[D] What comes first, math, or algorithm in research?

I'm learning meths behind diffusion right now (DDPM, Score-based, and other approaches). I'm wondering how exactly did researchers come up with the idea?

Does inventing new approaches go something like this?
1. We want to make better image generator.
2. Oh, the data will never be enough...
3. Let's multiply data - by adding some noise corruption
4. This this works well, what if we make a denoising network?
5. What if we make network that makes an image from …

algorithm data ddpm diffusion generator image image generator machinelearning math research researchers something will