Introducing (finally!) Box Arithmetic | Math Foundations 237 | N J Wildberger

[@ungarlinski7965]

I'm sure this will have a deep, profound effect on the world and your genius will eventually be recognized. I mean not in the next 10^10^10^.......^10 + 23 years, but it will happen. Hang in there!

[@brendawilliams8062]

Dr Wildberger, Thankyou.

[@christophergame7977]

This new arithmetic is exciting and most admirable. Though G. Spencer Brown's 'Laws of Form' is perhaps vague, perhaps scarcely intelligible, and is evidently far short of this new arithmetic, one can say that, to an extent, 'Laws of Form' hints at anticipating the spirit of this new arithmetic.

[@santerisatama5409]

Kauffman's interest in Eigenforms and iterants originates largely from 'Laws of Form. This article looks interesting: 'Cybernetics, Reflexivity and Second-Order Science' - Constructivist foundations, vol. 11.

[@christophergame7977]

@@santerisatama5409 Thank you for that. I will look it up. It may be over my head.

[@paulwary]

Yes, I also was reminded of LOF from the previous videos grasping for a path from the simplest possible foundation. "To cross again is not to cross" and "The value of the sign stated again is the sign", if I remember correctly.

[@farhadtowfiq6767]

Thank you, Norman! When you invoke symmetry there must be a group behind it. Would you elaborate?

[@theoremus]

Thank you for the video. I see the commutative property in action here. In my geometry videos, I try to illustrate the commutative property in the context of affine geometry.

[@NikolajKuntner]

So if I got e.g. the box corresponding to x^a := (2+alpha^{-1})^a in 14:50. To put this x^a into another black box supposedly gives me another, deeper box, and thus another expression (in terms of + etc.) Just thinking of this "put whole box into another black box" in terms of the algebraic expressions, there will be some recursive rewriting going from one expression to the "deeper" one. Is this a known mapping that has been worked with? Does it have a viewpoint beyond the making-longer of rooted trees with colored vertices?

[@njwildberger]

Its a great question: I don't think we have much idea of this mapping that you suggest. The meaning of the hierarchy past the multinumbers is to me unclear.

[@santerisatama5409]

@@njwildberger If nesting in boxes is a top down mereology, as it looks like from this presentation, then perhaps a good heuristic strategy is a full mereology with both top down and bottom up directions, which meet in their middle zone in the form of multinumbers or something like that, which are centered both horizontally and vertically. That's the big picture which intuition has been suggesting, but dunno what comes forth when trying to translate this intuition into coherent language.

[@JoelSjogren0]

I wonder if it might be possible to represent the independent variables α₀, α₁, α₂ as x^1, x^y, x^(y^2). This would be in accordance with a general principle that "box = exp", only refined by the idea that separate bases of exponentiation should be used at different levels in a tower of polynomial expressions. Note that if y is allowed to be a "complex number" then even if the magnitude of x^y is greater than that of x^1 it may still be the case that the magnitude of x^(y^2) is less than that of x^1. At least in this very limited sense, it is true of this idea for a representation, that x^1, x^y, x^(y^2) are "independent" -- as they should be, in order to properly represent the independent variables α₀, α₁, α₂. However, it would be necessary for y to vary with x in order to maintain the illusion, and even so, the mapping (x, y) |-> (x^1, x^y, x^(y^2)) maps 2 coordinates to 3, so won't be able to literally cover its codomain completely, although one can still hope for it to be pretty dense. Another strand of thought is that there might be not just "anti-boxes" but also "anti-boxing" (or "unboxing"), that is to say, an inverse to the operation of boxing. (Combined with "box = exp" from the previous paragraph, this would be a sort of logarithm operation.) There is then the question of how to devise a uniform system in which these two forms of opposition or "anti" would be properly recognized to be part of the same "stuff", as well as (correspondingly) how these two forms of composition (that is, the juxtaposition of msets and the composition of layers of nesting) are also made of the "same stuff". Perhaps it may be suitable when in need of an operation of "anti-boxing" to make a leap from trees to graphs. Perhaps graph rewriting procedures like those of Lamping and more recently Buliga could be reused in this situation. Certainly in the literature spawned by Lamping's procedure (e.g. the book by Asperti and Guerrini) there is a strong emphasis on the treatment of *boxes* and the extent to which boxes are duplicable by means of local rewrites, and indeed this does involve a meticulous treatment of unboxing. One faint hope is that this graph rewrite procedure, if adapted and generalized to a setting in which multivariate polynomials with rational number coefficients can be expressed (specifically, in a fashion based on the nesting of boxes as in these videos), would be eventually unified with other well-known algorithms, such as those found in the Gröbner basis framework for the manipulation of systems of multivariate polynomial equations. That is to say, my faintest hope would be that these algorithms, coming from somewhat disparate realms of mathematics, could turn out to also be "made of the same stuff", and that this could be realized once they shared a common unified syntax, in which the basic syntactic phenomena (such as opposition and composition; nods to the category theorists...) are recognized as being the same, and in which the multiplicity by which these phenomena then occur would be generated in a uniform way rather than mosaically assembled. A very, very faint hope, but still worth thinking about now and then.

