I understand your point: you are grouping them by a shared methodology, not claiming they are identical. My question is whether that shared feature is the most important one for classifying them as the same kind of knowledge, given that their justification works differently.
Because that's how hierarchical taxonomies work.
They're classified as the same kind of knowledge according to toolset. They're also classified as different kinds of knowledge within that parent classification, according to the context in which the tools are used. Both classifications are true at the same time.
I wouldn't normally presume to use an analogy for didactic purposes, but I'm all out of other ideas.
According to our hierarchical taxonomy of animals, cats and dogs are the same kind of animal. Cats and dogs are both carnivores, having that fundamental characteristic in common.
But according to the same taxonomy, cats and dogs are very different kinds of animals. Even though they both belong to the class of mammals, and the subclass of placental mammals, and even the same order of carnivores within the subclass, they're still considered different animals, one belonging to the canid genus, the other belonging to the felid genus. And at the level of genus, we see the same principle at work: Dogs and bears are the same animal: Canids. But at the species level of the hierarchy, they're considered very different animals. Same on the cat side. Cats and Hyenas are both the same animal: Felids. But when we step down to the species level, they're very different animals.
Same thing here. According to my preferred hierarchical taxonomy, logical proofs and empirical tests are the same kind of method, at the higher, more general level of toolsets. Within the category of the scientific toolset, my taxonomy provides for two distinct subcategories, where we recognize that the two methods are different from each other in fundamental ways.
That's the beauty of hierarchical taxonomies. They allow us to organize things according to nested categories, from broader and more general to narrower and more specific.
Cars and planes are the same thing: Vehicles.
Cars and planes are also different things: automobiles and aeroplanes.
Cars and trucks are the same thing: automobiles.
Cars and trucks are also different things: cars and trucks.
Men and women are the same thing: Humans.
Men and women are also different things: Men and women.
Bats and birds are the same thing: Winged creatures.
Bats and birds are also two different things: Bats and birds.
Americans and Brazilians are the same thing: Pan-Americans.
Americans and Brazilians are also two different things: Americans and Brazilians.
Logical proofs and empirical tests are the same thing: The toolset I call "the scientific method".
Logical proofs and empirical tests are also two different things: Logical proofs and empirical tests.
I really hope you understand now. If you can't understand how hierarchical categories work, and how things can be members of several nested categories that preserve both their similarities and their differences, then I don't see any point in continuing this conversation.
At this point, we're nowhere near your original question, which was whether there's ways of knowing that don't use the tools of the scientific method. I'm not going to keep answering questions that drag us further and further away from that, if you're not going to understand my answers anyway.
