Then I want to say that when we know what the sign means, what we know is how to use it. That means not only understanding the conceptual structure that gives is meaning, but what it requires us to do (and not to do). — Ludwig V
This is obviously incorrect. Use is fundamentally subjective. I can use a sign in a way which serves my purpose, and you might use the same sign in a completely different way to serve your purpose. To say that knowing the meaning is knowing how to use the sign is a very one sided way of looking at a multifaceted thing. This completely ignores the intersubjective (communicative) aspect of meaning. To include this supposedly "objective" aspect into your definition of meaning, we would need to consider "correct" usage.
And this exposes the real problem. What constitutes "correct" usage? Therefore referring to knowing how to use a sign, as an indication of knowing the meaning of a sign does nothing for us toward defining meaning, because "correctly" is implied by "knowing how to use", and this provides no guidelines for how to judge one's usage as "correct". So we really do not even approach the true nature of meaning in this way, because it is hidden by not including "correctly" within "knowing how to use". This is the issue exposed by Plato in The Republic, when he asked different individuals to describe how each would use the word "just". Each person had a different way of using the word, and debate was required to demonstrate that any individual's way was incorrect. This argumentation proved that the person really did not know how to use the sign "correctly", even though they could actually use it the way that they did. This indicates that "knowing how to use" implies some form of justification as implicit.
I don’t think so. “2 + 2 = 4” isn’t a statement about reality as such, but about a perceptual pattern abstracted by the mind. Numbers aren’t part of the world in the same way as, say, rocks or trees. They’re tools—mental instruments that help us structure and process sensory input. They emerge after perception, not before it. — DasGegenmittel
This is very doubtful, and that's the point of Kant's "a priori". Some form of abstractive power, or capacity, is necessary for, therefore prior to, sensory perception. And, since the difference between the thing-in-itself, and the perception of the thing (as a type of abstraction in the mind), is fundamental to the nature of knowledge, especially the fallibility of knowledge, we need to pay close attention to the nature of this difference in any epistemology.
This is the point of Aristotle's law of identity, "a thing is the same as itself". This law locates the identity of the thing directly within the thing itself rather than what we say about the thing, or how we symbolize the thing, thus creating a separation between the thing and the abstraction. When we adhere to this principle of separation, we notice that the so-called abstraction which corresponds with the symbol does not necessarily have any identity at all, as a thing. This allows for the reality of the fictive, imaginative, creative capacity of the mind. Therefore the real meaning of mathematical symbols such as "2" may be entirely imaginary, creations of the mind which are not at all based in perceptual patterns. And I really think that this is the true nature of what is known as "pure mathematics". The mind creates categories which are not based in abstractions produced from sensory perception, but based in its own intentions. The "empty set" for example.
Perception introduces difference. Without difference, there’s no concept of “two.” Numbers are thus not touchable objects, but operational categories—modalities of cognition. — DasGegenmittel
The problem with this approach is that we must allow that there is a means by which perception apprehends difference. This fundamental "mechanism" if we can call it that, determines which types of differences will be perceived. And, we need to take account of this mechanism, the a priori, to have a true understanding of the way that living beings come to know things. If we include this mechanism, then we see that the living being creates freely, through imagination, its own operational categories through the influence of forces such as intentions. Then through some trial and error process, experimentation etc., the successful, useful ones are maintained through time. If we do not include this fundamental principle, we wrongly presume that "correctness" is forced onto the living being by its environment, rather than something chosen by the being through its activities of application.
In other words, if we assume that difference is forced upon us by perception, as you propose, rather than something created by us for the purpose of judgement, we avoid having to understand the true nature of justification. We simply take justification for granted, as something given by the differences within perception. But this is fundamentally incorrect, as Plato demonstrates, justification is actually the means by which we get beyond the deceptions which the senses serve us. (Take the clock example for instance.) True justification requires that we establish a priori principles, real principles of difference, not just the apparent differences which the senses show us. Notice that we have five different senses. Within each sense there are differences which we notice, but we also need principles to account for the differences between one sense and another, which is a much deeper type of difference.
As for the broken clock case:
To say, “S cannot believe that a broken clock is working,” misrepresents the belief. “Broken clock” is an external diagnosis, not necessarily part of S’s belief content.
If S looks at the clock at 2:00 p.m., and the clock (stuck at 2:00) shows 2:00, S forms the belief “It’s 2:00.” That belief is justified (the clock appears fine), true (it is 2:00), and believed. The Gettier problem arises here. — DasGegenmittel
This is Plato's point with justification. The senses deceive us, and cannot be the source for true justification. The idea that sense evidence is what justifies, is itself misleading, guiding us toward faulty justifications. We must establish principles of comparison derived from the creative, imaginative mind, which form the real basis for justification. These principle are derived from concepts of sameness rather than concepts of difference. This is why it is very important to have a very rigorous definition of "same", to start from, as that provided by the law of identity.