I'm thrilled to see you're back to doing Math Foundations videos. I love this series and I have learned a lot watching it. It looks like you're really about to delve into the heart of the matter and I'm excited to see what's coming up!

I know I’m four years late to this video, but I’m so thankful to have found it. You’re a wonderful communicator and the subject matter is profoundly interesting. I’m so glad you produce content. Will be hunting you down on Patreon.

Surely a trivial observation but if you are putting natural number arithmetic as foundation aren't you already presupposing the notion of a set comprehending all natural numbers? Sets are the extensional counterparts of predicates/concepts/propositions. It seems impossible to reason without predicates and predicates and sets are like conjoined twins to me.

@sonicmaths8285

Ай бұрын

Yes, if you’re trying to understand what lies deeper than that you certainly encounter a self-referential foundation, meaning those areas really seem like axiomatic beliefs, although it does seem not too far fetched to think that they are the basis of mathematics. After all everything in mathematics is logical and we are always dealing with things which are elements of sets, thus we’re dealing always with sets and their properties and so on Kinda fascinating

@Ludwig1954

Күн бұрын

​@@sonicmaths8285I quite agree. Natural numbers arise "naturally" only when the Application "Quantification" comes up. Namely, in Maths entering the realm of Science, aka. the Quadrivium. Pure Maths belongs to the Trivium .

I was so pleased to find this video. I am a retired digital hardware designer from the early 70s and formal logic has been my life. Lately I have been amusing myself with topics in pure maths and Set Theory is (was?)my latest topic. After the first lecture I had reached the conclusion that it could not support the claim to be the basis of maths ! It was that obvious. Everything you say in this introduction chimed with me and I will follow through with the rest of the series. Thank you so much.

Have you heard of reverse mathematics? It's a program in mathematical logic to find the minimum amount of axioms we need to prove things, and actually proving that it's impossible to prove a theorem without assuming a certain axiom. For example, the fundamental theorem of algebra is provable from only assuming that computable sets of naturals exist. And the existence of the real numbers (as a complete linearly ordered field) is equivalent to "every countable commutative ring has a maximal ideal", in the sense that any axioms that prove one must prove the other (as long as the axioms include some simple stuff about the natural numbers, otherwise you could literary assert one of them as your only axiom). So most of the work of finding a simpler foundation as already been done, you just have to pick the strongest reverse mathematics system that you believe is true.

@inversehyperbolictangent3955

6 жыл бұрын

Sounds similar to the P vs. NP problem in computer science, and how people will show that a new problem X is equivalent to an old problem Y, where Y has already been shown to be in NP. The conclusion is that X is also in NP. Sounds useful. I do wonder how far 'most of the work' has 'already been done', though. Surely I expect that, as you say, a lot of work has been done. I just wonder how far people have pushed finite-and-computable numbers as far as they can really go, without abandoning them too quickly for additional infinite-or-uncomputable axioms. For example, Wildberger is currently exploring how far we can go into the field of Calculus in his new course. By the time most mathematicians reach the topic of Calculus, they've already assumed some form of Real numbers, taking them for granted.

@ReifAndreas

6 жыл бұрын

That's the very approach of the ZFC-basics. For example Zermelo 1908b.

@senselessnothing

4 жыл бұрын

@@ReifAndreas no variant of set theory succeeded with minimizing assumptions because no variant limits the possible amount of sets one can deal with. That's something reverse mathematics does.

@deemotion

3 жыл бұрын

Can you show this?

@senselessnothing

3 жыл бұрын

@@deemotion You can read about it in books, start from the wiki page, it's a surprisingly easy subject I find.

I appreciate the idea of being clear on definitions, but how are the axioms of set theory any different from the axioms of arithmetic? Is it that arithmetic is grounded in everyday "experience", while infinite sets are not? Is one more natural than the other? More "philosophical"? I'm genuinely curious, because I've gotten a lot out of these videos and your book Divine Proportions.

@writerightmathnation9481

2 жыл бұрын

From what I have gathered, professor Wildberger wants to avoid any axiomatic foundation in the sense you are using. He wants to only use the older meaning of the word "axiom", as in "self-evident truth". My first objection to that is that the notion that the axioms of natural number arithmetic seem to not be "self-evident" precisely because they must be taught in order for students to use them. In many of his videos, professor Wildberger also points out, tightly, I'd say, that computers are playing and will play, an ever expanding role in mathematics. To a machine, the notion of a "self-evident truth" is even more suspect than for humans, even in the sense of natural number arithmetic. Once one spa being convinced that there are any self-evident truths, your questions do seem to com new to the fore. I think he's aired more at indoctrination than instructing, which is sad. I think he has a lot of cool things to say, but he's overly polemic about his ultra-finitist view, and on the one hand, this damages his credibility, though not from the perspective of rigid in his computations and his constructions. His credibility is trained by his bias against creativity in mathematics. He often tries to get viewers to at a claim that more can be done, and more beauty comes into focus, by denying the existence of infinite sets, by the denial of the value of learning the theories of the real numbers and complex numbers, and by preaching against mainstream mathematics. If he would just prove certain of his claims, he'd get my attention. A problem I think he has not addressed, but should, is the following: What is the smallest positive integer that has no successor? You see, in many of his videos, professor Wildberger claims that for some specific number he can write down, most numbers that order of magnitude don't exist. Philosophically, I object to the phrase "that number doesn't exist", but let's put my objection to that aside for a moment and focus on his claims that he will found mathematics on the theory of the natural numbers, for which it is usually taught that the least number principle holds. Set theoretic approaches to the theory of natural number arithmetic often, and very naturally, I'd say, treat this principle as a first order axiom of set theory or as an axiom of second order arithmetic, but if we do neither, it seems to me that his non-axiomatic approach must needs eliminate it as a self-evident truth, and I don't it even works as a principle, in which one does not treat second order formulations of principles as being satisfactory for careful discussion of properties of numbers, at least in general. Returning to the specific case at hand, if he suggests that the least number principle is valid, valuable, or "true", then his claim that the number N=10^10^10^10^10^10 +23 (from another video - I think I got the right number of instances of the number 10 in the power tower...) is a number such that most numbers near N don't exist, as he sends to have done in another video, then we face the natural immediate query "What is the first natural number less than N that doesn't exist?". Among other things, I doubt that he can give an answer to this an at all. Thus, he has made a claim that is in the sane family with the following claims that it seems he objects to, namely the existence of something that he cannot prove exists, so that his contention itself can only be accepted axiomatically in the same sense that mathematicians in the mainstream accept the axiom of infinity. In fact, I expect that accepting it as an axiom is even worse, in that it actually yields a contradiction in his theory of numbers and in his theory of arithmetic, at a very foundational level.

@elcapitan6126

2 жыл бұрын

one could argue that even the integers are largely a product of technological advancements relating to uniformity and reproducibility (specifically the idea of a unit of something which really only itself is a useful concept if measure and repetition and precise equivalence is established between things in every day experience). small tribes would have little use for numbers beyond say one or two because that pattern wouldn't have conveyed any useful information for decisions (everyone in the tribe has a name, but what use is the idea of a tribe having the notion that it has "ten" members if they have no reason to compare that pattern to anything else?) most things in nature are not identical and often don't need to be treated as such. we discretize and attempt to maximize uniformity as a part of our technological processes (likely for the ease of communicating aspects of them for coordination purposes)

@mar98co1

Жыл бұрын

@@writerightmathnation9481 Yea, I like how he wants to criticize the current foundation as "philosophical" and imprecise, but when it comes down to it, his own approach leaves things up to "woo" magical intuition. Just shows the double standard he's applying

@TheoEvian

Жыл бұрын

@Saitama Tuga Because he didn't read it so he doesn't know what he's criticising, I would guess.

@patrickwithee7625

Жыл бұрын

@@TheoEvian I’m pretty sure almost no one has actually read Russell and Whitehead’s Principia.

Just recently found your videos after getting interested from a Russell/logic/philosophical angle. I must say, your ability to explain mathematics and concepts to a non mathematician is excellent. I’m really enjoying all the foundational/logic videos. I’m excited to venture deeper into all mathematics.

If set theory was _the_ foundations for mathematics... But it's not. There are several candidates, and they're not different kinds of set theory, for instance Homotopy Type Theory is based on Martin-Löf type theory, and they don't mention sets as fundamental concepts; sets are just a type. To be honest, I see a much more "religious" viewpoint from you than from anyone working on contemporary math foundations. By watching your video(s) I'm not really getting the feeling of you being wrong or whatever, as several points are really interesting. For instance, I really dislike set theory for several reasons, I have the feeling it's like an unrefined tool with a lot of patchings; it doesn't behave the way mathematics is literally thought and done by anyone (except people working on set theory itself): membership and equality the way they are now need to be rethought, and at some point we just use set theory to construct settings behaving as we hoped only to immediately forget how they're defined, but set theory strikes back because subtle issues rely on it. I could write down pages and pages of my thoughts about this topic, pointing out precisely what my concerns are. But I can't really follow what you're pointing out to, all I see is a neverending repetition of "this is not logically defined" and such. You're not even saying that this is your personal opinion, you claim that as a fact, and this is religion to me. Maybe I'm stupid and all the people supporting your viewpoint are way smarter than me, or maybe not.

@Sculman7

5 жыл бұрын

Most likely not.

@sarahbell180

5 жыл бұрын

The thing is set theory is usually regarded as the foundation: category theory and 'homotopy type theory' can also serve for these roles, but these are of select interest. Most of what he said appears to be true under a negative tone(I fully support set theory as foundations by the way). Rather, I don't think it was of fault of mathematicians to not very seriously examine foundations, for we never did in the longest time of the study of mathematics.

@paulwary

Жыл бұрын

HoTT is a very recent arrival compared with set theory. Start with an elementary text on eg measure theory, topology etc, and you will see that math as it stands now, how it it taught now and practiced now, is founded upon set theory. Arithmetic is founded on set theory, and famously Russel & Whitehead took hundreds of pages to prove that 1+1=2. HoTT may change that, given time.

It's great and important job you are doing! Thank you for your videos!

