Mathematician explains the limits of mathematics | Edward Frenkel and Lex Fridman
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- Опубликовано: 10 авг 2024
- Lex Fridman Podcast full episode: • Edward Frenkel: Realit...
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Edward Frenkel is a mathematician at UC Berkeley working on the interface of mathematics and quantum physics. He is the author of Love and Math: The Heart of Hidden Reality.
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Guest bio: Edward Frenkel is a mathematician at UC Berkeley working on the interface of mathematics and quantum physics. He is the author of Love and Math: The Heart of Hidden Reality.
I like the way Edward tries to keep it super simple
thats the secret, ney? using common words to say uncommon things.
So many do the opposite ^^
Still goedelized my poor neurons or rather fried them ;)
What a great conversation. Normally, an interview like this would seemed disjointed to me, and would use language or concepts above my comprehension. Here however, the conversation was so clear and focused, I felt I could join in without feeling like a dunce if we're there with them.
To summarize, there will always be a truth statement within a formal system that can not be derived from the existing axioms of the system. In the other word, mathematic system is always incomplete and ever expanding.
As math finds more truth, the range of human "free will" narrows. If something is true, no willful human can change it.
@@drbuckley1 that’s not what the interviewee is saying. He emphasized that as the maths expands, there will always be more room of human will not less. It’s like someone said, when my know more, I also know that my ignorance is even more. This only makes sense.
@@typicalKAMBlover21 If something is true, no human will can change it.
@@drbuckley1 exactly, therefore human’s range of knowledge will not change any mathematical laws that is running the universe. However, the more mathematical laws the human can COMPREHEND, the more unknown remaining he will PERCEIVE. Your model assumes a finite amount of math to be known, my model(and the incomplete theory the interviewee mentioned) are saying that the mathematical possibilities are literally infinite. The math world is literally an open world that can not be exhausted.
Could Edward please expain the limits of my understanding of mathematics 😅
Wonderfully explained!
utterly fascinating! and simultaneously wayeeeeeee over my head. I think I will have to watch this clip multiple times to begin to try to understand his explanation of the limits. The limit…. as it goes to to zero!😅
Great guest, love it....
Nobody seems to talk about statement size. That is, I would bet there are true statements that are nevertheless basically impossible to prove, not because they are independent, but because a proof, even a minimal one, is too large for a human to generate. For example, imagine that the Twin Primes Conjecture requires a billion statements to prove. I don’t know how you would even find the logical path to it, let alone live long enough to record it.
I want a poster of that cover shot! Thats an awesome example of geometry, perspective, and our light based existence!
You'll love the TensorFlow logo ;)
de.wikipedia.org/wiki/TensorFlow
@@Squidward1314 I will check it out as soon as I get off these endless stairs......
@@freeradical6390 I heard BLJs can help with that.
I like you so much lex ❤🎉👍🙏🙏thank for all your videos
Recommend Godel, Escher, Bach: An Eternal Golden Braid, which discusses this in great detail.
For anyone that cares about this topic.. Veritasium just did an excellent video about this a year ago called Math’s Fundamental Flaw.
i like the part where they simulated the game of life inside the game of life at around ~30:00
The way I see this is that a powerful enough formal mathematical system can not pull itself up by its own boot laces. It was mentioned at the start that Gödel produced two inconsistency theorems. What was the second one and how does it differ from the first?
Perfect video
I think and believe that his ending words describe not just mathematics but logic itself, the extent and limit of logic when intuitive truths are not dependent on the previous linear syntactic process even though the linear process is sufficiently dependable and useful. From a philosophical viewpoint, linearity and globally accepted intuitive truths are only relevant and applicable to the observer's reality. And within this reality, contradictions can not occur. However, conversely within another reality the truths will contradict the linearity of this reality. For example in our reality, it is assumed that there is a line that connects point A to point B on an infinite plane. However it may so happen that in another reality, lines can not exist. Lines are a derivative of direction. And direction requires space. What if simply by volition, one could be both at point A and point B? Or further still, what if the nature of this reality imposes that separate points can not exist because the infinite plane is in fact an endlessly curving and overlapping single point? And if I simply replace that infinite plane with the function of a quantum computer, quantum states would impose that point A and point B are the same and neither at the same time, which is inherently contradictory but true. Nice interview.
that's pretty abstract
@@thegaspatthegateway Logic and intuition are not abstract criterias. A philosopher uses the same theoretical foundation as a mathematician who derives facts without physically replicating them. You also seem to look-down on abstractions. By your logic, all emotion should be abstract and you should not feel them nor understand them because they lack intrinsic proof. The distinction between abstractions and verifiable physical proof is a human-construct. From a completely objective perspective, they are both the same illusory perception of reality.
who?
