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)

Did you try brew link --overwrite python?

It's not currently possible, but I believe it is possible in principle if the various Python formulae were fixed up to support this.

Dealing with the various executables (python3, pip3, etc) is fairly straightforward, but there are some conflicts that will require a bit more work:

❯ brew link --force --dry-run --overwrite python@3.10
Would remove:
/usr/local/bin/2to3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/2to3
/usr/local/bin/idle3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/idle3
/usr/local/bin/pip3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/pip3
/usr/local/bin/pydoc3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/pydoc3
/usr/local/bin/python3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/python3
/usr/local/bin/python3-config -> /usr/local/Cellar/python@3.9/3.9.9/bin/python3-config
/usr/local/bin/wheel3 -> /usr/local/Cellar/python@3.9/3.9.9/bin/wheel3
/usr/local/share/man/man1/python3.1 -> /usr/local/Cellar/python@3.9/3.9.9/share/man/man1/python3.1
/usr/local/lib/pkgconfig/python3-embed.pc -> /usr/local/Cellar/python@3.9/3.9.9/lib/pkgconfig/python3-embed.pc
/usr/local/lib/pkgconfig/python3.pc -> /usr/local/Cellar/python@3.9/3.9.9/lib/pkgconfig/python3.pc
/usr/local/Frameworks/Python.framework/Headers -> /usr/local/Cellar/python@3.9/3.9.9/Frameworks/Python.framework/Headers
/usr/local/Frameworks/Python.framework/Python -> /usr/local/Cellar/python@3.9/3.9.9/Frameworks/Python.framework/Python
/usr/local/Frameworks/Python.framework/Resources -> /usr/local/Cellar/python@3.9/3.9.9/Frameworks/Python.framework/Resources
/usr/local/Frameworks/Python.framework/Versions/Current -> /usr/local/Cellar/python@3.9/3.9.9/Frameworks/Python.framework/Versions/Current

This might require special handling of the contents of *.framework directories in brew.

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.

The error means those files already exist in /usr/local/bin

If you 'ls -lF' on that dir, are they sym links that point to a path with the word 'cellar' in it?

If so, then brew already put them in place for you. You're probably missing /usr/local/bin from your PATH

If not, then something else put them there earlier. On a fresh mac, that directory is empty. Brew is a better way to manage those files, so if it was my machine I would remove those files from /usr/local/bin, then you can run brew link. I'd probably keep them in another spot temporarily in case I later discovered a need for them.

Also, run brew doctor when you have a chance. It will tell you about any other problems like this.

Brew wants to own /usr/local/bin. I would let it, it's a great tool.

I presume you mean that you want the command python to start the Python3 interpreter, and pip to start pip3.

The clue is in the message:

Unversioned symlinks python, python-config, pip etc. pointing to python3, python3-config, pip3 etc., respectively, have been installed into /usr/local/opt/python/libexec/bin

That means... "if you want the command python to start python3 and pip to start pip3, you need to put /usr/local/opt/python/libexec/bin at the start of your PATH."

So, in your login script ($HOME/.profile or similar), you need to put:

export PATH=/usr/local/opt/python/libexec/bin:$PATH

Then log out and log back in for it to take effect.

As an aside, brew link python3 only means... "create a symbolic link in /usr/local/bin/python3 that points to /usr/local/Cellar/python/3.7.0/python3".

You can see that link, and where it points to, with:

ls -l /usr/local/bin/python3