This topic is very complex with a lot of ugly vocabulary. Just like this channel here, I also made a video trying to explain the insolvability of the quintic in the shortest, most intuitive path that I can think of, although I’m no expert. Here’s the best idea behind it I've got: 1) A quintic formula, if it existed, would have to be able to have all 5 of its roots interchangeable, since Vieta’s equations show that the coefficients in such a formula would be symmetric. ALL FIVE roots would be interchangeable. 2) It’s possible that a quintic equation may have a root with another root nested inside of it. Perhaps more than one. This could cause huge complications with interchangeability. Each radical root would have conjugates, and swapping a root that’s nested in another root would really mess up the conjugates of the outer root if done carelessly. 3) To avoid this potential issue, the set of all roots of the polynomial would have to be arranged in such a way that no matter HOW you permute the entire set of all five of them, the subset of the roots nested inside of other roots would have to stay in its own set. Any set of roots like this fits the criteria of a normal subgroup. There may be normal subgroups that do not have this property, but if a set DOES have this property, it will be invariant under conjugation, just like a normal subgroup. 4) The nesting of radicals can be thought of as the nesting of subgroups. And every time there’s a set of roots nested inside of another set of roots, it would have to obey the property of staying inside of its own set, no matter how the full group is permuted. 5) If a group of roots is not a chain of normal subgroups ending in the trivial group, the property of roots staying in their own group is not guaranteed. S5, the group representing all possible arrangements of 5 elements (roots in this case) can possibly have subgroups that are not a chain of normal subgroups. The A5 group is one such subgroup. Since this is a possibility, the quintic is not solvable in radicals.

he is a teacher gone professor +

I'm desiring to know if there exists any professor or academic scholar working in the physics area, that opposes the Theory of Relativity. I only heard of one from Germany, named Alexander Unzicker, but disappointingly it seems that he is a little bit far from the academic "circle". If there is someone he or she may be of great help to You. Anyone who knows any of them, may propose their names below my comment.

Gghh.

Ghhh

Do you have any video about the Zeno’s paradox?

Thank you, I needed this refresher - I am in my 70's. 😁

Excellent teacher. Thank you!

"a number even bigger than everything in the universe" : of course the observable universe is finite, BUT what if the whole universe itself is actually infinite in size ?

You bloweth my mind😊❤

I found this video very interesting. A few comments: 1. As a point of departure, let us consider the (quite natural) desire to define our concepts in terms of procedures we can do, which yield results we can write down on a piece of paper. I (and most American Pragmatists) are very sympathetic to that inclination. The problem comes when you try to pin that down. Do the fundamental laws of math and physics need to advert to literal pieces of paper? (Or perhaps tablets of clay?) What kind of thing counts as being capable of carrying out such a procedure? Does it need to be a human being? Can a computer suffice? Can any instrument suffice? I mean, if any instrument suffices, then, well, your wishes are fulfilled :-) because all measurements are made with some kind of machine or another. But it seems like all your wishes are *not* fulfilled :-) Can you clarity? 2. Remember when you first looked up the definition of a word in a dictionary...and that definition contained words you didn't understand, so you looked those up...etc etc until it either circled around to the first definition, or just bottomed out in undefined words? What would be wrong with the solution to definition which Euclid used....just try to winnow down the terms you need to a minimum, and declare those terms to be undefined? Point, line, and plane? We did not need any definitions of those to do geometry. Indeed, defining them would have set mathematics back---think of, say, projective geometry, where the roles of points and lines are interchangeable. Do you consider these concepts "metaphysical?" If so, what would be a suitable procedural definition for points, lines, and planes, which would enhance and not stifle the development of, say, projective geometry??? 2. One perspective I think you don't pay enough attention to is the third person perspective :-) We want the laws of math and physics to hold for any observer anywhere at anytime. This puts a fairly big constraint what we can pose as such laws: any two observers must agree that those laws hold for each of their individual observations and measurements. For example, this rules out what Leonard Siskind called "Aristotle's law", F=mv, because such a law will not hold, in general, between observers in motion with respect to each other. Particularly salient in this regard is your questioning of the legitimacy of acceleration, because velocity is suspect, because position is suspect, etc etc. F=ma....or better put in terms of p, momentum: F = dp/dv, is the simplest possible law of motion which all inertial observers have any chance of agreeing on. This requirement of intersubjective agreement is, I think, absolutely essential to any endeavor which calls itself "science." If nobody can reproduce the results, then those results are not scientific. But here's the kicker: can purely procedural definitions satisfy this requirement of intersubjectivity? Consider mass....is it always the case that any observer, anywhere, at anytime, could measure the mass of arbitrary object in the same way? When Newton unified the mass of the apple and the mass of the moon, does it really require that we are able to measure the mass of apples the same way that we measure the mass of the moon? Or the mass of an electron? Does *the same* procedure have to work for both? If so, what counts as "the same" procedure? If we weigh an apple, or if we throw it multiple times and trace its trajectory, is that "the same" procedure we use to measure the mass of the moon? What sort of criteria count when trying to determine when two procedures are "the same"? If you could address some of these questions, I'd be very grateful! Thanks again for a very thought-provoking video.

