Introducing (finally!) Box Arithmetic | Math Foundations 237 | N J Wildberger
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- čas přidán 27. 07. 2024
- Box Arithmetic is a new way of understanding what arithmetic, and much of associated mathematics, is actually truly about. The most important new understandings are that 1) the most powerful data structure for foundational work is not a set, and is not a list, but is rather a multiset, or mset: where elements are unordered and repetitions are allowed and 2) that the particle / anti particle duality famously discovered by Paul Dirac in 20th century physics has a deep and remarkable analog in the foundations of arithmetic. When we put these two together, we get Box Arithmetic.
Here we give an overview of how we propose to reconsider the developments arising from mset arithmetic in the light of this powerful box / anti box duality.
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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.
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.
@@santerisatama5409 Thank you for that. I will look it up. It may be over my head.
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.
I've been extremely busy in my own work and just recently got to this video. I know exactly what is going on here. I have known it for quite some time now.
This is NOT "voodoo" and you are definitely on to something!
In many respects your work here reminds me of G. Spencer Brown's work, but you have extended that work in more than just one direction. Specifically by hinting at the missing duality that his work didn't provide.
Your work is "dancing around" an important aspect of our universe: Duality.
All throughout history mathematics has been focused on, and almost exclusively wed to, an ontological view of our universe. There exists an epistemological view which has thus far been almost completely ignored.
One way to see this is by finding names for each complementary perspective:
1) "Mathematics of Exteriority" (ontological/external mathematics)
2) "Mathematics of Interiority" (epistemological/internal mathematics)
Gauss und Riemann (to name two) both came very close to discovering this duality, but the human mind itself was not sufficiently developed enough to grasp the framing which could make it "gel" and enter into our awareness.
There is no "Math / Anti-Math": instead, they are complimentary framings sharing a common "Origin".
I wish I could say more about this, but I don't want to do so in this public space. I don't even know if I have reached you nor that you even take me seriously.
Let me know in private if you would like to know more.
Dr Wildberger, Thankyou.
The black boxes are creation operators and the red ones are annihilation operators.
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
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.
Thank you, Norman! When you invoke symmetry there must be a group behind it. Would you elaborate?
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'.
Some amazing discrete mathematics, suitable for computers!
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. :)
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.
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'. :)
Even boxes with content require some sort of order: You should not "count" content twice. So there is still some old-fashioned ZFC-Order.
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.
"mathematic language design." Symbols provide mean for different mean for perspective constructions which could be helpful.
but like programming language.
All easy cases are equally easy to embed in a language, but all hard cases are unequally hard.
Anything from scratch is nothing, but a toy to form understanding.
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.
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
Yes!
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.
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.
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.
17:00 Looks to me, that if you add some a rule to this game:
aBox(sum element) = Box(sum element^a)
You'll get distribution law over negation.
I think this is quite natural and lets us equate both +0 and -0 as the same object.
@@thomassynths Not equating, just use the box rule (and extension).
put a red box & black box in a black box -> reduction to a black box;
put a red box & black box in a red box -> reduction to a red box.
0 is the only object the property is only convention. The only operation is the box rule: you can put boxes in boxes ( a & ~a) with cancelation/reduction. It is "additive" only, in that the process only goes one way, compounding.
Absolutely still working on it, sorry for the huge delay. I realized I needed to go all the way, and do not want to disappoint you.
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.
I'm going to imagine fractal tiling/space-filling multidimensional arithmetic and have a seizure while doing so.
brb
I thought you may have been headed in a different direction. I liked that, at the lowest level, the counting numbers were represented by the presence of absence of tokens (you called it [0], I would have just posited 'counter' or 'token' or 'unit') so that the proto-operation of concatenation underlied the system. After introducing your anti-zero (I would have called it perhaps n-token), I thought you may go ahead and introduce an i-token for complex numbers, and more token types for quaternions, and octonions, with different interactions within the mset. Does reifying the negative token as a box operator make such extensions impossible?
I though he would a a half token for rationals before i tokens
@@braden4141 He has the integers. Presumably rationals will be defined via an operation on 2 msets.
@@paulwary I just was thinking if he came up with a variant of mset to get the integers why wouldn't he do the same with the rationals.
@@braden4141 How would he do that? Rationals are defined in terms of the integers, so they would need to be layer above that I think. (Edit: that is, above the mset level where the naturals are defined. But of course if he does that, they would still be housed in an mset!)
@@paulwary I mean integer are defined via natural numbers in set theory in a similar way to getting rational numbers from integers. So him making a token for integer and not rational seems arbitrary.
yes yes yes. Similiar to the notion of Finite Simple Groups?
what about instead of a duality, a triality? what would that even be? what corresponding data structure?
is adding blue boxes arcane prohibited math?
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.
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 >
@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?)
@@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 >
@@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.
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)
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?
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.
@@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.
How do you count without ordering?
We don't use lists, we use trees ... [07:24 ff] Trees are lists with bells & whistles. "Tree" relies on the notion of a "list".
At 10:49 Norman says the are unordered. But they still are a list.
So, Dr Norbert, how does box arithmetic vary logically from Peano's set-formalized derivation of natural numbers and all that follows up to irrationals?
It's explicit.
