But was Tractatus really aimed to dispute the position of mereological nihilism?............................How does he actually do that, rather than simply asserting premises that he thinks is true? — schopenhauer1

In the Tractatus, language shows the logical form of the world, and the world is the totality of facts.

But where exactly is this world?

It is said that the Tractatus can be read from both the viewpoint of Idealism, where the world exists in the mind, and from the viewpoint of Realism, where the world exists outside the mind.

If the Tractatus is attempting to show that language can be analysed into elementary propositions, where each elementary proposition is independent of all other elementary propositions, and where each elementary proposition pictures a fact in the world, then whether this world exists inside the mind or outside the mind is irrelevant

Therefore, the Tractatus need not pay any regard to Mereological Nihilism, which is is a philosophical idea specifically about a world that exists outside the mind.

As I can assert that "evil is bad" as a self-evident truth, perhaps the Tractatus can also assert that "the world is the totality of facts" as a self-evident truth. Anyone disagreeing that "evil is bad" or "the world is the totality of facts" then has the opportunity to present their argument.

After all, most of our statements are assertions, whether "I walked to the supermarket", "stealing is bad", "I admire Monet's aesthetic", "the world is a complex place" or "the play starts at 9pm". Rarely is it expected that we need to justify what seems to be self-evident.

Also, you didn't answer my question.. "What philosophy DOESN'T think their understanding of the world comprises independent facts"? I have yet to meet a person, who thinks "This is morally bad, or this is good" is the same as "The cat is on the mat." What problem then is he solving? — schopenhauer1

The Tractatus and facts

"The cat is on the mat" is true IFF (the cat is on the mat), where "the cat is on the mat" exists in language, and (the cat is on the mat) exists in the world.

The fact that (the cat is on the mat) in the world is dependent upon there being a relation between the cat and the mat. What is the nature of this relation?

Mereological Nihilism, aka compositional nihilism, is the philosophical position that in the world there are no objects with proper parts, in that there are no metaphysical relations that connect parts to a whole.

For example see:
1) Wikipedia – Mereological Nihilism
2) Amie L. Thomasson's video "Do tables and chairs really exist"

The SEP article on Bradley's Regress discusses the ontological debate between particulars and universals, where FH Bradley specifically outlined arguments against the relational unity of properties.

The SEP article on Relations notes that some philosophers are wary of admitting relations because they are difficult to locate. As it writes "Glasgow is west of Edinburgh. This tells us something about the locations of these two cities. But where is the relation that holds between them in virtue of which Glasgow is west of Edinburgh?"

IE, if a fact in the world is dependent upon the ontological existence of relations between parts, then if there are no such things as ontological relations in the world, then it follows that there are no facts in the world.

As the existence of ontological relations in the world would lead to philosophical puzzles, my belief is that relations don't ontologically exist in the world, thereby agreeing with mereological nihilism and concluding that there cannot be facts in the world.

In the event that there are no facts in the world, then neither can there be independent facts in the world.

My point is this: which philosophies argue that the world, at least in terms of human communication, is not composed of facts or true propositions?
For example, the statement "the unicorn is on the mat" is a false proposition because it's an impossibility. — schopenhauer1

Yes, there is no dispute that what is important in language are facts and true propositions, but the dispute arises in deciding what is a fact and what is a true proposition.

I believe that both unicorns and mats can exist, and therefore it's possible that there could be a unicorn on a mat. The statement "the unicorn is on the mat" could well be both a fact and a true proposition.

Unicorns certainly exist in literature, taking as a an example the book Into the Land of the Unicorns by Bruce Coville. Unicorns certainly exist in language, otherwise we couldn't be talking about them. The fact that no-one has seen or photographed a unicorn in the world is not proof that unicorns don't exist in the world, in the same way that because no-one has seen a particular rock at the bottom of the Atlantic Ocean, this is not proof that such a rock doesn't exist.

You believe that the statement "the unicorn is on the mat" is a false proposition. I believe that the statement could equally be a true proposition.

If I say, "the cat is on the mat," and we observe a cat on the mat, we might call this a true proposition.........................It's a truism. Almost no one disputes it. Well done for stating the obvious. — schopenhauer1

I am sure most are in agreement that "the cat is on the mat" is true IFF (the cat is on the mat), where the proposition "the cat is on the mat" exists in language and (the cat is on the mat) exists in the world. This is a truism that no-one would dispute.

We know where language exists. What is disputed in where this (world) exists. Does it exist in the mind of the observer, the position of Indirect Realism, or does it exist outside and independently of the mind, the position of the Direct Realist.

Yes, it is obvious that the cat exists in the world, but it is not obvious where this world exists.

So, I'm puzzled as to why a philosophy would assert, "My knowledge is made up of independent facts," as if this were a profound statement. — schopenhauer1

When starting the Tractatus, Wittgenstein did think that knowledge was made up of independent facts, but later concluded that his reasoning was unsound. He wrote Philosophical Investigations on the principle that facts cannot exist in isolation from each other.

According to the Merriam Webster dictionary, "profound" means having intellectual depth and insight.

On the one hand the statement "my knowledge is made up of independent facts" is philosophically profound, but on the other hand the statement "my knowledge is made up of inter-connected facts". is also philosophically profound.

What we want to know is which statement is true.

1) Similarly, stating truisms in philosophy without delving into the mechanisms behind them adds little value.
2) My broader point is that non-empirical philosophies can also be considered true propositions
3) If Wittgenstein isn't explaining why a proposition cannot be true, why should we care if the broader claim, "The world consists of true propositions or independent facts," is correct? — schopenhauer1

Aren't thoughts 1) and 2) in opposition.

I cannot justify in words my non-empirical thoughts that "evil is bad" and "beauty is good", yet I believe them to be true. I believe them to be truisms.

The only mechanism I can think of to explain such beliefs is that they have been programmed by evolution into the human gene for the benefit of the survival of the group.

As there are only 75 pages in the Tractatus, primarily devoted to Linguistics, I don't think we should also expect a foray into Evolutionary Biology, even if that is what he believed.

It was not addressed by Witt, but it SHOULD HAVE if his goal was to show how propositional logic allows for mapping onto reality due to selecting out true states of affairs; the MECHANISM for doing so must be EXPLAINED. — schopenhauer1

There is nothing wrong in making an assertion and not justifying it by a mechanism, which, after all, is the basis of scientific modelling. From Britannica Scientific Modelling

Scientific modelling, the generation of a physical, conceptual, or mathematical representation of a real phenomenon that is difficult to observe directly. Scientific models are used to explain and predict the behaviour of real objects or systems and are used in a variety of scientific disciplines, ranging from physics and chemistry to ecology and the Earth sciences.

