@dbb and @laurel have explained things perfectly, I only wish to complement and expand on their excellent answers.
Upper belongs to a category of English comparatives that are either losing or have lost their comparative meaning and are mainly used as positive. Excluding old and elder, the next six examples also have more than one superlative.
c. 1300, originally comparative of up (adj.). Similar formation in Middle Dutch upper, Dutch opper, Low German upper, Norwegian yppare.
inner, outer, utter, and upper
POSITIVE. COMPARATIVE. SUPERLATIVE.
old older oldest
– elder eldest
_ former foremost, first
in (adverb) inner inmost, innermost
out (adverb) outer outmost, outermost
– utter utmost, uttermost
up (adverb) upper upmost, uppermost
low lower lowest, lowermost
Digging deeper, I found the following excerpt, edition 1865, which observes that upper is used differently from most comparative adjectives. Note that neither lower nor higher are included in this list.
Obs 4 – It may be remarked of the comparatives former and latter or hinder, upper and under or nether, inner and outer or utter after and hither; as well as of the Latin superior and inferior, anterior and posterior, interior and exterior, prior and ulterior, senior and junior, major and minor; that they cannot, like other comparatives, be construed with the conjunction than, introducing the latter term of comparison; for we never say “one thing is former, superior, etc., THAN an other”
Higher is not always the opposite of lower.
We can talk about the lower site and the upper site of an institution. In the UK, lower school will cater to children between the ages of 4 and 9, middle school from 9 to 11, while the Upper school will educate students between the ages of 11 and 18. (Wikipedia)
In medicine, we talk about the upper limb; the arm, forearm and hand, and the lower limb; thigh, leg and foot. In anatomy, "upper" refers to the POSITION above another part (i.e., superior), “my upper lip is sore”, while lower refers to the position below another part (i.e.inferior) “his lower lip was swollen”. If humans possessed four lips, we could say (I'm not saying we do) the lip positioned above all the others uppermost, whilst the bottom lip would be the lowermost (lowest).
Up and Down
You turn up the volume to hear better. (turn up =increase)
You turn down the sound when the neighbours complain. (turn down =decrease)
You climb up a mountain in the morning
You climb down before it gets dark
You go up to town if you live outside the city
You go down to town (or downtown) if you live nearby. (But there's some debate over its semantics)
In none of these instances is "up" or "down" an adjective. If I climb up something, it tells you in which direction I am climbing.
Cambridge Dictionary tells us
We use low for things which are not high, or which are close to the ground or to the bottom of something:
We have a sofa, two armchairs and a low table.
The wall is too low; we need to make it higher so the dog can’t get out.
Comparing low and high
Two mountain climbers are resting, they are at two different points. Mountaineer A is positioned at 2,000m while mountaineer B is at 2,500m.
- A is lower down.
- B is higher up.
But if the two were descending:
- A would be further down the mountain. (i.e heading down or downwards)
However, I should point out that the latter is more commonly used for horizontal distances, e.g.
further down the path
further down the stream
further down the line