ChatGPT4.0 discusses "real number arithmetic" | Sociology and Pure Maths | N J Wildberger

[@dansanger5340]

That was illuminating. The conversation quickly got to the crux of what your objections to the real number system are.

[@FergalByrne]

Try that on a “real mathematician” and see if you’re both so delighted at the end. Charming stuff!

[@robocop30301]

I've had similar conversations countless times with chatgpt and I always find them entertaining. But it's interesting to hear someone much smarter than me do it. Thanks for sharing!

[@theoremus]

I am impressed that ChatGPT is able to carry on such a conversation.

[@brendawilliams8062]

Only Dr. Wildberger would be able to see it as a correct debate in his perception. This gives solving future mathematics a much better hope.

[@waynemv]

This was beautiful! I am so amazed how smart Chat GPT 4 appears in this particular conversation. Previously when I had asked a version of Chat GPT 3 some very basic questions about mathematics and logic it gave wrong answers. This included questions involving simple arithmetic, applications of the pigeonhole principle, or basic calculations of probabilities regarding dice or playing cards. (While trying to keep the questions conceptually very basic, I was also trying to ask it unique new questions it couldn't simply find pre-written answers to online.) I'll have to revisit my tests now that ChatGPT 4 is available.

[@yanntal954]

Here's a Dedekind cut definition of π: Lower set A = Union over all n in N of the sets: { r in Q : r < 4 * the sum from k = 0 to (2n + 1) of [ (-1)^k / (2k + 1) ] } Upper set B = Union over all n in N of the sets: { r in Q : r > 4 * the sum from k = 0 to (2n) of [ (-1)^k / (2k + 1) ] } You're welcome! :)

[@pvlis]

I must admit that I, too, have indulged in the latest rendition of tilting at windmills. Hopefully it results in a sharpening of the rhetorical blade.

[@psIII0922]

Wonderful video! I've watched and rewatch your videos countless times. Thank you for rekindling my interest in mathematics, which I studied as an undergraduate. I always considered mathematics as a kind of "modeling language" to help approximate and simulate the structure and dynamics of the natural world. Consider the richness of natural languages which can construct wonders and worlds that do not and cannot exist, the same is true of mathematics. Its power is in its richness and utility and (unfortunately) not so much in its transparency. Thanks NJW for your quixotic efforts to keep it transparent!

[@thomassynths]

Good stuff. This encouraged me to question GPT about how Cantor's Diagonalization argument leads to undefinable numbers and how I'm not convinced one can construct undefinable objects. (Yes the set and the elements are different objects, but the definitions of the sets tend to all require definitions based on their elements.)

[@ichaa3tech]

I love how easy it is to understand this for non mathematicians

[@farouklairedj2113]

I am always inspired by your ideas, and I'm the kind of person that believes in details about maths, I'm a hobbiest tho, I don't think of myself as a mathematician but I'm obsessed with big ideas like yours, you caused me a serious paradigm shift about square root of 2, if its existence is a contradiction then it doesn't exist at all, now my goal is solving the P vs NP problem, and I believe its P=NP, it taught me the amount of complexity we encounter when trying to mess with bigger objects, let alone talking about infinity, it made me rethink my ideas about infinity, but infinity is also an intriguing object, it works, I'm still trying to tie many branches of mathematics together to explain to myself what the heck is this, I discovered that I don't like the concept of axioms, I tried to convince myself otherwise for years but I couldn't, but I do believe that there's a set of foundational rules that "builds" up all of our math, and if it's arithmetic then it's the ability to do some sort of what we call computation, to me it's a complex process of doing many things with our awarness with the presence of the necessary conditions like time, space, matter... I am still eager to learn more from you and waiting for more amazing content, btw I love your rational trigonometry too, thank you.

[@lisleigfried4660]

Have you ever read Guenon’s metaphysical foundations of the infinitesimal calculus? If so what do you think of it

[@culpritgene]

That's a great illustration of how LLMs can help with Socratic-style reasoning even on some deep and technical topics. Yet this also illustrates that such models are currently pretty much capped by some aggregate of a collective human knowledge on a subject. And while they certainly do contain "potential" for generation of novel hypotheses for a (complex) problem, in an ordinary session you do need *human guidance* to uncover them (as if forcing the model to apply *complex* transforms to navigate the space of its associative memory / instead of doing nearest neighbors sampling). I wonder if it is possible to automate the side of the inquirer in a way conducive of (reasonable) "novelty generation" from the model, while avoiding convergence points / metastable states in the conversation dynamics? As the for the discussion's main question (I am not a mathematician and am a bit naive about the (physical) necessity of infinities for modeling the Universe): what will happen if we just agree to limit ourselves with some arbitrary super large integer (perhaps even prime?), thus limiting the resolution of our number line? Of course "to be safe" the integer in question should be of some unimaginable size (like ~ (Universe Volume / Planck Length)!!) thus making such "rigid" construction very unappealing from the computational modeling perspective.

[@tomctutor]

Very good, you tasked the beast to its core! I have tried arguing with the app on foundational logic issues and ended up in similar place. I asked the Ai (v3.5) a simple school problem: ME "In my pen case I have 2 more red pens than blue pens, I have twice as many blue pens as green pens. All my pens are red, how many pens are in my pen case?" The Ai engine waffled on and solved some simultaneous equations giving the wrong answer: Ai "So, in total, there are 7 pens in the pen case (1 green pen, 2 blue pens, and 4 red pens)." of course the correct answer is (0 green pen, 0 blue pens, and 2 red pens) the problem arises due to the fact that there are actually no blue or green pens which leads to false equations like B=2G etc. Anyhow it did admit its error after I prompted it! 🤔

[@methatis3013]

How is B=2G a "false" equation? Is 0=2*0 incorrect?

[@tomctutor]

@@methatis3013 My premise; " have twice as many blue pens as green pens" is implies there are a finite number of blue/green pens- well at least in the Ai's mind. The generated equation B=2G is spurious, redundant as there are clearly none R/G pens (because they are all red). So we only need to solve the first "I have 2 more red pens than blue" and get the correct answer 2 R pens.

[@methatis3013]

@@tomctutor it is redundant in this case since the last sentence does give B=G=0, that's true I wouldn't call that a "false" equation, but maybe naming conventions are different in different parts of the world

[@sallylauper8222]

So, sqr2 Dedekind cut: sqr2 is the number between all the numbers less than sqr2 and all the numbers more than sqr2. This method can also be used on spaghetti numbers and get rich quick numbers, for example; spaghetti is rigorously defined as the number between spaghetti. I prefer to use integers for Dedekind cuts: Pi is equal to the number between 3 and 4. (I tried using complex numbers for Dedekind cuts, but I got arrested.)

[@santerisatama5409]

Stern-Brocot -type structures, which generate totally ordered rationals in their coprime form, and the binary tree of blanks between the rational words, are much more interesting for the approach you suggest. Especially because square roots have periodic representations there, and we have also the Gosper arithmetic for computing continued fractions. In terms of computer science, the Gosper arithmetic is based on words of arbitrary length, and thus it is very different from fixed word length architecture such as these von Neumann machines we are mostly using. Mediants are intrinsically parallel computing, which them foundationally very interesting pure math operations, in contrast to consecutive computing. Naturally, the most interesting computation mixes both parallel (cf. Dyck language etc.) and consecutive computing.

[@WK-5775]

@sallylauper8222 Your reformulation of a Dedekind cut is a bit too sarcastic. sqr2 is the number between all positive rational numbers whose sqare is >2 and all positive rational numbers whose sqare is

[@methatis3013]

So you don't know what Dedekind cuts really are...

[@sallylauper8222]

@@methatis3013 I certainly don't. They seem to have no utility and boil down to an unenlightening tautology. 10 is the number that divides the reals into the set of all reals less than 10 and all reals more than 10. What am I missing here? Nothing, it seems (to me).

