Sequence - Revision history

Lfox: /* The problem with Cauchy */

2024-07-08T20:21:12Z

The problem with Cauchy

← Older revision		Revision as of 20:21, 8 July 2024
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	== The problem with Cauchy ==		== The problem with Cauchy ==
	All those things are examples of sequences. But are they, or could they be, examples of ''Cauchy'' sequences? Unfortunately, will see that there can be no Cauchy sequences in reality.		All those things are examples of sequences. But are they, or could they be, examples of ''Cauchy'' sequences? Unfortunately, we will see that there can be no Cauchy sequences in reality.

	Suppose I identify a real life thing <math>R</math> as being a unit of a sequence <math>\{S_n \} </math>. In my specification of <math>\{S_n \} </math> I had to be very precise, so it makes perfect sense to ask whether or not <math>\{S_n \} </math> is Cauchy. However, to ''answer'' whether or not <math>\{S_n \} </math> is Cauchy, I have to go to arbitrarily large <math>n</math>; I have to think about parts of <math>\{S_n \} </math> that have no actual referent in <math>R</math>. So being Cauchy is only a property of the concept <math>\{S_n \} </math> itself, not a property of the actual physical concrete thing <math>R</math>.		Suppose I identify a real life thing <math>R</math> as being a unit of a sequence <math>\{S_n \} </math>. In my specification of <math>\{S_n \} </math> I had to be very precise, so it makes perfect sense to ask whether or not <math>\{S_n \} </math> is Cauchy. However, to ''answer'' whether or not <math>\{S_n \} </math> is Cauchy, I have to go to arbitrarily large <math>n</math>; I have to think about parts of <math>\{S_n \} </math> that have no actual referent in <math>R</math>. So being Cauchy is only a property of the concept <math>\{S_n \} </math> itself, not a property of the actual physical concrete thing <math>R</math>.

	This might seem like a trivial difference, but it's not. It renders the question "is <math>R</math> Cauchy?" meaningless. Worse than that, it also makes the concept of Cauchy useless. Indeed, since any real sequence <math>R</math> is finite, and since sequences are open-ended, I can, in ''all situations'', identify <math>R</math> as being a unit of <math>\{S'_n\}</math> and also as being a unit of <math>\{S_n \} </math>, where <math>\{S_n \} </math> is Cauchy and <math>\{S'_n\}</math> is not. <math>\{S_n \} </math> and <math>\{S'_n\}</math> agree for observable values of <math>n</math>, which is why they are both potentially valid identifications of <math>R</math>.		This might seem like a trivial difference, but it's not. It renders the question "is <math>R</math> Cauchy?" meaningless. Worse than that, it also makes the concept of Cauchy useless. Indeed, since any real sequence <math>R</math> is finite, and since sequences are open-ended, I can, in ''all situations'', identify <math>R</math> as being a unit of <math>\{S'_n\}</math> and also as being a unit of <math>\{S_n \} </math>, where <math>\{S_n \} </math> is Cauchy and <math>\{S'_n\}</math> is not. <math>\{S_n \} </math> and <math>\{S'_n\}</math> agree for observable values of <math>n</math>, which is why they are both potentially valid identifications of <math>R</math>.

Lfox at 20:20, 8 July 2024

2024-07-08T20:20:28Z

← Older revision		Revision as of 20:20, 8 July 2024
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	== The problem with Cauchy ==		== The problem with Cauchy ==
	All those things are examples of sequences. But are they, or could they be, examples of ''Cauchy'' sequences? ~~First we must figure out what~~ that ~~question means~~.		All those things are examples of sequences. But are they, or could they be, examples of ''Cauchy'' sequences? Unfortunately, will see that there can be no Cauchy sequences in reality.

	Suppose I identify a real life thing <math>R</math> as being a unit of a sequence <math>\{S_n \} </math>. In my specification of <math>\{S_n \} </math> I had to be very precise, so it makes perfect sense to ask whether or not <math>\{S_n \} </math> is Cauchy. However, to ''answer'' whether or not <math>\{S_n \} </math> is Cauchy, I have to go to arbitrarily large <math>n</math>; I have to think about parts of <math>\{S_n \} </math> that have no actual referent in <math>R</math>.		Suppose I identify a real life thing <math>R</math> as being a unit of a sequence <math>\{S_n \} </math>. In my specification of <math>\{S_n \} </math> I had to be very precise, so it makes perfect sense to ask whether or not <math>\{S_n \} </math> is Cauchy. However, to ''answer'' whether or not <math>\{S_n \} </math> is Cauchy, I have to go to arbitrarily large <math>n</math>; I have to think about parts of <math>\{S_n \} </math> that have no actual referent in <math>R</math>. So being Cauchy is only a property of the concept <math>\{S_n \} </math> itself, not a property of the actual physical concrete thing <math>R</math>.