I am in Crewe train station waiting to catch the train to London.
It is a fact that the train leaves from platform 4.
It is also a fact that the Eiffel Tower is 300m tall.
It is also a fact that the Great Northern is the highest selling beer in Australia.
It is also a fact that the capital of Nevada is Carson City.
My knowledge of the world is the set of these independent facts.

In the statement "my knowledge of the world is the set of these independent facts", whether the world exists in the mind (as proposed by Idealism) or exists independently of the mind (as proposed by Realism) is irrelevant to the truth of the statement.

That the sun rises in the east is explained by the model of the Earth rotating around the Sun. No mechanism for why the Earth rotates around the Sun is included within this model, as mechanisms are not part of models.

As Witt writes in the Tractatus:

2.12 "A picture is a model of reality"

and as a model the Picture Theory does not need to be justified by a mechanism.

That the fact "the train leaves from platform 4" and the fact that "the Eiffel Tower is 300m tall" are independent facts is true, independently of any mechanism that could be used to justify it. That they are independent is a primitive truth.

That the colour red is not the colour green is true is also a primitive truth, and as such cannot be justified by any mechanism.

Yes, Wittgenstein in the Tractatus may make statements not justified by any mechanism, such as:

6.3751 For example, the simultaneous of two colours at the same place in the visual field is impossible, in fact logically impossible, since it is ruled out by the logical structure of colour.

But this is one of the major aspects of the Tractatus, that some truths cannot be described in words, but can only be shown

4.1212 What can be shown cannot be said

We all know that the colour red is not the colour green, and there is no mechanism that could justify such knowledge. As the Tractatus says, it cannot be said, it can only be shown.

But was that even addressed by Witt? — schopenhauer1

No, because that was not the purpose the Tractatus. The Tractatus was addressing a specific problem, not trying to explain every aspect of language.

The Tractatus was an attempt to show that we can analyse ordinary language propositions,
such as "the car is red and the car is on the road", into elementary propositions such as "the car is red" and "the car is on the road" which can then be combined using truth-functions. A proposition is elementary when it is independent of all other elementary propositions.

The Tractatus was an attempt to prove in one particular instance that a whole may be understood by understanding parts that are independent of each other.

Even if the Tractarian project failed because of the colour-incompatibility problem, the question of the relationship of the whole to its parts remains of philosophical interest.

1) (Donald Davidson and his article What Metaphors Mean)...........I take it he means only in scientific applications, yes? Either way, I personally find this view unintelligible at face value. 2) A voltage is quite literally, not a pressure. 3) We seem to have a simile of sorts, based on the definition; but the definition merely reports usage. — 013zen

This leads into the question as to how a model of the truth, the metaphorical truth, the simile as an expression of truth and the literal truth relate to the Picture Theory of the Tracatus.

The Picture Theory of the Tracatus

Before the Tractatus, philosophers thought that language mirrored reality.

Wittgenstein in the Tractatus introduced the idea that language doesn't mirror reality, but has the same logical form as reality.

2.12 "A picture is a model of reality"

On the one hand there is thought

3 "A logical picture of facts is a thought."

On the other hand there is are propositions

4.12 "Propositions can represent the whole of reality, but they cannot represent what they must have in common with reality in order to be able to represent it – logical form"
4.001 - "The totality of propositions is language."

There is the relation of thought with language.

4 "A thought is a proposition with a sense."

We can have the thought that the grass is green, and we can have the proposition that "the grass is green". The thought is the proposition, in that any thought must be expressible as a proposition, and any proposition must be expressible as a thought. Both the thought and proposition picture not reality but the logical form of reality.

However, only substances can be pictured, as only objects make up the substance of the world. As concepts such as god, religion, ethics, good, evil and morality are not substances, they cannot be pictured.

2.021 "Objects make up the substance of the world".

7 "What we cannot speak about we must pass over in silence".

Thoughts and propositions are an amalgam of Formal Concepts and Concepts Proper. Formal Concepts include things such as objects, events, things, numbers, complex, fact, functions and make up the logical framework of a thought and proposition. Concepts Proper such as mountains, hills, tables and chairs are what are being represented.

4.1272 The same applies to the words complex, fact, function, number, etc. They all signify formal concepts.
4.126 ..............I introduce this expression in order to exhibit the source of the confusion between formal concepts and concepts proper.............

The logical part of a thought or proposition does not tell us how the world is. For Wittgenstein, there is no synthetic a priori, as knowledge about the world only comes from observation of the world, and for the Tracatus a priori philosophising has no use.

The difference between Direct Realism and Indirect Realism
The Direct Realist believes that thoughts and propositions picture reality, whereas the Indirect Realist believes that the thoughts and propositions picture the logical form of reality.

Indirect realism is broadly equivalent to the scientific view of perception that subjects do not experience the external world as it really is, but perceive it through the lens of a conceptual framework. Conversely, direct realism postulates that conscious subjects view the world directly, treating concepts as a 1:1 correspondence. (Wikipedia Direct and indirect realism)

The Picture Theory is a model
A picture is a model

2.12 "A picture is a model of reality"

The Picture Theory cannot be literal
A picture cannot be literal, as a picture has the same logical form of reality, not the same form as reality.

If we picture an apple, as the picture cannot tell us about the form of reality, the picture cannot tell us what exists in the world, meaning that what exists in the world is unknowable, as is Kantian noumena. If the picture could tell us about the form of reality, then we would know what existed in the world.

Therefore the apple we picture doesn't exist in the world, but it does exist in the picture, otherwise we wouldn't be able to picture it.

The difference between metaphor and simile
From the Merriam Webster dictionary
Metaphor = a figure of speech in which a word or phrase literally denoting one kind of object or idea is used in place of another to suggest a likeness or analogy between them (as in drowning in money)
Simile = a figure of speech comparing two unlike things that is often introduced by like or as (as in cheeks like roses)

To be a simile, the picture has to be like the logical form of reality, whilst to be a metaphor, the picture has to be the same as the logical form of reality.

For both the metaphor and simile there is a relation between two different things, whether Mr. S is a pig or Mr. S is like a pig. For a metaphor these two different things have a necessary property in common, whereas for a simile they only have a contingent property in common.