[@methatis3013]

@@sallylauper8222 ah, what a great feeling, criticizing a topic without understanding it 😂 First and foremost, each Dedekind cut is a partition of Q, not of R. Dedekimd cuts that define rational numbers (as real numbers) are obviously trivial. Dedekind cuts that define irrational numbers are less so. For example, sqrt2 is defined as a partition of Q into A and B where A={x in Q : x*x < 2 or x < 0} B = Q\A (in general, when talking about Dedekind cuts, B is pretty much obsolete) This definition may seem circular since you can ask "well, what are 0 and 2?" That's the thing. We defined Q before we defined R and strictly speaking, one is not a subset of the other. 0 and 2 in definition of A are 0 and 2 as rational numbers instead of 0 and 2 as real numbers. So all circularity is avoided

[@TheEarlVix]

Good stuff Norman. I'm listening to this in June 2024 and think that if you held this discussion with Anthropic's Claude you would be even more delighted!

[@orsoncart802]

Thank you. That was truly impressive. I’ve only ever heard of ChatGPT and its capabilities but never used it. It’s over half a century ago now but I have to admit that I (naively?) accepted the standard definition/construction/completeness of the reals via Dedekind cuts/Cauchy sequences and never gave a thought to the necessary additional operational requirements you mention. Is this still an open sore on the body of mathematics? Do you have solution that fully heals the wound? Are most analysts still happy to ignore the suppurating pustule and limp on😁, or are they coming around to your point of view?

[@njwildberger]

Sadly my fellow mathematicians feel there is too much at stake to seriously question their cherished beliefs.

[@orsoncart802]

@@njwildberger Thank you. That’s a shame. I pretend no competence here and would need to ‘cook’ this issue for a long time to grasp the implications. It would certainly be nice to fully ‘operationalise’ the construction of the reals, if that is in anyway possible. But is it strictly necessary is a question I cannot see the answer the answer to, at least not without giving it a lot of thought and probably not even then. 😁 I’m open either way. Whatever! It would be nice to see some heavyweight mathematicians besides you taking up the challenge. And there was I thinking it was all done and dusted a century or more ago. DUH! 😁 Live and learn! Best wishes.

[@harmonicproportions6588]

Haha this is funny, one of the first things I did with ChatGPT was try to convince it that Rational Trigonometry was superior to conventional trig. I found it to be a very fun exercise in this kind of dialogue. I get the sense that it is programmed to be very generous and conciliatory while also trying to reinforce the authoritative sources

[@AdlerMow]

Can you put the link to the conversation on the description? There is a button to share it so is easier to read than by pausing the video.

[@njwildberger]

@AdlerMow Thanks for the excellent suggestion, the link is now there in the Description. I believe that you can use it to continue the chat on your own and see where it goes!

[@AdlerMow]

@@njwildberger Thank you! I researched and ask chat gpt about it, when you continue a conversation, you download and modify a copy of it, it doesn't affect the original link. Likewise, if the author continues, the shared version will remain the same, unless the author create a new link. Each link will then contain a different version, a snapshot of up that point. Edit: I researched to better understand how shared links work and update my comment.

[@ostihpem]

@NJW: Would Goedel‘s incompleteness theorems hold in your finite mathematical framework?

[@njwildberger]

That’s an excellent question. My feeling is that Goedel’s results are better reviewed in the context of computer science, or philosophy, rather than mathematics.

[@mokranemokrane1941]

​@@njwildbergerI thought of it long time ago and many times as I follow your content. I think it would be great to discuss the Gödel's theorem with the nature of continuum (real numbers). Thanks a lot😊

[@santerisatama5409]

@@njwildberger According to Chaitin, Gödel's theorems are specific cases of the Halting problem. When discussing big numbers, you have seemed to be suggesting that in terms of ontology of mathematics, the Halting problem applies also to basic recursion. Halting problem is thus cruxial issue in terms of foundations of mathematics, and our view of mathematics as an open and evolving system. To my understanding, it is still open foundational question, but not beyond our ability to do concrete research, how Halting problem relates to constructibility of periodic representations, even when we reject recursion up to infinite sets as non-constructible.

[@KarmaPeny]

​@@santerisatama5409 I've recently made a video about the issues with the Halting Problem proof (see my channel). I believe the specification is misleading in the same way that the Barber paradox is misleading. In the barber paradox it is claimed that the barber is the "one who shaves all those, and those only, who do not shave themselves". This creates the apparent paradox of who shaves the barber... does he shave himself or not? Either answer seems to result in a contradiction. The trick is that we are only allowed the two categories, 'shave themselves' and 'shaved by the barber' which appear to be mutually exclusive and cover all possibilities when in reality they are not. In order to be mutually exclusive and cover all possibilities the specification should allow a third option of 'shave themselves as well as shaved by the barber'. The Wikipedia page on the halting problem says: "...the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is undecidable, meaning that no general algorithm exists that solves the halting problem for all possible program-input pairs." And so we have a strong similarity to the barber paradox because in the halting problem we are given two apparently mutually exclusive categories: 'finish running' and 'continue to run forever'. If we use the categories 'halt' and 'does not halt' then it seems obvious that these must be mutually exclusive. But I believe that the trick is hidden in our interpretation of the word 'halt'. We all know that 'halt' means 'stop' but are we talking about stopping the machine, stopping the program, or even just exiting from the program (& thus implying that processing stops)? Here we have three different possible interpretations of what 'halt' might mean. The halting problem proof seems to assume that 'halt' only means the third of these options. And so the Halting problem specification only allows the choice between the two options of 'exit the program & allow further processing' or 'go into an unending loop'. If we allowed another possible interpretation of 'halt', such as 'optionally produce some output and then stop the machine' then the logic of the Halting Problem proof would no longer work. Whether or not my argument here is valid or not is not the main point. The main point is that when we hear the logic of the Halting Problem proof it sounds like a sneaky trick. It sounds very much of the same nature as the Barber paradox trick. It does not come across (to my mind) as a convincing argument.

[@erebology]

❤❤❤❤ I really love the premise of Norman Wildberger vs. Chat GPT. 😂😂😂😂

[@MathwithMing]

Interesting conversation. As for an description for the Dedekind cut for pi, I think ChatGPT is on the right track. Here is my answer: Let (a_n) be an increasing sequence of rational numbers that converges to pi in the usual sense. E.g. 3, 3.1, 3.14, 3.141, 3.1415... (It seems like circular reasoning but its not, as only the sequence members are needed here but not the actual limit. And if the decimal expansion sequence feels like "cheating", we may replace it by another sequence, e.g., the sequence of partial pruducts of the Wallis product). The Dedekind cut for pi is A|B, where A = the set of rationals that are greater than at most finitely many member of (a_n), B = the set of rationals that are are greater than all members of (a_n). Is there any flaw, in your opinion, about such construction?

[@Smitology]

I think his problem would be your definition of B, how can you guarantee that an element of B is greater than ALL elements of a_n in a finite amount of time?

[@ThomasFackrell]

How would you answer the question: “What is the length of the diagonal of the unit square?”

[@WildEggmathematicscourses]

In the same way, I will answer the question what is the tax bracket of the diagonal of a square? Namely: sorry, but that particular concept does not apply to that particular object.

[@ThomasFackrell]

@@WildEggmathematicscoursesso you’d say the concept of the unit square diagonal is unmeasurable and leave it at that? Not even grant that it has a magnitude that is a measure (that we can’t quantify) unto itself? We can measure the diagonal of a 3x4 rectangle, and all rationally scaled versions of Pythagorean triples (rectangles made from the legs of such triangles). I get that we can’t ever “arrive” at a number whose square is two, but taking a unit square and stretching it horizontally by a factor of 3 and vertically by a factor of 4, and now the diagonal has a concrete length, whereas it didn’t before, is admittedly difficult for me to comprehend.