	~~A consequence of this fact~~ is that ~~if I had instead~~		This might seem like a trivial difference, but it's not. It renders the question "is <math>R</math> Cauchy?" meaningless. Worse than that, it also makes the concept of Cauchy useless. Indeed, since any real sequence <math>R</math> is finite, and since sequences are open-ended, I can, in ''all situations'', identify <math>R</math> as being a unit of <math>\{S'_n\}</math> and also as being a unit of <math>\{S_n \} </math>, where <math>\{S_n \} </math> is Cauchy and <math>\{S'_n\}</math> is not. <math>\{S_n \} </math> and <math>\{S'_n\}</math> agree for observable values of <math>n</math>, which is why they are both potentially valid identifications of <math>R</math>.

	is ~~some~~ finite ~~thing~~, I ~~could have instead identified~~ <math>R</math> as being a unit of ~~another sequence~~ <math>S'</math>.

Lfox at 20:09, 8 July 2024

2024-07-08T20:09:41Z

← Older revision		Revision as of 20:09, 8 July 2024
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	A '''sequence''' is a [[function]] which takes [[natural numbers]] as inputs, and returns [[numbers]] as outputs.		A '''sequence''' is a [[function]] which takes [[natural numbers]] as inputs, and returns [[numbers]] as outputs.

	== The Cauchy condition ==		== The Cauchy condition ==
			[TODO I now think this notion is problematic. See below. ]

	Suppose that <math>\{ a_n \}_{n=1}^\infty</math> is a sequence that computes [[π]]. [TODO what does this even mean? Does it actually make sense? If π is the ratio between a ''real'' circle's circumference and a ''real'' circle's diameter, then it's not a single number, but some class of numbers. Sometimes the ratio is 3.14, sometimes it's 3.18, etc.]		Suppose that <math>\{ a_n \}_{n=1}^\infty</math> is a sequence that computes [[π]]. [TODO what does this even mean? Does it actually make sense? If π is the ratio between a ''real'' circle's circumference and a ''real'' circle's diameter, then it's not a single number, but some class of numbers. Sometimes the ratio is 3.14, sometimes it's 3.18, etc.]

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	* a C++ program which takes in a uint32, ''n'', and outputs 2''n''+1		* a C++ program which takes in a uint32, ''n'', and outputs 2''n''+1
	* the following table on my screen [a screenshot of a few rows of an excel table where the entries are 2*row + 1 ]		* the following table on my screen [a screenshot of a few rows of an excel table where the entries are 2*row + 1 ]
	* ~~etc~~.		* a succession of locations of a bouncing ball at various times

			== The problem with Cauchy ==
			All those things are examples of sequences. But are they, or could they be, examples of ''Cauchy'' sequences? First we must figure out what that question means.

			Suppose I identify a real life thing <math>R</math> as being a unit of a sequence <math>\{S_n \} </math>. In my specification of <math>\{S_n \} </math> I had to be very precise, so it makes perfect sense to ask whether or not <math>\{S_n \} </math> is Cauchy. However, to ''answer'' whether or not <math>\{S_n \} </math> is Cauchy, I have to go to arbitrarily large <math>n</math>; I have to think about parts of <math>\{S_n \} </math> that have no actual referent in <math>R</math>.

			A consequence of this fact is that if I had instead

			is some finite thing, I could have instead identified <math>R</math> as being a unit of another sequence <math>S'</math>.

Lfox at 22:21, 22 April 2024

2024-04-22T22:21:41Z

← Older revision		Revision as of 22:21, 22 April 2024
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	== The Cauchy condition ==		== The Cauchy condition ==
	Suppose that <math>\{ a_n \}_{n=1}^\infty</math> is a sequence that computes [[π]].		Suppose that <math>\{ a_n \}_{n=1}^\infty</math> is a sequence that computes [[π]]. [TODO what does this even mean? Does it actually make sense? If π is the ratio between a ''real'' circle's circumference and a ''real'' circle's diameter, then it's not a single number, but some class of numbers. Sometimes the ratio is 3.14, sometimes it's 3.18, etc.]

	The practical question to answer, for someone computing this sequence on a computer, is "after how many steps can I stop running the computer program?" The answer is that you should wait until successive steps of the sequence vary by amounts smaller than your desired accuracy.		The practical question to answer, for someone computing this sequence on a computer, is "after how many steps can I stop running the computer program?" The answer is that you should wait until successive steps of the sequence vary by amounts smaller than your desired accuracy.
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	then we can stop computing around step 9, because at that point the output of the programming is varying by amounts much smaller than 0.01.		then we can stop computing around step 9, because at that point the output of the programming is varying by amounts much smaller than 0.01.

