This guy has a deep and powerful understanding of mathematics and physics. I am basing this not just on this lecture, but others. I am just posting it here. Thanks for making these gems available in the public domain. I usually fall half asleep when I watch other lecturers, but this guy keeps me awake because there's so much food for thought here. Philosophically as well as mathematically/physically! No wonder Perimeter Institute hired him.
I've always admired people who can explain complicated and abstract ideas with easy and great clarity of thought - and this lecturer is definitely a person to be admired for such traits.
This guy is an exceptional lecturer. The way he seamlessly goes from the technical to the context of the subject is something most lecturers can't seem to do. All the while letting the student know where he is. The start where he outlines the layers of mathematics/logic and how the physics relies on these, the Venn diagram of algebra, analysis and geometry and DG at the centre. and the bit where he talks about how it's important to know "what we are not talking about" when explaining the fundamentals of predicate logic without getting lost in the details of the example X as an element of Y since we haven't defined "element of" yet. Exceptional teaching.
For clarity, between 1:07:10 and 1:12:25, the assumption (M) should be q_j can be written as the j'th step if and only if there is m,n such that for 1≤m,nq_j is true. For example, assume P and P=>Q are axioms. Then, a valid proof that Q is true is as follows: (1) P (A) (2) P=>Q (A) (3) Q (M). Remark: (P^P=>Q)=>Q is a tautology which allows us to invoke (M) at stage j=3.
Ok I just want to say that I am forever indebted to the amazing Mr. Schuller, who has practically and unwittingly taught me all the basic higher mathematics that I need! 😢
A clean class of computer science. Logic is exactly circuit theory among many other things. The beauty of it and importance deserves a permanent place in our hearts and the internet.
Wonderfully structured. You keep the audience engaged, you go at a pace that does not tire the student but keeps them glued to the blackboard. And you go to the deepest corners and leave no aspect uncovered.
After finishing these lectures, you can go through the freely available book "Differential Geometry Reconstructed" which I think is a good follow up and comprehensive.
1:13:09 "a computer can quickly verify a proof by this definition" This is almost true, except for the fact that you can insert any tautology as a valid step in the proof. The problem of recognizing whether a given proposition is a tautology is coNP-complete, meaning we don't know how to do it in a way that scales efficiently with the number of terms in the proposition (and it's commonly believed no such way exists)
Randomly suggested by youtube and such a fantastic approach to the theme discovered .... Unbelievable!!! Thank you for upload this classes :) Great job
This is the best lecture on the subject that I ever watch. Amazing knowledge of math and Physics. I am very excited to see the other lectures. Thank you so much.
Hello Frederic Schuller, thank you very much for making this public. This lecture series is extraordinary clear and really excellent for a self-study and an overview (personally being a computer musician who slowly is studying more and more mathematics) Is it possible to get to know which textbook you are using? And is it possible to access the problem-sets somewhere on the web? Vilbjørg Broch
sometimes a person can be an excellent professor or an excellent scientist but this guy is both. In my life I had the privilege to watch a professor like that and I thank God for this.
Dear Dr Schuller, Brilliantly delivered lectures, great clarity of thought. Beautifully presented. I just wanted to know if there's a website for this course taught by you. It will be great to have access to the problem sets that you mention in some of the lectures. This will reinforce our understanding of the material. Also, is there a particular textbook you are following? Thanks.
Mr. Schuller, these lectures are just amazing. Just wow. Thank you so much for sharing it, I wish I could attend your lectures. I am especially amazed how well you have explained the role of different math branches for understanding contemporary physics. I was looking for it for quite some time now. Thank you very much.
The first time I heard of "Modus Ponens" the computer science teacher said "All men are human, Peter is a man, therefore Peter is human". Theses lectures are great, thanks for sharing !
Hands down for this amazing introduction to everything. If I was in the class, I will applaud but I guess the students were super confused. I watched this lecture in 4 parts so I had enough time to digest it. Thank you so much. You've been an amazing teacher to me, Dr. Schuller!
Wonderful lecture. For those wondering, he does describe a consistent axiomatic system correctly but the definition he stated was for an incomplete axiomatic system.
All the lectures of you, are brilliant! It very rigorously clears ideas of mathematics and how it is used to interprete the dynamics and ontological causal-structure of our universe. I am very grateful that these lecture series are open for all in such this public platform. Please keep posted Sir. Thank You!
