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so i guess this boolean logic is behind AI validating which of the possible answers generated has logical sense so to present it as the answer requested.

It's bothering that distinct operations + and * are defined to have identical properties. Too set theoretical IMHO. A suggestion that came to mind, shift the perspective/direction. Writing < for zero and > for antizero: + < > < < > > > < > > < < < > * < > First column not included in the nesting operations. 1-nesting of 2nd and 3rd columns starts from the second row, 2-nesting from the first row. 1--NESTING From top to bottom +: <><> : <> (<> incudes <>, then >< cancels) *: <<>> : <<>> From bottom to top /: ><>< : [empty] -: <<>> : <<>> 2-NESTING From top to bottom +: <<><>> : <<>> (< > includes <> includes ><, then >< cancels) *: ><<>>< : <> From bottom to top /: <<><>> : <<>> - : ><<>>< : <>

There's this video that explains how they use this technique to get large numbers using lambda calculus: czcams.com/video/Mzgw6zMtipQ/video.html

I think this is missing from the box arithmetic playlist

Hi Dr. Wildberger, This video really fascinated me, because I have been having similar thoughts without ever coming across other peoples views such as yours. I do not want to speak for you, but I believe in a fakeness that possibly goes way beyond what you are suggesting. I believe the whole real number line is completely misunderstood through a deep social conditioning of our understanding of reality. The 'order' of numbers on the real line is fake, it is an assumption of our reality that we believe in because we are deeply socially conditioned to do so. I actually believe that every number is infinitely dimensional (even natural numbers), and they live in a chaos space where there is no order or structure whatsoever. The apparent 'order' of the numbers on the real line is a contrivance of our reality that goes onto create our reality as we experience it. Put another way, the order we believe numbers sit in, create the type of reality we experience. If we believed that the numbers sit in a different order say, 1,2,3,4,6,5,7,8,9, this belief would create an entirely new reality for us all. I believe the fakeness goes way beyond this. For example the equals sign is the biggest fakeness of all, nothing is ever equal to something else that looks different, i.e. 1+2 is not the same as 3, they are fundamentally different things. I think we need to abandon 'equals', and replace it with 'transforms into'. I am also with you on connecting, sociology, psychology, and pure mathematics, again I take this to the extreme, because I believe that when we peel off all the layers of our ego, we will see all of truth, and this is not just a belief, as I have had mystical experiences where I have experienced exactly this. I would love to talk more with you, if this is of interest to you, I have also put up a few videos on these sort of ideas in my CZcams Channel. Thank you so much for the wonderful videos you have published.

44:32

bivector und angular momentum 21:40

20:00

The notation seems to be off. Take the binomial theorem: The usual notation (a_1 + a_2)^4 should translate to [a, a^2]^4 = [a, a^2] * [a, a^2] * [a, a^2] * [a, a^2] = [1_(4a), 4_(3a+a^2), 6_(2a+2a^2), 4_(a+3a^2), 1_(4a^2)].

Given two numbers such that the two numbers are 'a' and 'b'. Where 3a=b. Derive a sequence of three consecutive numbers such that their sum equals ab consecutively. So, c,r,e where c+r+e=ab. Consecutive, (c+0, c+1, c+2)=(c,r,e). Assume c=12 r=13 e=14. m=1 c=3 r=4 e=5. m,c+m,r+m,e=39. 3,12->4,3 27+12=39, and 36+3=39. 63+15=78=2×ab. Methodology: 3+9 =12. 12+1=13 12+13=25 39-25=14. Thus (c,r,e)=(12,13,14)=ab=39. How can the sequence be completed of c,r,e in ab?// Ro9, 39->3,12 27+12=39. ÷3 9+4=13 9=1 45= sum of 1->9 in both positions and magnitude. 1,2 2 3 3 3 4 4 4 4...9 9 9 9 9 9 9 9 9. 45 positions and 1+2+3+4+5+6+7+8+9=45. Sum of squares 1->9=285 2 10 15 (2+1+1=4 5) =27 2+8+5=15->6 ->123 1+4=5 1+3=4 5+4=9 6+3=9 11+7=18 10+17=27 16+20=36 21+24=45

