MF 241: Euler's Product for the Zeta function via Boxes II | Box Arithmetic | N J Wildberger
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- čas přidán 27. 07. 2024
- We continue exploring elementary connections between Box Arithmetic and Number theory, centered on Euler's product formula for the Zeta function. The Fundamental Identity of Arithmetic that we introduced in our last video plays a big role. But now we introduce some interesting variants: when we replace the numbers occurring in this Fundamental Identity with suitable powers of them, including also negative powers, and then apply the Sum operator as before.
What makes this works is a crucial multiplicative property of taking powers.
We make contact with another of Euler's famous formulas, for the sum of the reciprocal squares of natural numbers.
In this talk, we necessarily extend our usual finite careful mathematics with unbounded analogs. But that means that we need to be mindful of the exact meaning of the statements we make. Our general orientation is that the meanings of expression can be reduced to finite expressions by using the natural ordering of the Natural numbers to apply truncation consistently.
Video Contents :
00:00 Introduction
2:42 Powers And The Caret Operation
5:25 Applying f To The FIA
10:45 A Variant ( replacing with cubes)
13:25 Another Variant (replacing with reciprocals)
17:51 Suggests An Identity
19:51 Squared Reciprocals
21:39 Examples
24:32 A calculation
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I love the distinction between invention and discovery. "Brought into being" as opposed to, "it's 2".
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
Fascinating!
What is the link between fractions of the type 1/1, 2/2, 3/3 etc. and shifting the exponent between nested boxes?
Coprime fractions a/b -> a/(b-1) seems a very fundamental relation. Also, perhaps we shouldn't think of 1/1, 2/2, 3/3 etc. totally identical with each other. The tautochrone property of cycloid can be given a simple anatomy by interpreting the non-reduced fractions as different positions on the cycloid arc, and the reduced form 1/1 as the same duration from each position... to where when? Between the boxes?
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?!
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.
I don't think that we still have a full story of what we are actually doing, when we are converging a series towards a rational number.
We can do some chains of deductions that seem intuitively valid, or at least very interesting, while not yet if ever in possession of a full story. That's why we are studying foundations and experimenting with various foundational constructions. The whole process as such is very interesting and... beautiful.
Heh. You got a mention by Terrence Howard in the Joe Rogan/Terrence Howard/Eric Weinstein interview.
Fun times!
There is an interesting link here between this video and your previous one about limits. If we forget about PI from Euler and think geometrically about a circle, we could perhaps see two types of convergence, one using sums and another one using products where primes would play a role of bridging them together (not too slow, not too fast)
We see the series with Power=1 does not converge. However we see the series with power=2 does converge. So the deeper question - Is there a value of power(index) between 1 and 2 that is the limiting value for convergence of the Euler series/product.
Thankyou
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!
I believe MF 240 is missing/not public?
Norman, we are missing MF240 ... when I saw the first slide I thought, "And where is this coming from?!?!" 😂
I watched the previous episode ...
@@tomholroyd7519 yes, it is available now!
good..
according to the Standard Wisdom, no further new operations are available to play with...
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.
What is your critique on Homotopy Type theory as an attempt for the foundation of mathematics?
MF240?
Thanks Dean, that is now up
isn't this just the Peano arithmetic? Numbers are sets... call them boxes...
These not sets and their subsets. The boxes are inclusions aka "strict subsets", which can't be identical witth their supersets, as ordrinary subsets are. Set theory can't do proper mereology because ordinary subsets are not inequivalences of their supersets, and mereology is defined by Euclid as inequivalency between part and the whole. The common notion 5: "The whole is greater than the part".
Also, in Wildberger's definition of finite datatypes, the word "set" has clear distinct meaning as a datatype, as do msets etc. For axiomatic set theories, "set" is an "undefined primitive notion" and the definition of set in the "naive set theory" of Cantor leads to the Russels paradox.
@@santerisatama5409 en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
@@Kraflyn It's worth noting that the von Neuman's construction is an eigenform of Dyck language.
That doesn't yet mean that ZFC is actually consistent even with the von Neumann construction. Set theorists are in the habit of lying a lot, mostly due to self-deception and cognitive dissonance.
Von Neumann's construct has been an important step forwards, and I happened to stumble on an even better way to construct the whole number theory from Dyck language. having learned math mainly from Wildberger.
@@santerisatama5409 Yes but.... the box finite arithemtic is no different than the finite Peano set arithmetic... No infinite formulation is consistent. Just consider division by 0. If a theory cannot get rid of division by zero, the theory is flawed in the end. But that is not my objection... In the box arithemtic, just keep everything finite, and it is exactly the same as the standard peano arithemtic. Which does extend to infinite measurers.
@@Kraflyn My perspective is very different. If a theory can't coherently handle 1/0, the theory is very incomple and very flawed.
Stern-Brocot type constructions start from 1/0 as the holistic direction, and are thus more coherent that the bottom-up constructions.
Peano approach does not have coherent mereology. Box arithmetic is based on the inclusion relation. It's a world of difference, and we experience world as inclusive mereology, as participation.
This is WIP, and I'm slow to intuit and internalize the deeper vision and implications of box arithmetics. Way too early to draw any definitive conclusions, and if we discard this based on false conclusions, we might miss something great.