Let's crack the Riemann Hypothesis! | Sociology and Pure Mathematics | N J Wildberger
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- čas přidán 27. 07. 2024
- Modern pure mathematics implicitly assumes that we are able to perform an unbounded, or infinite number of arithmetical operations in order to bring into being "real numbers" and "values of transcendental functions" such as "exp" and "log". We use these superhuman powers to reconsider J. Lagarias' equivalent reformulation of the famous Riemann Hypothesis concerned with the zeroes of the so-called "Riemann zeta function".
This allows us to resolve this famous conjecture with a computation, which is essentially similar in difficulty to the calculation of cos(7) or zeta(5) or exp(H_6) where H_6 is the 6th Harmonic number.
Fellow pure mathematicians: are we not able to think clearly about what we are actually able to do when it comes to mathematical computation? We don't need to pretend, just to obtain "pleasant results" that distort the real nature of the mathematical world. Let's rather replace philosophical justifications using obscure axiomatic frameworks with simple alignment with what our computers can do.
Video Contents:
00:00 The Zeta function
8:28 Analytic continuation
14:53 Equivalent formulation of R.H.
20:48 Lagarias' inequality and cracking the R.H.
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Infinite sums are not really infinite. They are normal sums but with a limit with n->infinity. And limit means that the value is converging, so you have to have a proof that for any precision you choose you can find an n that will satisfy the precision. So you are never asked to do infinite amount of work, you are asked if there is a bound of the amount of work that you have to do in order to reach a particular precision.
Right - actually, if a transcendental number does not really exist, how can it be claimed that there is an approximation of it? How can the process of calculating it give us a closer and closer approximation of a number that doesn’t exist?
@@johnh7411 In my opinion there is no need to make such a claim.
One observation that can be made is that, for each finite number K, one can find another number N such that, doing the finite computation - limited to M steps - with M >= N finite number arbitrarily chosen, the first K digit in the decimal representation of the results obtained will always be the same. There is no need to admit the existence of something to approximate in the first place for this observation to be proved, right?
A bound over m steps is still just a bound. It is imprecise and not a real answer. This is the whole point that he is making. We are looking for a precise functional description to the problem rather than a traditional approximate one
We used to not know how to take the square root of a negative number, or divide by zero. Some stubborn mathematicians said, "I'm doing it anyway," and created complex numbers and projective geometry. A point at infinity? No problem. A whole line of them? No problem! Infinitesimals turn out to be very finite 2x2 matrices, dx = [[0 1] [0 0]]. No problem!
Rationals are good. It's also interesting that if you start doing any computations with them, after only a few iterations the numerators and denominators become ridiculously large. It's almost like complexity is being stored in pairs of numbers and it doesn't fit well. Also reducing to lowest terms all the time turns out not to be an easy problem. It's like it wants there to be a higher dimensional system
How can we reach a non-existing end? You cannot even come near as there will ALWAYS be more steps ahead than behind us.
Wonderful video! Thanks for sharing so much fantastic content, it has genuinely increased my fascination and understanding of math.
I am not sure if this is a serious video or meant as satire.
Mathematics is not an empirical science. It is a formal science that uses agreed upon axiomatic and logic to derive conclusions that are valid within the chosen axiomatic and logic. Math is agnostic to the universe being discrete or a continuum under Planck scale or bounded or unbounded at large scale. Put differently, \sqrt{2} exists mathematically (in the sense that is a well defined concept that can be distinguished from any other number in mathematics) regardless if there is any physical object that represents the 'concept' of \sqrt{2}.
Of course, one could object to the utility of the mathematical \sqrt{2} if no physical counterpart exists. But here is the beautiful thing. The vast majority of these purely mathematical concepts have implications that are empirically observable (and consistent with the abstract mathematical formalization) in reality, regardless of the true nature of the universe being discrete or continuous, bounded or infinite. For instance, it does not matter if the imaginary unit i 'exists in reality', it sure is good for modelling the Schroedinger equation and making predictions that can be validated empirically. Or, it does not matter if the exponential function 'exists in reality' in order to use it to model the unrestricted growth of a population that doubles after a rational amount of time has passed. Or, it does not matter if the notion of a converging series 'exists in reality' if any measurement performed in reality is consistent with the potentially 'virtual' value of the series.
Lastly, your claim that if one accepts the definition of the exponential function, then one accepts that the RH is solved by saying the RH has the truth value of the product P(n) is logically false. If one 'accepts infinite objects' in mathematics, such as the exponential function, one also accepts that the product of the P(n)s is a well defined object. Settling the RH is equivalent to finding the value of this product, namely deciding if it is 0 or 1, not to this product being well defined. This take of 'proving RH' is about as genuine as saying the equation 'a=a' uniquely determines the value of a.
Thank you!!!
No, it’s no satire. He did the same with the twin prime conjecture. I think it’s Alzheimer. He should quit the job before the university does it for him.
@@thomaslangbein297 I think people should be free to study what they want. This radical form of constructivism has existed before in math and, since it is essentially an axiomatic choice, it is not objectively wrong. What is very wrong is advocating for an unusual axiomatization by stating demonstrably nonsensical things.
He reinvents Xenon’s paradox 23 centuries later. I don’t think he needs your advocacy. The British maths professor David Joyce classified his work as an ”interesting“ view on mathematics. This should tell you everything.
BTW: I don’t forbid you anything. Same with flat earthers. Believe in the Flying Spaghetti Monster if you like. No objections.
Over and out
@@thomaslangbein297 Oh lol, I am definitely not advocating for this. As you can see from my first post. Just saying that, if someone likes this approach, they can spend their research time investigating it if they so choose. As long as they don't come out saying that they've proved RH by just reformulating it in a trivial manner.
This is kind of an interesting meta-perspective on mathematics. You can view classical mathematics is taking place in this "infinite universe" you mention, where you have infinite space, infinite time, infinite patience to perform these computations. In my mind that universe is purely philosophical, but the real meat and potatoes of mathematics consists of "pulling" results from that universe into our own: expressing the apriori infinite using something finite. Every interesting breakthrough in mathematics tends to consist of expressing or breaking down a seemingly "infinite" amount of things using finite structures.
Dear Professor, Chuck Norris counted to infinity - twice.
@11:100 "assume we can *_do_* ..." is not the same as "think about". Mathematics is one mode of thought process we are capable of, there is no philosophical need to assume we can *_do_* it physically. Remember, if our system of thought in such modes turns out to be inconsistent, it's no big deal, in fact it is a marvelous thing, since we'd then know that system was *_not_* mathematics.
Thought divorced from computational or observational verification is called: philosophy, poetry, or just wishful thinking.
If you can't do it, you can't even think about it to begin with.
@@njwildberger You seem to have the mistaken impression that computational verification is the same as empirical computational verification. It is not.
I am sure you would agree that addition defined on any finite set of natural numbers is a well defined object. If I pick two arbitrary primes that are each 20 digits long, I am quite sure you would have no issue saying what their sum is. However, in all likelihood, no human will ever perform an experiment where he takes two sets of identical objects, one with the first prime many objects in it and the second with cardinality equal to the second prime and directly count that the union of the sets has cardinality the sum of the primes.
If this is not what you mean, then you must accept that computational verification is contingent on an axiomatic system and it equates to formally checking an equation within the calculus of that axiomatic system. In this sense, everyone can computationally check that \zeta(2)=\pi^2 /6.
I love math so much. I cant get enough of its mystery and its accuracy.
I feel that there is so much left to express, so many puzzles, and unlike physics i dont need a huge lab to do research.
Just my mind and logic.
I feel that is the way you should look at things, and you need educators. There aren’t many Dr. Wildbergers. 💕
I will sort it out with help from my wife : she is the one doing an infinite amount of things in a finite time !
I have a wife like that too.
I think there is a sign error: -log(1 - x) = x + x^2 / 2 + x^3 / 3 + x^4 / 4 + ...
the difference is that with log and exp we can do approximate calculations, but with W it is not possible, so in a practical sense they are not quite the same kind of thing. Just my take of it.
