Greek Pure Mathematics and the "Infinitude of Primes" | Sociology and Pure Maths | N J Wildberger

[@richardfredlund3802]

firstly if someone wrote down a formula -- you could still make the same objection, that dispite having a formula there's still no way to do the calculation because the numbers won't fit on the hard drive. While there are potential ambiguities involved with the notion of infinity and the real numbers . There's nothing ambiguous about the Euclid proof of primes. It's an existence proof, so it doesn't actually require doing the calculation. your same argument could be applied to the commutativity of multiplication for integers. Do we know that commutativity works when the numbers get too big to calculate? Of course we do.

[@TheRealSeus]

We need to distinguish between "exists" and "can be computed". Euclids proof shows the existence of more primes than a finite list can hold, it does not say anything about how to compute or even how to represent those numbers. In that way i am willing to except that there must be primes greater than 10^10^10^10. If you ask: So tell me, which prime is greater than said number? Thats a different question and needs a different proof, it raises more questions about computability and representability etc. Existence and computability/representability are different concepts, which should not be mixed up in my opinion.

[@njwildberger]

@TheRealSeus You bring up a very important point. However as Aristotle already pointed out more than 2000 years ago, the concept of "existence" is hugely problematic. Does the "prime factorization of Harry Potter" exist? How about the "the set of all fish in the universe"? Or "one more than the biggest number you can think of". Fun and games, but probably not mathematics.

[@njwildberger]

And I should add, when we stick to computability, we find a way to circumvent this philosophical ambiguity. Can you write it down or not? End of story.

[@Gabriel-iz5kd]

Computability is not what you think it is. The senile guy thinks that 10^90 doesn't exist because he can't put 10^90 snails inside a leather pouch, so this is far from any reasonable discussion.

[@njwildberger]

@@Gabriel-iz5kd Forget about "the senile guy". How about your computer: does it believe that 10^90 exists? How about 10^90^900 ? Or maybe 10^90^900^9000? At what point are we going to say ... OK maybe we are getting beyond ourselves here?

[@santerisatama5409]

@Amat Uccan It's much easier to trust computers than programmers and datainputs. What we should not trust is that Turing machine - classical or quantum - is the final word of computation theory. Hypercomputation is interesting possiblity and field, but as usual, it's necessary to learn to walk before you can run. And as demonstrated by this lecture, mathematics is a slow art.

[@NikolajKuntner]

Dear Prof. Wildberger, could you please tell whether you consider the following statement to be true, as stated: "𝐹𝑜𝑟 𝑎𝑛𝑦 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑛, 𝑒𝑖𝑡ℎ𝑒𝑟 𝑖𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑎𝑠𝑒 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑝 𝑤𝑖𝑡ℎ 1

[@njwildberger]

@ Nikolaj-K To properly answer this question I first have to understand it. That crucially requires us to ask what is your definition of "natural number". Is it a Hindu Arabic expression like 3,451? Or does it include arithmetical expressions involving possibly higher operations, like z=10^10^10^10^10^10^10^10^10^10+23 ? Or perhaps it is something else. Also, do you allow the possibility that whatever representation you choose, it has to actually be written down? Or are we also including "numbers" which fit your specification but are so big they don't fit into the universe?

[@NikolajKuntner]

@@njwildberger For the "𝐹𝑜𝑟 𝑎𝑛𝑦 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑛,..." statement above, let's say the scope of what we want to call numbers here includes your z (an expression at least specified in terms of the recursively definable exponentiation operation ^ and at least its initial arguments 10 and 23 translatable into your strokes on a blackboard). Now when you argue that there would be such expressions y (say some even vaster powers of z) for which we won't ever compute its factors, would you say that this translates to a mathematical claim that y simply has no prime factors smaller than itself? The interesting thing here of course being that most mathematicians would say that if it is established that a number y does not have any such prime factors, then it is itself prime. In summary, do you reject the decidability of this excluded middle statement about primarily I started with, or are a whole lot of naturals at large size (such as z or y) prime simply by the fact that to be prime means to "not have a prime factor" (which you make the case can't ever be computed and thus doesn't exist).

[@njwildberger]

@@NikolajKuntner You are asking: what does it mean for a number n to be prime? Computationally, it means that n is not divisible by any prime number less than it, or say less than an approximate square root of it. If we cannot perform such a calculation, then we cannot say that n is prime. The computation of "primeness" is much bigger than the number n itself, at least if n is big. So you can think of "computing primeness of n" as being exponentially more complicated than computing n itself. So long before we get to z, this exponential complexity of computing primeness goes beyond the computing power of the universe. Nothing to do with our computers. That is an aspect of the world we live in, as far as we know. If we lived in a huge digital world consisting of k registers, flipping around according to certain predetermined laws from 0's to 1's and back again, then there would be a maximum number 2^k of possible states. In such a world, there would be a necessary limit to the extent of "natural numbers", even to the digital creatures within it. Do we live in such a world? Maybe -- important thinkers like Stephen Wolfram are very interested in such possibilities. Are there an "infinite number of primes" in such a world? Not a chance.

[@NikolajKuntner]

@@njwildberger If I understand your correctly, for a notion of numbers like z, you'd not accept the initial statement above as true. Indeed, your "we cannot say" then applies to a lot of common mathematical statements as well as their refutations. For those expressions (e.g. involving exponentiation, as does z) I don't yet know how I could make a prediction on how you'd judge different statements of the form "𝐹𝑜𝑟 𝑎𝑛𝑦 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑛,...". A statement like that for all n>7, the number 7 is smaller than it is a simple tautology for all n. But if I'd say that for all n, the number n+7 is bigger than n, is this true almost by definition or would there be an unknowable but hard limit where this stops being true? How can I know which for-all statements about numbers you accept and which you reject?

