Box Arithmetic with Polynumbers | Math Foundations 238 | N J Wildberger
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- čas přidán 14. 10. 2023
- We introduce fundamental terminology and notation for Box Arithmetic, putting the duality between mathematical objects and their anti objects at the centre of the development. We also move to a more geometric, visual approach with actual boxes over inline representations.
Is this the arithmetic of the world?
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I really like how this approach is coming together in these videos
This elucidation is much appreciated: The complexities encountered when inculcating fundamental mathematical rules seems to revolve around an inadequate demonstration and this set of illustrations provide the necessary framework for apprehending the proper vantage, it is opined.
The profundity of this novel insight has a semblance of being outside the scope of most instructor's purview with there being an eerie, perhaps intuitive, sense that something is missing with our routine calculative thinking and this seems to parallel other crucial missteps in academia such as philosophers presupposing an understanding of Being and there being the psychological correlary of 'change blindness', such as with perceptual dynamics of experience not appropriately factored given that the subject is without means for discerning how an awareness of the aspect of their experience would relationally consist to provide perspective before methodically working to prove these elementary ideas within the context of structural type and character theories with an eye to concretizing your Finite Matrix Group theory with respect to computation.
With respect to the above opinion, I might add that it is suggestible that the mathematicians of the modern day (Particularly the applied Maths majors such as with the engineering and economically-relevant scientific fields) have an inadequate "tool-set", which is to say that their working knowledge is inept, although remarkably vast in the case of most accomplished professionals, in that they have a deficiency with regards rudimentary features of the mathemes (Third mathemata of the Psyche; dianoia/Reason).
Certain conventions seem arbitrary and discerning their rationale, especially with respect to annotation schemata or notational convention, can require extensive investigation. For example, the epsilon or small Greek E similar to the Euro with the dash terminating at the curve for the "element of" or membership operator symbol typically encountered in axiomatic set theory (ZFC) seems to only maintain its commonality due to or given that it's now common place, although the thinking process behind this choice or selection of a grapheme in tutelage or at least diagrammatics with respect to conveying the syntax of equations, relevant expressions and the necessary constraints is not typically readily coneyed or obvious, to someone not acquainted with the subject.
I note, for instance, the symmetric bilinear, quadratic form encountered in topological geometry and Borel sub-groups, to tangentize or digress.
The notion is, of course, that the science of the objective reality has a 'mathematical underpinning' and the methodology is deductive (Creation and annihilation operators not just finding a basis, to equivocate, in the theory, yet contrariwise their physical instantiation being exemplary of Nature which is known by way of mental adducing or conception) and that the conventional hypothetico-deductive model of reductionism is merely inadequate and the obvious candidate is the only choice when selecting a way in which one might go about knowing - namely the only one (By process of elimination [Reductio ad absurdum/Proof by contradiction] iteratively and recursively).
By this, all that is meant is that relying upon sense-object verification is asinine with regards the appropriate method of science where to scient is to know.
An example of convention with graphical representation of mathematical objects, such as a-object would be the use of the colour red for negative and to take this to an extreme, there is the combinator theory of early last century which gives means for constructing any object or relation between they or them with regards the s- and k-combinators which is quite convolutive, however computationally more exquisite and availing a more sophisticated repertoire from an architectural/design perspective (Again, to proffer beliefs, when here optimally we are concerned with rigorous deductive proving and working with these constructs in a logically sound/coherent manner which ought be overt to the fledgling student).
The "truth table" for combining Zero and anti-Zero is equivalent to XOR, which is functionally complete. This suggests box/anti-box arithmetic could be used as another basis for the theory of computation, like lambda calculus.
can't wait to see how fractions can be expressed in the box arithmetic. I guess it'll be like 5/3 = 5^1*3^-1 or something like that
This is fascinating, why can't zero box be represented as a straight vertical line? Then use tally or even numerals. Also rather than colour, anti boxes could have a second edge on any side of the boxes, preferably left as we read left to right. I love the rules developing from this and in particular the annihilation principal which is fundementally how algebra works when terms "cancel out". This feels like something really important for modelling physics and infinities.
In neurophysiology, my hero Sir Charles Sherrington introduced the 'principle of reciprocal innervation': 'in a coordinated movement, the muscle nerve cells that are needed for the movement are excited, while those that are not are inhibited'. Inhibition is not just absence of excitation: it is the action of inhibitory nerve cells. The main excitatory transmitter is the glutamate ion, while the main inhibitory transmitter is the gamma-amino-butyrate ion. The leader of excitation is entry of sodium ions into the cell, while the leader of inhibition is entry of chloride or potassium ions into the cell. In subjective sensation, pain is not just an absence of pleasure: it is an active unpleasure.
"The word passion originally comes from the Latin word passio, meaning to suffer or endure. Most commonly, passion referred to the Passion of Christ, or the last days of Christ's life."
@@Nah_Bohdi Are you replying to this CZcams or to the one where I used the word 'passion'? The word 'passion' in English isn't quite the same as the Latin word 'passio'. I am using the word in its ordinary general English sense. I don't know just how much the word is used in the restrictive sense, 'of Christ', and how much in the more general sense that I am using.
Can you expand on "reciprocal innervation"? Can it be described as (simultaneous) parallel computation, or more generally, what is character of parallel and consecutive processes involved?
With different notation, we are able to further expand the idea of positive zero > and negative zero are the positive and negative numerator elements. By concatenating mediants, we get the following structure:
< >
< >
< >
< >
etc.
By unary count of the countable elements in the words (when < or > already part of , not counted separately), the count gives a Stern-Brocot type structure.
