## Trying to clarify objects in Wittgenstein's Tractatus

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I agree that the expression "logical objects" may be read in two ways. It can be referring to either 1) objects that are logical or 2) logic can be an object.

An unhappy apple is an illogical proposition not an illogical object. An apple on the table or inside the sun is not a combination of objects it is a relation of the objects apple and table (on) or apple and sun (in).

As concepts can be simples, the concept "grass" could be a simple, and as words such as "grass" logically picture an object such as grass existing in a logical space, this suggests that objects such as grass are also simples.

I don't know if you are attempting to interpret the Tractatus or argue against it. He makes a distinction between proper concepts such as grass and formal concepts such as 'simple object'.

At 4.126 Wittgenstein introduces the term "formal concepts".
— Fooloso4

In the function T (x), where T is on a table, the function T (x) is true if the variable x satisfies the function T (x). For example, T (x) is true if the variable x is a book.

As I understand it, the variable x is what Wittgenstein is defining as a formal concept.

Book is not a formal concept. In a proposition it does not have both the name 'book' and the variable name for a formal concept 'x'.
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An unhappy apple is an illogical proposition not an illogical object. An apple on the table or inside the sun is not a combination of objects it is a relation of the objects apple and table (on) or apple and sun (in).

An object in logical space must be a logical object, meaning that its necessary properties must be logical. For example, if an apple was a logical object in logical space, it would have the necessary properties such as weight, colour and taste. An apple having the necessary property of happiness would not be a logical object.

In a state of affairs, objects are combined, necessitating a relation between them.
3.1432 – Instead of "The complex sign "aRb" says that a stands to b in the relation R", we ought to put "That "a" stands to "b" in a certain relation says that aRb"
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I don't know if you are attempting to interpret the Tractatus or argue against it. He makes a distinction between proper concepts such as grass and formal concepts such as 'simple object'.

There are proper concepts such as "grass" and formal concepts such as the variable "x".

Wittgenstein in the Tractatus never explains what a simple object is, other than there must be simple objects, and that they must exist necessarily not contingently.

As states of affairs exists in logical space, and a state of affairs is a combination of objects, this means that these objects exist in logical space. An object existing in logical space infers that it it is a logical object.

I'm suggesting that in the expression "grass is green" is true iff grass is green, objects such as grass are not referring to actual objects, which are divisible, but must be referring to logical objects, which can be indivisible, and are simples.
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Book is not a formal concept.

I agree. The variable x is the formal concept, not the book.
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For example, if an apple was a logical object in logical space, it would have the necessary properties such as weight, colour and taste.
.

The properties of objects in logical space are formal, internal, necessary properties. Weight, color and
taste are not necessary properties. 'Fact' is a formal concept. The facts in logical space are about the formal, logical structure of of the world. Facts in physical space are accidental, contingent. They are made possible by the necessary, logical structure of the world.

There are proper concepts such as "grass" and formal concepts such as the variable "x".

'x' is not a formal concept. It is the name used to refer to the formal concept.

I'm suggesting that in the expression "grass is green" is true iff grass is green, objects such as grass are not referring to actual objects, which are divisible, but must be referring to logical objects, which can be indivisible, and are simples.

The name "grass" as it occurs in a proposition refers to an actual complex object. I think what you are getting at is along the lines of what I said above:

As part of a propositional analysis apples and tables can function as simples. Whether they do does not depend on their being possible, but on whether further analysis is needed in order for the proposition to make sense, that is, to know what is the case if it is true.

Book is not a formal concept.
— Fooloso4

I agree. The variable x is the formal concept, not the book.

It is because a book is not a formal concept that Wittgenstein does not refer to it by a variable name. The variable is a name not a concept.
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In the function T (x), where T is on a table, the function T (x) is true if the variable x satisfies the function T (x). For example, T (x) is true if the variable x is a book.

As I understand it, the variable x is what Wittgenstein is defining as a formal concept.

You're close, but this isn't quite right, I don't believe.

In the function: "T(x)", both "T()" and "x" show that to each corresponds a different formal concept.

"That anything falls under a formal concept [such] as an object belonging to it, cannot be expressed by a proposition. But it shows itself in the sign of this object itself. (The name shows that it signifies an object, the numerical sign that it signifies a number, etc.)

