MF 240: Euler's Product for the Zeta function via Boxes I | Box Arithmetic | N J Wildberger
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- čas přidán 27. 07. 2024
- Box arithmetic allows us to reformulate some aspects of number theory and put them into a more combinatorial / data theoretic framework. In this video we consider the Euler product for the Riemann zeta function.
We review some basics of Box Arithmetic and counting operations and look at various pleasant algebraic relations using both powers of multiplication and also powers of the next Box operation, which is the caret operation.
Our reformulation of the Euler identity centres around what we call the Fundamental Identity of Arithmetic, a combinatorial analog of the Fundamental Theorem of Arithmetic, concerning the essentially unique factorization of a natural number into primes. We then introduce the Sum operator on Boxes, and show how Euler's identity can be re-interpreted in the Box Arithmetic world.
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 Sum Operator S
29:35 Summation Laws
32:40 A Naive Application
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Dr. Wildberger you are a real unsung hero. Your videos are phenomenal!
Thank you!
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!
Thanks very much for the good suggestions
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!🌞
I wonder what's wildbergers opinion of him is..
@@stevves4647I was a little worried to even mention it to him, but Terrance Howard certainly means well, and at least he's interested enough to listen (at least recently) to some criticism, and instruction. I was just excited to hear Professor Wildberger's name in the wild!🙂
I noted that too, and that's the spot where I understood what Howard meant by "1*1=2".
The metaphor "1*1=2" is the in-your-face way of stating the same criticism against analysts that Berkeley described as "The ghost of the vanishing magnitudes."
Weinstein the physicist got scared at that point and directed the discussion to less explosive areas.
The subtext of the discussion was the coherent ontology of pure mathematics. Weinstein was not there to discuss our hot topic directly, but to offer his perspective of academic sociology of physics and mathematics, and to help Howard to steelman his arguments first on issues of tone instead of content.
As Weinstein said and as we know from experience, it's not a nice crowd, and don't respond well to "Real numbers don't exist! You are all wrong!" With time and experience we can learn to steelman our arguments and win each single debate. However, winning debates is not psychologically sufficient, to win also hearts we need to be able to show also genuine kindness and something foundationally better instead of dearly held absurd belief in real numbers. Wholesale rejection of philosophy has not been helpful rhetorics in that respect. Of course that's not what Wildgerber has been doing in practice, his argumentation is often deeply philosophical. The rhetorics is mainy targeted against Hilbert's post-modernism and Cantor's paradise/joke, what Wittgenstein saw as and called as sophistry of arbitrary language games.
Thanks for that!
@@santerisatama5409 i only listened to the discussion for a couple of minutes but felt the problem stems from mathematical foundations that are not necessarily based on reality but can still provide useful insight (such as imaginary numbers) vs mathematics that function mainly as an intellectual curiosity (at least for now) such as topos theory; with howard's ideas falling into the latter.
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.
In LaTeX you can use curly brackets instead of boxes, "{{},{{}}}". Oh wait... that'd be standard Set Theory. 🤣 (Wildberger is using set theory, but excluding axiom of infinity. It is still set theory under the hood. Dropping an axiom is just limiting your options. Adding the axiom does not necessarily generate inconsistency.)
multisets / boxes remind me of types built up from type constructors for product types, with higher kinded types as the higher orders of multiset/boxes.
Thanks!
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.
The eigenform of box inclusions is not unnecessary overhead. You might get such appearance from looking only the side of black boxes and forgetting the red boxes. We need both sides to try to tackle the issue of mathematical inverses in a coherent manner.
We have seen only WIP glimpses of box arithmetic so far, and have no idea how far Wildberger himself is in the construction, and/or where he sees it going.
The mset inclusions are by definition a mereological theory, and we have desperate need to return to coherent mereology after the set theory experiment. What kind of mereology exactly, can't say and I keep wondering.
What I can say is that this is very INTERESTING!
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!
N(A) would be what you get when you replace all the elements with an empty box, ie 0. So you would get [0]=1
I think this is missing from the box arithmetic playlist
Yes, MF241 makes more sense now.
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 !
Thx for the suggestion, I don’t know about those arrays of Riordan. Will chase that up!