	~~What~~ I have done in the above paragraph is I have ~~inadvertently~~ assumed that the sequence satisfies the '''Cauchy condition'''. A sequence is said to satisfy the Cauchy condition if, for any level of desired accuracy, there is some point beyond which any two outputs of the algorithm differ by an amount smaller than the desired accuracy. More formally, <math>\{a_n\}_{n=1}^\infty</math> satisfies the Cauchy condition if for any <math>\epsilon : \mathbb{Q}</math> such that <math>\epsilon > 0, </math> there exists an <math>N : \mathbb{N}</math> such that if <math>n,m \geq N</math> then <math>\|a_n - a_m \| \leq \epsilon</math>.		But how do we know that later, after many more steps, the sequence won't change in such a way that the earlier digits are affected? E.g. what if the 11th element of the sequence is 3.15? The answer is that what I have done in the above paragraph is I have implicitly assumed that the sequence <math>\{a_n\}_{n=1}^\infty</math> satisfies the '''Cauchy condition'''. A sequence is said to satisfy the Cauchy condition if, for any level of desired accuracy, there is some point beyond which any two outputs of the algorithm differ by an amount smaller than the desired accuracy. More formally, <math>\{a_n\}_{n=1}^\infty</math> satisfies the Cauchy condition if for any <math>\epsilon : \mathbb{Q}</math> such that <math>\epsilon > 0, </math> there exists an <math>N : \mathbb{N}</math> such that if <math>n,m \geq N</math> then <math>\|a_n - a_m \| \leq \epsilon</math>.

			Note that the Cauchy condition is only an ''idealization'' of the property that we would actually want a sequence to satisfy in real life. Some Cauchy sequences are not practical: For example, someone could define a Cauchy sequence for which we must choose <math>N</math> to be at least a googolplex whenever <math>\epsilon = 0.1</math>. Conversely, there are some practical sequences which are not Cauchy: For example, some sequences of partial sums that arise in perturbation theory are divergent, but give accurate and useful results if we only go a dozen terms deep into the sequence.

	== Examples ==		== Examples ==

Lfox: Created page with "A '''sequence''' is a function which takes natural numbers as inputs, and returns numbers as outputs. == The Cauchy condition == Suppose that $\{ a_n \}_{n=1}^\infty$ is a sequence that computes π. The practical question to answer, for someone computing this sequence on a computer, is "after how many steps can I stop running the computer program?" The answer is that you should wait until successive steps of the sequence vary by amounts sma..."

2024-01-21T01:55:16Z

Created page with "A '''sequence''' is a function which takes natural numbers as inputs, and returns numbers as outputs. == The Cauchy condition == Suppose that <math>\{ a_n \}_{n=1}^\infty</math> is a sequence that computes π. The practical question to answer, for someone computing this sequence on a computer, is "after how many steps can I stop running the computer program?" The answer is that you should wait until successive steps of the sequence vary by amounts sma..."

New page

A '''sequence''' is a [[function]] which takes [[natural numbers]] as inputs, and returns [[numbers]] as outputs.

== The Cauchy condition ==
Suppose that <math>\{ a_n \}_{n=1}^\infty</math> is a sequence that computes [[π]].

The practical question to answer, for someone computing this sequence on a computer, is "after how many steps can I stop running the computer program?" The answer is that you should wait until successive steps of the sequence vary by amounts smaller than your desired accuracy.

For example, if your target accuracy is 0.01, and if the printout looks like
{| class="wikitable"
|+
!<math>n</math>
!<math>a_n</math>
|-
|1
|4
|-
|2
|3.0
|-
|3
|3.2
|-
|4
|3.09
|-
|5
|3.16
|-
|6
|3.139
|-
|7
|3.143
|-
|8
|3.1408
|-
|9
|3.1419
|-
|10
|3.14149
|-
|...
|...
|}
then we can stop computing around step 9, because at that point the output of the programming is varying by amounts much smaller than 0.01.

What I have done in the above paragraph is I have inadvertently assumed that the sequence satisfies the '''Cauchy condition'''. A sequence is said to satisfy the Cauchy condition if, for any level of desired accuracy, there is some point beyond which any two outputs of the algorithm differ by an amount smaller than the desired accuracy. More formally, <math>\{a_n\}_{n=1}^\infty</math> satisfies the Cauchy condition if for any <math>\epsilon : \mathbb{Q}</math> such that <math>\epsilon > 0, </math> there exists an <math>N : \mathbb{N}</math> such that if <math>n,m \geq N</math> then <math>|a_n - a_m | \leq \epsilon</math>.

== Examples ==
The sequence <math>\{2n+1\}_{n=0}^\infty</math> could be properly thought of as that concept which---in appropriate contexts---subsumes the following concretes:
* the method that a child uses to produce the ''n''th odd number
* a C++ program which takes in a uint32, ''n'', and outputs 2''n''+1
* the following table on my screen [a screenshot of a few rows of an excel table where the entries are 2*row + 1 ]
* etc.