The Picture Theory cannot be that of a simile
The relationship between the picture and what is pictured cannot be that of a simile, as this would lead into an infinite regression, similar to the homunculus argument against Indirect Realism. This argues that the mind of the Indirect Realist is directed at an object, such as an apple, that represents another object, another apple, which in its turn represents another object, another apple -etc.

Similarly, the picture of an apple cannot be a picture of a representation of an apple, but must be a picture of an apple.

The Picture Theory must be that of a metaphor
In order to avoid this infinite regression, the picture must be of what is pictured. The picture of an apple must be the apple that is pictured, sharing the same necessary properties, and as such, a metaphor.

IE, the Picture Theory within the Tractatus is a metaphorical theory.

@013zen @Wayfarer

1) Voltage is not pressure; we are using one mode of thinking to facilitate another.
2) I've been mauling over recently if this is, perhaps, a re-envisioning of Maxwell's use of physical analogy in science, with the added concern of how to better adapt thought to those analogies in order to eliminate supposed "pseudo-problems".
3) .....................because all that is meant by "truth" is the correspondence, and whether or not there is such a correspondence, reality will tell us.
4) ..................there must be a certain conformity between nature and our thought. Experience teaches us that the requirement can be satisfied...............
5) But, then one can ask, in what sense can P, a proposition made up of words, correspond with reality made up of objects?
6) The map isn't a "true" representation of Boston but each map I look at has some of the correspondences with Boston and those correspondences might be "true" while others not. Reality is what tells you whether or not this is the case.
7) Is a volt a pressure? No, but certain similarities in one manner of thinking correspond to the other and can help to elucidate the other.

Taking the above into account, certain concepts may be helpful in working through the Tractatus.

What is the relationship between a model of the truth, a metaphorical truth, a simile expressing the truth or the literal truth

The Correspondence Theory means different things to different people
Correspondence means to have a close similarity, to match or agree almost entirely (Oxford Language Dictionary).

For the Direct Realist, as regards thought, if the Direct Realist sees a red postbox and thinks about this red postbox, their belief is that in the world exists a red postbox, and they are directly seeing it. Therefore, for the Direct Realist, there is no correspondence between what they see and think and what exists in the world, as what they see and think entirely agrees with what is in the world.

For the Indirect Realist, as regards thought, if the Indirect Realist sees a red postbox in the world and thinks about this red postbox, their belief is that in the world exists something that is not a red postbox but is causing them to see and think about a red postbox. Therefore, for the Indirect Realist, there is a correspondence between what they see and think and what exists in the world, as what they see and think only agrees with what is in the world.

As regards language, for both the Direct and Indirect Realist, there is a correspondence between the word and what exists in the world.

Therefore for Randall and Buchler, it is true that as regards thought, the Direct Realist believes they directly know reality and therefore don't need to compare their belief with the world, whereas for the Indirect Realist, as they believe they don't directly know reality, they therefore do need to compare their belief with the world.

What is a model of the truth
In thinking about why the sun rises in the east various models can be proposed. One model is that the Earth revolves around the sun. Another model is that the Sun was put into a chariot and everyday the God Helios would drive the chariot all along the sky causing the Sun to rise and set. It has been discovered that the model of the Earth revolving around the sun has proved more predictive that the model of the God Helio.

As any model can be improved, it is not that case that a particular model is wrong but that some models are more useful in particular contexts than others. For example, the model of the Earth revolving around the Sun may be more suitable in a science context, whereas the the model of the God Helios may be more suitable in a literary context.

It can also be seen that applying the wrong model in the wrong context may well result in unwanted philosophical problems which may well disappear if a more suitable model had been chosen. For example, on the one hand, using the God Helios model in a science context will clearly create philosophical problems that wouldn't have arisen if the Earth rotating around the Sun model was used. On the other hand, using the Earth rotating around the Sun model in the context of Greek mythology, in trying to understand the relationships between Helios and his parents Hyperion and Theia, will be clearly unhelpful

IE, no model is wrong in itself, but may be wrongly used. The purpose of a model is to be able to predict changes to the context within which it is being used.

Models, metaphors, similes and the literal truth in the example of seeing the colour "red"

If one sees the colour red, then one is thinking about the colour red.

1) For the Direct Realist, if one sees a red postbox then a red postbox exists in the world. The thought of a red postbox is neither a model, metaphor nor simile but is the literal truth.

2) a) For the Indirect Realist, if one sees a red postbox, the belief is that there is something in the world that caused the perception of such a red postbox. But what exactly that something is is unknowable, in the sense of Kant's noumena.

b) As regards models, the cause of what is seen can be modelled, such as a model of red postbox. Such a model exists in the mind and not the world, as the world is unknowable. The usefulness of the model is demonstrated by being able to predict changes in what is seen, not changes to what is in the world, but changes to what is seen.

c) However, it is a human characteristic to equate effect with cause. The cause of a bitter taste is a bitter drink, the cause of an acrid smell is acrid smoke, the cause of seeing the colour red is something that is red, the cause of heat on the skin is a hot object and the cause of a screeching noise is a screech. The effect is seeing a red postbox. The cause is categorised by the mind as a red postbox.

d) Therefore, what is the relation between what is seen (a red postbox) and what the mind has categorised as its cause (a red postbox). This relationship cannot be literal, as the effect is not the same as its cause. The relationship is not that of a simile, as the effect is not like the cause. The relationship must be that of a metaphor, because an identity has been assumed between two different things having a necessary property in common. Different in that one is an effect and the other is a cause, but sharing the necessary property that the mind has categorised the cause after the effect.

IE, for the Direct Realist seeing a red postbox is the literal truth and for the Indirect Realist seeing a red postbox is a metaphorical truth.

Models, metaphors, similes and the literal truth in the example of voltage and pressure

Is pressure a model, metaphor, simile or the literal truth in explaining voltage.

When one reads "Voltage is the pressure from an electrical circuit's power source that pushes charged electrons (current) through a conducting loop, enabling them to do work such as illuminating a light. In brief, voltage = pressure, and it is measured in volts" (www.fluke.com) - in what sense does voltage = pressure.

In the Merriam Webster dictionary, the word "pressure" has several meanings, including "the action of a force against an opposing force" and "voltage" as "potential difference expressed in volts".

We can model voltage by thinking about the pressure of water causing water to run when a tap is opened, where the pressure of water in pipes becomes a model for the voltage determining current in a wire.

As a simile, the pressure of water in pipes is like the voltage determining current in a wire.

