No philosopher worth her salt would try to answer this question as formed because of the large number of buried assumptions in it that presuppose the form of the correct answer, and that answers not meeting that form are, by definition, incorrect. Your question is a sophisticated version of "have you stopped beating your wife?"
What "truth" has mathematics itself produced from that search which is well-defined, non-trivial, and correct?
Is it a loaded question? I think it isn't, and it has a lot of valid answers. Here are some recent ones: http://en.wikipedia.org/wiki/List_of_unsolved_problems_in_ma.... Why the corresponding question for philosophers should be loaded? Are they exempt from settling things?
First, Goedl's Incompleteness Theorem comes out of philosophy through the early 20th century's focus on logic and positivism. I'm sure you consider that well-defined, non-trivial, and correct.
Second, there is no requirement for philosophers to "settle things", and it's undemonstrated that, in order to settle things, the answer must be well-defined, non-trivial, and correct. One example of that is Wittgenstein's dismissal of a significant number of previous philosophical problems as empty language games. The field advances; it doesn't "settle things" in the sense that mathematics lays down proofs upon which the edifice later builds. Sometimes the advance is dismissing large portions of what went before as mistaken.
I don't consider Godel's theorem to be philosophy. Its status is similar to Cantor's theorem stating that reals are uncountable. Surely is inspired by philosophy and has philosophical consequences, but it is a part of mathematics. Trivia: Godel used the Chinese remainder theorem in his proof http://mathoverflow.net/questions/19857/has-decidability-got.... I acknowledge that philosophy can be inspiration for mathematics, but this is rather unsatiating, as very many things can be.
You might not, but philosophers certainly do--I learned it in my philosophy classes on logic. I also learned there about Cantor's diagonalization proof. More generally, in the early 20th century there was a huge overlap between mathematics and philosophy. You had Russell and Whitehead's Principia Mathematica, you had the Vienna Circle, you had Carnap and logical positivism... It's really not possible to cleanly categorize Goedl's proof into either math or philosophy--they weren't disjunct categories then, and they're not now, either. More importantly, if you approached any of those figures and asked whether they were separate things, they'd have thought the question nonsensical. Philosophy was, and still is, the home of logic in the academy, and the fact that logic and math frequently seem like different sides of the same coin doesn't settle the issue either way.
Many philosophers have questioned the scientific definition of "correctness". For example Heidegger who argued that scientific truth is an approximation and measurement of things that we already presume to exist. These discoveries may be correct, and they may be true. But what about the more primordial truth (i.e. being) which we take for granted before we even begin a scientific inquiry? This is what he investigates, so pretty much all scientific definitions of correctness don't apply here. But it is very hard to convince followers of scientism that this investigation has any meaning, because they've already closed their minds to this form of thinking.
> Many philosophers have questioned the scientific definition of "correctness".
This concisely and aptly summarizes the reason for philosophy's low standing among intellectual disciplines.
Philosophers are manifestly unqualified to debate the scientific definition of anything, much less "correctness". Beyond this, a suitable definition is too short to be of interest to a philosopher, someone for whom the number of words uttered is always ranked higher than the intellectual content of each word taken separately.
A scientific idea is "correct" if it can be successfully compared to reality.
How hard is that? I hasten to add that no scientific idea ever becomes true for all time -- all such ideas are subject to falsification by new evidence, by new comparisons to reality.
> But it is very hard to convince followers of scientism...
Ah, yes, the "science is just another religion" gambit. It speaks volumes about the depth of modern philosophical thought.
Philosophers compare their ideas to those of other philosophers. Scientists compare their ideas to reality.
> A scientific idea is "correct" if it can be successfully compared to reality.
But this is similar to saying what is real is what corresponds to reality. Do you see the circularity here?
The philosophers who engage in questioning the "real" are not doing it for the reasons scientists engage in discovering "correct" phenomena. The longing for a deeper meaning and clarity beyond scientific inquiry is a spiritual longing. These philosophers are trying to describe ways in which human beings fit in the world, how we can deal with the groundlessness of our existence, what choices we have in light of the anguish that comes from our mortality.
The problem is that many people view this as a competition against science or "exact" thinking. It is not.
I think this quote from Leo Strauss sums up my point:
"Men are constantly attracted and deluded by two opposite charms: the charm of competence which is engendered by mathematics and everything akin to mathematics, and the charm of humble awe, which is engendered by meditation on the human soul and its experiences. Philosophy is characterized by the gentle, if firm, refusal to succumb to either charm. It is the highest form of the mating of courage and moderation. In spite of its highness or nobility, it could appear as Sisyphean or ugly, when one contrasts its achievement with its goal. Yet it is necessarily accompanied, sustained and elevated by eros. It is graced by nature's grace."
> But this is similar to saying what is real is what corresponds to reality. Do you see the circularity here?