It's convention.. I noticed this a long time ago... maths at it's core is by convention
he mentioned alan watts!!! so thats why hes so charismatic!! nice👍
Please someone tell me if I got this right.
If you make a formal system based from the most basic axioms you can, there will be a true statement in it that cannot be derived from the very basic axioms the formal system was made from??? Someone please, my brain hurts..
It's how i understand it too. And it means that a computer can't compute all true statements there are in a formal system, given the computer starts with the axioms. Which means, in my opinion, that there is more to reality than just computation. It may also me an argument against the simulation hypothesis.
Okay so.. Remember that formal systems are formal because they are based on axioms in the first place. So if any sort of system that stems from axioms is applied, it's requires an initial axiom that acts as a one way key. Then from there whatever truths you form, they are truths relative to the form of your system. From there, one can actually utilize the logic
Not that it cannot be derived - but it cannot be proven true within that formal system IF the system is consistent.
And Gödel’s second theorem states that a consistent formal system cannot prove its own consistency.
Just commenting so I can get alerted to where the rest of this conversation goes.
I'm always curious about these ideas, and have my own laymans-hypothesis on how the universe seemingly sprang from nothing that seems relevant in this conversation
My idea is basically that the universe appeared because if it didn't then only nothing would exist, but if "nothing" existed it wouldn't be nothing, as nothing can't be anything, and even just existing means it isn't truly nothing as ud have to ask where did existence come from?
But anyway you guys carry on, I wanna understand more about what was discussed in this video as its really difficult for me to follow 😄
If I'm not wrong Axioms are essentially the foundational rules of a system? Not exactly sure what he means by "consistent" when he talks about functioning of the system tho
It CANNOT BE SAID to be a true statement. That's the whole point! It would be UNDECIDABLE.
Undecidable doesn't mean that something has a value, but it's somehow hidden. It has NO value within the given system.
I really hate to give bad analogies, but try this. Suppose I had some kind of lottery where people could bet on whether the total number of plastic balls in some tank was even or odd. (I'm trying to set this up so that it's clear that the only two possibilities are even or odd.)
Now there's a nuclear war, and a bomb goes off right in the middle of some random process that might or not have added some more balls to the tank.
Now, what are the number of balls in the tank, when there is no tank and there are no balls? Is it even or odd? Those are the only two possibilities, right? Well, so it once seemed but no. It's UNDECIDABLE. (Further, it's meaningless, but meaninglessness certainly entails undecidability.)
Please understand, I'm not trying to explain formal undecidability in terms of physical systems. I'm trying by analogy to get at a small component idea, which is that there's no such thing as a true statement which can't be proven to be true. That's a fundamentally INCOHERENT idea, roughly analogous to trying to count balls which have turned into incandescent vapor.
He may be trying to explain it the most simplistic way but as many times as I tried to grasp I just got lost more.
😂 that is ok! Who doesnt!
May I suggest Douglas Hofstadter's opus "Gödel, Escher, Bach" as a starting point. The discussion of self-referential statements and "arithmoquining" there is very accessible. There are more specialized and condensed books - I could suggest several - but all of those are beyond the level of this discussion.
Basically hes saying if a certain mathematical theorem is consistent in its proofs but a problem that uses the same theorem is hard to prove with that same theorem that means the theorem is still true even though it seems impossible to prove this basically correlates with god and our reality
½at² I think
The most simplistic way is something like - Given complicated enough formal system, you would always need more axioms to determine truths within it.
Often the statements that need new axioms are constructed with some kind of feedback loop (like 'does set of all set is a proper set' in set theory), which reveals, I would say, the creative quality of fractals.
Reality might have infinite complexity if it contain every possibility. Perhaps every truth instantly spawns a contradiction that disproves it.
