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[At the risk of answering contrary to my original question...]

"Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." - G.H.Hardy, A Mathematician's Apology (London 1941).

Mathematics not only raises question about itself about its completeness and consistency but attempts to diagnose the "paradoxes, circularity and relativity" through variety of apparatus. To view the subject in terms of hierarchy would be to miss the point. A logician goes one step further and asks that if the subject was based on beliefs, are there degrees of such beliefs and means to quantify them? Unlike other sciences mathematician must be wary of proof, disproof but if the hypothesis is unprovable. A possible reformulation would be if mathematics is based upon intuition, rather than beliefs, latter connoting a wish-washy epistemology of an universe where logic may break down. (Vladimir Voevodsky in his video: What if Current Foundations of Mathematics are Inconsistent? pointed out that airplanes would still not fall down from sky regardless).

Furthermore, the subject has also opened up many avenues (modal logic, information theory, uncertainty theory, neural networking, cellular automata, and of course metalogic)-that the umbrella term misses in the original question - helpedhelping shape our understanding and broaden our horizon.

[At the risk of answering contrary to my original question...]

"Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." - G.H.Hardy, A Mathematician's Apology (London 1941).

Mathematics not only raises question about itself about its completeness and consistency but attempts to diagnose the "paradoxes, circularity and relativity" through variety of apparatus. To view the subject in terms of hierarchy would be to miss the point. A logician goes one step further and asks that if the subject was based on beliefs, are there degrees of such beliefs and means to quantify them? Unlike other sciences mathematician must be wary of proof, disproof but if the hypothesis is unprovable. A possible reformulation would be if mathematics is based upon intuition, rather than beliefs, latter connoting a wish-washy epistemology of an universe where logic may break down. (Vladimir Voevodsky in his video: What if Current Foundations of Mathematics are Inconsistent? pointed out that airplanes would still not fall down from sky regardless).

Furthermore, the subject has also opened up many avenues (modal logic, information theory, uncertainty theory, neural networking, cellular automata, and of course metalogic)-that the umbrella term misses in the original question - helped shape our understanding and broaden our horizon.

[At the risk of answering contrary to my original question...]

"Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." - G.H.Hardy, A Mathematician's Apology (London 1941).

Mathematics not only raises question about itself about its completeness and consistency but attempts to diagnose the "paradoxes, circularity and relativity" through variety of apparatus. To view the subject in terms of hierarchy would be to miss the point. A logician goes one step further and asks that if the subject was based on beliefs, are there degrees of such beliefs and means to quantify them? Unlike other sciences mathematician must be wary of proof, disproof but if the hypothesis is unprovable. A possible reformulation would be if mathematics is based upon intuition, rather than beliefs, latter connoting a wish-washy epistemology of an universe where logic may break down. (Vladimir Voevodsky in his video: What if Current Foundations of Mathematics are Inconsistent? pointed out that airplanes would still not fall down from sky regardless).

Furthermore, the subject has also opened up many avenues (modal logic, information theory, uncertainty theory, neural networking, cellular automata, and of course metalogic)-that the umbrella term misses in the original question - helping shape our understanding and broaden our horizon.

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source | link

[At the risk of answering contrary to my original question...]

"Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game." - G.H.Hardy, A Mathematician's Apology (London 1941).

Mathematics not only raises question about itself about its completeness and consistency but attempts to diagnose the "paradoxes, circularity and relativity" through variety of apparatus. To view the subject in terms of hierarchy would be to miss the point. A logician goes one step further and asks that if the subject was based on beliefs, are there degrees of such beliefs and means to quantify them? Unlike other sciences mathematician must be wary of proof, disproof but if the hypothesis is unprovable. A possible reformulation would be if mathematics is based upon intuition, rather than beliefs, latter connoting a wish-washy epistemology of an universe where logic may break down. (Vladimir Voevodsky in his video: What if Current Foundations of Mathematics are Inconsistent? pointed out that airplanes would still not fall down from sky regardless).

Furthermore, the subject has also opened up many avenues (modal logic, information theory, uncertainty theory, neural networking, cellular automata, and of course metalogic)-that the umbrella term misses in the original question - helped shape our understanding and broaden our horizon.