The circularity is in your wording, not in the thing itself. A scientist has an idea, one expressed clearly enough that two or more similarly trained individuals can understand the claim. The idea is tested against reality, in a way (again) that similarly equipped observers can agree that the result means what it seems to mean.
The outcome is either that the original idea is supported by, or falsified by, the comparison to reality. And the distinction between the idea, and its test against reality, is nowhere confused -- not among scientists, anyway.
> The philosophers who engage in questioning the "real" are not doing it for the reasons scientists engage in discovering "correct" phenomena.
That's for sure -- philosophers much prefer arguing about the meaning of reality, to dealing with reality on its own terms. Many modern philosophers, following this trend, slide into deconstructive postmodernism without ever realizing that they've crossed the threshold of absurdity (by posing the argument that all experience is subjective and there is no objective reality as scientists claim, but without realizing that their argument justly applies first to the words they've just uttered).
Oh, come on. This is beneath you surely. Where are these "philosophers," this monolithic horde of abstraction-loving pinheads stuck in the 15th century, too benighted to see the Real Truth right under their noses, too stuck in debates over definitions to think practically about application and science??
You have a cartoon version of philosophy in your mind that is in sore need of remediation. These are precisely the kinds of questions that philosophers have been utterly preoccupied with for centuries. As if Bradley, Dewey, William James, never existed! As if the very idea of a pragmatic, "reality-based" science had not been proposed and debated rigorously for decade upon decade.
As if the subject-object problem wasn't at the core of the foundations of modern science!! Far from being some late, decadent conceit of a handful of disconnected postmoderns, the problem of objective knowledge is at the very core of modern science, at its very foundations, on to the present day.
Have you read a single volume of philosophy in the last decade? Are you not aware that the Grand Poo-Bah of modern philosophy of Science, Popper himself, redefines "Objective Knowledge" to deal with that very problem?
If you are content with a naive, self-contained scientism that remains dogmatically immune to philosophical critique, that's fine. You're certainly not alone. But let's dispense with the sweeping generalizations that have no bearing whatsoever on the reality of the history of philosophy, science or ideas in general.
Yes, I think you've gotten through to me. You're willing to hold strong opinions regarding the core discipline of the Western Tradition in a state of abject ignorance, on the grounds that actually educating yourself on the topic before forming such an opinion is analogous to committing suicide in order to understand suicide.
That's as bizarre a rationale for willful illiteracy as I can think of. You seem content with it, and I wish you well.
This is naive scientific triumphalism of the first order, utterly illiterate with regards to the philosophical foundations of science, the rigorous and multi-layered debates over the status of scientific statements that have been ongoing for centuries to the present day: from Bacon, Gallileo and Descartes (with deep roots in Lucretius, Plato, Aristotle, Thomas, Epicurus, Heraclitus) through Spinoza, Leibniz and Newton himself, on into such titans as Kant, Hegel, Husserl, Bergson, Heidegger, Russell, Whitehead, Popper, Kuhn, Lakatos and Quine.
And yet! And yet! It was all so simple, gads and forsooth!!
I give you the final word, gentle reader, it was all for naught, these men with their devotion to rigor and clarity of thought, their conversations spanning centuries, their volumes of pointless abstraction, Hegel with his historicism and Kant with his categories, cynical Kuhn and his paradigms, pragmatic Popper with his Worlds 1 and 2, Whitehead with his ingressions pseudo-Aristotelian speculations and Bergson teasing out duration and intuition.
Throw out your Rutledge Encyclopedia: and behold the simplicity they all failed to grasp:
A scientific idea is "correct" if it can be successfully compared to reality.
How hard is that indeed!! Not hard at all. Simplicity itself, devastating in its directness and comprehensive competence, lovely in its completeness. Centuries of debate quelled in a single utterance. Well done.
Look, I sympathize. Philosophy, as the core discipline of human thought, has established beyond any doubt that there are limits to empirical human inquiry, and scientists don't like that idea. The standard response of scientific cheerleaders like, for instance, Lawrence Krauss, is to simply sniff at the entire enterprise, and assume there must be some problem with "those philosophers," rather than entertain the notion that there may, in fact, be limitations to reason and scientific inquiry.
This is fine. If you're too lazy and content in your own unexamined assumptions to read a few hundred pages of the men who did the hard work that gave you your entire livelihood, it will not affect your ability to do practical research.
But let's be clear about some of the practical implications of that position:
* Science does not need to be rational in any meaningful sense. (Go ahead and try to reinvent that idea without falling ass-backwards into the same problems "those philosophers" have been working through for centuries. Go ahead, I'll wait.)
* It does not need to be consistent within its own assumptions.
* Science is de facto "true", simply by virtue of being practiced by someone who claims to be doing science.
* Since we've now abdicated the rigorous discipline of establishing secure definitions, the notion of empirical science is now free-floating, ad hoc, and vulnerable to redefinition at any turn. The only thing required is for a group of men who call themselves "scientists" who have a different agenda to take control of a majority of significant journals or the community at large and impose their definition by fiat.