[@njwildberger]

@@NikolajKuntner It helps to start by clarifying in your own mind the distinctions, if any, between 1) a natural number 2) an arithmetical expression for a natural number that can be evaluated, and 3) an arithmetic expression for a natural number that cannot be evaluated. Examples of these three are 1) 52 2) 4^5+3^4 3) 3^4^5^6^7^8^9+9^8^7^6^5^4^3. You can be sure that your computer is very much interested in distinguishing these three possible scenarios --- and students of mathematics ought to be too!

[@jryer1]

As a boy, we still studied Euclid Elements, which was obviously demonstrated using geometrical shapes. Today I shake my head at what is taught, and how it's taught. Most young adults today can't even contemplate basic geometry. Truly, a sad state of affairs.

[@mimzim7141]

how old are you? in which country?

[@jarrodanderson2124]

I'd really hate to disappoint you but the best mathematics is being done right now. Sorry

[@DnBComplex]

​@@jarrodanderson2124 he said something about teaching and you took it as an offense to math itself.

[@brendanward2991]

Euclid's logic seems sound to me. So the question is: are his premises well founded? To undermine his proof, you would have to look at his earlier result (The Fundamental Theorem of Arithmetic) and see if its proof is sound. And so on.

[@dsm5d723]

There was no data science emphasis on his dimensional understanding of Nature. I derived his proof dimensionally; the 1D algorithmic execution of modern "irrational" math resolves within the Euclidean plane. And a 2D square time-stacked computation space would have made him first laugh, then vomit. 180 degrees to an energy dipole, 360 to a matter monopole. Rotate dynamically to fill a sphere of "gravitational" influence mathematically. His Arithmetic is sound, it is fractal factorial embedding to infinity in self-similar scaling. NJ has it right, I did it by curating the language of math and closing the logical gaps. Monte Carlo machines and economics, and the physics of war, stole the virtues of pure mathematics. Wittgenstein's Ruler made a tape measure. Look at the stuff on self-locating loops and strings; that is the informatics of arithmetic IN 2D. In 3D, bioinformatics to simplexity. Two time directions makes magic.

[@dsm5d723]

@@gorkym8864 Einstein is a T-shirt to me; I just summed over reality. Compute 'em and weep. gab.com/23andMe24andYou/posts/105368388483523590

[@dsm5d723]

@@gorkym8864 With integrity in my Integrals, bruh. L0l.

[@mimzim7141]

i think prof. Wildberger also critisized the fundamental theorem of arithmetic for the same reason i explain below. But here logically you don't need the find a flaw in the premises. He is saying that it is not in general possible to do the computation requested by the proof. Example "take the least number measured by A, B and C: the critique is that you cannot take (or find) that number in general, you can only do it when your inputs are small. The logic that seems sound is saying: if you take that number than this and that... and NJW is saying you cannot take number. So he is requesting that you can actually do all the steps as specified by the proof for any given inputs. Not just that in theory given infinite time and energy you could.

[@dsm5d723]

@@mimzim7141 Thank you; analytical continuation. My only addition is that WE have a 70 year tradition of letting machines think for us in the big areas of inquiry. NJ did a video on the "issues" with modern math emphasis. Square computation space and data science-serving stacks to Reimann infinity in a square vector space shape the mind. Not a problem for the Ancients, as Taleb has noted. How does an Escherian scaling to Euclidean logic sound? I reject the Borgesian mindfuck completely.

[@hermannwacker1902]

I have been a fan of prime numbers since school time. I am especialy interested in patterns in prime numbers. Looking for patterns it is often very helpful to visualize structures. Very interesting are prime number patterns I have found in the first euclidien numbers: 2*3+1 = 7 2* 3*5 + 1 = 31 2*3*5*7+ 1 = 211 2*3*5*7*11+ 1 = 2311 I have written down the natural numbers in 4 different tables and have highlighted the prime numbers. The 1. table has 6 = 7-1 columns. The 2. table has 30 = 31 -1 columns. The 3. table has 210 = 211-1 columns The 4. table has 2310 = 2311-1 columns In the tables it is easy to see column ranges that contain primes and column ranges that contain no primes forming rectangular areas! To visualise these tables I have written prime number software in JavaScript and have visualised the tables as html tables. In doing so one comes soon to the limits of a html browser and the limits of Javascript. All my prime number pages are still in german (Primzahlen) but the language of numbers and patterns is international. Question: Is there a fundamental law in mathematics, that describes these patterns? What are the idears of other people when they see these patterns? Hermann P.S. The links can be found in my first comment.

[@hermannwacker1902]

An overview can found on my main prime number page. www.wackerart.de/mathematik/primzahlen.html 6 columns: www.wackerart.de/mathematik/primzahlen.html#rechteck 30 column: www.wackerart.de/mathematik/primzahlen_rectangle_2.html 210 columns: www.wackerart.de/mathematik/primzahlen_rectangle_3.html 2310 columns: Attention this may be a tough job for your html browser. www.wackerart.de/mathematik/primzahlen_rectangle_4.html

[@robharwood3538]

Nice website. Lots of interesting stuff on Geometric Algebra.

[@hermannwacker1902]

@@robharwood3538 Thanks. I am a fan of geometric algebra.

[@mimzim7141]

15:15 "DE the least number measured by A B and C". Isn't DE in this definition the least common multiple of A B and C rather than their product.

[@JoelSjogren0]

That's what I thought too but if they are primes then it doesn't matter.

[@volodyanarchist]

No. Let's say we have primes 2, 3, and 7. We can construct many numbers that are measured by these primes: 42 works, but so does 84. You spotted the mistake, but a wrong one. "Measured" doesn't mean "least common" at all, for primes it means p1^x * p2^y * .... pk^z, where powers are more than or equal to one.

[@mimzim7141]

@@volodyanarchist i think "least number messured by A, B and C" would mean the least commun multiple. But you are right that in thie particular case, since A, B and C are supposed prime then the least commun multiple is also their product.

[@volodyanarchist]

@@mimzim7141 There was no "least" word in the proof by Euclid as i understand.

