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.
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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
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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".
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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.
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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".
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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.
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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.
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But then I do refer to 'I' — TonesInDeepFreeze
In Kant's terms, we have a unity of apperception. The mystery is why.
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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.
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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.
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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?
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Meanwhile, I would not contest that formation of concepts relies on first approaching an understanding of words ostensively. — TonesInDeepFreeze
:up: