3

Background

This question has its origin in this post. More specifically, in giving answers to the following questions,

  1. In what way does the collection of all sets consist of sets?
  2. What are collections of sets actually?

It was written here that,

The answers to 1. and 2. is that you have to set up some theory that allows you to talk about collections. The standard theory do this with is first-order logic, in which every formula with free variables is a description of a collection (aka a class), and these collections may also be described by auxillary functions and relations.

answer it has been commented that,

The point is that you can view first-order logic itself as a primitive "Theory of Collections", hence avoiding infinite regress. We are not describing or postulating how actual collections behave, but we are describing the language by which we express or refer to the collections. The potential for regress lies in disagreeing on what it takes to specify a formal language (since of course, we never actually have access to ALL formulas unless there were only finitely many), but certainly specifying the rules for producing well-formed statements and the rules for judging them justified is considered sufficient description: most humans seem to be expected to apply some sort of "pattern-matching" to the rules and to comprehend from them how to form well-formed statements (i.e. how to speak) and how to judge whether a statement is justified (what a proof is). So you don't to actually go ahead and develop a formal theory of languages before you can specify a formal language, but there is an important philosophical shift from mental to social activity in the level of justification required.

However when I asked my teacher the same question (the question linked above) he replied that,

We actually even don't need to worry about a Theory of Collection at all. When you learn a language, say English, do you ever worry whether the alphabets a,b,c,... belong to some collection or not? Surely not.

But I think that we do not worry about that mainly because we have an explicit list of what my alphabets are and also we have a precise criteria (so, we are inclined to believe) to determine whether a particular symbol is our alphabet or not.

My teacher also said that the correct way to view the situation is, "We have some symbols, whose meaning we don't know and we don't need to know. We also don't know and neither need to know where the symbols exist, whether in a collection or not." But when I asked him what he exactly meant when he said that we "have some symbols", he said that it is simply a variant of saying that the symbols exist.

The Question

  • If the symbols exist, then they exist where? (In case of alphabets, if I want to be painfully philosophical, I may be tempted to say that the alphabets exist in some "imaginative world" of my mind - which will in this case be "collection".)

  • Is it even meaningful to ask 'where the symbols exist'?

I think that it is necessary to have a Theory of Collection because the phrase "[w]e have some symbols" is very much vague. Consider the following question for example,

  • Does the meaning (or sense) of "have" in "have some symbols" the same as that of "have" in "have a doll"? How can you say that?
  • There is no knowledge, science and mathematics without language. Maybe, there is no "intelligence" nor "humanity" without the possibility of communicating through it... For the aims of modern math, language is highly "symbolical". – Mauro ALLEGRANZA Jun 27 '16 at 13:38
  • @MauroALLEGRANZA: So symbols exist? – user 170039 Jun 27 '16 at 13:39
  • For sure they exists. – Mauro ALLEGRANZA Jun 27 '16 at 13:40
  • @MauroALLEGRANZA: So, where do they exist? – user 170039 Jun 27 '16 at 13:41
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    If you 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 your banck account..). If no "not-physical" objects exist, why care about money. Brexit, -10% of stock markets... – Mauro ALLEGRANZA Jun 27 '16 at 14:12
2

I am going to give the overextended postmodernist view here, springing originally from Wittgenstein and psychoanalytic thinkers like Lacan. I am doing so just so that both ends of this range are pinned down. In some sense I think this is 'the truth', but it is far too difficult to actually use, and may not be worth considering in the context of teaching logic at all.

If the symbols exist, then they exist where? (In case of alphabets, if I want to be painfully philosophical, I may be tempted to say that the alphabets exist in some "imaginative world" of my mind - which will in this case be "collection".)

From that point of view the symbol is its intended effect upon people, so in that sense it exists. They exist in the realm of human intention or expectation and are backed up in the realm of the human imagination, including our simplified memories of the real past.

The letters of the alphabet do not exist as physical or even mental 'objects', they take variant forms and can be substituted by any potential form, for instance in binary coding, without loss of their power to be the letters that they are. I cannot put that object into your mind, I can only intend to, and accept the degree to which I do or do not succeed. If you do not already have a sense of the intention behind 'A'-ness, I cannot show you an A.