It may be that originally pressure only referred to water, but the meaning of words is not predetermined, and the meaning of words change with time, sometimes becoming more generalised. As the Merriam Webster dictionary notes, pressure may be taken as the interaction between two forces, and with a change in meaning, rather than saying voltage in wires is like pressure in water, we can now use the metaphor that voltage is pressure.

However, this then takes us to Donald Davidson and his article What Metaphors Mean, where he argues that metaphors mean what their words literally mean and that there is no hidden metaphorical meaning. In effect, he is saying that what seem to be metaphors are in fact words being used literally. In the case of the seeming metaphorical expression "voltage is pressure", he is making the case that this means that voltage is literally pressure.

IE, even the expression "voltage is pressure" may be read not only as a model but also as a simile, metaphor and literally, all dependant upon one's point of view.

I do wonder what if anything Witt can tell us today - even if I am correct that he did have something relevant to say to his contemporaries. — 013zen

As I see it:

As Kant successfully combined in his Critique of Pure Reason two prior theories previously thought independent of each other (Empiricism and Rationalism) into one, the synthetic a priori, Wittgenstein successfully combines in his Tractatus two prior theories previously thought independent of each other into one, (the Positivism of Mach and Ostwald and the model picture of reality of Hertz and Boltzmann) into one, language and thought as a logical picture of reality.

Positivism, in Western philosophy, is generally any system that confines itself to the data of experience and excludes a priori or metaphysical speculations.

For Boltzmann there is utility in a model corresponding with reality, and for Hertz, the thought of an image or picture, or the language of a sign or symbol, conforms with reality.

The Tracterian approach to language and thought as a logical picture of reality is equivalent to a metaphorical picture of reality

There are many articles that describe the language used by Wittgenstein in the Tractatus as metaphorical.
1) Wittgenstein and Metaphor, Jerry H. Gill, Philosophy and Phenomenological Research
2) Wittgenstein and metaphors in the Tractatus, Patrizia Piredda, 2021, Academia Letters
3) Wittgenstein’s Metaphors and His Pedagogical Philosophy, A Companion to Wittgenstein on Education.

One consequence of language as metaphor is the revolutionary idea that substance is able to traverse the actual and passible worlds rather than only underlying the actual world as traditionally thought. This solves the philosophical puzzle about how we are able to think about things that don't exist. Logical space is not the source of material change but of modal change (Kyle Banick - Wittgenstein, Tractatus Logico-Philosophicus: Metaphysics and Ontology, YouTube)

As regards terminology, I wouldn't call science's current approach a "metaphysical picture of reality", rather a metaphorical picture of reality. I agree about the importance of metaphor in how we understand the world, and as you say "Pressures and currents correspond to voltages and amps, despite energy not being transmitted in any similar regard whatsoever".

In the light of the Tractatus as proposing a metaphorical picture of reality, does this cast light on 6.54

My propositions serve as elucidations in the following way: anyone who understands me eventually recognize them as nonsensical, when he has used them - as steps - to climb up beyond them (He must, so to speak, throw away the ladder after he has climbed p it)

Once I am able to metaphorically picture a voltage as a pressure, the metaphor becomes redundant. in that I now understand voltage as pressure. Not that voltage is like a pressure but rather voltage is a pressure. For Wittgenstein, the ladder is the metaphor, and can be thrown away as redundant once it has enabled understanding.

Many argue that language is metaphorical in nature.
1) Metaphors We Live By is a book by George Lakoff and Mark Johnson published in 1980. The book suggests metaphor is a tool that enables people to use what they know about their direct physical and social experiences to understand more abstract things like work, time, mental activity and feelings.
2) Andrew May in Metaphors in Science 2000 makes a strong point that even Newton's second law is a metaphor.

The Tractatus developed the Picture Theory of Language, language as metaphor, from which developed the modal notion of possible worlds. This was revolutionary in the 1920's, and as such relevant to his contemporaries. But this raises the question whether still of relevance today. I would say yes, for two reasons. First, language as metaphor and the modal notion of possible worlds are still relevant, and second, it appears that even today there are many who have not yet accepted these insights.

For example, Donald Davidson in his article What Metaphors Mean argues that metaphors mean what their words literally mean, in that they have no hidden meaning but can be explained by what they do within the context it is being used

In addition, those Direct Realists who believe that the world we see around us is the real world itself, things in the world are perceived immediately or directly rather than inferred on the basis of perceptual evidence.

The insights of the Tractatus have still not been fully accepted, important though they are.

Witt himself says: — 013zen

“... I think there is some truth in my idea that I really only think reproductively. I don’t believe I have ever invented a line of thinking, I have always taken one over from someone else.........................What I invent are new similes”.

According to the SEP article on Ludwig Wittgenstein, "Considered by some to be the greatest philosopher of the 20th century", and according to the IEP article on Ludwig Wittgenstein (1889—1951) "Ludwig Wittgenstein is one of the most influential philosophers of the twentieth century, and regarded by some as the most important since Immanuel Kant".

So there must be more to Wittgenstein than he says of himself " I don’t believe I have ever invented a line of thinking". So the question is "what?"

Without wanting to overlap with @schopenhauer1 current thread on Wittgenstein, as you infer, no-one lives in a vacuum, including Wittgenstein when he wrote the Tractatus. So it cannot be that he did no more than cut and paste what was around him at the time, but must have creatively added something of significant originality.

It cannot be the case that his insights in the Tractatus have been superseded by his later works, or by more contemporary philosophers, as the Tractatus is still being discussed by contemporary philosophers as being of contemporary philosophic importance.

So even if the Tractatus is only an incomplete and partial explanation of the relationship between thought, language and the world, what original insights did it add to the contemporaneous debate between Mach/Ostwald on one side and Hertz/Boltzmann on the other?

Analogously, I am sure that no mark Monet ever made hadn't been made by a prior artist, so Monet's originality was not in the marks he made but in the relationship between the marks he made. Similarly, as with a simile, the originality is in the comparison between two things, not the things themselves. So we know Wittgenstein may well have borrowed ideas from both the empiricist and metaphysician, but where in the Tractatus is hidden his unique insight into the relationship between the empiricist and the metaphysical?

Though perhaps this is not directly relevant to the OP about the history surrounding the Tractatus, although maybe the question of why the Tractatus is important must be part of the history of the Tractatus.

@Apustimelogist

There is only one Being, and it includes both sides of the Nature/Geist distinction — Count Timothy von Icarus

There is only one World
Yes, there is only one World. Humans are part of this World. From an Enactivist perspective, humans have evolved in synergy with the world, and the human mind has developed from its embodied interactions within the World.