Formal concepts cannot, like proper concepts, be presented by a function. For their characteristics, the formal properties, are not expressed by the functions. The expression of a formal property is a feature of certain symbols. The sign that signifies the characteristics of a formal concept is, therefore, a characteristic feature of all symbols, whose meanings fall under the concept. The expression of the formal concept is therefore a propositional variable in which only this characteristic feature is constant" (4.126).

So, we cannot, for example, say:

"Red is a color" or "C(r)"
or
"1 is a number" or "N(1)"

for this is senseless.

Nor can we say:

"Color is a formal concept" or "F(c)".

The formal concepts are presupposed by the objects that already contain the characteristic features of the formal concept under which they fall.

This is why:

4.1271 – Every variable is the sign for a formal concept

In: "F(x)"

"x" is a simple object in that it presupposes the general form, or general characteristics necessary of any input which can satisfy the function. This is to say, that there is a formal concept associated with it, but "x" is not itself a formal concept, nor does it name a formal concept.
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You're close, but this isn't quite right, I don't believe.

In the function: "T(x)", both "T()" and "x" show that to each corresponds a different formal concept.

If, as Russell stipulates, x is a book, then there are no formal concepts in "T(x)". I don't know what () on the table means.

Wittgenstein is doing propositional analysis not coding.

This is to say, that there is a formal concept associated with it, but "x" is not itself a formal concept, nor does it name a formal concept

The first part is correct. The second part needs clarification. Formal concepts are represented in conceptual notion by variables. (4.1272)
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Ah, yes, I think I've misread that bit, now that I've looked at it again.

I'll look it over again later :)
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Okay, so looking over it once more...

If, as Russell stipulates, x is a book, then there are no formal concepts in "T(x)". I don't know what () on the table means.

A formal concept defines how the variables "T" and "x" are to "behave" or perhaps a better way to say it, is how they are to be understood.

These aren't like "proper concepts", such as "red", "hard", etc. which settles the external properties of complex objects.

In the proposition:

"The grass is green"

We have the presentation of a complex object with the material property of being red.
If we analyze the proposition into the elementary proposition we can the presentation of a "proper concept", which takes as input a simple object:

Fx

Now, we can in some sense talk about the structure of the elementary proposition, and we can note that whatever can be taken as input for "x" must be of a certain type, or kind. To this type corresponds a formal concept. We cannot, for example, input a proper number to which corresponds the formal concept of number for say, a simple object.

So, while we can say:

"There are two red fruits"

this analyzes into:

∃x(P(x)) ∧ ∃y(P(y) ∧ (x≠y))

There is no sign corresponding to the formal concept "number" despite what appears to be a number presented in the proposition.
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@Fooloso4

I now no longer believe that the x in F (x) is a formal concept, but in fact represents a concept proper.

A formal concept defines how the variables "T" and "x" are to "behave" or perhaps a better way to say it, is how they are to be understood. These aren't like "proper concepts", such as "red", "hard", etc. which settles the external properties of complex objects.

Consider the proposition "grass is green".

If x = grass satisfies the function Green (x) then Green (x) is true.

Where grass and green are "concepts proper" (4.126)

From 4.12, a proposition can represent concepts proper, such as grass and green, but cannot represent logical form, ie "formal concepts" (4.126). A proposition can only "show" (4.121) logical form.

So, within the proposition "grass is green", where is the logical form? The logical form of the proposition must be shown by the word "is", which is a relation between concepts proper. For Wittgenstein, unlike Frege, relations are not object. Relations have no existence other than relating concepts proper, in that if the concepts proper were removed, no relation would remain as some kind of Platonic Form.

As the concept "is" can only be shown and not represented, it is a formal concept rather than a concept proper.

Similarly within the function Green (x), where is the logical form? The term "Green" infers the expression "is green", where "is" is the formal concept and green is the concept proper.

For both the proposition "grass is green" and the function Green (grass), in the first the formal concept "is" is explicit and in the second the formal concept "is" is inferred. The concepts proper remain grass and green.
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We cannot, for example, input a proper number to which corresponds the formal concept of number for say, a simple object.