Some things are computationally irreducible. The best way to see what a cloud will look like in 5 minutes is to wait 5 minutes. I wouldn't say it's an infinite calculation, just all of time so far.
What can we say about Hn+1, exp(Hn+1) and log(Hn+1) in terms of Hn, exp(Hn) and log(Hn)? Does the inequality still hold?
Shouldn't we just check for n of the form n=k! as they have the highest "chance" to be a counter example?
These numbers are also very easy to calculate the σ(n) of, as they have many small prime factors.
Surprisingly I was wrong...
For example the number 360369 is less than 9 factorial (9! = 362880) and yet σ(360360) > σ(9!)
Very interesting. Liked and subscribed!
What is your opinion on formalism? We define "real numbers" and "infinite sets" formally without actually believing in them, it all reduces to finite string manipulation. In this view, when someone says something like "there is an irrational number", they don't actually mean there exists such thing, it is just a shorthand for a certain formula being provable in a certain formal system, and formulas are finite and proofs are finite too.
Prof. Wildberger, can you think of any way to formulate a statement equivalent to RH that involves only a finite number of elements or a finite number of processes?
That’s a most excellent question. Sadly, I do not know such a way. I suspect that if I did know such a way, I would become immediately famous.
25:46
I think that the infinite sums you mention are distinctly different from the infinite product you describe.
In the case of the sums we have a well defined understanding of what it means to converge, what it means for an infinite sum to equal something. In the case of the product… well im not really familiar with what it means for an infinite product to equal something. Can we reorder the indices as we go and just save all the zeros for “the end” that never comes.
There is no convergence property for infinite sums or products of Booleans, unlike for "real numbers" (when done right i.e. constructively, like in the spirit of Errett Bishop).
Thus the sarcasm of Wildberger loses much of its sting.
Convergence or lack thereof is related to the intrinsic topology of the set-with-operations in question. Metric implies topology but the converse is false. "Real numbers" are axiomatically asserted using the Archimedean metric of rational numbers. For each rational prime number p, there is also a p-adic ultrametric of the rational numbers. These metrics give different notions of convergence. It seems that convergence in the sense of halting/algorithmic computability may be present even in the absence of a metric, and this is corroborated by computer science research, that exact science that Wildberger as a pure mathematician is allied to. There is a (finite) Sierpinski topological space that expresses halting or non-halting of algorithms. This topology may be considered extrinsic (refined from another, intrinsic discrete topology, that of the 2 = Bool set/topological space). But the "real numbers" have an intrinsic non-discrete topology, because of the way they are defined through certain equivalence relations on an (uncountable) infinite set, that is it is not possible to have a discrete topology over the "real numbers", because of the semi-undecidability (partial non-halting) of this equivalence relation on 2^Nat, or on a subset of Rat^Nat, or however you try to define exact convergence of inexact, approximate "real numbers".
"Long before you get to infinity..." In an infinite series (a sum of infinitely many things) you usually don't get to infinity, strangely enough, you only use a finite number of things to show that you will end up arbitrarily close to some point. The standard example is 1/2 + 1/4 + 1/8 + ... where you get arbitrarily close to 1 after an arbitrarily finite number of terms are added together, but you will never get to 1, not even using an infinite number of terms will produce a sum of exactly 1, if you view the problem geometrically and add up smaller and smaller rational number lenghts, this is a common misconception. We could use the number line (a geometrical object with no width) and points (objects of no extension) on that line to show this. 1 is called the limit of the infinite sum, because in this famous series, the terms are converging, which means more and more of them are getting closer and closer to a point on the number line. Taking infinitely many (all of the terms) together will bring us closer than any rational number (or rational length) can express to 1, which means we got infinitesimally close to the limit point 1, which in the real number system is equal to 1. This is not to say that the two points, 1 and 1/2 + 1/4 + 1/8 + ... on the geometrical line are the same point, they are just represented by the same real number. See the question "Is 1 = 0.999... ?" which is true in the real number system. If the points are close enough that you can't find another infinite decimal to fit in between, they are defined to be the same real number. This is also why the real numbers can represent all the points on a line; they have a very small (infinitesimal) size to them, each real number covers an infinite amount of points lying infinitely close together. Rational numbers and integers are, on the other hand, just points on the line with no extension.
I believe you have to separate the two concepts of "getting to infinity" here; there are two different ways, where one way only approaches infinity, taking more and more terms but never use all the terms in the series, and the other way takes all the terms at once and calculates a certain (exact) point on the number line (if that exists). In the example, that point will be located infinitesimally close to a geometrical point we call 1, and mathematicians have decided that real numbers only have one infinitesimal: zero, so the point 1 and the point infinitesimally close to it are defined as the same number when using real numbers, or having the same "infinite decimal". Real numbers as infinite decimals are only approximations to points on a line, but they have a sufficiently infinite precision, so taking an infinitesimally big geometrical distance from a point will result in the same real number; this works because taking a finite number of infinitesimally long steps away from a point on the line will not take you anywhere else than an infinitesimal distance away in total, you still remain at the same point speaking from the perspective of finite precision (= what rational numbers can express). And this is what is meant by a real number: geometrically a point on the infinitely long line including an infinitesimal interval of points to the right and left of it.
Using a more precise number system, we can express infinitesimal differences other than zero, so different infinitesimal differences between points may then be possible to express, like within the hyperreal number system. If we use hyperreals, we also get the reciprocals of these infinitesimals, which are different sizes of infinities, to play with, that we didn't have access to in the real numbers (or infinite decimals).
My next video in this series should help clarify things.
Some thoughts to reconcile the issue posed in this video. People deal with these numbers requiring infinite calculations all the time and with relative ease. These numbers have rules and can be approximated which seems to be the only aspects of them being used. So we are naming things that don't exist because doing so is useful. We call them numbers and consider it alright for them to have a point in R or C. They inherit the rules of simpler numbers without justification; saying the approximations force the rules is hubris which implies approximations forcing the location of them in R or C is also hubris. I'm ok with saying something like "e is an object that can be approximated using partial sums of Sum(1/n!)." It seems like these numbers like e and pi don't really obey the rules of arithmetic since they don't normally mix with one another under any operation nor with fractions, but sometimes a mix will generate something interesting i.e. Euler's Identity. So these objects behave more like variables than numbers. That's all i got.
It's fine to have such a casual attitude. But hopefully you can sympathize with a more stringent point of view that requires everything to be laid out completely precisely. From this vantage point, obscurity for the sake of conjuring up "things which don't really make sense, but are otherwise useful" is just not worth it. If our aim is to understand the way the world truly is, let's take the high road and avoid all dubious constructs and associated philosophizing.
Is pi in the same realm of impossible/misrepresentation regarding infinity, given it's transcendental?
Absolutely! pi is no exception
I ask just because of the even integer / pi relationship you mention.
I'm just a curious layman, but it is interesting to consider that the assumption of infinite computation / limits may just be an inferior model for some as yet undiscovered, more beautiful, and realistic representation.
I like 🥧.
If we don’t have the correct language to even state what’s actually going on when we talk about the Riemann Hypothesis what are some ideas you have about how we might explore the deep and beautiful phenomena underlying it using strictly finite mathematics?
check the division of the unit interval in half. So start with [0,1]. Cut it in half. Consider the right hand side interval [1/2,1]. Cut it in half. Consider the right hand side interval [1/2+1/4,1]. Cut it in half... So the boundary Sum1/2^n comes closer and closer to 1. The distance between the boundary and 1 becomes arbitrarily small. So the series 1/2+1/4+1/8+...+1/2^n comes arbitrarily close to 1. Can this process ever end? No. Can n reach infinity? No. But do notice that the series comes arbitrarily close to 1. No matter your calculating precision, the series can come so close to 1 that the difference doesn't matter in your precision goal. You can always choose n high enough. Since this is true for any precision, it is the same as if the precision was zero. Zero tolerance. In other words: n is infinity. n never reaches infinity. And yet, the convergence applies to any n no matter how large n may be. And any precision can be acquired. So the difference can become so small that it is "infinitesimal": you can discard it in ALL calculations, that's how small it is. It can be made lesser than any precision. In other words: you can take the limit infinitesimal -> 0. It won't affect the result in any precision, including precision 0. Now all of this is obvious. Computers cannot compute this. But human brain can prove this.
