Linux Mint expressly discourages upgrades in general. I highly recommend you read the very good and thorough post written by Mint's chief developer about why it is generally a good idea to do a fresh install instead of an upgrade. Mint comes with a very good tool specifically designed to make this painless.
In any case, you should never jump releases like that. That's why you are removing so many packages. Some of these will be replaced by newer versions of themselves, others may no longer be used. Still others are held back because of various dependency problems. Certain packages you have installed need specific versions of, for example, gcc so it is held back to preserve those packages. So, even if you do want to upgrade you would be better off upgrading first to Mint 14 and from there to 15. Upgrading from 13 straight to 15 is likely to end badly.
This is the recommended method of upgrading to a newer Mint release (for a nice guide with screenshots, see source):
Use mintbackup to make a backup of your data. Strictly speaking this is not necessary if you know what you are doing and have your /home on a separate partition. It is always a good idea though.
Use mintbackup again to save the list of installed packages. This will just export a simple text list of the packages you have installed, make sure you save this where it will not be overwritten, preferably an external drive. Anything you have installed without using the package manager (from source or external installers) you will have to manually reinstall at the end.
Install a brand new shiny Mint 15 overwriting your old system. If you have a separate /home, make sure you do not format it, this one of the reasons it is a good idea to follow step 1.
Use mintbackup once more to restore your data from step 1.
Use mintbackup (last time, I promise) to reinstall your packages, just select the file you had exported your list to and let it do the rest.
dist-upgrade