How can I install a previous version of Python 3 in macOS using homebrew?

Short Answer

To make a clean install of Python 3.6.5 use:

brew unlink python # ONLY if you have installed (with brew) another version of python 3
brew install --ignore-dependencies https://raw.githubusercontent.com/Homebrew/homebrew-core/f2a764ef944b1080be64bd88dca9a1d80130c558/Formula/python.rb

If you prefer to recover a previously installed version, then:

brew info python           # To see what you have previously installed
brew switch python 3.x.x_x # Ex. 3.6.5_1

Long Answer

There are two formulas for installing Python with Homebrew: python@2 and python.
The first is for Python 2 and the second for Python 3.

Note: You can find outdated answers on the web where it is mentioned python3 as the formula name for installing Python version 3. Now it's just python!

By default, with these formulas you can install the latest version of the corresponding major version of Python. So, you cannot directly install a minor version like 3.6.

Solution

With brew, you can install a package using the address of the formula, for example in a git repository.

brew install https://the/address/to/the/formula/FORMULA_NAME.rb

Or specifically for Python 3

brew install https://raw.githubusercontent.com/Homebrew/homebrew-core/COMMIT_IDENTIFIER/Formula/python.rb

The address you must specify is the address to the last commit of the formula (python.rb) for the desired version. You can find the commint identifier by looking at the history for homebrew-core/Formula/python.rb

https://github.com/Homebrew/homebrew-core/commits/master/Formula/python.rb

Python > 3.6.5

In the link above you will not find a formula for a version of Python above 3.6.5. After the maintainers of that (official) repository released Python 3.7, they only submit updates to the recipe of Python 3.7.

As explained above, with homebrew you have only Python 2 (python@2) and Python 3 (python), there is no explicit formula for Python 3.6.

Although those minor updates are mostly irrelevant in most cases and for most users, I will search if someone has done an explicit formula for 3.6.

As an update, when doing

brew unlink python # If you have installed (with brew) another version of python
brew install https://raw.githubusercontent.com/Homebrew/homebrew-core/f2a764ef944b1080be64bd88dca9a1d80130c558/Formula/python.rb

You may encounter

Error: python contains a recursive dependency on itself:
  python depends on sphinx-doc
  sphinx-doc depends on python

To bypass it, add the --ignore-dependencies argument to brew install.

brew unlink python # If you have installed (with brew) another version of python
brew install --ignore-dependencies https://raw.githubusercontent.com/Homebrew/homebrew-core/f2a764ef944b1080be64bd88dca9a1d80130c558/Formula/python.rb

I have tried everything but could not make it work. Finally I have used pyenv and it worked directly like a charm.

So having homebrew installed, juste do:

brew install pyenv
pyenv install 3.6.5

to manage virtualenvs:

brew install pyenv-virtualenv
pyenv virtualenv 3.6.5 env_name

See pyenv and pyenv-virtualenv for more info.

EDIT (2020/03/19)

I have found using the pyenv-installer easier than homebrew to install pyenv and pyenv-virtualenv direclty:

curl https://pyenv.run | bash

To manage python version, either globally:

pyenv global 3.6.5

or locally in a given directory:

pyenv local 3.6.5