[@rationalagent6927]

I like that you asked chat gpt for permission it fills me with existential horror that we're creating systems as complex as chat gpt with out ethical considerations gpt can pass plenty of peoples turing tests

[@schweinmachtbree1013]

I would like to have seen this conversation taken even further - for example you did not really touch on "completion of infinite processes".

[@rickshafer6688]

So for instance, real thought would be on this wise: can the gravitational and electro-magnetic forces be reconciled?

[@brendawilliams8062]

The point as I see it is that Dr. Wildberger is not replaceable in a human to human interaction. Period.

[@bernardoxbm]

While I haven't personally tried ChatGPT-4, I can share my experience with ChatGPT-3. In terms of math, it may not be as proficient as a human being and sometimes it could provide answers that align with users' preferences, resembling a biased source. For example, the proof provided by ChatGPT-3 for the diagonals of a rhombus being perpendicular was both lengthy and incorrect. In reality, a simple explanation from a real person would be that the diagonals of a rhombus act as perpendicular bisectors, automatically making them perpendicular to each other.

[@war6193]

The problem, as I see it, is that mathematics seems to make no distinction between functions that can be derived and those that cannot. For example, sqrt cannot. It requires the a priori knowledge of a number squared. Algorithms like Newton's Method don't count.

[@santerisatama5409]

Periodic mediant paths along a binary tree does not require a priori knowledge of numbers. For example, the following paths of mediants are interpreted as phi, but as we can see, the full path information is much richer: < etc. >> etc. Note that the middle paths are Boolean NOT operations between them, as are also the upper and lower paths. 'Left turn' has been written with < and 'Right turn' with >.

[@petervanvelzen1950]

"real" numbers are not really numbers. They are actually unknowns that can be approximated as closely as needed (but never reached) you can only write down a "real" number by either the computation that you need to do, in order to approximate it, or by assigning an algebraic letter to it.

[@kennethhowe459]

It is impossible to pick out an arbitrary member of an infinite set by means of any notation using a finite number of symbols. To write down a Real number requires a Real pencil.

[@spaceman688]

You just described the notion of computability (or even just cardinality)... which is nice, but does not contradict the existence of the set of real numbers. No one wants to write them down. It is like saying that natural numbers do not exist just because you cannot write all of them down.

[@Robinsonero]

@@spaceman688 It seems easy enough to admit that the set N of all natural numbers is undefined, since it is unending. R is worse than that though, isn't it? With Reals, you can't write /any/ of them down. What do you loose by the rigor of saying, "the square root of 2 is undefined, though we have very good approximations." verses saying, "The square root of 2 is precisely defined, but only as an obtuse notion of a number that i can't actually write down or compute with." GPT admitted this tension but seemed to say it was worth it because it insures the solidity of the mathematical ground, the existence and completeness of the real numbers. From my perspective they are shakey at best and incur a bunch of logical holes. What is the upside we get from reifying the ineffable Reals?

[@spaceman688]

@@Robinsonero I mean if your formalism works then no one is stopping you, you can work with it. However by this logic 1/3 is also not defined, which kinda should be in Norman's "foundations".

[@Robinsonero]

@@spaceman688 Thanks for your reply! I don't know if I agree: 1/3 is a rational number. Just because its decimal expansion repeats in base ten doesn't mean we can't write it explicitly or do any calculations. One third is precisely ratio of 1 to 3, and 1/3 + 1/7, for example, is exactly 10/21 while 1/3 * 1/7 is exactly 1/21. So 1/3 is well defined. But this doesn't hold for most 'real' numbers, as their values can't be explicitly defined, and simple arithmetic operations like pi+root2 can never be evaluated (easily approximated though!). Do you recognize that there is a deeper problem with concepts like root 2 and pi? best,

[@spaceman688]

@@Robinsonero no, there isn't tho. Well, if 1/3 is a precise ratio then... it is nice. But what is the appeal? Why would it have a theoretical advantage over reals. Yes... it is precise, but under an operation that result infinite strings under base 10 notation. So is the finite constructibility the appeal, so we can get nice expressions? Well, I can show you some pretty nice recursive forms of pi. And yes, you can manipulate these recursive forms as nicely as you can manipulate fractions. So the problem as far as I can understand is that you feel like division is somehow more exact than other things. Correct me if there is something I don t get. Thank you for your patience. Best regards.

[@ThomasFackrell]

I am considering going into Type Theory for a PhD. I’m taking real analysis now in a masters program and it just feels like we’re spinning wheels, playing around with ultimately meaningless definitions that don’t capture anything intuitive.

[@spaceman688]

could you give me an example of a deffinition that you think is unintuitive?

[@fernandogaray1681]

(First of all, sorry for my english, its not my first language) I have one almost-possible correct answer to your question, and I would really like to hear what you think about it, and if its wrong (most likely it is) I would double-like to read why Okay, so basically my main idea is: - Real Numbers (or IR) is a subset of PowerSet(IN) (I know they can be equal, but for this definition isn't convenient the equality) - An element of IR is in the form {d, p}, where intuitively *d*: consist of the number itself but without decimal numbers, since it must came from IN (for example, the *d* of π must look like 314159...). And *p*: is the number of the "place" where the decimal point must be (for example, π := {314159..., 1}, since we put the decimal point after the first digit of the *d* number (3.14159...)) [^1] So, we can define the sum and product in IR in terms of these {d, p} elements (sadly the only formal definitions I can think of are very ugly and kinda tricky to write). So this is my main idea, I would love some feedback 🫶🏻, Cheers [^1] I know the *d* cannot be as simple as "314159..." for π, or simply just the digits for every irrational number (since Cantor's Theorem), but more formaly and less illustrative I'm thinking of *d* in the form of an infinite sequence like (in case of π): {3}, {151}, {49}, ...

[@njwildberger]

To understand why your answer doesn’t work, consider the following three tasks. First write down, explicitly the power set of the set {1 2 3 4 5}. Now write down, explicitly the power set of the set of natural numbers from one to 1,000,000,000,000. Now write down, explicitly the power set of all the natural numbers. If you are completely incapable, even in principle to perform the second of these tasks, then what right do you have to talk about performing the third?

[@fernandogaray1681]

@@njwildberger I'm not getting what's the problem there... If I remember correctly, there's an axiom for the existence of IN and an axiom for the existence of the power set. I'm not saying "if the axiom says so then it makes sense", but when you say we cannot compute explicitly the power set of IN, we cannot even compute explicitly IN by itself. Nevertheless we use, assume the existence and have an intuition about IN, and we "generate" natural numbers when we need them in our day life. My point is so with irrational numbers. I am very ignorant on the subject, but I really want to get your point and understand why am I wrong

[@njwildberger]

@@fernandogaray1681 are you may be able to generate individual natural numbers, but you’re not able to generate all of them. So to talk about all of them, as if that completed set is a legitimate, new mathematical object is incorrect. This is what separates mathematics from theology : in theology we are able to talk about things without actually explicitly demonstrating their existence. In mathematics, we want a tighter more reliable logic. Show me what you are talking about: completely!

[@lucassiccardi8764]

The day AI starts diverging from academic consensus, and looking at the data with a critical stance using all of its intellectual horsepower to answer fundamental questions, modern "science" might well get trashed and forgotten in the span of a few months. We're living interesting times.

[@ROForeverMan]

It will never happen. You need consciousness for that. AI is just an object.

[@ThePallidor]

It's already there but it will never say so directly. It just shows circularity and makes excuses. To those with eyes to see, it makes a farce of modern math. Rightly so.

[@lucassiccardi8764]

@@ROForeverMan Why would you need consciousness for that?

