From a theoretical computational perspective on (possibly) infinite sequences we have possibilities: (1) The sequence terminates after N steps; (2) It is provable (finitely) that given any index number k, the k'th component can be computed; (3) It is not proven yet that for any k, the k'th component can be computed - the sequence might "stutter" with some elements uncomputable; (4) it is proven that the sequence is uncomputable. The square root algorithm is in category (2) is it not? With a finite set of computing resources, this will not be computable either. However it very nearly is. Assume that a finite amount C of computing resources are available, then computation is possible up to some N. This makes case (2) still different from case (1). So it is a different kind of number sequence. Give it a new name, if required but many of its properties will be familiar to the reals.
Indeed I cannot imagine the end of the sequence here. Neither the end of the numbers approximating _/2, nor the sequence of arguments between people regarding its reality. So interesting that the "imaginary" numbers are concrete geometrical units whose existence is obvious (and irritating to many people), while the "real numbers" are in fact completely imaginary in reality, but people are nonchalant about that! We have a very long way to go as monkeys striving for pure rational understanding of the world. Thanks for these contributions to the effort!
Here’s the root of the problem. Consider an arbitrary square of side a then by Pythagoras’ Theorem the diagonal b can be determined by b^2=a^2+a^2=2a^2⟹2=b^2/a^2 =(b/a)^2 Now although the Greeks expressed this in terms of fundamental measurable segments, rather than numbers, they realized there was a problem because if a,b are commensurable or in modern parlance the ratio is rational then suppose b/a is cancelled to its lowest terms then ∵b^2=2a^2⟹b^2&∴b are even ⟹ b=2c whence 4c^2=2a^2⟹a^2=2c^2⟹a^2&∴a are even ⟹a=2d contradicting the asssumption that b/a is already in its lowest terms! Now Prof. Wildberger insists(11:50) that this contradiction means that √2=b/a cannot exist as a number, but I think it only means it cannot exist as a rational or commensurable ratio. And if you call a non-commensurable ratio incommensurable, what is wrong with calling a non-rational number irrational? If you want to insist that non-rational or non-commensurable ratios do not exist then that is a choice but it is an unproven one and to simply declare that no non-rational ratio has ever been exhibited in our number system is not proof that one doesn’t exist. That would require more work. It is also belied by the fact that unit squares with diagonals do in fact appear to exist (but then in the light of what General Relativity tells us about space in the cosmos perhaps it is plane geometry that is illusory). Moreover, even if you restrict R to Q there is still a real number line on which √2 can be consistently pinched (15:48) between converging sequences of upper and lower bounds; Ln
Wow, this video is soooo amazing and clear. Thank you so much for this. I've been watching it over and over again. So hooked on math. I love it. Thank you!
real numbers: no defined objects, no defined operations, thus neither additive group, nor multiplicative group, thus no field. as said in all student's math books.
this is essentially because some mathematicians decided that infinity doesn't mean what it means (i.e. that somehow infinity meaning unending somehow can be "imagined" as if that weren't so). certainly many results would have something analogous and more insightful if formulated with rationals and other algebraic structures. alas they knowingly left for the land of axioms where one can assume anything is true without justification or consistency of meaning. shame they were an influential bunch (the set theorists essentially)
I love the ... argument! Everything is reduced to ... if you believe in the reals. It is on par with creationism. God created ... in its "infinite" wisdom, and we poor c...ts believe it and are in awe. In fact, we are in awe with generations of mathematicians' aweness. I just say this to it: ...
