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I am a high school senior, and I stumbled upon this thread. Although I have not read every page, I would like to comment and share my thoughts on Philosophy versus Mathematics. I believe they both have practical and abstract implications; I believe their focuses are not the same and that their roots are very different, so the schools of thought should not be competing. One is not "better" than the other. Not all knowledge is as strong as its counterpart, either. It is objective. I think the nature of this debate entangles something that philosophy addresses: that there are different ways to know things. Emotion, reason, perception, and language are taught under IB Theory of Knowledge, but this does vary. Our opinions on which is more applicable/valuable/hardest/most essential are influenced by the factors mentioned above. Do we have an emotional connection with one of the subjects; does studying a subject or being successful in that stud sway our response; are there language barriers that influence our understanding of each subject; how do practical and abstract theories relate; etc?
The nature of mathematics is based on intangible concepts and rules we cannot prove. It is imperfect by design. We assume things in mathematics to be true only because they have practical value in society. For example, we do not definitively know that 0 exists. Or that numbers exist. We think they are real or at least there because we use what they represent. But what would nothing be to the store keeper without 0? Therefore, it is useful so we except that 0 exists even thought we can have perfect knowledge of 0. As we know, there is no proof for an axiom, and every area of knowing (like math, english, the arts) have axioms, or assumptions, used to practice and understand that school of thought. The core of mathematics- or any subject- is based in axioms, not logic. This is different from philosophy (not necessarily better or worse, just different by design).
Philosophy is how do we know what we know, and how do we know that our knowledge is valid. Mathematics examines only one spectrum of that question: empirical knowledge. Because mathematics is a branch under philosophy, we can answer that question IN TERMS OF mathematics, but that is only one perspective of what we know as reality.
Philosophy is practical because it questions the world and what we know about it. It looks at the big picture, and all the shades in between. Just my two cents.
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