[@santerisatama5409]

Nesting in boxes is by definition a mereological operation. As presented in this lecture, the mereological direction at this stage seems to be decomposition from whole to parts, ie. from top of the exponent tower to logarithms. For a whole mereology, we need both top-to-bottom and bottom-up directions, with well-defined interactions in their middle zone. Polymultithinigies which are centered both vertically and horizontally, fusing together both top-down nesting and bottom-up addition.

[@santerisatama5409]

How is magnitude/ordering defined here? Can we find a way to combine box arithmetic with (operator based) Stern-Brocot type structure, as mediants can be considered foundational for ordering/magnitude, especially in reflexive/parallel domains? Operator based (cf "functional programming") approach is IMHO much more interesting and promising approach than object-oriented for building syntactic bridges between various arithmetics.

[@santerisatama5409]

Very nice. Look's like a return to mereology, with numbers, polynumbers, multinumbers etc. as whole-to-part decompositions. Box-and-antibox a version of Brouwer's 'twoity'. :)

[@ebog4841]

what about instead of a duality, a triality? what would that even be? what corresponding data structure? is adding blue boxes arcane prohibited math?

[@njwildberger]

It’s a nice idea. We should be prepared to think dare I say outside the box. The challenge of course is to create an arithmetic that is internally consistent and better yet also has practical applications.

[@santerisatama5409]

For a trinity I've been playing with 1) mark and 3) both-and . Symbols < and > for integer numerators and for denominator. The 4) >< cancels out. Also 5) blank/white space. This way we get Stern-Brocot type mediants from the outwards seed < >. And something very interesting from the seed >

[@ebog4841]

@santerisatama5409 1) seems interesting but please be more clear because I have zero clue what any of this means. Please spell it out- feel free to assume low-aptitude on my part. 2) it seems that this is still a duality. You wrote "mark" and "anti-mark" , which suggests Binary stuff. Also, the stern brocot tree is just a Binary tree. Wouldn't a higher order tree be required? Wouldn't an entirely new notion of child node be required? And wouldn't there have to be a sort of "fairness" to the triality? As in: neither of the three colors being given preference? (I don't believe this ought to be necessary, but why wouldn't there be balance?)

[@santerisatama5409]

@@ebog4841 Binary logic would be 'EITHER mark OR antimark´' (cf. Law of the Excluded Middle; LEM). An integer is a single value defined by negation; 1-1=0; either negative or positive or zero. Likewise >< cancel each other, analogically to black and red boxes on same level. From either-or to neither-nor. On the other hand, the BOTH mark AND antimark is related to intuitionistic logic without LEM (cf. superposition). Rather simplistic and basic interaction of different logics. Both-and is semantically natural interpretation for the denominator element. With these definitions, we get Stern-Brocot type structure from seed the < >: < > < > < > etc. There's a binary tree of blanks separating the strings into words, and by interpreting the words as multisets we get the second row numerical values 1/0 0/1 1/0. Like boxes, the chiral symbols for relational operators are here prenumeric. The symbolic language contains mirror symmetry pairs, which can be interpreted as positive and negative fractions, but the actual negation becomes issue only with similar numerical interpretation of the seed >

[@santerisatama5409]

@@ebog4841 Your question was so interesting that I finally typed "Clifford algebra for dummies" in my search bar. :) The spirit of our times seems to be seeking ways to generalize Clifford algebra aka geometric algebra into foundational theory of mathematics. Playing with relational operators (and deriving Stern-Brocot type number theory from them, among other things) could be considered an attempt to find a point-free approach to generalizing/simplifying Clifford algebra. Same with box algebra, perhaps. It's not at all far fetched to think of boxes and antiboxes as constituent triangles of parallelograms. This is still a rather fuzzy intuition for me, but the middle zones between 1D and 2D as well as between 2D and 3D seem more interesting and promising than strictly defined Cartesian coordinate systems. Perhaps Chromogeometry in it's way was inspired by same intuition.

[@Bunnokazooie]

Some amazing discrete mathematics, suitable for computers!

