I’ve always understood the indirect realist to say they directly perceive sense-data, representations, perceptions and the like. — NOS4A2

The indirect Realist directly perceives something in their field of vision, which they can reason to be the planet Mars. The word "sense data" should be taken as a figure of speech, not literally, in that no-one has ever found sense data in the brain. As the word "house" is a representation of an object in the world, the dot is a representation of the planet Mars.

I don't think there's any problem with an indirect realist saying "I'm looking at mars". — flannel jesus

Yes, the Indirect Realist can say "I am looking at Mars" as they can say "responsibility is a heavy burden", "Sally is as bright as the sun", "the whole world is a stage" or "the wind whispered in my ears".

The “directness” of perception refers to the relationship between the perceiver and the perceived — NOS4A2

The Indirect Realist says that they directly perceive a dot in the sky. The Direct Realist says that what they directly perceive is the cause of the dot.

But that is like saying that if I was stung, I would know just from the sting the cause of the sting, whether a thorn or a wasp.

the Indirect Realist would say that they are looking at mars — Ashriel

The Indirect Realist would say that they are directly looking at a dot which they reason to be the planet Mars.

I agree that in language, rather than say "I am directly looking at a dot which I reason to be the planet Mars" in practice this is shortened to the more convenient "I am looking at Mars"

But the statement "I am looking at Mars" should be taken as a figure of speech, not literally.

I assume that cognition means conscious awareness of. In that case, my view would be very similar to semantic direct realism. — Ashriel

Both the Indirect and Direct Realist would agree that they have conscious awareness of perception of things in their five senses, such as the colour red, a bitter taste, an acrid smell, a painful sting or a grating noise

As both believe in Realism rather than Idealism, both would agree that there is something in the world that has caused such perceptions in their five senses.

We look into the night sky and see a dot that we know to be Mars.

The Indirect Realist would say that they are directly looking at a dot. The Direct Realist would say that they are directly looking at Mars.

The Indirect Realist could ask of the Direct Realist, in what sense of the word "direct" is the Direct Realist "directly" looking at a mass of ${6.4 * 10^{23}kg}$ when all they can see is a dot?

"Transcendence" is inherently a debatable term — Gnomon

Definitions

What is Kant's "Transcendental Idealism" about, taking the term "Transcendental Idealism" more as a name than a description, as even Kant considered finding a better term.

Based on the Merriam Webster Dictionary, "Transcendence" in Kantian philosophy means the state of being beyond the limits of all possible experience and knowledge. "Transcendental" in Kantian Philosophy means going beyond the limits of all possible experience but not going beyond the limits of human reason.

Note that it is ""Transcendental Idealism" not "Transcendent Idealism". The CPR is not about "Transcendent Idealism", as this would lie beyond what the human can cognitively grasp and would move into the realm of the unknowable. Not only beyond human experience but also beyond human reason, because beyond the scope of empirical investigation. Included would be such concepts as God and the soul.

The CPR is about "Transcendental Idealism", which is about our prior intuitions of time and space and a priori concepts of the Categories. Necessary conditions for the possibility of experiencing and understanding the world, and which predetermine our understanding of the world.

The CPR is not about religion or the spiritual realism, but is about what we can practically know about the world using reason and observation. Kant wrote:

A 369 I understand by the transcendental idealism of all appearances the doctrine that they are all together to be regarded as mere representations and not as things in themselves, and accordingly that space and time are only sensible forms of our intuition, but not determinations given for themselves or conditions of objects as things in themselves.

The Transcendental Argument

Among Immanuel Kant’s most influential contributions to philosophy is his development of the transcendental argument, which he uses to show that we would not be able to understand our sensory experience if we were not able to impose a priori intuitions an a priori concepts on them. Thereby enabling the "conditions of the possibility of experience".

A Transcendental Argument begins with a strong premise, and then reasons to a conclusion that is a necessary condition for the premise. In a sense circular, but justifies its own truth through its own logical coherence.

Kant requires a transcendental argument because of his belief that it is not possible to abstract intuitions and concepts just from empirical experience, but transcendentally deduce them empirical experiences

Kant uses such a Transcendental Argument in his Refutation of Idealism in B275 against the Idealism of Berkeley in order not to prove that things exist independently of the mind, but only the possibility that things exist independently of the mind.

Kant chose a middle ground between Rationalism and Empiricism

Kant wanted to avoid the excesses of both the Rationalist and Empiricists by establishing a middle ground.

The Rationalist, such as Descartes and Plato, argued that humans can use Pure Reason to discover knowledge about the world without having first observed the world. They argued that Pure Reason can transcend experience and discover things like God and the soul.

The Empiricists, such as Hume and Locke, argued that humans can discover knowledge about the world just by observing the world.

Kant in the CPR made the case that on the one hand Pure Reason cannot answer metaphysical problems such as the existence of God or the soul but on the other hand can go beyond sensory experience in order to discover necessary truths about the world.

