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数学

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Andrius Kulikauskas

  • ms@ms.lt
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  • My work is in the Public Domain for all to share freely.

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Book.20180626Variables istorija

Paslėpti nežymius pakeitimus - Rodyti galutinio teksto pakeitimus

2019 sausio 22 d., 22:37 atliko AndriusKulikauskas -
Pakeistos 11-22 eilutės iš
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į:
What I want to show is how I do cognitive investigation and how I think that can be fruitful for all manner of subjects including math and logic.

For students it's extremely baffling, it's a great mystery.

My investigation is simply to collect the ways that we talk about them.

For every epsilon greater than zero there is a delta greater than zero.

It could be specified unspecified, dependent independent, it could have a place, it could have a value.
2018 rugpjūčio 28 d., 14:26 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

Presented at the session [[http://www.uni-log.org/ss6-LAN.html | Language and Semiotics]] at the [[http://www.uni-log.org/start6.html | 6th World Congress on Universal Logic]], June 21-26, Vichy, France.
2018 liepos 17 d., 22:52 atliko AndriusKulikauskas -
Pakeista 8 eilutė iš:
[+A Structural Semiotic Study of How We Use Variables in Logic and Mathematics+]
į:
[+A Structural Semiotic Study of How We Use Variables in Mathematics and Logic+]
2018 liepos 17 d., 22:52 atliko AndriusKulikauskas -
Pridėtos 7-8 eilutės:
---------
[+A Structural Semiotic Study of How We Use Variables in Logic and Mathematics+]
2018 birželio 27 d., 15:04 atliko AndriusKulikauskas -
Pridėtos 2-3 eilutės:

'''[[Attach:Variables.wav | Audio]]'''
2018 birželio 27 d., 14:37 atliko AndriusKulikauskas -
Pakeistos 1-19 eilutės iš
[[20180621Variables | Abstract]], [[20180626Variables-Draft | Draft]]
į:
[[20180620Variables | Abstract]], [[20180626Variables-Draft | Draft]]

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Attach:VML-00-Title.png
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Attach:VML-01-Variables.png
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Attach:VML-02-Variables2.png
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Attach:VML-03-AllVariables.png
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Attach:VML-04-Foursome.png
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Attach:VML-06-EquilateralTriangle.png
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Attach:VML-07-WaysOfFiguringOut.png
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Attach:VML-08-SetTheoryAxioms.png
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2018 birželio 27 d., 14:33 atliko AndriusKulikauskas -
Pakeistos 1-34 eilutės iš
[++A Structural Semiotic Study of How We Use Variables in Math and Logic++]


Outline
* Examples
* System
* Six kinds of signs.
* Conclusions - creating and solving problems
* Ways of figuring things out in mathematics


--------------------------

[+Examples+
]







---------------------------

One of the profoundest hurdles for students of mathematics is learning to think of a given mathematical symbol in a variety of ways. The variable X may be thought of as a "constant" which may be "specified" or "unspecified", and which may "vary" or be "fixed", and may be "independent" of other variables or "dependent" on them.

What is a variable? Difference between symbol and value.

The options which our minds have for interpreting a variable may be dictated by a cognitive structure. We collected a variety of terms which characterize variables. Indeed, we found 12 pairs such as "known" and "unknown", "free" and "bound" or "input" and "output".

Such pairs indicate how a variable may be reinterpreted. Semiotically, a variable is a sign. What can we say about what different kinds of variables signify? A logical or mathematical expression contains variables from some alphabet A, B, C... If these variables are "free" or "unknown" then they refer to nothing more and we may think of them as uninterpreted. But if they are "bound" or "arbitrary" or "independent" then we imagine them indicating a range of possibilities. If they are "known" or "particular" or even "unspecified" then they refer to particular possibilities, which we ourselves may not yet know, however. And if they are "specified" or "dependent" or "values" then they are explicitly related to other variables so that we may say that we ourselves do know them.

In summary, we may think of variables as refering to unrelated originals (in an alphabet), interchangeable copies (in a multiset), particular elements (in a set) or prioritized items (in a list). They manifest a recurrent cognitive framework of levels of knowledge: whether, what, how and why. Each pair of terms refers to two such levels. Structurally, we find six pairs which our minds use to enrich the content of a variable, for example, to reinterpret a free variable as a bound variable, in order to define a problem by adding information. We also find six pairs which our minds use to emphasize the form of a variable, for example, to reinterpret an output as an input, in order to solve a problem by removing information. The way that our minds recast variables to create and solve problems is meaningful in exploring the cognitive foundations of logic.

Attach:variabletypes.png
į:
[[20180621Variables | Abstract]], [[20180626Variables-Draft | Draft]]
2018 birželio 25 d., 23:32 atliko AndriusKulikauskas -
Pridėtos 25-26 eilutės:

What is a variable? Difference between symbol and value.
2018 birželio 25 d., 23:12 atliko AndriusKulikauskas -
Pridėta 7 eilutė:
* Six kinds of signs.
Pakeista 10 eilutė iš:
* Six kinds of signs.
į:
2018 birželio 23 d., 15:33 atliko AndriusKulikauskas -
Pridėta 9 eilutė:
* Six kinds of signs.
2018 birželio 23 d., 11:19 atliko AndriusKulikauskas -
Pridėtos 9-20 eilutės:

--------------------------

[+Examples+]







---------------------------
2018 birželio 20 d., 14:53 atliko AndriusKulikauskas -
Pridėtos 2-8 eilutės:


Outline
* Examples
* System
* Conclusions - creating and solving problems
* Ways of figuring things out in mathematics
2018 birželio 20 d., 14:51 atliko AndriusKulikauskas -
Pridėtos 1-11 eilutės:
[++A Structural Semiotic Study of How We Use Variables in Math and Logic++]

One of the profoundest hurdles for students of mathematics is learning to think of a given mathematical symbol in a variety of ways. The variable X may be thought of as a "constant" which may be "specified" or "unspecified", and which may "vary" or be "fixed", and may be "independent" of other variables or "dependent" on them.

The options which our minds have for interpreting a variable may be dictated by a cognitive structure. We collected a variety of terms which characterize variables. Indeed, we found 12 pairs such as "known" and "unknown", "free" and "bound" or "input" and "output".

Such pairs indicate how a variable may be reinterpreted. Semiotically, a variable is a sign. What can we say about what different kinds of variables signify? A logical or mathematical expression contains variables from some alphabet A, B, C... If these variables are "free" or "unknown" then they refer to nothing more and we may think of them as uninterpreted. But if they are "bound" or "arbitrary" or "independent" then we imagine them indicating a range of possibilities. If they are "known" or "particular" or even "unspecified" then they refer to particular possibilities, which we ourselves may not yet know, however. And if they are "specified" or "dependent" or "values" then they are explicitly related to other variables so that we may say that we ourselves do know them.

In summary, we may think of variables as refering to unrelated originals (in an alphabet), interchangeable copies (in a multiset), particular elements (in a set) or prioritized items (in a list). They manifest a recurrent cognitive framework of levels of knowledge: whether, what, how and why. Each pair of terms refers to two such levels. Structurally, we find six pairs which our minds use to enrich the content of a variable, for example, to reinterpret a free variable as a bound variable, in order to define a problem by adding information. We also find six pairs which our minds use to emphasize the form of a variable, for example, to reinterpret an output as an input, in order to solve a problem by removing information. The way that our minds recast variables to create and solve problems is meaningful in exploring the cognitive foundations of logic.

Attach:variabletypes.png

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Puslapis paskutinį kartą pakeistas 2019 sausio 22 d., 22:37
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