A concept proper cannot be a number.

From 4.0312, "logical constants" such as “and,” “or,” “if,” and “then” are not representatives. This makes sense, in that logical constants are not Platonic Forms.

Suppose there is a horse in a field and another horse enters the same field. We can say "horse AND horse". But if one horse left the field, the AND would not remain in the field as some kind of Platonic Form. AND only exists in the relationship between two concepts proper.

3.1432: We must not say, “The complex sign ‘aRb’ says ‘a stands in relation R to b;’” but we must say, “That ‘a’ stands in a certain relation to ‘b’ says that aRb.”

Similarly, we could "horse 2 horse". But if one horse left the field, the 2 would not remain in the field as some kind of Platonic Form. Numbers only exist in the relationship between two concepts proper.

As numbers only exist in the relationship between concepts proper, and relations in the Tractatus are not objects, they are formal concepts.
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So, while we can say: "There are two red fruits" this analyzes into:
∃x(P(x)) ∧ ∃y(P(y) ∧ (x≠y)) There is no sign corresponding to the formal concept "number" despite what appears to be a number presented in the proposition.

However, isn't it the case that the logical symbol ∧ (which means AND), is where number is introduced into the expression. For example, "horse AND horse" by its very nature has introduced the concept of number, in that if I wanted to explain the concept of the number 2 to someone, I could say either "horse ∧ horse" or "horse AND horse" and then show them the field with 2 horses in it.
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See 4.12721. The concept of a number is a formal concept. Particular numbers are not. They fall under the concept of a number.
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See 4.12721. The concept of a number is a formal concept. Particular numbers are not. They fall under the concept of a number.

4.1272 "The same applies to the words "complex", "fact", "function", "number", etc - They all signify formal concepts............."1 is a number", "There is only one zero" and all similar expressions are nonsensical

Why do you think that particular numbers, such as the number 1, are not formal concepts?
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I think Wittgenstein is saying that an "object" like the number 1 has a sense if it is an object or a description. So, "There is one horse". Or "Look, 1 plus 1 is 2." Or "Here is one". But once you juxtapose 1 with a class of objects that it belongs, (a number), that cannot be "shown" in the state of affairs, so fails to make "sense" (literally, because obviously the phrase resounds as true to us in grammatical form).

I just don't buy this distinction which he was trying to make, which seemed to be following Russell's own paradox about the logic of the logical structures themselves, and the incompleteness theorem, etc.

Ideas exist in the world, and therefore, ideas can be objects. All this seems to stem from a need to lock things into "states of affairs of the world" that can be true or false. Since "One is a number" cannot be true or false as a state of affairs in the world, it fails to be a sensical sentence, at least for what he thinks is proper as to what can be said clearly.
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Why do you think that particular numbers, such as the number 1, are not formal concepts?

See the statement I put in bold:

A formal concept is given immediately any object falling under it is given. It is not possible, therefore, to introduce as primitive ideas objects belonging to a formal concept and the formal concept itself. So it is impossible, for example, to introduce as primitive ideas both the concept of a function and specific functions, as Russell does; or the concept of a number and particular numbers.
(4.12721)

If I say: "There are a number of horses" that is expressed by the variable x. This does not tell us how many horses. If, however, I say: "There are three horses" then the number of horses is not expressed as the variable 'x', which could mean any number of horses, but as '3'.The logical structure of the proposition is the same, but in this case I am not talking about the formal concept 'number'.
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I think Wittgenstein is saying that an "object" like the number 1 has a sense if it is an object or a description.

From Bertrand Russell's Introduction:
It follows from this that we cannot make such statements as “there are more than three objects in the world”, or “there are an infinite number of objects in the world”. Objects can only be mentioned in connexion with some definite property. We can say “there are more than three objects which are human”, or “there are more than three objects which are red”

3.1431 The essence of a propositional sign is very clearly seen if we imagine one composed of spatial objects (such as tables, chairs and books) instead of written signs. Then the spatial arrangement of these things will express the sense of the proposition.

It seems that an object like the number1 is a formal concept, and being a formal concept, can never be the sense of a proposition and can never be described by a proposition, but only shown.
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It seems that an object like the number1 is a formal concept, and being a formal concept, can never be the sense of a proposition and can never be described by a proposition, but only shown.