An interesting idea - but what he is saying is that axioms are a choice to build on top of, and the axioms/space that you build on top are built to be self-consistent e.g you don’t build with axioms which will ultimately be falsified . That being said after you have made the choice in axiom then you can have theorems that are true or false (like Riemann hypothesis could be true or false) but once proven then we must assume we can build on top of that using the logical inference (etc). So we can have infinite complexity across these various choices e.g Euclid geometry for small features on earth, but then you need to move to non-Euclid geometry where the axioms don’t hold about like angles don’t need to sum to 180 for a triangle - these can co-exist as long as we are consistent in the areas applied.
That is really well said! I'm building a statistical model for my master's thesis and your sentence encapsulates any statistical model's 'core problem'. Like George E.P. Box said "all models are wrong, but some are useful". This is a little beside your point but it really applies to many things in life.
Metaphysics
What would be the nature of the very concept of contradiction in that scenario?
probably for our own good honestly
Impressive. Eloquence is a god-given gift.
It’s not God given, you just have to work hard
AI has come far. Two great robots talking about math 😅
Humans actually are robots of sorts. Just not artificial..
😂
@@MichaelSmith420fu And yet some fraction of our intelligence is the artifact of the society we live in.
Just had the thought. Is there kind of, zero direct reference to errr, sexuality in Lex's podcasts. Despite the way it motivates all life. It's kind of a nice break from all that in a way, like reading Tintin again
Lmao
The conversation applies to many things
But honestly. HAAA HAAA!
@@nicolaslacombe1979 bro isn't this scary nobody know shit. we just assume and continue
My working theory since ~1st grade, is that math is consistent and therefore worth learning. But of course, Goedel says I can't prove it. I derive some Bayesian comfort though, as I see more and more mathematical propositions added to the pile of propositions proven true, and as they obligingly keep on not contradicting each other (or at least not in a glaringly obvious way).
saying big words doesnt make you smart
@@nicholasc.5944 Most professional mathematicians are intimately convinced that math is consistent (otherwise, they would have chosen another profession) although they know full well they can never prove it. Why do you think that is? Do you think that math's 2500+ year history of generating non-contradictory statements has anything to do with it? See Nicholas - first you ask the right question, then you get to the big words.
Thinking about it - it's an underappreciated and rather nice fact about human nature: smart people generally aren't that interested in telling others they aren't smart.
In my humble opinion there will be a time in a far future when mathematics will become a tool for an abstract modeling of abstract abstractions abstracted from abstractions. It will somehow merge into some kind of operational poetry or instrumental philosophy. We may get some fruition from all of these abstractions but also eventually we might lose track of it.
That thumbnail is surprisingly profound
There is nothing but our motion from infinity to infinity, and the infinite possibilities of infinity in between.
I think I saw him at numberphile
"We still feel the tremors of this discovery..."
I believe you're referring to "A Treatise of Formal Logic: its Evolution and Main Branches, with its Relations to Mathematics and Philosophy by J. Jørgensen?
But was Georg Cantor to begin the game by opening something everyone thought was closed, i.e. the different infinities. David Hilbert, the top of all genius mathematicians, presented a structure of logic that appeared truly great - up until Kurt Gödel proved the incompleteness theorem. It was a sad day for everyone involved; it looked more a court sentence than a theorem, truly.
In my humble opinion, of all modern thinkers, Bertrand Russell - with the antinomies and paradoxes - was the one that come closer around the "truth" involving formal logic. Something that Niels Bohr, with his concept of complementarity, approached by another angle, something the human mind can't easily grasp - that a logic statement can include, and be coherent with, its own opposite.
How can computer solve hyper-euclidean geometry, whose axioms involved straight line(s), but which never subjected to any "Field". Can computer simulate the effect of Gravitation onto the ray/path of light, (and make it curved/deviated/non- straightened). The proof will also effect/show the irrelevant of Artificial-Gravity (by Centrifugal or uplifting effect of (simulated space-elevator)...the two will never make Compatible nor Equivalent to each other..(the real gravity and the Artificial one(s)..are not compatible.)