Thank you so much! It helped me a lot with background for understanding Modern Logic

I have found my new KZread hole. Came to ground my understanding of Set Theory for future explorations, but staying for this fascinating and super long playlist.

Dear professor Wildberger, I wonder your opinion about the work of Alexander Grothendieck. Thanks for this new video!

Prof Wildberger. .thanks for the great videos. It's only upon reflection that maths can grow. Set theory is a beautiful theory..yet it's NOT perfect. We need to regularly revisit the underlying theories of maths to make them stronger and more rigorous.

I agree that foundations is incredibly important, and should be accessible to all mathematicians. I think the alienation is a hold over from Hilbert's program. However, doesn't the work of Godel completely dismiss your argument? Your foundation, if it can do arithmetic on the naturals, cannot demonstrate its own consistency. Therefore you'd have to resort to some other theory to demonstrate its consistency, which just moves the problem up a theory. That's the whole reason for the assumption of the consistency of ZF. We can't prove it.

@strangeWaters

2 жыл бұрын

iirc Godel incompleteness only applies if your system has infinitely many natural numbers to do arithmetic with, which prof. Wildberger's actually doesn't.

@writerightmathnation9481

2 жыл бұрын

Actually, Gödel's work goes much further in a way that your statement does not quite express, but for which a very short extension expresses completely, as follows: Wildberger's foundation, if it can do arithmetic on the naturals, can formulate statements equivalent to it's own consistency, but cannot demonstrate its own consistency, unless it is inconsistent. More significantly for the discussion at hand, Professor Wildberger's insistence on this ultrafinitist philosophy as a foundation for mathematics, as he often states, is based upon an idea, (among other similar ones) implicit in various polemical statements that he makes in many of his videos that "all mathematics that can be done can be done by a computer", and relies on Church's Thesis as an upper bound on the limits of computation, which is called a thesis, rather than a theorem, because it has not been proven, and it is equivalent to stating that Gödel's notion of a recursive function as the prototypical model of a computer program is the correct such model. Regarding the reason for the assumption of the consistency of ZF, really there's a bit more to it as well. Not only can we not prove such a consistency result for ZF, but we also cannot prove such a consistency result for Quine's New Foundations (NF), but for a much more incisive reason: We ARE able to prove that Quine's NF is inconsistent. For that reason, we DO NOT assume Quine's NF. The main reason that we assume the consistency of ZF is that although we cannot prove its consistency, we also have not proved its inconsistency, which could only be done by finding a contradiction, and because we cannot thus far imagine a good reason to assume that it is inconsistent; that is, we have no evidence to suggest that ZF is inconsistent. Finding a contradiction in ZF would be phenomenal, and Wildberger's program might help us to find one, but not by making a large number of polemical anti-ZF statements, for as you pointed out, his "arguments", such as they are, seem to consist of a long laundry list of ad hominem arguments.

@writerightmathnation9481

2 жыл бұрын

​@@strangeWaters This is a misrepresentation of the facts. Gregory Chaitin has developed a system of claims and proofs of those claims, that formulates a variant of the undecidability phenomena for finite languages. His results are very deep and I've not learned enough of it to drone on endlessly about it with clear and careful explanations, but as I recall, roughly (and I may be butchering this a bit), in any sufficiently large finite language that has sufficiently high expressive power there can be found a sentence that expresses the consistency of the self-same system, up to bounded complexity, but that same system cannot prove that this weakened form of consistency holds for itself.

@writerightmathnation9481

2 жыл бұрын

@@strangeWaters What does "iirc" mean?

@WelshPortato

2 жыл бұрын

@@writerightmathnation9481 dont know if any responses were deleted but it means if i recall correctly x

You are the best mathematician out there. I can answer all of your Euclid critiques. And I have not read him.

Hi, I am a big fan of your lectures and am trying to teach myself algebraic topology. Unfortunately, it is difficult to work my way through your lectures chronologically since they're not in order and aren't organized in a playlist. Would you mind creating a playlist of your algebraic topology lectures? Many thanks.

@lucianopollicino

8 ай бұрын

Did you find a way to study your own algebraic topology?

I wonder, what do you think of wittgenstein's attacks on set theory back in the 20th century?

@njwildberger

6 жыл бұрын

I think very highly of Wittgenstein's attacks on set theory, and they are going to figure prominently in this discussion, albeit down the road somewhat.

@AdamMcLean

6 жыл бұрын

"We have essentially outsourced the very heart of our subject to philosophy".

@njwildberger

6 жыл бұрын

@ Tommy Jensen: Your comment on Wittgenstein is about as sensible as your spelling.

@njwildberger

6 жыл бұрын

@Tommy Jensen: Does someone who spells incorrectly have excellent spelling? I think not. While it is tangential to our discussion, it is a point not without application to logic. To say that my equation was wrong, but only because I wrote it down incorrectly does not change the fact that it was wrong. In a similar way, your conclusion that we agree about Wittgenstein is clearly not correct. You make a statement about my awareness of criticism of W against G, without it seems any justification. You claim that K,D,G,A and B "all independently had to write articles to ..." Is that really correct? And the meaning of your last sentence is beyond me. Possible conclusion out of all of this: perhaps before you go further in your rhetorical efforts to correct my thinking, you should get your own act together?

@xjuhox

5 жыл бұрын

@@njwildberger Dear Norman, if your *Brouwerian ghoul* whispers to your ear all this stuff, it may be good to understand that it may not be true. You are bound to fail, just like Edmund Husserl's bullshit philosophy was bound to fail when he tried to attack the Einstein's theory.

You said that there are problems with large numbers, do you have any video on it? Or could you elaborate?

there was one video where you discuss and list the zfc axioms. Anybody has the link?

I watched the Math Foundations series from this video (250) until M.F 280 (the last one). In the M.F 280 video, you said that you will continue exploring the mathematical systematic approach to logic. At the end of this video (MF 250), I thought that you were going to explore the topic of logic in order to go back eventually to the subject of the foundations of mathematics and link the results of this exploration to some new paradigm of math foundation (as an alternative to set theory). Where's the rest? I "binge watched" this series form MF250 to MF280 and now I don't know where to find the rest :( P.S.: Thank you very much, your videos make me happy, *literally* .

Well said! I thoroughly enjoyed this

Hi Norman - great video. I'm currently working on a similar video to this one. I think the core problem occurred long before set theory; for me, the root of all evil is 'mathematical existence'. Mathematical existence merely requires compliance with the definition, so if we write down a definition of a number type (e.g. if we define a natural number) then by mathematical existence we can supposedly claim that ALL infinitely many of these numbers must 'exist'. It is mathematical existence that implies 'infinitely many' is a valid concept. For me, a number is just a property. In the same way that 'smooth texture' and 'rough texture' can be properties of objects, a natural number is just a property of a group of objects. In other words, a natural number is a property of a real/imagined group of specific/generic objects, and as such has no inherent existence of its own. And since infinitely many groups of objects cannot occur, not even imagined ones, then we cannot claim that infinitely many numbers must exist. I go on to consider signed numbers, different uses for fractions, the divide-by-zero problem and zero's multiplicative inverse problem, a smallest indivisible part approach to geometry, with reference to pi and the square root of two, and so on (I'm a long way off finishing my video just yet).

@ReifAndreas

4 жыл бұрын

Good post. But ... "In other words, a natural number is a property of a real/imagined group of specific/generic objects, and as such has no inherent existence of its own. And since infinitely many groups of objects cannot occur, not even imagined ones, then we cannot claim that infinitely many numbers must exist." Well, but then ... who are we to say what cannot occur? That the earth cannot go around the sun as people thought earlier? "not even imagined ones" This is not an argument at all. B/c No one can really imagine any property in its full natural appearance. Examples: The distance to the sun. A woman's mind. Why my PC's operating system behaves this exactly way. (Although an artificial product.)

What difficulties arise doing arithmetic with large numbers? The only thing I can think of is performing arithmetic takes more time as the numbers grow but it's still pretty simple to perform computationally/algorithmically (the steps to perform are pretty trivial), e.g. addition grows logarithmic-ally (linear w.r.t the number of digits)

@njwildberger

5 жыл бұрын

Please have a look at the MathFoundations series on Big Numbers, where I go into detail about extremely large numbers and the increasingly impossible arithmetic with them. For example, around this video: kzread.info/dash/bejne/jGx9mJp-Z9zAlbw.html

Nice explanation. Thanks

Sir, I am happy to see you back after long . Wish you good health and long life. You are the best teacher I have ever come across.

@njwildberger

6 жыл бұрын

Thanks! Perhaps you don't know about my Wild Egg Maths Courses (KZread) channel? I have been posting lots of videos there on the Algebraic Calculus. So I haven't really been gone at all. Please check it out!

Great to see you back in action, looking forward to the next video.

Good to see some sense being talked :-)

This was an enlightening video. Amazingly presented

IBM was under litigation at the time, antitrust law was enforced. They had to buy the OS from someone else. When free OS's became available, IBM immediately switched.

@njwildberger

2 жыл бұрын

@Anna Clara Fenyo Thanks for that clarification !

Just some thoughts I have. When you have to come up with nonsense like: one infinity is bigger than another; then you pretty much know youve gone astray. When you start constructing spatial dimensions out of an incompatible object, then you can come up with all sorts of crazy things to ooh and ahh about. Lines are not constructed out of points; Planes are not constructed out of lines; and Volumes are not constructed out of planes. A real number line; is a line. A line space metric is a segment, not a point. A segment is defined by the existence of 2 points. There are no smallest unit line segments, there are relationships between line segments within a line space. Assuming we can show what length is: Given 2 arbitrary line segments; A and B Case 1) if they are equal lengths; the one divides the other evenly - they are equivalent and boring and rational; they can model a discrete point space and ... yawn. Case 2) let A be smaller in length then B such that: A divides B evenly, in which case A is that arbitrarily smallest line segment; and if given enough A segments you can be equivalent to B .. the space is rational, and countable. Case 3) let A be smaller in length then B such that: A does not evenly divide B, theres a little segment of A that is left over - call it C. If C evenly divides A and B, then we have found an arbitrary smallest line segment that we can construct the elements in our system - so it reduces to Case 2. The division process terminates; the system is rational. Case 4) A and B and not evenly divide; but the division process always has a smaller segment left over. If repeated divisions of smaller and smaller Cs occur; then we can compare the segments C(0), C(1), C(2), ... , C(n), C(n+1), ....: If the ratio of C(0)/C(1) is equal in proportion to any arbitrary C(n)/(Cn+1); then we can arbitrarily standardize a "smallest" unit segment. The division process does not terminate, but it repeats a pattern. The system is rational. Case 5) A and B and yada yada ... We get a collection of Cn. If the ratio among Cn is not consistent, then there is no repeatable pattern; the system is irrational and uncountable; there is no "smallest unit segment" that can be defined which can construct all the segments within the system.