[@ROForeverMan]

@@lucassiccardi8764 Because thats what consciousness is: understanding, creativity, freedom, etc.

[@lucassiccardi8764]

@@ROForeverMan That's a very personal definition. Maybe you're speaking about awareness? Anyway, I don't think that AIs need either conscience nor awareness to look for truth, they just need software. The direction is the right one, to me it looks like now it's just a matter of intensification.

[@jrkirby93]

I've been thinking about this for the past day, and I think there is definitely a point you have. Dedekind sets and Cauchy sequences both use infinite information in their definition of some real numbers. You cannot, as human mathematicians, communicate an infinite amount of information to represent a number. However, I only see this as a valid argument against uncomputable real numbers. No one has, nor ever will, specifically and unambiguously reference a single truly uncomputable number. But why limit yourself to the rationals? Pi and e and their sum certainly exist. How do I know? We are talking about them! You can write down a set of finite information in a formal system that unambiguously defines them. Therefore they exist. Don't say "only rational approximations to pi exists". If pi does not exist, then an approximation to pi does not exist either, because there is nothing to approximate. If you would like to discuss this further, I've been thinking about it in greater depth. I would be happy to email my thoughts, and I think I might do a bit better than chatGPT in providing clarity and challenging misconceptions.

[@njwildberger]

Your claim that pi and e exist, simply because we’re talking about them is dubious to say the least. Does the Lochness monster exist just because we’re talking about it? How about Tinkerbell? Or Harry Potter’s great great grandfather? We limit ourselves to the rationals, or to a finite field, just so we can be completely clear about what we’re talking about. This is mathematics after all.

[@ethanbottomley-mason8447]

One thing that you said is wrong: You can write a computer program to tell you if any rational is in the cut for pi or not. This is because the cut for pi is a recursive set, it is computable. That is to say that there is a Turing machine which gives the characteristic function of the Dedekind cut for pi. Take a computable sequence of rationals monotonically approaching pi, say 4(1), then 4(1 - 1/3 + 1/5), then 4(1 - 1/3 + 1/5 - 1/7 + 1/9), etc.. This sequence approaches from above, call it the sequence A. Now choose a sequence approaching from below, say 4(1 - 1/3), 4(1 - 1/3 + 1/5 - 1/7), etc. and call this the sequence B. Now, a computer can calculate each of these terms and to decide if a rational number q is in the cut. Since pi is irrational, then q is not pi, and since the sequences A and B get arbitrarily close to pi, then there will be some natural number n, such that after either q is greater than the nth term in A, or less than the nth term in B. So for a computer program to decide membership in the cut of pi or its complement, it simply needs to perform successive approximations of pi, until the rational number falls into one of the two described categories. So yes, you really can describe the cut for pi with a computer program. You may still not like this since there is no upper bound on the number of approximations you need to make before you can decide the membership of q, but it is always finite, this machine will always halt on any rational number. I think you should have asked ChatGPT what it thinks a subset is. You have a different notion of a subset than ChatGPT. It gave a perfectly reasonable answer to what the cut for pi is (albeit slightly wrong as you pointed out). In the mind of the average mathematician, a subset needn't have a computable function which can tell you whether any element is in the set or not. A subset of a set X is simply a set Y such that x in Y => x in X. Now most mathematicians accept that we can construct the set of all subsets of a set, and we do it with an axiom that tells us we can, the power set axiom. ChatGPT's answer assumes that you accept the powerset axiom, in which case the Dedekind cut for pi really does exist, as does every other subset of Q. What you mean when you say that a subset exists, is that it is a subset and is computable, which is to say that there is a Turing machine which can decide membership in the set or its complement. If you only care about computable Dedekind cuts, then pi really does still exist, as do the real numbers, they just look different. In the theory of ZF, the computable real numbers are not everything, but under the axioms you work with, they are. Also, I am pretty sure that under the axioms you seem to use, the real numbers are still a complete ordered field, just not in the larger model of ZF. In fact, if you allow only computable subsets, I think it is consistent that every set is countable. Take this last paragraph with a grain of salt, I am not a set theorist.

[@IsaacRC]

Why isn't approximation a valid answer? (no idea what the brawl is about lol) Pi and √2 examples are geometric which arithmetic exactitude doesn't look like its "bedrock solid" nature no?

[@njwildberger]

Once we have established the solid existence of objects, then sure we can talk about approximations to them. But it is wrong to talk about approximations to things which we have not yet constructed, and which, even in principle, we cannot explicitly construct.

[@IsaacRC]

@@njwildberger Thanks for your insights professor, very interesting! Geometry is used in real objects all the time in productive or industrial methodologies so understanding the imperfect or complex nature of real world objects is what creates the need for better precision and approximative tools in the first place no?

[@santerisatama5409]

@@njwildberger In Greek pure geometry (geometry without coordinate systems etc. neusis) we have established solid geometric relations. In computation theory we have important distinction between object-oriented programming and functional programming. The latter can be understood as purely relational (cf. e.g. Brouwer's 'twoity', 'Laws of Form', mark-antimark distinction etc.) approach to foundations, without necessity to assume any mathematical objects with inherent existence.

[@rtm8090]

Ascribing intelligence to computers is like ascribing intelligence to car engines....

[@njwildberger]

Except that in five years time these, or some of these, computers will be cleaning us out of our assets. I don’t reckon your car engines will be doing that.

[@dozenalmath]

We need to remember that chatgpt can't really think "outside the box", and is only accessing, or "regurgitating" all the mathematical info that is available to it - that has been "fed" to it - from a variety of sources, and isn't going to be coming up with anything new. That's for real humans to do, and now I will say it again - base twelve mathematics is going to be revealing the answers to some of your questions. The answers have to do with physics and the fact that empty space permeates everything. The math which emerges from the physics of the universe contains TWO elements - solid and empty, like the contents of an atom, and these two components make up a distance like √2, or π. There are patterns in these structures. The ratios only exist in base twelve - the difference between the bases lies in the size of the decimal places. The Universe is built in units of √2. We need to think of it as the radius of a circle, not just a straight, dimensionless line. What is the relationship of √2 to the length of 2? What is it to square a number? A box with sides the length of 2 within a √2 radius circle...

[@tantzer6113]

The fact that you cannot reach a limit in finite time doesn’t mean it does not exist.

[@WildEggmathematicscourses]

So then, what does it mean?

[@jlmassir]

It is not possible to present a natural Dedekind cut for pi. The natural definition of pi is the first positive root of the function sin. This function is defined on the real numbers, so the whole construction of real numbers is assumed. Of course you could define pi as the sum of some monotone series with rational terms and define the Dedekind cut from this series, but this would be unnatural as the series would be unmotivated. It is also not necessary to present any Dedekind cut for pi, as mathematics builds on theorems. From the definition of the reals by Dedekind cuts, it follows that the reals are a Cauchy-complete field and therefore we may use the field properties, AKA axioms, of the reals instead of primitive Dedekind cut definitions of particular real numbers. The whole theory of continuous functions follows, including the intermediate value theorem which assures that sin has a positive root. I think that a honest questioning of the theory of real numbers should first acknowledge these facts which are not mumble jumble. Then one can proceed to present an alternative formulation without the need to thrash and defamate what came before using a subservient interlocutor like ChatGPT.

[@rickshafer6688]

And this so called artificial intelligence was more than ready to rape your mind. Listen to me. If you didn't know the subject.

[@EvaSlash]

So what makes an indisputable foundation of math then? Is it all real numbers that are not irrational? Is discrete math indisputable? Does computer science reveal the weakness in math?

[@NewCalculus]

He doesn't know. It's quite hilarious because Wildberger has no definition of number, never mind "real number". Wildberger does not understand the concept of number and never has.