Dear professor, you demonstrated that " s ∆ ~s => q" is a tautology, while discussing the idea of the definition of "consistency" of axiomatic system (at 1:20:00 approx. in the video). A tautology should be " s v ~s => q", isn't it? Please help me understand this.
Must watch lecture if you are into any level of mathematical logic and believe me he identifies the core principals of math logic in a precise manner. I could finally know how a proof was structured and finally didn't need to memorize any way to logic, but used logic itself to identify the content itself.
The Role Model Academic Lecturer! I have seen most of your videos on UA-cam, I appreciate the rigour also as a mathematician...Thank you and very best wishes!
There was a lot of information to process in each lecture but I haven't lost my interest neither for a second. That also holds for the lectures in Quantum Theory and the ones given in the International winter school on gravity and light. An amazing lecturer. Thank you very much for your effort. It would be very helpful if you may upload lectures also in QFT course for instance, with this kind of mathematical clarity. Is there any way that we might get the problem sheets?
39:02 That's the only part which appears to be not as concise as the rest, because you would want to be able to explain (or give an intuition for) this without using set theory (or really anything you build on top of logic). The reason why we don't say what x and y are is because we want to study very general properties that should not rely on the specific structure we will assign to them later (e.g. being a set). Not specifiying them should therefore be considered a strength of our approach (because it will work whatever they might be) and not as a weakness (implying that we don't know what we are talking about).
Hi Dr. Schuller, I am afraid I have to object that contraposition implies proof by contradiction at 31:00. The basis of proof by contradiction is p || ~p, i.e. the law of excluded middle, or LEM in short. So the reason is, if a statement is tautology, then it's negation is false; so proving negation being false proves the proposition itself being tautology. On the other hand, in other logic, i.e. those non-classical logic which refuse LEM, admits contraposition. Contraposition is admissible by axioms, while LEM is required to be an axiom. If it's not, or its equivalence is not, then it's simply unusable. For example, in constructive logic, the proof of contraposition goes following: (~q -> ~p) ((q -> False) -> p -> False) p -> q by discharging (q -> False) into False.
Thanks for this (second-youtubed) amazing course. So much appreciated! Constructive logic allows you to assume that A exists and reach a contradiction and by this prove that A cannot exist. But, it disallows you to assume that A not exists and reach a contradiction and by this conclude that A exists.
37:56 What is a function however? At the starting we dicussed, set theory is built upon logic, then how are we using a concept from set theory (that of functions) in logic?
Watching this really makes me miss in person lectures. Sitting there and watching the board slowly fill up with proofs and implications was a bit much at times, but this online learning just doesn't hold a candle to face to face.
1:21:07 Why define consistency in terms of there existing a q which cannot be proven? Wouldn't it be simpler to define it directly in terms of there not existing any 2 axioms which are the negations of each other?
Very nice. I am working through How To Prove It by Velleman and Sets For Mathematics by Lawvere. Found my way here by way of "The Portal" Discord group. Very helpful videos to supplement that work. This is my starting point on the long road to understanding differential geometry, the mathematical language of physics.
1:26:30 Another way to put it is: "An axiomatic system is consistent if from the axioms cannot be proven a formula and the negation of the formula. (Cannot be proven that )"
These are absolutely brilliant. Very clear exposition. Only thing...I would rather use 1 and 0 instead of t and f as they appear nearly the same on the board...
1:17:00 Can someone clarify? There's something quite fishy about the claim that, the "axiomatic system for propositional logic is the empty sequence". Although I'm no big expert on logic, this sounds almost certainly incorrect. Propositional logic has its own set of axioms, and rather, many, alternative ones-each producing the same propositional calculus. E.g. en.wikipedia.org/wiki/List_of_Hilbert_systems lists some of these axioms for propositional logic. Am I missing some nuance that differentiates what Prof. Schuller is saying, and what I'm juxtaposing it with?
His first two axioms of set theory (on Element relation and Empty set) are different from two of the ones on Wikipedia Feb 2023 (Extensionality and Restricted comprehension). Restricted comprehension follows from the Empty set and Replacement axioms. Am I correct in saying that his axiom on the Element relation "x\in y is a proposition iff x and y are both sets" follows from Extensionality or some other combination of axioms? If not, then this would be mildly disturbing since both he and Wikipedia claim to axiomatize the same system (ZFC). (Side note: EE PURP IC F is not ruined by replacing the axiom on the Element relation with Extensionality, both of which begin with an "E.")
