Like you, I enjoy browsing the 'unsolved problems' bits in Wikipedia and solving them in an evening. Don't we all?
The so-named Münchhausen trilemma proves that proving a truth is impossible. We can assume that the authors didn't notice the irony in this statement! That aside, this philosophical argument is goes along the lines of every fact, on its own, can only be shown as true when relative to other facts that are equally suspect. There are a few variations on the number and type of exact possibilities, but the fundamental argument is that truths are either self-proving or depend on other truths that depend on others ad infinitum. It is a bit like an enquiring child answering your every answer with "Why?" and you discovering that you can continue forever.
I wanted to examine this and considered mathematics. Consider 1+1=2. How do we know that sum is true?
Mathematics is an abstraction of reality. When we say "one plus one", each "one" there referred at some point to one actual object, and the maths is true if and only if one real object partnered with another makes two objects (which it clearly does, but of course this text is also abstract, so you'll need to find two actual things to confirm my bold claim). In a pure abstract sense, one plus one only equals two if it is related to reality.
However abstracted mathematics has become, it exists only as a tool to describe or utilise reality. If maths never did this, so that it became purely abstracted, then any conclusions it made would be untestable, and any conclusions would be as true or false as any other, that is, meaningless! But maths is never purely abstracted, and even the most obscure theorems ultimately relate to the world or are used in describing the world in some way.
there were and probably still are today highly abstracted forms of maths that are probably useless nonsense right now, that one day might be useful. George Boole's logic was perhaps too abstracted in its day to relate to any form of reality, and perhaps meaningless and its truths self-dependant, until the invention of actual real-world computers which use boolean logic. At that point the truth of Boole's logic snapped into actual reality.
Ultimately, the truth validity of mathematics is related explicitly to the truth validity of the universe as a whole, as much as we can each personally test it. This argument can be applied to the Münchhausen trilemma too. Truth is only as valid as it can be personally tested, and as such is tied up with belief.
Now we come to truth itself, and what we mean by it. Often, as in much of philosophy, the answer comes down to the word definition of "truth" and "belief".
Knowledge is relative and unique to every perspective because it is a collection of data from different points (the universe) to a singular point (us), and this data might change en-route. As such, information about the universe is different for each observer, and so always personal. Remember that information about the universe that we have only needs to be slightly different, any different at all, to be unique, and if knowledge unique then knowledge is relative, not absolute.
The notion of truth generally implies absolute knowledge, rather than a personal belief, and that's because humans are social creatures and generally groups of us know things and share information. As as result of this gossip, a consensus emerges of what is true and factual. This is convenient, as it saves us testing everything, but the consensus is only an approximate social belief, not intended to be an exact reflection of everything, or even an exact reflection of anything. If our experiences are unique then the only complete truth is our personal belief. If we see a ghost and nobody else does then society confirms that ghosts don't exist, but to us they do.
What of machines that can test things? Are these not independent of humanity and so judgeless infallible tools that measure what is true and what is not? Can't we determine what is true using a machine? A machine that analyses any aspect of the universe is no different from another person that also does this (people can be reliable judges too!). The result of a machine might reinforce, or destroy, a particular belief we have, but it can't define truth any more than another person could, merely offer an opinion to contribute towards our beliefs. Like any observer, a machine's sensor has a unique and limited view of the universe.
So the ultimate solution to the Münchhausen trilemma problem (also known as Agrippa's trilemma problem) is that a certain truth is true if we believe it to be, and that no further proof is necessary.