We learn this notion of 'A'-ness by playing out the effects by rote until they are natural to us: the notion is the collection of memories of its effects and imaginings about its potential deployments. So I can have a symbol in my mind only if I know it has an intended purpose, and I use that symbol to evoke a complex of memories and images in another person's imagination (or in my own).

Is it even meaningful to ask 'where the symbols exist'?

In order for the question to make sense, it has to be stripped of the basic assumptions we have about mental objects, or it becomes circular. Mental objects as real things are themselves simply symbolic and any realization of an idea is made up of symbols. Symbols are the things that symbolize, and that is not helpful. So the idea of a mental or symbolic realm becomes baseless and circular. One simply has to insist it into existence by fiat, which is unsatisfying.

Instead, you need to back off from the situation within your mind into the social realm. Once you see symbols or thoughts in terms of their usage in communication, then it becomes clear that we abstract expected usages into symbolic representations using some kind of 'cleaning up' mechanism that is largely similar among human beings. That is how you the above positions ends up saying that symbols are 'expectations' and nothing more.

I think that it is necessary to have a Theory of Collection because the phrase "[w]e have some symbols" is very much vague. Consider the following question for example,

We have an intuition of collection that empowers us to span the gap between parts and wholes without thinking about it too much. We can take that as our basic 'Theory of Collections', but it is not very much of a theory. Attempting to express it logically turns out to be extremely difficult, and to lead into conflict with very basic concepts like universality or negation. All of this is tied up in one neat package by Russel's Paradox.

Does the meaning (or sense) of "have" in "have some symbols" the same as that of "have" in "have a doll"? How can you say that?

No, it is the same as that in "we have laws" or "we have nations". By the laws we do not mean their impressions upon paper, we mean the expectation they encode. Likewise, by a nation we do not mean its people. When all the Romans had passed away, there was still Rome as an exemplar 'nation' upon which other nations built their legal systems, and even their languages.

These things are not physical objects, they are abstractions that hold our expectations of how things should happen -- they reify social processes. They are 'the content of language-games', which occupy one side of a natural duality between things and the rules we insist upon applying to them.

  • Excellent! I agree with the first paragraph too. Perhaps this is the reason for which my teacher (who is a logician) don't consider my questions seriously. – user 170039 Jun 27 '16 at 16:15
  • I read this answer once again and after reading your first paragraph I wonder why you wrote that the points 'may not be worth considering in the context of teaching logic at all'. Can you elaborate this a bit? – user 170039 Jun 30 '16 at 13:38
  • Logic is not ontology or epistemology. Although those boundaries are kind of arbitrary, and may let us oversimplify things, they also help us focus. Postmodernism is a very specific way of looking at what is real, and how we know things -- it is full of negative postures (like that symbols themselves are not mental objects) and striking discontinuities from what we consider normal (like the idea 'I have a nation' is not like the idea 'I have a doll'). To introduce it to people who really just want to focus on our shared intuitions about words and reference would just complicate their world. – user9166 Jun 30 '16 at 16:23
  • The basics of logic can be agreed upon from many different ontologies and epistemologies (from Platonism to Grammatical Formalism to Modal Realism to Functional Intuitionism to this one), and introducing any of them in detail might also just sow dissension and distrust among those from different camps than the professor. – user9166 Jun 30 '16 at 16:26
8

What is a variable ?

A variable is a "syntactical" object:

it lives in formalized languages.

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

When we have a propositional function like "x is a red" or "x is odd" we have a statement like "it is red" or "it is odd".

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.

The same for a mathematical propositional function: if I assert "x is odd", its meaning (its truth value) depends on the context.

In this case, a context is specified by an interpretation: if I focus on the domain N of natural numbers and I assign to the variable x (a term of the language) as "temporary" denotation the number 3, then my assertion become meaningful (in this case, a true one).


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

Thus,

a variable 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" 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.

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...

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:


Regarding the question

What are collections of sets ?

(but what is the link with the title ?), it is a very complex question: we have to dig deep into Philosophy of mathematics, and I'm not able to do it.