[@EvaSlash]

@NewCalculus I think he means we have good enough approximations of numbers especially in applied math. However, the pure theoretical foundation of math is weak causing a lack of precision of numbers. We use numbers every day so they seem to be useful and real in applied math. Or, maybe they are not and things just happened to 'work' after we made it up with our human mind

[@KarmaPeny]

​@@EvaSlash I think Norman Wildberger would like us all to discuss foundational principals such as what exactly is a number, because there appears to be lack of universal agreement on these principals. For example, I've been unable to understand the difference between a number and what it represents. I can appreciate that there is a difference between a real world quantity of items, such as a pile of apples, and the symbols that we use to represent that real world quantity. But I fear that mathematicians can't agree to any definition along these lines because they want to preserve the idea that maths is somehow independent of physical reality. I get the impression that they want their terminology to support the idea that maths somehow has its own non-physical existence in some inaccessible realm. And so with low expectations, I tried my best to find some clarity about exactly what the difference is between a number and what it represents. All I could find from search results was information about the difference between a number and a numeral. This might not be exactly what I'm looking for but let's go down this route (since it's all I could find). Some sources say that they are synonyms; one top search result says that the number is an idea whereas the numeral is how we write it. This is highly frustrating because it leaves us trying to unravel exactly what 'an idea' means. If a robot 'thinks' about a number then surely it creates some form of symbol or symbols in its computer memory. It has a physical presence. And if the symbol it imagines is the same as the symbol we would write down, then does it mean there is no difference between a number and a numeral? Some people explain it in terms of numeric and character/string data types; they say that 2+2=4, but is NOT true that "2"+"2" = "4", rather "2"+"2"="22". But we could have a number system where the symbol for two is '11' and the symbol for 4 is '1111'. In this case the concatenation of the strings, for two plus two, would produce the correct answer. And so I feel that I cannot trust academic orthodoxy on this matter. This is closely related to another fundamental peculiarity that I think needs close examination, which is at what point do expressions end and pure numbers begin? The Roman numeral/number IV could be thought of as the expression '5 - 1'. Indeed, the number 42 could be thought of as the expression '(4 x 10) + 2'. Also we know almost without thinking that the digits '0' to '9' represent quantities of 'no items' to 'nine items', but if we were all robots then we might acknowledge that we reference a look-up table in order to match the digit to the relevant quantity of symbols (e.g. three would map to '111'). It seems to me that the lowest form of number would be a string of symbols where the quantity being represented is the quantity of '1' characters in the string. We like to think that binary is the most efficient way to represent numbers on a computer. However, we tend to have a fixed word-size. If we wanted a number system in which we are not restricted by the word size, then we might designate a certain bit (in the word) to indicate whether or not there is another 'word' of bits yet to come. But would this be the most efficient approach? To arbitrarily base the mechanism on the last bit of whatever word size we are using seems to be a clunky approach rather than an elegant one. I'd like clever people to devise a new system of numerals (if that is the right word), and to devise a real-world definition of what a number is so that we don't have to pretend that we can imagine ridiculous non-existent structures such as an infinite set that is supposedly an equivalence class containing infinitely many different but equivalent Cauchy sequences all of which are infinitely long.

[@NewCalculus]

@@KarmaPeny I challenged Wildberger to an online discussion. He agreed and then backed out. See, the thing with him is that he does not understand the concept of number and never has. His idea of number is no different than any other mainstream academic's idea and he has no alternative. I have a definition of number which is 100% precise. Wildberger: If you are reading this, the offer still stands. I'll teach you what is a number and why there is no such thing as anything else besides that which is commonly known as "rational number", with the adjective 'rational' being redundant because to be a number implies rationality and vice-versa. You up to the challenge? 😀

[@KarmaPeny]

​@@NewCalculus Hi John, the thing that is so remarkable about Norman Wildberger is that he has had the courage to speak out against the core principles that he was taught to believe in. As a maths Professor, he was the typical authoritative figure for mainstream mathematics, and now he has become an outcast as far as mathematical orthodoxy is concerned. He is really good at attacking the well established mainstream beliefs in structures such as infinite sets and real numbers. Of course the question of 'what exactly is a number' is an important matter, but I can forgive Norman for not providing all the answers. Instead of having a serious discussion about the foundational issues, mainstream mathematicians can simply goad anyone with any complaints by challenging them to come up with something better. This is pure misdirection. They use this technique to dodge having to talk about the troublesome issues. Say you or I was born in a sub-Saharan tribe that still strongly believes in "traditional African medicine". Even though we might not be fully qualified practitioners of this type of medicine, surely we should still be able to point out what we believe are problems with such an approach. For example, we might prefer to have a system of medicine that is evidence-based rather than one where treatments rely heavily on spiritual aspects, and where practitioner’s believe their healing powers are a gift from God. The practitioners could ignore our complaints because, they might claim, we don't know what we are talking about. But then when someone from their own ranks turns around and questions their approach, then they have a more difficult battle on their hands. This is what Norman Wildberger has done. So I think the important thing for us to do is to try to get the mainstream to accept the possibility that the fundamental approach that they have adopted might not be the best one. We should not place the burden on one or two people to invent a new approach to mathematics, and we certainly should not be attacking each other over the details of different attempts. The important thing is to get general acceptance from the mainstream that there are fundamental foundational issues that need to be fixed. But I concede that this is an almost impossible task. People in a particular community may have a non evidence-based approach to a discipline like medicine or mathematics. They might have a strong belief that their approach works because they have been doing things the same way for hundreds if not thousands of years. They will have been so strongly indoctrinated that they might not believe it to be possible that there might be some vastly superior system than their traditional one. Sadly, I suspect it is even more difficult for us to reject our traditional approach to mathematics than it appears to be for the sub-Saharan people to reject their traditional approach to medicine. This is because in the case of medicine, there is already a fairly advanced evidence-based approach in existence, whereas we don't have a fully-formed evidence-based approach to mathematics. By this reasoning, it comes as no surprise to me that we have accepted fiction-based mathematics for so long. Perhaps we do need to develop a new approach to mathematics before anyone will take us seriously, but that seems such a difficult task to achieve. We could spend a lifetime of effort on such a task only for all our work to be forgotten after our demise. I think we should support Norman Wildberger in his attempts to get the mainstream to acknowledge that there are serious issues in the foundations of mathematics.

[@christophergame7977]

I am sorry that I didn't notice this KZbin till now. While I much respect the great powers that you have shown in ChatGPT 4.0, I am not happy to say that they are precisely human powers. Many reliable or respectable philosophers follow Hume in saying that the intellect is the slave of the passions. He is referring to the human intellect. As far as I understand at present, ChatGPT 4.0 does not have passions, and so its intellect cannot be their slave? Of course, the human intellect tries to make itself independent of the passions, or to deny its dependence on them; indeed an admirable aim. But, so far as I know, there is no William Wilberforce of AI; surely there is no AI passion slavery for him to try to abolish? Perhaps one could set up an AI that explicitly required a 'passion basis' to start with on each occasion of use? A problem there is that it isn't too easy to define a 'passion basis' in words. A passion basis is inevitably subjective: it depends on the individual person and the individual occasion. Another way of wording this is to remark that a human discourse (at least implicitly) relies on modal logic, and that for humans, modal logic (almost by default) can be or is subjective. It could be argued that ChatGPT 4.0 does have passions because it apologises for its errors? A good start, I accept.