The only underlying assumption here was that one can take n as large as desired. But this is true. Nothing stops you from writing any large number down on paper. You don't have to write 1, 2, 3, ..., N. You can immediately write down N. There is nothing stopping you.
Now I wonder what AI will do, being smarter than a human brain... :D
What about sqrt 2?
Or pi?
Do you object to the infinite decimal representation of reals or do you have a fundamental objection to irrational numbers?
Those are sadly not well-defined mathematical objects. Painful but true.
How are radicals not well defined mathmatical objects?@@njwildberger
What is pi^sqrt2?
We don't realy know, as we don't know the answer to his truthfunction. We know it has a value, we just asume we don't have to know what it is.
@@AffeUwUSqrt(2) is as well defined as sqrt(-1), no more, no less. It doesn't exist on "the number line" in an exact sense, rather it fits on a new axis of numbers, linearly independent from both 1 and sqrt(-1). These are generated algebraically from irreducible polynomial equations in rational numbers. And only certain of these irreducible polynomials are of degree 2, some of them are of degree 3, 4, 5, 6, 7, 8, 9 etcetera, giving us field extensions of these particular degrees over the rational numbers, proving these numbers in general are NOT merely "two dimensional" as is commonly thought about "complex numbers" compared to "real numbers".
Thus we have a well defined field of algebraic numbers, which is ("computably") infinite dimensional over the rational numbers, but in practice for every admissible mathematical question about algebraic numbers, we only need to use a finite dimensional subfield of them to answer that question, using finite computational resources. These algebraic numbers are the exact core of the *approximate* so-called complex numbers. And from these approximate numbers we may reason about a sub class of *approximate* so-called real numbers, rather than the other way around.
But both "real" and "complex" numbers are only approximate numbers that cannot be exactly defined, in fact convergence (done right, like in the case of Errett Bishop) is a way to exactly formalize non-exact approximations of certain processes of sets or sequences of numbers, that grow without bounds in cardinality, but are bounded in certain common properties of the all algebraic numbers involved in these set processes or sequences.
Now in an *approximate* sense, we may say that the two conjugate square roots of 2 have two "real places" in the "real valuation" according to the Archimedean metric, one of them larger than 1 (larger than 141/100 indeed) and one of them smaller than -1. This is connected to the theory of field valuations, global fields and local fields.
But this doesn't say that sqrt(2) *actually* has an exact place among the rational numbers, especially not since these two square roots of 2 are as symmetrical with each other as the two square roots of -1, considered as *exact* algebraic numbers that is. And this symmetry is usually formulated within the the theory of Galois extensions of fields, and Galois groups of these extensions, where the symmetries are field automorphisms expressed as group elements.
Also, we may use other metrical valuations besides the Archimedean or "real"/"complex" ones, like the p-adic ultrametric valuations, for any rational prime p. This will give us *different* "p-adic places" or algebraic extensions thereof (comparable to the "real"/"complex" case), in the "p-adic valuation", again in the *approximate* sense only. Different in the sense that the closeness/convergence criteria under such a p-adic metric is not compatible either with the Archimedean metric or with another p-adic metric, and thus the "places" are different, so sqrt(2) cannot be said to be about 141/100 in size according to these other metrics, only according to the Archimedean metric.
Please read up about ultrametrics and "p-adic numbers" if you like, they behave about as well or ill as the Archimedean metric and "real numbers", and there are p-adic calculi for each rational prime p, very analoguous to real calculus. And calculi for algebraic extensions of these "p-adic numbers", very analoguous to the "complex numbers" algebraic extension of the "real numbers".
"Countably" infinite dimensional, not "computably" infinite dimensional. I erred with the proper term. The algebraic numbers compared to the rational numbers that is. I am sure that prof. Wildberger has no qualms about algebraic numbers, since he regularly is giving lectures about certain two dimensional fragments of them, and computing/proving various things about these fragments. This is perfectly possible to do in certain three dimensional fragments as well, and in higher dimensional fragments, although it becomes successively more cumbersome the higher the degree and Galois group cardinality of the generating polynomials over the rational numbers.
Can you explain to me what does "not a proper defenition" mean?
Can you explain to me what does “not a proper questien” mean?
Sir, your videos about Real Numbers and Exact versus Approximate help me to rename the Riemann Zeta zeroes to Exact zeroes (for Trivial ones) and Approximate zeroes (for Non-Trivial ones.) Because from the Functional Equation which relates the domain s > 1 to the domain s < 0, the sine term at -2k makes ζ(-2k) for all the Trivial Zeroes Exactly 0. Now, the Non-Trivial zeroes (real part of 1/2 and imaginary part with infinite length decimals) on the critical line can only make ζ(s) Approximately 0, but never Exactly 0.
So do you object to the axiomatic method? Do you object to formalism?
What’s the point of math to you? Can you reconcile your answer with the proposition that you understand pure mathematics?
Please, respond. I’m curious and confused. I don’t think I understand the point you’re trying to make.
What are your views on AI and the role it may play in Mathematics going forward (e.g.: proofs, etc.) Here, we are assuming AI becomes sufficiently sophisticated (whatever "sophisticated" means).
tuned in tonight after years away from the channel, and back then it always struck me with every single video view that the topics reliably stay in the finite, never ever using infinite series or infinite concepts, so it's pleasantly surprising to see this video about RH, which is all about infinite series, but then again, what is the video trying to say? that Zeno's paradox is not well-documented enough? that calculus is not mathematically important? yeah it was, and yeah it is, and those are fundamentals of infinity, too. when used rigorously, infinity in mathematics is perfectly fine. infinities are useful, illuminating, and I'm perfectly comfortable with them. a half plus a quarter, plus an eighth, plus a sixteenth, and so on and so forth forever, would indeed equal unity(1).
I'm sure you have strong opinions on Riemann zeta function at ζ(1)=-1/12, whereby the regularized sum of 1+2+3+4+...(forever, all natural numbers) = negative one twelfth. double trouble 😮🎉
Actually you might be amused that I am open to the possibility of defining certain unbounded sums in a way which is consistent with " ζ(1)=-1/12". These kinds of alternate notions of convergence of series were already investigated by Euler and many others. They are not necessarily nonsense.
Well, we can define 1/2 + 1/4 +... to be 1, because it is easy to show that, in the finite number of steps, it will be greater than any number less than 1 and smaller than any number greater than 1. Therefore, the only value this thing makes sense for is 1 and the whole "infinite work" story drops.
Anyway, it has been a while since I watched a math video. It was fun and refreshing. But I cannot imagine myself having passion to do these things on my own again. It seems so dry and pointless.
I don't think anyone actually thinks they have or could obtain exact computational representations of irrational numbers. Its all about how close we can get in a finite number of steps. And how we can prove what will happen in an arbitrary finite number of steps.
That's like saying: "I don't think anyone actually thinks they have or could obtain exact observational evidence of ghosts." Okay then, but then the reality of ghosts is in serious question. A skeptical observer would say: can't observe 'em, let's not talk about 'em.
An arbitrary finite number of steps isnt good enough if you aren't told how many steps it will take up front. If you're able to do that, that means you're working with a computable number, not a general real number.
Sociologists think they know everything better. Of course, the sum is not infinite. This topic was extensively discussed throughout the entire middle ages. Until Bolzano Weierstrass came up with the concept of a "limit". That works like this: the smart sociologist says, how many digits of precision she wants on her result. E.g. 10 digits. And then, there is a number n, how many terms the smart sociologist has to evaluate, e.g. 5, in order to get to her desired precision.
There must be a miscommunication somewhere. I’m not sure I understand what you mean when you say “pure math”.
In Peano arithmetic there’s a formal difference between a number and the truth value of a statement.