What about square numbers? Ok, a sequence of square numbers: (0,1,4,9,16,25,36,49,64,81,100)=x. (n+1)-n=dx (positional) 1-0=1 4-1=3 9-4=5 16-9=7 25-16=9 36-25=11 49-36=13 would expect this pattern to continue indefinitely. 1,3,5,7,9,11,13,15,17 etc. So the inverse: (n=position) n+1+n=1+3=4 4+5=9 9+7=16 16+9=25 25+11=36 36×13=49 49+15=64 etc would expect this pattern to continue indefinitely. Odd numbers now have an origin and have an intimate connection with squares. Lets assign a poisition to the odd numbers with n+1 the counting function. 1,1 2,3 3,5 4,7 5,9 6,11 7,13 8,15 9,17 10,19 11,21 12,23. So we appear to have developed the 5 sequence with these pairs and a trangle sequence for p. 5+5=10 7+8=15 9+11=20 11+14=25 13+17=30 15+20=35 17+23=40 19+26=45 21+29=50 23+32=55 25+35=60 27+38=65 29+41=70 i do not see any impediment for this not to be true indefinitely. So the addition of the 2 and 3 sequence begining at 5 sums to the 5 sequence. P has a pattern of appearing 3,4 1,2 1,2 3,4 at 10 or 11 p switches to sums with a similar pattern. Ok. Notes: may not be immediately obvious how the numbers were extracted from the 2 3 sequences to make the 5 sequence.// 11+23=34 2+5=7 3+4=7 5+9=14 1+4=5. Cyclical. Very close. P is an intimate relation to othe types of numbers. Very efficient. Simple relation but difficult to articulate.

I can now try to point my question. Let me vaguely define a 'sheer rotation' as a 'rotation of a three dimensional object that preserves nothing but the object itself'. Is that enough to try to say what I mean? My question is 'does a quaternionic rotation preserve anything other than the object itself?' Or 'is a quaternionic rotation a sheer rotation?' Can a quaternionic rotation be expressed by exactly three numbers? I am asking why are four numbers used for a three dimensional rotation? Are three numbers not enough to rid us of gimball lock? What does the fourth number in a quaternionic rotation tell us?

The Riemann Zeta isn't special actually. Take any function representable as a sum over primes. Any such function will do. Now rewrite the sum over primes as a Riemann-Stieltjes integral; it is the integral where the measure is d pi(x). Here it is: the Prime Counting Function pi(x). So any sum over all the primes is connected to pi(x). Now.... the interesting question is...: is there such function, but simpler than the Riemann Zeta? 0o :D

YES YES YES. No boolean AND function of a "free variable" e.g. 1/B(x) < 1/2^181 < 1/(2*(the order number of the Monster Group)) ---> is greater than 2/(2^360). Hint: the binary representation of order number of largest Finite Simple Sporadic Group: the Monster Group is exactly 180 boolean bits. The Sporadic Groups have NO PARAMETERS. They are Whole Number "CONSTANTS" and form an informational framework and a Total Order on the Whole Number line. I.e, forms a Framework for Finite Physics. Finite Information Limits via the 15 SuperSingular Primes and the Supersingular Values of the Primes.

This is the most important understanding in pure and informational mathematics at least in the last 100 hundred, and maybe in the last 200 years.

So if I understand correctly, if a series "converges" to pi, it is a "type 3" series? since pi only exists as an iterative computational process... We can call our 2.1297 number "zi" and say our series converges to zi.

Thank you

4:52 Affine: geometry of parallel!?!