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 ?).

Arithmetic is about numbers: the theorems of arithmetic are true of all those structures (possibly: only one) that satisfy the arithmetical axioms.

Set theory is about sets: the theorems of set theory (unfortunately, we have more than one set theory...) are true of all those domains that satisfy the axioms.

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.

The "safe" collections are those that the theory asserts (axioms) or proves (theorems) that exist; we call them : sets.

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:

x = x

is the concept identifying the collection of all sets.


See e.g. by Penelope Maddy:

  • Can you tell what exactly is the answer to (1.) and what exactly is the answer of (2.) in your post? – user 170039 Jun 27 '16 at 14:07
  • In short, I think that the variables exist in their domain of discourse, in much the same way as that of elements existing in a set. – user 170039 Jun 27 '16 at 14:18
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    This answers the questions much better than the accepted answer, which just sounds more 'sexy'. – Eliran Jun 27 '16 at 21:15
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    @jobermark but Mauro explicitly includes the "social" in his notion of "world", so the symbols are in the world as well. Furthermore, I fail to see why an individual is prevented from interpreting the expressions they produce themselves, thus promoting them to symbols. Of course the person needs to learn how to interpret, which might require social interaction, but the act of interpretation itself does not require social interaction when learnt, e.g. reading and writing and revising. – Vladimir Sotirov Jul 1 '16 at 15:56
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    I'm saying that the distinction you feel he did not make in his answer was already taken into account by what he wrote and provided a quote. Instead of explaining how my interpretation might be incorrect (e.g. how your interpretation of the particular quote is different), which I specifically ask you about, you just double down on proclaiming he did not make the distinction (and that you don't misunderstand! oh the hubris!). I've no idea what you mean by "dimensions in... ontology" or by "No answer". – Vladimir Sotirov Jul 2 '16 at 5:09
2

I frankly think Mauro's answer is, if not wrong, then at least misleading. I think the set theory part of it is mostly fine (but irrelevant the OP's question), but I very strongly disagree with the idea that sets are merely special kinds of collections: rather, each set gives rise to the collection of its elements, but the set is not the collection and the collection is not the set.

On the other hand, I think his analysis of what variables are is very misleading and possibly outright wrong.


First, the meaning of a statement is not its truth value. Obviously whether the natural language sentence "it is red" is meaningful does not depend on whether the thing referred to by "it" is actually red. Similarly, a mathematical formula with a free variable like "x is red" can be meaningful even though it does not (and cannot) have a truth value (only propositions, i.e. formulas without free variables, can have truth values, although undecidable propositions also don't have truth values but are still meaningful). Whether "x is red" is meaningful depends on whether the type of x supports redness as a property it may or may not posses.

This runs somewhat counter to the usual formulation of first-order logic that mathematicians use, in which formulas with free-variables are only auxiliary constructs on the way to formulating propositions, that is, formulas with no free variables, which are always meaningful and can be true or false when given semantics in the sense of model theory. However, ascribing such an auxiliary status to formulas with free variables is merely a mathematical convention, not a philosophical position. In particular, in other formal languages, like type theory or fragments of first-order logic, assertions with free variables play a fundamental role and must be considered meaningful for the mathematics to even get of the ground.

Second, a statement with pronouns is only meaningful if the pronoun has a(n implicitly or explicitly) specified referent. Reference is the fundamental grammatical function of pronouns. Variables, on the other hand, do not have referents, they do not refer. To suggest that variables are like pronouns may misleadingly encourage the misconception that variables are the same thing as "unknowns" – they are not.

Rather, they key function of variables (in logic, not programming) is not that they refer, but that they can be substituted for by other expressions (of the same type as the type of the variable). Thus, the "x" in the statement "x is red" can be substituted for by any expression of the same type as "x", to produce a new statement. This natural language example is a bit difficult to deal with, because the type of "x" is unspecified, hence is implicitly everything. This means that I can substitute anything for "x", e.g. "horse" for "x", or "unicorn", or "banana", or "two", or "Mauro" to get various other statements: "unicorn is red", "horse is red", "banana is red", "two is red", "Mauro is red".


