One such, known as Xeno's Paradox, says that Achilles (a legendary Greek runner) should not be able to overtake a tortoise if the tortoise is given a head start in a race. By the same reasoning, it should be impossible for an arrow to hit a running rabbit.
This neatly illustrates how 'philosophy' can be used to confuse people and explains how it can be used with equal success to 'prove' whatever dishonest (or maybe, to be charitable, just not very bright) philosophers want you to believe, usually for money, and often to 'prove' diametrically opposite conjecture simultaneously, especially in different cultures. More of that later. Now back to Xeno...
Xeno's reasoning was this:
Let's assume the tortoise is given 100 meter start and Achilles can run ten times as fast as the tortoise. (The actual figures don't matter so long as the slower thing starts ahead of the faster one).
When Achilles has run to the point where the tortoise started from, the tortoise will have travelled 10 meters and will still be ahead of Achilles, now by ten meters. When Achilles has run the next ten meters, the tortoise will be one meter ahead. After the next meter, the tortoise will be one tenth of a meters ahead. And so on. In this way, Achilles can never overtake the tortoise because every time he gets to where it was, the tortoise will have moved on.
The gap continues to close but never reaches zero.
But, we can see that any runner can overtake a tortoise. We can also see that an arrow can hit a running rabbit. Why does the observation not match the theory? Is it the observation which is wrong, or is it the theory?
This problem taxed the brains of philosophers and mathematicians for centuries. There seems to be nothing wrong with the theory; the maths looks impeccable; the logic appeared to be sound. Yet runners can overtake tortoises and arrows can hit running rabbits.
Well, although no mathematician could prove it, because they lacked the mathematics, the theory is clearly wrong. A theory which produces a different outcome to reality is clearly wrong. But it wasn't until calculus was discovered (independently by Newton and Leibniz) that it could be shown mathematically.
The fundamental error in the theory is now obvious: Achilles and the tortoise are moving independently through space-time. Achilles' position is not dependent upon that of the tortoise. Achilles overtakes the tortoise when his trajectory crosses that of the tortoise. The only problem was in calculating the precise point in time when that happened. The wrong math was being used to calculate it so it could not give the correct answer. In fact, things were more complicated than Xeno was allowing for, and this shows the danger of relying on intuition to assess reality. The logic seemed sound because it was intuitive. The maths tells us intuition was wrong.
So what has this to do with philosophers 'proving' to people whatever they want to prove?
Let's take one of the philosophical arguments often used to justify belief in a god or gods: that of the origin of morality. The argument goes that there must be a god (or gods, according to the culture in which the argument is being used) otherwise there could be no origin for human morality. The argument goes that humans have no way to tell right from wrong unless a god (or gods) have told us what it is; that there is no objective right and wrong, only a set of rules handed down on the arbitrary whim of a capricious god. To behave 'morally' all we need do is learn the rules and obey the commands. The consequences of our actions are no concern of ours since they are the 'will of God'.
But, as with Xeno's Paradox, where does this lead us? Is the theory supported by what we can see for ourselves? Let's assume for the sake of argument that the one true god is the Christian god of the Bible, and that this god is perfect, omni-benevolent and inerrant, like Christians claim. If the theory is correct we should expect to see all non-Christians, and non-Christian cultures behaving in a chaotic and inhumane way towards one another, with no sign of any morality or ethics, whereas all Christians and especially Christian cultures should all be paragons of virtue with everyone behaving with impeccable morality and ethic and everyone would be doing unto others only and exactly what they would have others do unto them.
Is this what we see? Well, is it?
It would take an extreme form of parochial ignorance to believe that this is indeed the real situation. Anyone who has been to another country, or even seen television pictures of life in one, or met someone with another faith or no faith at all, could not fail to notice that, by and large, they behave at least as well, and often much better, towards others than do very many Christians.
Indeed, a moments thought, let alone seeing with your own eyes, should tell you that no society could succeed without the morals and ethics which ensure a more-or-less cooperative society which operates according to accepted rules of inter-personal behaviour and the necessary mechanisms for dealing with those who transgress them.
Any objective observation will tell you that Christian societies are not more moral than non-Christian ones and are frequently actually worse. Any reading of history will show you that Christian countries did not behave any more morally than non-Christian ones, and often behaved far worse. It will also show you that acceptable standards of behaviour have changed over time; that Christian societies changed their minds about right and wrong - slavery, female emancipation, burning heretics, etc. Generally we can see that the more fundamentally religious people and societies are, the LESS morally they tend to behave towards others.
The other possibility, which you've probably thought of already, is that a god has handed down morals to all societies, just in different ways. If THAT were true, we would all share exactly the same moral codes wouldn't we? And yet we can barely find two countries, or even two areas in the same country, which have exactly the same customs and traditions of behaviour towards one another, to women, adolescents and minorities; to ideas of appropriate punishment for crime, of political freedom and emancipation; of the age of consent, of contract and hospitality, or of the boundary between individual and collective interest and freedom.
Manifestly, we do NOT share a common set of detailed ethics and yet manifestly we have many morals in common. Our moral codes are like a wide-spread biological species - subject to regional variation and varieties - just like the human species.
And is it really the height of human morality to just obey orders? Does being moral really mean we have no concern for the effects of our actions on others, as long as we behave like a Nazi Auschwitz guard and obey the rules?
Just as with Xeno's Paradox, religious philosophers through the ages have debated this conundrum ad infinitum and never reached a consensus. And none of their different conclusions has managed to come close to describing observable reality - not that that has been seen as much of a problem.
So what's happening here, and what has this to do with Xeno's Paradox?
The theory is wrong because, like Xeno's Paradox, the basic assumption behind it is wrong: we do NOT get our morals from a god or gods. We get our morals from our cultures where they have evolved and developed over time.
If we apply THIS theory we can easily see why morals and ethics in different places and different cultures have many things in common yet differ in detail, and why they have changed over time.
Because the theory is correct it now equates to observable reality. And this is how we know the theory is correct.
Morality is not an argument for gods; it is an argument against them.