Heard about prof. Wildberger first time by Howard. Cool stuff here. Babylonian math, Riemann Zf, infinity critics etc. Is sqrt2 lesson any good?
Prof. talked about new foundations. Is there any literature to read more about it? Central concept Box Theory?
Please watch more videos, there is my book on rational trigonometry and an online course called algebraic calculus one at OpenLearning
Thankyou
Has anyone made a language parser for box arithmetics yet so far?
Hey, I've been watching a ton of your lectures, and wonder what your answer would be to my question. Are real numbers just the tangent space to complex projective space?
There is no valid arithmetic of “real numbers”. The objects are not properly defined, and neither are the operations.
Sorry for the wall of text, I got excited. You've become kind-of a hero to me despite my not-really-knowing how much I agree with the "controversial positions" which you propose I truly stand behind.
I understood that you find the real numbers problematic logically, I was just wondering if my interpretation of the tangent space of projective complex space matching closely if not identically with what is generally meant by "real" numbers is accurate.
I figure you're one of the best people to ask this question to, as your ability to apply skepticism is something I find laudable, and I was hoping that your abilities could find errors in my interpretation if there are any, not just a rehash of your issues with real numbers. I appreciate your input and thanks for your time.
Its honestly very refreshing to see someone who embraces skepticism philosophically and actually knows things. (beginning tangent ha, tangent.) I love Matt Dillahunty (but he doesn't really talk about anything outside philosophy or some science which intersects with many concepts related to philosophy) and have listened to an absurd amount of content by him but his focus is entirely on things which don't really flex his skeptical muscles. The topic of whether or not people are making valid and/or sound arguments in academia is far more interesting to me than if they're doing it for their preferred random beliefs.
I have no idea if you know who I'm talking about but the dude's helped me not feel alone in my embracing skepticism and trying to apply it to everything. But eventually I had my fill of philosophy and moved over to maths and found you about a year or so ago while continuing my mathematical explorations post-college.
I didn't get my degree for personal situational reasons but I've continued learning every single day, and watching a ton of lectures from as many worthwhile sources I can find and trying my best to apply my knowledge and test myself to find as many interpretations of the same things as possible to have a more well rounded view of everything I learn. That's what lead me to this question I have for you.
(more tangent below)
I've been playing with representing as many algebras and geometric forms, and whatever else I manage to study, but in a limited calculator like desmos, so I can test how much I really know. Limitations breed ingenuity. I also go to of my way to try and create art out of every new concept I learn. I've been playing/learning on desmos so long I know it doesn't mean anything but I've got literally 3,500 ish graphs saved. Yeah most of them are trash but it's been an amazing journey.
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.
After the failure of PM, Whitehead chose the point-free path (ie. coordinate independent), and Russel passed his magic wand to Wittgenstein. Wittgenstein's actual mathematical teaching teaches 'beginners mind', how to have fresh insights to foundational questions and to become a foundational thinker.
Msets are a good approach to number theory, the most coherent approach as I can see when we define numbers as tally operations.
As msets are not ordered, they are naturally "entropic" in the sense that the potential ordered information of the same structure gets hidden from the numerical interpretation of the structure.
This observation is not a small thing. If numbers as such are inherently entropic, that offers a natural explanation to the measurement problem of QM.
Yes, more generally modern maths has sadly limited itself with this delusion that set theory is the proper foundation for all of pure maths. A definite NO!!
@@njwildberger Yes, this box arithmetic exercise seems to establish that. On a rather eccentric further question, is there a useful relation between Spencer Brown's calculus of distinctions ("Laws of Form") and box arithmetic? While we are on eccentric stories, I am a fan of Jaakko Hintikka's new logic, with two different kinds of logical negation. He believes in the axiom of choice, which worries me, for the reasons that you have taught me. But I like very much his idea that a sentence is not valid if it has only syntactic validity without semantic validity.
@@christophergame7977 From my perspective, there is a kinship between Spencer Brown and Box arithmetic. There is a kinship between Box Arithmetic and Louis H. Kauffman's idea of iterants, and Kauffman gets his main foundational influences from Spencer Brown.