Nothing about variables has anything to do with the language being "formal" or "formalized", and even though I've expressed all of the above with literal symbols, variables themselves are not literal symbols, anymore than an actual tree is the literal word "tree". I don't think either Mauro's or jobermark's answers mean symbols in the sense of literal, written-on-the-page symbols; instead they seem to mean symbols as references to social experience, which is fine (though somewhat tautological I'd say) insofar as language itself is a social construct.

But I take exception in the determination of what variables are by declaring that variables are symbols and then determining the ontology of symbols. This is because to declare variables are symbols is merely to declare that variables are a part of language, and to determine the ontology of symbols is to determine the ontology of language. But the ontology of variables itself does not get specifically determined (because there are many more symbols other than variables!)

In other words, granting that language is made up of symbols, and that variables are symbols, it is important to say what kind of symbols variables actually are, i.e. what kind of social experiences variables refer to (if we buy the idea that social reference is the ontological status of symbols). I think they are much closer to common nouns than they are to pronouns, but the translation from logic to natural language is not direct. The reason the translation is indirect is that common nouns in natural language are almost always specific, that is, already substituted for. A better (but still poor) rendition of "unicorn is red" in natural language would be "Consider a (hypothetical) unicorn. This unicorn is red".

Here the common noun (a variable) by its very nature comes with its type (unicorn) and remains unsubstituted for because of the introductory "Consider a (hypothetical) unicorn". Note that in the second clause "This unicorn is red", "this unicorn" can be understood to either act like a pronoun in that it has a referent, which is the "(hypothetical) unicorn" of the first clause, or better: it can be undestood as asserting that the variable "unicorn" in "this unicorn" is equal to the "(hypothetical) unicorn" you were being asked to consider earlier. If instead the introductory clause said "Consider Charlie the unicorn", then the variable "unicorn" is asserted equal to "Charlie the unicorn". The fact that Charlie the unicorn is not red but white has to do with the fact that the proper noun (phrase) "Charlie the unicorn" has a referent by which we can determine the truth of "This unicorn is red".


My actual position would be that the whole symbol discussion is a red-herring: variables are a concept auxiliary to the concepts of substitution and hypotheticals; variables are a part of language (hence symbolic in the sense of the given answers) only insofar as concepts are expressed and possibly learned in language.

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    Set, class, type, collection, etc. etc. etc. He never said the meaning of a statement was its truth value, only that the meaning determines the truth value in different contexts, etc. etc. etc. The formalism is not enriched by indirection and empty wrappers. More vocabulary doesn't help. If I ask you for a set of bowling pins, you do in fact give me the members of a collection of bowling pins. We don't need to identify the collected whole constituted by the parts of a chicken... – user9166 Jul 1 '16 at 16:28
  • Moving the meaning of variables out of syntax and into pyschology puts you basically in line with my position that they express the intention to communicate. Rather than making up a language barrier to haggle over, why not simply define symbols in a way that bypasses the red-herring and gets to the point. – user9166 Jul 1 '16 at 16:31
  • 1) he literally says "... mathematical propositional function: if I assert 'x is odd', its meaning (its truth value) depends on the context." 2) I've no idea what "indirection" and "empty wrappers" you are referring to. If you are not going to back up such judgments with specifics, then I can only assume you're discussing this in bad faith. 3) Since the context is mathematical set theory, the distinction between sets and collection is exactly that sets are the collections that can be treated as objects themselves, i.e. as collected or completed wholes – Vladimir Sotirov Jul 1 '16 at 21:53
  • 4) the reason not to redefine symbols in a way that bypasses the red-herring is because it's invalid to critique a theory by redefining its terms. – Vladimir Sotirov Jul 1 '16 at 21:54
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    Where am I "largely approving" of Mauro's answer? I'm amused you end up repeating what I wrote in my opening paragraph; makes me question your ability to read on top of questioning whether you're aware that dressing up opinions as argument does not make them valid (seeing as you refuse to back up any of your claims so far). – Vladimir Sotirov Jul 7 '16 at 3:27

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