Hello Ken,
You say: "However, in the arithmetic paradigm, the proposition 2=2 is true and what makes it true are the rules of arithmetic that have been invented by humans and nothing else."
Depending on the exact meaning of "invented" I may agree or disagree with you on this one. The rules of arithmetic have been "invented" many time over the course of human history. Egyptians and Sumerians may have learned it from each other, but it is likely that the Mayan "invention" of arithmetic is independent of the "invention" that happened in the Middle-East/North Africa.
Instead of calling it an invention, we could also call it a discovery.Arithmetic is based on a set of patterns that are useful to count things. Entities with the ability to discern patterns and find it useful to count, may therefore discover the usefulness of these patterns.
Is the rule that the sum of the angles of a triangle in Euclidian space has a constant value, a discovery or an invention?
The people that first learned this rule may have called it a discovery. Maybe they were playing with sticks and ropes, and found out that no matter how they positioned their sticks, something remained invariant (that which we call the sum of the angles).
Certainly there are human elements in how we symbolize these rules. In most of the world we tend to write angles in some base 360 notation, a convention borrowed from the Sumerians, but the invariance of the sum of the angles under transformation of the triangle is just as true had we used a base 10 notation for angles.
Even some artists go so far as to call their creation a discovery. There have been sculptors who claimed that the statue they created, was already present in the marble, and the artist merely unveiled that presence. I don't necessarily subscribe to that idea, since the exact shape of the statue is defined by the statue itself, unlike physical representations of mathematical objects like circles and rectangles, which are always an approximation of some ideal.
The usage of the word ideal in this specific context doesn't necessarily make me a Platonist. A mathematical circle can be precisely defined and when measuring a physical representation of a circle we can, up to quantum uncertainty, at least in principle state how that physical representation deviates from that ideal. The same can not be said about categories. It is impossible to define an ideal chair, let alone measure a physical chair and state how the two differ from one another.
Later on you state: "Your phrase "but for the meaning they express" is worded in a way that seems to carry the hidden proposition that signs themselves somehow "contain" or "carry" meaning. "
If it comes through that way, then I have not been clear enough. Signs do not contain or carry a meaning, but they do by convention map to a particular meaning. The human element in this is the convention. In fact, I am inclined to view your notion that "killed" and "vermoordde" are not equivalent in the statements used earlier in this discussion, smells like a hidden proposition that signs themselves "contain" or "carry" meaning. If not, then how does meaning come into play in your thinking, or is meaning an entirely meaningless concept. And if that is the case, how can we have a meaningful discussion?
Near the end of your message you say: "As I see it, Logic is a method of reasoning according to certain rules.".
I don't necessarily disagree with you on this one, but if we look more closely at what those methods of reasoning are, then we see that substitution plays an important role, and substitution is only possible if we accept equivalence relations.
Finally, I'd like to return to the start of your message. You say: The concept of an "object" is something that humans have invented for the convenience of talking about their experiences.
While the word "object" is certainly a human invention and not a discovery, the universe we live in somehow seems to behave in such a way that two things that we call objects have the property of never being in the same place at the same time. Again, this notion breaks down at the quantum level, where probability waves can overlap, but for macroscopic objects we generally deal with, this seems to hold true. In that sense, two cups of coffee are discernible from one another by the fact that they are not located at the same place at the same time. Even ants seem to acknowledge this rule of nature, since they do not regularly have the habit of trying to walk through cups of coffee, even though they might be tempted to get in your coffee, if it contains sugar.
Niels