He doesn't believe in irrational numbers. So he will stop at rational numbers. He said in this series and other that he doesn't believe infinite operations are valid. He is some sort of constructivist don't know which kind though. He went into better details in other series why he thinks unending processes are invalid. This series is him making natural numbers upto rational as of now that is valid with his view on math
The goal is to constructivist version of Peano's arithmetic and beyond using msets instead of sets
Watching some of your other videos, i was thinking on a similar concept earlier today. How do we draw or imagine a negative area?
x^2 + 5^2 = 8^2 - 10x
One could draw a square with sides 8, comprising two smaller squares with sides 5 and x, filled in. Then the remaining area would be two empty rectangles, each with areas 5x, empty. In other words, on the lhs of our eqn., we can imagine two physical squared areas. On the rhs, we have a larger theoretical squared area mysteriously merged with two "anti" (or negative) rectangles that are equivalent to the lhs. I suppose one can physically imagine negative area (-10x) as bounded emptiness or just a perimeter with "nothing" inside.
I think i'll imagine your anti-boxes as similar bounded emptiness until further revision. :)
It’s an interesting thought. In geometry negative areas are usually associated with opposite orientations.
Instead of just squares of multiplication by self, areas of parallelograms can be thought as multiplication of any two multiplicands. Stern-brocot type structures have a very interesting property (which we learned in this series!) in this regard. Given a fraction a/b, the form 1/ab is called 'simplicity'. On each new row of generation, the simplicities of new mediants add up to 1 by standard field arithmetics.
Instead of limiting ourselves to the standard parents 1/0 and 0/1, we can generalize the Stern-Brocot algorithm to the centered form 1/0 0/1 1/0, from which we get left and right side mediants L1/1 and R1/1. As consequence, if we define the L and R sides as positive and negative, then the area-like simplicity sums of new mediants cancel each other in the manner of 1-1=0.
The structure becomes more visible with the following version of Stern-Brocot algorithm, with < and > as numerator elements and as the denominator element:
< >
< >
< >
< >
etc.
The simplicities are defined for the numerical multiset interpretation of the above structure, but it's very interesting open question, if and how they can be defined on the level of formal language of chiral symbols.
Geometrically, the area of an parallelogram is the concatenation of two inverse triangles.
Now I have to watch the earlier parts
My stumbling with numbers is the axiom of choice. The construction here is inducing this at a definitional level. Can these constructions formal avoid this glitch?
The axiom of choice is evident nonsense
@@carly09et Professor Wildberger said something very early on in this series of foundational thinking, which I found very liberating and inspiring. He said, in paraphrase, that foundations of mathematics is an open field of innovation and investigation, with rigorous loyalty to what you establish. Instead of Formalist post-modern games of arbitrary axiomatics, in this series foundational thinking is an ongoing process of investigation, where intuition and cumulative observations lead to rechecking initial definitions. Which makes this so fun and inspirational!
AFAIK, historically AoC was invented as a blatant absurdity in order to claim that "real numbers" can be imagined as well-ordered, even though no well-ordering can be demonstrated. A claim of well-ordering is necessary for the blatantly absurd and counterfactual claim that "real numbers" form a field.
@@njwildberger Agreed, But I find your inverse/anti idea here ambiguous.
It is the formal resolution without choice that is the problem.
Hmmm. . . I'm thinking of John Horton Conway and his games-based approach leading to his enunciation of the hyper-reals, which I suspect Dr Wildberger would find abhorrent.
Conway's game of surreal numbers (a version of generalized Dedekind cut) is defined as a purely consecutive 2-player game. Which can't handle trinities etc. without abhorrent infinite processes.
What we are doing here is parallel foundations instead of purely consecutive games.
Riddle me this: Just for the fun of it, recapturing problems of set theory more than 100 years ago.
Is there a box which contain all boxes? Does an all-containing box exist?
Does a box exist, which holds all boxes which do not fit into a box?
to all three questions is no. because he barred the use of infinite boxes in his box arithmetic.
Greetings, prof. Wildberger.
Have you seen the recent work by the folks over at the Ramanujan Machine?
This video they published is fascinating: czcams.com/video/Uk04gfIt8yM/video.html
It seems that they're laying the foundation for a proper theory of irrational/real numbers that even a skeptic like you might be willing to consider.
I think the 21st century might be when mathematicians finally tame the real numbers once and for all.
I'd love to hear your thoughts about it!
Interesting, thanks for sharing. Stern-Brocot type analogue of continued fractions has been discussed in this series, and there's lots of exciting things to to be found when representing continued fractions as paths along the binary tree of blanks between totally orderee SB-fractions.
Still trying to wrap my head around Gosper arithmetic, hoping perhaps some day to find a simpler bitwise arithmetic for the L and R paths.
Centered and fully palindromic Stern-Brocot structure (meaning 1/0 0/1 1/0 as numerical seed string) has many benefits, as it allows to represent both positive and negative square roots etc. as bitwise NOT-operations.
Small but IMHO very important comment on the video. Calling vertices/nodes of a graph/lattice "points" is an endless source of confusion.
So called Real Numbers have been a blatant Zeno absurdity attempt to play the Formalist post-truth language game in order to give point-reductionist definition to number theoretical continuum. That absurdity can't be and should not be saved, Stern-Brocot type structures and binary tree for paths between fractions offer a coherent point-free continuum. The real problem of real numbers is the Zeno machine of point-reductionism; constructible numbers and algebraic numbers are OK and very interesting - as long as we do our best to keep math a coherent whole and e.g. stop violating the undecidability of the Halting problem by applying quantifiers 'there is' and 'for all' to induction by arbitrary axiomatics.
I'm just trowing something against the wall to see if it will stick, maybe they are p-adics?
you can teach high level math to a child because it's just applied logic..