I cannot get over the synchronicity of this moment. May i share with like minded folks: i was just reading a book by graham framolo. “ the universe speaks in numbers” and was just reading about Godel and einstein!Why ? Because another very extraordinary podcast- “ surreal physics” had taken the time to answer me about her video- and in turn another guest ( viewer) offered up a question in response about godel. So , I re find the passages about Gödel in the book i am reading , and am quite happy , i will reply with confidence , that godel was quite quantum in his thinking in my opinion. well i have the book still open- page 103! and realize i need to get dinner started. I put on the kitchen tv ( youtube is on it) and this is the suggestion!!! Lex i was looking forward to this broadcast! The clip is ridiculously relevant! Wonderful. I am a few minutes in and feel like you and your lovely guest Professor Frenkel, have stopped in to give me even more perspective as i cook up this chicken!!!I am humbled by this fractal , amazing, synchronistic world we live in!!! Thanks Lex!!!
enjoy the chicken !
@@emielfull a perfect response😂
The imagination of synchronicity is a cognitive bias.
@@IsomerSoma or is cognitive bias the synchronicity of imagination? If you catch my drift.
There’s a lot of people in the world. Just about everything has happened to somebody.
Didn’t get :(
Many thanks!
Alexander Zenkin: "the truth should be drawn ..." ("SCIENTIFIC COUNTER-REVOLUTION IN MATHEMATICS")
Galileo Galilei left on the philosophical and mathematical testament to “draw the truth” - the limits of mathematics and its ontological basis - framework, carcass, foundation: “Philosophy is written in a majestic book (I mean the Universe), which is constantly open to our gaze , but only those who first learn to comprehend its language and interpret the signs with which it is written can understand it. It is written in the language of mathematics, and its signs are triangles, circles and other geometric figures, without which a person could not understand a single word in it; without them, he would be doomed to wander in the dark through the labyrinth. ("Assay master").
A.N. Whitehead: "A precise language must await a completed metaphysical knowledge."
Mathematics❤
For somebody who only has a basic understanding of logic and maths, can you recommend a good read about Gödel's incompleteness theorems from a metaphysics perspective?
It's not metaphysical.
It's a formal proof.
@@starfishsystems Yes, at least the second thing I know. BTW what happens when one tries to formally prove the correctness of the incompleteness theorems with a computer? Do such attempts lead to variants of the halting problem?
What are the fundamental differences between what computer algorithms and axiomatic systems can 'make out of' Gödel's work on one hand, and what a mathematician's experience and intuition can do on the other hand? Can this non-formalizable (perhaps metaphysical) difference be further characterized - is it unique to creatures like us - perhaps bound to physicality (including what lies beyond, to enable physicality)?
With the progress of AI, such questions even come into the center of mainstream media & debate: the fundamental differences between machine learning or formal languages VS the experience, will, qualia and reasoning of humans.
Of course, all entities can be contemplated from a metaphysics perspective. Ranging from the most simple-looking entities to the most abstract, vast, enigmatic, imaginary, bottomless, remote... and so on.
However, Gödel's incompleteness theorems look like very special entities!
Edward Frenkel: "It's really incredible. So, this was a revolution. 1931 [...] and we are still feeling the tremors of this discovery."
Introducing technical terms like metaphysics, epistemology, ontology alone will not explain or simplify anything - there's just this intuition that Gödel discovered something very deep about the world, be it for the logical, mathematical, mental, physical... - perhaps even platonic or divine world.
There's an old question, designed as a somewhat peculiar thought experiment: "Do the Gods stand above Pi?". No apparent connection to the above, just the mind-bending part.
Intuitively Gödel's discovery can be assumed as intimately related to the nature of the world, we so happen to have emerged in, perhaps also intimately linked to its personality and legislative bodies.
@@DarkSkay
The proofs of the incompleteness theorems were provided as an essential part of publishing the theorems. They ARE demonstrably correct.
Therefore an automatic proof of correctness WILL halt.
And in this sense there's no essential difference between a human proof of correctness and an automatic one. It seems to me that you're trying too hard to go down a rabbit hole that in fact doesn't exist.
When I was studying automata theory back in the 1970s (not very long after the development of ZFC, so these were exciting times for the field of mathematical logic) my prof said something memorable on the subject of the human element in mathematics.