What's the Wildberger version of cyclic groups, roots of unity, does anyone know where to find that?

6:57 It does not exists any video named "WT15"

Worthlesss

"For most of history, most mathematics was applied mathematics" FALSE. The oldest form of mathematics discovered was number theory. People have been talking prime numbers for thousands of years, with the only application ever of such a thing came up recently in cryptography. Whether or not there are odd perfect numbers is the oldest unsolved problem in mathematics. Throughout most of history, mathematics was mostly pure.

This should have 0 likes. Just a clickbait, nothingburger, anti-intellectualist video of a man past his prime going insane. Ultrafinitism, intuitionism and constructivism are pseudoscience.

mr wildberger, you are a mathematician in the truest sense of the word. you remember something that other mathematicians of the modern day have forgotten - that math is linked to reality only by proofs of abstract concepts through simpler ones, and the stronger the proof, the more real the mathematical concept is. i appreciate the educational videos that you make and you better believe that i will be watching all of them (or as many as i can)

I appreciate his positive ideas but not his strawmanning.

I notice that there is no quotient rule right off the bat. Is this because the dual numbers are not a field?

Dear Professor, thank you very much for your insightful review - it was truly fascinating. If possible, I would greatly appreciate your thoughts on more recent textbooks, particularly on mathematical analysis, functional analysis, real variable theory, and infinitesimal calculus. I am especially curious to hear your opinion on Terence Tao’s textbook. Thank you again for your valuable contributions!

Dear Norman Wildberger, you are a new mathematical star to me! I am theoretically-physics trained measurement specialist who made serious contributions in Cryo-EM (Google: Wiley prize 2017). During the pandemic I went back-to-the-basics while spending much more time at home rather than being distracted by daily lab life. I was somewhat shocked by my conclusions seeing that Shannon’s sampling theorem pre-assumes determinism of the data and of the results. My conclusion that the maths of one of my scientific heroes was full of inconsistencies was shocking to me as you will appreciate, but worse was that those misconceptions survived . By the way, that determinism spreads over all of Physics... I have just put out a series of four papers that I think you will appreciate. (Google: Symmetric Sampling and the Discrete Laws of Physics). I now associate New South Wales not longer just with Bob Dylan’s lovely song (kzbin.info/www/bejne/m4jak4mkrbWShsk) on being condemned to go to the end of the world but also with the mathematically impossible reachability of infinity. Thank you for your insights! Marin van Heel

Not only that but since all motion and its quantification can only be relative then the speed of light can only be relative. Thus, the postulate of c as a constant ignores this most fundamental of physics axioms, which is that all motion can only be relative. So you cannot speak about the speed of light while ignoring the frame of reference context of its motion. This is a rookie mistake that exists at the core of fundamental physics.

Fascinating. I came to a lot of the same conclusions in a book I am writing. Especially the problems the 1/(1+t^2) fixes, i.e., the mapping of the zero to 1.

You see the flaws in SR in the same exact way I do.

Please make a lengthy course on galois theory... Please professor. This topic is very dangerous for all students..

No way you can justify Choice as a symmetry argument. I'm pretty much convinced Norman is right to be skeptical about R.

I love this idea so much. Encoding sets in a higher dimensional space without any of the set theory problems. It makes so much sense and reminds me of an algebraic version of a hypergraph or something. Thank you for building this!

Would really love to see this amazing series continued. I'd also be interested to know where you have learned about financial systems, and whether there are any specific resources you would recommend, Professor Wildberger. It seems surprisingly difficult to find clear and precise expositions of a lot of economics topics as a mathematically minded newcomer to this topic. It has definitely been a lot of fun to follow the Little Cactus Bank up to this point :)

There's a big problem that I haven't seen mentioned yet. As stated in the video, an algorithm is "a finite list of well-defined instructions that give a step-by-step procedure for a calculation". This would mean that even if algorithms are viewed as strings of keystrokes on a computer, they are countable, in the sense that you can create a bijection between algorithms and the natural numbers, where "real numbers" are uncountable. There are so-called non-computable numbers that this algorithmic approach doesn't cover. I would also argue that definitions in general are countable, thus in some sense there are "more" real numbers than definitions. So most real numbers are undefined, unless you also allow algorithms or definitions themselves to be infinite. This would sort of leave us with this vague circular logic where definitions are just as poorly defined as real numbers, which defeats the purpose of a definition in general.

You’re misreading Einstein’s point about the constancy of c in all inertial frames. It is not the motion of the emitter that is crucial, but rather the movement of the inertial frames of reference. The point is that if a whole body of water is moving relative to an observer we _will_ see a different speed od propagation than when the water is stationary relative to the observer, and the same is true of sound when the air is moving relative to the observer. This is important because in models which use a mechanical aether, the relative motion of frames of reference would yield different observed speeds of light. But light does not move in a medium which itself can move; therefore its speed is constant in all inertial frames and all inertial observers will report the same speed of light from any given emitter.

Next: one plus one does not equal two. This is sophomoric stuff.

