From The Aquarium Wiki
The Fish Bowl
This article is designed to dispel the myth that a basic glass bowl is a good place to keep a fish. It is not.
History of the Bowl
The fish bowl origin starts first with the Romans and then in 1300AD in China. Back then the Chinese were just developing the Goldfish we know today.
They would keep ponds full of Goldfish in their houses. They would have large ceramic bowls with pretty fish patterns inside and out. When they had visitors or for special occasions they would take the Goldfish out of the pond, and put them in these bowls. When the event was over, those fish went back to their nice big pond.
Over the decades, especially popular in the 1800s, these bowls turned into glass ones and having a little fish in your house sat on a table became common decorations.
So why can't I have a fish in a bowl?
This is the 21st century. After years and years of analysing and studying we know what happens when a fish is in a container of water. We know how nitrifying bacteria behaves. We know why the water goes cloudy and smells if the water is not maintained. And we have invented nice big fish tanks, filters and pond filters to improve these animal's lives. We know about the Nitrogen Cycle.
It is with this knowledge that we know that maintaining the water quality fish need to thrive is near impossible in a small container without a proper mature filter.
Typical fish sold for bowls
The most common fish housed in a bowl is the Goldfish. The traditional Goldfish bowl. However we know that these fish are high waste producers, they grow very large and should live a very long time. Bowl kept Goldfish often end up stunted by poor water quality (not by tank size) and end up dying long before they're even sexually mature.
The second most common fish housed in a bowl is the Betta splendens. This is backed up with the myth they live in puddles, which is false. They originate from vast, warm, densely planted rice paddies and swampy water. A glass bowl cannot be heated easily and in cold water a poor Betta will be lethargic and eventually die. They do need good water quality in order for those long fancy fins to not develop bacterial infections too, which means they need that mature filter.
There is one exception to the no-fish-in-a-bowl rule. Those that follow the el natural/walstad method. This involves using earth as substrate and very dense planting. However, any living organisms in these set ups are best being low waste producers as it is a very delicate balance. Goldfish do not fall into this category and eat live plants anyway! Often Walstad tanks still have mechanical filtration and are not completely without a equipment.
The moral of this story
A fish bowl is not a suitable home for fish. It is too small, too unstable, and will result in fish that are sickly, lethargic and eventually die.