[@mimzim7141]

@@volodyanarchist 13:49

[@id3655]

Professor, please read the (controversial) first volume of Spengler's "Decline of The West : Form and Actuality," which is available for free on archive.org . Chapter 2 discusses the conception of number!

[@jasonc0065]

Does it discuss Antifa and white self-hate, too?

[@Zebedex]

​@jasonc0065 no no no, you see.. you can't question the actions and intentions of antifa. Why you ask? You see they call themselves antifacist. Duhh it's in the name. You see because their name means something good, the whole group can never questioned! Obviously it's true that If I make a group and call it the antibadguy group it means it can never do wrong and should never be questioned. In fact anyone who opposes my new group must be a bad guy. Why? Well because my groups name is antibadguy, duhh. It's impossible for my group to do anything wrong. Why? Well because the name means anyone who goes against us must be a bad guy. It's in the name, duh! The name determines everything! You can never question an organisation if it has a nice name! How dare you?!

[@coshvjicujmlqef6047]

Professor, do you have a series about abstract algebra and algebra geometry?

[@KarmaPeny]

Excellent description of how the Ancient Greeks equated natural numbers to measurable lengths. We use 'numbers' all the time without giving much thought to what a number actually is. Another up-vote from me. It makes a lot of sense to me to think of a natural number as representing a measurable quantity. Having a tangible meaning removes a lot of the weirdness of having to imagine that numbers are transcendental objects with their own mystical existence. I am a huge fan of your work and I agree with what you say about big numbers. I agree with you that a lot of mathematics needs to be re-worked especially where the concept of infinity is concerned. Being an atheist computer scientist, I view the brain as being nothing more than a finite biological computing device. Even though we don't fully understand how it works, there is no reason to believe that it is anything more than a finite machine working with a finite amount of data. Norman, where we differ appears to be on the existence of numbers. A computer can work with numbers without any ability to access the 'infinite' because numbers are created as-and-when needed, they do not have their own out-of-computer existence. I believe the same argument should be applied to humans working with numbers. In other words, numbers are 'in-brain' objects that are created as-and-when required. Numbers do not already exist out there in the universe. At any time there are a finite number of brains and hence only a finite amount of numbers can ever exist. Do you agree? If not, then how do you explain the 'existence' of numbers?

[@ROForeverMan]

Brain doesnt exist. "Brain" is just an idea in consciousness.

[@NuYiDao]

There was an historical interlude, possibly in the enlightenment era, when Mathematics was looked to for notions of absolute truth which turned out not to be the place to look. Infinities are still marvelled over today as if they are a secular replacement for deprecated notions of the divine. Really enjoying this fresh perspective which brings much needed humility to the discipline of math's which I think has become constipated thru accumulation of dogma. Especially like the fact that in a Nietzschean way this New Thunder is actually classically sourced. Not to say that infinity doesn't exist, I think the Romantic perspective is correct, and the incommensurable is everywhere, but it is certainly not always helpful in mathematics, in the age of generalised computing, to reserve forbidden spaces & to consider the ineffable in a meaningful way. Maths should be grounded, where at all possible we must seek the ground truth.

[@MrCmon113]

>when Mathematics was looked to for notions of absolute truth which turned out not to be the place to look That's exactly what you are doing and why you have a problem with the infinite.

[@NuYiDao]

@@MrCmon113 Give me a bit more than assertion and I could reply to you..

[@santerisatama5409]

Lovely.

[@leandrobastos6962]

According to your thinking it's possible to obtain a mathematical law that breaks down when applied to bigger elements of the same kind. Is there any record of this? Can you prove the feasibility of YOUR assumption? Maybe it's a stupid question, but I'm a complete layman in mathematical terms.

[@spiveeforever7093]

Does subordinate mean non-constructive? Or is there more to it than that?

[@thea.igamer3958]

Please make a video on your take on non standard analysis

[@ChrisDjangoConcerts]

The music matches the topic of the series very well

[@xxlabratxx01]

Pythagorean scale?!?!?

[@brendawilliams8062]

Professor Wildberger you could lead Helen Keller to the path. Thankyou so much for leading me to a definition of how to see the math I love.

[@strangeWaters]

Hellen Keller was actually an intellectual fyi.

[@jaanuskiipli4647]

Actually the term 'infinite' just states that it is not bounded by any finite amount, without stating anything about the actual amount

[@njwildberger]

@Jaanus Kiipli We must be careful to recognize that even though we might like to use a term in a certain way, others do not necessarily follow our example. Pure mathematicians like to discuss "infinite sets", and in fact these are the basis for entire disciplines, among them topology and analysis and differential geometry and much of algebraic geometry. So it comes down to the old distinction between "potential infinity" and "actual infinity" laid out already by Aristotle, the latter also sometimes called "completed infinity". If we are not careful about the distinction, we allow ourselves to slip from one meaning to another perhaps even without noticing.

[@santerisatama5409]

@@njwildberger There's also a nuanced distinction between potential infinity (noun like) and unbounded process (verb like), the latter closely associating with undecidability of Halting problem.

[@cunningham.s_law]

is the an equivalent of prime numbers in the rationals or in matrices?

[@thomassynths]

en.wikipedia.org/wiki/Prime_element en.wikipedia.org/wiki/Irreducible_element

[@kylemoulton1948]

I was so excited to see the continuation of the famous math problems on infinitesimals, and this is what I get. Also, I have a question about Euclids proof. When ‘Euclid’ says, “take the least number DE measured by A, B, and C”, is that length the modern day lcm of A, B, and C or is it the product of said lengths

[@njwildberger]

@Kyle Moulton I have clarified the interpretation at the end of the video description, as several people have mentioned this.

[@kylemoulton1948]

Thanks for the clarification! Also, my roommate and I get into many passionate (both mathematicians) debates about your claim that there is a physical limit (notable speaking the size of our universe) to our ability to factor large numbers. They say that mathematics shouldn’t be refined the limitations of our universe. Meaning that these large numbers have unique factorizations up to permutations even if they are not accessible to us in this universe. Their claim follows for other mathematical objects that are transcendental to our universe such as irrational numbers. My question is should mathematics be defined inside the constraints of our physical universe, or can it’s domain be much larger?

