This inconsistency has always been in my mind, but Norman communicates it with such clarity that it gives it meaning. It seems to me that mathematics has two main procedures buried under the notion of numbers: algorithms for construction and deconstruction. Vaguely, construction is to build something out of basic building block, and deconstruction is to deconstruct something into basic building blocks. The results of these do not always coincide and the results are not always complete. Perhaps that should be the next frontier in math, researching what can or cannot be constructed/deconstructed. Thanks for the amazing videos!
I'm so glad to see someone address the distinction between quantities and (how I think of it) position. It irks me how some things in mathematics are labeled with numbers, such as elements of groups, despite having barely any shared properties. In a sense hiding various other properties which are cumbersome to express arithmetically, but intrinsic and fundamental to the actual objects. Thank you for your diligence in scrutinizing your 'tools' as much as your 'work'.
Supporting your resistance to the embedding of various number systems within each other, there is a fairly strong modern consensus among practitioners of Object Oriented programming and software development to 'prefer composition rather than inheritance'. Inheritance is the analog of embedding: Once we have developed Integers, the Naturals become 'just' a specialized type of Integer; once we have Reals, the Rationals become 'just' a special type of Real; etc. Composition is (generally) less rigid, more flexible, and in some cases more powerful, and it is defined by the 'has a' relationship versus inheritance's 'is a' relationship: E.g. if we build up Integers as being *composed* of two Natural numbers, and the Integer is their difference (the internal 'less' operator), then you could say that each Integer 'has a' Natural number (actually, two of them, to be precise), rather than saying that suddenly Naturals have changed status and a Natural 'is an' Integer. If the Reals actually functioned, then they might be able to be expressed in a similar way: E.g. if it were really possible to concretely define Dedekind cuts by specifying the 'gap' between two rationals (analogous to the 'less' of two Naturals in Integers), then perhaps it could be said that a Real 'has a' Rational (actually, two, again), rather than saying that Rationals have changed and now a Rational 'is a' certain kind of Real (i.e. there are Reals and the Rationals inherit from the Reals -- it is also conceivably possible to reverse the inheritance such that Reals inherit from Rationals and 'extend' them, but I'm just portraying the situation as orthodox modern mathematics would have us believe is the case). However, something like Dedekind cuts (and Cauchy sequences, and all other current manifestations of Reals) require not just one 'has a' relationship, or two, or some finite number, but an infinite completion more like a 'has all possible' relationship, which is actually impossible to program. It's yet another way of thinking about the fundamental problem with Reals.
Very lucid presentation of the foundational challenge between discrete and continuous. My suggestion for the role of "unit segment/area/etc" is the expression , open interval defined as both more and less / both increases and decreases. To my astonishment taking relational operators as a formal language and concatenating mediants with the symbols according to simple procedures, leads to structures with obvious and natural Stern-Brocot type discrete interpretations, ie. a very natural and foundationally meaningful link form new(ish) language of continua to rational numbers. With lots of new and interesting structure to explore. Note that 'concatenating mediants' is internal process of generating new rows with more fractional resolution, and thus inherently "revercimal" perspective of division/partition instead of additive perspective of discrete counting. Closely related to what are called p-adic numbers, but with this approach we connect directly with coprimes of Stern-Brocot type structures instead of prime numbers of p-adic formalism.
Video Content 00:00 Introduction 04:50 Origin of numbers: Counting and measuring 15:35 A bridge from counting to measuring requires a unit segment 17:19 Fractions as tools for measurement 25:03 Decimal,binary,sexagesimal PVS
It never really hit me that there was this 'great divide' between numeric systems until I started working at an electrical company and a friend their tried to explain the difficulty he had of trying to get the latest invention (a digital microprocessor - then cutting edge technology) to control an analogue system of motors. Imagine having a drum of tissue paper slowly unwinding at one end whilst at the other end a drum is winding the paper up. He likened it to a cassette tape reel (a common device at the time but extinct now). The problem he pointed out to me that the speed of the two drums were always different; The big drum unwinding had to go slower (at first) than the winding up drum until the the amount of paper balanced and then the winding up drum got bigger and it had to be slowed down whilst the winding down drum got faster. All the time the tension between these two motors had to be kept sufficient so that the tissue thin paper did not get ripped or tear in the middle. It blew my mind to discover that this connection between digital & analogue systems was so fragile & paper thin.