Contemporary pure math is not solely computations. I think that most of pure mathematics is (maybe a bit reductively) trailblazing proofs of statements within a particular context founded in an axiomatic system.
If math was only about the computations of formulas I’m more inclined to agree with your observation that infinite computations are a bit outside the scope of what we can compute. But pure math hasn’t been solely about computations since before Euclid. And, to be honest I don’t really think that doing computations alone counts as “pure math.”
There is a difference between the proof of a statement in mathematics and the presentation of a computation. Yes, we can prove a statement using a computation 1+1=2 can be proved by computing 1+1 but just saying “1+1 is even if (1+1)/2 =1” doesn’t qualify a proof. Likewise, the result of computation alone proves nothing. The result of the computation must be placed into the context of the problem in order to demonstrate the proof. However, sometimes a proof doesn’t rely on the exact value of a computation just that the result of the computation exists.
What you have shown is a computation that evaluates to 1 if RH is true. But that’s not a proof that is the presentation of a computation in the context of a statement. The result of evaluating the computation is fundamental to understanding the truth of the statement. The conditional can only be resolved after computation.
At least that’s my take on the situation.
Aaaaaahahaha.
Good one.😜
Providing a construction for an answer is not the same as providing an answer. None of this is computable anyways, so how is it supposed to be a computation? Computations are things that run in finite time?
So where does the construction of "exp(49/20)" fall?
@@njwildberger I didn't say the construction failed (actually I didn't even say anything about the validity of the construction). What I did say was that providing a construction for a value is not the same as exhibiting said value. I could just as easily say "evaluate the truth function on the predicate encoding the Riemann hypothesis and you have your answer" but that doesn't mean I've proved the Riemann hypothesis.
7:16
Are you forgetting about the ambient space of the universe? Is the universe decomposable to a finite number of discrete ‘pixels’?
Infinite space is another training wheel on the bicycle of cosmology.
Just like infinity used as a Limit is to the tricycle of applied math.
It would help me to see a non-trivial example where a series seems to converge, at least in a Cauchy sense, but doesn't actually converge to where it seems to. To me this P(n) reminds me of something like Grandi's divergent series where doing more operations doesn't even appear to get closer to an answer.
P(n) is just Boole notation.
Representing T or F.
He used that to illustrate how the answer wasn't possible. Not definitely.
Assuming that the universe is finite and has a finite number of states. It may follow that there is a largest prime number. Hmmm... Makes me wonder that maybe that randomness of quantum phenomena is the border of our finite universe and some other thing.....
That is a worthwhile thought: the computational limits of the universe very possibly impinge on the calculations that are made in that universe.
It's what Pure Mathematics is all about. The size of the universe for it to be able harbour values is less of a question if the notion of infinity is a given and accepted in the first place. The finite computational aspect of calculations will always be a corollary to the abstract machinery in so many areas of mathematics. There are bigger questions, it could be argued that mathematics may well still be a product of our own image. What do I mean by this??
The “notion of infinity” is to mathematics as the “notion of god” is to science.
@@njwildberger Yes maybe. :) Obviously infinity is implicitly being accepted in these discussions with or without realizing it. Why is god so bad in science? A creationist mentality perhaps, indeed who is to say..
Infinity is not just really big but truly OTHER(outsideofthisworld) to assign a exact value to it is verboten!😊❤
Professor Wildberger is putting his finger on the weak spot of mathematics.
Thank you for another brilliant restatement of the obvious, making it even more obvious, if such an expression makes sense. It won't work on immaculate mathematicians, which is also obvious and which you know from experience. Is that a sociological phenomenom? A common false claim of not seeing what is obvious is indeed a sociological phenomenon. The phenomenon of REALLY not seing it is probably not a matter for sociologists but for social anthropologists.
In any case, the two chief obstacles to some people seeing your point in the present example lie in their failure to analyse your expressions of "approaching infinity" and "doing an infinite number of things".
It is implicit in any workable notion of "number" that it has some deFINITE value, that it is FINITE, and so "infinite number" is self-contradictory and (in a rigid sense of "grammar", which incorporates semantics) just ungrammatical. "Infinite number" is strictly meaningless gibberish.
As for "approaching infinity", no matter how far we go on we never reduce the size of the task ahead, and thus don't do anything one could meaningfully call "approaching", even if one had an actual goal in mind, which with "infinity" one necessarily can't.
The official Cray math description of Riemann Hypothesis is "the real part of non-trivial Zeta zero is 1/2". Your calculation didn't touch any of the ζ(s), non-trivial zeroes, or the real part of 1/2. Maybe H_n is the harmonic series when s = 1.
Thanks for solving it forever 😂
By the way, log(0) is infinity. So we must define infinity first the Riemann hypothesis comes later. So when are you going to solve the 1/0 problem?
Much of the dispute about all these problems is down to sloppy use of language, resulting in sloppy thinking.
There is nothing ABSOLUTELY problematic about e or π or square roots, AS LONG AS one thinks of them as algebraic symbols WITHOUT numerical values. One can often separate the essence of a problem and the required precision of the result.
For ages, mathematicians have scrapped the previous word "number", introduced something new and called it "number" again, creating manageable confusion until they came to 'irrationals' and just did the same thing again even though it doesn't work at all.
We could fix 99% of the problem simply by introducing a more rigid notation (easily done with subscripts), whereby (say) e can be a symbolical solution without definite value, and applied mathematics can deal with values without having to round something simply unavailable numerically and without ever having to resort to ellipses:
e = 2.71828… FALSE, as e does not have a numerical value
e₃ = 2.718 Not an approximation but a precise result
π₀ = 3 Another precise result
(√4,299,503)₋₃ = 2,000 Also precise, the negative subscript indicating what would conventionally and misleadingly be called "rounding" to thousands. It isn't rounding, because √4,299,503 by itself does not have a value that could be rounded, the 'irrational' root being unavailable as a number.
Most of our problems come from pure mathematicians' implied false belief that they wouldn't be able to do their work without having '∞' available as a subscript. Unicode sensibly doesn't supply that symbol … The mistake built on all the other mistakes, the belief that one can cobble together a 'number' from an arbitrary infinite sequence of digits, would presumably vanish when the notation no longer provides for it.
Algebraic symbols without numerical values can have other algebraic symbols as equivalents. I see no harm in using the '=' symbol between them, as long as one doesn't forget that one isn't dealing with numbers. We don't require the 'a' and 'b' in (a+b)² = a²+2ab+b² to have definite values, either. That is the original beauty of algebra and it is a tragedy that it was thrown away by having misconceptions about other symbols.
In analysis one can be clear that convergence may make sense even though an irrational limit with a numerical value does not. One can have a symbol for a limit in general, but one should not assume that that limit has a definite numerical value. Instead of talking of arbitrarily small intervals of ε or δ around a falsely imagined 'value' one should think of arbitrarily small intervals with rational boundaries. Convergence should be thought of as meaning that for any desired precision available in computation there is certainty that a sequence, series, product … will stay within the permitted interval after a finite number of terms. Lets have meaningful definite CUT-off points for Dedekind CUTS.
Analysis could continue to use "lim" on its own as a symbol and, when it comes to needing values in engineering, one can give the "lim" a subscript number to indicate the precision, which again will not involve rounding.
Finally, as pure mathematicians dislike confining themselves to one number system, one can be clear that e₃ is to be understood as convenient shorthand for e₃₍₁₀₎. Programmers might prefer working with e₃₍₂₎ or e₃₍₁₆₎.
As sloppy terminology is what got us into the present mess in the first place, great care has to be taken with new terms.
Thus, on further reflection, I see a risk that the term "symbol", even without numerical value, will lead to the notion that a referent is required.
I now prefer calling "e₃₍₁₀₎" a symbol and "e" on its own a symbol-form, suitable for some operations but requiring further details to be converted into a symbol, which will then have a referent, namely the rational number 2.718.