I see this topic was discussed 100 years ago by Godel Hilbert Russell etc. Will type theory replace the set theory as the foundation？

Love the way you use those balls, drumsticks and carton. Thanks for the lesson

Found this: OEIS A360569: a(n) = floor(Product_{k=1..n} log(prime(k))). Gives 0, 0, 1, 2, 5, 14, 41, 122, etc. Came up when I looked into nesting depths of Stern Brocot type constructs from generators < > (interval 1/0 1/0) and: a) <>< <> ><> b) <<> <> <>> c) ><> <> <>< d) <>> <> <<> which are the operator language combinatorics for the numerical interval 1/1 0/1 1/1 Procedure: concatenate mediants from the generators, then delete the word separating blanks from the generated rows, then delete substrings >< from the row strings. The results for a) and b) look the same as for the generator < >: <> 0 <<>> nesting depth 1 <<<>>> nesting depth 2 <<<<<<>>>>>> nesting depth 5 <<<<<<<<<<<<<<<>>>>>>>>>>>>>>> nesting depth 14 <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> nesting depth 41 In case of c), the string contraction procedure deletes the whole string, no nesting. In case of d) each row gets reduced to <>, nesting level stays 0. Hope this helps. :)

We should return to foundations and start cautiously fixing errors.In my opinion the foundation on which calculus is built is illogical and vague.But it is not beyond the ability to make corrections.

Preferably without dropping them😂

Video Contents : 00:00 Introduction 2:42 Powers And The Caret Operation 5:25 Applying f To The FIA 10:45 A Variant ( replacing all numbers in FIA with their cubes) 13:25 Another Variant (replacing all numbers in FIA with their reciprocal) 17:51 Suggests An Identity 19:51 Squared Reciprocals 21:39 Examples 24:32 A calculation

Video Contents: 00:00 Introduction 5:47 Boxes 10:34 Critical Operations 14:20 Counting Laws 17:21 Binomial Theorem (Caret Form) 20:19 An Ongoing Calculation 23:56 The Fundamental Identity Of Arithmetic 27:32 The Sun Operator S 29:35 Summation Laws 32:40 A Naive Application

Trying to understand more of the context. Fundamental theorem of arithmetic is said to be a corollary of "Euclid's lemma:" Euclid's lemma - If a prime p divides the product ab of two integers a and b, then p must divide at least one of those integers a or b. Here's the original, translation from the Heath edition. "BOOK VII, PROPOSITION 30. If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers. For let the two numbers A, B by multiplying one another make C, and let any prime number D measure C; I say that D measures one of the numbers A, B. For let it not measure A. Now D is prime; therefore A, D are prime to one another. [VII. 29] And, as many times as D measures C, so many units let there be in E. Since then D measures C according to the units in E, therefore D by multiplying E has made C. [VII. Def. 15] Further, A by multiplying B has also made C; therefore the product of D, E is equal to the product of A, B. Therefore, as D is to A, so is B to E. [VII. 19] But D, A are prime to one another, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore D measures B. Similarly we can also show that, if D do not measure B, it will measure A. Therefore D measures one of the numbers A, B. Q. E. D." What do we mean by division? Euclid means something different than what is currently taught in schools. Euclid means by "measures" continuous partitions of pure geometry constructions, for which equivalence relation 'neither more nor less' holds. Euclid's definition of mereology (aka inclusion relations") comes from the common notion 5 "The whole is greater than the part", ie. mereology is defined as nested inequivalence relations. Proof of proposition 30 first evokes the previous proposition 29, which defines coprimes: "PROPOSITION 29. Any prime number is prime to any number which it does not measure. Let A be a prime number, and let it not measure B; I say that B, A are prime to one another. For, if B, A are not prime to one another, some number will measure them. Let C measure them. Since C measures B, and A does not measure B, therefore C is not the same with A. Now, since C measures B, A, therefore it also measures A which is prime, though it is not the same with it: which is impossible. Therefore no number will measure B, A. Therefore A, B are prime to one another. Q. E. D." Can Box arithmetic derive the related Fundamental Identity of Arithmetic somehow "more directly" from the principle of inclusion? If that can be demostrated, it would be very exciting!

Sehr gut

I love the distinction between invention and discovery. "Brought into being" as opposed to, "it's 2".

In a recent viral Joe Rogan podcast episode, Terrence Howard tried to convince Eric Weinstein that 1x1=2 because of an "identity problem" in mathematics that supposedly you discussed. I'm curious which video or lecture it was that he picked up that notion from (though I'm quite sure it's not this one). It amuses me to know that Mr. Howard and I have both been a part of your online audience.

