I would say it's mostly a matter of scale, complexity, and motive. A farmer farms a farm and a gardener gardens in a garden. Farms are usually bigger than gardens, and farming often connotes a bit of animal husbandry, whereas a gardener is likely to feel that the chickens are not part of the garden the way they would be part of a farm.
These days most farmers want to make money and most gardeners do not, but that's far from universal. Plenty of "hobby farmers" and "gentleman farmers" out there still, and lots of gardeners trying to squeeze a marketable surplus out of their gardens, too.
This is a really good question, and as such seems to have several correct answers. I know many "farmers" who also garden. These fellows tell me the real difference is two parts; number of acres crops are being grown for sale on and the growing methods used.
I do not agree with their assessment of the methods used being a part of the description difference between Farm and Garden. I do agree with them on the quantity of acres commercial crops are grown on being the deciding factor. I know farmers in California that grow every thing from onions to artichokes to wheat, and I know some gardeners in that same state that grow the same crops, but almost all of their produce goes into their own freezers or is canned for the winter. Here in Arkansas, there are just a few crops being Farmed; Corn, Soybeans, Milo (grain sorghum), Rice, Oats, Wheat. Most of the farmers I know also have a "garden" which they differentiate from the farm by the statement "the garden feeds us, the farm feeds everyone else".
I would encourage people to think much more broadly about this. I discuss this issue with people all the time. I call myself a farmer to most people. Yet, I only work 10 acres part time. Here is some of my thinking on this.
For those who think scale is the determining factor, I point to herb farms that make $100,000s annually all confined within a greenhouse in the middle of an urban center.
For those who think crop is the determining factor, I point to fish farms which are lucrative farming.
For those who think cultivating the land is the determining factor, I point to commercial mushroom operations in concrete bunkers and poop racks which seems more about being an HVAC technician than anything else. They are farms.
For those who think economic return is the determining factor, I point to many row crop farms that end in the red without a net income for the farmer, but they are still considered farms.
For those who think the amount of time spent on the operation is the determining factor, I point to truffle farmers who harvest that strange fungus while on a pleasant walk with a dog or pig.
For those who think primary occupation of the farmer that is the determining factor, I point to many farmers who have town jobs and work their farming activities around that.
I could go on, but I think my point is made. All of these examples are farms. So, I would simply let the person self-designate what and who they are regardless of their practice, crop or growing medium.
Now, if we're talking about a government classification, there is some definition of that in each state and each federal government. However, in Missouri, my 10-acre place that I only work part time was assigned a "farm number" last year.
If someone prefers to call themselves a farmer or a gardener and they plant an acre of sweet corn and sell it out of their trunk on the side of the road, I say "right on, brutha!"
Dan, I think you clearly expressed the significance of the economic factor in a distinction between farm and garden . That some farms fail highlights the point.
I also think it is a largely arbitrary distinction and that it can only be made in cases where there is a "gardening" form.
I have never heard of anyone gardening fish
Oh gosh, there are plenty of basement aquaponics people that are, in effect, gardening fish. Which is why I think it's best to let people self designate whether they are a farmer or a gardener or some other designation.
I understand that small scale fish farming goes on all over. That really is the point. Lots of people doing it and calling it fish-farming, whatever their scale. Essentially, English does not make the distinction.
in a language with deeper history in the practice there may be a distinction. I don't know
Most of what I've learned from gardening comes from Monty Don, and walking in gardens.
I've worked on farms in Martha's Vineyard, Ohio and a ranch in Texas.
It seems a garden is a lot of things to a lot of people, and takes many forms, but the intention is always to find serenity.
A French formal garden, one such as Ville D'Este, was serenity through orderly control of nature. They and the Italian gardens might have food plants, but they were usually exotic and for ornamentation, and to show off their wealth they would never sell the produce. English Romantic-period gardens were peace with nature, a return to animism, the 'noble savage,' and began landscape architecture.
Farms have taken many forms and many scales, but the intention is always to survive or thrive.
Permaculture seems to pose the suggestion - when in the forest, why not do as the other creatures do?
Provided you set up a system that makes abundance, if you can pull that off and still feel serenity, then you've got a garden. When it's so productive that you're selling stuff you can't eat to others, it's a farm. The definitions seem unnecessary to me, they might become effectively arcane, if you can survive, thrive, and find serenity at the same time. You might call that place an "Eden."
Unless that stinks of Christianity to you. Then I suggest we call it,
In my opinion it generally comes down to a few factors.
If you're trying to make money, it's usually called farming.
If you have livestock of any sort, it's usually called farming even if it is only for personal use.
If you're just growing vegetables (or ornamental plants) for personal use it's generally called a garden.
A loose definition might be that farming is a profession and gardening is a hobby.
So in a very general sense if you're doing it just for yourself you're a gardener. Unless you have animals, then you get bumped up into the farmer title as you rarely hear anyone say that they 'garden' animals. If you're doing it attempting to make money, its a farm. Though some small farms (a few acres or less) are sometimes called a market garden.
You could take 2 identical small gardens and the person doing it for just their family would likely say they are a gardener. If the other person was doing it to sell what they produced they would possibly call it a market garden, or refer to themselves as an urban farmer if they were doing it in town.
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