Mathematics is precisely not a science, because we made it all up; that's why it's knowable. Ask Bertrand Russell.

Science is a matter of observing, and then trying to explain what was observed. Mathematics begins with axioms, and proceeds to explore the implications of those axioms. Science begins with observation and proceeds to explore the implications of the data. The difference isn't just whether you invent your raw material or go out and find it; imagine starting with a page full of math based on unknown axioms, and having to deduce what the axioms are from what you see there. Now imagine that most of what's on the page is irrelevant, and half of what you really need is on another page, which might be hidden somewhere in France or Outer Mongolia. Naturally, you won't even know enough to realize that anything's missing until you've already spent ten years beating your head against what you've got. That's science.

Mathematics is, of course, a very useful tool for explaining observations. That's why we invented it.

Your point (as I see it) is like saying that grep alone is "better" than grepping text files, because grep is deterministic and fully understood in advance, while text files could contain anything. But is it even meaningful to compare the two? A tool that's not used is of interest only to people with freakishly long attention spans. The only way grep can tell you something you don't already know is if you push some messy, unpredictable crap through it. Crap + analysis == science.

I think you're conflating two different kids of truth. Absolute certainty is only possible regarding things that we ourselves have created (analytic propositions, if you will); truth about the world outside ourselves (synthetic propositions, if we must) is more slippery and (to a lot of people) more interesting. Physics is "inaccurate and limited" because that's what "science" means (language, e.g. the word "science", is analytic too).

Another way to put it: The parts of mathematics which aren't known, are necessarily implied by the parts that are known; this is not true of a herd of buffalo.

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