For anyone interested in the Corollary to the theorem at about 31:23, really it should be that we can prove things by using the contrapositive. Contradiction is related, but slightly different. Here is a nice answer from math stack exchange math.stackexchange.com/questions/262828/proof-by-contradiction-vs-prove-the-contrapositive. Very good lectures though, and intending to keep watching!
glad to hear that there are professional mathematicians that dont trust the proofs by contradiction. I am no mathematician, just an enthusiast, but this kind of proof always feel sketchy to me :)
If you master the subject there is no need to follow a textbook. But it would be great if F. Schuller would write a textbook. All his lectures are really awesome.
The statement about the barber is neither true nor false, so according to the definition of a proposition given at the beginning of the video, namely that a proposition is something that is either true or false, wouldn't that just imply that the statement given was not considered a proposition? Then the question about whether or not it could be proved would not even arise.
صحيح اتابع من البيت بالرغم من تخرجي من الجامعه منذ ١٠ سنوات واكتب معاه واتابع كل المحاضرات وشريت دفتر خاص للمحاضره . Thanks I watched these lecturers frome my home in Saudi arabia i graduates frome university since10 yars
Could someone help me prove the statement about hairdressers mentioned around 50:00. What a hairdresser can do only a hairdresser can do implies that a hairdresser can't do anything that a non-hairdresser can do.
Take any action that a non-hairdresser can take as a counterexample. Assume that there is a non-hairdressing action that the hairdresser can take, and call it "fishing". From the statement "what a hairdresser can do, only a hairdresser can do", it follows that because our hairdresser can fish, a non-hairdresser cannot fish. In particular, the fisherman cannot fish. This is a contradiction, proving that our assumption about our hairdresser's capabilities was flawed.
Thank you, Frederic, for the lectures, these and the many others. Regarding this one: I think it is better to start with sets, because {T,F} is a set, the concept of a variable needs the set, quantifiers need the set notion, and so on. So first elementary set theory, then logic, then more advanced set theory, spaces, and so on,...
Actually, the real logic doesn't need any sets. It doesn't use {T, F} set, just deductive rules which describe how to get any tautology. There are no real variables in there, just formal symbols. And why do quantifiers need sets?
Schuller you are a great mathematics teacher, I think you should write a book I would think it'd have the possibility to become a classic, something that focuses on the logical, set axiomatic, and proof theory aspects of fundamental mathematics with are I think some of the hardest concepts for beginning math students, with the most room for improvement in the current literature in structural writing and exposition. Thank you for the videos, very useful and well done.
Someone has compiled notes to these lectures, which are quite good and available online. I agree with you; he seems to construct a very elegant 'big picture' of concepts and relations between them. Even though I'm not a beginning Maths student at all anymore, I still find some fresh and pleasing ways to think about certain concepts in Prof. Schuller's lectures.
29:00 is proof by contraposition not contradiction (everybody mixes this up). Beautifully presented lectures, thanks! en.wikipedia.org/wiki/Proof_by_contrapositive
I know I'm replying 3 years late, but for anyone looking for an elementary understanding of Gödel's incompleteness theorem, there is a great book by the name Gödel's Proof by Nagel and Newman. I'd recommend going for the revised edition as there is an excellent foreword and a few corrections/clarifications to the original by Douglas Hofstadter.
This guy has a deep and powerful understanding of mathematics and physics. I am basing this not just on this lecture, but others. I am just posting it here. Thanks for making these gems available in the public domain. I usually fall half asleep when I watch other lecturers, but this guy keeps me awake because there's so much food for thought here. Philosophically as well as mathematically/physically! No wonder Perimeter Institute hired him.
how ca I find the problem sheet for lecture 8 Tensor Theory? Anyone?
@@drlangattx3dotnet did you find it?
@@drlangattx3dotnet were you able to find problem sets for the other lectures?
@@michealmclaughlin429 did not find it. Can you help please?
indeed --- indeed -- indeed
The internet needed this lecture. Thank you.
Yeah i really enjoyed it
I've always admired people who can explain complicated and abstract ideas with easy and great clarity of thought - and this lecturer is definitely a person to be admired for such traits.