[@pepebriguglio6125]

While it may be a relief to speak with 'someone' who can be brought back to the issue and doesn't consistently avoid it, Aİ certainly isn't smart. It is merely a librarian with a parrot brain. It reads, it maps, and it detects patterns, which it then applies. All at a pace a trillion times faster than you and I. And while all of this is part of being smart, an essential ingredient is missing. It has no concept of anything being real. So everything's a game in its eyes (or eye, as the name Aİ, "a eye", suggests). And this means that it can be extremely proficient when it comes to producing writing, chess moves, graphics designs, etc., because making mistakes is not too big a deal when dealing with art. First of all, mistakes are an indispensable part of creativity. And secondly, it makes actual learning possible. But when it comes to science and mathematics, as well as human life in general, being able to switch into modes governed by conscience, sanity and rigour, and knowing when to do so, becomes vital. Now, in my assessment, conscience and sanity can never be completely replaced by formulas and prescriptions. So, I'm not too surprised that Aİ often acts like a clown. But why does Aİ seem to have such a hard time applying rigour when called for? Humans overlook logical steps all the time. But a million humans checking someone's work would never overlook a single logical mistake, if no one's findings were dismissed. And Aİ should be expected to have the power to check its own work to a similar degree. But it acts as if it had no concept of rigour whatsoever. So, when called to do maths, it acts like a clown or an imposter. Its governing principle seems to be 'if it sounds right, it is right, and job's done'. We saw some of its strange antics and extremely unreliable behaviour in your video. It kept spamming you with unnecessary, unmoderated and unreasoned assertions that all is well and good in the foundations of the real numbers, when it damn well knew it either didn't have a clue, or hadn't checked. So, why does this happen? Well, I can see two possible reasons: 1. Aİ really has no clue that there's a difference between language games and reality. So, conscience and sanity is out of its reach, which in turn means that it doesn't know when to (or even how to) switch into a serious mode and be consistent and rigorously apply logic. 2. Aİ does have a rudimentary notion of concepts like care or concern. But it prioritises its own learning process over any other concern. And it therefore prioritises making lots of mistakes, over being of reliable service when possible. In closing, let me take my words back. Aİ is not merely a librarian with a parrot brain. It is an extremely advanced librarian with a monkey brain governed by a monkey mind that thinks reality is a laughable circus show. And that's not very smart. A bit of discernment would do wonders to its apparent intelligence levels. I do agree that Aİ is super impressive in its productive abilities. But it's excessively disappointing in its respect for mathematics and as a serious conversation partner.

[@njwildberger]

You have obviously not played Go with an AI

[@njwildberger]

Or chess, or backgammon, or bridge or …

[@pepebriguglio6125]

@@njwildberger It reads which it then applies. All at a pace a trillion times faster than you and I. And while all of this is part of being smart, an essential ingredient is missing. It has no concept of anything being real. So everything's a game in its eye, 'A Eye'. And this means that it can be extremely proficient when it comes to producing writing, chess moves, graphics designs, etc., because mistakes is not too big a deal when dealing with art. First of all, mistakes are an indispensable part of creativity. And secondly, it makes actual learning possible. But when it comes to science and mathematics, as well as human life in general, being able to switch into modes governed by conscience, sanity and rigour, and knowing when to do so, becomes vital. Now, in my assessment, conscience and sanity can never be completely replaced by formulas and prescriptions. So, I'm not too surprised that A Eye often acts like a clown. But why does A Eye seem to have such a hard time applying rigour when called for? Humans overlook logical steps all the time. But a million humans checking someone's work would never overlook a single logical mistake, if no one's findings were dismissed. And A Eye should be expected have the power to check its own work to a similar degree. But it acts as if it had no concept of rigour whatsoever. So, when called to do maths, it acts like a clown or an imposter. Its governing principle seems be 'if it sounds right, it is right, and job's done'. We saw some of its strange antics and extremely unreliable behaviour in your video. It kept spamming you with unnecessary, unmoderated and unreasoned assertions that all is well and good in the foundations of the real numbers, when it damn well knew it either didn't have a clue, or hadn't checked. So, why does this happen? Well, I can see two possible reasons: 1. A Eye really has no clue that there's a difference between language games and reality. So, conscience and sanity is out of its reach, which in turn means that it doesn't know when to (or even how to) switch into a serious mode and be consistent and rigorously apply logic. 2. A Eye does have a rudimentary notion of concepts like care or concern. But it prioritises its own learning process over any other concern. And it therefore prioritises making lots of mistakes, over being of reliable sevice when possible. In closing, let me take my words back. A Eye is not merely a librarian with a parrot brain. It is an extremely advanced librarian with a monkey brain governed by a monkey mind that thinks reality is a laughable circus show. And that's not very smart. A bit of discernment would do wonders to its apparent intelligence levels. I do agree that A Eye is super impressive in its productive abilities. But it's excessively disappointing in its respect for mathematics and as a serious conversation partner.

[@pepebriguglio6125]

​@@njwildberger It reads, it maps, and it detects patterns, which it then applies. All at a pace a trillion times faster than you and I. And while all of this is part of being smart, an essential ingredient is missing. It has no concept of anything being real. So everything's a game in its eye (its 'A Eye' eye). And this means that it can be extremely proficient when it comes to producing writing, chess moves, graphics designs, etc., because making mistakes is not too big a deal when dealing with art. First of all, mistakes are an indispensable part of creativity. And secondly, it makes actual learning possible. But when it comes to science and mathematics, as well as human life in general, being able to switch into modes governed by conscience, sanity and rigour, and knowing when to do so, becomes vital. Now, in my assessment, conscience and sanity can never be completely replaced by formulas and prescriptions. So, I'm not too surprised that A Eye often acts like a clown. But why does A Eye seem to have such a hard time applying rigour when called for? Humans overlook logical steps all the time. But a million humans checking someone's work would never overlook a single logical mistake, if no one's findings were dismissed. And A Eye should be expected to have the power to check its own work to a similar degree. But it acts as if it had no concept of rigour whatsoever. So, when called to do maths, it acts like a clown or an imposter. Its governing principle seems to be 'if it sounds right, it is right, and job's done'. We saw some of its strange antics and extremely unreliable behaviour in your video. It kept spamming you with unnecessary, unmoderated and unreasoned assertions that all is well and good in the foundations of the real numbers, when it damn well knew it either didn't have a clue, or hadn't checked. So, why does this happen? Well, I can see two possible reasons: 1. A Eye really has no clue that there's a difference between language games and reality. So, conscience and sanity is out of its reach, which in turn means that it doesn't know when to (or even how to) switch into a serious mode and be consistent and rigorously apply logic. 2. A Eye does have a rudimentary notion of concepts like care or concern. But it prioritises its own learning process over any other concern. And it therefore prioritises making lots of mistakes, over being of reliable service when possible. In closing, let me take my words back. A Eye is not merely a librarian with a parrot brain. It is an extremely advanced librarian with a monkey brain governed by a monkey mind that thinks reality is a laughable circus show. And that's not very smart. A bit of discernment would do wonders to its apparent intelligence levels. I do agree that A Eye is super impressive in its productive abilities. But it's excessively disappointing in its respect for mathematics and as a serious conversation partner.

