Q Series via Box Arithmetic | Math Foundations 239 | N J Wildberger
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- čas přidán 27. 07. 2024
- We have a look at the interesting topic of q-series from algebra / analysis / combinatorics / number theory from the point of view of our new "box arithmetic" which prominently utilizes anti-boxes along with boxes. This is a chance to get some more familiarity with this curious new arithmetic in which the role of "nothing" is different from what we are used to.
To interpret Euler's pentagonal formula we introduce an unbounded extension of the current Box arithmetic.
This new way of thinking appears to have obvious computational advantages, in that the notation is capturing some of the arithmetic more efficiently. It also finesses the need for philosophical discussions about "variables".
Video Contents:
00:00 Euler's Pentagonal Number theorem
3:10 Box Arithmetic
7:18 More arithmetic with boxes
10:51 Unbounded extensions
15:10 An identity of Euler
21:44 Notation for multiplicities
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Norman, thank you for your teaching.
Digging little deeper in the issude (including watching again MF202, MF203 and MF204, I'm starting to think that Euler's original mistake was to map the coprime fractions of the general continued fraction into nats, and thus hide what is going on under the hood.
This lecture has made some good progress in lifting the blanket.
Very interesting a and timely for me personally. Just a reminder I don't think this video has been placed into a playlist yet.
Actually I think it has been (after checking your page), but I don't see it come up in the vid desc on mobile for some reason
I appreciate the extra lighting!
Respected sir , I am an MSc student in Mathematics from India . It is a humble request to you to discuss a little bit about measure theory (Lebesgue and Product measures , Carathéodory extension and Radon Nykodym derivative) . I feel it really difficult . It is very difficult in fact as i can see . To much rigorous .
I watched your all videos on algebraic topology and it was really amazing . I have algebraic topology in next semester . It will certainly help me . Thanks sir .
I am sorry but I do not think that classical measure theory is a correct subject in mathematics. Your difficulties with this subject is fully justified.
Cool idea!
I haven't been a regular viewer but immediately on seeing the box arithmetic and your use of void, I *must* refer you to have a look at a kindred spirit in William Bricken's Iconic Arithmetic and the James Algebra (a demo of it here!) czcams.com/video/CyeZX21auq0/video.html
I would be delighted to hear your thoughts, or arrange a conversation for you with those who've investigated this much more deeply e.g. Louis Kauffman or William Bricken!
I've been learning about the discourse surrounding Laws of Form and iconic formalisms for over a year now, and I'm super excited to see (presumably) another source, this time from you! Makes total sense given your interest in mathematical history having made clear the things we take for granted in contemporary math.
Can't wait to queue up your box arithmetic and do a comparative analysis between it and what I've learned so far about James Algebra ^~^
Forgot to mention - I think you'll especially find the investigation of the additive imaginary J very fascinating, where J + J = 0; J =/= 0; and J can be described as log(-1). In Volume 3 of Iconic Arithmetic, Bricken makes some very interesting forays into a purely reflective trigonometry without angles - rather, partial reflections.
@@weavermarquez1271 Sounds fascinating -- thanks for the links and references
Thanks for the link, very interesting. I've been a fan of Louis Kauffman for a long time and had the same reaction that it would be great to see a discussion between Wildberger and Kauffman.
My own main interest has been process oriented instead of object oriented (cf. funtional programming vs. object oriented programming) and for process notation I've been using < 'increases' and > 'decreases', which is interesting in relation to James Algebra notation utilizing Dyck pairs [ ] and ( ). The basic operation of Stern-Brocot type structures, namely 'concatenating mediants', has been revealed to be very fundamental, as Wildberger suggested and predicted.
To understand this better, let's allow ourselves to give Void an anatomy of distinctions based on basic concepts of theory of formal languages. When we concatenate the mediant of blank characters of white space, the mediant is the concatenation! There is a very fundamental distinction in the "void" of white space, even though we don't usually see it as attenction is focused on written characters instead of the background. I just realized couple days ago that in terms of formal language theory, Dirac's delta function is the concatenating mediant of white space! Even though by themselves blank white space and concatenation can be considered void without form, in relation to each other and to the whole context of formal language, they form a distinction. Expressions and < > are also visually clearly distinct.