This guy is an exceptional lecturer. The way he seamlessly goes from the technical to the context of the subject is something most lecturers can't seem to do. All the while letting the student know where he is. The start where he outlines the layers of mathematics/logic and how the physics relies on these, the Venn diagram of algebra, analysis and geometry and DG at the centre. and the bit where he talks about how it's important to know "what we are not talking about" when explaining the fundamentals of predicate logic without getting lost in the details of the example X as an element of Y since we haven't defined "element of" yet. Exceptional teaching.
For clarity, between 1:07:10 and 1:12:25, the assumption (M) should be q_j can be written as the j'th step if and only if there is m,n such that for 1≤m,nq_j is true.
For example, assume P and P=>Q are axioms.
Then, a valid proof that Q is true is as follows:
(1) P (A)
(2) P=>Q (A)
(3) Q (M).
Remark: (P^P=>Q)=>Q is a tautology which allows us to invoke (M) at stage j=3.
Ok I just want to say that I am forever indebted to the amazing Mr. Schuller, who has practically and unwittingly taught me all the basic higher mathematics that I need! 😢
A clean class of computer science. Logic is exactly circuit theory among many other things. The beauty of it and importance deserves a permanent place in our hearts and the internet.
This is the true act of love and compassion, Thank you!
Wonderfully structured. You keep the audience engaged, you go at a pace that does not tire the student but keeps them glued to the blackboard. And you go to the deepest corners and leave no aspect uncovered.
Let it be known
Frederic is a King above all!!!
We appreciate you!
After finishing these lectures, you can go through the freely available book "Differential Geometry Reconstructed" which I think is a good follow up and comprehensive.
By whom?
@@noditschi Alan U. Kennington, freely available online
@@burakcopur3841 thanks
a very hard book to read
@@antoniomantovani3147 he did say it's a follow up after finishing all these lectures
1:13:09 "a computer can quickly verify a proof by this definition"
This is almost true, except for the fact that you can insert any tautology as a valid step in the proof. The problem of recognizing whether a given proposition is a tautology is coNP-complete, meaning we don't know how to do it in a way that scales efficiently with the number of terms in the proposition (and it's commonly believed no such way exists)
Then you have the contradiction that a tautological one form is not a tautology
Amazing!, not just the content but the way he delivers it with such calmness and clarity, incredible!
This series of Lectures is pure pleasure! Sometimes I come back here just to be amazed again. Thank you!
Randomly suggested by youtube and such a fantastic approach to the theme discovered ....
Unbelievable!!! Thank you for upload this classes :)
Great job
This is the best lecture on the subject that I ever watch. Amazing knowledge of math and Physics. I am very excited to see the other lectures. Thank you so much.
Absolutely brilliant, breath-taking and addictive!
If I cloud only develop such addictions!
Great and clear content. We need more of these guys on the web.
This is amazing sprinkled with a great sense of humor, thanks for sharing!
Hello Frederic Schuller, thank you very much for making this public. This lecture series is extraordinary clear and really excellent for a self-study and an overview (personally being a computer musician who slowly is studying more and more mathematics) Is it possible to get to know which textbook you are using? And is it possible to access the problem-sets somewhere on the web?
Vilbjørg Broch
Did you find out the book they are using?
drive.google.com/file/d/1nchF1fRGSY3R3rP1QmjUg7fe28tAS428/view
@@ArponPaul Thanks for the class note with so many details. But it's not the book Prof. Schuller is using. Do you know the book he is using?
@@vinbo2232 I do not know about the textbook. I will let you know if I can get any information.
@@ArponPaul Thanks
sometimes a person can be an excellent professor or an excellent scientist but this guy is both. In my life I had the privilege to watch a professor like that and I thank God for this.
Dear Dr Schuller,
Brilliantly delivered lectures, great clarity of thought. Beautifully presented. I just wanted to know if there's a website for this course taught by you. It will be great to have access to the problem sets that you mention in some of the lectures. This will reinforce our understanding of the material. Also, is there a particular textbook you are following?
Thanks.
+007bibhuti
+1
I'd love to have access to problem sets.
+007bibhuti +1
Remarkable lectures !!!I I wish we had some problem sets to solve , so that we could test our concepts.