[@santerisatama5409]

Not at all sure if this is reducible to these Von Neumann architecture machines, because of the inherent parallelism of box and antibox. Von Neumann architecture seems as purely consecutive as can be and hides the necessary parallel aspect from sight rather efficiently. In the classic definition of a Turing machine, the 'tape' symbolizes the parallel aspect, as it continues both left and right. The 'both-and' is the prerequisite for the head to make ''either left or right' choises. We could say that this reflexive/chiral approach is an investigation of more a general parallel theory of computation, giving the Turing's 'tape' much richer anatomy. Which kinda means that we start directly from quantum computation, instead of trying to define quantum parallelism through consecutive classical filters. BTW nowadays it seems that the biggest practical difference between computation and pure math is that the former largely adopt the methodological restriction of writing ASCII code, while the latter use the classic method of manually drawing on plane. :)

[@KipIngram]

The black boxes are creation operators and the red ones are annihilation operators.

[@santerisatama5409]

Yes! In alternative complementary construction, creation operator is < > and the annihilation operator > < > The resulting concatenation becomes numerator element symbolizing duration, and the numerator elements can thus be seen as accelerators. Coprime fractions with duration in denominator and acceleration in numerator are thus unique frequencies. When annihilation operator concatenates > < > >< < the concatenation annihilates, and we can also define rewrite rules for the annihilation: DelX: Xa: concatenation Xb: whitespace Xc: equivalence; copypaste etc. based on if A is neither more nor less than B, then A = B. Xa and Xb turn marked processes into unmarked processes. Dirac delta is here the unmarked definition: Concatenation is the mediant of whitespace! To mark the unmarked, we can write whitespace as "_" and concatenation as "|", and thus mark this prenumeric formal language definition of Dirac delta as _|_. The unmarked Dirac delta can further define pixelated white space of affine space, the tape of Turing machine, etc. White space is prenumeric continuum, and concatenation concentrated continuity. "Discreteness" is purely perspectival separability, like whitespace partitioning strings into words, etc. Mereological fractions are constructed in the following manner, from whole to parts < > < > like before, giving < and > the numerical value 1/0 and their concatenation the numerical value 0/1. Thus, the next row of concatenating mediants is < > and corresponds numerically with 1/0 1/1 0/1 1/1 1/0 We can see already that the operator language algorithm and this numerical interpretation of reading the words as mset tri-tally operations generats coprime fractions in total order, Stern-Brocot style. Box arithmetic can be seen as a different numerical operator, starting from interpreting the operators < and > analogical to Kleene's star function

[@christophergame7977]

This admirable work makes one think also of the branched logic of Jaakko Hintikka (and others) considered in his book 'The Principles of Mathematics Revisited'.

[@KaiseruSoze]

If you have a need for a unique label you make one up. Unique positions along a line are a few good examples. And those labels inherit the order of the first, second and ... place along the line. That is useful. I.e., the numbers don't matter. What they refer do does. The universe has been debugged. It's a pretty good place to start with the fundamentals of math.

[@njwildberger]

It’s interesting that the labelling space that we use goes up in dimension as we go down the hierarchy so somehow naturally geometry ends up figuring in the organisational framework for this arithmetic

[@jamesyeh2677]

For negation, I wonder if it makes sense to negate all the boxes within the outer box, rather than negating the outbox only. The benefit would be as follows. If I want to see if one poly number is the negation of another, and annihilate both, in the original presentation I would have to look inside the box to see if they are the same. If they are the same then annihilate both. In the new scheme, you would simply have the rule such that 2 different color outer boxes annihilate each other, exposing their contents. We then successively annihilate boxes to see what is left.

[@nikbl4k]

is that a 5B Vic Firth wood tip?

[@cogwheel42]

The presence of anti-boxes implies that there cannot be a Box that contains all Boxes, since it would have to contain all pairs of boxes and ant-boxes, which would mutually annihilate.

[@YawnGod]

I'm going to imagine fractal tiling/space-filling multidimensional arithmetic and have a seizure while doing so. brb

[@ReifAndreas]

Even boxes with content require some sort of order: You should not "count" content twice. So there is still some old-fashioned ZFC-Order.

[@braden4141]

the boxes are multisets as he stated in previous videos and what he means by unordered is that multiset containing {1,1,2} is the same as a multiset containing {1,2,1}. unlike lists which the order does matter to differentiate lists with same contents. why his approach to multisets does not fall in to the same traps as old-fashioned ZFC. because he barred the use of infinite boxes(multisets). the only boxes he allows in his box arithmetic are boxes that are constructed from existing boxes starting from the empty box and anti empty box.

[@fransroesink3784]

Great work again! But I miss something... What is the sum of a Box and a antiBox? I propose: a antiBox containing the contents of the Box and the contents of the antiBox together. What is the sum of two antiBoxes? Proposal: a Box containing the respective contents! Obviously, opposites with the same contents themselves, annihilate each other. The same holds for multiplication (and the caret-operation). Examples: Box plus antiZero = antiBox (with the contents of the Box) antiBox plus antiZero = Box (idem)