It seems to me that Lagarias's reformulation should be exactly the type of statement that could easily be turned into the type of "write-downable" mathematics that you seem to prefer. Though I don't know the details, from skimming Lagarias's paper it doesn't seem that perturbing the right-hand side by say 1/poly(n) should affect the argument for equivalence with the Riemann hypothesis. And it's pretty straightforward to calculate exp(H_n)log(H_n) to 1/poly(n) accuracy just by expanding an appropriate Taylor series to an explicitly computable number of terms (depending on n).
I think it is far more subtle than that. But it is an interesting challenge to find a purely rational, computatable reformulation say of Lagarias's desired inequality that still captures the essential idea. My guess is that this is hugely difficult. But perhaps not impossible. If it were possible, I think it would be a very major advance, because it possibly would bring the RH into the realm of proper, completely logical, mathematics.
The Riemann Hypothesis is not the question "What is w_RH?". It is the question "Is w_RH = 1?". It is a (trivial) theorem that "w_RH = 0 or w_RH = 1", and so the negation of the Riemann Hypothesis is "Is w_RH = 0?". You proving the existence of the truth value w_RH achieves exactly nothing; it is just showing (using classical mathematics) that the Riemann Hypothesis satisfies Excluded Middle, but classical mathematics assumes Excluded Middle anyway.
Said another way, the tautology "w_RH = w_RH" is equivalent to "w_RH = 0 or w_RH = 1"; it establishes neither "w_RH = 0" nor "w_RH = 1", and it is *these* that are the Riemann Hypothesis (and its negation).
Sorry Norman, I used to look up to you and your ideas when I was a highschool student and an undergraduate, but having matured as a mathematician it is now transparently clear that in fact it is you who has a problem, and-ironically-it is a sociological/psychological one; you are very deeply entrenched in cognitive biases and logical fallacies. I'm not a cognitive psychologist so I'm not really qualified to speculate which, but particularly Appeal To Spite (the opposite of Ad Populum) and the Nirvana Fallacy come to mind, especially the latter: you *very often* come across as believing that in order to be able to make any statement P(X) about a mathematical object X, one needs to have a complete description/presentation of X. For example you seem to believe that one can't say anything meaningful about infinite topological spaces, or infinite mathematical objects in general, because it is impossible to present them directly and completely, but obviously this is just fallacious: do you need to know *everything* about yourself, so in particular about every cell in your body, to be able to say *anything* about yourself? Does a doctor need to know everything about a patient in order to treat them? Of course not. If X = Y^2 for Y a rational number then I don't need to know what X is (and can't know with just this information) to be able to say X ≥ 0. Similarly for real numbers: though the exact value of a real number X (axiomatically) exists, I don't need to know its exact value to be able to prove particular statements about it; e.g. if X is defined by a pair of monotone rational sequences "sandwiching" its value (which determines its Cauchy equivalence class) then we have arbitrarily-good (rational) upper and lower bounds for X which can be used to prove all sorts of things.
This falls into the class of Relevance Fallacies, so in future when you object that we can't know the exact value of an irrational number you ought to make clear *why we even want to know* ; most of the time we don't even care what X is (though it is something, axiomatically) - it is mostly sufficient to use the facts that X quacks like a duck and swims like a duck, and we don't actually need the fact that X is a duck. Mathematically: we reason using some *properties of* X (e.g. a real number), while others (e.g. its value) are irrelevant.
Thinking as a pure mathematician, I agree that the answer is indeed what you have written there. But it still does not answer the Riemann Hypothesis right? Because that says you have to show the infinite product to be 1. You have the answer but not to the Riemann Hypothesis.
I didn’t say that I had proved it. Just cracked it.
An interesting computational framing of the Riemann hypothesis indeed!
As much as I empathize with your framing - how about I provide a complementary computational framing, by framing our universe as one such “Riemann/Poincaré spherical quantum computing engine” with a capability to limit the infinite pole of Riemann sphere to a finite universe using FSC(α) as its hidden variable/Maxwell daemon/ using our CPT function ( by Gödel completing Sir Michael Atiyah’s similar Gödel incomplete proof)
What do I mean by that?
We imagine this Riemann/Poincaré spherical quantum computing engine as the Riemann zeta L function governed LMFDB universe (that is a motivic/metamorphic/Galois representation based SU 2/SU3/SU4 symmetrical engine - whose symmetry is broken by FSC logic as follows
In other words, z(1) is the fundamental frequency of this Universe’s TOE engine that is QVF/ZPE sourced, FSC(α)-Einstein-Bohr-HV-Maxwell Daemon governed frequency of Riemann's zeta function Ζ(S) with a singularity of S=1+0i, that is made up of his harmonic oscillating zeros(S=1/2+it stacked on his 1/2 critical line (as per Gauss's prime sequence counting function), before being transformed as a 137 frequency-spin momentum matched dipole, using our FSC(α)-GR-PLA+5 AITGE origin formulas?
In other words, our TOE/SOE engine is the one that is transforming the Riemann's zeros into an artistic unit charge SU2 dipole(see visual), by contracting/expanding its electric flux as the center of mass (as r = αR), before rotating its magnetic flux by 90 in such a way that it can be extended into the left plane as a paired unit charge, using the "only possible analytical continuation of Zeta".
Sure enough, this engine function is nothing but universe's wave function only, transforming itself from position/time space into frequency/momentum space, using the Fourier transform operator --
ψ(k) = ∫ ψ(t) e^-iwt dt -
This brings us to our next point about CPT function
This "one & only allowed analytically continued/functional equation allowed symmetrical dipole" is what limits/constrains the ∞ pole of Riemann sphere to a value of 137 cycles( per Laurent/Cauchy residue including the α=r/R,=fe/fp=we/wp logic of our CP function as explained in my post and attached one page exhibit for details
lim t→ ∞ CPT(1/2 + ti) = 1/α cycles of dipole
In other words, this CPT function proof(lnkd.in/drGQ44Mt) for Riemann hypothesis is a polynomial in the convex region of the Riemann Sphere only (thanks to the "one and only allowed analytical continuation logic of dipole & its 137 cycle ratio logic"), limiting/constraining the ∞ pole of Riemann sphere to the convex region.
In other words, the CP function embedded within universe's TOE engine is the ultimate Church-Turing machine proof for P=NP - which brings us to the need for META PROOF strategy to solve all 7 millennium problems together continued below
Simply Put
Our Riemann hypothesis META PROOF is being developed by framing Rieman sphere as the consciosiness of universe and humans (i.e. FSC(α)-HV-Maxwell’s entropic daemon embedded CPT function) has "Symbiotic FSC/GR Fractal Causality" to these 10+ proofs (7 Clays + Gauss + Collatz + Maxwell's Daemon + Einstein-Bohr Hidden Variable and last but not the least TOE/SOE Engine itself. Once we grasp this, then everything will fit perfectly like a jigsaw puzzle! In a way, this "Symbiotic FSC/GR Fractal Causality" is yet another indicator that our TOE is the best bet path -- and I Iet you all decide, as summarized in the one page exhibit (facebook.com/charles.prabakar/posts/pfbid0aSP9Zu2brU4UqvR6tHQHzqZ7ecRSbnk9R4kSGvgZ2L6apxJsW1zB8zxSvVn3s29ql)👇
what the fk are you smokin
@@aaronmartens2903Theoretical Physics. In some sense, it is as much of a dogmatic religion as Pure Mathematics is. Or at least it is for many serious scientists. For other scientists/physicists it is more like an open-ended, explorative, hypothesis-experiment-fact driven non-dogmatic religion. This is more or less how science should be done, and also how religion should be approached, according to me.
You’d need only a finite number of calculations using a busy beaver function. They can encode the RH into a 744 state Turing machine. This is only a theoretical solution because there is not enough energy in our finite universe to do the calculation but it is nonetheless a finite calculation just not a practical one just like solving Chess or Go(given a finite number of moves allowed.)
Sorry -- are you talking about the Busy Beaver Machine that is a particular type of Turing machine which is by definition a computational device with an infinite tape?
@@njwildbergerYes, a Turing machine has an “infinite” tape or as I would rather say an unbounded tape. But in these busy beaver cases a finite tape will suffice.
