How can I connect multiple microphones to my computer (more than 4) and process the outputs independently?
Depending on what kind of application is that and what kind of microphones (dynamic, condenser, home-made electret ones) solutions can be different. Are you going to process everything in real-time or to make your analysis offline?


If your microphones doesn't require Phantom power and can be plugged into mic-in, then you can buy few of cheapest USB sound cards, plug them to your laptop, run your linux, create virtual device with JACK and you can record. But do not expect to run in real time - most likely there will occur lot's of problems with synchronisation (had them on my own).  So it is good for not a real-time application. Here is some article about that: Using Multiple Devices with JACK and topic from Audacity guys.
If you want to use microphones that require a phantom power, then you should buy/borrow some 4+ channel USB/FireWire audio interface. You plug it into your computer and record whatever you want. No problems with latency, everything is synchronised. What program can you use? That's up to you - on linux I suggest Ardour on Windows I am using REAPER a lot. But for scientific applications just use the ASIO drivers and record tracks.