I´ve been watching several of the series up till now, and this one remindes me of fractals. It´s like measuring a length with different rulers. Which let thoughts come in, like, what if the intervall of [1,2] has in some sense a fractal character. Whether for RAT or even the reals, when going to smaller partitions. But maybe I´m just melancolic late in the evening and in the same time pleased to feed my "math neurons". For they didn´t do any real work in math for 6 years when I studied a few semester. Even today I´m looking for new insights for example in prime number theory. Or fractals, as they are found in the ecg-waves of the heart. Meanwhile I became an occupational therapist working with stroke patients. So, for the love of mathematics, thank you for sharing and exploring other shores in the field. Alex (27), germany
Thank you sir for very clear explanation. I do agree that we have a big problem of making definitions or maybe understanding the Definition. Anyway I do believe that not getting a real number and only approximation is the secret and the essence of the universe. Without approximation the universe WON´T be even able to move. The ERROR should be there in order to this MACHINE (everything that moves) to be able to work). I am following this series with JOY. Thank you.
The method of reasoning described by Doyle is actually "abduction" rathar than "deduction". Now to the point I'm trying to make: The formal foundations of mathematics were introduced to make these concepts precise. Dr. Wildverger is yet to mention and discuss those in detail, let alone refute them, whatever that's supposed to look like.
11:39 - Um, 22/7 is MORE than pi. It's 3.1428... But it's given here as Archimedes's *lower bound* on pi. 221/70 is yet larger, so pi is actually not in that range. This website: itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html says that what Archimedes actually found was 3+10/71 < pi < 22/7. That works.
Thanks again for this great video! What do you think about the fact that the summing of all natural numbers equals -1/12? Have you heard about this? There are even applications in quantum physics that rely on this fact somehow. I don't know the details but yeah, you might know what I'm taking about.
Keni Angervo I was wondering the same thing. Without having studied it my impression is the view of the unfounded (or dodgy axiom) status of much of mathematics is incorrect. Shouldn't it just be called "theoretical mathematics"? Maybe this is dealt with in a video I haven't seen yet.
I think , if planck constant really exists, which tells us, lenght as a dimension is discrete and it can be only smaller as planck constant, Calculating length or area by approximation can give the exact result if only that approximations is as fine as planck constant.
There is something to say "God made the natural numbers, all else is the work of man" Kronecker. I appreciate your lectures, thanks a lot professor Wildberger.
I think "existense" is more of a philosophical arguments. Yes real lines can be constructed by approximations, but also the real numbers can be formulated as axiomatic system, similarly to geometry. The "length" of the square root of 2 "exists" geometrically, if you accept that (plane) geometry is some kind of idealisation of our perception of the plane (you can't find a "line" or a "point" in rea life can't you? It has not length or width. You can accept the fact that the "geometrical image" you draw is approximation of the ideal generalization called "plane geometry" then you can associate a "length" in your mind to that line because you can actually see it. You can accept that this "length" can not exist because we are idealizing, or that it does exist (how do you prove that you have a string with irrational length? What's the unit length to begin with?). Anyway, whatever fits for you doesn't impact the beautifull theories of real analisys that continue to serve us. Anyhow thank you a lot for devoting you time, even if I disagree (or do not accept?) with your view I still think it's important to bring up things like that. Unfortunately it doesn't happen a lot in university. Moreover it's really unappreciated nowadays that you can get remarkebly far without resorting to defining things as debatable/dubious such as real numbers, and I truely love the fact that you show these kind of things.
Math has left the real world long ago, It has attained sort of nirvana and is detached to all constrains and boundaries of real world. It contains just beauty and bliss... The sole purpose of "Pure" mathematician is to look for beautiful mathematics. you can use their ideas, distort and apply them in real world I don't think they will have any problem. We don't need approval of computers and calculators for the number system. I guess computers just can't get the mathematical ideas our brain can perceive. They can't appreciate the beauty iota contains.
around min 15 ,you said , that physical intuition of having a curved line straighten up gives the length of that line is a big assumption for mathematics. why is that? I thought you have said several times that the more mathmatics is closer to real world, more concrete it is. with that being said, Why shouldn't we assume something that is highly likely valid in physical world is also valid in mathematics??