Mr. Schuller, these lectures are just amazing. Just wow.
Thank you so much for sharing it, I wish I could attend your lectures.
I am especially amazed how well you have explained the role of different math branches for understanding contemporary physics. I was looking for it for quite some time now.
Thank you very much.
The first time I heard of "Modus Ponens" the computer science teacher said "All men are human, Peter is a man, therefore Peter is human". Theses lectures are great, thanks for sharing !
Hands down for this amazing introduction to everything. If I was in the class, I will applaud but I guess the students were super confused. I watched this lecture in 4 parts so I had enough time to digest it. Thank you so much. You've been an amazing teacher to me, Dr. Schuller!
Amazing! He explains things very well. I can understand more than 90% materials. See you guys at the final lecture.
28:28, 35:20, 49:56, 1:07:25, 1:10:40, 1:22:00, 1:24:08, 1:28:25
This is pedagogy; not whatever it is my mathematics professor was attempting to do.Thank you, Prof Schuller. ❤
You are a great teacher. Thank you for sharing these lectures.
wow...the first 2 minutes and I am completely captured. The interpretation and reflection on those two quotes by Wittgenstein
Amazing first lecture!
I will try to follow the next ones until my levels allows me. Greetings from Colombia.
How is school there?
Wonderful lecture. For those wondering, he does describe a consistent axiomatic system correctly but the definition he stated was for an incomplete axiomatic system.
He is clear and concise which in turn enables learning. Love it
6:17 why is Statistical Physics at the intersection of geometry and algebra? I thought statistics is more about analysis?
Thank you Dr Schuller for uploading this lecture! Hope to see many more!
One of the best first 15min I've ever watched!!
All the lectures of you, are brilliant! It very rigorously clears ideas of mathematics and how it is used to interprete the dynamics and ontological causal-structure of our universe. I am very grateful that these lecture series are open for all in such this public platform. Please keep posted Sir. Thank You!
Dear professor, you demonstrated that " s ∆ ~s => q" is a tautology, while discussing the idea of the definition of "consistency" of axiomatic system (at 1:20:00 approx. in the video). A tautology should be " s v ~s => q", isn't it? Please help me understand this.
Must watch lecture if you are into any level of mathematical logic and believe me he identifies the core principals of math logic in a precise manner. I could finally know how a proof was structured and finally didn't need to memorize any way to logic, but used logic itself to identify the content itself.
The Role Model Academic Lecturer! I have seen most of your videos on UA-cam, I appreciate the rigour also as a mathematician...Thank you and very best wishes!
is there anybody having access to the problem sheets?
There was a lot of information to process in each lecture but I haven't lost my interest neither for a second. That also holds for the lectures in Quantum Theory and the ones given in the International winter school on gravity and light. An amazing lecturer. Thank you very much for your effort.
It would be very helpful if you may upload lectures also in QFT course for instance, with this kind of mathematical clarity.
Is there any way that we might get the problem sheets?
39:02 That's the only part which appears to be not as concise as the rest, because you would want to be able to explain (or give an intuition for) this without using set theory (or really anything you build on top of logic). The reason why we don't say what x and y are is because we want to study very general properties that should not rely on the specific structure we will assign to them later (e.g. being a set). Not specifiying them should therefore be considered a strength of our approach (because it will work whatever they might be) and not as a weakness (implying that we don't know what we are talking about).
Hi Dr. Schuller, I am afraid I have to object that contraposition implies proof by contradiction at 31:00. The basis of proof by contradiction is p || ~p, i.e. the law of excluded middle, or LEM in short. So the reason is, if a statement is tautology, then it's negation is false; so proving negation being false proves the proposition itself being tautology.
On the other hand, in other logic, i.e. those non-classical logic which refuse LEM, admits contraposition. Contraposition is admissible by axioms, while LEM is required to be an axiom. If it's not, or its equivalence is not, then it's simply unusable.
For example, in constructive logic, the proof of contraposition goes following: (~q -> ~p) ((q -> False) -> p -> False) p -> q by discharging (q -> False) into False.
I wish I discovered this series earlier, so so good!
yes, I wish I could have watched this about 30 years ago.
this guy is genius
Is there a lecture notes or reference book to this course ? Thank you very much.
Thanks for this (second-youtubed) amazing course. So much appreciated!
