BUG: spearman correlation doesn't work on non-numeric data

[Desiment]

Pandas version checks

• I have checked that the issue still exists on the latest versions of the docs on main here

Location of the documentation

https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.corr.html#pandas.DataFrame.corr

Documentation problem

Since rank correlation coefficients make sense also for ordinal data, documentation for option numeric_true is a little confusing.

Currently, it's saying the following:

numeric_only : bool, default False
    Include only `float`, `int` or `boolean` data.

When ordinal data (categorical dtype with defined order) presented in dataframe, it seems natural to expect that rank correlation still be computed when using numeric_only = False. For example, something like this should work:

import pandas as pd
df = pd.DataFrame({'workweek' : [4, 5, 6, 4], 'income' : ['low', 'middle', 'high', 'low']})
df['income'] = df['income'].astype('category').cat.set_categories(['low', 'middle', 'high'], ordered=True)
df.corr(method='spearman')

Yet it throws ValueError, since it cannot convert string low to float. Moreover, using numerical categories with the specific order results in incorrect behavior. For example,

import pandas as pd
df = pd.DataFrame({'a' : [1, 2, 3, 4], 'b' : [4, 3, 2, 1]})
df['b'] = df['b'].astype('category').cat.set_categories([4, 3, 2, 1], ordered=True)
df.corr(method='spearman')

returns that Spearman's correlation between a and b is equal to -1, while real value is equal to 1.

Suggested fix for documentation

I believe, that following additional sentence makes things a bit less confusing.

numeric_only : bool, default False
     Include only `float`, `int` or `boolean` data. If value is False, method will try cast non-numerical 
     columns to float (note that ordinal data, if possible, will be converted ignoring specified order)

[rhshadrach]

Thanks for the report. It appears to me that these (spearman on non-numeric data and categorical order not being respected) are bugs. I think the latter is covered in #33245, so we can have this issue be above the former.

[micheleuap]

Supporting (ordered) categorical variables should be relatively simple -
i.e. just a matter of replacing them with their codes/orders before converting them into a numpy array

Let me know if I should go ahead and make a PR to this effect and add some tests

[rhshadrach]

@micheleuap - PRs to fix are welcome.