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.

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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Here are the Insights into Mathematics Playlists:

[jimmyraconteur]

I think this is one of the first series of Dr. Wildberger's that I've sat all the way through and taken notes!

[fcvgarcia]

Thanks, Professor Wildberger, for bringing such great insights.

[taliesinbeynon]

This is all very pleasant and exciting! Projecting forward a bit, one might wonder whether we can ultimately use these ideas to produce finite field versions of ODEs and PDEs, and obtain something like harmonic analysis on discrete manifolds, recapitulating some of the beautiful theory we have there from the past two centuries.

[narek]

Just subscribed. This channel is awesome! I will use it as a supplement for my classes.

[pickeyberry]

Video Content

00:00 Introduction
02:24 Evaluation of polynumbers
08:33 Semi-cubical parabola (Leibniz, Niele)
14:53 The curve y²=x³ is a decasteljau Bezier curve with parametrization
29:35 The curve x³=y² over a field IF has tangent line 3xa_2y=a³

[willjennings]

This was a great series and I approached it from Umbral Calculus.

[rebusd]

At first I was annoyed at your cheek but upon further reflection, if Cantor was attempting to model a continuum which later evaluated empirically to be nothing more than a series of discrete Plank-units, then there is no real world underpinning as of yet from which to derive or construct an actual infinity. So I get your point, and am approaching with a lot more grace, humility, and curiosity, I must say.

[TupperWallace]

This alternative to thinking in terms of functions is very exciting. Your choice of the elliptic curve over finite fields for an example is tantalizing, given the extensive use of elliptic curve cryptography these days, cf. Bitcoin. I’m highly motivated to try to understand this approach, but it’s hard for my old brain!

[Catrice_]

Great job. Now I am interested in how differential equations could look like and how to solve them. Best regards

[PeterHarremoes]

Nice :-) The relation between derivatives defined in terms of infinitisimals and notion of double roots is important, but I think much more could be said about it. I hope there will be covered in course Alg. Calc. 2.

[abdonecbishop]

again ..well done Norman....seems finite field Calculus does not need Berkeley's “ghosts of departed quantities” to establish consistency

[tricky]

19:05 that's beautiful! A natural definition of calculus as the position and momentum vectors of a point particle following a path! Beta is a special-relativistic dilation parameter? A sort of 'density' of the curve?

[eytansuchard]

Dear Norman, here are questions ? 1) Is math consistent if only recursive numbers are valid ? In this case the power set of Aleph0, "2^Aleph0" is also of cardinality Aleph0 and only recursive sets are defined as legit objects of math. 2) Are you familiar with chapters 11 and 12 of Neural Networks and Analog Computation Beyond the Turing Limit by Hava T. Siegelmann ? If numbers in the physical world are recursive, then an analog machine can solve problems that a digital computer can't. For example, Turing's halting problem is solved! Quantum states can encode recursive numbers if their probability function yields recursive numbers. 3) Can it be proven that if events in space-time obey a probabilistic set of equations then they can be embedded in a deterministic manifold ? A deeper theorem would be that every stochastic process is valid if and only if it can be observed by a deterministic observer. The Gauss bell lives in deterministic coordinates. The dice that is thrown by a player manifests "a probabilistic law" of distribution in the sense that there is an observer that writes down the results and performs the "probabilistic experiments" from a deterministic point of view. This principle is at the heart of describing the world as realization events and not as particles. Where these events do not make geodesic curves, forces and thus "matter/particles" appear. Gravity is simply a controlling response of an observer space-time. 4) Please look for arXiv:1806.05244v14 . Is there any way to get 4/Pi in (43) not from Ettore Majorana's notebook but directly from the theory ? There is a more advanced theory in researchgate. (43) is the mass ratio Muon/Electron, (43.7.1) Weinberg angle, (43.8) W+/Tau, (43.10)-(43.12) Tau/Muon, (43.10), (43.11), (43.17), (43.17.2) inverse Fine Structure Constant. In researchgate: (43.17.3)-(43.17.6) the mass gap. Please note that (29.1), (30) offer an equivalence of charge and gravitational mass in additional to ordinary inertial mass. The Bullet Cluster hot positively ionized gas must manifest a "Dark Matter" effect. So not all the DM effect is particles. There is a more advanced unpublished theory that shows an exact relation between the Bottom Quark and the Muon. So lepton universality must be broken and so is charge symmetry. 5) Are math and physics, the product of our mind as neural networks ? Basically, NNs model datasets by weights. If the model is too complex, the generalization is weak and faces overfitting. If it is too simple then the model underfits the datasets. As Einstein said "simple but not any simpler". More advanced NNs generate models through language. So as NNs we may impose our structure on "reality".

