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What is the menaingmeaning of the statement "it is red" ? It depends on the context: if I utter it pointing at my (red) book on my desk, I'm uttering a true sentence, that someone being present to my utterance can verify.

A variable is part of the language; specifically: of the formalized language of mathematics. 

Thus,

And symbols are the "stuff" otof our social life: when we drive a car and we stop at the crossroad seeing the traffic light blinking red, we do that becuasebecause we interpret a symbol: the color of the traffic light.

When we play chess, the pieces we are playing with, no matter of their color, or the fact that they are made of wood or chalk, are symbols; we can play with them also if they have different shapes from usual, prvidedprovided that we can recognize the different "roles".

And we can play without physicals pieces at all, on a TV screen or by mail. We can do it because the peicespieces of chess are symbols that receive their "meaning" in a certain context: the players share the rules of the game.

Symbolic manipulation is there from the beginning: we learn counting quite early, and then we learn "playing"playing with geometrical shapes, and then we go on, until derivatives, tensors and more.

period. 

If so,

In our "world", made of individuals, physical objects and social life: without language and symbols no social life is possible, and without a "social reality"social reality no language is needed.

But in a nutshell, the "common view" is that mathematical theories "speak"speak about some abstract "furniture" of the world (our actual one or only a possible one ?).

Set theoritheory is very very peculiar, due to the fact that the concept of set is quite "basic"basic : for mathematicianmathematicians and philosophers of 19th Century (end of) to every "concept"concept we can "manage"define (define, imagineimagine, conceive of) there must be the corresponding set of objects satisfying that concept.

This "philosophically"philosophically natural point of view does not work mathematically [see: The Early Development of Set Theory]; thus, the need of a more rigorous set theory [see: Zermelo's Axiomatization of Set Theory].

According to the framework of modern set theory, a concept define a "collection"collection, but not every collection is an object existing in the domain (or universe) of set theory.

Having said that, the collection of all sets consistconsists of what? Every colelction is "identified"specified by a concept (expressible in the theory). If we consider the "usual" first-order language of set theory, the concept:

What is the menaing of the statement "it is red" ? It depends on the context: if I utter it pointing at my (red) book on my desk, I'm uttering a true sentence, that someone being present to my utterance can verify.

A variable is part of the language; specifically: of the formalized language of mathematics. Thus,

And symbols are the "stuff" ot our social life: when we drive a car and we stop at the crossroad seeing the traffic light blinking red, we do that becuase we interpret a symbol: the color of the traffic light.

When we play chess, the pieces we are playing with, no matter of their color, or the fact that they are made of wood or chalk, are symbols; we can play with them also if they have different shapes from usual, prvided that we can recognize the different "roles".

And we can play without physicals pieces at all, on a TV screen or by mail. We can do it because the peices of chess are symbols that receive their "meaning" in a certain context: the players share the rules of the game.

Symbolic manipulation is there from the beginning: we learn counting quite early, and then we learn "playing" with geometrical shapes, and then we go on, until derivatives, tensors and more.

period. If so,

In our "world", made of individuals, physical objects and social life: without language and symbols no social life is possible, and without a "social reality" no language is needed.

But in a nutshell, the "common view" is that mathematical theories "speak" about some abstract "furniture" of the world (our actual one or only a possible one ?).

Set theori is very very peculiar, due to the fact that the concept of set is quite "basic": for mathematician and philosophers of 19th Century (end of) to every "concept" we can "manage" (define, imagine, conceive of) there must be the corresponding set of objects satisfying that concept.

This "philosophically" natural point of view does not work mathematically [see: The Early Development of Set Theory]; thus, the need of a more rigorous set theory [see: Zermelo's Axiomatization of Set Theory].

According to the framework of modern set theory, a concept define a "collection", but not every collection is an object existing in the domain (or universe) of set theory.

Having said that, the collection of all sets consist of what? Every colelction is "identified" by a concept (expressible in the theory). If we consider the "usual" first-order language of set theory, the concept:

What is the meaning of the statement "it is red" ? It depends on the context: if I utter it pointing at my (red) book on my desk, I'm uttering a true sentence, that someone being present to my utterance can verify.

A variable is part of the language; specifically: of the formalized language of mathematics. 

Thus,

And symbols are the "stuff" of our social life: when we drive a car and we stop at the crossroad seeing the traffic light blinking red, we do that because we interpret a symbol: the color of the traffic light.

When we play chess, the pieces we are playing with, no matter of their color, or the fact that they are made of wood or chalk, are symbols; we can play with them also if they have different shapes from usual, provided that we can recognize the different "roles".

And we can play without physicals pieces at all, on a TV screen or by mail. We can do it because the pieces of chess are symbols that receive their "meaning" in a certain context: the players share the rules of the game.

Symbolic manipulation is there from the beginning: we learn counting quite early, and then we learn playing with geometrical shapes, and then we go on, until derivatives, tensors and more.

period. 

If so,

In our "world", made of individuals, physical objects and social life: without language and symbols no social life is possible, and without a social reality no language is needed.

But in a nutshell, the "common view" is that mathematical theories speak about some abstract "furniture" of the world (our actual one or only a possible one ?).

Set theory is very very peculiar, due to the fact that the concept of set is quite basic : for mathematicians and philosophers of 19th Century (end of) to every concept we can define (imagine, conceive of) there must be the corresponding set of objects satisfying that concept.

