A very small fraction of experimental science is about just doing something and seeing what happens without a strong prior. You are expected to do your homework first. As a scientist, your role is typically to read and understand all the relevant prior work in the area first, then using this theory to derive a hypothesis for the outcome of your experiment. If you are lucky, the way the experiment unfolds might be outside your understanding, given that your understanding of the prior theory was correct, you have made a new scientific discovery which will be used to create new theory.
The fragment of uncertainty in this is typically very small and the process is very far from guessing your way through, hoping to find something new (whatever you found using such a method will most likely be either already known or incorrect).
Yes, I agree with you. My comment looks dull with your explanation. Of course, I also don't advocate doing something without any hypothesis. Though, I think hands-on practice on a subject can help you learn things better. For example, I find it very useful for linear algebra (maybe this is not applicable to other subjects). I can analyze my flawed intuitions.
When I first heard about the Monty Hall problem, I didn't understand it and tried it myself. It was way easier for me to understand the flawed intuition by analyzing each line as opposed to, say, for example, the explanation of Judea Pearl (which is also good).
What I wanted to emphasize is that it is not bad guessing your way through a problem you don't understand. But yeah, of course, you should have some knowledge.
I wish I could say I have been doing science already, not yet (hopefully), but I was referring to (say) programming problems like (for example) one recently where I could see that it boiled down to asking whether there existed a linear combination of a input set of vectors with a single second vector - I was able to spot the connection, I'm not convinced I would've been able to get it without the mathematics under my belt to spot the pattern.
Regardless, I plan to be learning until the day I drop, science or not.
Mathematics is the exploration of the a priori. Historically some axioms have been deemed more real than others. I think Gauss rejected non-eulidean geometry for instance. But this point of view has changed. With abstract algebra, and I believe also computer science, modern mathematics is about exploring connections between structures such as they emerge from stipulated axioms and rules of inference. It is science in the sense that ideas and hypotheses can be tested experimentally. But a proof requires more than non-falsification. Then there is the complication of potentially irreducible computation problems, where essentially a kind of mining of the computational space is the only way forward. This is the new kind of science Stephen Wolfram speaks of.
The fragment of uncertainty in this is typically very small and the process is very far from guessing your way through, hoping to find something new (whatever you found using such a method will most likely be either already known or incorrect).