There is no electrical engineering without the fictitious sq2. In 3D, the math is absurdly large for the identity matrix of a solid, the Skin Effect in light as it renders in math to our eyes. The Lyman Alpha Line Split, .006 angstroms, and beta radiation is permeable to the eyes. Stick with this. e(c) is 1D error correction for matter-energy transfer, and at three decimals in base 10, 0.577. Split the difference right between the eyes, and get .003 plus 0.577, or 58% matter-energy transfer efficiency. We are molten salt nuclear fusion reactors.
Being a forever Mathematics student I am mesmerized by explanation of very basic concept of Mathematics in a way that it should be so Mathematics be more enjoyable. Thanks professor
Thanks for the video. I have got a feeling that equality (=) is intrinsic in the counting system, while comparison (< or >) is for the measuring system. I might be wrong, but the video let me think of "why" we count or measure (before indulging into the how side of the operation)
@Stephen Yes that is a very good point. The notion of equality is a problematic aspect with the Greek arithmetic of line segments. How do you actually tell when two segments are equal? Do you have to keep magnifying them endlessly to check that they never fail to line up?? With the counting system this is much easier. I think Eudoxus has something to say about that.
@@njwildberger I have been following your videos and your analysis about "realness" is very inspiring. For me, R looks like a philosophical substratum on which we can construct different types of number systems. It is just like "car" is a thing, "elephant" is a thing, but "thing" is not a "thing". (I have been naïve in describing my thoughts - laughable from set theory experts). I want to say this because I want to give R a position in my mind.
@@chjxb Haha 😄, that phrase '"thing" is not a "thing"' made me chuckle, but actually it's a pretty darn good point! In computer programming, particularly in Object Oriented Programming, there is often the paradigm that 'everything is an 'object'', and in this case, the term 'object' has a precise definition in terms of being a certain kind of data structure combined with a certain set of operations (indeed the .equals() operation is usually one of them!). But at the end of the day, an 'object' in object-oriented programming isn't actually an 'object' in the everyday sense of the word -- although, interestingly there *is* (usually) an 'object' data structure representing the 'class' of 'objects' from which all other 'classes' implicitly inherit; thus, all objects inherit the .equals() operation from the Object class, and you can subclass various other concepts like 'Car' or 'Elephant', and 'override' .equals() for your particular subclass, e.g. so that Elephant.equals() compares the names of the two elephants in question, or Car.equals() compares the serial numbers or the makes of the cars or the license plates, for instance. Not sure how this relates to your desire to give R a position in your mind, but maybe it can help by giving some clues about how others have tried to tackle similar problems.
@@robharwood3538 Thanks. I do not fully understand your response but I appreciate your effort. Just like real number is a kind of substratum for us to build constructable number systems, the computing automaton itself is not a specific object. Does it make sense?
Math wll eventually change professor.You're heading in the right direction for years in my opinion but i think the last piece must be given from our understanding of our ability to make concepts.That's gonna end this confusion and justify 100% of what you 're saying all those years.Also on your point about proportions.Humans have built in a close relationship on recognise proportions for survival reasons.Biology backs you up on this prof.
Sir, thank you for your videos. Sir, Do you have any plans to make videos on mathematics formula & how to write mathematics proof (I still don't get math proof) ? How to bring our imagination into reality with the help of mathematics (how to think in mathematics) ? Thank You 👍👍👍
But then you don't get a consistent arithmetic with the usual rational numbers, and certainly not with other "irrational numbers" in general. For example, my usual challenge: what is pi + e + sqrt(2) cannot be (easily) expressed or tackled from such a point of view. It is not enough to "define" the occasional "irrational number". What you want is a consistent arithmetic with these things that can be checked and is falsifiable. In other words, it should be so clear that the usual mumbling about "oh just take a point wise sum of representative Cauchy sequences" or some such becomes an obvious obfuscation.
