Or fifth, fourth, whatever. Omitting the article would make sense if it were "gear five". But with gear being a countable noun, I'm totally stumped.
Whenever I start my car I switch from neutral to first gear
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"First gear", "second gear", "third gear" are really conditions or settings of the transmission, not sequence numbers.
Early in automotive use, "first gear" was "low gear", "second gear" was middle gear" and "third gear" was "top gear". Those were not countable things, but transmission settings. When numbers were applied to the gears, they continued being used grammatically as before.
Likewise, golf clubs stopped having names. "Drivers" didn't change, but "brassie" became "two wood", "spoon" became "three wood", etc. Those clubs had always been "a brassie", or, "the spoon" and continued as "a two wood", etc.
One reason for all this is that "gears" have been defined with ordinal numbers: "First", "second", etc. Clarity requires that one not refer to the first gear, as, the first gear one may want to use might not be "first gear".. So, the article cannot be used with clarity.
As to golf clubs, a player might have three woods, but only one "three wood", Golf clubs are identified with cardinal numbers. The article is needed to identify a club in question. Generally a player will only have one club identified by a number and (historically) material. "Give me the four iron", or, "give me the three wood" are clear as to what is wanted.
According to this English Club article, one of the things that we don't use an article for is routine places- home, work.
We could usefully extend this to cover standard positions, for example in various sports and in music: half back in soccer and football, fly half in Rugby, silly mid-on in cricket, small forward in basketball.
As positions, these never take an article:
He plays at centre half
but when we use exactly the same term to define the role of a person, we are in the same position as we are with captain: the article is optional
He is centre half
He is the centre half
He is a centre half
When we look for positions that contain numbers, we get examples such as:
Third base - baseball
Third position - ballet
Second oboe - orchestra
None of these take an article when used as a position. Note that neutral in a gearbox also does not take an article.