There is absolutely nothing "ambiguous" about the syntax and proof theory of first-order logic, or ZF (or ZFC or any other popular veraion of set theory specified by a set of sentences in the first-order language of a single binary relation ("element of"). If you do not like fact that ZF requires an infinite collection of axioms, you can choose one of the finitely axiomatizable set theories. From a strict formalist viewpoint, we can consider mathematics to be a game played within this syntactic world, in which we identify "interesting" sentences and provide proofs of these sentences from the axioms. We may or may not choose to ascribe any meaning to these sentences, but the subjective notion of which sentences are "interesting" to prove would likely require that we do so. Within this theory, functions that exist ultimately correspond to provable sentences of the form ALL x. EXISTS ! y. P(x, y). There is nothing "ambiguous" or "problematic" about this. If we do not wish to permit every such sentence as corresponding to a "real" function, we are perfectly free to impose additional reestriction, such as that the predicate P(x, y) is recursive; a notion that (since Goedel) is already definable even if we have available only basic amount of arithmetic. In thie above setting, sequences are merely functions on the set of natural numbers, and as Ihave already said, functions just correspond to sentences of a particular syntactic form. There is nothing "ambiguous" about this. We do not have to do "an infinite amount of work" to verify that a particular sequence is Cauchy -- all we have to do is express that statement as a sentence in the logic and give a proof of it. It is not a requirement that we do an infinite amount of calculation to check it. Yes, of course there is (most likely) no decision procedure that is capable of inputting the specification of a sequence (i.e. as a sentence of a particular form) and determining whether or not that sequence is Cauchy, but the same thing is true of many, many other properties of interest in mathematics, so what is the big deal here? You are welcome to reject the notion of provability associated with ZF, ZFC, first-order logic, or any other formal system and only accept results that are proved using a system that somehow satisfies your algorithiic or finiitistic requirements (I have been watching a number of your videos, trying to get an understanding of exactly what you would reject and accept, so far without success). However, it is misleading to say that standard mathematicsl concepts, such as real numbers, or Cauchy sequences, which have perfectly good definitions within a theory such as ZFC, have "problems" or are "ambiguous". The problems lie only in the intuitive interpretation that one might try to ascribe to formulas in such a theory, not with the theory itself. I anxiously await a video in which you present a formal system that you do find acceptable. I think the amount of mathematics you will be able to do in such a system will be quite limited, but it would be interesting to be shown otherwise.

Hi Dr. Wildberger. Thank you so much for showing this construction. It is quite interesting! I assume that you are more comfortable with infinite boxes / multisets representing the sum of an infinite number of things than "traditional" infinite sums since the former can be written as a collection with an indication that its elements go on infinitely, while traditional infinite sums directly invoke an operation being done infinitely many times. I'm curious if you feel that this is a meaningful distinction between the two perspectives. One could argue that box arithmetic is just another way of conveniently "disguising" the fact that we've constructed an object which invokes an operation being done infinitely many times (using box addition an infinite number of times on finite boxes to construct an infinite box), similarly to how you claim modern mathematics has constructed an entire language to disguise its weak logical foundations.

In addressing the expansive representation of the zeta func, the corresponding boxes are also of infinite extent. The box representation seems an unnecessary overhead. It's an added level of abstraction. Much like Box arithmetic we do of-course represent irrationals in some "finite" form all of the time in mathematics. But without spelling the decimal digits out. The representation of irrationals may not be much of an issue after all. In a computer the situation is rather different. The level of abstraction in a computer is rather crude, a machine can only work with binary data (we know this). Until a computer is able to represent mathematical abstraction in the way people do, this situation isn't going to change. So there isn't anything inherently wrong with mathematics. You could well argue that with the advent of AGI, the task of abstracting natural laws may be a step close. The news is that even that may be impossible without an understanding of the laws themselves. Having said that, mathematics may yet not be the universal language that we believe it to be because it is still cast in terms of a formulation of human understanding and perception. Where there is perception, there is ambiguity and where there is ambiguity there is subjectivity.