Constructive logic allows you to assume that A exists and reach a contradiction and by this prove that A cannot exist.
But, it disallows you to assume that A not exists and reach a contradiction and by this conclude that A exists.
The way he writes t & f makes him always correct.
I searched up a German name I made up, and not only does he exist, he teaches what I needed. Badabingbadaboom
37:56 What is a function however? At the starting we dicussed, set theory is built upon logic, then how are we using a concept from set theory (that of functions) in logic?
Watching this really makes me miss in person lectures. Sitting there and watching the board slowly fill up with proofs and implications was a bit much at times, but this online learning just doesn't hold a candle to face to face.
Finally, a mathematician who introduces the subject the correct way, via a wider philosophical picture. Excellent!
1:21:07 Why define consistency in terms of there existing a q which cannot be proven? Wouldn't it be simpler to define it directly in terms of there not existing any 2 axioms which are the negations of each other?
Dr. Eastlake : Thanks a lot, for your excellent lectures!
I'm Physicist. Thank You For Great Lectures. Love From India 🇮🇳🇮🇳🇮🇳🇮🇳
One of the few professor which remembers that his lecture is being recorded
thank you so much professor Schuller. The lectures are very very good.
Wow! I'm awed. He lectures, writes and explains everything at the same time without any notes. How does he do it?
Because he loves what he does and master it
his notes probably are on a desk outside of the fov of the camera.
How are these lectures not more popular? Fantastic.
Difficulties of proofs, with translations:
1) "Easy" = Axiom
2) "Difficult" = Unprovable
3) "Hard" = Left as an exercise for the reader
Very nice. I am working through How To Prove It by Velleman and Sets For Mathematics by Lawvere. Found my way here by way of "The Portal" Discord group. Very helpful videos to supplement that work. This is my starting point on the long road to understanding differential geometry, the mathematical language of physics.
This is such an incredible series
1:26:30 Another way to put it is: "An axiomatic system is consistent if from the axioms cannot be proven a formula and the negation of the formula. (Cannot be proven that )"
helpful. Thanks.
"it is always important to know what a subject is NOT talking about" very insightful.
Not understanding the implication at 50:00. Can someone clarify?
These are absolutely brilliant. Very clear exposition. Only thing...I would rather use 1 and 0 instead of t and f as they appear nearly the same on the board...
No mathematician does that
What a wonderful summary.
Amazing! Why is a lecture like this (and the next one) not required for all students of math or physics?
It probably is
@@sereya666 it isn't sadly
This part has 78k views (07.13.2018), second one has 50k and the last - 4.5k.
This saddens me.
I'm enormously grateful. Thank you.
We need more lectures. I love the way you are explain Matematics. Are you planning to do some lectures about H - Function? A huge respect for you!
1:17:00 Can someone clarify? There's something quite fishy about the claim that, the "axiomatic system for propositional logic is the empty sequence". Although I'm no big expert on logic, this sounds almost certainly incorrect. Propositional logic has its own set of axioms, and rather, many, alternative ones-each producing the same propositional calculus. E.g. en.wikipedia.org/wiki/List_of_Hilbert_systems lists some of these axioms for propositional logic. Am I missing some nuance that differentiates what Prof. Schuller is saying, and what I'm juxtaposing it with?
Are there problem sets to accompany this lecture series?
His first two axioms of set theory (on Element relation and Empty set) are different from two of the ones on Wikipedia Feb 2023 (Extensionality and Restricted comprehension). Restricted comprehension follows from the Empty set and Replacement axioms. Am I correct in saying that his axiom on the Element relation "x\in y is a proposition iff x and y are both sets" follows from Extensionality or some other combination of axioms? If not, then this would be mildly disturbing since both he and Wikipedia claim to axiomatize the same system (ZFC).
(Side note: EE PURP IC F is not ruined by replacing the axiom on the Element relation with Extensionality, both of which begin with an "E.")
I have the same question too. But it is not obvious to me.
For anyone interested in the Corollary to the theorem at about 31:23, really it should be that we can prove things by using the contrapositive. Contradiction is related, but slightly different. Here is a nice answer from math stack exchange math.stackexchange.com/questions/262828/proof-by-contradiction-vs-prove-the-contrapositive. Very good lectures though, and intending to keep watching!