This said, I don't think we are yet in the positions to formalize the kinship. The current situation is more like growing and nurguring a new forest of many distinct seed and sapplings that may complement each other in coherent manner.
Maybe somebody following intently the creative edge proceeding on many fronts might be able to cook together a homotopy theory or something like that. For me the nlab jargons are way too obtuse and I don't think time is ripe for such effort. I'm more interest in seeing where Norman's construction is leading to and trying to get what intuitive meaning I can gather from it, and that's already challenging enough.
@@christophergame7977 That said, we can clearly see that Box arithmetic is a kind of what Kauffman calls "Eigenforms".
In that respect I think that the term "fixed point" is rather unfortunate historical terminology for a constant in change, and I use the term Y-combinator instead. The point of the thought experiment of "Archimedean point" is that there is no such point.
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)].
or (1 + a)^4 should translate to [0 1]^4 = [0 1] * [0 1] * [0 1] * [0 1] = [1_(0), 4_(1), 6_(2), 4_(3), 1_(4)]
or (1 + a_1)^4 should translate to [0 a]^4 = [0 a] * [0 a] * [0 a] * [0 a] = [1_(0), 4_(a), 6_(2a), 4_(3a), 1_(4a)]
I don't see why you need the 'caret' operation for the binomial theorem as a^4 = a*a*a*a is not equal to a^{\^4}=a\^a\^a\^a.
A series of addition can be translated to a multiplication: 1+1+1 = 3*1 or 1+1+1 = 1*3. However, a series of multiplication cannot be translated to the defined caret operation: 1*1*1 1 \^ 3 = 3 nor 1*1*1 3 \^ 1 = 3. Hence, your definition of the caret operation should probably be called differently to not confuse it with the usual exponential: 1*1*1 = 1^3.
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
I have this functor in my pocket: ☐ → {} ... _et voila!_ standard Set Theory less AoI. 🤣
You have to ask yourself why a multiset is not a structure definable in standard set theory?, even ZF. The answer is that multisets are definable in ZF, so it was a dopey question. It is damn hard to escape setization. Many have tried. None I know about succeed. Not even Category Theory and HoTT alphas. (This is not to say ZFC is true and consistent in a platonic sense beyond Tarski, but it _could_ be.)
To my understanding, box arithmetic does not have subsets (which very peculiarly are equivalence relations, yet somehow "distinct" superset-subset relations). Boxes are just inclusions, which set theory confusingly calls "strict subsets" instead of "subsets", if it bothers to make that distinction at all.
I must admit that the moving of exponents between nested boxes seems pretty wild to me (in a good interesting way), very hard to mereologically internalize. Where/how/when are the logarithms then?
@@AchrononmasterWhy do you claim that multisets are not definable in usual set theory? Can't one just identify them with the counting of how many times an element appears in a multiset? In set theory, these correspond to counting functions, i.e. functions from finite sets to the natural numbers (including 0). All these notions are well-defined in set theory. If one identifies two such functions if they differ only in parts of their domains where their value is 0, then one can encore the same information as in multisets.
"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.
"up to infinity if you wanted"???
You write as though you regularly have counted to infinity. How long did that take?
@@fluxpistol3608 Peculiar interpretation of that comment. How awesome is it that we live in time we do? Your thoughts? I was searching my comment, many many times: I am searching for the noun "I", or a grouping of letters that create a grouping of phrases that a person could maybe parse together some idea that: Sometime in my life I was dumb enough to start something (counting to infinity) that I knew would never finish. All good here, maybe English is not your first language.
@@Dystisis 🤣 Hiliarious when I typo (and leavespelling errors), I love it! "up to "dat real real that is abritary close to the unspeakable '\inf' "
@@DOTvCROSSa lot of dodging but no actual ability to answer one simple question. Interesting. Don't worry, tends to happen with passive aggressive individuals. They hear what they want and not what's being said and have a tendency to attack phantoms that don't exist. You don't know how to communicate like an adult in good faith so you can return to the kids table and I'll answer the question for you. It's no. You've never counted to infinity. It's not something that exists to be counted to and the video doesn't suggest anything like your passive aggressive interpretation insists. Check your ignorance at the door next time you watch a video to learn something.