Thus, it seems to be the case for Witt’s theory, 1 + 1 = 2 is formal as it is not a state of affairs per se, but a description of a category of sets that may occur as a state of affairs. It’s a description of a class not of a particular state of affairs that could be true or false.

I might argue that more Kantian discussion is needed here which is completely ignored it looks like.
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(4.12721) "A formal concept is given immediately any object falling under it is given. It is not possible, therefore, to introduce as primitive ideas objects belonging to a formal concept and the formal concept itself. So it is impossible, for example, to introduce as primitive ideas both the concept of a function and specific functions, as Russell does; or the concept of a number and particular numbers."

As I understand it, a proposition cannot express a formal concept, ie the logical structure of the proposition, but it can only be shown by the proposition.

From Bertrand Russell on Something by Landon D.C. Elkind

In their monumental Principia Mathematica, Russell and his co-author Alfred North Whitehead attempted to create a logically sound basis for mathematics. In it their primitive proposition ∗9.1 implies that at least one individual thing exists. It follows that the universal class of things is not empty. This is stated explicitly in proposition ∗24.52. Whitehead and Russell then remark: “This would not hold if there were no instances of anything; hence it implies the existence of something.” (Principia Mathematica, Volume I, 1910, ∗24). Here then, logic seems committed to the existence of something.

Whereas the early Bertrand Russell thought that a pure logical structure wasn't possible, Wittgenstein believed that a pure logical structure, one of formal concepts, was possible.

IE, 4.12721 is saying that the concept of a number and particular numbers cannot be primitive but are both formal concepts.
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If, however, I say: "There are three horses" then the number of horses is not expressed as the variable 'x', which could mean any number of horses, but as '3'.

I could say "there are 3 horses".

As the number 3 cannot be described but only shown, this makes it a formal concept.

For Wittgenstein, numbers are not objects having Platonic Form that can be described in the absence of the horses that they are quantifying.
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Thus, it seems to be the case for Witt’s theory, 1 + 1 = 2 is formal as it is not a state of affairs per se, but a description of a category of sets that may occur as a state of affairs. It’s a description of a class not of a particular state of affairs that could be true or false.

As the elementary proposition "1 + 1 = 2" asserts the existence of a state of affairs, the logical structure of the elementary proposition "1 + 1 = 2" must be mirrored in the state of affairs.

As numbers are formal concepts, I think I am right in saying that Wittgenstein would call this proposition meaningless.
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As numbers are formal concepts, I think I am right in saying that Wittgenstein would call this proposition meaningless.

Yes, but Kant would simply classify it as analytic a priori. It is a truth that can be grasped through purely reasoning and not experience (equivalent to Wittgenstein's "state of affairs in the world"). But I am perplexed why with all this epistemological history he could have drawn from, he ignores it.
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What falls under a formal concept is not another formal concept.

When something falls under a formal concept as one of its objects, this cannot be expressed by means of a proposition. Instead it is shown in the very sign for this object. (A name shows that it signifies an object, a sign for a number that it signifies a number, etc.)
(4.126)

The sign '3' signifies a number, not the concept 'number'. '3' falls under the formal concept number. If '3' was a formal concept then every number would be a formal concept. In that case we would have the formal concept 'number' and the formal concepts '1', '2', '3' .... and so on.

Every variable is the sign for a formal concept.
For every variable represents a constant form that all its values possess, and this can be regarded as a formal property of those values.
(4.1271)

'Number' is the constant form. 1, 100, and 1,000 are variables that have as a formal property this formal concept.
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'Number' is the constant form. 1, 100, and 1,000 are variables that have as a formal property this formal concept.

I am unsure what you are saying here. I don't see how a mathematical statement like "1 + 1 = 2" is NOT a formal concept, UNLESS it was about a "state of affairs of the world". It isn't. It is Platonic non-sensing according to this view. If it IS something that is a state of affairs, tell me "where" it is found, other than the concept itself (the set of 1 and the set of 1 combining). It is a purely logical statement, not a state of affairs in the world. It's not saying THIS 1 plus THAT 1 = 2. Just 1 + 1 = 2, a generalized statement. As I said to @RussellA:

Yes, but Kant would simply classify it as analytic a priori. It is a truth that can be grasped through purely reasoning and not experience (equivalent to Wittgenstein's "state of affairs in the world"). But I am perplexed why with all this epistemological history he could have drawn from, he ignores it.
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Mathematical equations are pseudo-proposiitons , but this does not mean the equation is a concept, either proper or formal. 1+1=2 is not concept, it is a calculation.