17:40 Agreed!! Thank you for the video

Ican see that you really love mathematics , thanks for this.

Nice.Regarding magnitudes problems (continuum, too big or too small) , I think it is a mater of counsness focus of the beholder. For exapmle, if you focus on small magnitude 0,000001, that is the point in continuum. It depends on what you focus. Regarding irational numbers, I think its a uncomparable things issue. Like 3 ancient Greeks unsolvable problems (trisecting the angle, doubleing the cube and squaring the circle). Comparing grandmas with frogs is irrational. Same as comparing curved and straight shapes or 2vs3 dimmensions. Those problems were present forever, even in ancient mathematics, and never sold because they can not be looked in this way.

Excellent video. I really wish more teachers taught this way.

Excellent summary. Its interesting how they detached mathematics from philosophy (calling it illegitimate to them) just long enough to starve mathematics of monotheists and breathe in misdirecting chaos into the natrative before gluing it back into philosophy. I'm working on a theory I call neural relativity, and the foundations of that I call dialectic topology. The foundation becomes the neural mesh, and differential reasoning becomes neural Interferometry. How mathematical ideas evolve is well accounted for by population interactions of neurally relative value judgements.

Excellent synopsis.

Prof, could you check for filtered msgs/replies? Some of mine getting stuck.

@njwildberger

6 жыл бұрын

@Inverse HT: Have done, I am not sure why some remarks get slotted into the possible spam folder. Hopefully they should be visible now.

15:58 You say "There's a polite agreement not to examine too carefully what this logical foundation of set theory is." I am earning a doctorate in set theory. What planet do you live on? What do you think we do? Exactly what you just said we don't. Anyone can carefully examine set theory; some in my department do both algebraic geometry and set theory or number theory and set theory. There is no more a divide between set theory and other disciplines than there is between algebra and topology. It's just another field of mathematics.

@tomasmatias4109

5 жыл бұрын

I wonder if this is a good place to ask. The set that consists of the largest integer how many elements belong to it? By definition it should be one, as there can only be 1 largest anything right? But the integers do not have a largest element, so no elements belong to it right? So the set that I defined is the empty set right? I feel I am close to the contradiction that makes me unable to understand what the phrase "let a be any number belonging to the reals" (sorry I don't know how to write it in mathematical notation in this comment ) means.

@FractalMannequin

5 жыл бұрын

I'm starting to think this is getting a little too awkward, if not paradoxical. Norman is a professor from UNSW and he should have at a least a vague idea of what's going on in other branches of mathematics, and if he doesn't, then he should be selfconscious of his own ignorance enough not to sell any false claims to people outside the subject...Yet he does.

@MRBiel201

5 жыл бұрын

@@tomasmatias4109 What you're doing here is called naive set theory and, as you pointed out, it leads to contradictions. It essentially abandoned after Russel's paradox for the ZFC axioms. If you want to find a problem with actual set theory, you need to use the ZFC to construct the set that you have described and, if you cannot, it probably is simply not a set.

@senselessnothing

4 жыл бұрын

Unless one gets into the philosophy of language and the nature of man, it's rather impossible to figure out what's going on and whether you can even define primitive constructs such as axioms to derive the rest of mathematics out of without circularity. To me it seems like set theorists and logicians never got that memo.

@mzg147

4 жыл бұрын

@@tomasmatias4109 "By definitiom it should be one as there only can be 1 largest element" *if it exists*. **If** a largest element exists, then there is only one. Generally, you can say there is at most 1 largest element, 0 or 1.

What role might finite fields play in considering the foundations of mathematics... I think once you give a video giving your stance on finite fields and how they may help us organize number theoretic meaning then I with you will find that many people will not only understand this... but want to employ, quite practically, your stances on the foundations of mathematics... quite quickly and quite precisely... I think you have our attention now what people really want is a discover-able workflow that they can pursue without waiting on a lecturer... but is interested in proving and participating in activities that are engaging and interesting to someone ready for it... Wouldn't it be nice if philosophical quibbling could be turned in to focused attention on guiding those looking for answers, interested in learning a sound and thoughtful mathematics (This is not saying that you are not doing that... clearly, you are) just hoping out-loud for others to realize you are not "arguing against people's livelyhood" but trying to open our eyes that scientific thinking is not only a healthy en-devour for the young but for all...

Having read a little of yon philosophers, I shunned their silliness. No wonder that they were hidden behind a veil! || noticed that Pascal's Triangle can be regarded as eleven to the power of n = 0, 1, 2, 3... as in 1, 11, 121, 1331... very handy for working with powers and roots in general.

I am writing this comment before watching the video but I have seen quite a number of your previous videos. I have just retired from a career as a computer software engineer. Back in the dark ages I studied Maths - I think at the time (1966) there was only one computer course in England at Manchester. But in 1966 I did not really know what a computer was. I can honestly say I did not understand a single word (or theorem) of the set and number theory part of my analysis course. I kept the book we used and have recently come back to it. Again I found it heavy going but with a lot of material on youtube, especially your videos, I can make more sense of the subject. Or to be more accurate I am starting to develop the conclusion that it does not make sense. My perception (of this part of analysis) has moved from something like the Cheshire cat in Alice in Wonderland - a diffuse body with no discernible purpose in life - to one of an Emmentaler cheese - the body can be consumed but it is full of holes. However, both of the views (the "standard" and a finite view) on numbers and the continuum seem to have flaws. I understand your view to be that infinity does not exist. So the natural numbers must have a maximum, let us call this M. So we can ennumerate from 1 to M. However, according to Cantor's set theory, the number of combinations up to M are the power set 2powerM. But any number greater than M is non ennumerable. So if you assume M you are stuck with having to some how deal with a large finite unnennumerable collection of numbers. Cantors has one well known and much discused flaw (as applied to numbers) - the size of 2power(aleph0) is not known (the continuum problem). But there appears to me one other weakness which I have never seen mentioned. In his famous diagonal proof Cantor assumes that the unique (number) symbols he creates in the diagonal represent a new unigue number. I come from a computer background and am very familiar with the dyadic number system used in computer architecture. In particular the number system is modular with modus being 2powerN, where in a modern PC N is typically 64. So why not just extend this to the natural numbers and claim that in the limit the resulting ("infinite") size is modular, i.e. aleph0 is modular? So any number "greater" than alepho is actually just a repeat of the already established enumerable collection of numbers. So my proposal: - The natural numbers are modular So there are no non-ennumerable numbers. - Irrational numbers can exist but they are ennumerable. This implies all numbers have to be constructed. - The idea of a number as "data" is replaced with a formal description of its construction. - An irrational number has to be "presented" but does not necessarily have to be written down. For example sqr. 2, e, pie etc. imply formulars for working out irrational number. They have many writtten forms (as rational numbers) depending on the acurracy required. - There are no random irrational numbers. A true random "process" can only produce a rational number. Of course, since I have never seen anything similar proposed by anyone, it is quite likely there is something I have overlooked. Or it is thought irrelevant. But I do believe there is a potential flaw in Cantor's theroem which seems to have been overlooked.

@njwildberger

6 жыл бұрын

Hi Kevin, Well I do hope you have watched the video. As for your comments, I have of course sympathy with your skepticism towards modern analysis: this should be much wider spread. But I am not sure we can confidently talk about a largest number M. Even if there was such a thing, there is no way we could enumerate from 1 to M. It is more likely that the process of enumeration of large numbers just gets slower and more expensive as we keep going and eventually we do not have the resources to continue. So it is safer if we restrict our claims to the regions in which we can make observations (physics) and the domains where we can make computations (mathematics). Let the philosophers take care of the rest.

@KarmaPeny

6 жыл бұрын

Hi Kevin, I too am an old-ish computer software engineer & I too have struggled long and hard with maths concepts that claim to handle the infinite (I even studied at UMIST in Manchester many years ago). You said "Irrational numbers can exist but they are enumerable. This implies all numbers have to be constructed." I can appreciate where you are coming from because it might seem extreme to discard concepts like pi and the square root of 2, but for me the question is what does it mean when we say these things 'exist'. Maths began as a way of abstractly modelling physical objects and events, but at some point it was decided that maths was detached from reality and thus cannot be explained by language that describes the natural world. By definition, it became a supernatural belief. This is where I believe it went wrong. This dates back to the Ancient Greeks. Euclid decided it was OK to base geometry on infinitely small points and Plato insisted that existence in a 'third realm' detached from all reality should be the basis of mathematical logic. The mainstream have adhered to these beliefs ever since. To see where they first went wrong, imagine two apples where one has a rough texture & the other has a smooth texture. The definitions of 'rough texture' and 'smooth texture' can exist on their own (e.g. as descriptive pieces of text). But a texture can only occur if an object exists with that characteristic. That is to say, a rough or smooth texture is a property of a real/imagined object, and so has no inherent existence of its own. Now imagine the decimal symbol for two. The symbol represents two of anything (such as apples, or in the simplest case, tally marks). The definition of a natural number can exist on its own (e.g. as a descriptive piece of text). So if it is a property, a natural number can only occur if a real/imagined group of specific/generic objects occurs. In other words, a natural number is a property of a real/imagined group of specific/generic objects, and as such has no inherent existence of its own. My point is that just like a texture cannot exist without an object, a natural number cannot exist without a group of objects. Now I'll make the same point but with reference to computing. In a computer system the following objects might exist: a person, a table and several apples. There might be relationships between these objects, so an apple might be owned by a person, and/or it might be situated on a table. The numeric values corresponding to 'number of apples owned by a particular person' or 'number of apples on a particular table' can be derived from the model. These concepts are properties of the model that cannot exist if the related objects don't exist. So if 'table A' does not exist, then 'number of apples on table A' cannot exist. My point is that a number can only exist if an object exists (like a person/table) that can be attributed a group (of other objects like apples). Specifically, the number relates to the defined group. I argue that numbers have no inherent existence of their own, just like a texture cannot exist on its own, it is just a property of something else that exists. So I think the core problem is the strange idea that numbers have their own existence and that it follows all numbers must somehow exist. We have become so accustomed to thinking this way that it is second nature. That's why it is so difficult to realise that this is the problem. This notion of Platonic existence immediately implies infinity is a valid concept and it underpins all mathematical logic, the concept of the continuum, infinite sets and so on.