Summary
"Transcendental Idealism" uses the Transcendental Argument to make sense of the world given our sensory experiences.

(SEP - Kant’s Transcendental Idealism)

It might be considered weird, that someone describing his philosophy as “transcendental idealism” writes a paragraph called “Refutation of Idealism”. — Pez

The word "idealism" has more than one meaning (Merriam Webster - Idealism)

The term "transcendental idealism" should be thought of as a name rather than a description, as the Champs-Élysées is a name and not a description.

In fact, he proposed renaming his transcendental idealism with the more informative name of "formal" or "critical idealism," (Introduction to Critique of Pure Reason)
===============================================================================

The sole point of interest here hinges around the notion “outside” — Pez

In B275, Kant wrote "I am conscious of things outside me".

In the Introduction to the Critique of Pure Reason is the paragraph:

In the second edition, however, Kant inserted a new argument, the "Refutation of Idealism" (B 2 74-9), which attempts to show that the very possibility of our consciousness of ourselves presupposes the existence of an external world of objects that are not only represented as spatially outside us but are also conceived to exist independently of our subjective representations of them.

I may have the concept of an apple sitting on a table, and it may well be the thing outside me bears no relation to my concept of it.

Kant attempts to prove in the Refutation of Idealism that I can be conscious of things outside me.

Even if the thing outside me bears no relation to my concept of an apple sitting on a table, this doesn't detract from the fact that I am conscious of a thing outside me.

I suppose my thought is that "perception" can't separate us from the real in the manner I perceive indirect realists to say. — Moliere

We perceive a stick bent in water. Is the stick in water really bent?

I'm not tempted to define realism in opposition to idealism, for instance. — Moliere

There is realism and idealism as concepts and there is Realism and Idealism as proper nouns. As concepts they overlap, but as proper nouns are distinct. For example, the SEP articles on Realism and Idealism.
===============================================================================

I think a realist is open to non-causal relations, as long as they are real. — Moliere

Yes, the Eiffel Tower and Empires States Building are related, but non-causally.

My perception of an object and the object causing my perception are related, and causally.

What are the underlying beliefs you think are the same? — Moliere

Both the Indirect and Direct Realist believe in i) Realism rather than Idealism, ii) that there is a long and complex causal chain of events between the object in the world and our eventual perception of it and iii) there is a causal indirectness and a cognitive directness between our perception of an object and the object itself.

However, whilst both the Indirect and Direct Realist agree that there is a cognitive directness, they differ in what that actually means.

feedback — Ashriel

Some Direct Realists are Phenomenological Direct Realists believing in causal directness, and other Direct Realists are Semantic Direct Realists believing in cognitive directness.

However, it seems that the vast majority of Direct Realists are Semantic Direct Realists.

But there does not seem to be much difference between Semantic Direct Realism and Indirect Realism, meaning that it is only the labels "direct" and "indirect" that people are disagreeing with, not their underlying beliefs, which are probably the same.

But philosophy might be considered as an attempt to understand why we can know what will happen (if at all). — Pez

The most that philosopher Kant claimed about our knowledge of what exists outside us is in his Refutation of Idealism in B275 of the Critique of Pure Reason. He concludes "I am conscious of things outside me". IE, he can neither know what these things are or even why these things are as they are.

I am not a Mathematician, and have limited knowledge about set theory.

As a very broad generalization, I think of at least these two categories: (1) Matters of fact. (2) Matters of frameworks for facts. — TonesInDeepFreeze

:up: For me, the statement "Monet's Water-lilies is an example of beauty" is a fact and is true. However, I am speaking within the framework of a European Modernist. Within a different framework, say that of a Californian Post-Modernist, the statement, may be neither a fact nor true.
===============================================================================

That ordinary mathematics says "1+1 is 2" is matter of fact. But whether ordinary mathematics should say that 1+1 is 2 is a matter of framework. — TonesInDeepFreeze

:up: Within a different framework, say that of binary numeral system, 1 + 1 = 10
===============================================================================

But whatever we take mathematics to be talking about, at least we may speak of abstractions "as if" they are things or objects. — TonesInDeepFreeze

Is this an example of Putnam's Modalism, the assertion that an object exists is equivalent to the assertion that it possibly exists?

If I said "I am going to buy an apple", I am not referring to "an apple" as a particular concrete thing or object, but rather referring to "an apple" "as if" it were a particular concrete thing or object.

Whilst the definite article refers to a particular concrete thing "a house" "a mountain" or "a cat", the indefinite article, "a house", "a mountain" or "a cat", doesn't refer to a particular concrete thing, but rather refers to a particular concrete thing that possibly exists.

In language also, we can refer to things that exist, "I want this cat", and refer to things that possibly exist, "as if" they exist, such as "I want a unicorn".
===============================================================================

The 'it' there must refer to something — TonesInDeepFreeze

What does "it, the knight on a chess board, refer to?

