Is all knowledge dependent on culture? how to define knowledge to our own perspective? if we don't exist does the knowledge disappear?
If you are a strict dualist or an outright idealist, the answer to the last question is 'no', the ideal forms like the ideas in mathematics and logic remain, independent of material reality, and even if all intelligent species went extinct, so there were no thinkers in the world, the idea of a thinker has been established and would always exist. In fact since it ever happened, it was possible, and so it always existed even before there were any thinkers...
The answer to the first question, then, also has to be 'no'. From an idealist point of view there are underlying forms for perfect concepts, even if we will never meet one that has not been compromised by the interference of a culture.
Of course, there is a strong trend away from idealism in modern thought, because we keep finding our investment in material reality pays off in ways we did not expect. And that makes the concept of a realm separate from it seem less and less worthwhile or logical.
I tend to favor the compromise form of pantheism, a la Hegel, which suggests that intelligence is part of the structure of the world in such a way that if all thinking beings were eradicated other thinking beings would arise. That position is neither idealist, nor anti-idealist, but is largely opposed to 'substance' dualism. It presupposes that information is an aspect of material reality and that complex enough matter will always develop intelligence, because of the nature of time and resources. You can look at that inclination as an extension of Darwinism, or an assertion that intelligence is more basic than matter. The difference is largely without meaning.
In that framing, your answer is ambiguous. The knowledge may be gone, but any truly important part of it will return whenever it is lost. So it is lost? And if any important idea returns constantly in various forms, there is surely something behind the various forms making them versions of the same thing. But that 'something' can never be expressed or embodied except through being incorporated into beings with a culture. So it exists through cultures and across cultures, but is never observed independent of a culture.
if we don't exist does the knowledge disappear?
Yes. Knowledge is a subjective concept, and so without subjects to do the "knowing", there is no knowledge.
Is all knowledge dependent on culture? how to define knowledge to our own perspective?
Most educated "laymen" would put forward the intuitive argument that empirical facts are objective and therefore culture/subject independent, and value and ethical statements are subjective and dependent on cultural background.
Consider however the Pirahã, who don't have concepts of cardinal and ordinal numbers, and who, because of their language, might not be capable of numeracy at all. For someone from the Pirahã people, a lot of seemingly empirical and objective facts to someone from another culture might appear completely different.
This extreme case illustrates the Quine-Duhem thesis, that all observations are ultimately theory laden, including those we think are indisputably objective. Although the idea that everything is theory laden was developed mainly within the context of the underdetermination of scientific theories, one could easily extend the principle to show that knowledge in general is always language and theory dependent, and therefore culture dependent.
Derrida offers a similar idea, but using a different approach, with his concepts of différance and deconstruction. Roughly speaking, Derrida establishes that terms in a language have meaning only relative to other terms, they never have inherent meaning. Again it is straightforward to extend this to the context of culture: Knowledge is based on meaning, and meaning is relative and therefore culture dependent, and so the same applies to knowledge.
Some in the scientific community have taken issue with this, feeling that the whole "Knowledge is culture dependent/everything is a social construct" idea has been taken too far by various thinkers and philosophers. Things came to a head with Alan Sokal's publication of a parody paper mocking such viewpoints, in what became know as the Sokal Affair.
how to define knowledge to our own perspective?
A good start for a working definition: A person knows that a certain fact holds, if
- he believes that the fact happens
- he can argue why he believes that the fact happens
- and his belief is correct.
Is all knowledge dependent on culture?
In general, scientific knowledge does not depend on a certain culture. Any international conference of scientists collects people from all nations and cultures. And they argue and decide independently from their culture.
if we don't exist does the knowledge disappear?
The facts continue, but the knowledge in our mind disappears.
On the opposite, much knowledge has been written down in books or journals. As soon as nobody is more able to read and understand those texts, one can discuss whether this knowledge has disappeared too.