Dear Fredric, can you please advise on any reading list associated with this course? Thanks
Where he comes a bit short in mathematical rigor and clarity, he makes up in making the physics roar.
Thank you for your wonderful lectures!
What are the prerequisites for this lecture series?
glad to hear that there are professional mathematicians that dont trust the proofs by contradiction. I am no mathematician, just an enthusiast, but this kind of proof always feel sketchy to me :)
Which book is being followed? Or what are the recommended books? This must be mentioned right in the start of the series
If you master the subject there is no need to follow a textbook. But it would be great if F. Schuller would write a textbook. All his lectures are really awesome.
Almost certainly he is following his own notes. This is too clean for him to be following the thoughts of another.
This is really fantastic. Thank you so much!
The statement about the barber is neither true nor false, so according to the definition of a proposition given at the beginning of the video, namely that a proposition is something that is either true or false, wouldn't that just imply that the statement given was not considered a proposition? Then the question about whether or not it could be proved would not even arise.
صحيح اتابع من البيت بالرغم من تخرجي من الجامعه منذ ١٠ سنوات واكتب معاه واتابع كل المحاضرات وشريت دفتر خاص للمحاضره .
Thanks I watched these lecturers frome my home in Saudi arabia i graduates frome university since10 yars
خونة آل سعود
Could someone help me prove the statement about hairdressers mentioned around 50:00.
What a hairdresser can do only a hairdresser can do implies that a hairdresser can't do anything that a non-hairdresser can do.
Take any action that a non-hairdresser can take as a counterexample.
Assume that there is a non-hairdressing action that the hairdresser can take, and call it "fishing". From the statement "what a hairdresser can do, only a hairdresser can do", it follows that because our hairdresser can fish, a non-hairdresser cannot fish. In particular, the fisherman cannot fish. This is a contradiction, proving that our assumption about our hairdresser's capabilities was flawed.
Does the hairdresser dress his own hair, that is the idea...
@@bumpty9830 that reasoning seems ok, but I still don't see why this follows from that corollary - -||
Did you figure it out?
Thank you, Frederic, for the lectures, these and the many others. Regarding this one: I think it is better to start with sets, because {T,F} is a set, the concept of a variable needs the set, quantifiers need the set notion, and so on. So first elementary set theory, then logic, then more advanced set theory, spaces, and so on,...
Actually, the real logic doesn't need any sets. It doesn't use {T, F} set, just deductive rules which describe how to get any tautology. There are no real variables in there, just formal symbols. And why do quantifiers need sets?
I could listen to him forever
Are you high or just a brown nose????
... And still not understand anything.
Schuller you are a great mathematics teacher, I think you should write a book I would think it'd have the possibility to become a classic, something that focuses on the logical, set axiomatic, and proof theory aspects of fundamental mathematics with are I think some of the hardest concepts for beginning math students, with the most room for improvement in the current literature in structural writing and exposition. Thank you for the videos, very useful and well done.
Someone has compiled notes to these lectures, which are quite good and available online. I agree with you; he seems to construct a very elegant 'big picture' of concepts and relations between them. Even though I'm not a beginning Maths student at all anymore, I still find some fresh and pleasing ways to think about certain concepts in Prof. Schuller's lectures.
29:00 is proof by contraposition not contradiction (everybody mixes this up). Beautifully presented lectures, thanks! en.wikipedia.org/wiki/Proof_by_contrapositive
is ¬x a predicate? it is proposition valued and dependent on x. By proposition valued I assume it takes values true or false like a proposition.
¬ is an operator, not a proposition though.
you know shit's gonna be fire when Wittgenstein is credibly referenced multiple times in the introduction to the lecture.
Can anyone explain to me the Gödel's theorem proof? Or anyone has a (not so complicated) reference where I can check out the proof for this theorem?
I know I'm replying 3 years late, but for anyone looking for an elementary understanding of Gödel's incompleteness theorem, there is a great book by the name Gödel's Proof by Nagel and Newman. I'd recommend going for the revised edition as there is an excellent foreword and a few corrections/clarifications to the original by Douglas Hofstadter.
Which book do you use in your lectures?
An axiomatic system is consistent if it discriminates propositions
Is it that the theorem should be "the equivalence operator between the two be True" rather than just write down the equivalence operator?
No
😭😭
i want to study this course, thank you
ye man
what a gift to the world!!