"It" must refer in part to a physical object that exist in the world and in part to rules that exist in the world.

The game of chess is played between two people, and as neither player can look into the other's mind, the rules must exist in the world in order to be accessible to both players. For example, "the knight either moves up or down one square vertically and over two squares horizontally, or up or down two squares vertically and over one square horizontally". However, as rules cannot refer to themselves, in that rules cannot be self-referential, they must refer to something external to the rules, in this case, a physical object.

IE, if there were no rules there would be no game of chess, and if there there were no physical objects the rules would have nothing to refer to.

There are therefore two aspects to "it". The extension, the physical object of a knight, and the intension, the rules that the knight must follow.

Such an approach to understanding "it" is supported by Wittgenstein's Finitism. Wittgenstein was careful to distinguish between the intensional (the rules) and the extensional (the answer). Mathematics is the process of using rules contained within an intension to generate propositions displayed within the extension. For example, the intension of 5 + 7 is the rule as to how 5 and 7 are combined, and 12 is the extension. (Victor Rodych - Wittgenstein's Anti-Modal Finitism - Logique et Analyse)

Such as approach to understanding "it" also follows from natural language. The intension of the word "beauty" is a rule that determines what is beautiful and the extension of the word "beauty" are concrete instantiations, such as Monet's Water-Lilies or a red rose in a garden.

There may be many possible rules for what is beautiful. Francis Hutcheson asserted that “Uniformity in variety always makes an object beautiful.”. Augustine concluded that beautiful things delight us. Hegel wrote that “The sensuous and the spiritual which struggle as opposites in the common understanding are revealed as reconciled in the truth as expressed by art” .

It is impossible for a finite mind to have a list of all beautiful things in the world, yet can recognise when they see something is beautiful. The human mind has the concept of beauty prior to seeing a beautiful thing. Such a rule is probably innate, the consequence of millions of years of evolution existing in synergy with the outside world.

Such a rule is the intension of the word "beauty" and physical examples, such as a Monet Water-Lily are the extensions of the word "beauty".

IE, "it" refers to the intension and extension of the word "knight". The intension being the rule the "knight" follows and the extension being the physical object ,whether made of wood or plastic.
===============================================================================

And the number 1 in mathematics is an abstract mathematical object that we speak of in a similar way to the way we speak of concretes, but that does not imply that the number 1 is a concrete object. — TonesInDeepFreeze

However, if there were no concrete objects in the world, there would be no concept of the number "1".
===============================================================================

And as 'experience' and 'occurring' are the notions I start with, I must take them as primitive. — TonesInDeepFreeze

:up: Yes, there are some concepts, such as "beauty", that we cannot learn the meaning of by description from the dictionary, but are probably innate within us. Innatism is the view that the mind is born with already-formed ideas, knowledge, and beliefs.
===============================================================================

Notice that I didn't say 'experiences' plural — TonesInDeepFreeze

In my terms, thinking about the concept "beauty", which is probably innate, and therefore primitive within us, there only needs to be one intrinsic rule able to generate numerous extrinsic examples.
===============================================================================

But then I do refer to 'I' — TonesInDeepFreeze

In Kant's terms, we have a unity of apperception. The mystery is why.
===============================================================================

As I go on, I find that certain other notions such as 'is', 'exists', 'thing' or 'object', 'same' 'multiple'. etc. are such that I don't see a way to define them strictly from the primitives I've allowed myself. — TonesInDeepFreeze

Certain words such as "house" can be defined as "a building for human habitation, especially one that consists of a ground floor and one or more upper storeys". We can learn these concepts from the dictionary using definitions. But sooner or later, we come across other words, such as "is", "exists" and "thing" that are primitive terms, cannot be defined, but only learnt from acquaintance.
===============================================================================

But the very determinations of fact, let alone the conceptual organization of facts, are vis-a-vis frameworks, and it is not disallowed that one may use different frameworks for different purposes. — TonesInDeepFreeze

:up: For me, a Modernist, the statement "Monet's Water-lilies is a beautiful painting" is true, but for others, the Post-Modernists, the same statement is false.
===============================================================================

For me, the value and wisdom of philosophy is not in the determination of facts, but rather in providing rich, thoughtful, and creative conceptual frameworks for making sense of the relations among facts. — TonesInDeepFreeze

But how can there be wisdom in the absence of facts. How can we understand the wisdom of Kant without first knowing those facts he applied his wisdom to?
===============================================================================

Meanwhile, I would not contest that formation of concepts relies on first approaching an understanding of words ostensively. — TonesInDeepFreeze

:up:

So, at least in a mathematical context, "Infinity is unknowable" doesn't have an apparent meaning to me. — TonesInDeepFreeze

It seems that "infinity" as an object is more a problem of natural language than mathematics and set theory.