[@njwildberger]

@@kylemoulton1948 Physicists have also in this past century had to wrestle with similar philosophical conundrums. Can we have a theory about things which we can't measure? Do such "things" even have a right to be talked about? I think the consensus now is that you want to stick with concepts that are supportable by actual experiment and observation. That might be a narrow position, but it does remove a lot of auxiliary chit chat about extra planes of consciousness, astral forces, underlying deterministic theories of QM etc. So maybe we can frame the discussion into two broad possibilities: Type A) mathematics supported by computational reality involving explicit examples and concrete spcifications Type B )"mathematics" supported by claims about "objects" which have no possible manifestation on our computers, on our pads of paper or in any other physical form, and which encourages armchair "theorizing /talking". Clearly Type B "mathematics" extends for beyond Type A mathematics. Then we can let the new generation make some decisions on their own about which kind of mathematics most interests them.

[@kylemoulton1948]

@@njwildberger If type A mathematics is contained in type B mathematics, and the computability of type A is derived from type B via accessibility with the constraints of our universe, would that imply that we could directly compute, interact, or observe type B-A mathematics if we were not constrained by our own universe, biology, or other limitations? In other words, can the propositions proposed in type B mathematics and proved to be true in the domain of type A mathematics, inductively imply the propositions in type B-A mathematics to be true as well? Or does the logical systematization of mathematics fail because us humans cannot observe them over the horizon? In order for something to exist, does it have to be observable by a human. If so, I think that is a highly anthropocentric view. Consider the limits of observability in physics with this case study: “Is space discrete or continuous?” This question is outside the domain of the scientific method as of now, because there is no known way of observing the smallest increment of space, if it were there. Or as our technology increases to look more closely at space, we cannot assume it is continuous because we haven’t found a discrete unit. Nonetheless, said question has an answer, even if it is not as simple as one or the other. Maybe the question needs more information or refining (like what exactly is space, what are its constituents, how are all these properly defined, etc), but after the question is posed in way such that it has an objective answer, it is still beyond the capabilities of science to answer it. This is an example of a universal existence that does not meet our ability to observe so directly.

[@santerisatama5409]

@@njwildberger I don't think primary empirism - intuition - should be kept apart from pure mathematics. That's how you get arbitrary axiomatics of Formalism, and mathematics as nothing else but language game. Going intuitionist all the way to the rabbit hole of idealism is a coherent position that unites (introspective) empirism with rigorous construction. Pure math should not be a-priori self-limited by any sort of physicalism/realism. Math is more about relations than objects. Which doesn't of course justify postulation unreasonble concepts such as bounded unbounded aka actual infinity while claiming to subscribe to LNC. Current limits of computation should not be limit of mathematics. Hypercomputation is also a field of study of pure math and computation theory.

[@brendawilliams8062]

I thank you for clearing the prime theory up

[@CandidDate]

(Triangle #10)^(Triangle #10) = (Triangle #10 000 000 000)? Assume all rationals (Triangle #R) are triangle #s. Example (Triangle #3) = 3^3^3 = 3^9 = 19, 683. Now, can you do any sort of arithmetic with the triangle #s? Like (Triangle #25) / (Triangle #10) = (Triangle #25 /10) = (Triangle #2.5) This way you could represent astronomically huge numbers, but do regular or semi-regular arithmetic with them. Like the laws of exponents where 3^2 * 3^3 = 3^(2+3) = 3^5. Here, addition of exponents is used when multiplying powers with a similar base. You would never actually "cash out" the huge numbers, but have a representation of them which can be compared by size. Just an idea. Of course, primes are integers and fractional triangle numbers are decimal expansions. In this way, you could define all non-primes as triangle numbers and what you have left over is the list of possible huge primes.

[@tjejojyj]

Thank you. The difference between Euclid’s to show you can find another prime and the variant that “shows” there are infinite primes is significant and new to me. Would proof of infinite primes would be correct if it concluded “so if we had infinite time and infinite computing power we could create a set of infinite primes”? ie. it’s not a proof but a conjecture or hypothesis that is unobtainable. The idea of iterating Euclid’s proof indefinitely seems intuitively possible but just runs in to an infinite process. The fundamental issue seems to me to be about the Platonic postulate of all numbers already existing. If this is true then we can derive something about their properties. If it’s not true all we can do is speculate about what might be calculated.

[@FisicoNuclearCuantico]

Stay healthy professor. Best regards.

[@apolloniuspergus9295]

Professor Wilberger, I have seen many of your videos, and seen your point of view, and now I ask you to do something that would prove your point once and for all: try to derive a contradiction from the assumptions that you are opposed to using propositional logic and publish a paper about it. I long for seeing it

[@palmtoptigeri9797]

It may be the case that philosophy is more than just propositional logic. It may be the case that what you are asking for is impossible. It may be the case that Wildberger abuses language when he refers to his criticisms of mathematics as "logical difficulties".

[@apolloniuspergus9295]

@@palmtoptigeri9797 Seems like if he is right, we wouldn't be able to model continuity algebraically

[@santerisatama5409]

@@apolloniuspergus9295 Mereology of continua is very doable modeling. Unless by modeling you mean reduction of continua to the idea of discrete quantification. But it seems continua are deeper than discrete quantification. Empirical and logical observation! Fine, a firm foundational ground to start to construct from. :)

[@suvarnasuvi20]

Thank you

[@paperell]

I was reading a book on number theory, the moment I saw the proof I knew if I came to the professor's channel l I would find criticism of it 😁

[@theboombody]

I love how every video on this channel gets so many comments.

[@dsm5d723]

Euclidean-Minkowski Place-Time. You are a fine purity tester in mathematics, sir. And it IS back to the floating 60 point system, or base 10 X the 3x3 intersubjectivity of language; LOOK at the geometry of Cuneiform. Hammurabi indeed.