Mathematics is a logical method.
The propositions of mathematics are equations, and therefore pseudo-propositions.
(6.2)
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Yes, but Kant would simply classify it as analytic a priori. It is a truth that can be grasped through purely reasoning and not experience (equivalent to Wittgenstein's "state of affairs in the world"). But I am perplexed why with all this epistemological history he could have drawn from, he ignores it.

It has been said that Wittgenstein never studied philosophy as such, although he may have learnt from certain other philosophers he was in direct contact with, such as Bertrand Russell. So he did ignore epistemological history as he was not interested in the history of philosophy as a field of knowledge.

There may be a difference between Kant's analytic a priori and Wittgenstein's formal concept, in that Kant's analytic a priori is knowledge prior to any knowledge about the world, whereas Wittgenstein's formal concept straddles on one side language and thought and on the other side the world.
4.1272 - "The same applies to the words "complex", "fact", "function", "number" etc. They all signify formal concepts"
4.1274 "To ask whether a formal concept exists is nonsensical"
6.22 "The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics.

In the Tractatus, the formal concepts existing in language, which cannot be described but only shown, are mirrored by formal concepts that also exist in the world
4.21 - "The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.

IE, for Kant, the analytic a priori exists prior to any knowledge of the world, whereas for Wittgenstein the formal concepts in language are mirrored by formal concepts in the world.
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What falls under a formal concept is not another formal concept..................If '3' was a formal concept then every number would be a formal concept.

Mathematical equations are pseudo-proposiitons , but this does not mean the equation is a concept, either proper or formal. 1+1=2 is not concept, it is a calculation.

'Number' is the constant form. 1, 100, and 1,000 are variables that have as a formal property this formal concept.

The Tractatus mentions three kinds of concepts: formal concept, concept proper and pseudo-concept.

Formal concepts
The logic that ties elementary propositions together and states of affairs together cannot be described but can only be shown.
4.1272 "The same applies to the words "complex", "fact", "function", "number" etc. They all signify formal concepts"
4.1274 "To ask whether a formal concept exists is nonsensical"

Pseudo-concepts
Nowhere in the Tractatus does Wittgenstein describe what an object is, other than they are necessary for the substance of the world. Objects are pseudo-concepts.
4.1272 "This the variable name x is the proper sign for the pseudo-concept object.

Concepts proper
Concepts proper are things in ordinary language such as apples, tables and books
4.126 "the confusion between formal concepts and concepts proper"

Two types of elementary propositions can be considered, Tractarian and ordinary language. Problems arise when ordinary language elementary propositions are used to illustrate Tractarian elementary propositions.

Ordinary language elementary propositions
Ordinary language elementary propositions must include both formal concepts and concepts proper, such as "grass is green", where "grass" and "green" are objects and "is" provides the logical structure.

Tractarian elementary propositions
Tractarian elementary propositions must include both formal concepts and pseudo-concepts.

Elementary propositions mirror states of affairs in the world
4.21 "The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.

For example, in the elementary proposition "F (x)", x is the sign for the pseudo-concept object. F is the sign for the internal property of x, and as an internal property is a necessary part of the object x. As objects are pseudo-concepts, then F is also a sign for a pseudo-concept. The logic is shown in the function F (x) itself, signifying a formal concept.

Properties are internal if necessary to the object
4.123 "A property is internal if it is unthinkable that its object should not possess it"
4.124 "The existence of an internal property of a possible situation is not expressed by means of a proposition: rather it expresses itself in the proposition representing the situation, by means of an internal property of that proposition".

As Bertrand Russell writes in the Introduction, objects can only be mentioned in connexion with some definite property
"Objects can only be mentioned in connexion with some definite property."
"It follows from this that we cannot make such statements as “there are more than three
objects in the world”.................the proposition is therefore seen to be meaningless.........We can say............“there are more than three objects which are red”"

The number 3
There is the universal concept of number and there are particular numbers, such as 3.

