Yes! Those are exactly some of the alternative frameworks I had in mind. Also things like using differential algebra instead of classical epsilon-delta analysis and using Liouville's theorem to calculate integrals gives similar vibes.
Large parts of constructive mathematics seem to also be aligned well with physics and engineering (using only things that can be explicitly constructed) and hopefully could lead to interesting mathematical physics results. Same with, for example, quaternion analysis, which is very rarely used (contrarily to complex analysis), due to difficulties in operating quaternion valued functions, or computable analysis with its surprising result about differentiation of real an complex functions. I get similar feelings about non-classical logics. Somewhat more fringe examples that come to my mind, but are also interesting are for example non-Diophantine arithmetics or Holors theory (tensor generalisations).
Thanks for the "receding shoulders climbing problem" it’s a very nice way to frame the problem. I agree that there is a lot to be done to make shoulder-climbing process faster, easier and more widely available. It should also be possible, to make it more easy by for example storing not only results, but also derivations of many mathematical relations and computations. E.g. nowadays, there is no reason to put only results of integration in the integration tables, but whole derivations should be available as supplementary materials (preferably even in some form of symbolic computing code; this is somewhat realised with RUBI – rule based integration package). Size of the paper book is no longer the problem, so such “interactive derivations catalogues” should be extended to as many possible branches of mathematics, physics and engineering as possible.
Large parts of constructive mathematics seem to also be aligned well with physics and engineering (using only things that can be explicitly constructed) and hopefully could lead to interesting mathematical physics results. Same with, for example, quaternion analysis, which is very rarely used (contrarily to complex analysis), due to difficulties in operating quaternion valued functions, or computable analysis with its surprising result about differentiation of real an complex functions. I get similar feelings about non-classical logics. Somewhat more fringe examples that come to my mind, but are also interesting are for example non-Diophantine arithmetics or Holors theory (tensor generalisations).
Thanks for the "receding shoulders climbing problem" it’s a very nice way to frame the problem. I agree that there is a lot to be done to make shoulder-climbing process faster, easier and more widely available. It should also be possible, to make it more easy by for example storing not only results, but also derivations of many mathematical relations and computations. E.g. nowadays, there is no reason to put only results of integration in the integration tables, but whole derivations should be available as supplementary materials (preferably even in some form of symbolic computing code; this is somewhat realised with RUBI – rule based integration package). Size of the paper book is no longer the problem, so such “interactive derivations catalogues” should be extended to as many possible branches of mathematics, physics and engineering as possible.