[@robharwood3538]

Prof. W, I think your argument regarding slide 9 is not stated with enough clarity/rigour. It seems to me that you are really trying to argue that we cannot use Euclid's argument to prove that *the* generated prime *is guaranteed* to be greater than 10^(10^10). However, I think it may still be possible to generate *a* prime that just *happens* to be bigger than 10^(10^10) using Euclid's methodology, if we just happen to get lucky in our initial choice of a smallish collection of primes. Consider a happy accident where we happen to choose just the right set of initial primes such that: p1p2...pk + 1 = 2^m - 1 Let's call 2^m - 1 as M, and suppose that it just so happens that M is Mersenne prime. And that M is greater than 10^(10^10), but not much greater; perhaps it's less than 10^(10^11). And further suppose that we can computationally confirm that M is indeed a Mersenne prime. In other words, we are able to find the prime factor q in the third step, and it just so happens that q = M. Thus, in this happy accident, q, the new prime found by Euclid's methodology is a prime greater than 10^(10^10). We didn't prove that q was absolutely *guaranteed* to be greater than 10^(10^10), we just got 'lucky' and it was. After all, Euclid's proof has no guarantees about the size, it just covers the *case* when the size happens to be as big as the product of the primes + 1. Lastly, I think this is not *really* that unreasonable of an idea. Pretty unlikely I admit of course, but if you start with the lowest known primes such that their product is just barely above 10^(10^10), and start your search from that point, at least you could maximize your probability of stumbling upon such a prime. Also, it doesn't have to be restricted to a Mersenne prime: There are other kinds of primes that can be more quickly confirmed than through brute factorization. All that said, I *do* understand your intended point/argument that we can't use Euclid's proof to *guarantee* that whatever generated prime we get will be greater than 10^(10^10). It's just that the way you laid out the argument didn't seem to make this aspect of your point clear enough, in my opinion, opening it to the objection I just raised above. Cheers!

[@palmtoptigeri9797]

This reminds me of the intro to the television show The A-Team: "If you have a problem, if no one else can help, and if you can find them - maybe you can hire... The A-team" So if you have a collection of primes, if these primes are represented in such a way that you can do arithmetic with them, and if you can actually perform such and such computations - surely (not maybe!) you can find a prime not of the collection you started with.

[@njwildberger]

@Boaz K Sorry but it is not so sure at all. The point is that the presciption involved in Euclid's argument is impossible if the number of primes is large. You can't always form a product: for example you cannot form the product of all the numbers from 1 to 10^10^10. It is impossible. So what that means is that it really does not make sense. To say "take the product of all the numbers from 1 to 10^10^10" is like saying "walk to Andromeda". It becomes an empty phrase with only poetical or religious meaning.

[@MrCmon113]

@@njwildberger Maths is all about making general statements about things you don't compute. You can often show that an equation has one solution without providing one solution.

[@njwildberger]

@@MrCmon113 This is the kind of thing that sociologists can sink their teeth into. Various practitioners or students might make a wide range of claims about "what mathematics is about". This is sort of similar to the many debates on the relation between "science" and "religion". No doubt religious practitioners of science historically have been more inclined to include explanations of natural events involving supernatural causes (reference of course needed!) Could it be that mathematicians who fervently want to believe in mathematical objects/things that you can't actually compute are in a similar position?

[@palmtoptigeri9797]

@@njwildberger I know a mathematician who had founded a political party based on atheism in my country, and he said to me something like "one who does not believe in (mathematical) infinity must be a miserable person".

[@njwildberger]

@@palmtoptigeri9797 Most of us say silly things at some point.

[@brendawilliams8062]

Thankyou

[@danielmilyutin9914]

Euclid proof of infinite number of primes doesn't allow to generate primes. Counter example is: 2,3,7=2*3+1, 43 = 2*3*7*43+1 = 1807 = 13*39. The proof of Euclid is based on contradiction and is not constructive. But I can agree with your other arguments about "philosophy of finity".

[@brendanward2991]

A Boolean analysis of these Euclidian proofs could be enlightening!

[@Kraflyn]

p1 p2 p3 ... pN + 1 isn't divisible by any pn for 1

[@hermannwacker1902]

That is not so easy as it seems to be on the first view! Not all euclidean numbers (p1 p2 p3 ... pN + 1) are prime. This is not an algorithem for calculating huge primes when you know all primes up to prime n! Here is a list of euclidean numbers some of them are not prime: www.wackerart.de/mathematik/big_numbers/euclid_numbers.html E6 = 13# + 1 = 30031 = 59 × 509

[@Kraflyn]

@@hermannwacker1902 I didn't say it's an algorithm for calculating primes :D It's just a proof of infinitude of primes. Assume only N primes existed, and here it is, the one single sentence proving that there exists a prime larger than the largest prime, which is a contradiction. Already Greeks knew this. So why would he claim that we don't know if there are infinitely many primes? oo

[@hermannwacker1902]

@@Kraflyn Your first sentence stated: "p1 p2 p3 ... pN + 1 isn't divisible by any pn for 1

[@Kraflyn]

@@hermannwacker1902 Hey. He says somewhere in video that we today don't know if there are infinitely many primes. I don't know if he forgot to mention that he meant primes "of that particular form" maybe... But his statement was that we don't know if there are infinitely many primes... Which is proven wrong by Greeks in a single sentence... Namely, "assume there are N primes and consider number...." So, by this assumption, there are no more than N primes, and so the last prime is by assumption pN. But with this assumption, p1p2...pN+1 is clearly not divisible by any prime, of which there are N altogether, and so p1p2...pN+1 is by definition prime itself. And it's also larger that pN, obviously. So... I'm not sure We're talking about the same thing here... :D Then, as a physicist one cannot measure infinity, and yet the entire standard model lies on a continuous background :D Which is, well, continuous, and so there are infinitely many points in any interval. And yet, without geometry there is no physics. So we do measure infinity. That is, if the background is continuous. Which is not necessarily so: spacetime could fuzzy up at Planck length for instance. I don't see how this helps with primes exactly, though... I do know that he wants to put everything on a finite background, but I fail to grasp his argument completely. I wouldn't mind if someone found time to explain in detail... Btw, I do believe background is discrete and finite. Some extremely interesting "miracles" and "coincidences" appear if one assumes just this. Now, if pN is the last prime, then... :D