Number is described as a formal concept.
4.1272 "The same applies to the words "complex", "fact", "function", "number" etc. They all signify formal concepts"
4.126 - "When something falls under a formal concept as one of its objects, this cannot be expressed by means of a proposition. Instead it is shown in the very sign for this object. (A name shows that it signifies an object, a sign for a number that it signifies a number, etc.)"

Objects are pseudo concepts because they exist in the world and make up the substance of the world. Mathematical equations, which show the logic of language and the world, are, as you say "a logical method", and as part of the logical method are formal concepts.

As the Tractatus uses the term pseudo-proposition in a negative way, mathematical equations cannot be pseudo-propositions
4.1272 "Whenever it is used in a different way, that is as a proper concept-word, nonsensical pseudo-propositions are the result"
5.535 "This also disposes of all the problems that were connected with such pseudo-propositions"
6.22 "The logic of the world, which is shown in tautologies by the propositions of logic, is shown in equations by mathematics.

What is the number 3 in the Tractatus? It cannot be a pseudo-concept as it doesn't exist as part of the substance of the world. It should be treated as any other logical function, such as "and", "or", "if" or "then", which make up the fabric of logical structure, and are formal concepts.

Logical constants don't represent, but show.
4.0312 "My fundamental idea is that the "logical constants" are not representatives; that there can be no representatives of the logic of facts."

The number 3 is a sign that signifies a number. Numbers are formal concepts. Therefore, the number 3 is a sign that signifies a formal concept, in the same way that the logical constant "and" also signifies a formal concept.
4.126 "A name shows that it signifies an object, a sign for a number that it signifies a number, etc"
4.1271 "For every variable represents a constant form that all its values possess, and this can be regarded as a formal property of those values."

IE, within the Tractatus, the number 3 cannot be a pseudo-object as it doesn't make up the substance of the world, but because it is part of the logical structure of both elementary propositions and state of affairs, it must be, as with all particular numbers, and as with all logical constants, a formal concept.
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Mathematical equations are pseudo-proposiitons , but this does not mean the equation is a concept, either proper or formal. 1+1=2 is not concept, it is a calculation.
@RussellA

Right, and that makes no sense to me to break that apart from the notion of "formal concepts". Being this is his own system, he can do whatever, I guess.

Why is "One is a number" a formal concept and "1 + 1 = 2" not a "formal concept"? I can break the world up in any number of ways, and make exceptions for everything that doesn't fit quite right.
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It has been said that Wittgenstein never studied philosophy as such, although he may have learnt from certain other philosophers he was in direct contact with, such as Bertrand Russell. So he did ignore epistemological history as he was not interested in the history of philosophy as a field of knowledge.

And sure, why should a philosopher NOT consult past philosophers who were discussing similar themes :roll:. Let's reinvent the wheel!

There may be a difference between Kant's analytic a priori and Wittgenstein's formal concept, in that Kant's analytic a priori is knowledge prior to any knowledge about the world, whereas Wittgenstein's formal concept straddles on one side language and thought and on the other side the world.

He cared about thoughts? He sure seems to give it short shrift....Just read some Kant, and bring some thought back (and stop pretending everything is "language") :wink:.

In the Tractatus, the formal concepts existing in language, which cannot be described but only shown, are mirrored by formal concepts that also exist in the world
4.21 - "The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.

Why can't they be described but 1 + 1 = 2 can be so?

IE, within the Tractatus, the number 3 cannot be a pseudo-object as it doesn't make up the substance of the world, but because it is part of the logical structure of both elementary propositions and state of affairs, it must be, as with all particular numbers, and as with all logical constants, a formal concept.

So, are you agreeing with me? That was what I said, that numbers (or rather equations) are formal concepts because they are not abouts states of affairs of the world. Again, Kant is informative here, it is an analytic a priori statement. You don't need to know a state of affairs of the world, for this to be true (i.e. one doesn't need experiential evidence, and it is not contingently true on a state of affairs in the world).
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The Tractatus mentions three kinds of concepts: formal concept, concept proper and pseudo-concept.

