Halving the ingredients in a recipe is arithmetic, and it works. Halving the recipe is not, because several things that govern the outcome do not scale with quantity.
Surface area does not halve with volume
Cutting a quantity in half does not halve the surface exposed to heat and air, particularly if the same pan is used.
A halved braise in the original pot presents proportionally more surface, so it evaporates faster and reduces further in the same time.
The result is a dish that is drier and more concentrated than intended, which is often mistaken for a seasoning error.
Cooking times shorten unpredictably
Heat penetrates food from the outside, and the time required depends on the thickness of the piece rather than on the total mass in the pan.
Halving the number of pieces therefore changes cooking time very little, while halving the size of each piece changes it considerably.
Recipes state a single time, so a scaled-down version needs the time re-derived from what the food is doing rather than adjusted proportionally.
Small quantities are harder to measure
Halving a teaspoon of a potent ingredient introduces proportionally larger error than halving a cup of flour, and leavening agents are particularly sensitive.
Some quantities do not divide cleanly at all, most obviously eggs, and the usual workarounds change the ratio of white to yolk.
Because egg contributes structure, moisture and fat in different proportions from each part, that substitution alters more than the volume.
Pan geometry has to be reconsidered
Baking a halved batter in the original tin gives a shallower item that bakes faster and browns differently, usually drying out before the centre sets.
Matching the depth requires a tin of roughly half the area, which is rarely the same tin cut mentally in half.
On the hob, the equivalent adjustment is a smaller pan, so that the liquid depth and the food's contact with the base stay similar.
Some recipes resist scaling entirely
Preparations that depend on a minimum volume to behave correctly, including deep frying and certain candy stages, do not scale down usefully.
Anything relying on a long slow reduction also suffers, since the smaller batch reaches the endpoint before the flavour development the time was providing.
Where a dish resists halving, cooking the full quantity and storing the remainder is usually the better answer than fighting the arithmetic.