This philosophically natural point of view does not work mathematically [see: The Early Development of Set Theory]; thus, the need of a more rigorous set theory [see: Zermelo's Axiomatization of Set Theory].

According to the framework of modern set theory, a concept define a collection, but not every collection is an object existing in the domain (or universe) of set theory.

Having said that, the collection of all sets consists of what? Every colelction is specified by a concept (expressible in the theory). If we consider the "usual" first-order language of set theory, the concept:

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A variable is a "syntactical" object.:

it lives in formalized languages.

In athe formalized language of logic and mathematics, it "works" like a pronoun.

Thus, aA variable is part of the language; specifically: of the formalized language of mathematics. Thus it is a symbol.,

a variable is a symbol.

If we can play with them, they exist... Someone can bet on a chess play: he can bet "real" money (and also bitcoins...) and loose it: he can do all thorugh the web. So what ? the chess pieces do not exist ? neither the web? not the money (it is only a string of bit on our banck account..). If no "not-physical" objects exist, why care about money, Brexit, -10% of stock markets...

So,

symbols exist,

period. If so,

they exist where?

In our "world", made of individuals, physical objects and social life: without language and symbols no social life is possible, and without a "social reality" no language is needed.

Maybe relevant:

A variable is a "syntactical" object.

In a formalized language it "works" like a pronoun.

Thus, a variable is part of the language; specifically: of the formalized language of mathematics. Thus it is a symbol.

If we can play with them, they exist... Someone can bet on a chess play: he can bet "real" money (and also bitcoins...) and loose it: he can do all thorugh the web. So what ? the chess pieces do not exist ? neither the web? not the money (it is only a string of bit on our banck account..). If no "not-physical" objects exist, why care about money, Brexit, -10% of stock markets...

A variable is a "syntactical" object:

it lives in formalized languages.

In the formalized language of logic and mathematics, it "works" like a pronoun.

A variable is part of the language; specifically: of the formalized language of mathematics. Thus,

a variable is a symbol.

If we can play with them, they exist... Someone can bet on a chess play: he can bet "real" money (and also bitcoins...) and loose it: he can do all thorugh the web. So what ? the chess pieces do not exist ? neither the web? not the money (it is only a string of bit on our banck account..). If no "not-physical" objects exist, why care about money, Brexit, -10% of stock markets...

So,

symbols exist,

period. If so,

they exist where?

In our "world", made of individuals, physical objects and social life: without language and symbols no social life is possible, and without a "social reality" no language is needed.

Maybe relevant:

5 added 1855 characters in body
source | link

Thus, a variable is part of the language; specifically: of the formalized language of mathematics. Thus it is a symbol.

Symbols are everywhere in "our world": they are in the language we use to "speak of" the world: the world of everyday experience as well as the world of more abstract objects, like black holes, quarks, GDP, numbers, mathematical structures.

And symbols are the "stuff" ot our social life: when we drive a car and we stop at the crossroad seeing the traffic light blinking red, we do that becuase we interpret a symbol: the color of the traffic light.

When we play chess, the pieces we are playing with, no matter of their color, or the fact that they are made of wood or chalk, are symbols; we can play with them also if they have different shapes from usual, prvided that we can recognize the different "roles".

And we can play without physicals pieces at all, on a TV screen or by mail. We can do it because the peices of chess are symbols that receive their "meaning" in a certain context: the players share the rules of the game.

When we communicate, we are playing a game with "linguistic pieces" (symbols) according to the shared rules we have learned when we learned how to speak and write... and count.

Symbolic manipulation is there from the beginning: we learn counting quite early, and then we learn "playing" with geometrical shapes, and then we go on, until derivatives, tensors and more.

If we can play with them, they exist... Someone can bet on a chess play: he can bet "real" money (and also bitcoins...) and loose it: he can do all thorugh the web. So what ? the chess pieces do not exist ? neither the web? not the money (it is only a string of bit on our banck account..). If no "not-physical" objects exist, why care about money, Brexit, -10% of stock markets...


Regarding the question

Regarding the question

Thus, a variable is part of the language; specifically: of the formalized language of mathematics. Thus it is a symbol.

Symbols are everywhere in "our world": they are in the language we use to "speak of" the world: the world of everyday experience as well as the world of more abstract objects, like black holes, quarks, GDP, numbers, mathematical structures.

And symbols are the "stuff" ot our social life: when we drive a car and we stop at the crossroad seeing the traffic light blinking red, we do that becuase we interpret a symbol: the color of the traffic light.

When we play chess, the pieces we are playing with, no matter of their color, or the fact that they are made of wood or chalk, are symbols; we can play with them also if they have different shapes from usual, prvided that we can recognize the different "roles".

And we can play without physicals pieces at all, on a TV screen or by mail. We can do it because the peices of chess are symbols that receive their "meaning" in a certain context: the players share the rules of the game.

When we communicate, we are playing a game with "linguistic pieces" (symbols) according to the shared rules we have learned when we learned how to speak and write... and count.

Symbolic manipulation is there from the beginning: we learn counting quite early, and then we learn "playing" with geometrical shapes, and then we go on, until derivatives, tensors and more.

If we can play with them, they exist... Someone can bet on a chess play: he can bet "real" money (and also bitcoins...) and loose it: he can do all thorugh the web. So what ? the chess pieces do not exist ? neither the web? not the money (it is only a string of bit on our banck account..). If no "not-physical" objects exist, why care about money, Brexit, -10% of stock markets...


Regarding the question

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