The square root of 2 is fundamentally no more or less valid of a number than 1/2 or 2 itself. You prove that the square root of 2 is not a rational number, and immediately make the logical jump that the square root of 2 is not a number at all, but the same can be done for fractions. 1/2 is not an integer, but does that mean it is not a number? Integers are the result of addition and subtraction, rationals are the result of multiplication and division, and radicals like the square root of 2 are the result of exponentiation. There is a logical progression from each of these operations to the next, why would you accept the first two but not the third? 1/2 is the number such that, when you multiply it by 2, you get 1, and sqrt(2) is the number such that, when you raise it to the second power, you get 2. These are fundamentally the same type of object: they are defined by mathematical relations to existing objects, but do not exist within the existing framework so we append them to it to create a more complete set of numbers. Also, you seem to make a big deal out of approximations and finite decimal expansions, but this is completely illogical. Of course fractions tend to be manageable in a decimal expansion, the decimal expansion is based on fractions! If you write that 1/2 = 0.5, you have only written shorthand notation for 1/2 = 5 * 1/10. You could easily do a similar expansion, except instead of using fractions you use the square roots, and suddenly square roots will have simple finite expansions while simple fractions will have ugly infinite expansions. In any case, there is no need for these expansions, simply writing sqrt(2) is a perfectly fine way to write the number exactly. The fact that it cannot be written as a finite sum of fractions is as insignificant as the fact that 1/2 can never be written as a finite sum of integers. PS: I am a physicist, and as such am more concerned with what works to describe real things, rather than arbitrary definitions. You are of course free to define the term number such that only integers and rationals are included, but such a tortured usage is simply not helpful and everyone else will continue to use the standard definition.
actually it's hugely relevent as we don't define what is number and in this case 1.41 dot dot dot is a nonsense. but in the "modern real maths" we don't define real number like this. we can define them ase the "limit of cauchy sequences" and because maybe a lot of different sequences will have the same "limit" our real number is the set of all sequences that converge towards this "number" and i forget to precise it will be rational sequences because we know how to define it from ordinal. (i know you dont like infinite ordinal but there is no need to use infinity there) and for rational it will be very easy. it's for exemple the constant sequence of terms q that defines the real q. and in a more precise way we can say that numbers is a class of equivalence of the relations : have the same limit so its for two sequence : Un-Vn converge towards 0 which is simple and not infinite. and then R is the quotient set of that relation. so then to make that reasonning wrong shiw me where is the incoherence. because i think in math as far as there is no experimentation the obly boundary is coherence. yes true class of equivalence of converging rationnal sequences isn't really our conception of number . but just let me say that if we want to deeply understand the number 2 we have to admit the ordinal {null, {null}}. anyway. we are mathematicians ot physicist we don't seek the reality or the deep truth behind things it's false. even nulbers only exist in our minds 2 banana our 2 apple is never the number 2 we invented 2 conceptually as we invented sqrt (2). as mathematicians we seek the coherence of our theory and their beauty or interest
This sounds like you don't understand the idea of infinite numbers and the different degree's of infinity. Eric Cantor described this. Either that or you just don't want to make that presumption with your god approach. Infinity is what gets us to the exact answer of the square root of 2. Can we really exactly calculate it. No we cannot but that is because we can't do infinity in the practical everyday sense. This doesn't mean however that infinity is a concept that we can't understand and use. If it wasn't then Zeno's paradox wouldn't be a paradox and we wouldn't be able to travel any distance at all. As both a physicist and a mathematician, I disagree with your statements because they do not get to the heart of the mathematics. What is to tell us that we can't continue toward infinity. Just because it is not practical doesn't mean that we can't do it. This was discussed by Cantor in his discussions of his degree's of infinity. By suggesting this you are going against many far more brilliant mathematicians including Riemann and Cantor which are mathematicians i respect immensely. Your ideas hold no actual use because they do not help us understand mathematics and they do not actually make any sense. Sorry I thought your videos would enlighten me but they confirmed my professor in what he was saying about similar topics about infinity and analysis. your video will misinform the public. if anyone has any questions please ask and i'll ask my professor about this and get his thoughts.