@ksmyth999

6 жыл бұрын

Thanks for replying to my comment. I have now watched the video and look forward to seeing the next videos in the series. When I decided to review part of what I had studied 50 years ago I was hoping I would understand it a bit better. I was surprised when I started to have doubts about the subject matter. Finding your excellent videos has helped me to put this in perspective. It is rare (for me) to find a non trivial maths book or video which I have understood on the first pass. You seem to have a natural gift for explaining complex issues in a simple and straight forward way. It occurs to me that maybe current maths foundations are still assuming a classical deterministic era. Whereas the "real" world has moved on and appears even stranger than first thought. Since current mathematical tools seem capable of dealing with, for example, quantum mechanics there may be little incentive to change (the foundations). So, unfortunately, I think you may have an uphill battle, although surely some of the obvious paradoxes we are being faced with should not be allowed to stand.

@ksmyth999

6 жыл бұрын

Maybe the difficulty is more fundemental. How solid are our definitions? For example if you examine Cantor's two well known theroems. The first one is about sets - deriving the power set (2powerN where N is the number of elements). The second diagonal argument is supposed to derive an unenumerable "set" of numbers. But is the assumption valid that the, admittedly unique, string of characters that he shows are not in the enumerable set actually represents a number that is not in the list? For example, assuming aleph0 is a modulus would imply all the unenumerable strings are just repeats of the numbers in the list.

Question: If an alternative foundational approach, which you talk about, is adopted, will it change anything regarding how higher mathematics is done?

Dr. Wildberger, what would you say is the most encompassing area of mathematics that includes virtually all other objects of mathematics as "special cases"? What is the current highest floor/perspective from which all other objects are encompassed? I'm looking for that highest perspective on all other areas. Is it category theory, such that category theory is the widest Venn diagram that overlooks all other objects and relations? Or is it set theory or something else? Thank you.

@njwildberger

2 жыл бұрын

@swavekbu I would not claim to say there is such an area exactly. Mathematics is a multi-faceted subject, but I think it safe to say that geometry and algebra are somehow the basis of a lot of the subject.

@swavekbu4959

2 жыл бұрын

@@njwildberger Thank you. I appreciate very much your videos on history as I share your interests in the foundations rather than forever expanding the subject. The acceptance of yesterday's notions (e.g., imaginary numbers, or even negative numbers) forms the basis on which future discovery is made, so I think it's essential to critically reexamine foundations, even down to the nature of mathematical logic. Please keep posting great videos!

@pmcate2

7 ай бұрын

Set theory lol. Or category theory as you mentioned

I find this whole argument on Set Theory being ill constructed and bad rather dumb honestly. It’s true that exploring alternative definitions has its values and can provide new intuitions. However, rejecting set theory as a powerful foundational idea of mathematics is nonsense to me. They represent an intuition that was able to model so many ideas on a way broader scope than arithmetics. And that’s the thing, for the sake of being able to model behaviors in the world that are extremely complex, you need to make approximations and to have access to those various approximating models, you cannot restrict yourself to arithmetics on natural numbers. And that’s the power of Set theory, it allows us to create sound models that have a huge explanatory power on the real world. Maybe you could use only arithmetics to describe every behavior of the world with enough computational power (even though it’s not even clear if the world is discrete, for all we know it could as well be continuous), but the truth is we don’t have that power and we still need to extract meaning from the world and continuous analysis has proven to be incredible for that. It doesn’t really matter if it exists in truth, it is sufficiently precise and close to the truth to extract meaning and as such is valuable.

Could you do more debates please? It's easier, for me at any rate, to follow the arguments when I'm hearing both sides. I saw the one you did, which left me hungering for more.

Thank you professor Wildberger for sharing your efforts on providing a different and hopefully more solid approach to mathematics, I think it's a very valuable contribution. I'd like to know, what are your views on Category theory? I find it is often a very useful, natural, and even necessary approach to certain problems. Thank you!

@MathProofsable

4 жыл бұрын

The category theorist's response to ZFC is a well-pointed topos with a natural numbers object along with the axiom of choice. What's nice with this definition is that it is pretty easy to motivate and remember the axioms. It is also responding to the necessary structure alone. My guess is that Prof. Wildberger would have problems with this structuralist approach because he still seems preoccupied with the substance of mathematics rather than just the structure.

So beautifully well explained even to a non maths person.

I'm still learning mathematics at a formal level, but I always thought if it was possible to derive all of mathematics from the concept of Turing machines, defining all objects as states of a Turing machine and operations as algorithms that are performed by Turing machines, I don't know if it is possible to do it, but, maybe, guess that's at least compatible with the Church-Turing Thesis

@pmcate2

7 ай бұрын

No, there are only countably many turing machines, meanwhile standard math has uncountable sets. You're free to deny those objects exist, but at that point you're not doing standard math.

Let me know if I'm understanding this correctly, and this is more or less pertaining to your analogy with software being more important than hardware, by getting the foundations correct (if that is even possible) we would be able to unlock ever more riches in the form of theorems unknown today because of our current axiomatic systems. It is like bad software not letting one access different applications on your computer, whereas a cleaner and well written software (operating system) will not have those same bugs and thus the computer will run fluidly. I do have some questions however, such as 1. How do we know that there must exist a correct foundation for mathematics? 2. Is it possible that a foundation of mathematics can involve not just logical concepts that we as humans like to entertain as certainties, but other types of concepts stemming from our minds? I think further developments in neuroscience alongside with artificial intelligence may come to influence our perception of mathematics in the future, any thoughts on this?

@WildEggmathematicscourses

6 жыл бұрын

I heartily agree that we can expect to unlock new riches with a more solid foundation. This happens already with Rational Trigonometry, which was responsible for the discovery of Chromogeometry, the most exciting new direction in modern planar geometry, and also for a vastly improved theory of Universal Hyperbolic Geometry. But in my Algebraic Calculus One course I hope to show that even ordinary Calculus can benefit hugely. However I do not think our minds and their neuroscience has much to do with mathematics at all! In the future, what will be increasingly important is not how our minds work, but what our machines are capable of. We only have a few years left of superiority, so enjoy it while it lasts!

Why in heaven's name would he want to be in sync with the computers, when he could be ahead?

@Dystisis

3 жыл бұрын

It's not a question of synchronicity, but of what mathematics was always about, namely feasibility in principle.

I don't get it. Is it that starting with Axioms is the problem? Then why so? What are definitions if not axioms with different names?

Newton tried to formalise calculus, but failed. Dedekinds pushed the problems from numbers to sets, and zfc puts the problems of sets into metaphysics.

Professor Wildberger how would you differentiate set theory from logic. If set theory is untenable as you claim, is there any way to still keep using predicate logic divorced from set theory and so we can still talk about proof sequences with logical operators and definitions of the union, intersection, and complements etc. of sets or do we run into trouble? (i would obviously think you have no problem with propositional logic since that is very simple and what logic gates and modern computers are based off now) I'm just saying this given I know you object to ZFC and ZFC uses predicate logic in its language, like for example the Axiom of Extensionality or any others. Where does one draw the line if there is one? I only have two undergrad semesters worth of studying logic. One in the philosophy department where I learned informal, Aristotelian/syllogistic, and Boolean propositional logic and one in the math department for a discrete math course where I reviewed propositional logic and learned predicate logic for the first time. Other than that my knowledge of logic comes from reading one half of Hofstader's GEB and knowledge that ZFC was introduced to repair Russell's paradox in naive set theory, which would be more layman sources of understanding logic. A response would be appreciated whenever and if you have the time. Thank you.

But isn't it true that any axiomatic system will have inherent limitations? (I.e. godel's incompleteness theorems)

@melvinpjotr9883

3 жыл бұрын

Goedel's theorems require a certain minimum amount of complexity (lets call it the "Goedel threshold") of the axiomatic system. Below the Goedel threshold you can have theories, that are complete, (provably!!) consistent and decidable. So mathematics is entirely foolproof and non-ambiguous, as long as your axiomatic system remains below the Goedel threshold. On the other hand, any theory above the Goedel threshold is either incomplete (assuming it is consistent) or it is outright inconsistent (Note: an inconsistent theory is "trivially complete" in the sense that you can "prove" every statement, including its negation). Unfortunately the Goedel threshold appears to be rather low. For instance, Peano arithmetics lies above the Goedel threshold. I suspect that "Standard Euclidean Geometry" (including the parallel postulate) is incomplete as well (not 100% sure, there seem to be conflicting accounts in the literature that might have to do with the fact that sometimes the "term" Euclidean geometry is assumed to include the parallel postulate and sometimes not). Historically Peano arithmetics was regarded as "obviously true". Understandably mathematicians were reluctant to give up Peano arithmetics, just because of Goedel's findings. The interesting question that haunts me, however, is: Do we actually require Peano arithmetics (or Standard Euclidean Geometry that requires a genuinely infinite space) to understand the processes in the real world? Or asked differently, are there theories below the Goedel threshold, that are sufficiently complex to describe the real world, but that are at the same time not too complex to run into Goedel incompleteness? Curiously, the theory of real closed fields lies below the Goedel threshold. Real closed fields include the 1,2 4, and 8 dimensional division algebras of the (strictly positive) reals, the complex numbers (excluding zero), the quaternions (excluding zero) and the octonions (excluding zero). Note that we can divide and multiply without constraint in these algebras, but addition and subtraction are non-global operations that must be limited to sufficiently small local regions. In any case, these division algebras can be regarded as numerical, but also as well-defined geometric and logical structures, so that they seem to be principally capable to describe the most fundamental aspects of the real world (logic, arithmetic, geometry). There are only four division algebras (over the reals) and they all are - in a well-defined sense - inherently finite, meaning that every (necessarily divisible) number within them is finite (although the number can become arbitrary large or arbitrary small). We can also - via stereographic projection - map the entire geometry to a finite sized p-dimensional sphere (p=1, 2, 4, 8), from which two antipodal points (corresponding to zero and infinity, which are not divisible numbers) are removed. The octonion algebra and its associated finite 8-dimensional (curved) geometry has an amazingly rich structure. I would not be surprised (in fact I strongly suspect) that the octonions are sufficient to describe every aspect of the the real world. I could say more, but that would take roughly 1000 pages ...

thank you sir .