As you also say:

In set theory, there is no constant nicknamed 'infinity' (not talking about points of infinity on the extended real line and such here). Rather, there is the predicate nicknamed 'is infinite'. — TonesInDeepFreeze

feedback — Ashriel

May be useful to consider the Pierre Le Morvan article Arguments Against Direct Realism and How to Counter Them

Also Phenomenological Direct Realism (aka causal directness) may be described as a direct perception and direct cognition of the object "tree" as it really is in a mind-independent world. Semantic Direct Realism (aka cognitive directness) may be described as an indirect perception but direct cognition of the object "tree" as it really is in a mind-independent world.

Infinity is unknowable by the finite human mind, yet we know the meaning of "infinity"

Anyway, to say that thinking of abstractions requires thinking of a concrete examples does not say that we don't think of abstract objects — TonesInDeepFreeze

The human mind can discover both abstract and concrete concepts in physical object

I agree. The ability of being able to think about abstract concepts, such as independence, beauty, love, anger and infinity, is a crucial part of the human mind.

When travelling to a foreign country, and hearing the word "hasira" for the first time, how in practice can we learn its meaning. If it were a concrete noun, the local could point out several physical examples that could be described by it in the hope that the foreigner was able discover what they had in common. However if it were an abstract noun, would the same approach be possible, as in pointing out several physical examples that could be described by the abstract noun?

If I visited a university with a guide, and the guide pointed at the buildings and people moving between the buildings carrying books, and said "this is a university", would I understand the meaning of "university" just as the concrete concept "a set of buildings" or the abstract concept "a place of teaching"?

I believe that the human mind is inherently capable of discovering when looking at a physical object both concrete and abstract concepts

IE, the human mind knows the meaning of words such as "beauty" and "infinity" and can think about the abstract concepts beauty and infinity, and could have learnt what concept is connected to what word by being shown several physical examples of each.
===============================================================================

What is a concrete example of the concept of 'does not exist'? — TonesInDeepFreeze

The human mind can discover abstract concepts in physical objects



You are a foreigner in a foreign land learning the language. You have already learnt that the word "tufaha" means "apple", and you are now trying to learn the concepts expressed in the words "ni" and "sivyo". To make life more difficult, these are abstract concepts. But what is your best guess as the the meanings of "ni" and "sivyo"? The fact that are able to make an educated guess shows that abstract concepts can be instantiated in concrete examples.
===============================================================================

Also, the lines I'm thinking along is that certain utterly basic abstractions, such as 'object', 'thing', 'entity', 'is', and 'exists' themselves presuppose abstraction no matter what concretes are involved or not. — TonesInDeepFreeze

Abstract concepts don't of necessity refer to physical things, but wouldn't exist without physical things

Yes, the meaning of the abstract concept "exists" is independent of any particular physical thing referred to, whether an apple, a house or a government. However, the abstract concept "exists" would not exist in the absence of physical things.

The concept "Beauty" would not exist if there were no beautiful things. The concept "anger" would not exist if there were no angry people. The concept "governments" would not exist if there were no societies of people. The concept "infinity" would not exist if there were no physical objects.

IE, an abstract concept does not refer to a particular concrete entity, but abstract concepts wouldn't exist if there were no concrete entities.
===============================================================================

Yet, the notion of 'concrete instantiation' is itself an abstraction made of the the two abstractions 'concrete' and 'instantiation'. — TonesInDeepFreeze

New words are learnt either within language or within a metalanguage

On the one hand, words can be learnt by description, where a new word is learnt from known existing words. For example, if I know the words "a part that is added to something to enlarge or prolong it", then I can learn the new word "extension".

However, if only words were learnt by description, there would be the problem of infinite regress. Sooner or later some words must be learnt by acquaintance, where a word such as "beauty" can be learnt by looking at particular "concrete instantiations", such as a Monet painting of water-lilies or a red rose in the garden.

Similarly, I can learn the meaning of the word "concrete" by looking at particular "concrete instantiations", such as a bridge over a river or a skyscraper in a city.

But, as you point out, this suggests an infinite regression, in that it seems that I can only learn the word "concrete" if I already know the meaning of "concrete instantiation".

However, this is not the case, as whilst words by description are learnt within language, words by acquaintance are learnt outside language and within a metalanguage.

IE, the new word "beauty", although it becomes part of language does not require language to be learnt. This avoids the infinite regress problem of learning the meaning of the word "concrete", if in order to learn the meaning of "concrete" I must already have to know the meaning of "concrete instantiation"

Kant was rightfully claiming that we can never attain to a knowledge of things surrounding us per se i.e. independent of us? — Pez

Kant believed that a world exists outside the mind and independent of the mind, and tried to prove this using the Transcendental Argument "Refutation of Idealism" within section B275 of Critique of Pure Reason.

We observe things in our five senses, such as a stone falling from rest

We have discovered that the equation ${d = 0.5 * g * t^{2}}$ can accurately predict the distance a free-falling object falls from a position of rest, which we can describe as one of our "Laws of Nature".