@@njwildberger Fair enough, but that does not mean we cannot talk in general about geometric rations. 'the square root of to' has connotive baggage, but I see no reason why it is otherwise unsuited to denote the entity. I agree that there are limits on what we can to with these entities (in comparison to the rations of commensurate intervals, which which we can do a lot), but we can say *some* things about them.
@Gennady Arshad Notowidigdo If by 'encapsulate' you mean what I more commonly hear spoken of as 'include', sure. And we can to this in a countable but unending way to include the algebraic numbers and beyond, without offend our host I think. :) I understand his issue with uncountable infinity more than his issue with countable infinite sets, as I understand the latter to me modelled finitely by generators (as in programming -- finite code that will generate the sequence 0, 1, 2, ... without end. At no time will the entire sequence have been generated (i.e. the generator will never be _exhausted_ ) but he generator stands as a finite, well defined, in short _real_ mathematical structure as any other.
Which "they" are you talking about? The rational numbers that can be expressed with current technology, or those that could potentially fit inside our universe when expressed in Hindu Arabic notation, or those that could fit inside our machines, or perhaps again the universe, if we allowed any expressions in Mathematica? Maybe they end with a whimper, not a bang.
@@biblebot3947 Please watch my videos on the Dihedrons. That will no doubt clarify that i is not really a problem at all if you are willing to extend your arithmetic to matrices. In that context, you can also find an algebraic solution to x^2=2. It just doesn't have the property that it is a "number" somewhere on the number line between 1.414 and 1.415. But that is what the analysts pretend. The important distinctions between algebraic and analytic approaches to sqrt(2) were clearly set out in my video on "Inconvenient truths about the square root of 2" which is Math Foundations 80 at th-cam.com/video/REeaT2mWj6Y/w-d-xo.html
@@njwildberger so everything except the location is good? Well, if the algorithm is correct in the digits it generates, then you can take any rational number and compare it with the results of the algorithm, and you’ll find a difference in a finite amount of time. This allows you to compare the rationals with the rationals.
No rational number can measure the diagonal of a unit square. However, you can make a cauchy sequence of rational numbers that get progressively better at approximating the true value. The limits of all such cauchy sequences are the real numbers! They do exist!
@Ethan Jensen No you cannot "make a Cauchy sequence that gets progressively better at approximating the true value (of "sqrt(2))", as you can find out yourself by trying to do it. Not just the first few terms and then three dots mind you. Have a go at making the first trillion terms and then the three dots; that will focus your mind on the difficulties.
that's pretty much what being incommensurable means I believe NW is against the idea of calling limits numbers, although you can devise an arithmetic for them relying on abstractions that go far beyond what numbers used to be considered for the vast majority of history - in that sense you could consider polynomials to be numbers just as easily as series or limits or Cauchy sequences
I guess you thought the 3 dots after the 8 terms you solved meant dot 3 considering you might have dyslexia, so you just manufactured an answer and poured some dot 3 brake fluid into your car. Very smart.
This inconsistency has always been in my mind, but Norman communicates it with such clarity that it gives it meaning. It seems to me that mathematics has two main procedures buried under the notion of numbers: algorithms for construction and deconstruction. Vaguely, construction is to build something out of basic building block, and deconstruction is to deconstruct something into basic building blocks. The results of these do not always coincide and the results are not always complete. Perhaps that should be the next frontier in math, researching what can or cannot be constructed/deconstructed. Thanks for the amazing videos!
I have just stumbled across this channel and had to sub, 13 years of content! I cant wait to devour it all.
I'm so glad to see someone address the distinction between quantities and (how I think of it) position. It irks me how some things in mathematics are labeled with numbers, such as elements of groups, despite having barely any shared properties. In a sense hiding various other properties which are cumbersome to express arithmetically, but intrinsic and fundamental to the actual objects.