@santerisatama5409 It is challenging to describe the reciprocal innervation curcuitry in words. A diagram would make it easy. Sherrington's principle is set out in his classic, 'The Integrative Action of the Nervous System' (1906). I guess you can find it in any textbook of neurophysiology. It is clearly evident in the mammalian spinal cord, but I guess also in most or all nervous systems. Usually, a nerve cell has either exclusively inhibitory or exclusively excitatory outputs. This may be called a version of Dale's principle.
The nervous system or brain is parallel processing incarnate, one might say in 'pure' form?
@@christophergame7977 I think I get it. There's synchronous symmetry between contracting and relaxing muscles, etc.
Both < > and >
Well this has been a wild ride. I started watching these math foundation videos in 2019, and now I'm caught up. These videos have given me an appreciation for the rational numbers and the difficulties that arise from using real numbers.
I'm wondering what are your thoughts are on proof assistants like lean4 and the language of type theory? It seems to me like codifying existing theorems that rely on real numbers with rational variants in the language of type theory would be important and useful.
Hi Norman! I love the direction this is going. As @cogwheel42 points out the "anti box" addition is defined using an XOR gate, so I wonder what would happen if we define other operations using other boolean truth tables such as NAND. Maybe these new operations would be useful in any way? It seems like the most fundamental would be to start with NAND (instead of XOR) since it is functionally complete and then define other operations, e.g. XOR (which is used to define addition in this video) from NAND. From that, we should be able to use it as a more rational foundation of mathematics and computation (since all digital computers are just built from combinations of NAND gates in pracitice).
I really miss that days ❤️
I've been thinking about the box arithmetic in the context of programming, the different types as Types in the program, and implementing the operations in a parametric way. What has come to be is the idea that at the core of multiset is the idea of insertion being primary.
Insertion is represented addition, where something like " + [ [] [] [] ] " is inserting 3 []'s. What the higher order operations represent, then, are some form of higher order insertion (data base operations).
For example lets take
" * [ [ [] ] ] "
what this does when applied to M is it inserts 1 [] inside of each element of M, now take
" * [ [] [] ] "
what this does when applied to M is it inserts into M the contents of M once essentially duplicating the data, now for an even more complicated one
" * [ [] [ [] ] ] " what this does when applied to M is it duplicates the contents of M where you also insert 1 [] into the duplicated elements.
Now with ^ we have even more depth to the insertion for example
" ^ [ [ [] [] ] ] " what this does when applied to M is it duplicates the contents of the contents of M.
In this partial application way of thinking these boxes represent both an action and data in some sort of dual way. This can be thought of the two ways you can do the partial application, take M ? N where ? is some arbitrary operation in this arithmetic, you can partial apply to extract ?N or ?M which are to be applied to M or N respectively. Here either is data or the insertion.
Yes!! Enough said.
Thank you Norman. I like the name Box Arithmetic. I have a series of videos on a subject called Box Operators.
This is what it’s like to step beyond the “complex” numbers.
This mathematics really makes you FEEL like you’re Alexander Hamilton, and he had a torrent affair (his quarternions!), and he wrote it down, right there (on the bridge!), highlights!
@@imnimbusy2885 wrong h
The Box magic of path from unnested 0 to nested 1 seems closely associated with 'Devil's staircase'. According to Wolfram alpha, there's "amazing" connection with continued fractions, phi and fibonacci.
The litmus test is to define polynomial composition.
I would love to have a book on this, do you have any other resources or collections of notes available Norm?
The sum of zero and anti-zero is like mod-1 arithmetic (clock arithmetic of 0 and 1), with zero=0 and anti-zero=1. I don't know if this has any significance. Just something I noticed.
Does this relate to Hintikka's two kinds of negation?
Are you implementing this approach anywhere? C++ is a good place to demonstrate your arithmetic, because it can represent Concepts as in abstract algebra.
@derekpmoore That is a good question. I am not much of a programmer. But who knows, maybe I can get some AI to do it for me??
@@njwildberger do you know Alexander Stepanov’s work on the foundations of programming? Believe it or not, but your work is in the same vein.
@@njwildberger czcams.com/video/fanm5y00joc/video.html
@@njwildberger elementsofprogramming.com/eop.pdf
Will you please make a video on fermats last theorem ,its attempted proofs and valid proof .
Since it has been proved by andrew wiles ,but if anyone have new another very elemntary proof of it based on like mathematical induction ,will it be celebrated
I’m sorry, but my interest in that topic is not sufficient to motivate me to understand the proof
1:56 my head hurts already
Thanks. It's very magical that 0->1, when put in a box. Boxing gives countability to what is contained. Return to mereology with relational approach is the right path, I feel so very strongly.
There's also practical but no less important challenge of notation, ASCII representation of Box Arithmetic. But perhaps better wait for formulating a theory of rationals for that.
The connections of nerve cells in general will not obey the box rules.
I think you've re-invented math about an infinite number of times!
This should lead to a new way of doing calculus?
360 ×360
Antiparticles rank DOWN there with quantum mechanics, relativity, and we might add axiomatic set theory, in the dustbin of historical silly ideas.
Not to say anything against this approach to arithmetic.
Black and red boxes are difficult to distinguish for your colour-blind viewers
Hi Richard, Thanks for letting me know. I'll have to think about a way of possibly overcoming that.
Cajas negras tal como se muestran, cajas rojas -> cajas negras giradas 90 grados = rombos. Tal vez sea útil si solo se trata de dos colores.
There are colorblind filters that adjust such colors automatically.
@@njwildbergerovals and rectangles or something, lots of hat boxes are oval, so you can keep the name!