Title: Batched mapping of existing string column casts boolean to string
State: closed

### Describe the bug

Let the dataset contain a column named 'a', which is of the string type.
If 'a' is converted to a boolean using batched mapping, the mapper automatically casts the boolean to a string (e.g., True -> 'true').
It only happens when the original column and the mapped column name are identical.

Thank you!

### Steps to reproduce the bug

```python
from datasets import Dataset

dset = Dataset.from_dict({'a': ['11', '22']})
dset = dset.map(lambda x: {'a': [True for _ in x['a']]}, batched=True)
print(dset['a'])
```
```
> ['true', 'true']
```

### Expected behavior

[True, True]

### Environment info

- `datasets` version: 2.18.0
- Platform: Linux-5.4.0-148-generic-x86_64-with-glibc2.31
- Python version: 3.10.13
- `huggingface_hub` version: 0.21.4
- PyArrow version: 15.0.2
- Pandas version: 2.2.1
- `fsspec` version: 2023.12.2

Comments:
This seems to be hardcoded behavior in table.py `array_cast`.
```python
if (
            not allow_number_to_str
            and pa.types.is_string(pa_type)
            and (pa.types.is_floating(array.type) or pa.types.is_integer(array.type))
        ):
            raise TypeError(
                f"Couldn't cast array of type {array.type} to {pa_type} since allow_number_to_str is set to {allow_number_to_str}"
            )
        if pa.types.is_null(pa_type) and not pa.types.is_null(array.type):
            raise TypeError(f"Couldn't cast array of type {array.type} to {pa_type}")
        return array.cast(pa_type)
```
where floats and integers are not cast to string but booleans are.
Maybe this should be extended to booleans?

Thanks for reporting! @Modexus Do you want to open a PR with the suggested fix?

I'll gladly create a PR but not sure what the behavior should be.

Should a value returned from map be cast to the current feature?
At the moment this seems very inconsistent since `datetime `is also cast (this would only fix `boolean`) but nested structures are not.

```python
dset = Dataset.from_dict({"a": ["Hello world!"]})
dset = dset.map(lambda x: {"a": date(2021, 1, 1)})
# dset[0]["a"] == '2021-01-01'
```
```python
dset = Dataset.from_dict({"a": ["Hello world!"]})
dset = dset.map(lambda x: {"a": [True]})
# dset[0]["a"] == [True]
```

Is there are reason to cast the value if the user doesn't specify it explicitly?
Seems tricky that some things are cast and some are not.

Indeed, it also makes sense to raise a `TypeError` for temporal and decimal types.

> Is there are reason to cast the value if the user doesn't specify it explicitly?

This is how PyArrow's built-in `cast` behaves - it allows casting from primitive types to strings. Hence, we need `allow_number_to_str` to disallow such casts (e.g., in the [scenario](https://github.com/huggingface/datasets/blob/a3bc89d8bfd47c2a175c3ce16d92b7307cdeafd6/src/datasets/arrow_writer.py#L208) when we are "trying a type" to preserve the original type if there is a column in the output dataset with the same name as in the input one).

PS: In the PR, we can introduce `allow_numeric_to_str` (for floats, integers, decimals, booleans) and `allow_temporal_to_str` (for dates, timestamps, ...) and deprecate `allow_number_to_str` to make it clear what each parameter does.

Would just `allow_primitive_to_str` work?
This should include all `numeric`, `boolean `and `temporal`formats.

Note that at least in the [ C++ implementation](https://arrow.apache.org/docs/cpp/api/utilities.html#_CPPv410is_numericRK8DataType) `numeric `seems to exclude `boolean`.
[](https://arrow.apache.org/docs/cpp/api/utilities.html#_CPPv410is_numericRK8DataType)

Indeed, `allow_primitive_to_str` sounds better.

PS: PyArrow's `pa.types.is_primitive` returns `False` for decimal types, but I think is okay for us to treat decimals as primitive types (or we can have `allow_decimal_to_str` to be fully consistent with PyArrow)

Fixed by:
- #6811