To paraphrase, he said, "Humans have vast imaginations, and cognitive processes which we don't ourselves understand. But what ultimately matters in mathematics is not where we get our ideas. They could be handed to us by space aliens. What matters is whether we can prove them correct."
@@starfishsystems Thank you very much!
Well, from a practical or utilitarian standpoint we want to first discover interesting logical statements (resp. "discover them with our minds" aka "invent them") and then classify them as either true or false. "Every logical statement must be either true or false" (*).
Except the sentence (*) is already false; not the case! Since even in very basic languages there is (at least) one more (revolutionary) category: undecidable.
Arguably countless great thinkers over millenia of discoveries thought (*) to be the case. Not exactly a detail! And yet, this historical detail or observation alone reveals something about the nature of reality - or how reality likes to appear to our minds - resp. instanciate our minds resp. how our minds like reality (to appear).
Philosophical questions aren't bound to the utilitarian approach outlined in the first paragraph, can contemplate "rabbit holes" and chose directions freely. Philosophy can do without axioms, its directions and constraints are either freely chosen or come from the outside context it evolves in. And this context includes Gödel's revolutionary discoveries, which can be assumed to be connected to the fabric, the many echelons of reality... up to the imagination or reality of the Gods, and other, yet undiscovered entities we lack words and concepts for.
Gödel's work has a unique metaphysical touch, a flavour other formal results usually don't.
Thought was gonna talk about calculus limits haha
Ohhhhhh, now I get it. 😂
As I understood it the point was a theory can be true without it being proven. So at some point an idea can be true but we cannot confirm this. Seems to tally with real life..
Yes it has analogy to quantum physics doesnt it
It’s important to note that Gödel’s theorems apply solely to mathematical formal systems and the truth value of theorems within them. The possible similarities found in scientific theories and real life examples are not subject to Gödel’s theorems.
I love this Russian teacher, but wow, I have to stop and replay so many times because of his accent! I am grateful the Russian Lex speaks so clear English!
Love the wig 😆
I think math can be obejective and should be split into objective and subjective classes.
Example, audiotory talented people can find perfect pitch using no known concepts our mathematics can calculate. They're using math but, we don't have the technology to comprehend it so we call it math. In my neuro/physics research into visual marketing of music, I discovered a few formulas to emulate natural talent. These formulas led to me making 5000% roi with my first business as Teen.
I would suggest we begin to study the math these people like Haruka Fukuhara use and learn their math vs calling it a gift. To me and based on my research, it's math. Just a different unknown way.
I think it's interesting to think about,you know, how if if like there really was a singularity that is everything, then 1=∞. Is there a symbol for such a paradox? Yes and it is called light.
I believe that Geodel showed there were infinitely many true statements that cannot be proved thusly. Mystery is a property of the universe.
Pretty close. But we can't suppose those to be true statements. That would be dreadfully begging the question. Your claim is "not even wrong," as the saying goes. Gödel said no such thing. He would have thrown up his hands in despair that you got so close and then utterly missed the point.
To borrow your language, Gödel showed that mystery - or to be more precise, undecidability - is an innate property of formal systems such as mathematics.
To the extent that mathematics is expected to be the EASIER, more general, more tractable, less circumstantially constrained, representation of objective reality, we expect reality to be necessarily a HARDER problem.
In other words, if the "toy" version of reality has essential features of undecidability, then the "real" version will, if anything, have even more of them.
It's not that some true statements can't be proved. They are NOT true statements UNTIL they are proved. That's what truth is, in formal terms.
Now, concerning the mapping of the set of statements or propositions to the set of proofs, consider this:
1) A nonempty set of propositions P can be made concerning every real number. As one example, a proposition Pr can be made that some real number Xr is positive. (The statement, clearly, may be true, false, or undecidable.)
2) There are an uncountably infinite number of possible values for Xr, therefore the set P is UNCOUNTABLY infinite.
3) The set of proofs Q contains discrete proofs Qi each composed of some finite number of discrete steps. By Cantor diagonalization, therefore, all possible proofs are enumerable. They constitute in total a COUNTABLY infinite set.
4) Thus, even in this simple example, we have an UNCOUNTABLY infinite number of questions and a COUNTABLY infinite number of proofs.
@@starfishsystems Second statement was my own with no attribution to Geodel.