As you rightly say " If we don't fall into the trap of thinking that relationships between knowers and objects are in some way "less real," than relationships between objects and other mindless objects, I think we avoid a lot of the problems of this distinction".

The mind is different to what is outside the mind
Within our minds we have the concept of "object", such as apples and tables, and in our minds we have the concept of number, such as the number 1 and the number 7. The question is, accepting that the human has evolved as part of the World, because we have in our minds the concepts of number and object, do numbers and objects of necessity exist in the World outside the mind.

The question can be extended. Because we have the concept of the colour red in our minds, does the colour red exist in the world outside the mind. Because we sometimes experience angst, does angst exist in the World outside the mind. Similarly, does pain, anger, fear, disgust, joy, surprise, anxiety, sadness and happiness exist in the World outside the mind.

It is true that there is one World, of which humans are a part, but it does not follow that what exists in the mind of necessity also exists outside the mind, otherwise, to make my point, the mining of anxiety would be as common as the mining of lithium. This is clearly not the case.

As you say "For example, it's impossible to explain the natural, physical properties of something without any reference to how it interacts with other things, the context it is situated in, etc." This is true, but as with angst, any such interaction is not of necessity between a world outside the mind and the mind, but may well be contained within the mind.

So we know that what exists in the mind does not of necessity exist outside the mind. This leaves the question as to whether our concepts of numbers and objects also exist outside the mind as numbers and objects.

Numbers and objects
Numbers and objects are Formal Concepts, in the sense as introduced by Wittgenstein in the Tractatus 4.126, and are to be distinguished from Proper Concepts such as "apple" or "table". Formal Concepts are part of the syntax of language rather than its semantics. Other Formal Concepts include the existential quantifier Ǝ "there exists", also a part of the syntax of language rather than its semantics. For this reason, as Wittgenstein notes, we cannot meaningfully say "there exists", "there are 100", "there are objects" as one can say "there exists a mountain", "there are 100 books" or "there are grey objects".

The concept of number is intimately linked with the concept of object, in that we cannot say "there are 100", as for the expression to be meaningful we must say "there are 100 apples", where the number 100 refers to the object apple. Any Formal Concept within a grammatical expression must involve a reference, ie, a Concept Proper.

I can say "I see one object, an apple". I can also say "I see two objects, the top of the apple and the bottom of the apple". I can also say "I see four objects, the top of the apple, the left of the apple, the bottom of the apple and the right of the apple". I can continue dividing the apple up, and increasing the number of objects I can see at each time.

But, as you say "That we can imagine a vast horizon of potential concepts does not entail their historical actualization". For practical and pragmatic reasons we just say "I can see one object, an apple".

The human mind can judge that a particular set of atoms exists as a single form, in this case, as a single apple. But the question remains, in the absence of a human mind, what determines that a particular set of atoms existing in space exists as a single object or not. What determines whether for example the loss of a single atom from an object causes the object to disappear from existence. What determines whether that atom was a necessary or contingent part of the object. The human mind can make such a judgement, but what what in the absence of the human mind can make such a judgement.

How can objects exist in the absence of the human mind if there is no mechanism for differentiating between different particular sets of atoms. What determines whether an individual atom is a necessary or contingent part of that object.

If objects cannot exist in the absence of the human mind, then numbers, which are intimately linked to the existence of objects, neither can exist in the absence of the human mind.

Quantifier Variance
As regards Quantifier Variance, we can consider the expression Ǝ n; n * n = 25. As the number n doesn't exist in the absence of the human mind, the expression cannot be referring to existence in the absence of the human mind but must be referring to existence in language and thought. Therefore, in this particular instance, it is not the existence quantifier E that is varying, but rather the predication of the existence quantifier that is varying.

Expanding on your thought that "There is a fundamental sense in which, conceptually, things can be defined in terms of "what they are not", I believe that we can better understand numbers when we appreciate that they have no existence in the absence of the human mind.

Likewise, an explanation of counting seems to require some mention of the fact that the world already has things that we can count in it. — Count Timothy von Icarus

In the world we see an object on the left and we see an object on the right, and we say in the world there are two objects. From this we conclude that numbers exist in the world, as the object on the left and the object on the right exist independently of our observing them.

However, as objects in order to exist in the world have to be extended in space, the object on the left has both a top and a bottom. We can then justifiably say that in the world are three objects, the top object on the left, the bottom object on the left and the object on the right.

So are we looking at two objects or three objects? It depends on what an object is. Numbers cannot exist independently of the objects being numbered.

However, there is no means independent of the human mind to determine what an object is. IE, there is no means independent of the human mind to determine what makes a discrete object a discrete object. As @Srap Tasmaner pointed out, the problem of the edge case, and as you pointed out, the Ship of Theseus problem.

As there is no means independent of the human mind to determine what an object is, and as numbers are dependent upon the existence of discrete objects, numbers cannot exist independently of the human mind.

IE, it is a problem of circularity, in that there are two objects provided we have already determined that there are two objects.

I don't see how DNA would explain that, rather it might explain why we see things in the same kinds of ways. — Janus

Humans have a general commonality, in that all the self-reproducing cellular organisms on the Earth so far examined have DNA as the genome (https: //onlinelibrary.wiley.com).

I agree that even though humans share a commonalty because of their shared DNA, in that both the Nominalists and Platonists accept that numbers exist, they may differ in their particular beliefs. The Nominalist's belief that numbers are invented and exist in the mind, and the Platonist's belief that numbers are discovered and exist in the world.

It is true that two identical objects may behave very differently when subject to different environments, whether a pebble moving down a slope or a pebble stationary on flat ground, whether a human living in Reykjavík or a human living in Pretoria. It is therefore hardly surprising that humans, even though they share a general commonality, may differ significantly in their particular beliefs and actions. It may well be that someone who is now a Nominalist who had had the life experiences of someone who is now a Platonist may well have turned out to be a Platonist and vice versa. As regards general commonality between humans, perhaps nature outweighs nurture, and as regards particular actions and beliefs, perhaps nurture outweighs nature.

As regards Quantifier-Variance, as Hale and Wright wrote: "[it may be] a matter of their protagonists choosing to use their quantifiers (and other associated vocabulary, such as ‛object’) to mean different things – so that in a sense they simply go past each other".