Philosophy may seem to be arcane and needlessly obsessed with definitions. And when its obscure language intrudes into the hard work of real empirical research, it can seem to be a giant non sequitur that can be easily ignored, as science moves forward (whatever "forward" means, now that we can't establish what is truly salutary and what is not).
But such willful ignorance not only skirts the fact that it was such philosophical examination of the limits and range of human knowledge that actually established science to begin with, that sort of ignorance sets enormously dangerous precedents, a sort of intellectual stare decisis for the future of human inquiry. It opens the door to sophistry, demagoguery and ultimately pure irrationalism.
History is long, and intellectual tyranny is opportunistic. Whatever integrity and above-board intentions you think scientists can maintain in the long-term, on good will alone and not deep self-awareness of core philosophical discipline is absurdly naive. Without a community committed to rationality (which is what philosophy does even if it only sets up negative limitations that seem unsatisfactory to scientists who want carte blanche to do whatever they want), then the entire discipline in the long term most likely will be taken over by non-rational concerns: commercial, military, governmental or even religious.
Absent the basic language provided by philosophy that can give at least some basic definitions and intellectual rigor for what is and isn't "science" (again, good luck reinventing all that), welcome back to the pre-socratic age of sophistry, demagoguery and a new Dark Ages. Just give it a century or so -- buy hey, you won't be around, so why worry?
"These discoveries may be correct, and they may be true."
This is the problem with Philosophy. You can take something correct and true, and disregard it. Worse, it is disregarded in favor of what is essentially an opinion.
Looking for philosophical truths that are well-defined and correct is a bad starting point. If something can be described of as correct or incorrect, then there would not be much of a philosophical discussion since you would be able to find the answer.
Philosophy is looking to ask the deeper questions: the questions that do not have a yes/no answer, and cannot be well defined (or to put it more accurately: truths that can be well defined, but in so many different ways that it is difficult to choose one definition).
However, just because these questions do not have yes/no answers and are not well-defined, does not make them trivial!
A great example of where philosophy affects your life daily is political philosophy. Ask yourself if you are liberal, conservative or something else? None of these schools of thought can be described as 'right' or 'wrong'. They are all different ways of looking at the question of how a society should function.
Yet discussing this question is incredibly important, and the discussions that political philosophers have trickle down into your mainstream politics (a fantastic recent example that we are all aware of is the Tea Party Movement, which itself is saying that we should use the philosophy of hardline conservatism). Political parties did not just sprout out of nowhere: they involved people having incredibly thoughtful discussions.
You may be saying that although some philosophy affects your life (eg. politics), other stuff just seems to be pointless. For example, you might argue that discussing the meaning of 'right or wrong' is pointless and that we should just get on with life. However, the way that we treat someone that has done a 'wrong' will depend on our political system (for example, you may follow a school of thought that says we should place rehabilitation before punishment, or vice versa), and therefore to ask how our political system should treat these people that has done a 'wrong', we should ask what it means to say that something is 'wrong' in the first place (eg. if you do not intend an outcome, have you done a wrong?), or what it means to place 'blame' on someone (eg. does blame mean that we do not approve of their action, or that they should not approve of their own action?). These are not just abstract thoughts, but practical considerations that filter through to the very political system that you live in.
If you would like some other examples of philosophy affecting your day to day life, then I would recommend looking up these topics:
* Medical ethics: when does life start (conception? a heart beat? birth?)? Is it right to take the life of a featus after life has started? What about when rape is involved? What about when the mother is at risk of death if a pregnancy is to go ahead?
* What is a person? Should we follow someones directives after a personality change (what change would a person need to make before they invalidate their own do not resuscitate order)? If a person has changed, are their contracts or will still valid?
* What does it mean to say we are free? Is freedom something we should strive to achieve? Is freedom a human right? Is privacy something we should strive to achieve? Is privacy a human right? What about where these rights conflict?
* What is law? When do we say something has lawful authority? Does law need to be moral? Does law need to be fair? Is it ok to follow the law, even if we think it will do something that is morally wrong?
I'm happy to give more examples, or to go into more detail about why these questions are important.
NOTE: I studied law at university and decided to take a module on moral philosophy and it was one of the best decisions I ever made. Philosophy is about having discussions on difficult questions and gives you the skills to do so; this has affected everything I have done since. Once you start asking philosophical questions, they will start popping up in everything you do. I regularly find myself asking questions about nature and philosophy when coding just because I can see how it can trickle down to have a tangible impact on what I'm creating. If there is one thing I would urge you to do this christmas holiday, it would be to read a philosophy book (Zen and the Art of Motorcycle Maintenance is a great place to start), it might just change your life.
What "truth" has philosophy itself produced from that search which is well-defined, non-trivial, and correct?