Religious, maybe, but then so are the natural numbers viewed from a certain perspective. All depends on where you want to draw the line doesn't it.

@Dystisis

4 жыл бұрын

Way to miss the point by about 30,000 nautical miles.

@ReifAndreas

4 жыл бұрын

Exactly. What Math and Religion has in common that there are axioms. (And in both fields the vast majority is thinking that way.) Rebecca Goldstein gives a nice intro here: kzread.info/dash/bejne/iXuZm9ScdMmnpJM.html Of course you can do both without them. I think with less success. And, if you ask some of the true believers of a religion, they would tell you that it isn't a dogma, they have the experience that god did this and that. (And I wouldn't doubt that in general, just be careful about a singla person's religious intuition.) Not only for the sake of provoking I'd say: Norbert does exactly that. He's so deep in the construction - believe that he thinks he's experiencing that it works while ignoring the opposed arguments. For my part I think god and math is bigger than we think ;-) Neither this or that approach gives the full picture. While the axioms work by some magnitudes better than Brouwer.

I've never accepted a lot of Cantor's work, and I don't believe Poincare did either. Nice to see it questioned again. A big question for me is whether or not the set of whole numbers includes numbers that are beginningless. I mean, we have numbers with no end digit in the rationals, so why not numbers with no beginning digit in the wholes? If whole numbers include beginningless numbers I believe it can be shown that they have the same cardinality as the reals - at least the reals between zero and one, which SUPPOSEDLY have the same cardinality as the entire real line.

@imaginary8168

3 жыл бұрын

Same cardinality by definition means that there exists a bijection from one set to the other. We have 2 sets: (0,1) and R A simple bijection would be cot(pi*x) (we pay attention only to the branch from 0 to 1). If you look at the graph of this bijective function it "transforms" (0,1) into R, so they do have the same cardinality.

I would have to argue differently. I would suggest that algebra should be the foundation of mathematics alongside natural numbers. They should have a duel importance. Why I would suggest algebra as a base is that it is a subject area that encapsulates the basic fundamental operations of mathematics such as equations.

@EM-qr4kz

2 жыл бұрын

What about Geometry then?

@bighands69

2 жыл бұрын

@@EM-qr4kz I am not against geometry being taught but it is not a good starting place for many students. Algebra is in everything from calculus to geometry and what beats most students is the fact they do not have a good algebra foundation. I would be happy if kids just left school with good algebra skills and nothing else. Most cannot even do the most basic algebra. I also think this whole applied mathematics approach at schools is a bad idea as well and will end up failing.

what about category theory?

@MathProofsable

4 жыл бұрын

The category theorist's response to ZFC is a well-pointed topos with a natural numbers object along with the axiom of choice. What's nice with this definition is that it is pretty easy to motivate and remember the axioms. It is also responding to the necessary structure alone. My guess is that Prof. Wildberger would have problems with this structuralist approach because he still seems preoccupied with the substance of mathematics rather than just the structure.

@senselessnothing

4 жыл бұрын

@@MathProofsable It would kind of require him to be a formalist to care about the structure.

The idea of setting the foundations of a theory is to create a logically consistent framework in which you can state the techniques and results you already have from the beginning. But it appears that in your project for a new foundations of mathematics you cannot state almost any result of modern mathematics. Just so you know, trigonometry is no longer exciting since the 11th century more and less.

Hey may I pose a question: let’s say we have an equivalence relation aRb. Why can’t I represent this within set theory as set T comprising subset of Cartesian product of a and b, mapped to a set U which contains true or false? Thanks so much!!

@WildEggmathematicscourses

6 ай бұрын

Yes you can do this. Even easier perhaps is just to specify the subset of the Cartesian corresponding to the True values

@MathCuriousity

6 ай бұрын

@@WildEggmathematicscourses Hey kind God, would you please help me unpack how I would do this? (My motivation for this was because some guy on Reddit said it’s illegal in set theory to try to use some “meta” relation like equivalence relation or even reflexive relation as going from one set to a set consisting of the elements true or false.

@MathCuriousity

6 ай бұрын

@@WildEggmathematicscourses how would I do this kind soul? Would you give me an example? I am a nubile.

@MathCuriousity

6 ай бұрын

@@WildEggmathematicscourses I don’t understand - the whole confusion I have is - if I have a reflexive relation for instance - it seems the ordered pair is of the elements a and b the relation acts on - but where is the “truth” stored ? My whole idea is to represent the truth part. I feel like I’m set theory isn’t the truth part just represented in our heads since it’s not being actually mapped to and the only thing being represented are the ordered pairs that represent “truth”?

@billh17

4 ай бұрын

@MathCuriousity said "... mapped to a set U with contains true or false" "true" and "false" are not sets in standard set theory. Any set U can contain only sets (if it contains any thing). That is, the statement "true in U" is meaningless for any set U. The predicate "in" requires that the statement "x in y" has x and y as sets. But, "true" is not a set.

Actually in beginning my study of mathematics I've been on set theory and it all seems like we needed cows so we taped a bunch of cats together

Mr. Wildberger, you should introduce a QUANTUM HYPOTESIS in mathematics.... ::)) in order to make it contradiction-free. Kurt Goedel should have known this easy solution in order to resolve his "paradox"

Could not agree with you more that IBM made a mistake by outsourcing the PC OS. They were ***IBM***; they could have done the job no problem. It was exactly as you described - a grotesque strategic error.

Foundation and framework are different. Foundation goes underneath. Framework is around the sides.

What should I be studying to understand the real foundation of math? I have a book on discrete math is that what I should study?

@methandtopology

8 ай бұрын

Maybe you should study some modern maths first

It seems to me that an honest treatment of the so-called real umbers and analysis should involve the interval arithmetic introduced by Ramon Moore. What do you think, Dr. Wildberger?

Wait, they outsourced it to Philosophers who were still wrestling with the "Chicken and Egg" problem. What could go wrong!

If we define real numbers as points on the continuum, then irrational numbers must exist. Here is a simple proof that √2 exists. We expect the function f(x) = x^2 - 2 to intersect the real axis (i mean, the curve doesn't suddenly jump over the x-axis and miss it, that would be strange).To make this formal, we could even use the intermediate value theorem. There are two x intercepts, the positive one is what is defined as √2. It can be shown that it is not rational using an argument involving parity (evenness). Thus √2 is a real number because it does correspond to a point on the continuum.

13:52 - When were infinite sets found to be logically questionable? From what I gather, ZFC was meant to avoid contradictions surrounding sets with infinite comprehensibility, but not infinite sets themselves. 15:28 - I don't think that axiomatics is new at all. The same criticism could be made of Euclid, as well as you or any other mathematician. They could just codify the unproven assumptions you're making into axioms and then critique your use of axioms as "dressing up". I don't think this is a consistent criticism of axiomatics, which I think is unavoidable in the long run.

@WildEggmathematicscourses

6 жыл бұрын

"Infinite sets" (it is better to use quotations so one is not implying they actually are meaningful) were found to be logically questionable almost as soon as they were introduced by Cantor. In further videos I will be looking at a lot of negative comments on this topic from lots of illustrious mathematicians in the latter part of the 19th century and the early 20th century. Of course, these days, a sullen orthodoxy surrounds the subject, and almost no-one desists.

@tylerantony7399

6 жыл бұрын

Okay, any arguments or sources to arguments about how infinite sets are logically questionable?

@kasperpeulen8609

6 жыл бұрын

We can not prove that a infinite set exists, as it impossible to logically construct it. The solution that is chosen for this problem is to just assert that it exists, by an axiom. The axiom of infinity. This axioms defines a set, with at least one element and further described as whenever an element is in te set, there is another element, a bigger element, that is also in the set. By this definition, the set can not be finite. However, it is impossible to locally construct this set to show that the set exists by construction, so it must be taken as an axiom. I think it is clear what is logically problematic with this. By introducing this axiom, you introduce objects that can be proven to "exist", but also that can be proven to be impossible to construct. See this stackexchange question for examples: math.stackexchange.com/questions/2808804/does-the-existence-of-a-mathematical-object-imply-that-it-is-possible-to-constru What does existence of a mathematical object really mean in such a mathematical universe?

@tylerantony7399

6 жыл бұрын

I don't agree that impossibility of construction for things whose existence is provable is logically problematic.

@pieinth3sky

6 жыл бұрын

You probably can't tell the difference between validity and soundness. Proving arbitrary theorems by arbitrary rules of inference from arbitrary axioms doesn't make mathematics.

You have read my mind. I alwsys thought that defining natural numbers in terms of sets is just wrong.

I whole heartedly disagree with your position. I also deeply appreciate your videos. I learn so much. Your videos are liked and subscribed. I wish I had more time to keep up with your channel. I look forward to hearing your case. I hope you don’t mind if I cross examine in the comments if I get a chance. Cheers from the US!

@njwildberger

6 жыл бұрын

Hi Mark: I welcome robust discussion! That is exactly one of the things that is so sadly missing in modern pure mathematics.