@@zTheBigFishz
That's fine. But I was addressing your first statement, about undecidability.
@@starfishsystems It's not there is just a finite list of statements that are true but not provably true. There are infinitely many of them. That's it.
Which boils down to a some point you have to accept the statement on faith?
Consistency means that all statements that can be proved are true.
Completeness means that you can prove all true statements.
Godel proved that an axiomatic system can’t have both properties simultaneously. Because it has to be consistent to make sense - it means it's not complete, which means there will always be an undecisive statement within the system - neither true nor false.
Basically, he constructed a statement similar to: “I am now lying”. If I’m telling the truth I’m lying, and if I’m lying I’m telling the truth.
Boom, whole Hilbert’s positivistic philosophy exploded. Btw Godel was 25 when he did that.
Nicely said.
I would like to suggest Gregory Chaitin or Cristian Calude as future guests of your podcast. Either one could provide your subscribers with a much better overview of Gödel's incompleteness theorems, their consequences, as well as a deeper insight into the limits of mathematics. Frenkel's presentation is far too superficial and anecdotal.
I think his aim was to give a cursory overview and emphasize its implications rather than provide a technical explanation. Definitely agree that having someone who goes into the detail of it would be great!
No comment comparing him to Jaime Lannister?
Godel uses an incomplete system to prove its own incompleteness?
Uses "an incomplete system"? No. He provides a theorem based on mathematics and logic that to logically assert that proofs require more than the sum of their parts; To show that systems can not be completely defined beyond locally.
His theorem(s) works the same way others' do.
Fair point. The system he uses may well be formally incomplete.
It's important to understand, however, that an incomplete system can give rise to entirely complete and PROVABLY correct results. Just not in all things.
It would be like discovering that every old castle has some uninhabitable rooms. You can still live comfortably in the remaining rooms.
We might, very hypothetically, be living in a reality where an incompleteness proof happens to be one of those which is undecidable. This is an entirely plausible scenario. But, happily, we have the proof, so it turns out that mathematical reality is not opaque on this point.
What's especially interesting is that this is a result about mathematical reality, unconstrained by the particulars of physical reality. Gödel incompleteness is true not just in our universe but in every universe. It's true even in universes which don't or cannot physically exist.
You know, I always preferred fairy chess to chess
Saved by zero❤ Sorry I couldn’t resist.
Good one.
I don't get it. I probably shouldn't be here.
@@shockmarkets7384 it’s the Thomas Dolby song I thought it would fit there
What is mathematics ? This is the question mostly students ask me on my page and channel.
Aristotle defined it as the study of quantities. Euler also gives this definition in his Elements of Algebra.
@@ocean_0602 thank you for the definitions. 👍👍
A useful tool.
Gödel (fancy that, how convenient) provides a pretty good answer. A simplified version is that mathematics is the study of any set of symbols and rules for operating on them.
We apply the rules and watch what happens. Sometimes interesting patterns seem to emerge, which leads us to make and test propositions in order to determine how the rules predict these patterns.
Oh, and axioms are representation. It doesn't matter what system you place the axioms, they are still representations of their counterparts.
They needn't have counterparts. They needn't represent anything but themselves. This is absolutely basic to the concept of "axiom" in formal systems.
Ok. So, there are mathematical truths that do not derive or are born from other subordinate math truths or axioms. These truths are true by themselves, they do not need axioms. They are fundamental math truths.
Can someone give an example of such math truths? How does this transfer to non-math truths, like just real life or physical truths?
At around minute 10 he says that there are theorems that cannot be proven by a system of axioms and that implies that everything in life is not computational {min 8:55}. So, if life is not computational is he saying life is not deterministic or that life is deterministic but just that there are some truths that are fundamental, and life still is deterministic?
Apologies for the long response but hope this helps. By "fundamental math truth" I am assuming you are referring to a mathematical proposition that is non-derived and is accepted as self-evidently true. But that is exactly what an axiom is. For example, Euclid's first postulate, "a straight line segment can be drawn joining any two points." This is an axiom, fitting your description of a "mathematical truth", since it is assumed to be true without requiring any inference or deduction. Many axioms, particularly in geometry, can be observed in real life experience - making them in a sense objectively true. For example, no one would disagree that one cannot draw a straight line segment between two points - because we can easily verify the truth of this ourselves by drawing a straight line between two points. Even Euclid's 5th postulate, which cannot be proven, is accepted as true since we can verify its truth in real life (parallel lines). However, mathematical topics which are more abstract in nature contain axioms that are unobservable or even unrelated to real life. In these cases mathematicians construct their own axioms as a means to build towards a theorem or proof, and these axioms are often altered or reconstructed as needed. This is where the subjective nature of math that Dr. Frankel is speaking of comes into play.
