Science, or natural science, is generally considered to be about the quantifiable and measurable empirical world. Maths and certainly logic are generally outside science. That is why it is generally referred to as Maths and Science, and logic is a part of philosophy departments.
Science is not philosophy.
You are correct that science needs to quantifiable or measurable, because that the mean of acquiring knowledge, and testing if that knowledge is true or false. And yes, science does use maths and logic, some with greater degrees than other fields.
Science is a methodology of acquiring knowledge.
That's how science differ from philosophy, the mean of testing any statement. The hypothesis is tested, and if the test succeed "quantifiably", then it is possible that the hypothesis will eventually become a scientific theory. But if the hypothesis failed in repeated tests or there are lack of evidences to support the hypothesis, then that hypothesis has been refuted, and should discarded.
No hypotheses and theories in science are immutable. Any one of them can discarded if the evidences don't support the them, perhaps because there are better alternative (explanations or predictions, or both) to replace existing theories or hypotheses.
Philosophy is different in that certain people will accept this or that philosophy, regardless if it is false or wrong, or if there are better alternative. Philosophy don't require methodology of empirically testing. Philosophy is based on logic alone.
That's how science differ from philosophy.
And you are wrong. Yes, science can use logic and maths, but it can also rely on logic and maths alone, without evidences. These fields or branches of science is called theoretical physics.
Theoretical physics don't rely on evidences that can be verified or tested. What can be tested in theoretical physics is the mathematical models or equations. Many areas in theoretical physics are untestable, but it is "provable" through mathematical equations.
Among scientists and mathematicians, science relies on "evidences" for verification, but mathematics relies on "proofs", and proof means mathematical equations or mathematical models, or logic. However, theoretical physics does rely more on proofs than evidences.
But in the real world, I would rather rely more on science than on mathematics alone, because that's the engineering and technological side of me.
So, for you to say that maths and logic are "outside of science", you are wrong when you are ignoring theoretical physics.
Also, I have less patience with philosophy and because it has very little real-world application and not often practical than science.
I am qualified civil engineer and computer scientist/programmer, so for me I preferred my science and maths to be applicable, practical, usable. Philosophy is useless to me.