Formal concepts are pseudo-concepts.

Objects are pseudo concepts because they exist in the world and make up the substance of the world.

'Object' is a pseudo-concept because it says nothing about what is the case, not because it makes up the substance of the world.

The number 3 is a sign that signifies a number. Numbers are formal concepts. Therefore, the number 3 is a sign that signifies a formal concept

'3' signifies the value of the concept number. A particular number falls under the concept number in a way analogous to 'table' falling under the concept 'object'. That does not mean that 'table' is a pseudo-concept.
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Why is "One is a number" a formal concept and "1 + 1 = 2" not a "formal concept"?

In the Tractatus, there seem to be formal concepts and pseudo-concepts. Pseudo-concepts are the objects which are necessary for the substance of the world. The rest is logic, which cannot be described but must be shown.

I think that the propositions "one is a number" and "1 + 1 = 2" should be treated in much the same way, as being part of the logical structure. Numbers are not objects.

As Bertrand Russell writes in the Introduction
"It follows from this that we cannot make such statements as “there are more than three
objects in the world”.................the proposition is therefore seen to be meaningless.........We can say............“there are more than three objects which are red”"

Numbers are not Platonic Forms that remain after the objects have been removed. Numbers play their part in the logical structure, not in providing any substance to the world.
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That was what I said, that numbers (or rather equations) are formal concepts because they are not abouts states of affairs of the world. Again, Kant is informative here, it is an analytic a priori statement.

Kant knows "1 + 1 = 2" prior to observing the world.

For Wittgenstein's Picture Theory, elementary propositions mirror states of affairs in the world
4.21 "The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs.

The Tractatus is saying that the logical part of the proposition "1 + 1 = 2" in language mirrors the logical part of the state of affairs 1 + 1 = 2 in the world.

One difference between Kant and Wittgenstein is that Wittgenstein's Picture Theory in the Tractatus does not engage with the possibility of knowing that 1 + 1 = 2 prior to observing the world (as I understand it).
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One difference between Kant and Wittgenstein is that Wittgenstein's Picture Theory in the Tractatus does not engage with the possibility of knowing that 1 + 1 = 2 prior to observing the world (as I understand it).

I think he might say something like, "1 + 1 =2" might be non-sense if it doesn't have a state of affairs in the world that it is discussing that can be true or false. But you used an interesting word here- knowing. Knowing is something the mind does. Wittgenstein seems to not care to discuss mind, but language limits. If signs are not signifying a possible states of affairs, they are not picturing anything, and thus cannot be communicated with any sense.

However, clearly, analytic and a prioricity statements exist. "All bachelors are male" is not a state of affairs. 1 +1 =2 is not derived from empirical evidence, but as a functioning of how numbers work. What does he think of such things that are not "states of affairs" in the world, but are nevertheless statements that can be communicated.

Let's say that his idea is, "These analytic a priori statements cannot provide information about the world itself", what does this add to the philosophical school of ideas? Kant already elucidated that it is only empirical and synthetic statements that the "world" can inform us, but that a priori and analytic statements are truths that our reasoning can inform us. Both can be communicated using symbols. One tells us about the state of affairs of the world, whether the case is true or false (synthetic-contingent, and experiential-empirical), and the other is necessary for language itself to function. Certainly, this could lead to a regress (definitions of definitions of definitions), and surely, at some point, it is simply just a matter of "knowing" the object is the object without any further explanation, but then we are getting into psychology, and NOT the "limits" of language. Surely I can point to these processes that account for object formation in the mind, and how we attach meaning to objects. And then, I have a "state of affairs" about how the mind KNOWS objects, and is not an infinite regress of definitions of the concept, but a theory of meaning that accounts for the concept-formation, and thus where language ends definitionally, I can continue on explanatorily with the psychology of concept-formation.
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Wittgenstein seems to not care to discuss mind, but language limits.

For Wittgenstein, thought was language and language was thought. I may disagree, but that seems to be his position. As he said, the limits of my language is the limits of my world.
Notebooks 1914-16 – 12/6/2016 – page 82.
Now it is becoming clear why I thought that thinking and language were the same. For thinking is a kind of language. For a thought too is, of course, a logical picture of the proposition, and therefore it just is a kind of proposition.
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If signs are not signifying a possible states of affairs, they are not picturing anything, and thus cannot be communicated with any sense.