In a similar manner, the human "Laws of Nature" can be used as a Transcendental Argument for the existence of a world outside the mind and independent of the mind.

Theorem

The mere, but empirically determined, consciousness of my own observations proves the existence of a world outside me.

Proof

We observe a stone falling from rest and develop an equation that is able to predict its motion, accepting that: i) the equation is a human invention, ii) we know it has been successful in the past, iii) we don't know why it works and iv) we don't know whether it will stop working in the future.

However, we do know that what we observe is beyond the control of the human mind, in that by thought alone we can neither slow down nor speed up the rate of fall of the stone and by thought alone we cannot change the equation that successfully describes what is being observed. It follows that as what we observe is not within the control of the mind, it must be within the control of something outside the mind and independent of the mind.

IE, the consciousness of my own observations is at the same time an immediate consciousness of the existence of other things outside me.

Now the question is: are we in the position of these chicken or can we rely on being fed every day? — Pez

Knowing what will happen does not mean we know why it will happen

We are in the position of the chickens. For example, we have discovered that the equation ${d = 0.5 * g * t^{2}}$ can accurately predict the distance a free-falling object falls from a position of rest. This equation describes what will happen not why it will happen. Until we know why the equation is able to predict what will happen, we cannot know whether that one day the equation will no longer work.

If the chickens knew why they were being fed, they would know that one day the feeding would stop, and would be slaughtered together with 140,000 of their comrades every minute..

It's not required that each concept, each abstraction itself corresponds to a particular concrete. — TonesInDeepFreeze

I appreciate your post.

Yes, the mind has a framework of concepts, such as beauty, infinity, pain, mountain, house, government, addition, multiplication, sky, salt, which we use to organise our concrete experiences.

On the one hand, for example, our concept of mountain refers to an abstract object, in that it does not refer to one particular mountain, but mountains in general. But on the other hand, there must be an intentionality to our concepts, in that the mind is not able to comprehend a concept without thinking about something concrete, whether a particular object, such as the Mont Blanc, a particular process, such as adding more height to a high hill or the particular word itself, "mountain".

That is not to say that each concept can only have one concrete instantiation, but rather each time I think of a mountain my concrete instantiation may be different, and different again for anyone else who thinks of a mountain.

My point is that I agree that it is not the case that an abstract concept corresponds to one particular concrete instantiation, but rather we can only understand an abstract concept by thinking of some concrete instantiation of it, which may be a concrete object (Mont Blanc), a concrete process (addition) or a concrete word ("mountain").

Immanuel Kant presents us with a surprising and seemingly absurd alternative: we ourselves are the source of physical laws. Seemingly absurd, because we cannot influence the laws of nature. — Pez

It depends what the expression "the laws of nature" is referring to. It could be referring to "the laws of nature" existing as concepts within the human mind, or it could be referring to the laws of nature existing outside the human mind and independently of the human mind.

If the first, we can influence them because they exist within the human mind. If the second, there is the problem of how we can know about them if they exist outside the human mind and independently of the human mind.

"Infinity of infinites" in natural language

In natural language, we know the meaning of the word "infinity", yet infinity is unknowable by the finite mind, meaning that "infinity" must refer to something knowable.

Something very large is knowable, such as the number of grains of sand on a beach, the number of water molecules in a glass of water or the number of people living in a city.

We are able to understand "Infinity" as like "the number of grains of sand on a beach".

Therefore, our understanding of "infinity" is not literal but rather as a figure of speech, specifically, a simile.

If "infinity" means "like the number of grains of sand on a beach", then the expression "an infinity of infinities" becomes an "like the number of grains of sand on a beach" of "like the number of grains of sand on a beach". This is ungrammatical

IE, in natural language, the term "infinty of infinites" is ungrammatical.

We can imagine things without a concrete instantiation. — Metaphysician Undercover

I find that hard to believe. How is it possible to imagine a unicorn by not picturing a unicorn? How is it possible to imagine the number 6 without picturing six things. How is it possible to imagine beauty without picturing something beautiful?

It is true that when I imagine a unicorn I could picture the word "unicorn", but this is still a concrete instantiation.

When you imagine a unicorn, if you are neither picturing a unicorn nor the word "unicorn", what exactly are you imagining? What exactly is your "Intentionality" directed at?
===============================================================================

We can imagine things without a concrete instantiation. That's how artists create original works, they transfer what has been created by the mind, to the canvas — Metaphysician Undercover

Problematic. If an artist, no matter whether Monet or Michaelangelo, could create something that hadn't existed before, this would be the same problem as to how something can come from nothing, the same problem as to how there be an effect without a cause.

An artist may reorganise existing parts, a blue line, a tree, a sky or a yellow mark, into a new whole, such as a painting of Water-lilies. An artist may change the relationship between parts that already exist, but the artists cannot create the parts out of nothing.

In fact, your challenge would be to find an artwork which included a part that did not already exist in some previous artwork.

The artist finds new relationships between existing parts. They don't create the parts.