Thank you for your diligence in scrutinizing your 'tools' as much as your 'work'.
Supporting your resistance to the embedding of various number systems within each other, there is a fairly strong modern consensus among practitioners of Object Oriented programming and software development to 'prefer composition rather than inheritance'.
Inheritance is the analog of embedding: Once we have developed Integers, the Naturals become 'just' a specialized type of Integer; once we have Reals, the Rationals become 'just' a special type of Real; etc.
Composition is (generally) less rigid, more flexible, and in some cases more powerful, and it is defined by the 'has a' relationship versus inheritance's 'is a' relationship: E.g. if we build up Integers as being *composed* of two Natural numbers, and the Integer is their difference (the internal 'less' operator), then you could say that each Integer 'has a' Natural number (actually, two of them, to be precise), rather than saying that suddenly Naturals have changed status and a Natural 'is an' Integer.
If the Reals actually functioned, then they might be able to be expressed in a similar way: E.g. if it were really possible to concretely define Dedekind cuts by specifying the 'gap' between two rationals (analogous to the 'less' of two Naturals in Integers), then perhaps it could be said that a Real 'has a' Rational (actually, two, again), rather than saying that Rationals have changed and now a Rational 'is a' certain kind of Real (i.e. there are Reals and the Rationals inherit from the Reals -- it is also conceivably possible to reverse the inheritance such that Reals inherit from Rationals and 'extend' them, but I'm just portraying the situation as orthodox modern mathematics would have us believe is the case).
However, something like Dedekind cuts (and Cauchy sequences, and all other current manifestations of Reals) require not just one 'has a' relationship, or two, or some finite number, but an infinite completion more like a 'has all possible' relationship, which is actually impossible to program. It's yet another way of thinking about the fundamental problem with Reals.
I have found your analogy very insightful!
Very lucid presentation of the foundational challenge between discrete and continuous.
My suggestion for the role of "unit segment/area/etc" is the expression , open interval defined as both more and less / both increases and decreases. To my astonishment taking relational operators as a formal language and concatenating mediants with the symbols according to simple procedures, leads to structures with obvious and natural Stern-Brocot type discrete interpretations, ie. a very natural and foundationally meaningful link form new(ish) language of continua to rational numbers. With lots of new and interesting structure to explore.
Note that 'concatenating mediants' is internal process of generating new rows with more fractional resolution, and thus inherently "revercimal" perspective of division/partition instead of additive perspective of discrete counting. Closely related to what are called p-adic numbers, but with this approach we connect directly with coprimes of Stern-Brocot type structures instead of prime numbers of p-adic formalism.
Video Content
00:00 Introduction
04:50 Origin of numbers: Counting and measuring
15:35 A bridge from counting to measuring requires a unit segment
17:19 Fractions as tools for measurement
25:03 Decimal,binary,sexagesimal PVS
It never really hit me that there was this 'great divide' between numeric systems until I started working at an electrical company and a friend their tried to explain the difficulty he had of trying to get the latest invention (a digital microprocessor - then cutting edge technology) to control an analogue system of motors. Imagine having a drum of tissue paper slowly unwinding at one end whilst at the other end a drum is winding the paper up. He likened it to a cassette tape reel (a common device at the time but extinct now). The problem he pointed out to me that the speed of the two drums were always different; The big drum unwinding had to go slower (at first) than the winding up drum until the the amount of paper balanced and then the winding up drum got bigger and it had to be slowed down whilst the winding down drum got faster. All the time the tension between these two motors had to be kept sufficient so that the tissue thin paper did not get ripped or tear in the middle. It blew my mind to discover that this connection between digital & analogue systems was so fragile & paper thin.
There is no electrical engineering without the fictitious sq2. In 3D, the math is absurdly large for the identity matrix of a solid, the Skin Effect in light as it renders in math to our eyes. The Lyman Alpha Line Split, .006 angstroms, and beta radiation is permeable to the eyes. Stick with this. e(c) is 1D error correction for matter-energy transfer, and at three decimals in base 10, 0.577. Split the difference right between the eyes, and get .003 plus 0.577, or 58% matter-energy transfer efficiency. We are molten salt nuclear fusion reactors.