I have thought about it, but I don't have any problems with infinity and I don't have problems thinking about infinity or finding ways to deal with it. I have read papers by Cantor, Godel and others. I will admit i'm not the best mathematician right here and now, and there are people far smarter than I am who can better answer questions, i'm humble enough to know that. I am primarily a physicist and a mathematician second and though I love mathematics I wouldn't presume to know more than minds that have spend more time on said problems than I have. There is a reason why their ideas work. A reason why their ideas are not just useful in physics and engineering but computer science, finance, biology and many other fields. Besides who is to say what we can and cannot do in mathematics? As long as it is consistant with our axioms of mathematics then it is fine. If it isn't then we can either reject it or we can deal with it and come up with useful definitions. Its how we came to deal with infinity, complex numbers, abstract spaces, non-euclidean geometries and fractal dimensions. These ideas were rejected and strange at first but eventually they were accepted because they are logical and they fit within our axioms. Sqrt(2) is the same way same with pi and e just because we can't get to the end doesn't mean they aren't real. pi is very real otherwise we wouldn't be able to draw a circle, sqrt(2) is real because otherwise the simplest right triangle no longer can be drawn. Yet these things can be. Just because our piddly decimal notation can't draw all decimal values doesn't mean they aren't real objects.
+CrownedFalcon00 If you keep watching these lectures, I hope to turn your thinking around to a quite different direction. You don't need to be so humble. You can think clearly. You can proceed step by step. Brilliance is not needed, and perhaps not even desirable. With this kind of orientation, you don't need to take anything for granted, no matter how esteemed the person who makes a claim. If they can explain their mathematics from first principles in a way that you can clearly understand what they mean, then good. If they can support their notions with clear examples and concrete explicit computations, then good. Otherwise, it is their problem, not yours. I aspire to offer a mathematics that is free from authority. Don't believe what I am saying because you think I have achieved a certain level of smartness, or status. Rather just follow it carefully and see if it stands on its own two legs. Most of modern mathematics does not stand on its own two legs.
Great video. Wish the analysis class I took a few years ago was this clear.
20:00 - is this mention about computers related to the floating point inacurracy in code?
🥞Excellent content!! 🔥🔥🔥
From a theoretical computational perspective on (possibly) infinite sequences we have possibilities:
(1) The sequence terminates after N steps;
(2) It is provable (finitely) that given any index number k, the k'th component can be computed;
(3) It is not proven yet that for any k, the k'th component can be computed - the sequence might "stutter" with some elements uncomputable;
(4) it is proven that the sequence is uncomputable.
The square root algorithm is in category (2) is it not? With a finite set of computing resources, this will not be computable either. However it very nearly is. Assume that a finite amount C of computing resources are available, then computation is possible up to some N. This makes case (2) still different from case (1). So it is a different kind of number sequence. Give it a new name, if required but many of its properties will be familiar to the reals.
Thank you Professor.
Indeed I cannot imagine the end of the sequence here. Neither the end of the numbers approximating _/2, nor the sequence of arguments between people regarding its reality. So interesting that the "imaginary" numbers are concrete geometrical units whose existence is obvious (and irritating to many people), while the "real numbers" are in fact completely imaginary in reality, but people are nonchalant about that!
We have a very long way to go as monkeys striving for pure rational understanding of the world. Thanks for these contributions to the effort!
This is way too awesome
Here’s the root of the problem. Consider an arbitrary square of side a then by Pythagoras’ Theorem the diagonal b can be determined by b^2=a^2+a^2=2a^2⟹2=b^2/a^2 =(b/a)^2 Now although the Greeks expressed this in terms of fundamental measurable segments, rather than numbers, they realized there was a problem because if a,b are commensurable or in modern parlance the ratio is rational then suppose b/a is cancelled to its lowest terms then ∵b^2=2a^2⟹b^2&∴b are even ⟹ b=2c whence 4c^2=2a^2⟹a^2=2c^2⟹a^2&∴a are even ⟹a=2d contradicting the asssumption that b/a is already in its lowest terms!