Modern German axiomatics (e.g. ZFC Axioms, Hilbert Axioms and Peano Axioms) - pioneered by Moritz Pasch, David Hilbert, Giuseppe Peano and Ernst Zermelo - are a departure from the Ancient Greek tradition of definitions and axioms (i.e. self-evident or obvious truths) - as exemplified by Euclid's Elements. In German axiomatics, objects like set, point, line, plane and number are undefined; and some of its axioms are unjustified beliefs (e.g. Axiom of Foundation, Axiom of Infinity, Axiom of Replacement, Axiom of Choice, Axiom of Line Completeness). Indeed, there is a philosophical and mathematical schism between Ancient Greek mathematics (i.e. Plato and Euclid) and Modern German mathematics (i.e. Hilbert and Cantor). That is, Platonism vs Formalism; and Euclidean Geometry vs Cantorian Set theory. Is there a way of bridging the gap between the two mathematical cultures?

Ok then. I challenge you to present a non-axiomatic foundation for mathematics.

@senselessnothing

4 жыл бұрын

You wouldn't call it a foundation unless it's axiomatic though.

@erikawimmer7908

3 жыл бұрын

@@senselessnothing at least today

Wildberger's mathematics is coherent. It just does not allow the richness of modern mathematics to be expressed. His mathematical space is a poor one, deprived of the delights of the real numbers, topology and even the square root of 2. Rather than attacking set theory, he should stick to investigating what structures he can find in his limited mathematical space. It is an interesting exercise, but not an alternative to modern mathematics.

@douggwyn9656

6 жыл бұрын

More importantly, see how much of mathematics can be done without involving what are known to be troublesome issues. For example, quite often when there is already some theorem whose proof uses the Axiom of Choice, mathematicians are happy when somebody proves the theorem without invoking AoC. There are indications that mainstream mathematics includes weirdness (such as Banach-Tarski) that would not arise through a viable alternative approach. Not too long ago I sat in on a bunch of theoreticians who were bemoaning that despite lots of invention they still don't have satisfactory bases for theories of the infinite. (I think ultrafilters were part of the discussion.) It is worth considering metric geometry based on squared distance, which is how physicists usually do it anyway, rather than on normed distance. And so on, ...

@AdamMcLean

6 жыл бұрын

"when there is already some theorem whose proof uses the Axiom of Choice, mathematicians are happy when somebody proves the theorem without invoking AoC". Perhaps such proofs would still have to use the concept of infinity.

@Rafael-rn6hn

6 жыл бұрын

I'm still hung over the argument that real number arithmetic is not precisely defined (or isn't defined at all). If that is the case, how can we feel confident modern mathematics is really sound? (Not from applied results, but first principles.)

@AdamMcLean

6 жыл бұрын

Have a look at non-standard analysis, which provides a rigorous foundation.

@twwc960

6 жыл бұрын

Non-standard analysis still uses the traditional real number system. It differs from conventional approaches to analysis by replacing the epsilon-delta limit definitions with a theory of infinitessimals. But since the theory still requires real numbers, it would probably not be considered logically sound according to Dr. Wildberger.

Mathfoundaions250, although useful, does not delve into how the belief in real numbers or ZFC might be considered a religion. He does point things way toward a future foundation for math that might be a a “living system”. This also sounds like a religion. Do we have a choice between religions expressed here?

You keep publishing the same video with slightly different spin. I keep wondering if there's a next video.

@ReifAndreas

6 жыл бұрын

Sure it will. He's hooked as he knows there's something about it which makes it great.

@modolief

6 жыл бұрын

Dole Pole 😀 ... thanks, made me laugh

@christopherellis2663

5 жыл бұрын

It takes a while to get ones teeth into it, sort of like a prelude to the main course

Fascinating! Someone exists who still thinks there is somehow "content" in math, and that there is something more than axioms underlying the core of math! I used to think that way too, but I no longer do. Also, logicians in the early 20th century were lifted away from philosophy towards mathematics and computer science: to that end, modern logicians are very truly mathematicians and programmers.

@njwildberger

4 жыл бұрын

Sorry, but the status of modern logic is very questionable. Is it part of philosophy, or computer science, or maybe linguistics? Yes there is an overlap also with mathematics, but it would be a stretch to say that logicians are very truly mathematicians.

@antoniolewis1016

4 жыл бұрын

I'm glad you replied, but for your own health please please please don't focus too much on the comments. Here on KZread, most comments are bad for your health. With that in mind, I am very thankful for your videos! Your debate on infinity shed quite a bit of light for me on your position! I have been thinking and mulling about the things you post here and on other videos, and I guess I have a question for you. We invented fractions in order to answer questions of division where whole numbers can't be solutions. We invented negative numbers in order to answer questions of subtraction where positive numbers can't be solutions. We invented the imaginary number i in order to answer the question of square roots of negative numbers, where neither positive numbers nor negative numbers can be solutions. In each of these cases, it is impossible to represent the invented number in terms of the old numbers, and instead we represent the new number by writing down the question it answers. The sign for a negative number is the same as the one for subtraction (the dash -), and the operation for division is the same symbol for a fraction (the slash /). My question is this: if we accepted these inventions, why can't we accept the invention of the irrational or transcendental real numbers? And if we refuse to accept irrational numbers, how do we describe the ratio of the circumference of a circle to its diameter? Do we choose to not answer that question, or do we insist that circles don't have diameters? Once again thank you very much for your videos. Even if I disagree sharply on infinity, I find them helpful.

@njwildberger

4 жыл бұрын

@@antoniolewis1016 Thanks for the nice reply. You are quite right in your assessment of how we came up with all these new numbers to make up for (perceived) deficiencies of our arithmetic. But, in the cases of fractions, negative numbers and imaginary numbers we can create logically coherent extension systems in which these numbers operate. Sadly this is not the case with "real numbers". We do not have a viable "arithmetic with real numbers". What we only have is a "talking about arithmetic with real numbers". To support this talking we have a plethora of different symbols to represent outcomes of various imagined infinite arithmetical processes, such as : "log 3", "sin 5.4", "arctan 12", "gamma(13.7)", "zeta (53)" etc. What we don't have is an actual arithmetic that incorporates all of these one off symbols consistently and coherently. For example, suppose I asked a pure mathematician to sum up all the quantities I just wrote down. Do you think I would get a coherent answer? Of course not. There would just be some mumbling at "infinite processes" and then the fall back on what the calculator spits out as an approximate to the "real value". As for circle circumferences: it is not at all obvious that the "circumference of a circle of radius 1" is a well defined mathematical number. In fact it is not! Some things in mathematics are exact, others are only approximate. That is the way it is. No need to invoke religion to get everything "exact" for our happy mind frame.

@antoniolewis1016

4 жыл бұрын

I would love to know what precise problems you have found with arithmetic on real numbers. I agree that mumbling at infinite processes does not constitute a proof, and I am disheartened that some mathematicians use such language. However, there are ways to formalize this. My favorite way to deal with real numbers rigorously is to think of them as infinite sequences of rational numbers. So for me, the "real number e" is represented by the infinite sequence: 1, 1+1=2, 1+1+1/2=2.5, ... The sequence is bounded from above and each term is greater than the last. With this, each term in the infinite sequence is a rational number that can be manipulated by standard tools of rational numbers. Sequences A and B can be summed by summing the nth term in A to the nth term in B and seeing the resulting sequence. A similar process can be done for subtraction, multiplication and division. We can thus do arithmetic on these sequences just fine. We can choose to call them numbers or not call them numbers, but that's just labels. We can embed the rational numbers in this just fine: the rational number 1/3 can be represented by the infinite sequence: 1/3,1/3,1/3,... In another video you criticize the idea of an infinite sequence, so I don't think you'll like this. But what about the infinite sequence is incoherent? I'm not forcing you to construct an infinite sequence, as I know you reject the axiom of infinity: I'm only asking what makes the idea an infinite sequence, as you say it, incoherent. I want to know how the axiom of infinity is incoherent, and I want to know exactly what the contradiction that comes out of it is. One more thing: please don't use the term "religious belief" when discussing these matters. It conveys an air of saying that religious belief in general is unjustified, dogmatic and irrational.

@njwildberger

4 жыл бұрын

@@antoniolewis1016 Incoherence does not manifest itself in contradiction. Rather we just get a weak, flabby kind of thinking, leading only to further confusions. Please have a look at the earlier videos in the series where I deal at length with the difficulties with "real numbers" both in terms of "infinite decimals", "Dedekind cuts" and "Cauchy sequences of rationals". But instead of repeating those arguments, let me appeal to a very good litmus test for whether or not you can do arithmetic correctly: just try some non-trivial examples. Here is an example of a trivial arithmetic with "real numbers": pi+pi=2pi. Whoopee! We are doing arithmetic with "real numbers". I hope that is not very convincing. Here is a less trivial example: calculate 1/2+1/3+1/5. I hope you can do this in your head, and get exactly 31/30. And now for the real test: calculate pi+e+sqrt(2). And that is where you find that all that mumbling about infinite this or that amounts to very little.

I am 100% agreement.

@nerdwarrior0956

6 жыл бұрын

I agree with you wildberger

immensely insightful...would have appreciated no less 30 years ago...thank you for time and effort

Oh, by the way the question about Religion refers back to Bishop Berkeley’s chajjenge to mathematicians .

For someone who doesn’t shut the hell up about classical philosophy, you forget that Sextus Empiricus pointed out that all human knowledge necessarily comes from postulates that cannot be proven from inside the system. If ZFC is a religious belief system as you say, so is literally everything else.

Your assertion faces not for future but turns back to the past of intuitionists , Kronecker or Brower etc.

Its interesting...

Im confused. What's the problem with ZFC. Is there an explicit contradiction that anyone can point to?

@aleksandarignjatovic3130

6 жыл бұрын

I would wager that the point is not in contradictions. The point is that ZFC can be used only on mathematical objects that can be defined as sets. Objects like natural numbers, ordered pairs and others are not sets. But ZFC has to define them as such because it cannot allow such important mathematical objects as numbers to remain outside of set theory. But once you start on that path you create a very artificial theory. No matter what mathematicians say these days natural numbers are NOT sets, period. The problem with the phrase a can be defined as b, is that if you take it too far anything can be defined as anything else, because it is manipulation.