Being a forever Mathematics student I am mesmerized by explanation of very basic concept of Mathematics in a way that it should be so Mathematics be more enjoyable. Thanks professor
Pray for your good health.
Love you sir.. From India.
Jay shree Ram...A indian slogan 🙏
I always enjoy your videos. Thank you so much as this one changed my viewpoint about mathematics once again.
Thank you so much for this lecture. Your presentation is simple and yet philosophical. I appreciate that.
Thanks for the video. I have got a feeling that equality (=) is intrinsic in the counting system, while comparison (< or >) is for the measuring system. I might be wrong, but the video let me think of "why" we count or measure (before indulging into the how side of the operation)
@Stephen Yes that is a very good point. The notion of equality is a problematic aspect with the Greek arithmetic of line segments. How do you actually tell when two segments are equal? Do you have to keep magnifying them endlessly to check that they never fail to line up?? With the counting system this is much easier. I think Eudoxus has something to say about that.
@@njwildberger I have been following your videos and your analysis about "realness" is very inspiring. For me, R looks like a philosophical substratum on which we can construct different types of number systems. It is just like "car" is a thing, "elephant" is a thing, but "thing" is not a "thing". (I have been naïve in describing my thoughts - laughable from set theory experts). I want to say this because I want to give R a position in my mind.
@@chjxb Haha 😄, that phrase '"thing" is not a "thing"' made me chuckle, but actually it's a pretty darn good point!
In computer programming, particularly in Object Oriented Programming, there is often the paradigm that 'everything is an 'object'', and in this case, the term 'object' has a precise definition in terms of being a certain kind of data structure combined with a certain set of operations (indeed the .equals() operation is usually one of them!).
But at the end of the day, an 'object' in object-oriented programming isn't actually an 'object' in the everyday sense of the word -- although, interestingly there *is* (usually) an 'object' data structure representing the 'class' of 'objects' from which all other 'classes' implicitly inherit; thus, all objects inherit the .equals() operation from the Object class, and you can subclass various other concepts like 'Car' or 'Elephant', and 'override' .equals() for your particular subclass, e.g. so that Elephant.equals() compares the names of the two elephants in question, or Car.equals() compares the serial numbers or the makes of the cars or the license plates, for instance.
Not sure how this relates to your desire to give R a position in your mind, but maybe it can help by giving some clues about how others have tried to tackle similar problems.
@@robharwood3538 Thanks. I do not fully understand your response but I appreciate your effort. Just like real number is a kind of substratum for us to build constructable number systems, the computing automaton itself is not a specific object. Does it make sense?
Thanks you Teacher NJ Wildberger.
@Gaurav You are most welcome.
Excellent video, as always. Have you ever done any vids on the methods used in the Rhind papyrus ? ( Regula falsi, etc )
Are there covectors and other tensors in the affine space, and what do they do?
I always come back and learn something new. You are the most awesome.
@brenda williams Thanks Brenda!
Thank you too Professor! I have felt myself so limited in the universe :D Looking forward to new video!!!
Bravo, my Go/Chess formulated as two entire number systems combined on a grid.
If we can step away from our conditioning, we may be able to arrive at a more complete understanding.
Math wll eventually change professor.You're heading in the right direction for years in my opinion but i think the last piece must be given from our understanding of our ability to make concepts.That's gonna end this confusion and justify 100% of what you 're saying all those years.Also on your point about proportions.Humans have built in a close relationship on recognise proportions for survival reasons.Biology backs you up on this prof.
Sir, thank you for your videos.
Sir, Do you have any plans to make videos on mathematics formula & how to write mathematics proof (I still don't get math proof) ?
How to bring our imagination into reality with the help of mathematics (how to think in mathematics) ?
Thank You 👍👍👍
How about defining 'the square root of two' as the ratio of the two segments, and 'is not rational' as 'the segments are incommensurable'?
