Do you think math and philosophy are very similar or do you think they are two different things? If you think they are similar, why? And if you think they are different, why?
Let's reframe the question: what do math and philosophy have in common?
Definitions
What do philosophy & math mean? Such
concepts are names for the benefit of university deans. Otherwise, they're not really separate. And meanings change — only nowadays do we call Newton a physicist; back in the day, he was
a natural philosopher
What's philosophy? Investigating what underlies things, one meta level down. You can also think of it as "thinking about why we're doing what we're doing"
What's math? Fairly recently, it was the investigation of numbers and three-dimensional shapes. Nowadays, it's the investigation of patterns in general
We'd expect math to be similar to philosophy, when we model something one meta-level down, mathematically
Examples
Math investigators often work with fuzzy concepts, that they define more as needed. Defining things can be philosophical
For example, "function" was a commonsense object, no precise definition, until investigators felt the
pain of not having one. So they built frameworks like set theory one meta-level down
An investigator can keep any crazy imprecise definition in mind, when it's cognitively more effective to do so. But the more precise definition can strip down the concept into a form that can be used in a zillion places. And give you more confidence in your ideas
Major benefit of philosophy: see hidden connections between things. If you make something more abstract, it can apply in all sorts of situations. If you build a
general framework for (say) sadomasochism, it can be used to describe any dominance hierarchy — which permeate most current societies
Warning
People philosophize all the time, for things big and small. Each has sweet spots where they're more useful. They're all fallible and vary in quality — like anything else. Even dominant ones. Some were made for surprisingly limited purposes. Some others started as mad experiments that broke out of the lab and won hype to lure more research money
Random thought
Speaking of calculus — isn't it weird how we downplay the intelligence of people under 25? Many countries teach calculus (the hard way) much younger, like 16-18. (Not to mention
teaching math at 5)