The artist imagines new relationships between "concrete instantiations" of existing parts. The artist pictures new relationships between existing parts, and if successful, then applies them to the canvas.

Anyway, the point is that this "picturing" does not require a "concrete instantiation", which I assume implies a physical object being sensed. — Metaphysician Undercover

I can only imagine a unicorn by picturing a unicorn. A picture requires a "concrete instantiation". A "concrete instantiation" can be on a screen or a piece of paper. Both a screen and a piece of paper are physical objects existing in the world. As physical objects in the world, I can sense them.



You can only imagine a unicorn if you know what a unicorn is. How can you know what a unicorn is without having first seen several "concrete instantiations" of it as physical objects in the world?
===============================================================================

But if "1=1" is meant to signify that the thing identified by "1" on the right side is the very same as the thing identified by the "1" on the left side, then it is not a mathematical expression. It is an expression of identity. — Metaphysician Undercover

Identity is a valid part of mathematics.

Wikipedia -Identity (Mathematics)

In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables within a certain range of validity.

The key point here, is that imagination does not require sensation of whatever it is that is imagined. — Metaphysician Undercover

I imagine a unicorn by picturing a unicorn.

How do you imagine a unicorn if you don't picture a unicorn?
===============================================================================

The point though, is that in the case where you used "=" to signify identity, it is not a mathematical usage. — Metaphysician Undercover

"1 = 1" is a mathematical expression. The expressions "Twain = Clemens" and "sugar = bad" are not mathematical expressions.

Similarly, the word "infinity" has one meaning in a formal set theory and a different meaning in everyday natural language

From Frege's "Context Principle", the meaning of "=" and "infinity" depend on their contexts.

A "fictional object" is not an object,............. OED #1 definition of object "a material thing that can be seen or touched". — Metaphysician Undercover

A fictional object sounds like an object.

The OED notes "There are 14 meanings listed in OED's entry for the noun object, four of which are labelled obsolete" and then says "purchase a subscription".

The Merriam Webster includes "something mental or physical toward which thought, feeling, or action is directed".

I think Cervantes would have had great difficulty in writing "Don Quixote" without being able to describe objects.
===============================================================================

I suggest that your "position" is not consistent with common understanding. — Metaphysician Undercover

I agree that if one stopped one hundred people at random in the street, only a few would know about philosophical Monism.

Examples of modern philosophers who were monists include Baruch Spinoza, Georg Wilhelm Friedrich Hegel, Arthur Schopenhauer, and Bertrand Russell (https://study.com)
===============================================================================

There is no such thing as a "concrete instantiation" of a concept......................show me where I can find a concrete instantiation of beauty, — Metaphysician Undercover

How can one learn a concept in the absence of a concrete instantiation of it?

For example, as a test, suppose you thought of a concept. In practice, how can you teach me its meaning without using concrete instantiations of it?
===============================================================================

Just point out this 6 to me, so i can go see it with my own eyes — Metaphysician Undercover

===============================================================================

I mean, you presented me with "a horse with a single straight horn projecting from its forehead", and i understand this image without seeing a concrete instantiation — Metaphysician Undercover

Exactly, you understand the concept using images.
===============================================================================

Furthermore, there is no practical advantage to designating "=" as meaning identical in the case of "1=1" — Metaphysician Undercover

There are two different cases.

The first a case of identity where the two 1's refer to the same thing. The second a case of equality where the two 1's refer to different things.

The practical advantage of using identity rather than equality is to distinguish two very different cases.

Of course, that's known as fiction. — Metaphysician Undercover

"A mythical animal typically represented as a horse with a single straight horn projecting from its forehead" describes an object, even through the object is fictional.

In fact, from my position of Neutral Monism, all objects, whether house, London, mountain, government, the Eiffel Tower, unicorn or Sherlock Holmes are fictional, in that no object is able to exist outside the mind and independently of the mind.
===============================================================================

How do you think of a number as a natural concrete object? Are you talking about the numeral, or the group of objects which the numeral is used to designate, or what? — Metaphysician Undercover

The problem is, how does the mind understand an abstract concept, such as beauty, ${\sqrt{2}}$, ngoe, or the number 6?

My belief is that the mind cannot understand an abstract concept in isolation from concrete instantiations of it, in that, if I am learning a new word, such as "ngoe", it would be impossible to learn its meaning in isolation from concrete instantiations of it.



IE, I see no possibility of learning an abstract concept, such as "ngoe" or the number "6" without first being shown concrete examples of it.
===============================================================================

There is no such natural concrete object which the symbols refer to, in theory. only abstract concepts. — Metaphysician Undercover

If I wanted to teach you the meaning of the symbol "ngoe", which I know is a concept, how is it possible for you to learn its meaning without your first being shown particular concrete instantiations of it?
===============================================================================

And in application the concrete situation referred to by the right side of the equation is never the same as the concrete situation referred to by the left side. — Metaphysician Undercover

Given 1 and 1, if the second use of 1 refers to the same thing as the first use of 1, then the proper equation should be 1 = 1. The symbol "=" means identity

Given 1 and 1, if the second use of 1 refers to a different thing as the first use of 1, then the proper equation should be 1 + 1 = 2. The symbol "=" means equality.

