergrad.

Hilbert spaces are rare beasts in the infested jungle of Banach spaces.. The parallelogram law can break down for some spaces and hence is not an inner product space).

Thinking out loud: norms for ML, Hilbert, Banach, complete etc etc.

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

Cuchulainn wrote:
The construct is one of the Euclidean space generalisations - the Hilbert space. The fact that it contains all such possible states of the system stems from its completeness, and the completeness relies on the Cauchy convergence.

Good remark. In Hilbert spaces H you can define an inner product and then a norm induced by that inner products. All kinds of norms can be defined. So, a Cauchy sequence will converge to an element _in_ H,

There are spaces that are not complete.

Some spaces only have a metric and then induced  norm. Banach spaces are complete metric spaces. These with Sobolev spaces) are the heavy-duty spaces for applied functional analysis and standard fodder for PDE/FEM. This was my favourite stuff as undergrad.

Hilbert spaces are rare beasts in the infested jungle of Banach spaces.. The parallelogram law can break down for some spaces and hence is not an inner product space).

Thinking out loud: norms for ML, Hilbert, Banach, complete etc etc.

The question of whether something is real or in the imagination is not a well posed one. It is impossible to distinguish the two. Even what you consider as seeing is the mental construct from the photons hitting your retina having already filtered by the retina and its photosensors and neurons not to mention the layers of neural networks in the brain.  As Einstein put it well "what you observe is determined by your theory."

Is an integer real or imaginary, is a fraction real or imaginary, is a real number, is a complex number? Consider the present example of a complete space.

Banach space is a complete normed space. However, a norm is not necessary for the characterization of completeness. So long the space is a topological vector space, a Cauchy sequence can be defined. Whether a topological vector space is complete or not, is only a technicality, rather than an essential feature of a space. For an incomplete topological space, we can construct a new topological vector space by defining all Cauchy sequences of the original space as elements of a new space. The new topology consists of the closed sets each of which is the union of an original open set and all of its Cauchy sequences. We preserve all the binary operators like addition and multiplication. This newly constructed vector space is complete and preserves all the previous operations.

In fact, completion is used wider than just the topological vector space extension. We complete a set in many myriad ways just so that an operation can be carried out. The very notion of fraction comes from the completion of the division of integers, the negative numbers from the completion of subtraction, the real transcendental number from the completion of solving integral polynomial equations, the irrational number from the Cauchy sequence on the real line, and the complex number from making up the roots of real polynomial equations. These are all mental constructs. Are they real or imaginary?
Last edited by lovenatalya on May 15th, 2018, 4:39 pm

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

Philosophy is a battle against the bewitchment of our intelligence by means of language.

Ludwig Wittgenstein

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

Cuchulainn wrote:
Philosophy is a battle against the bewitchment of our intelligence by means of language.

Ludwig Wittgenstein

Very well put. I would add "buttressed and constrained by logic". I see many a sophistry unconstrained by logic dressed up as philosophy.

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

Everything sounds like philosophy if there's no context. If we poke our heads out of Plato's cave, the question is whether we experience phenomena which can be described in terms of the Cauchy sequence. Yes - whenever we measure/estimate something.

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

katastrofa wrote:
Everything sounds like philosophy if there's no context. If we poke our heads out of Plato's cave, the question is whether we experience phenomena which can be described in terms of the Cauchy sequence. Yes - whenever we measure/estimate something.
Very true!

And yet, it is not the physical measurements themselves that form a Cauchy sequence. It is the mathematical estimation process in the mind of mathematician and chosen under mathematical assumptions about the measurement process (e.g., IID) that create a Cauchy sequence. And if the assumptions in the mind of the mathematician/physicist that were used for the estimation process are untrue for the physical system (e.g., the mean is non-stationary), then even the sequence of mathematical estimates might be non-Cauchy.

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

katastrofa wrote:
Everything sounds like philosophy if there's no context. If we poke our heads out of Plato's cave, the question is whether we experience phenomena which can be described in terms of the Cauchy sequence. Yes - whenever we measure/estimate something.

Very true!

And yet, it is not the physical measurements themselves that form a Cauchy sequence.  It is the mathematical estimation process in the mind of mathematician and chosen under mathematical assumptions about the measurement process (e.g., IID) that create a Cauchy sequence.  And if the assumptions in the mind of the mathematician/physicist that were used for the estimation process are untrue for the physical system (e.g., the mean is non-stationary), then even the sequence of mathematical estimates might be non-Cauchy.

Your knowledge of Cauchy sequences and their applications is not deep enough to allow you to make sweeping statements like these. You can't know everything. so best just to admit it. As Wittgenstein said, your use of language has bewitched you.

Terms like 'measurement process' and 'estimation process' mean nothing because you haven't defined them..We cannot proceed. And mixing maths and physics is unfortunate. The text is not telling me anything about Cauchy sequences.

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

non-Cauchy.
Nice guess. But what is a non-Cauchy sequence and is it interesting?

e.g. is this ${(1 + 5/n)^n}, n = 1,2,3 ,...$ a Cauchy sequence? Does it converge and to what?

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

Measurement amounts to constructing a density matrix of the statistical ensemble (trivial - diagonal in classical case), which you investigate. Actually, the description of the quantum measurement *process* (involving the quantum evolution) also generally requires the Cauchy sequence. Any measurement or optimisation, I would add - if you think about it long enough, they mean the same