Science can only tell us a posteriori synthetic truths, can this category extend to all propositions about the world. Surely the only things which can by known ONLY by a priori analytic are tautologies and relations of ideas. Could you give me examples of things about the world which cannot be known through science, or rather whether you agree that science in the ideal conditions could tell us everything? Also as a second question - can science tell us anything? Tell with certainty. Also as Hume believes that causation isn't real can we really trust science?
Science consists of empirically testable explanations about the world. So any idea that's not empirically testable is not science. For example, the idea that there is a real world as described by science is not empirically testable. If parts of the world stopped existing when you aren't interacting with it, but the rest of the world acted exactly as if it did exist, you couldn't test that. That's not ruled out by any experiment, it's ruled out by being a bad explanation. In substance the "non-existent" part of the world still appears in your explanations you have just labelled it as not real. See "The Fabric of Reality" by David Deutsch, Chapter 4.
Hume's argument against causation is wrong. He claimed that you only see one even succeeding another, but that doesn't prove that they are causally connected and so causation is bunk. Hume, like most philosophers even today, thought that knowledge has to be justified: shown to be true or probably true. Knowledge is never justified because justification is impossible, unnecessary and undesirable. If you assess ideas using argument then the arguments have premises and rules of inference and the result of the argument may not be true (or probably true) if the premises and rules of inference are false. You might try to solve this by coming up with a new argument that proves the premises and rules of inference but then you have the same problem with those premises and rules of inference. You might say that some stuff is indubitably true (or probably true), and you can use that as a foundation. But that just means you have cut off a possible avenue of intellectual progress since the foundation can't be explained in terms of anything deeper. And in any case there is nothing that can fill that role. Sense experience won't work since you can misinterpret information from your sense organs, e.g. - optical illusions. Sense organs also fail to record lots of stuff that does exist, e.g. - neutrinos. Scientific instruments aren't infallible either since you can make mistakes in setting them up, in interpreting information from them and so on.
Rather, knowledge is created by noticing problems with current ideas, proposing solutions to those problems, criticising the proposed solution until only one is left and there are no criticisms of it, and then looking for a new problem. Causation is an idea that solves problems, so it is knowledge. The idea that X causes Y can be tested by looking for instances where X occurs and Y does not.
For more on epistemology, see See "The Beginning of Infinity" by David Deutsch, Chapters 1, 2 and 10, Chapter I of "Realism and the Aim of Science" by Karl Popper and
For more on causation see "The Beginning of Infinity" Chapter 5.
If Quine is right there is no clear distinction between the linguistic and factual components of language, both are interrelated (caricatural example: if someone else utter a false sentence such as "this cat is blue", maybe he is wrong on the meaning, maybe on the facts, but you can never be sure). Quine emphasizes that no statement can be tested in isolation, you always put your whole web of belief to test (including the meaning you implicitely attach to words, your mataphysical presupposition and so on). This is why meaning and facts are interrelated, and this is why for Quine there is no analytic synthetic distinction.
Science could tell us anything if there existed an ideal language perfectly suited for the world and devoid of any ambiguity, in which to express this anything. But finding the most appropriate concepts to express truth is itself subject to scientific and philosophical inquiry.
Without a clear analytic synthetic distinction, there is continuity between science and metaphysics (facts and conceptual analysis), there is always room for metaphysical disagreement, and you can't say anymore that science can tell us anything.
Well, to steal blatantly from Wikipedia:
"Science (from Latin scientia, meaning "knowledge"1) is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe."
In other words, science is all about using testing, experimentation, and observation to build models which we can use to make educated guesses for how the universe will act in the future (based on how it has been observed to act in the past).
As such, there are plenty of things science will never tell us. Events that occurred before humankind began to rigorously test and record our observations of the world/universe will never be "proven" by science. Events which cannot be measured by scientific tools also cannot be proven (e.g. psychic abilities), although this can sometimes be mitigated by using available tools to measure the indirect effects of such "unmeasureable phenomenon."
To use a specific example, the "big bang theory" only covers the expansion of matter as we have observed. While the scientific models may show that the rate of expansion we have observed is consistent with all matter rapidly expanding from a single point, that is not the same as science telling us that is what actually happened. Such conclusions fall into the realm of conjecture.
Conversely, what CAN science tells us? Scientific models can be used to predict how matter and energy will act in the future, based upon known conditions and our observations on how it has acted in the past. Given your 'under ideal conditions' caveat, I would say science could give us 100% accurate information on the future behavior of the universe once we assemble a model of sufficient robustness.
There are things within the realm of science which science cannot tell you. For example, in quantum physics there are things which are in principle unpredictable. It is impossible in principle to predict the time when one specific radioactive atom will decay. Slightly weaker are chaotic systems, where things are so hard to predict that science will never be able to predict them.
There are unfalsifiable statements. For example, someone claims that an infinitely powerful being created the earth 6,000 years ago and hid dinosaur bones that look to any scientist as if they were 200 million years old. That's a statement that cannot be proven false (except if scientists can prove that these bones are forgeries). So science cannot tell us if that statement is write or wrong.
And there are plenty of questions which are just outside the realm of science. For example anything related to ethics, what is right and what is wrong (questions about mental states can in principle be answered by psychology).
I suppose science can give you an approximation to how the world functions, but it cannot tell you the fundamental nature of the world.
A likely candidate for your question is qualia: the "redness" of red (i.e., its experience, not wavelength or physical effects) is neither a tautology nor objectively derivable using objective methods, it must simply be experienced. Much like you have to taste an apple or banana to really know what it tastes like...no theory can provide that - qualitative experiences are a totally different class of phenomena.
Yes, there are reasons to believe that there are things that science cannot tell us. There are at least two major "sections/areas" that science can not "tell us" anything about them. 1) Anything outside of its domain (ethics, arts, etc.). 2) Anything outside of the "current observable" universe (future scientific discoveries, advances, etc.).
It is supposed that Science is trying to reveal Truth/s. That science cannot elude without revealing the 'revealer'/witness for it will not complete without this 'process'. What will happen then? Can the 'result' be a 'nothingness' without consciousness? Who'll be 'experiencing' 'it'? Would that be explainable or understandable by words as a mere experience?
Here Science can't do anything with any device.
I have the tendency to believe that a human brain, can't design a human brain since the human brain only allows for conscious access to itself. Leaving all the unconscious parts obscured.
Making it totally impossible to fully understand.
We might come close to it with trial and error, though.
Also there is no such thing as exact science. Science is the art of approximation.