Continuing:

Given a horse's body and a horse's head, as a horse's body is different to a horse's head, the proper equation should be horse's body + horse's head = horse

This raises the question as whether a horse as a whole is more than the sum of its parts, a horse's body and a horse's head.

Has the whole emerged from its parts, or is the whole no more than the sum of its parts?

IE, by knowing the parts, can I of necessity know the whole?

Referring back to Kant, by knowing the parts, for example, the number 5 and the number 7, can I of necessity know the number 12?

In your first question, "a person who can speak English" is a description, not an object. — Metaphysician Undercover

Can there be a description without an object being described?

Isn't "a person who can speak English" a description of the object (a person who can speak English)?
===============================================================================

It is not a true representation of how we use numbers, to think of a number as itself an object. — Metaphysician Undercover

The word "object" has different meanings. In mathematics, a mathematical object is an abstract concept (Wikipedia – Mathematical Object). In natural language, it can be something material perceived by the senses (Merriam Webster - Object).

For example, we can think of the number ${\sqrt{2}}$, the number ${2 * 10^{100}}$ and ${\infty}$ as abstract mathematical objects but cannot think of them as natural concrete objects.

However we can think of the numbers 1. 6 and 10 as not only abstract mathematical objects but also as natural concrete objects.

That raises the question as to how we are able to think of something that is abstract, disassociated from any specific instance (Merriam Webster – Abstract). For example, independence, beauty, love, anger, Monday, ${\infty}$, ${\sqrt{2}}$ and the number 6.

George Lakoff and Mark Johnson in their book Metaphors We Live By propose that we can only understand abstract concepts metaphorically, in that we understand the concept of gravity by thinking about a heavy ball on a rubber sheet.

Thereby, we understand the concept of independence by remembering the feeling of leaving a job we didn't like. We understand the concept of beauty by looking at a Monet painting of water-lilies. We understand the concept of infinity by thinking about continually adding to an existing set of objects. We understand the concept of ${\sqrt{2}}$ by thinking about the number 1.414 etc etc. We understand the concept of 6 by picturing 6 apples.

IE, we can only understand an abstract concept metaphorically, whereby 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 (Merriam Webster – Metaphor).

If that is true, then even more reason why one would then consider the question in regard to mathematics. If it's meaningless in context C but defined in another context D, then it wouldn't make sense to say that then it is inapposite to context D. — TonesInDeepFreeze

That raises the interesting question that if an expression such as "infinite infinities" has no meaning in a natural language, the everyday spoken and written language used to describe the world around us, but does have meaning in the formal language of set theory, then what exactly is the relationship between a formal language such as set theory and the world around us?

The principal problem with set theory..............is that set theory is derived from a faulty Platonist premise, which assumes "mathematical objects" — Metaphysician Undercover

In a random web site is set a problem that can be solved by set theory:

In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can speak English only? How many can speak French only and how many can speak both English and French?

Doesn't this problem, soluble by set theory, assume "objects", such as the object "a person who can speak English"?

If the number "1" does not refer to an object, what does it refer to?

along with its fantastic representation of "infinite" — Metaphysician Undercover

I would say that "infinite number" does not refer to an object, because unknowable by a finite mind, but does refer to a process along the lines of addition, which is knowable by a finite mind.

There are infinite sets that have sizes different from one another. — TonesInDeepFreeze

I take the OP as asking the question "are there an infinite number of infinities?"

The answer would depend on whether looked at from set theory or natural language.

Set Theory is a specific field of knowledge with its own rules, and as the Scientific American noted: As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.

However the terms infinity and infinite sets are also used in everyday language outside of set theory, such as "I have an infinity of problems" and "I have an infinite set of problems".

As the OP doesn't refer to the very specific field of "set theory", having its own particular rules, I think the OP should be considered as a problem of natural language.

Within natural language, the question "are there an infinite number of infinities" is meaningless, as not only is "an infinite number" unknowable, it follows that whether there is one or more infinite numbers must also be unknowable.

On the assumption that the OP refers to a problem in natural language, otherwise it would have specifically referred to "set theory", as it refers to that which is unknowable, although syntactically correct is semantically meaningless.

That reminds me of intuitionists or at the very least of psychologists in the ontology of mathematics, where the number 2^100 does not exist until it is thought up. — Lionino

But then again, the number 2 does not exist until it is thought up.

