Rainbow Troll - Please give me a specific example of something that can be known without science.
There are many examples I could offer, but the arguments are complex and I fear they might lead to off topic detours, so instead of discussing Leibniz, I will use a simple example that anyone can understand and agree with: mathematics.
It would be a mistake to say that mathematics is exclusively a priori, since some very simple mathematical operations can be demonstrated empirically (add one apple to a basket of seven and you have eight apples); but excluding the simplest arithmetical operations, most mathematical truths are forever beyond empirical demonstration and must be arrived at deductively through reason alone. A common example that everyone deals with everyday are negative numbers: no one can demonstrate their ontological reality - you cannot remove apples from an empty basket - and yet negative numbers are invaluable in physics, which is a branch of science that is devoted to studying what most people would consider the real world. The roots of negative numbers, known as 'imaginary' to mathematicians and the fact that by performing operations on them with 'real' numbers gives us 'complex numbers' which scientists use every day to solve problem in the real world is yet another example of demonstrably real entities that can never be detected with any physical instrument for the simple reason that they are not physical entities. How could any scientist - no matter how brilliant - prove through experiment that some infinities are 'bigger' than others? Yet the German mathematician Georg Cantor proved it without any laboratory equipment, through simple deduction.
Now, some physicists (and even mathematicians) have argued that mathematics is only a human invention and that there are no 'laws' of physics, only generalizations drawn from many observations over time and translated into mathematical terms. In other words: the 'laws' of physics do not dictate the behavior of the real world, they only describe that behavior in terms that the human mind can comprehend. I admit that I see no way of really refuting this line of thought, but I nontheless see it as absurd. It is simply an unacceptable coincidence that an invention of the human mind should be able to not only describe physical phenomena so precisely, but also predict it with little to no error. Pythagoras was right: the world is composed of numbers and equations. What we know as physical reality is only our experience of a mathematical, metaphysical reality filtered and organized through our imperfect sense organs. The 'real world' is scentless, colorless and insubstantial; all those latter qualities being (to use Aristotle's terms) accidental, not essential. Though I think it is unlikely that our world is a computer simulation, I believe that The Matrix is a pretty good analogy for reality.
But it is not just that logic can prove things to which science has no access to, it can also disprove erroneous ideas which science could never refute; ideas than have plagued humankind for centuries, resulted in untold death and misery, and held us back for long enough. I think the worst of these errors is the monotheistic God concept. Even Richard Dawkins has admitted that he cannot prove that God does not exist. On a scale of belief with 10 representing implacable atheism and 1 perfect faith, he places himself at 9 or perhaps 9.9999... His theist opponents have wrongly and unfairly interpreted Dawkins humility on this point as a weakness and have used it to argue that theism offers certainties that atheism could never provide.
Well, I am not as humble as Dawkins. On Dawkin's scale of belief/unbelief, I place myself squarely at 10. I am absolutely certain that God, as he is understood by Christians and Muslims, does not exist; that he cannot exist. And I know this NOT through omniscience, but through simple logic. In a former post I gave a summary of my position which anyone is free to try and refute if they wish. But they will not because they cannot. There is no such thing as an omnipotent, omniscient creator of the universe. There can't be because the very qualities of this alleged being are incompatible with each other. He is a logical impossibility with no more probability of existing than a five sided triangle. Behold the power of logic!
Again, I don't want to dis science. I think that science, properly understood, is applied mathematics which is the mother of science, logic, and philosophy. But as the daughter of math, science should no her place and not defy her mother or ignore her sisters. But some recent scientific theories have proceeded to just that; a perfect example being Quantum Mechanics. Heisenberg's uncertainty principle rightly states that when a scientist uses an electron microscope to discover the position or velocity of a particle this very act of observation (the electron beam) causes the particle to deviate from the trajectory it would have taken had he not attempted to observe it. Therefore we cannot know, in principle, the simultaneous position and velocity of any given particle. Okay, I agree; but then scientists apply logical positivism to Heisenberg's principle and say that, since we cannot measure one without changing the other, the particle has no position/velocity until we measure it and it is in fact our act of measurement which "collapses the particle's eigenstate". Once I accept this non-sequitur, I open myself to all sorts of madness: parallel universes, cats that are simultaneously dead and alive, observer created reality.
It's ironic that scientists who use the scientific method, with its double blind experiments that are specifically designed to protect the subject(s) of the experiment from being affected by the observer, have now rejected the reality principle in favor of solipsism! Niels Bohr, one of founders of quantum physics, famously argued that there is no way to prove that the moon exists when we are not looking at it and, if one takes logical positivism seriously, he's right! But all this madness could easily be avoided if scientists embraced the logical (but empirically unprovable) assumption that particles DO have both a specific velocity and position even if we are incapable of measuring both simultaneously. This would not be a wild leap of faith, it would only be an acknowledgement of the reality principle upon which the very philosophy of science depends. Otherwise, if reality is not objective but generated by the observor, then the scientific method becomes useless.
I could give you other examples but I trust I have made my point. Strict empiricism is a dead end road, a cul de sac, that ends in madness. The only way that science can save itself is by retracing its path back to the point just before it accepted Wittgenstein's error, and then go from there.