Now Prof. Wildberger insists(11:50) that this contradiction means that √2=b/a cannot exist as a number, but I think it only means it cannot exist as a rational or commensurable ratio. And if you call a non-commensurable ratio incommensurable, what is wrong with calling a non-rational number irrational? If you want to insist that non-rational or non-commensurable ratios do not exist then that is a choice but it is an unproven one and to simply declare that no non-rational ratio has ever been exhibited in our number system is not proof that one doesn’t exist. That would require more work. It is also belied by the fact that unit squares with diagonals do in fact appear to exist (but then in the light of what General Relativity tells us about space in the cosmos perhaps it is plane geometry that is illusory). Moreover, even if you restrict R to Q there is still a real number line on which √2 can be consistently pinched (15:48) between converging sequences of upper and lower bounds; Ln
Thanks for clarifying that. As I said, I have virtually no analysis experience, so I appreciate this!
Thank you for your work, Professor Wildberger.
Thank you sir for your wonderful lecture
I don't know why some people are disliked this wonderful speech
Wow, this video is soooo amazing and clear. Thank you so much for this. I've been watching it over and over again. So hooked on math. I love it. Thank you!
real numbers: no defined objects, no defined operations, thus neither additive group, nor multiplicative group, thus no field. as said in all student's math books.
@Inalambrico That captures a big aspect of the sad state of modern pure mathematics pretty well.
this is essentially because some mathematicians decided that infinity doesn't mean what it means (i.e. that somehow infinity meaning unending somehow can be "imagined" as if that weren't so). certainly many results would have something analogous and more insightful if formulated with rationals and other algebraic structures. alas they knowingly left for the land of axioms where one can assume anything is true without justification or consistency of meaning. shame they were an influential bunch (the set theorists essentially)
I love the ... argument! Everything is reduced to ... if you believe in the reals. It is on par with creationism. God created ... in its "infinite" wisdom, and we poor c...ts believe it and are in awe. In fact, we are in awe with generations of mathematicians' aweness. I just say this to it: ...
This is amazing :)
It looks like baloney to me.
The square root of 2 is fundamentally no more or less valid of a number than 1/2 or 2 itself. You prove that the square root of 2 is not a rational number, and immediately make the logical jump that the square root of 2 is not a number at all, but the same can be done for fractions. 1/2 is not an integer, but does that mean it is not a number? Integers are the result of addition and subtraction, rationals are the result of multiplication and division, and radicals like the square root of 2 are the result of exponentiation. There is a logical progression from each of these operations to the next, why would you accept the first two but not the third?
1/2 is the number such that, when you multiply it by 2, you get 1, and sqrt(2) is the number such that, when you raise it to the second power, you get 2. These are fundamentally the same type of object: they are defined by mathematical relations to existing objects, but do not exist within the existing framework so we append them to it to create a more complete set of numbers.
Also, you seem to make a big deal out of approximations and finite decimal expansions, but this is completely illogical. Of course fractions tend to be manageable in a decimal expansion, the decimal expansion is based on fractions! If you write that 1/2 = 0.5, you have only written shorthand notation for 1/2 = 5 * 1/10. You could easily do a similar expansion, except instead of using fractions you use the square roots, and suddenly square roots will have simple finite expansions while simple fractions will have ugly infinite expansions. In any case, there is no need for these expansions, simply writing sqrt(2) is a perfectly fine way to write the number exactly. The fact that it cannot be written as a finite sum of fractions is as insignificant as the fact that 1/2 can never be written as a finite sum of integers.
PS: I am a physicist, and as such am more concerned with what works to describe real things, rather than arbitrary definitions. You are of course free to define the term number such that only integers and rationals are included, but such a tortured usage is simply not helpful and everyone else will continue to use the standard definition.
actually it's hugely relevent as we don't define what is number and in this case 1.41 dot dot dot is a nonsense. but in the "modern real maths" we don't define real number like this. we can define them ase the "limit of cauchy sequences" and because maybe a lot of different sequences will have the same "limit" our real number is the set of all sequences that converge towards this "number" and i forget to precise it will be rational sequences because we know how to define it from ordinal. (i know you dont like infinite ordinal but there is no need to use infinity there) and for rational it will be very easy. it's for exemple the constant sequence of terms q that defines the real q.