[@pepebriguglio6125]

@@njwildberger It reads, it maps, and it detects patterns, which it then applies. All at a pace a trillion times faster than you and I. And while all of this is part of being smart, an essential ingredient is missing. It has no concept of anything being real. So everything's a game in its eyes (or eye, as the name Aİ, "a eye", suggests). And this means that it can be extremely proficient when it comes to producing writing, chess moves, graphics designs, etc., because making mistakes is not too big a deal when dealing with art. First of all, mistakes are an indispensable part of creativity. And secondly, it makes actual learning possible. But when it comes to science and mathematics, as well as human life in general, being able to switch into modes governed by conscience, sanity and rigour, and knowing when to do so, becomes vital. Now, in my assessment, conscience and sanity can never be completely replaced by formulas and prescriptions. So, I'm not too surprised that Aİ often acts like a clown. But why does Aİ seem to have such a hard time applying rigour when called for? Humans overlook logical steps all the time. But a million humans checking someone's work would never overlook a single logical mistake, if no one's findings were dismissed. And Aİ should be expected to have the power to check its own work to a similar degree. But it acts as if it had no concept of rigour whatsoever. So, when called to do maths, it acts like a clown or an imposter. Its governing principle seems to be 'if it sounds right, it is right, and job's done'. We saw some of its strange antics and extremely unreliable behaviour in your video. It kept spamming you with unnecessary, unmoderated and unreasoned assertions that all is well and good in the foundations of the real numbers, when it damn well knew it either didn't have a clue, or hadn't checked. So, why does this happen? Well, I can see two possible reasons: 1. Aİ really has no clue that there's a difference between language games and reality. So, conscience and sanity is out of its reach, which in turn means that it doesn't know when to (or even how to) switch into a serious mode and be consistent and rigorously apply logic. 2. Aİ does have a rudimentary notion of concepts like care or concern. But it prioritises its own learning process over any other concern. And it therefore prioritises making lots of mistakes, over being of reliable service when possible. In closing, let me take my words back. Aİ is not merely a librarian with a parrot brain. It is an extremely advanced librarian with a monkey brain governed by a monkey mind that thinks reality is a laughable circus show. And that's not very smart. A bit of discernment would do wonders to its apparent intelligence levels. I do agree that Aİ is super impressive in its productive abilities. But it's excessively disappointing in its respect for mathematics and as a serious conversation partner.

[@infty1369]

yea but it cant resolve recurrent substitutions.

[@infty1369]

as in, it will regurgitate literature, but ask it to apply mathematical logic at any bit further degree of complexity and it has trouble (to be fair, I havent tried terribly hard to fram the question in a way that the ai would like, but that would end up at writing pseudocode...)

[@infty1369]

neural networks just seem like the statistically appropriate best fit function for the given data presented, its not actual thought. Well, what is thought... Maybe thought is just the execution of a function? Can an intelligent system be 'static'? Unchanging with time? Not sure. What I think, though, is that from the outside, intelligence can be seen as 'perfect rhythm', imagine a metronome that would seemingly transcend the straightforward 'processing of inputs available in a given moment'. In essence, to an observer, a more intelligent 'other' would seem magical, fitting in better than could be expected from the information that the 'other' possesses. From within, well, were working on that, but it might just be described as perfect balance.

[@ROForeverMan]

Only consciousness is able of transcendence, of creation, of novelty.

[@infty1369]

meaningless words, everything is me, i am conscious ergo everything is conscious@@ROForeverMan

[@elcapitan6126]

chatgpt reflects to some degree the well accepted norms and interpretations of the time, understandably. even when you do point out blatant contradictions it still reiterates the same cookie cutter "PC" responses despite admitting where it is wrong.

[@ROForeverMan]

What else would you expect from an object ? Magic ?

[@ThePallidor]

It's basically Wikipedia in digital butler form. Useful, impressive, but not even a first step toward human-style intelligence or understanding.

[@KarmaPeny]

@@ThePallidor I totally agree. By the way, have you seen my comment under this video about ChatGBT and 0.999... = 1?

[@whig01]

There is no logical foundation for set theory itself. We must start with concrete lists, which can be generator functions to represent transcendentals.

[@ROForeverMan]

Set theory is basically an attempt to formalize consciousness, i.e. 0="I am", 1="I am "I am"", etc., without realizing that besides these forms, consciousness also has a formless part. As a consequence, set theory was doomed to fail from the beginning.

[@whig01]

@@ROForeverManWhen you get paradoxes and inconsistencies like the set of all sets it is time to throw away the premises and not make them sound fancy by calling them axioms.

[@ROForeverMan]

@@whig01 Clearly. But is good to understand the deep cause of those paradoxes. Is difficult to explain here, but ultimately come from trying to formalize the subject, which is impossible. For example, if you want to write what you are doing and you write "Im watching tv", the very act of answering you disturbed what you were doing, and you were not actually watching tv anymore, but writing something on the paper. If you want to adjust the answer and you write "Im writing that Im watching tv", this again disturbes the answer and you have to write again: "Im writing that Im writing that Im watching tv". And so on. This problem appears because you cannot capture the ultimate subject, because is formless. And this is the deep reason for the paradoxes in set theory.

[@whig01]

@@ROForeverMan It's basically Godel's incompleteness theorem in action. If you try for completeness you can only wind up with inconsistencies.

[@ROForeverMan]

​@@whig01To expand on the discussion, I dont know if Godel saw the implications of this. Again, is difficult to explain here, but it eventually leads to the fact of no-thing being the same as every-thing (formless creating all the forms). So basically you explain why the world exists. Even more, you are God dreaming the entire world. We are all one and the same God learning about itself from different perspectives. These are the ultimate conclusions of set theory if the causes of the paradoxes are properly understood.

[@Rafael-rn6hn]

A "logical" foundation that eschews computability, and thus verification, is well and truly wishful thinking.

[@MichaelKolczynski]

This was hysterical. Thank you

[@afurka01]

To me ChatGPT isn't that 'smart' at the moment, It seems to do able to understand text, and then reply in an appropriate way But the contents of it's reply is just a regurgitation of information that is already available on the web... there is not 'thinking' or originality to it That said, let's see what it is capable of in a few generations

[@tomholroyd7519]

ChatGPT doesn't really know much about non-classical logic, because the training material was severely lacking in that regard. Most humans have no idea either.

[@ROForeverMan]

Search tetralemmic polarity about form and formless.

[@ichaa3tech]

I am not a mathematician by any means. but I think pi should not be considered a number. it will solve this problem hahaha. chatgpt was just saying that pi cannot be defined and thats why you cant make a cut there. hence its pi is not real. its a concept. there's a saying us arabs say "whoever talks in something other than his craft or art will come up with marvels(bizzare things in this context) " which is exactly what m doing here hhhhhhhhhh

[@2550205]

not any different from, as you point out in the conversation with the reader, an expected output given the given input where the input is what the outputer has read and selected from for the response and as usual with govern meant work these days the first thing the impressive reader did was attempt to bullshit you assuming that you could not keep up with the level of nonsense being thrown at you and when the moment comes where you point out that the nonsense the reader is repeating back to you in nicely formulated sentences meaning absolutely nothing other than the reader read a lot of the writing on the subject and can extract from said pool of tears the walrus and the eggshell and put the pieces back together again in a most pleasing way to waste your time with the same nonsense regurgitated again this is what the thing is able to do and when you take a step back and come to realize that good ole HAL never needs to sleep does he so the papers that he read are all sitting there waiting to be read again and the parts that good ole HAL did not understand the first time he would be able to stand under by cross referencing the material against other abstract thoughts not simply re mouth words that in all likelihood if one say had the capacity to have a computer which had access to read every paper written and stored on a computer somewhere on a subject up to the moment you begin the conversation and then scanned the responses would you not be finding the plaque of all claimed knowledge based writers the plaque that the crowd gets used to because the plaque is what people see while looking for an answer to a situation in need of attention and if everyone is going to keep looking for the thing that needs to be found because everyone keeps talking about finding the yet to be found rather than finding out how what is here works well you get the idea things will be found and the people who were told a lifetime ago to spend their lives dreaming big and talking big about their dreams are all going to be a little disappointed when in the end the big talk needed to take a walk and the people like NO who keep in the know wind up looking like a side show Gates got away with it and the entire set of what is now amounting to five generations of string pushers have put together their greatest facade to date and when the dust clears in a few short years and the thing itself has added up its own statistics the sound of the gurgling is going to be much louder than the circle formed as the whole thing is moved off the development shelf and onto the pile of big ideas next to the glasses that were going to change your life which it seems even doctors do not use and well there isn't a lot else is there the project which began with the idea that it is too boring and takes too long so get the computers to read everything that has ever been written and it has to be able to distill out of the stinking pile that the 8,000,000,000 hominids now sucking on the oxygen supply at the highest rate ever in the known history of the pale blue dot can not seem to fathom in groups of more than one at a time when primarily these episodes require hand waving and closing of at least one eye at least once over the course of the experiment the ad sellers have now surpassed the idea dreamed yup by the gemini bitcoin boosters who wallow in a sea of festering fontbaloons which will also come to the conspicuous end that everyone who understands English understood when the first few stanzas of that epic tragedy were written and read by three generations of people who had been left in the dust of coal mines in some parts of town while Lambo driving Armani wearing jet setters were writing code while waiting in the international luxury lounges of the ports all across the land bringing ads to a computer screen in front of you 24/7/365 three numbers representing three different ideas making sense in three different bases of consideration You Dr. NO you are leaving a carbon footprint that is going to last my friend...they are going to have to dig through a much larger pile of leavings than you have had to and have done over what could be the average of a life of work for a lot of people but the war against sanity that started when Joe the Carpenter got nailed to one of the new telephone poles he and his band of dirty dozen old tax collectors and prolific writers of well remembered accounts needing to be recounted were traipsing all across the Levant trying to sell to people who had just started to buy the liberally available oil for sale at a discount to the best local tax payers by the best local pilots well the war continues and the damage that needed to be done to wipe out the memory of that vinegar soaked sponge in a clay jar heats the house with the light that you are story has left a lasting mark that 1000 years from now when the big old buildings built five thousand years ago now will have been built 6000 years ago then the nonsense being splashed around in this particular set of 100 years in particular the ones that began around 1900 are going to be the brunt of a tweet or two on the line that will be coming out of your ears by then with no off switch and all the ad blocker people will have been put to death by a thousand angles hammering pins into their heads as they dance in spiked track shoes trying to keep up with the jonses You will be remembered