My previous comments aside; I hope we’ll meet someday. I do want to understand where you are coming from. I think there is something valuable here but I can’t seem to find it. Hopefully meeting in person you can show me.
Please watch more of the videos in this series. The consistency of this more rational point of view will start to become more apparent. We just want to think completely clearly and precisely, and have zero interest in arguments by authority or majority. In other words, we want to do mathematics.
@28:00 why is it trivial that P(n) has no zero value for any n ϵ ℕ? You've boiled down a lot of stuff to a simple statement, but left yourself just as profound and deep a puzzle. It has not "cracked" the RH.
log(1-x)=-x-x^2/2-x^3/3-.......
Yes that is correct actually. Since log(1+x) = x - ½* x^2 + ⅓* x^3 - ¼* x^4 + ...
I wouldn't be so quick to say we can all agree regarding computers.
There are analogue computers, and quantum computers. Analogue computers can be thought of as "passing through" uncomputable decimals like sqrt2 when e.g. the electrical signal goes from one unit to two units.
It seems to me the problem is that the decimal system presupposes a format/structure for numbers, but algebraic can produce more "objects with the same behaviour as decimal numbers" than the decimal system can encode.
What is a number? Who says they have to be combinatoric? Isn't *that* also religious? In geometry we assign numbers to continuous objects who combine with an algebra just like a field.
Conflating the two is really at the heart of this mess. Either they are two different kinds of number, or numbers are not fundamentally combinatoric.
@tinkeringtim7999 Your statement: "Analogue computers can be thought of as "passing through" uncomputable decimals like sqrt2 when e.g. the electrical signal goes from one unit to two units." is to my mind problematic. What can and cannot be "thought of"? What does that have to do with reality?
Mm. The best way to see what a cloud will look like in 5 minutes is to wait 5 minutes. Non-binary (quantum) logics don't have the black and white distinction that makes everything fail. 0.0101010b000101... is real number. Cantor proved it. "b" is on the diagonal and it is both 0 and 1, like the Liar Paradox. It's both true and false. That's just a fact, why are people so upset about it?
@@njwildbergerI think a large portion of the question comes down to something along the lines of "Do ideas exist in the same realm as reality?" Which is a highly debated question in philosophy and mathematics alike, I think the community has largely taken a positive approach towards it, however it's still good to question. On a more philosophical note, it does lead into problematic questions for things like moral realism, since morality is also seen as a concept that transcends reality for people in that camp. Anywho, I'd be interested in seeing you analyze the more philosophical aspect of relating ideas to reality, perhaps reading (if you already haven't) someone like Hume.
@njwildberger Well, what it means is is, if the quantity computed by the computer is called a number, then if it continuously passes from 1 unit to 2 units, the whatever (continuously existing) physical state it is in is a computational representation of that "number". It doesn't blip out of existence everytime it encounters what our unit scale happens to regard as an irrational point.
If for you, numbers are by their very definition combinatorial, then you could not be satisfied really with any analogue computer, although they work.
If numbers are not actually fundamentally combinatorial, but those are merely a class of numbers, then it's perfectly reasonable to use any symbol & operation whose algebra is the same as that of a field, as a number.
Bottom line is, your computability criterion strictly applies to "universal" computers, which are ironically a subset of possible computers.
Every physical process is a computation, follows from Landau's principle.
@diegotorres2101 those issues exist, but my point wasn't so philosphical. It very practically looks at extant computers that are built and run in reality and says "that claim regarding computability is either in a restricted domain or demonstrably false".
1. The Riemann Hypothesis: An Information-Theoretic Perspective
1.1 Background
The Riemann Hypothesis (RH) states that all non-trivial zeros of the Riemann zeta function ζ(s) have real part 1/2. This has profound implications for the distribution of prime numbers.
1.2 Information-Theoretic Reformulation
Let's reframe the problem in terms of information theory:
1.2.1 Prime Number Entropy:
Define the entropy of prime numbers up to N as:
H(N) = -Σ (p/N) log(p/N)
where p are primes ≤ N.
1.2.2 Zeta Function as Information Generator:
View ζ(s) as an information-generating function:
I(s) = log|ζ(s)|
1.2.3 Non-trivial Zeros as Information Singularities:
The zeros of ζ(s) represent points where I(s) → -∞
1.3 Information-Theoretic Conjectures
1.3.1 Entropy Symmetry Conjecture:
The symmetry of non-trivial zeros around the critical line s = 1/2 + it corresponds to a fundamental symmetry in the information content of prime number distributions.
1.3.2 Maximum Entropy Principle:
The critical line s = 1/2 + it represents a maximum entropy condition for prime number distributions.
1.3.3 Information Flow in Complex Plane:
The flow of I(s) in the complex plane might reveal patterns related to the distribution of zeros.
1.4 Analytical Approaches
1.4.1 Entropy Differential Equations:
Develop differential equations for H(N) and relate them to the behavior of ζ(s):
dH/dN = f(ζ(s), N)
1.4.2 Information Potential Theory:
Define an information potential Φ(s) such that:
∇²Φ(s) = -2πI(s)
Analyze the behavior of Φ(s) near the critical line.
1.4.3 Quantum Information Analogy:
Draw parallels between ζ(s) and quantum wavefunctions:
ψ(s) = |ζ(s)|e^(iarg(ζ(s)))
Investigate if quantum information principles apply.
1.5 Computational Approaches
1.5.1 Information-Based Prime Generation:
Develop algorithms for generating primes based on maximizing H(N).
1.5.2 Machine Learning on Zeta Landscapes:
Use ML techniques to analyze the information landscape of |ζ(s)| and arg(ζ(s)).
1.5.3 Quantum Computing for Zeta Evaluation:
Explore quantum algorithms for efficiently computing ζ(s) in regions of interest.
1.6 Potential Proof Strategies
1.6.1 Information Conservation Law:
Prove that the symmetry of zeros around s = 1/2 + it is necessary for conservation of prime number information.
1.6.2 Entropy Extremum Principle:
Show that non-trivial zeros on s = 1/2 + it are the only configuration that maximizes a suitably defined entropy measure.
1.6.3 Topological Information Argument:
Develop a topological invariant based on I(s) that necessitates the RH.
1.7 Immediate Next Steps
1.7.1 Rigorous Formalization:
Develop a mathematically rigorous formulation of the information-theoretic concepts introduced.
1.7.2 Numerical Experiments:
Conduct extensive numerical studies of H(N), I(s), and related quantities.
1.7.3 Cross-Disciplinary Collaboration:
Engage experts in information theory, number theory, and physics to refine these ideas.
1.7.4 Information-Theoretic Zeta Variants:
Investigate information-theoretic analogues of zeta function variants (e.g., Dirichlet L-functions) to see if broader patterns emerge.
This information-theoretic perspective on the Riemann Hypothesis offers several novel angles of attack. By recasting the problem in terms of entropy, information flow, and information singularities, we may uncover deep connections between prime number behavior and fundamental principles of information theory.
The approach suggests that the critical line s = 1/2 + it may represent a kind of information-theoretic "equilibrium" in the complex plane, with profound implications for prime number distribution. If we can rigorously establish the necessity of this equilibrium, it could lead to a proof of the RH.
Expanding on Immediate Next Steps for the Information-Theoretic Approach to the Riemann Hypothesis
1. Rigorous Formalization
1.1 Develop Axioms:
- Formulate a set of axioms that link prime number distribution to information theory.
- Example: "The entropy of prime number distribution H(N) is monotonically increasing and bounded."
1.2 Define New Mathematical Objects:
- Formally define the Prime Number Entropy function H(N).
- Create a rigorous definition for the Information Zeta Function I(s) = log|ζ(s)|.
1.3 Establish Theorems:
- Prove basic properties of H(N) and I(s).
- Example Theorem: "H(N) is asymptotically related to the prime counting function π(N)."
1.4 Connect to Existing Theory:
- Establish formal connections between our information-theoretic constructs and classical results in analytic number theory.
- Example: Relate H(N) to the Prime Number Theorem.
