Are current mathematical definitions of derivatives what we need in physics?

[@MrSlugmuffin]

very interested in those updates

[@AssumptionsofPhysicsResearch]

Will do! May I ask what is your background?

[@MrSlugmuffin]

@@AssumptionsofPhysicsResearch i don’t have a background. by wich i mean i have no accreditation. that being said i’m very much a math person.

[@MrSlugmuffin]

@@AssumptionsofPhysicsResearch also i don’t mean this as a criticism, but i felt like the last part of this video was to short.

[@AssumptionsofPhysicsResearch]

@@MrSlugmuffin Yeah... I cut it short because I realized it was getting too long. I thought I'd divide the description of the problem from the attempt of a solution in two different videos. As for you background, you must have some if you are able to follow and interested in more details! 😀 It could even be "I studied engineering 20 years ago but always like math" type of thing... 😁

[@MrSlugmuffin]

@@AssumptionsofPhysicsResearch around 18 i became obsessed with math. became acquainted with two mathematicians who definitively told me i was not in fact crazy. one of whom told me about homotopy type theory, univalent foundations, lots of other stuff as well. i’m particularly interested in apollonian circle packings. proofs without words 1 2 and 3 by rodger nelson. visual complex analysis tristan needham. every single thing john stillwell has ever written. john conways books. indras pearls. (proofs without words series are the only ones i don’t hit a wall on btw) after my freind attended a htt workshop he told me that statements in arithmetic, logic, and geometry are all “homeomorphic” he didn’t say homeomorphic i’m just abusing the term to wrap this up.

[@MrSlugmuffin]

more please

[@Scapeonomics]

I think that Geometric Algebra might offer some useful formalisms for derivatives as multivectors. It seems very difficult to talk about what derivatives are without a notion of angle, which leads you straight into complex algebras. I suppose you could also use hyperbolic angles instead, but complex exponentials seem to be more appropriate as a fundamental way to map angles onto numbers. You also mention a position-momentum phase space, but find it lacking a way of defining a norm that has physical meaning, but if there are any sort of wave mechanics going on, or in other words, an oscillation or rotation in the phase space, we could use whatever norm gives us a simple sine wave or unit circle.

[@AssumptionsofPhysicsResearch]

>seems very difficult to talk about what derivatives are without a notion of angle, You can! And that's exactly why I can't simply use geometric algebra! :-D All of the geometry of classical phase space (i.e. symplectic geometry) does not have a notion of angle. There is no angle in the position/momentum plane. You only need directions for derivative (i.e. the vector space structure).

[@Scapeonomics]

@@AssumptionsofPhysicsResearch Why is it that orientation and angular momentum are less privileged degrees of freedom than their linear counterparts? Considering that both linear and angular components are necessary to fully describe the proper isometries of real objects in Euclidean space, and in fact, without the inclusion of rotations, the very notion of Euclidean distance breaks down.

[@AssumptionsofPhysicsResearch]

@@Scapeonomics Not sure what you are asking... it has nothing to do with my previous comment. Angle/angular momentum is a separate DOF from position/linear momentum. You still have the same problem: there is no "angle" between angle and angular momentum. You only have the symplectic structure, with angle/angular momentum conjugate quantities.

[@Scapeonomics]

@AssumptionsofPhysicsResearch when we create a position-momentum phase space on R2, we at least imply there is some useful interesting information gained from this structure or relationship. Indeed, we can see in physical systems a conserved quantity (energy) as we observe them evolve in time, most trivially seen in the simple harmonic oscillator, and along with this we can specify the time coordinate using phase angle.

[@wwkk4964]

You don't need angles for differentials, you can use just lengths and ratios. In fact, Brahmagupta, the personwho came up with the negative numbers and zero and rules to do arithmetic with them, did it specifically for geometric purposes of calculating sines using half chords. he also came up with what's now called newtorn sterling approximation and second order interpolations. it was proto calculus without any notion of angles and he gave the area of the quadrilateral of 4 sides using only semi perimeter and sides, never ever invoking any notion of an angle. discrete maths is possible without angles.