Insights into Mathematics | Infinitesimal Calculus with Finite Fields | Famous Math Problems 22d | N J Wildberger @njwildberger | Uploaded 3 years ago | Updated 1 day ago
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 )

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My research papers can be found at my Research Gate page, at 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 openlearning.com/courses/algebraic-calculus-one Please join us for an exciting new approach to one of mathematics' most important subjects!

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