Particular differences are especially noticeable. The Nominalist may say that numbers exist in the mind and the Platonist may say that numbers exist in the world. But perhaps QV is pointing out a hidden commonality in seemingly different beliefs, in that an individuals actions and beliefs are determined as much by their lived life experiences as by innate characteristics, as much as by nurture as nature

An individual's actions and beliefs should not be considered in isolation at one particular moment in time, but should be thought of as part of a process stretching back many years. If someone does say "‛there exists something which is a compound of this pencil and your left ear’ and someone else says ‛there is nothing which is composed of that pencil and my left ear’, perhaps QV is making the point that these statements should not be considered in isolation as necessarily contradictory, but rather should be considered as a glimpse of an ever-changing process stretching back in time, embodied not just in one or two individuals but in the multi-various life experiences of whole communities.

IE, perhaps QV is saying that individual statements, such as "numbers only exist in the mind", should not be judged as true or false in isolation from the wider community out of which it has emerged, in that particular words only have meaning within a wider context.

How would that work? — Janus

How does commonality between humans work because of their shared DNA?

For the same reason that there is more commonality between humans who share 99.9% of their DNA than commonality between humans and chickens who only share 60% of their DNA (https://thednatests.com)

Other than positing some hidden connection between all minds, there is no way to explain the commonality of human experience, a commonality that extends even to some animals. — Janus

That humans share 99.9% of their DNA (essential for development, survival and reproduction) with other humans may explain the commonality of human experience (https://thednatests.com)

Which theory of time is the most evidence-based? — Truth Seeker

Perhaps the same could be said about space. In a similar way to Presentism, it could be said that only the space that I exist in is real, and any space outside is simply a construct of my consciousness.

This approach has the advantage of treating space and time as two aspects of the same thing, ie, space-time.

Perhaps any theory of time is dependent upon a prior theory of space-time.

You are confusing "simulate" with "is like". — Ludwig V

According to Merriam Webster, "like" means the same or nearly the same, whilst "simulate" means to give the same or nearly the same appearance often with the intent to deceive. Therefore, simulation involves some form of deception.

The world as experienced by the human mind can only ever be a pale representation of any real world that may or may not exist outside the mind, yet we deceive ourselves that we can directly know such a world. It is inevitable that any world in our mind can only ever be a simulation of any world that may or may not exist outside our minds.

"We are living in a 'simulation' and such a virtual world is the same as the 'real world' in every respect, except that it is simulated and therefore 'not real." I have a few arguments against this notion: — jasonm

From the viewpoint of fish in an aquarium, is their existence a simulated life, in that the aquarium simulates the ocean, or is it a real life, in that their environment is all they know. As with every life-form, their lives are both simulations and real. Takeaway pizzas simulate real food, social media simulates real life, sports events simulate medieval battles, surveillance cameras simulate the nosy neighbour in a small village, modern government tries to simulate the parents in a traditional family (whether father of the nation or mamala to the people) and corporate employers simulate the process of having to work for essential food and shelter.

As regards argument 1), the designers do pop out at times, but the life-form is not aware because of their necessarily limited intelligence.
As regards 2), similarly, the life-form doesn't notice violations in the laws of physics also because of their necessarily limited intelligence
As regards 3), the same amount of computing power would be required to house a "simulated" world as a "real" world.
As regards 4), the simplest explanation for fish in an aquarium is that they exist in a simulated world

You may respond that humans are the supreme intelligence in the universe and are all knowing, yet humans have only been around for 2.8 million years whilst fish have managed to successfully survive in a hostile world for more than 500 million years, regardless of whether this world happens to be "simulated" or "real".

The intention isn't built into the AI models, it's the user that forms the intended use and guiding principle of creation. — Christoffer

HG Wells in his book The Time Machine wrote “It sounds plausible enough tonight, but wait until tomorrow. Wait for the common sense of the morning.”

This is data. Two writers may use this as a foundation and inspiration for their own works

If the first writer did no more than repeat the same material and wrote “It sounds plausible enough tonight, but wait until tomorrow. Wait for the common sense of the morning”, then this would be plagiarism and considered theft.

However, if the second writer in using the same material was able to discover a deeper structure, and was able to base their writing on such a deeper structure, then their writing would not be considered as either plagiarism or theft.

For example, Wells i) draws attention to the concept of time by following the idea of tonight by the idea of tomorrow, ii) repeats the same word "wait" but gives it two different meanings when he writes "wait until tomorrow" and "wait for the common sense", and iii) contrasts opposites, when he infers that an idea first thought successful may in fact not be so.

If the second writer did no more than copy this deeper structure discovered in Well's writing and wrote "One started by questioning the importance of training data in AI but ended by becoming more confused than ever, confused about the role of the algorithm and confused about the role of the engineer creating the algorithm, striving to find the nuances but only discovering the noise", then I am sure no there would be no question of either plagiarism or theft.

There is data, and at a deeper level there is what the data means. Can an AI algorithm ever discover what data means at this deeper level?

Numerous research studies have found links between how the human mind generate new ideas to that of how AI models do it. — Christoffer

As Pablo Picasso said: "good artists copy; great artists steal”, which also references the difference between art and craft.

As Claude Lorrain painted figures and trees in the foreground with boats on water in the background, Andre Derain 300 years later painted figures and trees in the foreground with boats on water in the background. But was Derain a good artist copying Lorrain or was he a great artist stealing from Lorrain?

Both the good artist and the great artists generate new ideas, but there is a difference. The good artist copies what is immediately visible in a great painting whilst the great artist discovers what is hidden beneath the surface of a great painting.

It seems that at the moment AI is copying what is immediately apparent in existing texts, making it a good source, rather than being able to discover the hidden structure behind existing texts, potentially making it a great source, and then possibly rivalling the best of humans.

The idea of infinity can't be properly expressed using language, but then again, infinity is a word. — Scarecow

It depends on what you mean by "infinity".

If "infinity" means continually adding one to an existing set, then the idea of infinity can be properly explained in words as "continually adding one to an existing set".

If "infinity" means something outside of what can be known by a finite brain, then the idea of infinity can also be properly explained in words as "something that is outside of what can be known by a finite brain".

But how to explain the seeming contradiction that we can know that there is something that cannot be known?

For example, I know that I am seeing colours and shapes. However, I can only infer that I am looking at an oak tree, and if in fact there is an oak tree, then I can infer what mass it has. Therefore, I shouldn't say that "the mass of the oak tree is something that I don't know", as both the oak tree and its mass are inferred, but rather "the inferred mass of the oak tree is something that I do know".

Therefore, rather than say "I know that there is something that cannot be known by a finite brain" it would be more correct to say "I infer that there is something outside of what can be known by a finite brain".