The whole confusion resulted from the wrong premise that infinite numbers do exist. No they don't exist at all. So it is an illusion. — Corvus

:up: :up: :up:

Of course, but I'm saying that in context of sets in mathematics, 'infinity' as a noun invites misunderstanding, especially as it suggests there is an object named 'infinity' that has different sizes. — TonesInDeepFreeze

In the Merriam Webster dictionary infinity is classed as a noun, and within mathematics is the infinity symbol ∞. But as you say, this is problematic as it suggests that infinity is an object, such as a mountain or a table, which can be thought about. But in one sense this is impossible, as it impossible for a finite mind to know something infinite, where infinity is an unknowable Kantian "ding an sich" ("thing in itslef").

So what does the word "infinity" refer to, if not a noun inferring an object?

As the Wikipedia article on Infinity writes: Infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.

As Literature as a noun refers to the study of books, perhaps Infinity as a noun refers to the study of infinite sets. Both Literature and Infinity are nouns, but refer to a process, not the intended conclusion of the process. This makes sense, as processes are comprehensible to finite minds. A finite mind can comprehend the process of adding to an existing set, even if not able to comprehend the eventual conclusion of continually adding to an existing set .

IE, "infinity" is a noun and refers to a process rather than any conclusion of that process.

'infinity' is not an adjective. — TonesInDeepFreeze

True, infinite is an adjective and infinity is a noun

But it can get complicated.
Music fills the infinite between the two souls - Rabindranath Tagore
Infinity pencil with eraser - Amazon

If that is the case, then it seems barmy to talk about different size of the infinite sets. — Corvus

There cannot be different sizes of infinite sets

As you say: "Infinity is a property of motion or action..............Infinite number means that you keep adding (or counting whatever) what you have been adding (or counting) to the existing number"

What does "infinite set" refer to?

It cannot refer to an object, an infinite set, as comprehending an infinite set is beyond the ability of a finite mind. It can only refer to the process of being able to add to an existing set.

In other words, "infinite set" refers to "a set that can be added to", where "that can be added to" qualifies the object "a set".

As a "set" is an object it can have a size, and therefore there can be different sizes of sets.

However, as the qualifier "that can be added to" is not an aspect of the size of the set, whilst the expression "different sizes of sets" is grammatical, the expression "different sizes of infinite sets" is ungrammatical.

What is infinity

On the one hand we have the concept of infinity within the symbol ∞, but on the other hand a finite mind cannot comprehend an object of infinite size. So what does our concept of infinity refer to?

As the Wikipedia article Extended Real Number Line notes, the infinities are "treated" as actual numbers, not that the infinities are actual numbers.

As our concept of infinity cannot refer to an object, as comprehending an infinite number of things is beyond the ability of a finite mind, it can only refer to the process of adding to an existing number of things until it is not possible to add any more, which can be comprehended by a finite mind.

IE, "infinity" refers to a process not an object.

Infinity is a property of motion or action — Corvus

I agree. "Infinite" is a property attached to an object, such as "large house" or "infinite number".

As "large" doesn't exist as an object, "infinite" doesn't exist as an object.

As I wrote before: ""infinity" as an adjective means something along the lines "any known set of real numbers can be added to"".

The fact that you can stack a property onto a substance to make an object does not mean that that object is instantiated in real life, — Lionino

Yes, for example as in "infinite number" where "infinite" is a property of "number".

Can there be infinite infinities?

Can there be an infinite set of (infinite set of numbers)?

The word "infinite" is not a noun but an adjective qualifying the noun "set".

Therefore, there can be infinite infinities because the word "infinity" is an adjective.

does that mean that there are infinitely infinite infinitely infinite infinitely infinite infinitely infinite infinitely… (etc.) infinities? — an-salad

Hopefully this doesn't contradict what @TonesInDeepFreeze has said. It seems that if the word "infinity" was being used as a noun, then yes, there would be an infinite number of infinities. However, the word "infinity" is not being used as a noun, but rather is being used as an adjective, in which case there is only one infinity. IE, "infinity" as an adjective means something along the lines "any known set of real numbers can be added to".

But he is not denying the outside empirical world where you see all the daily objects and interact with them. — Corvus

There are several things in your posts that I don't agree with, but as I am off on holiday, I won't be able to tackle them.

However, I think you are misusing the term "Empirical World".

The IEP article Immanuel Kant: Metaphysics differentiates between an "empirical world" in the mind and a "mind-independent world" outside the mind

Kant responded to his predecessors by arguing against the Empiricists that the mind is not a blank slate that is written upon by the empirical world, and by rejecting the Rationalists’ notion that pure, a priori knowledge of a mind-independent world was possible. Reason itself is structured with forms of experience and categories that give a phenomenal and logical structure to any possible object of empirical experience. These categories cannot be circumvented to get at a mind-independent world, but they are necessary for experience of spatio-temporal objects with their causal behaviour and logical properties. These two theses constitute Kant’s famous transcendental idealism and empirical realism.

IE, the "Empirical World" is the world as perceived via the senses. That we may perceive tables and chairs in this "Empirical World" does not of necessity mean that tables and chairs exist in a "Mind-Independent World".