This is really really interesting. On the second watch, now both 240 and 241, I started to get a little peak into FIA having some confluent connection with the deeply fascinating simplicity property of Stern-Brocot type constructs that numerically generate coprime fractions in order of magnitude. I first heard of the simplicity property from Wildberger, of course. His presentations of SB-tree are also really worth a watch! :) The simplicity of coprime fraction a/b is 1/ab. When summing together (in standard field arithmetics summation) all new mediants of a new generation sum up to 1/1 in the reduced form. If we put the generations on same row instead of the tree form of only new mediants, the sums of the rows become 1, 2, 3 etc. What are the non-reduced forms of those rows?!

Dr. Wildberger you are a real unsung hero. Your videos are phenomenal!

Dear Professor, Fantastic as usual ! At 23:38 the Harriot/pascal array presented is often used as an introductory example to Rordian Arrays (for the pair of power series (1 / (1 - X), X / (1- X)). So I was wondering : have you ever studied Rordian Arrays ? Do you have any interest in them ? These objects seem to me quite close to the general "mathematical attitude" one gets of your work. Thank you, again for your great content !

nice video but i have to say that your last formula does not "solve newcomb's paradox". it just describes one of the two major approaches in more mathematical detail. The classic numbers of 1.000.000 and 1.000 are intentional to make it obvious that 1 boxing gives you better results which is what your expected value calculation underlines. however, the whole point of newcombs problem is one boxing is mathematically better while making no causal sense. By the time you whip out the calculator the predictor has already placed the money in the boxes and can't take it out anymore. so you might as well take it all.

Professor Wildberger, thank you for your invaluable mathematics videos. They're incredibly informative and rational---pun intended. If I may offer a suggestion to enhance the viewing experience: consider improving the lighting. A simple ring light or softbox in front of you could brighten your face significantly. Adding a fill light on the opposite side would reduce shadows. Balancing the lighting between you and the whiteboard, and perhaps stepping slightly forward, could also help. These small changes could make your excellent content even more enjoyable to watch. Thank you again for sharing your knowledge!

I have this functor in my pocket: ☐ → {} ... _et voila!_ standard Set Theory less AoI. 🤣

I am a fan of Alfred North Whitehead, but this makes me think that Whitehead and Russell crippled themselves by trying to work with set theory, when they could have worked with msets.

Question, just to make sure I get the boxes counting: N(A)=? where A is an empty box. In other words, N(Z(B)) = ? where B is any given box. My guess is that it's equal to 0, am I right? Thanks so much!

isn't this just the Peano arithmetic? Numbers are sets... call them boxes...

Professor Wildberger- you were mentioned on Joe Rogan the other day! Eric Weinstein took the time to gently (extremely gently) refute the basic mistakes made by Terrance Howard (an actor who has taken a lot of well-deserved criticism lately regarding his maths related assertions). He mentions you in passing (Mr. Howard does), and- more or less- as an appeal to authority during an early back and forth. Again, Eric was trying hard to help Terrance understand his mistakes, and so its not as if your ideas were being debated- but I just thought it was cool to hear you come up!🌞

according to the Standard Wisdom, no further new operations are available to play with...

Thankyou

Thanks!

"An ongoing version" Oh you MEAN repeat calculations up to infinity if you wanted. Oh that's right, you do not believe in infinity, sad.

May I suggest, using more conventional notation, perhaps with other symbols for addition, multiplication etc, instead of boxes? I work mostly in lyx (latex), and using symbols that is there, instead of boxes, would be beneficial, from a logistical perspective. I love the idea of box arithmetic, just thinking about the logistics, and notation aspects.

Yes, MF241 makes more sense now.

MF240?

Schools can't teach mathematics. You'll find a variety of approaches and corresponding mistakes in the material presented to students. Then you'll find undergraduate schools competing against each other with inconsistency. Having reached the top I've seen the worst. There really needs only one textbook on foundational/graduate level mathematics.