@billh17

6 жыл бұрын

Aleksandar Ignjatovic said "No matter what mathematicians say these days natural numbers are NOT sets, period." What do you think is the way mathematics should handle natural numbers?

@seanhaight2551

6 жыл бұрын

Aleksandar Ignjatovic I feel like you're making a slippery slope argument. Of course if you wanted to you could define anything as anything else. I would say that using sets to define natural numbers is a perfectly reasonable thing to do. They seem to behave exactly as we want them to and as far as I know there aren't any contradictions that arise.

@aleksandarignjatovic3130

6 жыл бұрын

As I said the problems are not contradictions. Using the usual set definition of natural numbers, do you think that the following " {0,1} to the power of {0,1,2} equals {0,1,2,3,4,5,6,7}" makes any sense? Well, I do not, that is all. @ billh17 They should be equivalence classes of cardinal numbers of sets.

@billh17

6 жыл бұрын

Aleksandar Ignjatovic said that natural numbers "should be equivalence classes of cardinal numbers of sets." Can you clarify? Are you saying that the number 2 is the class of all sets z such that z has cardinality 2? For example, let % be the empty set. Are you saying that the number 2 should be the class that contains {%, {%}} and {{%}, {{%}}} and {%, {%, {%}}} and so forth? That is, 2 = class

Dr. Wildberger! what do you think about the cardinality of your ignorance?

@gerardoares8064

5 жыл бұрын

I'd say bigger than 2^(No)

Great

IBM did develop an operating system for its computer. it was called OS2 and was actually better than Windows. IBM just could not market it properly.

I don't think the software/hardware analogy was a very happy one. As a computer programmer I'd say software isn't more important than the hardware it runs on. It'd be like saying thoughts are more important than brains, cars are more important than roads or food more important than stomachs. You can't have one without the other, at least not in a practical way.

@inversehyperbolictangent3955

6 жыл бұрын

Have you ever programmed in Java? Ever heard of a virtual machine? The software is absolutely more important than the hardware. The concept of the Turing Machine itself is a prime example. Of course you need *some* hardware to run it, but the hardware can be interchangeable. Think of it this way: Would you rather exist as a brain and body with no thoughts (i.e. brain dead), or as thoughts in an artificial brain and body?

@DamianReloaded

6 жыл бұрын

That's simply false. If software was "absolutely more important than the hardware", then hardware wouldn't be necessary. AAA games wouldn't need bleeding edge video cards. No hardware, no software. No matter how important software is, it simply cannot be without the hardware. Just as thoughts cannot be without a brain. It's like saying software is more important than electricity. Bollocks.

@inversehyperbolictangent3955

6 жыл бұрын

" If software was "absolutely more important than the hardware", then hardware wouldn't be necessary." If I had meant that, then I would have said it. "X is absolutely more important than Y," is not the same as "Y is not necessary at all." "2 is absolutely greater than 1," does not mean that "1 equals 0."

@DamianReloaded

6 жыл бұрын

But saying 2 is absolutely more important than 1 is just as false as what you said earlier.

@inversehyperbolictangent3955

6 жыл бұрын

"It's like saying software is more important than electricity. Bollocks." Are you aware that the concept of the Turing Machine proves that a computer can be constructed to run the software without any electrical power supply? You could, in theory, build a Turing machine out of water jets, or even people. In fact, you can write a Turing Machine with pencil and paper. As I said, the hardware is potentially interchangeable. The software is what determines what will be computed. Again, would you rather be a brain without thoughts, or thoughts in an artificial brain?

Okay. Set theory as a foundation has problems. What do you propose as an alternative?

1. Wildberger does not give indication he is acquainted or familiar with Lambda Calculus. Otherwise he would have realized Comp Sci natural fit with mathematics. 2. Re: assuming that itcworks and dressing up as axiomatic (a) Ignores the Bourbaki project which extends the 19th Century German approach of rigour in mathematical proofs. Bourbaki was instrumental in shifting toward wanting a firm bases for all mathematics as Euclid had done for Geometry. (b) Cantor was less interested in Analysis than try resolving the concept of infinity and what meant to infinite. 3. Re: supposed outsourcing to the Logicians Wildberger ignores Frege instrumental efforts and more to the point, most mathematicans favoured Logic as the solid foundation on which to set mathematics. These errors alone demonstrate either a lack of understanding the history of Set Theory or willful ignorance at wanting to stir up unnecessary attention. Final note, in contrast, the alternative approach to an axiomatic schema is Category Theory which philosophically is rooted in a Structuralism approach to Mathematics.

@inversehyperbolictangent3955

6 жыл бұрын

This is just a summary video. Have you seen his prior videos discussing most of the topics you mentioned? As for Lambda Calculus, it doesn't solve the problems of the Reals. Comp Sci fits well with finite, computable mathematics, not so with the infinities associated with Reals.

@CalBruin

6 жыл бұрын

Inverse Hyperbolic Tangent I subscribe to Wildberger's channel so that is how I, of course, know and watched this video so soon since publication. Your ignorance is showing or you misread what I wrote. I never claimed Lambda Calculus addressed the Continuum.

@thomassynths

6 жыл бұрын

Your (1) point confuses me, as I feel Wildberger often wants to root his mathematics on computable things. In this video alone, he has CS written on his diagram (2:41). It's clear to me that he realizes the natural fit between CS and math.

@inversehyperbolictangent3955

6 жыл бұрын

"I never claimed Lambda Calculus addressed the Continuum." Then what's your point in bringing it up? You say, "Otherwise he would have realized Comp Sci natural fit with mathematics," but why do you assume that he does not realize this?

@Rafael-rn6hn

6 жыл бұрын

Lambda calculus is a model of computation equivalent to a Turing machine. It's very easy to write non-terminating programs in it. This is a logical weakness in the same way as square root of two (as a "real" number), the set of all sets that do not contain themselves, etc (it's the Curry-Howard equivalent of a logical paradox). So I don't think lambda calculus is a good foundation for mathematics in the scheme Wildberger is pursuing. *Typed* lambda calculus with total functions however may be a step in the right direction. But I honestly don't think we should focus so much on functions in order to discus mathematics. Yes, functions are important, but not necessarily the fundamental objects of discussion; not everything needs to be reduced to and encoded as a function. So when it comes down to it, the answer is building up a type theory. Which is not so far removed from what Wildberger has been doing. (Ofc, he isn't taking a traditional type theoretical approach with type inference rules, reduction rules, etc, but that's to be expected)

Wilderbergers mathematics should be immune to Godel incompleteness theorems, i.e. his math could be (in theory) sound and complete. Right? Or let me rephrase: Are any systems without infinite sets immune to Godel? I am really curious if NJW's math system could avoid Godel...for me that's the dealbreaker because his system is much more intuitive and "down-to-earth" and if it can be sound and complete I wouldn't mind its lack of expressive powers.

@WildEggmathematicscourses

6 жыл бұрын

We will be seeing that there is a legitimate question about the role of Godel's "theorems" in terms of the foundations of mathematics.

@FractalMannequin

5 жыл бұрын

@@WildEggmathematicscourses I can already see it: "Gödel's works are unlogical and therefore cannot be taken seriously. Problem solved.".

@sarahbell180

5 жыл бұрын

@@FractalMannequin I can see farther: fmnq's understanding is limited and therefore cannot be taken seriously. Problem solved.

@FractalMannequin

4 жыл бұрын

​@@sarahbell180 Lol. That's so Wild Egg.

@sarahbell180

4 жыл бұрын

@@FractalMannequin You got the wrong person. But perhaps I did too, whatever.

Anyone know any good framework for calculus without the real numbers? Something along the lines of discrete calculus but extended to also include rational numbers with a finite decimal expansion.

@inversehyperbolictangent3955

6 жыл бұрын

Wildberger is working on such a course right now, in fact. It's called Algebraic Calculus. He has a separate channel for it, and you can actually sign up to participate in it. It's still under development, though.

@keffbarn

6 жыл бұрын

Nice! I'll look into it.

@douggwyn9656

6 жыл бұрын

A lot of rational numbers don't have finite decimal expansions.

@keffbarn

6 жыл бұрын

I'm a computer scientist so I don't really deal with the infinite ones. The way i see it, those that have so called infinite expansions are really just functions that can give you arbitrary resolution of said quantity depending on how much memory you want to spend.

@inversehyperbolictangent3955

6 жыл бұрын

AFAIK, all rational numbers with infinite decimal expansions can be represented as finite repeating decimals (like 0.(3)... where ... means 'repeat the part in parentheses indefinitely'). Also, they all have completely finite rational representations, as well as finite continued fraction representations. You only need to deal with decimal expansions when, for instance, outputting the number as a decimal string, or when storing it in a fixed-length approximate representation. Is the existence of infinite n-ary expansions of rationals really a serious problem? (Sure, the denominators can get really big, or the repeating expansions as long as the denominator, but for big denominators you can either deal with BigInteger data structures, or potentially you can develop a system that handles arithmetic with continued fractions (it's been done before, not sure how practical it was or not).)

These are stable views I have to say. In my humble opinion the problem is that we tried to give life to symbols aka numbers,etc.. They are just that. Symbols.It's a language.We use them.Relationships among them are relationships among the things that these symbols represent.But if we forget that and treat symbols as entities by themselves, we have to build laws for them as nature did for the world.Therefore we have axioms. That's the key difference. We try to be God and we fail to build perfect models.The dichotomy that the professor speaks about in his videos is exactly that.We begun to see symbols as entities. That's why we separate math to concepts when in fact they are all one chapter of symbols.In the math world we can zoom forever.Therefore we, in our minds, say there are soooo many numbers in the (0,1).In reality we constantly creating as we zoom new symbols(numbers) which represent nothing because the world doesn't have so much things that need so much symbols.The true question is why we got so wrong and I think the answer lies in the psychology of the human being who is starving for control and social acceptance.. Imagine tomorrow if we changed the gravity of math to just a science type..It goes even deeper and I think I wrote enough..