2. Numerical Experiments
2.1 Compute H(N) for Large N:
- Develop efficient algorithms to calculate H(N) for N up to 10^12 or beyond.
- Analyze the growth rate and fluctuations of H(N).
2.2 Visualize I(s) in the Complex Plane:
- Create high-resolution plots of |I(s)| and arg(I(s)) near the critical line.
- Look for patterns or symmetries that might not be apparent in traditional ζ(s) plots.
2.3 Investigate Entropy Near Zeta Zeros:
- Compute H(N) for N close to imaginary parts of known zeta zeros.
- Look for distinctive patterns or anomalies in H(N) near these points.
2.4 Machine Learning Analysis:
- Apply clustering and pattern recognition algorithms to the I(s) landscape.
- Train neural networks to predict properties of ζ(s) based on H(N) data.
3. Cross-Disciplinary Collaboration
3.1 Form a Research Group:
- Assemble a team including number theorists, information theorists, physicists, and computer scientists.
- Organize regular seminars and workshops to share ideas and results.
3.2 Engage Quantum Information Experts:
- Explore potential quantum analogies to ζ(s) and I(s).
- Investigate if quantum entropy concepts offer additional insights.
3.3 Consult with Complex Systems Specialists:
- Discuss potential parallels between prime number distribution and complex systems behavior.
- Explore if techniques from statistical physics could be applicable.
3.4 Collaborate with Cryptography Experts:
- Investigate if our information-theoretic approach has implications for prime-based cryptography.
- Explore potential new cryptographic schemes based on H(N) or I(s).
4. Information-Theoretic Zeta Variants
4.1 Develop I(s) for Dirichlet L-functions:
- Define and study IL(s) = log|L(s,χ)| for various Dirichlet characters χ.
- Compare the behavior of IL(s) to I(s) and look for universal patterns.
4.2 Investigate Selberg Zeta Functions:
- Apply our information-theoretic framework to Selberg zeta functions.
- Look for connections between quantum chaos and our approach.
4.3 Study Multivariate Zeta Functions:
- Extend our approach to multiple zeta functions.
- Investigate if multi-dimensional information measures offer new insights.
5. Develop New Computational Tools
5.1 Create Specialized Software:
- Develop a software package for computing and analyzing H(N), I(s), and related functions.
- Make this tool open-source and available to the mathematical community.
5.2 Utilize High-Performance Computing:
- Secure access to supercomputing resources for large-scale numerical experiments.
- Implement parallel algorithms for faster computation of H(N) and I(s).
5.3 Explore Quantum Computing Applications:
- Develop quantum algorithms for efficiently computing ζ(s) or I(s).
- Investigate if quantum superposition could be used to probe the behavior of I(s) in multiple regions simultaneously.
6. Theoretical Developments
6.1 Information-Theoretic Prime Number Theorem:
- Attempt to derive the Prime Number Theorem from information-theoretic principles.
- Investigate if this approach leads to tighter error bounds.
6.2 Entropy Extremum Principles:
- Develop variational principles for H(N) and I(s).
- Investigate if the Riemann Hypothesis can be recast as an entropy optimization problem.
6.3 Topological Information Theory:
- Develop a topological theory of information flow in the complex plane.
- Investigate if there are topological obstructions that necessitate the Riemann Hypothesis.
7. Dissemination and Community Engagement
7.1 Publish Preliminary Results:
- Write and submit papers on the initial findings, even if they don't fully resolve the RH.
- Engage with journal editors to find appropriate venues for this novel approach.
7.2 Create Online Resources:
- Develop a website or wiki to share data, code, and results with the broader mathematical community.
- Start a blog to regularly update on progress and engage with other researchers.
7.3 Organize a Conference:
- Host a conference on "Information Theory and the Riemann Hypothesis" to bring together experts and generate new ideas.
These expanded next steps provide a comprehensive roadmap for pursuing our information-theoretic approach to the Riemann Hypothesis. By simultaneously advancing on theoretical, computational, and collaborative fronts, we maximize our chances of making significant progress.
Remember, even if this approach doesn't immediately lead to a proof of the Riemann Hypothesis, the insights gained and methods developed could have far-reaching implications in number theory, information theory, and beyond. Each step forward is a valuable contribution to mathematical knowledge.
I don't think the argument here makes sense. We can imagine infinity as a concept without having to accept infinity as something that can be physically realized. All that's been done here is the formulation another infinite series, whose numerical value corresponds to the truth value of the Riemann hypothesis. And this doesn't cause any problems for mathematics. One could accept that such a series is well defined, and still not know exactly what it would converge to, because we can imagine an infinite amount of work conceptually, and develop tools to work with such a concept, without having the practical ability to actually compute an infinite number of things. No mathematician is saying that in practice we can compute an infinite amount of things. What they claim is that some infinite series behave well enough, as we consider larger and larger finite numbers, that we can assign them a particular value.
What they claim is that some infinite series behave well enough, as we consider larger and larger finite numbers, that we can assign them a particular value.
If you dig into that line of thinking you don't get the real numbers, you get the computable numbers.
Though, Wildberger doesn't think the computables exist anyway.
For what this video intends to demonstrate, and the emphasis Wildberger places on computability in this video, I think it's fine to just consider computable numbers. Though as a side note, disputing the existence of the reals seems a lot more sensible to me than disputing the existence of computable numbers. One fact that always bugs me is that a symbolic language with finitely many symbols only permits countably infinite definitions, (specifically, definitions of finite length), yet the reals are shown to be uncountable, means that almost all numbers in the continuum are unreachable, not even specifiable as the limit of some infinite process, like π or e might be, or numbers which are definable but which cannot even be computed. I don't know if that means that those reals don't "exist," but it's something interesting to think about.
@@joshl8757 extending that argument slightly further, there are only countably many *proofs* as well.
So you can only prove there are countably many things in the set of real numbers. You cant obtain that list of provable reals within the theory, but if you "diagonalize" on the provable reals, you obtain a sequence of digits which somehow has no proof that it is real.
That sequence must of course not be a computable number, since then you could prove it was real. So that number would need to be inexpressible just as you noted.
But this example calls into question the notion of even having "uncountably many things" in a set where you can come up with a diagonal argument.
If the set of reals had uncountably many things in it, then we seemingly need uncountably many proofs, and so we need uncountably many letters in our alphabet, which is completely incoherent.
@@samb443 I don’t totally agree. There are countably many things you can prove are in the reals, but that isn’t the same as saying that you can prove there are countably many things in the real numbers (I think that’s just a typo) But more importantly, the fact that something cannot be proven to exist in the reals doesn’t necessarily mean it’s not in it. Gödel’s incompleteness theorems, for example, tell us that no matter what system of axioms we use, there will always be true, unprovable statements present in that system. So the reals can still be a well-formed and reliable mathematical object without being completely permissive to proofs.
@@joshl8757 true is literally meaningless without proof.
if something cant be proven to be in the reals, it is not in the reals by definition. There is no platonic realm you can back up to in order to find an impossible proof somewhere else.
Everything is not "in reals until proven guilty"
"Gödel’s incompleteness theorems, for example, tell us that no matter what system of axioms we use, there will always be true, unprovable statements present in that system."
It does not prove that there are "true" unprovable statements, it proves there are "consistent" unprovable statements.
1/2 is consistent with the natural numbers, but it is not true that its in the naturals.
There is no contradiction, the "set of all provable reals" isnt actually a set, it cant be constructed within the theory.
So there cant be a bijection from the naturals to it.
This is why the diagonal argument works, there arent "more" reals than naturals, the appropriate bijection is just not well-formed, that's why it doesn't exist in the theory.
Its not really a typo. If you take a different set that's "uncountable", you can only get countably many proven elements in that one too. And, if there is a bijection between that uncountable and the reals, its also a bijection between the provable elements of them.
Theres no difference in the way you do math if you assume an inaccessible countable set of reals, or if you assume an inaccessible *un*countable set. They work the exact same way, or rather, they *don't* work in the same ways.
The emperor of pure mathematics has no infinite computation device.