@all: Here's an idea that continues from what NW said and that could lead to a contradiction by using real numbers: Let's take a real number like Pi. Now, "Pi" is just a symbol for a number, defined to be between 3,14... and 3,15... (I do simplify heavily). By this definition 3.142... and 3.147... are both !!! the number Pi. But then Pi wouldn't be equivalent to Pi anymore, because e.g. in our example we'd have two equivalent numbers (Pi and Pi) and not (because they have clearly two different properties: 3.142... and 3.147...) and that'd cause a contradiction and absurdities like that Pi - Pi is not necessarily zero. Am I mistaken that you can always find at least two different numbers within a certain intervall of a Cauchy sequence that defines Pi? Isn't Pi a "foggy thing" where you can never be sure that the Pi you use is equivalent to the Pi you used a minute ago? What does modern math say about this argument, how is it solved or is it a problem for them at all? (Please bear in mind if you answer that I am not a mathematician.)
Thanks for interesting video(s). Although I see many practical uses for this careful approach (especially when dealing with computers), I don't see this way as being the pure mathematics prove or some kind of persuasion that it is impossible to work out what for example sqrt(13) is. Ok, sure, infinite processes can't be done using computers, but that doesn't mean there is no other way (not that I know of any as for now). But actually what I see as basic problem is in a definition of the idea of a 'number'. It fits nicely into your quest of debunking mathematical concepts (like sequences, functions...) and in my opinion idea of a number is core to many problems. How one can say or deny there are 'real numbers' if definition of a 'number' is more or less intuitive. Sure, real numbers would be tremendously more complex then natural numbers, but in a way even integer are. thanks again
on sl.4 despite len of CB is sqrt(13) we actually will get a rational area of a square build on this side of a triangle, so its working somehow. assuming we are just inventing a language to define some objects and if our defs is unambiguous maybe we can use them sometimes. nevertheless limits are seems to me not so unambiguous indeed (and .9...==1 as well!)) ).
Start with this 2x3xsqrt(13) right triangle, So the hypthothenus has no length. Now you draw a new cartesian grid that aligns with the longest side of the triangle, and call its length 1. Then of course, the right angle sides no longer have (rational) length. So by choosing a grid, we can decide whether a side has a length or not.,
Ok, so what if we take a piece of string that is exactly 12" long. That's a nice rational number, so it exists. Now change the string into a curved shape. Now the usual calculation of an arbitrary curve segment will run into irrational number issues. However, the string is STILL 12" long - the length of the string still exists. The string did not change just because we re-arranged it (actually it probably did, if it's a real piece of string, but you get the idea). How do we cogitate on that? Now, maybe is we set this all up with rational numbers there's a way to do this exactly - that would be quite cool.
This is a good question, which needs to be thought of in the right way. We want our mathematics to model the real world, and to provide answers to real life questions, but we should not insist that the foundations of the subject be based on physical reasoning. While you have an intuitive idea of the "length of a piece of string", it is in fact not a precise notion. Magnify the string, and you will see lots of little wisps at the end --which one are you using? And any measurement system has only a limited range of precision (perhaps up to 0.001 millimeter, but no further). We however want a pure mathematics based on precise notions. So it is not necessarily true that curves in mathematics have prescribed "lengths", rather we require to DEFINE what we mean by the length of a curve. Unfortunately, this is very difficult to do in a precise sense. In fact it is even problematic for a line segment in the plane! Fortunately the quadrance of a line segment is always defined, and that is one simple reason (there are quite a few others) why Rational Trigonometry is a more powerful theory than current trigonometry.
That's good - I definitely think we should use different thought processes for the basics vs. the practical applications. The "bounds" you discussion the video fits in extremely well with science and engineering, as you noted when you talked about interval arithmetic.
Nobody disagrees, that sin(1/4) needs to approximate, if the value needs to compute and expressed with digits or as an ratio. But using approximations in a mathematical transformation with many steeps would result in an increasing inacuracy, which needs to take care of, which would be much work. But remaining in the symbolic expression remains exact. And just that is meant by using real numbers. Using pi, e, sqrt(2) or sin(1/4) in the equation as long as possible frees from the need to consider the accuracy all the way before the last step. The class of cauchy-sequences for a real numer hast the nice function, that it didn't specify the value of epsilon, it is true for all epsilon greater 0. And because of that it didn't need to be defined all the way through the calculation, only at the end, when we need the digits, than we must give it a certain accuracy, but not before. And for the steps before there is the world of symbolic computation, which is also doable from computers today. After watching 10 hours of your videos, I have still not seen any good arguments where you proof your point against real numers. It would help if you could give your arguments somewhere in a condensed form without all those very long explanations for undergraduated.