What if there was a way to approach this using rational volumes. Many different shapes with identical volumes? It’s possible to simply construct these. For instance using a cube side 1 it is possible to add fractions of the shape to itself to form 4096 different shapes of the same volume.

Honestly how much improvement would be if every student learn and do a recap every year at the beginning of these basic concepts. Fundamentals should be learn and relearn all the time.

I get Archimedes' method, but wonder how the heck the rest of them buggers came up with them goddamn formulas.

But with music not stopping, you can say the same with Ponzi scheme? Is there a difference?

You are awesome, Prof. Wildberger. These are issues I have struggled with in mathematics my whole life but not in such a coherent fashion.

100% agree with you Profesor

I have loved mathematics since I was a child and have been plagued by just these sorts of questions re infinity and existence of limits. Thanks for showing that the Emperor has no clothes!

I was halfway inclined to give this finitist farrago the benefit of the doubt but this video really takes the biscuit. Absolute nonsense.

Absolutely ridiculous. You haven’t proved anything. You only take an infinite number of steps if there are an infinite number of twin primes, which you haven’t proved. Unbelievably stupid.

To be frank I can't see a single thing your change in foundation solves for mathematics. I have heard many mathematical philosophers express a desire to find new axioms to expand math or solve issues for example the famous joke about Choice, Zorn's and well ordering. You are though the fisrt I have heard who wants to discard infinity itself. I think you need to clarify in this video actual problems rather than gesture at what seems to amount to "bad vibes" you get from the reals and infinity. You say this is where the difficulties begin but you haven't named them. I am genuinely enthusiastic for someone to attempt to reconstruct modern mathematics from a finitist point of view but you still have all your work set out for you to say that this is necessary. Modern mathematics has grown corn so to speak (one you may dislike perhaps), you're vision still needs to prove that it can. Lastly, I have a philosophical issue with your framing of axioms as somehow inherently bad. It's my perspective that all mathematics must necessarily come down to an axiomatic sense, you can hide it behind intuition but that in my opinion is just a sloppy handing off of them and not an actual solution.

Who else is here taking notes? Recon!

Do we can get something well known if we play the same game, but for a square matrix that is not necessarily an adjacency matrix of a graph ?

How can you approximate something that, in your view, doesn’t exist? Saying that you get closer and closer to the number which, when squared, yields 2 is to admit to the existence of such a number as the limit of the sequence of approximations. Your use of quotes won’t make the problem go away.

I think you said somewhere, something along the lines that whenever there is a symbol near a number that this denotes an operation taking place. That makes sense to me in terms of understanding the number systems outside of the natural numbers. Another factor is that most everything, if not everything, outside of the natural or whole numbers occur by choosing an arbitrary zero. So negative numbers result from selecting an arbitrary starting point. A step that is usually unavoidable in many cases, mapping for instance. Another problem is that the actual aspect, and thereby the appropriate units, of things being measured isn't always clear. Unit squares and lengths are considered equivalent but the process for determining square roots for each should/could differ because the results could differ based on the unit be used. Maybe having symbols denoting which unit is in play would unmuddy the waters. My fourth grade teacher (After speaking to the town's high school math teacher I think) made the point that the reason that we don't cut the pie into 7 pieces instead of 10 to get 1/7 is because all the operations have to maintain a common unit in order to get an accurate result. Not quite in those words. Still, thereafter whenever infinity appeared it registered as an annoying convenience. Sue

infinite series is not problem with dedekind cuts because rudin define real numbers to be a union of cuts which will pesent an upper bound and result as a least upper bound represents a real number. So in the end there no any infinite set involve to constuction of any real numbers assocoitaed

sides are scalar infinites while internals are 360 degree skewed which hopefully you fixed by some method but reality forces encapuslation and qunatization of even distances - love your video - love herons formula infinity inward collapse leads to same thing which root function inverts or does 2nd derivative to bring 4D to 2D area

There are an even number of odd and even numbers 1-9 (2,4,6,8). When you introduce zero, it gives the number 5 the power of an even number by constituting the number 10. Start at 3. You'll go from 3 to 9, from 9 to 10, from 10 to 5, from 5 to 15, from 15 to 16, 16 -> 8 -> 4 -> 2 -> 1. Draw a five pointed star and begin counting the points until you reach 4, then start over. Notice the next point is #1, regardless of which direction you've counted. See the three points between the 4th point and the 1st? Look at 10 and 11; how many 1s are there between these two numbers? 3. Each point on the star is 1/5, or 1 five, no? Zero gives access to odd numbers above ten and those below ten. We went from 3 to 9 back down to 5 - all odd numbers under ten, in our first exercise of 3n+1. To go any further is pointless. It's "heads I win, tails you lose" with numbers. The problem is zero, mate. It creates a new set of numbers: from 1-9 to 0-10. How many numbers is that, 0-10? Eleven. How many extra numbers are there between the two sets? 2. Does this need further explanation or? It's pretty evident.

You are starting to sound like an angry old man Norman.....

This is definitely one of the best videos on the topic, thank you.