[@hermannwacker1902]

@@Kraflyn As far as I have understood the infinity of prime numbers is proved by contradiction. "tertium non datur" As it is said in latin. (The third case does not exist!) We only have two states true and false. In our case this kind of logic is used to construct a contradiction for a finit set and then make a prediction for an infinit set. That is my understanding of the proof. If a boolean logic is implemented by a electronic device often a three state logic is used: high, low and not connected.(When the potential of the logic element is undefined). The basic question is how do we prove, that this prove technic can be applied in this case and the result is acceptable. I am thinging a about way to prove the statement in a constructive way: Example: We have the finit set of primes: {2,3,5,7,11,13} given the Euclid number: E6 = 13# + 1 = 2*3*5*7*11*13 = 30031 = 59 × 509 which is not a prime! This is a contradiction! => (p1p2p3...pN)+1 is not always a prime. E6 is not divisible by 13 and primes less then 13. But E6 is divisible by 59 and 509. In the range between 13 and (E6 = 30031), are primes! The prime density shrinks for huge numbers. What happens for huge N in the range between pN and En? Apear more primes or thrinks the number of primes? Implementing calculations with En becomes difficult because the number grows extremly fast with n. It is very probable, that it is not possible to preform a proof in this way. But thinking in this way gives me an understanding in the underlying machinery. A proof by contradicion does not give this view into the underlying machinery. It is also always a challange to transfer mathematics into software and then prove that the software works. In software a proof is substituted by test. But this is another complex theme.

[@maynardtrendle820]

Good man!

[@whig01]

It's a choice between incompleteness and inconsistency. If you say there are more primes than can be contained within any set, then you admit incompleteness. If you say that you have an infinite set of primes, then you are going to produce inconsistencies.

[@whig01]

@@gorkym8864 Your comment contained no content.

[@whig01]

Incompleteness is not a bad thing. It is proven by Godel's theorem. The issue is that by using infinities we have inconsistencies, as a consequence, in order to try to get around incompleteness. A set of primes which does not contain all primes is an incomplete set. A set of all primes is infinite, which is equivalent to dividing by zero.

[@whig01]

@Amat Uccan Infinity = 1/0. Have a nice day.

[@whig01]

@@gorkym8864 You're so very intelligent. Maybe you can make a video and people will watch it.

[@whig01]

Professor Wildberger is correct to seek a concrete basis for math, and his detractors are adding nothing to the conversation here.

[@robinbrowne5419]

Thank you :-)

[@turdferguson3400]

The problems you outline about the arithmetic proof are that the computation is very nontrivial and very difficult. However, this is not an advantage for the geometric proof, because it would be at least as difficult to construct the analogous segment and try to find a factor for that segment. Conclusion: the alleged advantage of the geometric proof over the arithmetic proof vanishes. Just as the arithmetic proof involves a difficult computation, so too the geometric proof involves an equally difficult computation. If you want to talk about real world cost, the geometric proof is not better, because it requires ink and paper, and if the line segment representing 1 unit is 5 mm, then there's no sheet of paper that can fit 2×3×5×7×11×13 + 1 = 30,031 (150meters), so we can't compute the proof. And yet, I can try to factor this number by hand using arithmetic. This shows that, for at least some proofs, performing the computations arithmetically is superior to performing the computations geometrically.

[@cunningham.s_law]

do you belive in induction?

[@njwildberger]

@honeyspoon That is actually a good question. Induction is a key aspect of natural numbers, where we establish mathematical truths by proceeding from 1 and stepping thru the natural numbers one by one till we get to whatever number we are interested in. However this argument breaks down with "numbers" as big as m=10^10^10^10 since we CANNOT count from 1 to this number. As I explain in my MathFoundations series on Big Numbers, the vast majority of numbers betweeh 1 and m are so big that they don't fit into the universe, no matter what system of nomenclature you have to represent them. So induction is not really a valid argument in this range. Here is a physical analogy: I can walk from Sydney to Melbourne, but I cannot walk from Sydney to Andromeda. So any thought experiment / argument that requires my trip to our neighboring galaxy bumps into the problem that it is contemplating the impossible.

[@billh17]

@@njwildberger said " I can walk from Sydney to Melbourne, but I cannot walk from Sydney to Andromeda." But this is not mathematical induction, but just regular induction (which is known not to always lead to correct statements when generalized). Mathematical induction says: if you have a statement P(n) and you know P(1) is true and you know P(k) implies P(k+1) is true, then you can conclude that P(n) is true for all n in N. In your example, you have shown only the base case is true and you haven't shown that the inductive step is true.

[@njwildberger]

@@billh17 But what is not usually stated, and should be, is the requirement that you can actually get from 1 to n step by step. Not by exponential leaps, because that is not what the inductive argument covers--- only one step at a time. That is why induction does not work neither to 10^10^10^10 nor to Andromeda. But Andromeda is closer.

[@MrCmon113]

@@njwildberger Does the set of all combinations of phenomena in the universe have a number? Or the set of all combinations of elements from that set? At what point do numbers stop?

[@njwildberger]

@@MrCmon113 If the universe was a petrie dish, and we amoeba swimming around in it, then our aim would be to try to understand the situation from where we are. The "set of all permutations" of all the atomic constituents of the petrie dish is not something that would be accessible to us. However to a larger creature, say a not very hairy ape who was looking at the petrie dish universe from outside with an electronic microscope or such, yes perhaps the above set of permutations can be something that could be analyzed. And if you were a slimy galactic super octopus looking down on the world of the not very hairy ape with your infinitely large eyeballs, yes maybe the "set of all real numbers" has also a well defined meaning. But suppose we do not want to pretend to be god? Suppose we just want to honestly study the actual world in which we live, to the best of our abilities -- but no more?