When we wish to increase distinctiveness and do more formal language than Dirac delta/concatenating mediant in white space, the natural choice is relational operator < for more than white space.
@@weavermarquez1271 Log(-1) and/or hyperlogarithm(-log) as the inverse of hyperoperations tetration, pentation etc. is very interesting, and IMHO we need to be very clear that in this case were discussing mereology and mereological foundations of mathematics, where we start holistically from the whole and take nesting algorithms as more fundamental than additive algorithms. In the holistic approach It's better to stick with pre-numeric formalism (e.g. boxes and arrows) and derive number theory as a side effect.
Worth also remembering, that in Euclid's Elementa the concept of angle is not limited to straight lines and also ovals have an angle. Curved angles can be described in terms of more-less relation, but not given a definite numerical value. At least in some case it seems possible to give them description in terms of a repeating forms of continued fractions, which are by standard definition constructive numbers.
Norman, this is really beautiful. I share though your hesitation in the definition of the "open to the right" box. Would not be simpler to define a Big-O box to indicate a box that contains terms of higher order than some fixed `n` ? (perhaps equal the degree of the partial dotted expression?)
@acortis Yes I agree that we will have to be careful about such "ongoing" structures, much the same way as we should be careful about formal power series and related maths.
@@njwildberger In this regard constructing box arithmetic would benefit much from more clear conceptualizing of it's mereological aspect. Are we constructing.
A) additively from parts to whole in the additive direction
B) nesting from whole to parts in the logarithmic direction
C) both
D) neither
Recursion as such can be neutral at least in some aspects in terms of mereological direction, whether we are composing or decomposing. If and when the construction proceeds to fractions, perhaps then the mereological character becomes more lucid, as fractions intuitively express a mereological relation. .
Norman, I made an interesting discovery; perhaps you can tell me if it's well known. We generalize the geometric series into a multiseries in an unbounded number of variables with all the coefficients unity:
M[u1,u2,u3,...] = sum_{k_j >= 0} u1^k1 u2^k2 u3^k3 ...
Now we do what I call the Schroeder move and it turns out:
M[v,v^2,v^3,...] = 1v^0 + 1v^1 + 2v^2 + 3v^3 + 5v^4 + 7v^5 + 11v^6 + 15v^7 + 22v^8 + 30v^9 + 42v^10 + ... = sum_n p(n) v^n
where p(n) is the partition function. If we want the actual partitions, we do:
M[u1 v, u2 v^2, u3 v^3, ...] = 1 + v^1( u1 ) + v^2( u1^2 + u2 ) + v^3( u1^3 + u1 u2 + u3 ) + v^4( u1^4 + u1^2 u2 + u1 u3 + u2^2 + u4 ) +v^5( u1^5 + u1^3 u_2 + u1^2 u3 + u1 u2^2 + u1 u4 + u2 u3 + u5) + ...
Thanks. Computation guys and others would also very much enjoy ASCII representation for notation of Box arithmetic. I don't have any definitive suggestion in that respect, what comes to mind first are different notations of Dyck-language which allows nesting hierarchies, e.g. [ ] for black and ( ) for red. Another, perhaps more novel and interesting option could be a pair of Dyck language symbols and "inverse Dyck-language" pair, ie. ( ) for black boxes and ) ( for red boxes. A further ASCII distinction that can be utilized is concatenated pairs () vs. pairs non-concatenated pairs which include blank character of white space ( ).
It's very interesting mathematical question as such, which kind of Dyck (and inverse Dyck) language would be mininal requirement to represent Box arithmetic in non-ambiguous way.
Minimal languages based on notational chiral symmetries is a very interesting topic also in relation to quantum theory, because e.g. relational operator forms < > and > < are invariant whether read from L to R or R to L, and thus satisfy the reversible computing condition of QM. The chiral notation of the superposition of both < > "outwards" and > < "inwards" (which as Boolean NOT relation also satisfies the reversibility condition) offers a more expressive notation than the standard O-I notation for qubit interval, which as such is not visually symmetric and reversible, as the symbols O and I are already as such palindromic.
I think you mean unicode representation. ASCII only has 1 byte per character, thus 256 options and is quite well-established and set since decades ago. If you're on a unix machine you can type "man ascii" in a terminal (or type that into google) and view the manual.