Am I walking uphill when I walk on an inclined treadmill? (part 1)?

You are not walking uphill as your height is constant.

But you are simulating walking uphill - instead of increasing your potential energy, you are transferring energy to the treadmill system. The amount of energy transferred (hence the work you do) may not be as much as walking up a real hill at the indicated incline - it depends on the treadmill design and calibration.

Long discussion on this in the link.

(part 2)

Last night I visited the gym and I walked 10 km (6 miles) on a treadmill, with the treadmill set to a 5% uphill gradient. But was I really walking uphill?

If it were a real 5% gradient I would have climbed 500 metres as well as walking 10 km. I certainly feel like it today. I am more tired than I would be if I had simply walked 10 km.

But I didn't actually gain any altitude. At the end of the walk I was in the same place I was in when I started. So it can't be the same as climbing 500 metres. I didn't gain any gravitational potential energy.

On the other hand, within my moving frame of reference I certainly did climb 500 metres. If I were in a box moving downwards on a very long treadmill, and there were no peep holes in the box or any other way of telling that I wasn't stationary, how would I know? Anything I do inside the box has to be the same as if the box were not moving, doesn't it?

Clearly these statements can't both be true. But which statement is true, and why?

Apologies for the long answer, I find this fascinating though.

With each step on the inclined treadmill, you really do gain some gravitational potential energy, but the treadmill track takes you back down to where you were before you took a step. In other words, the treadmill is moving you backwards and downwards, so to remain on the treadmill, you have to move forward and upwards.

Suppose the treadmill is flat. After walking for some time, you could ask if you had really walked any distance, even though you remained stationary relative to the earth. The answer is yes here as well, as you have propelled yourself against the action of the belt.

This is stupid. If you did clime a real mountain and when you got to the top, someone in a helicopter flew you back down to the base... you are right back where you started having put no more energy into the downward travel, yourself. The energy spent by the helicopter is like the energy spent by the treadmill. You think if the helicopter brought you back and all that gravitational potential energy went away, you wouldnt feel tired the next morning? Take another physics course, man. This time at an accredited school.