It's very important to note that Godel's theorems apply solely to mathematical axiomatic systems. While there may be real life analogies and parallels, the implication of Godel's work does not extend outside of math and into real life; into non-math or physical truths. Therefore it would be a mistake to apply Godel's theorems to free will and determinism. To answer your question though, Dr. Frankel does state somewhere in the full podcast that he does not believe in determinism, that life cannot be a predictable chain of events since experiments such as the double-slit in quantum mechanics demonstrate otherwise.
No, you have it exactly backwards. There are NO possible "truths" in mathematics which do not derive from axioms and subsequent constructions built on them. This is basic.
It's axioms, and axioms ONLY, which are taken to be true on their own standing. And that's only because we have to start somewhere. If we could do without them, we would, I think. But we can't.
It's a delicate business even to call these axioms "fundamental math truths." They in fact are not "truths" but PREMISES: statements provisionally taken to be true, but which we don't KNOW are true. They just seem like nice ideas. Again, we have to start somewhere.
If you want examples of truths such as you supposed in your question, of course there are none.
Axioms however are easy to find. They're basically on Page One of any formal system, so you'll have no trouble looking them up. Just pick the system and go from there. Try Set Theory or Number Theory for relatively easy starting points.
The material can be kind of dry. There are excellent textbooks available at any university bookstore, but I think there's nothing better than to sit in on an introductory course in these subjects. The physical enthusiasm of the instructor is a big factor in helping to animate the concepts and derivations.
And do the exercises! You won't want to, but they're essential for cementing your understanding of the material. You'll sit in a lecture and go, I totally get this. And then a few hours later when you turn to even the easy exercise it's like, I totally don't get this. You must close that gap or you will soon fall behind. To fall behind is terribly demoralizing. Stay on top of the material, right away. Take notes. Do the fucking exercises. Review the chapter, get a tutor, but practice it until you know it and trust that you know it. Then it's easy.
Fifty years I've been doing this. I kind of know what I'm talking about here.
As humans, we know next to nothing about anything. Still, that’s all that’s been required to get where we are in the reality we feel.
insightful. very paradoxical isnt it? shocking we have survived and thrived based on these 'truths'
My mathematics is limited 2.
This was very educational, but it stops where it's beginning to get interesting.
For example: What are the implications of Godel's theorem for science, our perpection of the world. Etc.
Mathematics rarely makes claims about physical reality.
However, as a practical matter, if something is mathematically impossible, then in a physical system which closely corresponds to that mathematical model, we expect something interesting to happen.
It's very strange listening to the left hemisphere of the brain explaining an obvious problem it manufactured through it's inherent data-computational limitations. You can't bite your own teeth, it appears that simple by analogy. Assuming that you could through technical gymnastics was the manufactured part. Living in paradox is the existential entrapment we are necessarily faced with, out of which comes the creative process associated with phenomenology and auto-poiesis.
Divide by zero. Checkmate, math.
An open coke bottle at the edge of a table... is like sticking ones balls into an unplugged blender.
Pray that nobody comes by and plugs it in.
According to Godel's incompleteness theorem, Godel's incompleteness theorem is incomplete.
That could possibly have been the case. Then we'd be truly fucked. But in fact it isn't, it's a good solid mathematical result.
So basically we're not living a simulation?
AI is good at theorems not axioms
Even the walls on this pic are blue and yellow.
Put the cap back on the coke bottle!
More accurately, mathematician explains how mathematics is LIMITLESS
He actually said it’s not limitless
Physics is NOT "based on" mathematics. Physicists use mathematics as a tool but physics itself is based on observations about the natural world.
So maths then
@@JR-iu8yl yeah, observations soon become measurements
Quite so. And also for all of the physical sciences. The math is just a useful DESCRIPTION or model.