As will come as no surprise, I disagree with the notion that there is no exact length, no number V13, only approximate values also in theory (not only practically). It would be very hard to do math without these exact values. The argument that "an infinite amount of work etc is required" is not, I think, a valid one, because we have certain smart methods to get around those infinities and get the precise answer anyway. For example, adding up the series of rationals, where the next term is the last one divided by 2: 1/2+1/4+1/8+... is easy. Even though it never ends, you only have to use the whole object (an infinite number of terms) in your solution method at once to solve it. Let's leave the solution to the reader (it isn't approximate although the series is infinite), have fun!
Finally an answer to the "how long is a piece of string" question. A. There is no such thing as length. Seriously though, I'm a computer programmer and have little trouble believing in all these concepts of approximation as they are very familiar to me. I find it weird that pure mathematicians believe that they are finding exact solutions to some of these problems.
Come on! In computer science "arbitrary accuracy" is of course practically impossible, BUT in the mental game of mathematics there are no physical boundaries. The definition of exact value is given in the epsilon-delta sense, and for every epsilon (say ε = 10^(-7000) ) you can mathematically find appropriate approximation and we call that the exact value. But physically (e.g. with computers) it is next to impossible to find appropriate approximation with very small epsilon.
Dave Poo Pure mathematics is based on axioms and logical rules. ("Pure mathematics is the magician's real wand." -- Novalis). On the other hand, computer math is also based on (arithmetical) axioms but computing power is limited by physical structure of the machine. Intelligent human can proof that √2 is irrational but computer just keeps computing decimals.
Juho x A cow can not represented as a ratio a/b of natural numbers. Therefore a cow is not a rational number. A cow is irrational number! "Rigor and intelligence", my ass.
pieinth3sky You can't approximate a cow by rational numbers for every rational epsilon. A set of cows is also bad for usual arithmetics: if you split one cow or add tho cows the result might be inconsistent with all the other cows.
The concepts are fantastic, but the presentation is ingenious:?) Thank you again for sharing.
these are awesome, many thanks
I´ve been watching several of the series up till now, and this one remindes me of fractals. It´s like measuring a length with different rulers. Which let thoughts come in, like, what if the intervall of [1,2] has in some sense a fractal character. Whether for RAT or even the reals, when going to smaller partitions.
But maybe I´m just melancolic late in the evening and in the same time pleased to feed my "math neurons". For they didn´t do any real work in math for 6 years when I studied a few semester.
Even today I´m looking for new insights for example in prime number theory. Or fractals, as they are found in the ecg-waves of the heart.
Meanwhile I became an occupational therapist working with stroke patients.
So, for the love of mathematics, thank you for sharing and exploring other shores in the field.
Alex (27), germany
Thank you sir for very clear explanation. I do agree that we have a big problem of making definitions or maybe understanding the Definition. Anyway I do believe that not getting a real number and only approximation is the secret and the essence of the universe. Without approximation the universe WON´T be even able to move. The ERROR should be there in order to this MACHINE (everything that moves) to be able to work). I am following this series with JOY. Thank you.
The method of reasoning described by Doyle is actually "abduction" rathar than "deduction".
Now to the point I'm trying to make: The formal foundations of mathematics were introduced to make these concepts precise. Dr. Wildverger is yet to mention and discuss those in detail, let alone refute them, whatever that's supposed to look like.
11:39 - Um, 22/7 is MORE than pi. It's 3.1428... But it's given here as Archimedes's *lower bound* on pi. 221/70 is yet larger, so pi is actually not in that range.
This website: itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html says that what Archimedes actually found was 3+10/71 < pi < 22/7. That works.
I noticed that.
Interesting! What is your position with respect to the usage of the axiom of choice? Havé you Made already a related vidéo?
Thanks again for this great video! What do you think about the fact that the summing of all natural numbers equals -1/12? Have you heard about this? There are even applications in quantum physics that rely on this fact somehow. I don't know the details but yeah, you might know what I'm taking about.
Keni Angervo I was wondering the same thing.
Without having studied it my impression is the view of the unfounded (or dodgy axiom) status of much of mathematics is incorrect. Shouldn't it just be called "theoretical mathematics"? Maybe this is dealt with in a video I haven't seen yet.
I think , if planck constant really exists, which tells us, lenght as a dimension is discrete and it can be only smaller as planck constant, Calculating length or area by approximation can give the exact result if only that approximations is as fine as planck constant.