and in a more precise way we can say that numbers is a class of equivalence of the relations : have the same limit so its for two sequence : Un-Vn converge towards 0 which is simple and not infinite. and then R is the quotient set of that relation. so then to make that reasonning wrong shiw me where is the incoherence. because i think in math as far as there is no experimentation the obly boundary is coherence.
yes true class of equivalence of converging rationnal sequences isn't really our conception of number . but just let me say that if we want to deeply understand the number 2 we have to admit the ordinal {null, {null}}.
anyway. we are mathematicians ot physicist we don't seek the reality or the deep truth behind things it's false. even nulbers only exist in our minds 2 banana our 2 apple is never the number 2 we invented 2 conceptually as we invented sqrt (2). as mathematicians we seek the coherence of our theory and their beauty or interest
This sounds like you don't understand the idea of infinite numbers and the different degree's of infinity. Eric Cantor described this. Either that or you just don't want to make that presumption with your god approach. Infinity is what gets us to the exact answer of the square root of 2. Can we really exactly calculate it. No we cannot but that is because we can't do infinity in the practical everyday sense. This doesn't mean however that infinity is a concept that we can't understand and use. If it wasn't then Zeno's paradox wouldn't be a paradox and we wouldn't be able to travel any distance at all.
As both a physicist and a mathematician, I disagree with your statements because they do not get to the heart of the mathematics. What is to tell us that we can't continue toward infinity. Just because it is not practical doesn't mean that we can't do it. This was discussed by Cantor in his discussions of his degree's of infinity. By suggesting this you are going against many far more brilliant mathematicians including Riemann and Cantor which are mathematicians i respect immensely. Your ideas hold no actual use because they do not help us understand mathematics and they do not actually make any sense. Sorry I thought your videos would enlighten me but they confirmed my professor in what he was saying about similar topics about infinity and analysis. your video will misinform the public. if anyone has any questions please ask and i'll ask my professor about this and get his thoughts.
+CrownedFalcon00 Or you could think about it yourself for a few weeks.
I have thought about it, but I don't have any problems with infinity and I don't have problems thinking about infinity or finding ways to deal with it. I have read papers by Cantor, Godel and others. I will admit i'm not the best mathematician right here and now, and there are people far smarter than I am who can better answer questions, i'm humble enough to know that. I am primarily a physicist and a mathematician second and though I love mathematics I wouldn't presume to know more than minds that have spend more time on said problems than I have. There is a reason why their ideas work. A reason why their ideas are not just useful in physics and engineering but computer science, finance, biology and many other fields. Besides who is to say what we can and cannot do in mathematics? As long as it is consistant with our axioms of mathematics then it is fine. If it isn't then we can either reject it or we can deal with it and come up with useful definitions. Its how we came to deal with infinity, complex numbers, abstract spaces, non-euclidean geometries and fractal dimensions. These ideas were rejected and strange at first but eventually they were accepted because they are logical and they fit within our axioms. Sqrt(2) is the same way same with pi and e just because we can't get to the end doesn't mean they aren't real. pi is very real otherwise we wouldn't be able to draw a circle, sqrt(2) is real because otherwise the simplest right triangle no longer can be drawn. Yet these things can be. Just because our piddly decimal notation can't draw all decimal values doesn't mean they aren't real objects.
+CrownedFalcon00 If you keep watching these lectures, I hope to turn your thinking around to a quite different direction. You don't need to be so humble. You can think clearly. You can proceed step by step. Brilliance is not needed, and perhaps not even desirable.
With this kind of orientation, you don't need to take anything for granted, no matter how esteemed the person who makes a claim. If they can explain their mathematics from first principles in a way that you can clearly understand what they mean, then good. If they can support their notions with clear examples and concrete explicit computations, then good. Otherwise, it is their problem, not yours.
I aspire to offer a mathematics that is free from authority. Don't believe what I am saying because you think I have achieved a certain level of smartness, or status. Rather just follow it carefully and see if it stands on its own two legs. Most of modern mathematics does not stand on its own two legs.