Node your homework. This essay was written for an IB Theory of Knowledge course; I feel obliged to apologise for its high BS quotient. Sections referring to Areas of Knowledge and Ways of Knowing are specifically related to the ToK curriculum and rubrics, and may be clarified by a quick review of the information at Theory of Knowledge. Use of the word "knower" is again a ToK quirk.
If "to know" means "to grasp in the mind with clarity or certainty" (www.dictionary.com), proof (or lack thereof) seems to have very little impact on the ability of an individual to "know". One can grasp a concept or fact with "clarity or certainty" without being able to provide proof of it. Knowledge can be acquired through others, through inference, or through faith; the knower need not personally have proof of an item or even suppose it has been proven to know it. The conviction or understanding and, according to this definition, therefore the knowledge, of an individual does not necessarily rely upon proof. This definition has certain limitations: it is not consistent with the generally accepted definition of knowledge; and it relies on the assumption that grasping a fact with clarity or certainty does not include having proof of that fact. Neither of these limitations, however, make the definition less intrinsically valid in this case.
One conclusion about the relationship between knowledge and proof cannot be universally correct. Some Areas of Knowledge must necessarily have answers subtly different from others. However, in general it is not necessary to prove something in order to know it. The many ways of acquiring knowledge of all areas do not all involve explicit proof of what is being learned.
For example, in mathematics, there are specific proofs for each formula. The accuracy of any answer is an absolute. However, one can still know a mathematic formula without being able to prove it. I know that Snell's Law is true, even though I am incapable of proving the equation by deriving it. I know that all triangles have angles amounting to 180 degrees, though I may not have a protractor on hand to prove this, or know why it is so. This is because the knowledge has been passed to me by people in positions of authority, or because I have gathered it from other trustworthy sources. The absolute nature of mathematical facts means that gained second-hand, they retain accuracy. It is possible to know something about mathematics without being able to prove it; proof is more clear-cut in mathematical knowledge than it is in many other areas.
I once had a science teacher who proposed that it is impossible to prove anything in science; only to disprove things. For example, theories about the structure of matter continue to be edited from one decade to the next, disproving old hypotheses and creating new ones. The statement that all scientific theories will be eventually disproved is likely true; scientific theories and facts are changing all of the time, as new scientists disprove the old theories. It would be impossible to assume that science as a whole is unproven and therefore impossible to know; thus it must be possible to know what is unproven. If one only grasps a scientific theory clearly, they have knowledge of it. If we believe the statement that it is impossible to prove scientific facts, it must be true that we know unproven scientific theories.
In psychology, the same principle holds, except that it is as impossible to disprove a psychological hypothesis as it is to prove one. In part this is due to the fact that extended testing upon the human mind is considered unethical in most modern societies, and in part due to the extreme complexity of the human mind. At any rate, knowledge of psychology is largely conjecture, based upon patterns in human culture and theories like the archetypes of Jung or the Id and Ego of Freud. The logic behind such theories can be proven, but the theories themselves cannot. Psychological theories are backed up by observations and conjecture. Many of the things we know about psychology are things we cannot provide concrete proof for, yet it would be impossible to argue that we do not know anything about psychology.
History is yet more nebulous. Since so much of the study of history depends on second-, third-, fourth-hand, and even more remote sources, actual proof of historical fact is highly relative unless one happens to be an eyewitness; and even then, David Hume argues that "remembered events differ only from imagined events in being more vivid" .
Many such remote historical facts seem the only logical possibilities. Columbus must have landed in 1492, because there is overwhelming evidence to that effect. Is this logical conclusion real proof, if the arguer cannot cite or otherwise provide the "overwhelming evidence"? Defining "proof" as "the evidence or argument that compels the mind to accept an assertion as true", it is. This leaves facts with no particular evidence to support or deny them, though. One can have a great deal of this simple factual knowledge of history (or, indeed, any other topic) without ever being able to explicitly prove any of it or knowing how it could be proven.
Many aspects of religion are seriously impacted by history, or historical figures -- Jesus, Mohammed, the Bahá'u'lláh. However, clearly proven detail about these figures and their lives is on the whole absent, and often not entirely necessary. Followers of religions which rely on historical events as parts of the belief system are generally convinced through trusted sources or "overwhelming evidence", two methods of knowing previously discussed. In the main, the problems between religion and proof arise elsewhere, for the existence of a supreme being, that most important principle of most religions, is impossible to prove objectively. Like psychology, religion cannot count even on disproof. Some people must have individual spiritual experiences to convince them of the reality of a higher being, else we are making astounding leaps of faith. Indeed, in the context of religion, such leaps are acceptable --as long as we realise that they do not constitute absolute truth.
Morality (often an important aspect of religion) is another thing that is very difficult to prove. One can have clear, certain knowledge that it is wrong to do some act, but no "proof " to back up that belief. For example, I do not belong to any particular religion, or subscribe to any particular school of moral thought. However, I "know" that it is wrong to kill another human being. If you were to ask me why this is so, I would have no proof to offer; it is simply what I believe. Many people have this kind of belief (knowledge), whether it is on the subject of morality or not. This internal knowledge may be one of the most "certain" forms of knowledge, yet is impossible to back up with proof.
The "truth" of many religious principles is known, deeply and in some cases passionately, by many people throughout the world? but many of them disagree on it! Is this, perhaps, because all of them lack proof? Still, their lack of proof does not call into question whether or not they know their religions. In fact, they most definitely do grasp them with "clarity or certainty". Religious people know many things, but cannot prove that these things are true. Religious knowledge is based upon faith, and impossible to prove.
One could argue that when we do not truly have proof for our knowledge, we only perceive that we know it; that is, that while you may think you know something but cannot prove it, you truly do not know it at all. This argument revolves around the definition of "to know" as "to regard as true beyond doubt". What, exactly, is "truth"? If we assume that something that is known by this second definition must be true ("true" defined as "not false or erroneous" - dictionary.com), proof is surely required. However, the vast quantity of knowledge which would, under this definition, no longer be knowledge, makes it an unreasonable one.
The further a discipline of knowledge leans towards subjectivity, the harder it is to prove the things that you know about it. Mathematical knowledge is the most easily proven, and religious knowledge the hardest to prove. Through all areas of knowledge, though, it is impossible to restrict what one knows to what one can prove. There will always be thoughts, ideas, and even facts that people have knowledge of, but cannot prove --whether it is individual inability to prove something, or complete lack of proof for it, it can still be known.
Different possible definitions of "to know" and variations in the nature of knowledge in various disciplines make "Can we know something that has not yet been proven true?" an interesting question to answer. However, we must consider that something does not have to be true in order for someone to know it. Many people knew, at one point in time, that the world was flat. Now, we know that it is not --but does that make their knowledge any less real? Perhaps not. Presence or lack of knowledge and proof of knowledge are not inextricably intertwined; they are linked by the subject they concern, but otherwise separate.
It is clearly possible to know something without being able to prove it.
Sources are class materials now lost.