As with all inferences, as we can only infer infinity, we don't know whether infinity exists or not.

First, we do not only picture facts to ourselves. — Fooloso4

I look and see a fact in the world such as "the apple is on the table". As no-one else can see into my mind, in that telepathy is not a thing. I can only picture facts to myself.

2.1 "we picture facts to ourselves"

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The picture that comes to mind need not be the result of conscious choice. With regard to the model of the accident the color of the car has no bearing on what is being depicted. What a picture represents is a logical relation: — Fooloso4

In the model is a red piece of wood and in the world is a red car.

I agree that when I see the colour red, I have no conscious choice as to what colour I see, in that I cannot choose to see another colour, such as blue or green.

I agree that there is a logical relation between the red piece of wood in the model and the red car in the world.

But who knows what these logical relations are. These logical relations cannot be determined by the picture alone.

For example, is the logical relation between the red in the model and the red in the world (in that the red in the model pictures the red in the world) or is the logical relation between the red in the model and the car in the world (in that the red in model pictures the car in the world).

Within the same picture can be innumerable logical relations.
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'Object' is a pseudo-concept but not all objects are simple objects. Spatial objects such as a chairs tables, and books ( 3.1431) are not simple objects. — Fooloso4

Objects such as chairs, tables and books are not Tractarian objects. These are objects in ordinary language. and the Tractatus is not dealing with ordinary language.

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.

The key word is "imagine". Wittgenstein is using an analogy. He is not saying that tables, chairs and books are Tractarian objects.
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'x' is the variable name for the pseudo concept 'number'. (4.1272) Substituting "a number" for 'x' gives us: "Number is a number" which is nonsense. The variable name 'x' cannot be used for both the pseudo-concept 'number' and 'a number'. — Fooloso4

Exactly. As Bertrand Russell writes in the introduction, to say "x is an object" is to say nothing. As he says, meaningless.

To say “x is an object” is to say nothing. 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”.

The proposition "x is a number" is a Formal Concept. Formal means logical. It is the whole proposition that is the Formal Concept, because the Formal Concept establishes the relations between its parts, "x" and "number".

4.127 "The propositional variable signifies the formal concept"

"Number" is a pseudo-concept

Introduction "This amounts to saying that “object” is a pseudo-concept."

The variable name "x" is a sign that signifies a number, such that "x is a number".

In the proposition "x is a number", we could substitute "x" by "x is a number" giving the proposition "x is a number is a number". Continuing "x is a number is a number is a number". But this becomes meaningless. Therefore, in the proposition "x is a number", there must be an identity between "x" and "number", meaning that if "number" is a pseudo-concept then "x" must also be a pseudo concept.

Therefore both "x" and "number" are pseudo concepts. The proposition "x is a number" is a Formal Concept.

I am off on holiday, but have appreciated the conversational research about the Tractatus.

Even this on the face of it seems odd to call a "pseudo-concept"...And what about "concepts" like "processes"? — schopenhauer1

There are two distinct worlds. There is our ordinary world with concepts proper and objects like books, tables and mountains. There is the Tractarian world with pseudo-concepts and objects that are simples and indivisible.

Objects such as blue, unicorn and hat and concepts such as processes and evolution are not referred to in the Tractatus as they are concepts proper in ordinary language and the Tractatus is not dealing with ordinary language.

Another reason that concepts such as process and evolution are not referred to in the Tractatus is that they are abstract concepts, such as angst and beauty, which can neither be described nor shown. Only concrete concepts such as blue, unicorn and hat that can be either described or shown.

A proposition such as "unicorns wear hats" is an ordinary language proposition. In the Tractatus, the propositions Wittgenstein is referring to have the form fx

4.24 "I write elementary propositions as functions of names, so that they have the form "fx", ∅(x,y)", etc."

It may be convenient to try to understand the Tractatus using ordinary language propositions such as "grass is green", but only as an analogy, as one tries to understand gravity by picturing a ball on a sheet of rubber.

However the Tractatus is philosophically important in beginning to develop the idea of modal worlds, possible worlds, where we can sensibly talk about non-existent things like unicorns and Sherlock Holmes, something that Bertrand Russell had problems with.

Anyway, I am off on holiday, but want to say that I have learnt a lot from this thread.

If so this itself would be an illustration of a "psychological theory" that goes beyond simply "acquaintance" (showing) the object, thus refuting that "acquaintance" or "showing" is where it must stop. — schopenhauer1

I agree that explaining how the mind can learn the meaning of the world "ngoe" from just five pictures is beyond my pay grade. All I know is that it works, and is in principle very simple.

The Tractatus only begins after I have learnt the word "ngoe", and only then, does the word "ngoe" in language mirror the "ngoe" in the world.

'Object' is a pseudo-concept. A particular object is not. — Fooloso4

'Object' is a pseudo-concept because it says nothing about what is the case, not because it makes up the substance of the world. — Fooloso4

Right, but the issue is whether something that falls under a pseudo-concept is a pseudo-concept. — Fooloso4

Tractarian objects are pseudo-concepts

Introduction - "This amounts to saying that “object” is a pseudo-concept. To say "x is an object" is to say nothing".

Why is a Tractarian object a pseudo-concept? Things can be said about concepts proper, such as book and tables, but things cannot be said about Tractarian objects because they are simples. They are pseudo-concepts because they are simples.

2.02 "Objects are simple"

Why are Tractarian objects simples? If they weren't simples, propositions could not picture the world.

2.021 "Objects make up the substance of the world. That is why they cannot be composite
2.0211 "If the world had no substance, then whether a proposition had sense would depend on whether another proposition was true
2.0212 "In that case we could not sketch any picture of the world (true or false)
2.023 "Objects are just what constitute the unalterable form"

Therefore as Tractarian objects (ie, pseudo-concepts) are simples (ie, indivisible), there cannot be anything that falls under them.
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Formal concepts are pseudo-concepts. — Fooloso4

Tractarian objects are pseudo-concepts

Introduction - "This amounts to saying that “object” is a pseudo-concept. To say "x is an object" is to say nothing".

The proposition "x is a horse" is a concept proper and the proposition "x is a number" is a formal concept.
(Spark notes Propositions 4.12 – 4.128)

4.127 "The propositional variable signifies the formal concept"

Examples of propositional variables could be: "the sky is blue", "the sky is purple", "grass is green", "grass is orange".
Being propositional variables, each has the value either true or false.
In the truth table, "the sky is blue" is true, "the sky is purple" is false, "grass is green" is true, "grass is orange" is false.
Therefore the propositional variable "x is a number" signifies a formal concept.