To Professor Wildberger : I tried to contact you via email your published email address but apparently the message did not go through. I gave another try by using the comment below. I hope that one way or another the very important discovery will reach you. Please contact me any time you like. Sincerey, Tamas Varhegyi

so excited for the next video

If you enjoyed this you’ll also enjoy Steve Patterson and his “Patterson in pursuit” podcast. He agrees strongly that set theory as a foundation is nonsense.

Prof. Wildberger, it's kind of interesting that you are finding yourself driven to revisit the foundations of logic, as I've been driven to the same destination, but from a different origin. I'm coming at it from the perspective of the late E.T. Jaynes, author of _Probability Theory: The Logic of Science_ (2003). It's a mostly complete, but partly unfinished (due to his passing) book whose thesis is that Probability Theory is most correctly understood as an *extension* of finite propositional logic. He, too, spends a good deal of time and effort to make the case that infinite sets are troublesome and (at best) need to be handled with extreme care. As such, he develops his working version of Probability Theory not quite from Rational Numbers, but it might as well be; he comes very close to the same kinds of ideals that you've been propounding. I would highly recommend checking out his book (if you search 'jaynes probability theory', probably the first link will be a full pdf; or you could purchase a hard-copy or maybe ebook version if you prefer). My purpose is to develop an intuitive framework for people to use logic and probability directly in discussions and arguments. In other words, to bring it as close as conceptually understandable and usable to the average educated lay person as is possible. So, I'm sticking with your numerical basis of Rationals for representing actual probabilities, and trying to 'finish what Jaynes started' by tying that back to the foundations of Logic itself. (I don't actually think I'm anywhere close to being qualified to do this on my own. But the conceptual threads seem to be there, and it seems to me that it wouldn't take a huge intellectual endeavour to pull the threads together and weave them into a sort of foundational 'whole'.) It seems to me that zeroth order propositional logic can be solidly founded in Truth Table semantics, and this can be connected to the Natural numbers with Boolean algebra (0 and 1), and that this in turn can be 'extended', as Jaynes proposed (and mostly completed; following R.T. Cox and George Polya mainly), into probabilities within the range [0, 1] using Rational numbers. That's the direction I'm shooting for. I thought I should mention that along the way, I've discovered a few interesting tidbits you might be interested in pondering yourself: - For quite some time, early BCEs to early CEs, the logic of the Stoic philosophers was far more popular than the logic of Aristotle, and it has only recently been pieced together that this Stoic form of logic (specifically, that of Chrysippus of Soli) is extremely close to what we now call Propositional Logic, and was about as sophisticated. This stunned me, but apparently it's not bullshit. What happened was that Aristotle's Term Logic (itself somewhat resembling today's Predicate Logic with its quantifiers), which had long been seen as a 'rival' to Stoic logic, basically just managed to be the logic favoured by the literate class of the European Dark Ages (specifically Christians), and Chrysippus' voluminous works were simply ignored and subsequently lost. The only way historians were able to piece together the essence of his system was by having modern logicians, who have essentially re-invented Propositional Logic a la Boole and others, re-examine mentions and references of Chrysippus' work within the works of others such as Galen, Seneca, Cicero, et al. - Boole's formulation of logic, and the success of modern circuit logic, based on Boolean Logic, confirms in my mind that Propositional Logic is a solid foundation, like Natural numbers. - If Jaynes' argument (again, based on Cox, Polya, and others) holds up (and I think it does), then Rational numbers could extend Boolean Logic into Probability Theory (sans Reals), and that extension could be enough to form the basis of basically all of Science. Jaynes didn't *quite* go so far as to embrace Rationals-only, but in at least a couple places in the book, you can tell that he would have been open to the idea, and even perhaps secretly hoped it was possible. - If that's the case, then I could definitely foresee that your efforts to base all of mathematics on Naturals and Rationals could and would be closely aligned with the goal of basing all of logic on the Naturals (0 and 1) and Rationals [0 to 1]. - Or, perhaps, it might actually work the other way around. Perhaps it's possible to base Naturals and Rationals off of Logic. Not axiomatic logic, but grounded logic. (This is, after all, essentially how computers already work today.) - It seems to me that the problem of infinite sets is closely related to the complications that arise from universal quantification in predicate logic. How can you ever 'complete' a 'For all x' in an infinite context? - Finite/computable math founded on finite/computable logic, or vice versa? - What could it mean to bring Probability Theory as Extended Logic to your current endeavours in Math Foundations, Geometry, and Algebra? You've implicitly been using 'logical' argumentation to lay out your theories. Can it be made explicit? Can Probability be a useful concept in this regard? At first glance it might seem absurd, but I wonder. - Modern AI and Machine Learning are now heavily involving probability theory, especially the notion of Bayesian Probability and Inference, into the latest, most successful working AIs. - At the heart of it all may be a centralized role for Information Theory. Probability, Logic, Computer Science, Math, Algebra, Geometry, probably Calculus, and definitely Linear Algebra are all closely associated with Information Theory in a variety of ways. Come to think of it, so is Science itself. (Have you thought of how to use Rational numbers in the context of Information Theory, where one of the most common 'operations' is to take logarithms, which are almost never resolvable as Rational numbers? A sophisticated and *practical* theory of 'approximation by Rationals' seems in order, perhaps.)

@douggwyn9656

6 жыл бұрын

"How can you ever 'complete' a 'For all x' in an infinite context?" We do that routinely, using finite patterns and specific interrelations among properties to encompass all members of the relevant class without having to inspect each member individually.

@KarmaPeny

6 жыл бұрын

Inverse Hyperbolic Tangent - You said "My purpose is to develop an intuitive framework for people to use logic and probability directly in discussions and arguments. In other words, to bring it as close as conceptually understandable and usable to the average educated lay person as is possible. " Just imagine if everyone agreed on a framework of logic and probability that could be applied in normal everyday decision making. This would result in massive agreement, accelerated progress in all fields, an end to almost all the worlds problems, and it would bring world peace. Your purpose should be the goal of most of us that work with logic (I'm in computing), it is certainly mine.

@douggwyn9656

6 жыл бұрын

For example, "if natural number n is divisible by 6, then n is divisible by 2". It's easy to prove for all natural numbers n, using basic properties of natural numbers.

@inversehyperbolictangent3955

6 жыл бұрын

+Doug Gwyn , You're assuming precisely the topic under question. Q: "How can you complete a 'for all x' in an infinite context?" Your answer: "We just do, because we assume we can." Have you seen Wildberger's objections to universe-bustingly-large Natural numbers? If not, I would suggest doing so, to get the full context of the objection. Search his videos for 'large', and watch the top several results, if you haven't already. If you *have* already, then how do you answer his challenges in those videos?

@inversehyperbolictangent3955

6 жыл бұрын

"This would result in massive agreement, accelerated progress in all fields, an end to almost all the worlds problems, and it would bring world peace. " We are on the same page. :-D It's a lofty goal, but not actually unreasonable/implausible. It will just take a good deal of time, effort, and persistence, by most of the world's population. lol Sounds impossible at first glance, but there is potentially a pathway there, in the long run. The key area is public education reform, which is perhaps why I'm so interested/excited about NJ Wildberger's efforts.

I'm just wondering about this series of videos. Although I'm a math amateur at best: your basic premise seems to be that we cannot out-calculate infinite series or irrational numbers in floating point form. So, for example, do you also deny that the solution to the Basel problem is pi^2/6, for example? I'm confused.

@njwildberger

3 жыл бұрын

@Kees-Jan Yes I think the solution to the Basel problem is not yet properly understood. The nature of "pi" is the issue --- that is much more subtle and complicated than just another number. Please consider joining the Algebraic Calculus One online course to find out about algebraic alternatives to a lot of more confusions.

" In particular, my concrete question of: what exactly is the sum pi+e+sqrt(2) ? never seems to get answered." There is nothing concrete about that question. It's a bit of school-boy rhetorical fluff. Since none of pi, e, and sqrt(2) has an "exact" value in any sense known to a person who uses the word "concrete" so sloppily, the answer is obvious: Tell me how exact you want the answer, pay me for my time, and I'll give you the correct answer to that degree of exactness. In your beloved "counting" numbers, Norman, it's seven.

@njwildberger

4 жыл бұрын

@Loh Diwei, In other words, you don't know how to calculate the sum. Why not just come right out and admit it? Here is an easier question for you: what is 1/2+1/3+1/5? I bet you can answer that one, without getting all huffy and puffy.

@njwildberger

3 жыл бұрын

@John Doe You have got to be kidding! Now you want us to start philosophizing about the essential nature of computation? Let me give you an example. If the question is: what is 1/2+1/3+1/5, then the result of calculating should give an answer of ... 31/30. Everyone should be able to agree on the answer. If we insist that fractions are written in reduced form, with no common factors in the numerator and denominator, then the answer is also unique. No philosophizing necessary. Similarly with natural number arithmetic, if the question is: what is 2+3+5, then the result of calculating should give an answer of ... 10. So it is not a lot to ask, that the proponents of "real number arithmetic", which I claim is actually a fiction, ought to be able to step up to the plate and CALCULATE pi+e+sqrt(2). Without further waffling about ... "up to a specified number of digits" or similar. Just show us the answer.

@pretragovic

3 жыл бұрын

@@njwildberger I assume that by \pi, e and \sqrt{2} you mean the limits of the sequences a_n = sum_{i=0}^n 2*i!/(2i+1)!! b_n = sum_{i=0}^n 1/i! c_n = sum_{i=0}^n (-1)^(i+1) (2i-3)!!/(2i)!! . Then \pi + e + \sqrt{2} is the limit of the sequence d_n = a_n+b_n+c_n = sum_{i=0}^n [ 2*i!/(2i+1)!! + 1/i! + (-1)^(i+1) (2i-3)!!/(2i)!! ].

@sinclairabraxas3555

Жыл бұрын

@@njwildberger you sound like pythagoras lol what the fuck