[@alexandartheserb7861]

Prime numbers formula is: (1x2x3)xN +- 1, i.e. 6N +-1, but first presentation is more beautiful since it shows first numbers: 1, 2 and 3 and is also compound of first primes (2 and 3) after which formula applies.

[@Cor97]

Why this lengthy introduction of Euclid? What Euclid said or meant or did not said and meant, why is that significant to this discussion? His authority? Alright, nobody is likely to come up with a prime that is larger than ..... But not everyone will therefore dismiss the idea that a new bigger prime can be constructed from 3 given ones. For instance many mathematicians claim the sum of the angles of a triangle to be 180 degrees (in the Euclidian case). However, nobody has ever measured this beyond a rather crude accuracy. We certainly cannot measure angles with accuracies of say 1/10^10 degrees. let alone 1/10^10^10 degrees. Nevertheless, many people seem to be convinced by the usual in my opinion rather compelling geometric proofs. Is mathematics in this regard so much different compared to other disciplines? For instance, nobody has visited Centauri Alpha. What does that prove about this star? Perhaps in future people will get there and find out that our current ideas are not entirely correct or even completely incorrect. We then will correct them, I suppose. Does mathematics have God-given theorems? Or are these actually man-made and can be proven to be wrong at some point?

[@Cor97]

@James Herndon Indeed, Euclid is not just interesting, many of his ideas have inspired and directed human development. I have studied many of his proofs which also modelled my own way of thinking. But how does that translate to the discussion about primes? Sure, there is a limit to what we can calculate. Does that mean that our mathematical concepts are useless? Imagine a rectangle with sides like 10^10^10+something and 11^11^11+something , does it not have an area because I cannot calculate it?

[@klammer75]

Almost sounds like a Newtonian vs einsteinian divide….for all practical purposes Newton works….but Einstein subsumes Newton and is the more general theory….I think this is a similar type of distinction here….IMHO🎓🤷🏼‍♂️😎

[@adolfninh23]

But why Gauss suspected the authority of Euclid ? First the peril of Greek reasoning lies in their dependency of facts, a.k.a axioms which does not reveal anything about the axiomatic objects themselves. Take Archimedes for example, he found rules of lever but know nothing about gravity force. The same situation with Simon Stevin in 17th cenutry for the stationary of chain of bead on inclined plane. For Euclid himself, declaring something is absurd because the shorter can not measure the bigger is quite convenient but is it ? Abstuse axioms can stil yield true result in reality, it's up to where and when you invoke those facts during the line of arguing. Second, Greek math didn't comprehend zero 0. So if Geogre Berkley argument about the failure of modern Analysts to construing arithmetic of zero holds, this must also be applied to Greek system. How do you make sense of 0 divided by zero. And that lead us to an historic fact, our current mathematic is not pure but a merge between Abacist and Philosophy

[@dsm5d723]

24:20 Ok, you have another way of saying it. "external computations" = gamma corrected computations in a Leibniz holomorphic ring. Analytical continuation in modern million dollar lies. From one POV, equations of motion can be compressed as mathematical fractions in some cases. Again, those self-locating strings and loops are what the Greeks might have done with supercomputers; then no conjecture. Leibniz thought many worlds, time was the problem. So he emphasized space, until Boltzmann wrecked both. Oh, so Rayo's Prime can't be toy computed? Do not tell Penrose, he might shit himself. A Penrose Prime Problem is an unpaid act of stupidity.

[@dsm5d723]

Nobody addressing this? CIAholes?

[@tomsvoboda2309]

Basically you're claiming that the property of being a prime, or having a prime factor, somehow depends on the current technological progress. Ridiculous.

[@njwildberger]

@Tom Svoboda That is not what I said. I rather implied that the situation is dependent on the size of the world we live in. In particular the "prime factorization of 10^10^10^10+1" is not a meaningful mathematical expression in our world. That might be disappointing, but it is what it is.

[@tomsvoboda2309]

​@@njwildberger is prime factorization of 10^97-1 a meaningful mathematical expression?

[@ronancarroll8554]

@@tomsvoboda2309 Yes, its factors are 12004721*846035731396919233767211537899097169*109399846855370537540339266842070119107662296580348039*3^2 Factored using Maple

[@ancalagonyt]

There's a problem with this point of view. It assumes that what we can calculate is precisely defined. So, for example, if we asked Euclid whether or not (2^82,589,933)-1 was prime, he wouldn't have been able to answer us, because he can't compute it. And if we asked the mathematicians of 1998, they couldn't either. But now we can. When our computational power increases (or decreases), pure mathematics doesn't change, only our access to it. And if you said otherwise, and tried to wall off not only the infinite and the potentially infinite, but anything "too big", and "too big" could be precisely defined as any natural number bigger than N, it still wouldn't work. You could have n= N/2, which is by definition small enough to calculate, and then take n+n+n, which is a small, simple, finite calculation, with a number which isn't too big to calculate, and yet the result is "too big". The only way to cut off calculations which are too big in a way that's not vague or fuzzy or inconsistent would be to stop calculating with variables like n, so as to disallow any expression like n+n+n. Too big to exist doesn't exist in theory. It's too vague and imprecise, and it changes when our computational power increases or decreases. If you don't have a computer or paper, your computational ability decreases to what you can do in your head. If you have a login to a supercomputer you can access over the internet, your computational ability increases past what you could do in your head or with pen and paper or with your cell phone or laptop. Those things all have practical effects on what you can compute, but when you can calculate is_prime((2^82,589,933)-1), that doesn't cause it to come into existence, and when you can't, it doesn't fade away or change the answer. And there are different ways of computing things, some of which are more efficient than others. If the function f computes more efficiently than the function g, and f(n) is within our computational power but g(n) isn't, but m = f(n) = g(n), does the number m slip in and out of existence when I alternately try to compute it with f and then with g? Perhaps we shouldn't say that there are infinitely many prime numbers, because actual infinities are dubious, and we should instead say that any finite collection of primes is incomplete. Perhaps we shouldn't say that real numbers exist because actual infinities and/or the axiom of choice are philosophically dubious, but if we're forced to abandon all symbolic computations because some of them are hard, or if we have to say that maybe there is a finite complete collection of primes even though we have a proof there isn't, or if numbers start slipping in and out of existence, that's worse.