What does Wittgenstein mean by formal? He refers to formal relations, which are logical relations between objects. He refers to formal properties, which are internal and logical. Formal means logical.

4.122 "In a certain sense we can talk about formal properties of objects and states of affairs, or, in the case of facts, about structural properties: and in the same sense about formal relations and structural relations. (Instead of "structural property" I also say "internal property": instead of "structural relation", "internal relation")

The variable name x signifies a pseudo-concept object

4.1272 "Thus the variable name "x" is the proper sign for the pseudo-concept object. Whenever the word "object" ("thing". etc) is correctly used, it is expressed in conceptual notion by a variable name"

On the one hand the propositional variable "x is a number" signifies a formal concept and on the other hand the variable x signifies a pseudo-concept object. Therefore, a formal concept cannot be a pseudo-concept.

That depends on the medium of representation, whether what is being pictured is intended to communicate something to someone else, and what it is that is being represented............2.1 "we picture facts to ourselves" — Fooloso4

If we were only picturing facts to ourselves, then we are using a Private Language, which Wittgenstein in Philosophical Investigations said was not possible.

Even if we were picturing facts to ourselves, we would have to make the conscious choice whether i) the red in the model is picturing the red in the world or ii) the wood in the model is picturing red in the world.

If that were the case, a picture wouldn't be a model of reality, a picture would be a model of an individual's conscious decisions.

2.12 "A picture is a model of reality"

Just as the car does not become the bicycle, it is necessary that whatever it is the represents the car in the picture does not become something else. — Fooloso4

In the model is a red piece of wood, and in the world is a red car.

From the Picture Theory, the red piece of wood in the model pictures the red car in the world.

But how do we know, just from the picture itself, whether i) the red in the model is picturing the red in the world or ii) the wood in the model is picturing red in the world?

From the picture itself, we cannot know. We need someone to come along and tell us which is the case i) or ii), and if that happens, this destroys the Picture Theory, which is meant to stand alone.

Certainly, this could lead to a regress (definitions of definitions of definitions)....................Surely I can point to these processes that account for object formation in the mind, and how we attach meaning to objects. — schopenhauer1

As I see it, some words we learn by description and some by acquaintance.

As regards learning by description, we can go to the dictionary and discover that a "tree" is defined as "a woody perennial plant having a single usually elongate main stem generally with few or no branches on its lower part". But then we have to look up the definition of "woody" and end up in an infinite regress.

Sooner or later, we have to learn words by acquaintance, as illustrated in the picture below. As the Tractatus notes, I cannot describe the meaning of "ngoe", I can only show it. Though I agree that this is not the same approach as laid out in the Tractatus.



What do you think "ngoe" means, now you have been "shown" the picture?

'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. — Fooloso4

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.

Wittgenstein seems to not care to discuss mind, but language limits. — schopenhauer1

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. — schopenhauer1

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 — schopenhauer1

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 — schopenhauer1

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)

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. — schopenhauer1

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).

Why is "One is a number" a formal concept and "1 + 1 = 2" not a "formal concept"? — schopenhauer1

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.

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. — Fooloso4

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. — Fooloso4

'Number' is the constant form. 1, 100, and 1,000 are variables that have as a formal property this formal concept. — Fooloso4

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.

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. — schopenhauer1

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.

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. — schopenhauer1

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.

(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." — Fooloso4

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'. — Fooloso4

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.

I think Wittgenstein is saying that an "object" like the number 1 has a sense if it is an object or a description. — schopenhauer1

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.

See 4.12721. The concept of a number is a formal concept. Particular numbers are not. They fall under the concept of a number. — Fooloso4

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?

@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. — 013zen

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. — 013zen

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. — 013zen

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.

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). — Fooloso4

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'. — Fooloso4

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. — Fooloso4

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

There are no ‘logical objects’ (4.441) — Fooloso4

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.

As regards sense 1), in logical space are objects in combination. In order for this to be the case, these objects must be logical and their combinations must be logical. For example, a blue apple is a logical object, whilst an unhappy apple is not a logical object. An apple on top of a table is a logical combination, whilst an apple inside the sun is not a logical combination.

As regards sense 2), Frege treated relations and universals as objects. FH Bradley in treating a relation as an object concluded that relations don't ontologically exist in the world. Wittgenstein disagrees with Frege, and concludes in 4.441 that "There are no logical objects". This means in particular that relations, such as "to the left of " or "on top of" logical cannot be treated as objects, and in general that logical form, such as i) all H are M ii) S is H iii) therefore, S is M is not an object, in the sense that "grass" is an object.
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As part of a propositional analysis apples and tables can function as simples. — Fooloso4

Apples and tables as names in a proposition are concepts. Some think that concepts are simples, including myself, and fall within the theory of Conceptual Atomism.

From the SEP article on Concepts

Conceptual atomism is a radical alternative to all of the theories we’ve mentioned so far is conceptual atomism, the view that lexical concepts have no semantic structure (Fodor 1998, Millikan 2000). Conceptual atomism follows in the anti-descriptivist tradition that traces back to Saul Kripke, Hilary Putnam, and others working in the philosophy of language (see Kripke 1972/80, Putnam 1975, Devitt 1981).

If concepts weren't simple, when we thought of a concept such as grass as a set of other concepts, such as a low green plant, we would have to think of each of these concepts, such as a plant, as a set of other concepts, such as a living organism. But sooner a later a concept must be a simple otherwise our thought would be never-ending.

Therefore, thought requires that some concepts must be simples.

In the Tractatus picture theory, a proposition such as "grass is green" pictures the state of affairs grass is green.

The state of affairs grass is green exists in a logical space.

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.

It is true that actual grass is divisible, for example into the top of the blade of grass and the bottom of the blade of grass, but objects aren't actual objects but rather logical objects, and logical objects such as grass can be simples.

Ir objects such as grass are not actual objects but logical objects, then logical objects such as grass can be simples.
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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.

4.126 (I introduce this expression in order to exhibit the source of confusion between formal concepts and concepts proper, which pervades the whole of traditional logic)...................so the expression for a formal concept is a propositional variable in which this distinctive feature alone is constant.
4.127 The propositional variable signifies the formal concept, and its values signify the objects that fall under the concept.
4.1271 – Every variable is the sign for a formal concept
4.1272 Thus the variable name "s" is the proper sign for the pseudo-concept object.