There is something to say "God made the natural numbers, all else is the work of man" Kronecker.
I appreciate your lectures, thanks a lot professor Wildberger.
Prof. NJW, what about the Narrative Numbers?
Sorry Jim, I do not know what they are.
I think "existense" is more of a philosophical arguments. Yes real lines can be constructed by approximations, but also the real numbers can be formulated as axiomatic system, similarly to geometry. The "length" of the square root of 2 "exists" geometrically, if you accept that (plane) geometry is some kind of idealisation of our perception of the plane (you can't find a "line" or a "point" in rea life can't you? It has not length or width. You can accept the fact that the "geometrical image" you draw is approximation of the ideal generalization called "plane geometry" then you can associate a "length" in your mind to that line because you can actually see it. You can accept that this "length" can not exist because we are idealizing, or that it does exist (how do you prove that you have a string with irrational length? What's the unit length to begin with?). Anyway, whatever fits for you doesn't impact the beautifull theories of real analisys that continue to serve us. Anyhow thank you a lot for devoting you time, even if I disagree (or do not accept?) with your view I still think it's important to bring up things like that. Unfortunately it doesn't happen a lot in university. Moreover it's really unappreciated nowadays that you can get remarkebly far without resorting to defining things as debatable/dubious such as real numbers, and I truely love the fact that you show these kind of things.
Math has left the real world long ago, It has attained sort of nirvana and is detached to all constrains and boundaries of real world. It contains just beauty and bliss... The sole purpose of "Pure" mathematician is to look for beautiful mathematics. you can use their ideas, distort and apply them in real world I don't think they will have any problem.
We don't need approval of computers and calculators for the number system. I guess computers just can't get the mathematical ideas our brain can perceive. They can't appreciate the beauty iota contains.
around min 15 ,you said , that physical intuition of having a curved line straighten up gives the length of that line is a big assumption for mathematics. why is that? I thought you have said several times that the more mathmatics is closer to real world, more concrete it is. with that being said, Why shouldn't we assume something that is highly likely valid in physical world is also valid in mathematics??
@all: Here's an idea that continues from what NW said and that could lead to a contradiction by using real numbers: Let's take a real number like Pi. Now, "Pi" is just a symbol for a number, defined to be between 3,14... and 3,15... (I do simplify heavily). By this definition 3.142... and 3.147... are both !!! the number Pi. But then Pi wouldn't be equivalent to Pi anymore, because e.g. in our example we'd have two equivalent numbers (Pi and Pi) and not (because they have clearly two different properties: 3.142... and 3.147...) and that'd cause a contradiction and absurdities like that Pi - Pi is not necessarily zero. Am I mistaken that you can always find at least two different numbers within a certain intervall of a Cauchy sequence that defines Pi? Isn't Pi a "foggy thing" where you can never be sure that the Pi you use is equivalent to the Pi you used a minute ago? What does modern math say about this argument, how is it solved or is it a problem for them at all? (Please bear in mind if you answer that I am not a mathematician.)
Thanks for interesting video(s). Although I see many practical uses for this careful approach (especially when dealing with computers), I don't see this way as being the pure mathematics prove or some kind of persuasion that it is impossible to work out what for example sqrt(13) is. Ok, sure, infinite processes can't be done using computers, but that doesn't mean there is no other way (not that I know of any as for now). But actually what I see as basic problem is in a definition of the idea of a 'number'. It fits nicely into your quest of debunking mathematical concepts (like sequences, functions...) and in my opinion idea of a number is core to many problems. How one can say or deny there are 'real numbers' if definition of a 'number' is more or less intuitive. Sure, real numbers would be tremendously more complex then natural numbers, but in a way even integer are. thanks again
on sl.4 despite len of CB is sqrt(13) we actually will get a rational area of a square build on this side of a triangle, so its working somehow. assuming we are just inventing a language to define some objects and if our defs is unambiguous maybe we can use them sometimes. nevertheless limits are seems to me not so unambiguous indeed (and .9...==1 as well!)) ).
Start with this 2x3xsqrt(13) right triangle, So the hypthothenus has no length. Now you draw a new cartesian grid that aligns with the longest side of the triangle, and call its length 1. Then of course, the right angle sides no longer have (rational) length. So by choosing a grid, we can decide whether a side has a length or not.,
Ok, so what if we take a piece of string that is exactly 12" long. That's a nice rational number, so it exists. Now change the string into a curved shape. Now the usual calculation of an arbitrary curve segment will run into irrational number issues. However, the string is STILL 12" long - the length of the string still exists. The string did not change just because we re-arranged it (actually it probably did, if it's a real piece of string, but you get the idea).
