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.
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! 😢
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.
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)
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.
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
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.
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.
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!
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.
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.
Wonderful lecture. For those wondering, he does describe a consistent axiomatic system correctly but the definition he stated was for an incomplete axiomatic system.
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 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 !
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).
Randomly suggested by youtube and such a fantastic approach to the theme discovered .... Unbelievable!!! Thank you for upload this classes :) Great job
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?
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.
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 )"
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!
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!
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.
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...
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.
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.
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.
29:00 is proof by contraposition not contradiction (everybody mixes this up). Beautifully presented lectures, thanks! en.wikipedia.org/wiki/Proof_by_contrapositive
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.
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.
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.")
Thank you. A masterful presentation of such a beautiful subject. I recommend "General Relativity for Cosmology" lectures by Achim Kempf and "Applied Differential Geometry" book by Ivancevic for the physicists out there.
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.
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 :)
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
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?
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?
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.
The Role Model Academic Lecturer! I have seen most of your videos on TH-cam, I appreciate the rigour also as a mathematician...Thank you and very best wishes!
14:39 "And the wavefunction in quantum mechanics is not a function at all, it's a section of a complex line bundle over the physical space" ahh quantum fields??
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?
The professor is technically correct. Establishing proof by contrapositive is all you need to do to establish proof by contradiction. This is because proof by contradiction is a special case of proof by contrapositive. You can demonstrate this quite easily: Lets say that you are trying to prove a statement p. Assume the opposite -p is true and prove -p => F, where F is some false statement. Then the contrapositive proves T => p, i.e. some true statement implies p. Thus by modus ponens, p must be true. In other words, T and (T => p) implies p. The contrapositive and modus ponens are all you need to execute a proof by contradiction.
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?
Excellent professor and way of conveying the subject, which shows deep understanding. What bothered me a bit is that this lecture shows the sentiment seven years back, where politically correct was all the rage, and the poor lecturer tries to constraint logic so that he does not offend any snowflake, 46:20
The internet needed this lecture. Thank you.
Yeah i really enjoyed it
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
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.
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! 😢
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.
This is the true act of love and compassion, Thank you!
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
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
Difficulties of proofs, with translations:
1) "Easy" = Axiom
2) "Difficult" = Unprovable
3) "Hard" = Left as an exercise for the reader
The way he writes t & f makes him always correct.
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
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.
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
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!
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.
28:28, 35:20, 49:56, 1:07:25, 1:10:40, 1:22:00, 1:24:08, 1:28:25
This series of Lectures is pure pleasure! Sometimes I come back here just to be amazed again. Thank you!
this guy is genius
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.
Absolutely brilliant, breath-taking and addictive!
If I cloud only develop such addictions!
This part has 78k views (07.13.2018), second one has 50k and the last - 4.5k.
This saddens me.
Wonderful lecture. For those wondering, he does describe a consistent axiomatic system correctly but the definition he stated was for an incomplete axiomatic system.
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 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 !
This is pedagogy; not whatever it is my mathematics professor was attempting to do.Thank you, Prof Schuller. ❤
wow...the first 2 minutes and I am completely captured. The interpretation and reflection on those two quotes by Wittgenstein
I'm Physicist. Thank You For Great Lectures. Love From India 🇮🇳🇮🇳🇮🇳🇮🇳
6:17 why is Statistical Physics at the intersection of geometry and algebra? I thought statistics is more about analysis?
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).
Randomly suggested by youtube and such a fantastic approach to the theme discovered ....
Unbelievable!!! Thank you for upload this classes :)
Great job
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?
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.
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 )"
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.
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!
Remarkable lectures !!!I I wish we had some problem sets to solve , so that we could test our concepts.
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!
Great and clear content. We need more of these guys on the web.
He is clear and concise which in turn enables learning. Love it
Amazing!, not just the content but the way he delivers it with such calmness and clarity, incredible!
You are a great teacher. Thank you for sharing these lectures.
Amazing! He explains things very well. I can understand more than 90% materials. See you guys at the final lecture.
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.
I could listen to him forever
Are you high or just a brown nose????
... And still not understand anything.
is there anybody having access to the problem sheets?
This is amazing sprinkled with a great sense of humor, thanks for sharing!
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
I wish I discovered this series earlier, so so good!
yes, I wish I could have watched this about 30 years ago.
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
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.
One of the few professor which remembers that his lecture is being recorded
Amazing first lecture!
I will try to follow the next ones until my levels allows me. Greetings from Colombia.
How is school there?
This is such an incredible series
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.
I searched up a German name I made up, and not only does he exist, he teaches what I needed. Badabingbadaboom
"it is always important to know what a subject is NOT talking about" very insightful.
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.
29:00 is proof by contraposition not contradiction (everybody mixes this up). Beautifully presented lectures, thanks! en.wikipedia.org/wiki/Proof_by_contrapositive
Take note, people: this is _the. most. important. subject._ to have a grasp on as you go into higher mathematics. Spend the time.
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.
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.
but he is right, from contrapposition you can get a contraddiction principle
Finally, a mathematician who introduces the subject the correct way, via a wider philosophical picture. Excellent!
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.
Thank you Dr Schuller for uploading this lecture! Hope to see many more!
How are these lectures not more popular? Fantastic.
Thank you. A masterful presentation of such a beautiful subject. I recommend "General Relativity for Cosmology" lectures by Achim Kempf and "Applied Differential Geometry" book by Ivancevic for the physicists out there.
One of the best first 15min I've ever watched!!
thank you so much professor Schuller. The lectures are very very good.
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.
Dr. Eastlake : Thanks a lot, for your excellent lectures!
I'm enormously grateful. Thank you.
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 :)
Where he comes a bit short in mathematical rigor and clarity, he makes up in making the physics roar.
Is there a lecture notes or reference book to this course ? Thank you very much.
A proposition it's a formula in the language L that has all its variables bounded
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
What a wonderful summary.
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?
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?
😭😭
i want to study this course, 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!
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.
Thank you Dr Frederic Schuller
what a gift to the world!!
The Role Model Academic Lecturer! I have seen most of your videos on TH-cam, I appreciate the rigour also as a mathematician...Thank you and very best wishes!
I wish you were my mathematics teacher; 40 years back❤
14:39 "And the wavefunction in quantum mechanics is not a function at all, it's a section of a complex line bundle over the physical space"
ahh quantum fields??
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 wish this becomes the first principles everywhere.
you know shit's gonna be fire when Wittgenstein is credibly referenced multiple times in the introduction to the lecture.
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.
Thank you for your wonderful lectures!
Is it that the theorem should be "the equivalence operator between the two be True" rather than just write down the equivalence operator?
No
The professor is technically correct. Establishing proof by contrapositive is all you need to do to establish proof by contradiction. This is because proof by contradiction is a special case of proof by contrapositive.
You can demonstrate this quite easily: Lets say that you are trying to prove a statement p. Assume the opposite -p is true and prove -p => F, where F is some false statement. Then the contrapositive proves T => p, i.e. some true statement implies p. Thus by modus ponens, p must be true. In other words, T and (T => p) implies p. The contrapositive and modus ponens are all you need to execute a proof by contradiction.
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?
This is really fantastic. Thank you so much!
Excellent professor and way of conveying the subject, which shows deep understanding.
What bothered me a bit is that this lecture shows the sentiment seven years back, where politically correct was all the rage, and the poor lecturer tries to constraint logic so that he does not offend any snowflake, 46:20
This professor is really really good. Hes a bit serious but damn he knows his stuff very well.