Okay, let us try some:
1) we use induction all the time (reasoning from example to general rule), without it the scientific method makes no sense (why Bacon is an icon in science). however, as Humne pointed out it can not be proved using logic. Philosophers have been trying for three hundred years, hence Whitehead's quip, "Induction, the Glory of Science, the embassment of philosophy" (or something like that). The paradox is that we use induction all the time in science (and in baseball, we know where to go to catch the grounder, hop and all, because we assume it will behave "as if" it were any of the thousands we have caught before) and it works all the time (within reason) but there is no reason to believe it should work even once! (wiki "the problem of induction")
2) we use higher mathematics all the time (anything more complex than simple arithmetic). Yet it is inconsistent, it cannot be proved (that is what Goedel's Theorem is all about wiki it if you need to). Yet these assumed to be true by faith structures "fit" the universe quite well (see Minkowski space and the Schroginger Wave equation or, the extreme case, Superstring or Twistor Theory). The paradox is that even algebra requires faith, we assume it will poduce the right answers even though we cannot prove it!
I really know you are not asking about that, so let's get to the Paradoxes of Physics.
3) In the beginning of the universe or under special curcumstances of gravitational collapse the density of matter at a point becomes infinate (it is not solveable with the math of General Relativity). This is a black hole (wiki it). Nothing can leave it (the gravity is so strong that light just stops at the surface) and anything that gets to the surface from the outside diappears into the black hole (think of a real hungry teenager). What we know of as space and time and energy and matter simply cease to exist in the black hole. Actually, it is a little more complex than that. The singularity (where mass density goes to infinity) exists at a point. It is surrounded by something (we do not and cannot know what, physicists make lots of guesses about it) and on the "outside" of this something is the event horizon (wiki it) where gravity is so strong that light cannot escape. The paradox is that a regaion of the universe exists (or existed in the past in the case of the big bang) that we simply cannot know in any physicalist way (this due to the equations of the untimate materialist, Einstein).
4) time keeps going for the entity aproaching the event horizon (a person would sense the passage of time and a clock would register the passing of time in seonds, minutes and hours). However, from the point of view of someone "outside the influence" of the gravitational field the entity approaching the event horizon slows down to the point where they freeze at a point in time. The paradox is that time can simultaneouly stop (yes, this is an approximation, but you tell ge how close to zero you need to get and I will tell you how long it takes the event to freeze) and flow on without noticeable change, depending on where you are (it stops for the entity away from the horizon and continues for the one nearing the horizon). (wiki "The Twin Paradox" for something real similar)
5) a quantum event "just happens" it has no cause in the sense that the cause has to preceed the event. You want to do this on a sunny day or infront of a very bright light. Take apart two pair of polarized sunglasses (the cheap plastic kind or the 3D glasses from the theatre). Hold one right lens up, then take the left lens and put it right on top of the right one (for 3D glasses they must be at 90 degrees). All that happens is the light becomes (about) twice as dim. Now rotate one lens ninety degrees. All light stops coming through. Now have someone else put a third lens at 45 degrees to the first two. Put the lens in front of the two, no change. Put it behind the two, no change. Put them between the two, ah, a little light comes through. What you have done is performed a quantum experiment. Imagine you could slip that third lens it really, really fast or that the two lenses were really, really far away. Putting the third lens in at some point (if the speed was near light speed or the distance was like between here and the moon) when the third lens is added the person watching the result would see the dim light "before" you got the third lens into place. The paradox is that causation can happen before the effect. (wiki "retro-causation")
6) imagine the double-slit experiment (wkii if you do not know what it is). A lightbeam shines on two slits in a priece of metal. If the slits are narrow enough and the right distance apart, the pattern on the wall will be a set of dark and light lines called the interference pattern (like the interference pattern between two waves on a still lake cuased by two rrocks impacting closely together). Conver one slit. The pattern on the wall is merely a patch of light going from dark to light to dark again. Now slow down the rate at which the lightwaves get to the slit until you know there is only one wave at any time between the light source and the wall (think of this as turning down the brightness in a television). The pattern now becomes a set of flashes. This is caused by particles passing through one slit and hitting the wall. Put a piece of old-fashioned filem on the wall where the light patch was before, come back after an hour or so and develop it and you will get an image that looks just like that bright patch. Open the second slit and you will see a set of flashes again. Put the film on the wall, come back in an hour and develop the film. You will have an image of the interference effect. Here is the paradox: you know the particle of light hits the wall as a particle (that is what causes the film to expose) but in the last case the only explanation (remember, we are only letting one particle at a time go through the apparatus) is that the particle interfered with itself to cause the interfernce pattern, which was caused by the particle hitting the film. You have just shown how a "piece" of light can be a particle and a wave at the same time.
I can keeep gpoing on. Schrodinger's cat (wiki it if you have to). The probability is 50% that the cat is alive and 50% that the cat is dead. If that cat is "mere material" (does not have a consciousness, since true materialism, what science does, assumes even our thoughts are mere by-products of brain chemical and electrical stuff, this is a no-brainer for a strict materialist like our scientist here) and the quantum event (usually the decay of some radioactive paricle) is known to have either happened or not happened, the cat is alive and dead until we open the box to "measure" the quantum event. The quantum event's "probability wave" "collapses" (called the "collapse of the wave function", wiki it) to either did not decay (in which the cat is alive) or did decay (in which the cat is dead). We cannot know which until we "measure" or "observe". In other words, it is our action (opening the box) that determines the aliveness/deadness of the cat.
Okay. Look up "Wigner's Friend" and "the EPR Paradox" and "Bell's Theorem or Inequality" on Wiki or Stanford's Encylopedia of Philosophy.
The bottom line: the world (if logic and relativity and quantum are correct) is a lot wieder than you think. In addition read the forgoing discussion with bakhtijan about most of the universe being hidden from us due to "redshift" and "the expansion of the universe".
The role of metaphysics in philosophy is to take these real paradoxes and resolve them by postulating a consistent and logical explanation. Science does not do this, we scientists take these paradoxes as "Reality".
An example of one such metapysic is "Bohmian Mechanics" (wiki). Where the physicist David Bohm postulates an additional quantum term (a "hidden variable") that allows us to say "the Cat is either alive or Dead, the problem is we cannot know which except by looking in the box". Warning there may be as many as ten or twelve metaphysical explanations for any of the paradoxes I mentioned or listed.
Phew!
Pax et amore omnia vincunt.