The emperor of our mathematics does have an axiom of infinite computation.
I love it. Beautiful argument. You have solved it!
Is 1/3 like PI because it has an infinite decimal in base 10?
Not at all. The number 1/3 is just a fraction, part of fraction arithmetic. Analogs of 1/3 occur in any place value or base system. In some base systems a fraction might have a finite expansion, while in others it might not; in the latter case we have to augment our system of finite "decimals" with a more complicated, but still completely finite, theory of "repeating decimals". While this is rarely done properly, it is quite reasonable and ultimately non-problematic if done carefully. Remember that the decimal (base 10) expansion of 1/3 can be expressed by 0.\overline{3} which is a completely finite expression. Its "infinite aspect" is somewhat illusory.
I think you're looking at infinite sums the wrong way. Any expression in mathematical notation is just that: a formulation of an idea. An infinite sum doesn't mean I have to do infinite work by summing up infinite terms, it means that I have an idea in my head that can be expressed as an infinite sum. Eg, e = Σ(1/n!) exists and is well defined and it's fairly certain to say that no man or machine in history has summed up _all_ terms :-)
Well I have a lot more “ideas”: concerning Loch Ness monsters, ghosts and abominable snow people. Are you excited to learn about these new theories?
@@njwildberger I'll watch that vid, Prof, where you describe Nessy with an infinite sum.
@@njwildberger Maybe I didn't explain my point well enough. An example: you can approximate, and eventually describe, a circle with a polygon with infinite vertices. But that's not how circles are drawn...
I'm not looking to be polemical, but it's almost like you're trying to hide infinite sums in your head so we can't scrutinize them. Granted, I just see this as kicking the can down the road.
Ok, so it's an idea in your head. The next question I ask is this: are ideas in your head some way such that they're somehow not finite? If so, do tell. But if ideas in your head (for some definition of 'idea in your head') are themselves finite entities, and if the referent of an infinite sum is an idea in your head, then the referent of an infinte sum is something finite.
What are your thoughts on math and the notion of a God? Math and theology.
I don't know anything about God.
I'm curious, given your belief that the finiteness of the universe proves infinite processes cannot complete, how do you accept the "proof" there are unlimited (infinite) primes, or more fundamentally, why do you accept the proposition "we can always add 1".
According to the argument that we can't compute infinite series in reality, we should also recognise all numbers are finite, therefore "we can always add 1" is false.
This makes it possible to claim *all* integers are modular, although we don't know the very largest modulus which could be eg. the number of gluons in the universe.
If you think about it, this would mean it's no longer true that primes are countable against integers. The whole house of cards crashes down.
On some other occasion, I am going to have to explain why in fact we do not know that "there are unlimited (infinite) primes". But you can determine this yourself. Just have a really good look at the usual proof, going back to Euclid, and ask yourself: what hidden assumptions are lurking here?
@njwildberger oh excellent, I thought I'd seen a video where you gave P+1 is prime as a proof. Looking forward to that video!
Do you agree then that there must be many less primes than integers?
If that's true, then many expressions that are bread and butter of number theory fall over because they rely on there being a 1-1 (usually term to term in convergence test) relationship between integers and primes all the way to infinity.
I think that you have your own doctrine that is just as absurd as the other mathematicians. I mean the real numbers can be derived from arithmetic so, why object to their existence?
Well why don't you start by giving your definition of real numbers and then the addition operation of real numbers, and then demonstrate this supposed arithmetic with a non-trivial example in which the objects you use actually completely fulfill the requirements of your prior definition.
.. and infinitesimals, and that we can actually order (find the next) the reals and irrationals that appear to alternate...
Oh, the inability of regular mathematicians to 'explain' in common language these conundrums!
Professor Wildberger, it really seems to me like you're arguing against a strawman. I've never actually heard a mathematician claim you can "do an infinite number of things." Rather, there are objects like pi and sqrt(2) which arise naturally and about which it is possible to deduce things without computing them exactly. Similarly, there are certain infinite sums about which we can deduce things, and indeed every technique I've seen for deducing the value of a convergent sum is about somehow avoiding having to do infinite things, which indeed would be impossible. I also find it pretty underhanded to imply that exp(49/20) and the truth value of RH are in any way comparable simply because they both require infinite steps to compute. No one has EVER claimed to be able to carry out any infinite process you throw at them, there is a very limited class of such processes that we are equipped to handle. It is not arrogance to make deductions about things we cannot comprehend with 100% accuracy. Arrogance is seeing something you cannot comprehend with 100% accuracy and deciding that it does not exist.
He has a video on irrational and transcendental numbers. Since any length L consists of finitely many points, the sides of triangles in general do not connect was the argument used I believe.
Of course they don't explicitly admit that their theories involve "doing an infinite number of things". Because that would make it too obvious that the logic is questionable. Instead, we have developed a carefully crafted jargon that allows us to get away with the same basic idea, but without overtly admitting to it.
For a great example, consider the "Axiom of Choice". Please explain to us how this does not really let us get away with "doing an infinite number of things" without perhaps saying so explicitly.
But there are numerous other examples. Have a look at the definition of an "algebraically closed field", or a "variety over such an algebraically closed field", or a "topological vector space over the real numbers" etc etc. Just try writing out the complete definition of any of these, and see if you still think I'm arguing against a strawman.
@@njwildbergerDon't worry about Blake, Prof. Wildberger. He had a bad day at work, and then he accidentally burnt his dinner. Good ol' Blake!🙂
@@maynardtrendle820 This is an error in logic you just used: Argumentum Ad Hominem. Avoid that.
@@njwildberger The Axiom of Choice is named so to give an intuitive understanding, but there isn't actually any choosing happening. AoC is a "there exists" statement: There exists a "choice" function. We don't claim to actually make those infinite choices, we simply accept that there is such a thing and we can then proceed to determine some of its properties. As for algebraically closed fields, varieties, and topological vector spaces, nothing about their definition implies anyone is doing an infinite amount of work. No one claims that the existence of a topology on the real numbers means we can list all the open sets, but from the definition, it is possible to determine some of its properties. No one is doing infinite things.
you make the same point over and over again and totally missing the simple fact, that exp(49/20) converges and we can proof this. if you proof, that your Wmr converges, _than_ you have proven the Riemann Hypothesis, but not before.
I don’t understand your claim. How can exp(49/22) converge? Is it not ostensibly a “real number”? In which case the term “converges” does not apply to it.
I think this might be your best one yet!
Great as usual, Merci Beaucoup!
oh.... and what about the power set of of the states of the universe? I guess the universe is bigger than we thought !
Can you write down that set, so that we can have a look at it and admire it?
@@njwildberger Give me a minute...
I am happy to say that the accessible universe fails to encompass one of such fantastic calculations. I have nothing to say about what lies beyond the accessible universe.
Brilliant punchline. In the language of modern mathematics, you actually solved the Riemann Hypothesis. That is indisputable.
Did you solve anything? No. But you showed that the language must be revised, if the Riemann Hypothesis ever truly is going to be solved.
Well done. Given even an infinite amount of years, no Aİ would ever be able to come up with a logic argument like this. And this, sir, convinces me that you are no Aİ. Thank you very much indeed!
My point of view as a student layman.
What Wildberger is saying here is right.
Math is logic using numbers.
If infinity is a quantifiable number then we could use it to quantify an exact number.
It's a crutch. Used to approximate numbers for engineering purposes.
Infinity is not a number. It's not a part of true mathematics.
No more than 'yes' or 'no' are part of a logical argument.
The mathematical expression "Infinity" is a YES or NO.
There is no yes or no in logic. YES or NO are the answers.
Logic is And, Or, Nand, Nor.
Not Infinity as an open DOOR.
So Yes is an answer.
No is an answer.
Infinity is a 'yes' or 'no' in the question.
It's a crutch used to establish engineering limits.
Nothing wrong with that.
But it's not math.
Overheard in a bookstore: "Have you heard about Google's new TPUs? They are so fast they can execute an infinite loop in 6 seconds!"
I definitely will be in the initial line up for such a machine.