Class 'app\routing' not found - php

Question

I'm currently building an application that will require a few different name spaces, however most of the application I want to put under the app namespace.
In my index.php file I declared to use the app namespace, however PHP still insists my class cannot be found like so...
<?php
define('PATH',strtok ( $_SERVER['REQUEST_URI'] , '?' ));
define("ROOT",$_SERVER['DOCUMENT_ROOT'].'/',true);
require 'vendor/autoload.php';
require 'app/class_loader.php';
use Carbon\Carbon;
use app\routing;
$routing = new routing;
$routing->router();
and inside my routing class I declare it as part of the app namespace like this
<?php
namespace app;
class routing
{
However I keep getting Fatal error: Class 'app\routing' not found in /var/www/my_app/index.php on line 15
This is my autoload file
<?php
function autoload_class_multiple_directory($class_name)
{
# List all the class directories in the array.
$array_paths = array(
'app/controllers',
'app/models',
'app/services',
);
foreach($array_paths as $path)
{
$file = $path.'/'.$class_name.'.php';
if(is_file($file))
{
include $file;
}
}
}
spl_autoload_register('autoload_class_multiple_directory');
I thought this is how namespaces worked? I want to ensure I make my application as easy to follow and code in from the get-go, could somebody please help point me in the right direction?
cheers!