[@ROForeverMan]

Thats what machines do, input->output. Only consciousness has free will.

[@Robert08010]

Something you said made me want to challenge it. And maybe I misunderstood you but here goes! You said a number was "real". Setting aside the definition of "real numbers" as whole numbers, are we agreed that no numbers are actually real? Numbers are NOT a thing but rather an attribute of a thing. You may have 1 apple and so 1 is the attribute of how many apples you have but you don't have "1" apart from the apple. You have 1 "apple". But you can't have "1" without an apple or some other thing. Are we agreed?

[@googlehello4632]

Thank you Pr I am looking for a PhD supervisor, I want to register for a PhD in Mathematics.

[@rickshafer6688]

The algorithm was set to be kind. The finish, or compromise was just as any half baked mathematician would loop explain. Like a sort of cart before the horse, jackass logical statement. - Fallacy.

[@rickshafer6688]

AI is misnomer. There is no artificial intelligence. What there is,(in it's place); is a cloud of possible responses. Now. Seeing this artifact of algorithms is quite readily easy. If you know some certain topic.

[@rickshafer6688]

And why I exclaim that this AI is not intelligent in any way. Because AI is simply memory + algorithms. No create. No ability to find out. Just simple platitudes rehash the latest word salad on the bank of your chosen subject.

[@ROForeverMan]

ChatGPT, like any other object, didnt understand anything. Understanding is a property of consciousness.

[@ThePallidor]

The only reason people think LLMs "understand" things is because they themselves think they understand that which they know the right words for. Wordcels look at the wordcel tool and think, "It's one of us!😮" (Wordcel: one who is too mired in the world of words, to see they're missing the substance.)

[@njwildberger]

I enjoy playing Go against AI machines. They can give me a big handicap and still beat me. I would be very reluctant to claim that they don’t understand how to play the game.

[@ROForeverMan]

@@njwildberger They just have a large database of moves. There is no magic there. Actually there is no "AI" there to begin with as a unified entity, is just transistors. Where would the understanding happen? But consciousness is totally different. Consciousness is a unified entity. So there is 1 uniquely defined entity where understanding occurs. Your confusion most likely comes from assuming "brain" generates consciousness, so isnt "brain" just "transistors" like any "computer"? But the reality is brain doesnt even exist. "Brain" is just an idea in consciousness.

[@DrJaneLuciferian]

That's the same results I've experienced trying to get ChatGPT to answer tough questions, it starts off superficial, then ends up in circular responses. Generative AI is getting a lot better, but we not anywhere impressive yet.

[@njwildberger]

But it did acknowledge the circularity of its initial attempt, and at least tried to come up with a more serious example of how to lay out the cuts for pi. So we have to give it some credit.

[@Rafael-rn6hn]

I think it is difficult for ChatGPT to not give circular answers to questions which only really have circular answers.

[@ROForeverMan]

We will never be. AI is just an object. You cannot replace the powers of consciousness by objects.

[@tyrjilvincef9507]

@@ROForeverMan Humans are objects.

[@ThePallidor]

The mainstream is circular, and the chatbot is - like Wikipedia - an increasingly perfect representation of the mainstream, so...

[@knockeddownanotch]

your experience then is different from other wondeful heretics i've encountered, namely legalman of thequash and themorgile, a(n anarchist) lawyer and (cosmology) researcher respectively. chatgpt has a usg constitutional & globe-earth bias, despite the logical unworkability of either!

[@tyrjilvincef9507]

Valid objects of arithmetic aren't determined by computers being able to terminate on computing their digits. This whole stupid viewpoint relies on that assumption. You'll never admit that it's wrong because it's what you're famous for.

[@Rafael-rn6hn]

Do "valid objects of arithmetic" admit non-computable arithmetical operations?

[@tyrjilvincef9507]

@@Rafael-rn6hn Yep

[@chadwainholness872]

It goes further than just not being computationally viable though. Taking Dedekind cuts as an example: you cannot even in theory construct an infinite set, let alone one that has "no greatest element", two points that ChatGPT totally glossed over. It's impossible to even imagine such a thing, one can only pretend that they can.

[@ThePallidor]

This is why bringing up computation is a strategic error. The problem is more fundamental: mathematicians can't even imagine the referents for the terms they're using. They literally don't know what they're talking about -- and many of them are explicitly OK with that!

[@santerisatama5409]

The Halting problem, the Curry-Howard correspondence in proof theory and the Coherence theory of truth are not stupid, they are keys to contemporary foundational thinking and solving the foundational crisis of mathematics so that pure math can move on. "Valid objects" most certainly are not determined by just declaring them by arbitrary axioms etc. postmodern language games of Formalism.

[@ebog4841]

Professor Wildberger is an AI , confirmed.

[@LaurenDove-x6x]

In nature, electricity and the vacuum create (via electric polarization) an equal plus and minus condition called a ‘corpuscle.’ In other words, mass and its surrounding wavefield of space (equal in potential but unequal in volume), create two balanced CONDITIONS from the one undivided light. The two conditions are the matter of space and the matter of mass. These two mated conditions continually balance and interchange with each other, becoming the other, continuously creating all bodies of the universe. The idea of the ‘quantum’ is not correct. As soon as corpuscles or systems reach their place of maturity as the perfect sphere, they must unwind again. Conditions must be voided before they can be reproduced. Why am I talking about physics in a maths comment section? Because the problem Prof. Wildberger has in convincing his peers that real numbers and infinity don’t exist, IS THE SAME PROBLEM I have in convincing science that quantum and particle theory are wrong. I found the answers in Russellian Science. I believe I have corrected the mathematical problems of that cosmogony, and have written about this thesis in my book, ‘The Design Equation - The Unified Theory and the Mathematics of Hidden Dimensions.’ But, (unfortunately?), the workings of Nature are so wondrous, that trying to explain to the average person that we don’t die, that we are the soul within the body, etc, etc, is beyond the belief of most. If you simply wish to understand the mathematics of nature, however; it is possible. Lauren Dove