Infinitesimal Calculus with Finite Fields | Famous Math Problems 22d | N J Wildberger

Is it possible to do Calculus over finite fields? Yes! And can infinitesimal analysis still play a part? Yes! This video will show you how, by working out explicitly some remarkable geometry formed by the semi-cubical parabola over the explicit finite field F_7.

It is helpful to realize that "real number analysis" is a red herring --there is no need to pretend to be able to "make an infinite number of computations" to set up tangents, derivatives or integrals. Another is that the algebraic theory of polynumbers rightfully takes over from the pointalist modern "function" viewpoint. And the importance of de Casteljau Bezier curves, sadly neglected by modern educators, takes centre stage.

All this is, and will be, developed at greater length in the Algebraic Calculus One course, now available, and the Algebraic Calculus Two course, in preparation, at openlearning.com. There are opportunities to advance undergrad education here, to make new discoveries, and to touch base with a more computational, logical tight mathematics. In this video we unearth a nice hexagon, both inscribed and circumscribed, to the semi-cubical parabola over F_7.

Video Contents: (thanks to Me Too)
0:00 Introduction
2:30 Retreat from the 'functional' POV.
3:10 A symmetrical POV. It makes 'at a glance' sense of the table of powers.
5:40 Polynumbers are elemental ("primary"), functions are not.
7:40 Polynumber formalism of Derivatives over [point-to-point] 'secantism'
15:20 Switch from 't '( 'variable' ) parameter to a ( polynumber ) 'α' := '| 0 , 1.. ' parameter dependence
16:00 Shift from a 'α' := '| 0, 1.. ' to 'α' := '| 1 , 0.. + 'ε' := '| 0 , 0.. ( bipolynumber ) parameter
'| 0 , 1.. '| 1 , 0..
17:45 'point' plus 'vector' Derivative description
31:10 ( see 13:20 )

************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger

My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.

Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/algebraic-calculus-one/ Please join us for an exciting new approach to one of mathematics' most important subjects!

If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at https://www.patreon.com/njwildberger Your support would be much appreciated.

Here are the Insights into Mathematics Playlists:

https://www.youtube.com/playlist?list=PL55C7C83781CF4316
https://www.youtube.com/playlist?list=PL3C58498718451C47
https://www.youtube.com/playlist?list=PL5A714C94D40392AB
https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
https://www.youtube.com/playlist?list=PLIljB45xT85D94vHAB8joyFTH4dmVJ_Fw
https://www.youtube.com/playlist?list=PL8403C2F0C89B1333
https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
https://www.youtube.com/playlist?list=PLIljB45xT85DH__ZzGQWQrVRxlbKh-Nsa
https://www.youtube.com/playlist?list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
https://www.youtube.com/playlist?list=PLIljB45xT85DSrlV6NX8RMBksZhdTHtwW
https://www.youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7