"science alleges that it’s intellectual carelessness to believe things that can’t be proven"

Science is not about "proof". (Mathematics is but that is unrelated to this discussion.) It is simply not possible to use the scientific method to prove that anything is true (although it can demostrate that some hypotheses are false).

Science is about accumulating evidence that certain hypotheses are true. The more such evidence that is gathered, the higher the probability that the hypotheses in question are true. However, science (when practiced correctly) is always open to new evidence that the hypotheses are false. Scientific revolutions occur when hypotheses previously believed to be true are shown to be false because of new evidence.

No scientific hypothesis is ever 100% established as true. However, some hypotheses are so close to 100% that we use them as if we are certain they are true. The question to be asked about any hypothesis or belief is this: based on the currently available evidence what is the probability it is true?