How do we cogitate on that? Now, maybe is we set this all up with rational numbers there's a way to do this exactly - that would be quite cool.
This is a good question, which needs to be thought of in the right way. We want our mathematics to model the real world, and to provide answers to real life questions, but we should not insist that the foundations of the subject be based on physical reasoning. While you have an intuitive idea of the "length of a piece of string", it is in fact not a precise notion. Magnify the string, and you will see lots of little wisps at the end --which one are you using? And any measurement system has only a limited range of precision (perhaps up to 0.001 millimeter, but no further). We however want a pure mathematics based on precise notions. So it is not necessarily true that curves in mathematics have prescribed "lengths", rather we require to DEFINE what we mean by the length of a curve. Unfortunately, this is very difficult to do in a precise sense. In fact it is even problematic for a line segment in the plane! Fortunately the quadrance of a line segment is always defined, and that is one simple reason (there are quite a few others) why Rational Trigonometry is a more powerful theory than current trigonometry.
That's good - I definitely think we should use different thought processes for the basics vs. the practical applications. The "bounds" you discussion the video fits in extremely well with science and engineering, as you noted when you talked about interval arithmetic.
Nobody disagrees, that sin(1/4) needs to approximate, if the value needs to compute and expressed with digits or as an ratio. But using approximations in a mathematical transformation with many steeps would result in an increasing inacuracy, which needs to take care of, which would be much work. But remaining in the symbolic expression remains exact. And just that is meant by using real numbers. Using pi, e, sqrt(2) or sin(1/4) in the equation as long as possible frees from the need to consider the accuracy all the way before the last step.
The class of cauchy-sequences for a real numer hast the nice function, that it didn't specify the value of epsilon, it is true for all epsilon greater 0. And because of that it didn't need to be defined all the way through the calculation, only at the end, when we need the digits, than we must give it a certain accuracy, but not before.
And for the steps before there is the world of symbolic computation, which is also doable from computers today.
After watching 10 hours of your videos, I have still not seen any good arguments where you proof your point against real numers. It would help if you could give your arguments somewhere in a condensed form without all those very long explanations for undergraduated.
As will come as no surprise, I disagree with the notion that there is no exact length, no number V13, only approximate values also in theory (not only practically). It would be very hard to do math without these exact values. The argument that "an infinite amount of work etc is required" is not, I think, a valid one, because we have certain smart methods to get around those infinities and get the precise answer anyway.
For example, adding up the series of rationals, where the next term is the last one divided by 2: 1/2+1/4+1/8+... is easy. Even though it never ends, you only have to use the whole object (an infinite number of terms) in your solution method at once to solve it. Let's leave the solution to the reader (it isn't approximate although the series is infinite), have fun!
Approximation with bound actually is kind of an exact value, but described in two terms.
Finally an answer to the "how long is a piece of string" question. A. There is no such thing as length.
Seriously though, I'm a computer programmer and have little trouble believing in all these concepts of approximation as they are very familiar to me. I find it weird that pure mathematicians believe that they are finding exact solutions to some of these problems.
Come on! In computer science "arbitrary accuracy" is of course practically impossible, BUT in the mental game of mathematics there are no physical boundaries. The definition of exact value is given in the epsilon-delta sense, and for every epsilon (say ε = 10^(-7000) ) you can mathematically find appropriate approximation and we call that the exact value. But physically (e.g. with computers) it is next to impossible to find appropriate approximation with very small epsilon.
Juho x
So if you can't program a computer to do it? how come you can do it in your "mental game"?
Dave Poo
Pure mathematics is based on axioms and logical rules. ("Pure mathematics is the magician's real wand." -- Novalis).
On the other hand, computer math is also based on (arithmetical) axioms but computing power is limited by physical structure of the machine. Intelligent human can proof that √2 is irrational but computer just keeps computing decimals.
Juho x
A cow can not represented as a ratio a/b of natural numbers.
Therefore a cow is not a rational number.
A cow is irrational number!
"Rigor and intelligence", my ass.
pieinth3sky You can't approximate a cow by rational numbers for every rational epsilon. A set of cows is also bad for usual arithmetics: if you split one cow or add tho cows the result might be inconsistent with all the other cows.