As far as I see (grep '#include .*\.c'
), they only do this in doublefann.c, fixedfann.c, and floatfann.c, and each time include the reason:
/* Easy way to allow for build of multiple binaries */
This exact use of the preprocessor for simple copy-pasting is indeed the only valid use of including implementation (*.c) files, and relatively rare. (If you want to include some code for another reason, just give it a different name, like *.h or *.inc.) An alternative is to specify configuration in macros given to the compiler (e.g. -DFANN_DOUBLE, -DFANN_FIXED, or -DFANN_FLOAT), but they didn't use this method. (Each approach has drawbacks, so I'm not saying they're necessarily wrong, I'd have to look at that project in depth to determine that.)
They provide makefiles and MSVS projects which should already not link doublefann.o (from doublefann.c) with either fann.o (from fann.c) or fixedfann.o (from fixedfann.c) and so on, and either their files are screwed up or something similar has gone wrong.
Did you try to create a project from scratch (or use your existing project) and add all the files to it? If you did, what is happening is each implementation file is being compiled independently and the resulting object files contain conflicting definitions. This is the standard way to deal with implementation files and many tools assume it. The only possible solution is to fix the project settings to not link these together. (Okay, you could drastically change their source too, but that's not really a solution.)
While you're at it, if you continue without using their project settings, you can likely skip compiling fann.c, et. al. and possibly just removing those from the project is enough – then they won't be compiled and linked. You'll want to choose exactly one of double-/fixed-/floatfann to use, otherwise you'll get the same link errors. (I haven't looked at their instructions, but would not be surprised to see this summary explained a bit more in-depth there.)