brew install python or brew info python output mentions:

Unversioned symlinks python, python-config, pip etc. pointing to python3, python3-config, pip3 etc., respectively, have been installed into /opt/homebrew/opt/[email protected]/libexec/bin

So running, as root (for Bash (executable bash), use the ~/.bash_profile file):

echo 'export PATH=/opt/homebrew/opt/[email protected]/libexec/bin:$PATH' >> ~/.zprofile
source ~/.zprofile

gets you those symlinks created by Homebrew, python=python3, pip=pip3, etc.

python --version

Output:

Python 3.9.10

And:

pip --version

Output:

pip 21.3.1 from /opt/homebrew/lib/python3.9/site-packages/pip (python 3.9)

Note: This applies to any versioned Python installation with Homebrew

After installing Python 3.11, you are given the location of the bin folder with Python 3.11:

==> Caveats
Python has been installed as
  /usr/local/bin/python3.11

Unversioned and major-versioned symlinks `python`, `python3`, `python-config`, `python3-config`, `pip`, `pip3`, etc. pointing to
`python3.11`, `python3.11-config`, `pip3.11` etc., respectively, have been installed into
  /usr/local/opt/[email protected]/libexec/bin

You can add this folder to be first in your PATH and python or python3 will pick up Python 3.11 (since those are in the folder):

So: add the following line to ~/.zshrc:

export PATH="$(brew --prefix)/opt/[email protected]/libexec/bin:$PATH"

brew --prefix ensures that the correct path is used, regardless of where your brew install resides.

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Note: You can find outdated answers on the web where it is mentioned python3 as the formula name for installing Python version 3.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

python, python3, and python@3 are all currently aliased to python@3.10. We are in the middle of a migration from 3.10 to 3.11.

Did you run brew install python@3.11? Or one of the aliases above?

After installing python3 with brew install python3 I was getting the error:

Error: An unexpected error occurred during the `brew link` step
The formula built, but is not symlinked into /usr/local
Permission denied @ dir_s_mkdir - /usr/local/Frameworks
Error: Permission denied @ dir_s_mkdir - /usr/local/Frameworks

After typing brew link python3 the error was:

Linking /usr/local/Cellar/python/3.6.4_3... Error: Permission denied @ dir_s_mkdir - /usr/local/Frameworks

To solve the problem:

sudo mkdir -p /usr/local/Frameworks
sudo chown -R $(whoami) /usr/local/*
brew link python3

After this, I could open python3 by typing python3 👍

(From https://github.com/Homebrew/homebrew-core/issues/20985)

I have done this sort of thing in two ways - however both do not coexist with homebrew although I suspect there should be a way similar to the macports one.

General package manager

The first is using Macports as the package manager. There are separate ports for several python versions 2.x and 3.x currently 2.4, 2.5, 2.6, 2.7 and 3.1, 3.2, 3.3 3.4 and 3.5. For example to install 3.4 do sudo port install python34 There is also a python_select port that running it allows you to choose which version of python is run by /opt/bin/python. e,g, to select macports python 2.7 port select --set python python27

Each package manager needs to run as the only package manager otherwise it will be confused with unexpected different versions of libraries and executables on its paths.

From comments the similar homebrew way is given in https://stackoverflow.com/a/4158763/151019 Short answer seems to be brew switch the equivalent to port_select

Python version managers

The other way is install direct from python.org This will install a python into /usr/local/bin I think this installs a python and python3 executable. The issue with Homebrew is that it assumes that things in /usr/local are installed by it and so brew doctor will report these python.org pythons and libraries as issues and as this question shows also the frameworks in /Library will be problematic.

EDIT 8/2018 *** I would now suggest using conda either miniconda or anaconda. This is another package manager but based and mainly meant for python, it also gives the equivalent of python's virtual environments and pyenv (as per other answer) and so each project can be in a different python and different libraries (Macports only switches python version for all projects)