But then you don't get a consistent arithmetic with the usual rational numbers, and certainly not with other "irrational numbers" in general. For example, my usual challenge: what is pi + e + sqrt(2) cannot be (easily) expressed or tackled from such a point of view. It is not enough to "define" the occasional "irrational number". What you want is a consistent arithmetic with these things that can be checked and is falsifiable. In other words, it should be so clear that the usual mumbling about "oh just take a point wise sum of representative Cauchy sequences" or some such becomes an obvious obfuscation.
@@njwildberger Fair enough, but that does not mean we cannot talk in general about geometric rations. 'the square root of to' has connotive baggage, but I see no reason why it is otherwise unsuited to denote the entity.
I agree that there are limits on what we can to with these entities (in comparison to the rations of commensurate intervals, which which we can do a lot), but we can say *some* things about them.
@Gennady Arshad Notowidigdo If by 'encapsulate' you mean what I more commonly hear spoken of as 'include', sure. And we can to this in a countable but unending way to include the algebraic numbers and beyond, without offend our host I think. :)
I understand his issue with uncountable infinity more than his issue with countable infinite sets, as I understand the latter to me modelled finitely by generators (as in programming -- finite code that will generate the sequence 0, 1, 2, ... without end. At no time will the entire sequence have been generated (i.e. the generator will never be _exhausted_ ) but he generator stands as a finite, well defined, in short _real_ mathematical structure as any other.
You do not like infinite sets? How about natural numbers where do they end?
Which "they" are you talking about? The rational numbers that can be expressed with current technology, or those that could potentially fit inside our universe when expressed in Hindu Arabic notation, or those that could fit inside our machines, or perhaps again the universe, if we allowed any expressions in Mathematica? Maybe they end with a whimper, not a bang.
And of course the square root of two can be expressed very easily as a dihedron.
If you’re willing to accept the incomputable sqrt(-1), then why not accept the incomputable sqrt(2)?
I’d argue that sqrt(-1) is a bigger leap of faith than sqrt(2) because the latter is (somewhat) computable.
@Gennady Arshad Notowidigdo but he rejects the real numbers.
He won’t use them, but he’ll gladly use i
@@biblebot3947 Please watch my videos on the Dihedrons. That will no doubt clarify that i is not really a problem at all if you are willing to extend your arithmetic to matrices. In that context, you can also find an algebraic solution to x^2=2. It just doesn't have the property that it is a "number" somewhere on the number line between 1.414 and 1.415. But that is what the analysts pretend. The important distinctions between algebraic and analytic approaches to sqrt(2) were clearly set out in my video on "Inconvenient truths about the square root of 2" which is Math Foundations 80 at th-cam.com/video/REeaT2mWj6Y/w-d-xo.html
@@njwildberger so everything except the location is good?
Well, if the algorithm is correct in the digits it generates, then you can take any rational number and compare it with the results of the algorithm, and you’ll find a difference in a finite amount of time.
This allows you to compare the rationals with the rationals.
@Gennady Arshad Notowidigdo
Proofs rarely, if ever, use explicit computations
The fact that there’s no exact answer to a computation is irrelevant
No rational number can measure the diagonal of a unit square. However, you can make a cauchy sequence of rational numbers that get progressively better at approximating the true value. The limits of all such cauchy sequences are the real numbers! They do exist!
@Ethan Jensen No you cannot "make a Cauchy sequence that gets progressively better at approximating the true value (of "sqrt(2))", as you can find out yourself by trying to do it. Not just the first few terms and then three dots mind you. Have a go at making the first trillion terms and then the three dots; that will focus your mind on the difficulties.
that's pretty much what being incommensurable means
I believe NW is against the idea of calling limits numbers, although you can devise an arithmetic for them relying on abstractions that go far beyond what numbers used to be considered for the vast majority of history - in that sense you could consider polynomials to be numbers just as easily as series or limits or Cauchy sequences
I guess you thought the 3 dots after the 8 terms you solved meant dot 3 considering you might have dyslexia, so you just manufactured an answer and poured some dot 3 brake fluid into your car. Very smart.