[ihebbendebba]

I swear I was about sleeping at 11pm, before seeing this video bumped in my recommendations. thank you sir <3

[tomallen]

Prof. Wildberger, I had to punt after 25 minutes on this video! I will have to review the split-complex numbers, I think! I always had this question about your extremely ambitious mathematical program.... You say mainstream analysis is "not logically valid." But do you really have the goal of creating "a more powerful system of analysis." Because if you do, I could understand the incredible time and effort you are putting into this area of mathematics. But to be fair, Abraham Robinson's system of non-standard analysis is interesting. But (I'm not sure about this) I don't think there is anything that can be proved in non-standard analysis that can't be proved in regular analysis! I wouldn't be surprised if that isn't true in your system! Tom A.

[sschmachtel]

Wow, yep same opinion that this might be used to cook up something very useful. Immeadiately I thought about triangles and what kind of shapes resulting from triangles you can apply this to.
Especially not only triangles but also shapes like the superelipse equivalent to triangles? Just a feeling though that there is maybe something where you dont have straight tangents??? Some kind of generalisation?
Anyway shapes are pretty important for discretising PDEs and I think that those round shaped triangle superelipse equivalents could be interesting.
Thinking about a curve filled triangle, on the outside it is a pure triangle but turns more and more circle or point like when approaching the centroid.
I tried to find some formula for such shapes.... which doesnt seem to work with anything like sine or cosine or alike supereliptic formula, since you'd need a sine and cosine that sort of repeat after 3 maxima, not 4.
Oh well I dont know now if this is undestandible ....
Anyway what I also sort of wanted to say that easier stuff with fields to describe polygons... you can almost be sure that there is some applications.
Like the parametrised curves for the circle ... which you can also use to create superelipses if I remember correctly... and they are much more accurate obviously, as sine and cosine involve those series calculations where it very much depends on how you calculate them and sum them.... preferably starting with the smallest numbers. And yet each multiplication or division step introduces inaccuracies due to floating point limitations

[SuperParaNatural]

Great video! A question that comes to mind: is there a canonical topology we can induce on a given finite field that is congruent with this notion of calculus, where functions are differentiable only if they are continuous? This would give rise to an analogue of the manifold for finite fields.

[kylemoulton]

I rlly enjoyed this math problem series in way that it has formalized, generalized, and broaden the scheme of classical differential calculus. Im having difficulties adjusting in sense that I can mentally visualize a polynomial as a curve on a Cartesian plane. What is the your geometrical intuition behind poly numbers or are they just a data structural algebraic tool to get away from variables. Also, what is the fundamental problem with variables? I’m not too clear on that. Lastly, this new notion of differential calculus on finite fields has me thinking a lot about its greater potential over classical calculus. Would it be possible to use these ideas in knot theory? In particular could this is be used to find minimum crossing numbers of general knots or finding minimum linking numbers? Just something I’ve been thinking about.

[john_g_harris]

At about 10 min Norman says that this drawing is the graph of y^2 = x^3. The drawing is misleading. The graph should be shown as a dotted line because there are holes in it. For instance, at x=2 there is no rational fraction whose square is 8 so there isn't a corresponding y value.

[jeanboyable]

U have cought my attention with the new way of doing math.