[@pedroth3]

Our minds, are able to reach truths, that aren't computable in finite time, and that fact is a non issue, I think. Godel's theorems comes to mind...

[@njwildberger]

@peterluigi Yes I know those kind of confusing unfathomable truths --- they often come to me in the middle of the night, and then I wake up.

[@rigansmontes]

It is false that geometry was more fundamental than arithmetic to the ancient Greeks. Plato, for example, defines magnitudes in terms of numbers. And numbers are not magnitudes but multitudes, as can be ascertained in all ancient Greek writers. Consequently, they did not represent numbers using lines. It is easy to debunk this widespread myth by reading Euclid's Sectio canonis, in which lines are used to depict the string of a monochord and its potential division in ratios of numbers. Length and position (properties of magnitudes) are-absolutely speaking-irrelevant. What matters is that the string (a magnitude) be divided into parts in accordance with numerical ratios so that concordant intervals may be produced. Thus, we read in the introduction that musical notes “are composed of parts,” and that “all things that are composed of parts are spoken of in a ratio of number with respect to one another, so that notes, too, must be spoken of in a ratio of number with respect to one another.” Clearly, this is not always true of magnitudes-but it is always true of measurable multitudes. As Wilbur Richard Knorr explains (in The Evolution of Euclidean Mathematics (Dordrecht: Reidel, 1975), 308-10), “number theory was always recognized as a discipline entirely separate from geometry.”

[@lafluth]

Happy to be convinced otherwise but 2 things really bother me: 1. Right from the start of the video, math is referred to as belonging to “scientific tradition”. This is just so wrong. Math is NOT a science. Physics, chemistry, biology, sociology etc. are sciences, trying to explain the world with reasoning and among others mathematical tools, using experimentation as validation. Mathematics is not observation and explanation of the world. A mathematical proof does not rely on experimentation for validation. It is a system of logic that people have developed. It is based on axioms that cannot be proven, they are just statements you start with and do not question, by definition. There is no epistemological process in math. 2. The proof that there are infinite primes does not need computation or observation. It is self containing and perfectly valid in and of itself: Assume there are only N primes p1 to pN. Now multiply them and add 1 to form the new number Q. Is Q not a prime? Can it be formed as a factor of other primes? Well, the only existing primes we assume here are p1 to pN. It cannot be divided by any of them because any division of Q by a given pk number would result in the number (p1 x p2 x…x pk-1 x pk+1 x…x pN) + 1/pk. The first term is an integer, the second term cannot be. Therefore Q is a new prime. I am neither a mathematician nor a sociologist, but even I can figure these things out. Inexactitude in popularisations really bother me, because they achieve the exact opposite of their goal. They actual contribute to spreading more misinformation than enlightenment.

[@zachbills8112]

We don't actually have to calculate the prime factorization, we just have to know it exists. You really hate the power of mathematical logic for some reason.

[@njwildberger]

@Zach Bills We don't actually have to observe something to know it exists?? Is this a power, or a sleight of hand? Why not ask a scientist?

[@volodyanarchist]

@@njwildberger Here is the thing. Let's say we ask "Is there a prime larger than 10^10^10^10?" There are two possibilities: 1) 10^10^10^10 exists as a number, and thus it should be possible to run Euclid's algorithm 10^10^10^10 times at which time we would have that many prime numbers, and since unit (1) is not a prime number, at least one of the numbers in our set would have to be larger than 10^10^10^10. 2) This number does not itself exist, which means that there are no primes larger than that, because there are no numbers larger than that. I believe that a line in eucledian space is defined as having no bounds the second possibility falls flat on its face. In some sense you are searching for a bound for numbers.

[@mlevy2429]

@@njwildberger there are many things I don’t observe directly but know to exist... also scientists use this as well. You are kidding yourself.

[@njwildberger]

@@mlevy2429 We are not asking that you need to observe them directly, but someone you trust should be able to, somewhere. Since we are aiming at explicitness here, please give us an example of something that you, or someone else, don't observe directly but know to exist.

[@rdubeau]

@@njwildberger You must have heard this many times but you are not doing mathematics when you make a physical argument such as “There can be no way to compute x in the known universe.” I love listening to your talks. Your ones about the Stern-Brocot tree were really enlightening to me for just one example! But to prove there are no completed infinities in mathematics, you need to produce a mathematical contradiction from assuming that there are. Not a hand wavey “physical impossibility!” If you can show a mathematical contradiction I’d love to see that. Infinities give me headaches. Hell, even the term “completed infinity” seems to have a purely linguistic contradiction in it to me. But my math profs assured me that English and math are different topics, as are physics and math. Modern set theory postulates an “infinite set” and so far as I know, no mathematical contradiction has been found to this postulate. Can you give one that does not rely on your idea of physics?

[@zachbills8112]

The proof clearly demonstrates that there cannot be a finite number of primes. By definition that means there are infinitely many. You put all your effort into contrived, tortured reasoning to be a contrarian.

[@njwildberger]

@Zach Bills Sorry, but Euclid does not prove that "there cannot be a finite number of primes". Please watch the video again carefully.

[@mlevy2429]

Euclid’s proof is an existence proof. He doesn’t claim to be able to compute which number is the missing prime. We know one exists from the proof, why do you think they would write the software for GIPS if there could only be finitely many. They know there is infinitely many from the proof that is why we are able to continue searching for larger and larger ones forever. Your philosophy is crowding out your ability to do math.

[@njwildberger]

@@gorkym8864 Please watch Famous Math Problem 1.