Within the Tractatus, an elementary proposition pictures a state of affairs. The state of affairs pictured may or may not obtain. If there were no states of affairs to picture, then there would be no elementary proposition. It seems that one important feature of the Tractatus is in developing the modal idea of possible worlds, allowing us to talk about non-existent things, such as Sherlock Holmes and unicorns. This was something Bertrand Russell had trouble with, how to think of something that doesn't exist. For Wittgenstein, the problem goes away, as objects always exist, and only their combinations change. This allows the mind to move between actual and possible worlds .
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1 +1 =2 is not derived from empirical evidence, but as a functioning of how numbers work

For Wittgenstein in the Tractatus, there is no synthetic a priori. We cannot know "grass is green " or "1 + 1 = 2" prior to observing the world. Our only knowledge comes from observation. We have no analytic a priori knowledge. The Tractatus is an Empiricist Theory.

I disagree, but that is the Tractatus.

The Picture Theory is limited to elementary propositions mirroring states of affairs in the world. However, "1 + 1 = 2" is only the logical part of an elementary proposition, not a representative part of an elementary proposition. The logic of "1 + 1 = 2" in language is mirrored by the logic of 1 + 1 = 2 in the world as a state of affairs.

There is no temporal consideration, in that knowing "1 + 1 = 2" in language is contemporaneous with 1 + 1 = 2 being the state of affairs in the world.

Personally, as I believe that the world is fundamentally logical - in that one thing is always one thing, if thing A is to the left of thing B then thing B is to the right of thing A and if event C happens after event D then event D happened before event C - then if language mirrors the world, then language also will be fundamentally logical.

I think that the Picture Theory is fundamentally flawed, in that it leads to as you say "an infinite regress", but that is another matter.
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Surely I can point to these processes that account for object formation in the mind, and how we attach meaning to objects

From Wittgenstein’s Picture Theory of Meaning by Christopher Hurtado

The picture theory of meaning was inspired by Wittgenstein’s reading in the newspaper of a Paris courtroom practice of using models to represent the then relatively new phenomenon of auto-mobile accidents (Grayling 40). Toy cars and dolls were used to represent events that may or may not have transpired. In the use of such models it had to be stipulated which toys corresponded to which objects and which relations between toys were meant to represent which relations between those objects (Glock 300).

Yes a red toy car can picture a real red car, but the flaw in the Picture Theory is the statement "had to be stipulated", which has to happen outside the Picture Theory.

Suppose in the world is a red car, a blue bicycle and a green truck. Suppose in the model is a red piece of wood, a blue piece of metal and a green piece of marble.

The Picture Theory assumes that the red piece of wood pictures the red car, the blue piece of metal pictures the blue bicycle and the green piece of marble pictures the green truck. But why should this be so?

Why cannot it be the case that wood pictures a truck, metal pictures a bicycle and marble pictures a car?

Or perhaps red in the model pictures a distance of 3 metres, blue in the model pictures a distance of 5 metres and green in the model pictures a distance of 10 metres.

There is no necessity that a red piece of wood pictures a red car, and yet the Picture Theory depends on this unspoken necessity, which seems to me to be a fundamental flaw in the Picture Theory.

IE, I agree that Kant's Critique of Pure Reason makes more sense than Wittgenstein's Tracatus, although the Tractarian idea of modal worlds is very important in philosophy.

(Kyle Banick - Necessity and Contingency - YouTube)
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'3' signifies the value of the concept number. A particular number falls under the concept number in a way analogous to 'table' falling under the concept 'object'. That does not mean that 'table' is a pseudo-concept.

There are two kinds of objects, concepts proper and pseudo-concepts. There are
concepts proper in our ordinary world, such as "furniture", and there are pseudo-concepts in the Tractarian world, of which the variable x is the proper sign.

In our ordinary world, a "table" is a particular instantiation of the concept proper "furniture". However it is also the case that "a table" is another concept proper, which mat be instantiated in its turn.

In our ordinary world, something that falls under a concept proper can also be a concept proper.
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