In short: even the most logical system we have-math-contradicts itself. Don’t be too hard on yourself when you or your arguments prove self-contradictory: it’s postulated to be inevitable.
It’s not that it contradicts itself, it is that there are statements which are true but algorithmically undecidable. So you just can’t prove/disprove them using the axioms and rules of inference.
Also these statements are (at least in Gödel’ sown proofs) always self-referencing statements.
@@daesi
You're right in that we do not conclude that the system contradicts itself. It's simply incomplete.
You're wrong in saying that there are true propositions that can't be proven within the system. Those are not true propositions, nor are they false propositions. They are UNDECIDABLE propositions. They do not have some hidden value, they have no value at all.
There's a basic error here though isn't there? We're given the argument that math is limited by the system within which it's organized and then forced to accept that those aren't just the limits that have been placed on it? "It works and it's what we know and anything that points out a contradiction with something we already believe true must be a lie" just doesn't scream scientific method to me
The limits of my tolerance are tested when Lex calls him "Gaydel".
A lot of people pronounce it like girdle, which is also wrong according to Germans lol. The proper pronunciation is actually in a strange combination that we English aren't used too but I believe its something in between Gih, goi, and gir. Lol Gödel
@@MichaelSmith420fu He was Austrian so the Germans can't lay claim the right pronunciation. All of those options you listed are perfectly fine. Lex's pronunciation makes it more like Gädel which is indefensible.
If you make duck lips as if to say "ooh" but you say a longish "eh" as in "feather" you'll be close.
It's a sound which appears in many languages. In French it shows up in the œ ligature: words such as coeur and fleur and noeud. In Germanic languages it's consistently written ö, though in Scandinavian languages ø may also be substituted.
The limits don’t start at 0, and don’t end at infinity.
There, your answer has been questioned.
Not trying to be a jerk, but why not take that true statement that cannot be proven, to be an axiom?
A true statement which cannot be proven with one set of axioms could be proven with another set of axioms. Axioms are more fundamental and can't be proven.
What Gödel proved was that once you make this statement an axiom, then you end up with another formal system that again there is a sentence which could not be proved and so on...
@@thanasiskanellakis2439 This means there are at least countably infinite statements that cannot proven to be true in any system.
I wonder if it is in fact uncountable, but suppose not.
@@diffgeo23
Oh, the set is definitely uncountable.
Consider for example questions such as "Is some real number X positive?" The set of possible values X is uncountable. Thus the set of all possible questions is also uncountable.
Meanwhile the set of all possible answers consists of many but enumerable finite sequences of discrete assertions. Clearly by diagonalization this set is countable.
Subtracting the countable set of answers from the uncountable set of questions yields an uncountable set of unanswered questions.
And that's why AI has nothing to do with human consciousness. It is an entirely different spices.
[Very spicy, yes]
Ima show you why math isn't limited. Just as we break down 2 to get two 1s we can break down 1 infinitely. There is infinity inside 1 but we don't use the infinite pieces instead we break it down and break down other things like 5 and 3289 and even breaking down .000072362 so that we can build using the pieces. If you limit the pieces, there is still room to grow. Math is a tool and it is limitless unless you aren't using it, then the limit is you. We just section things off and use the pieces in different ways.
God
After 10'33" still don't know what he was on about.
Most mathematicians could care less about the axioms. They do math, prove things, solve problems, etc, not worrying about the axioms. Also, axiomatization come usually at the end, after we do most mathematics.
You haven't spent much time among mathematicians, I take it.
You're describing engineers.
@@starfishsystems ...Ι am a mathematician. Are you?
Is maths limited, or are our interpretations of maths limited?
That depends, is math a discovery or an invention?
@@nicholascarter9158
It's an interesting puzzle, isn't it? And you'll get different interpretations from different mathematicians.
Here's a bridging concept which is fun to think about. Suppose that ALL POSSIBLE mathematical proofs are just lying around in conceptual space. (By "proof" here I include both valid and invalid proofs: all statements which are in proof form. They're just finite patterns of symbols: Gödel based his undecidability theorems on exactly this.)
So, there's a set of candidate proofs lying around waiting to be discovered. Okay, so they're DISCOVERED. Done.
Not so fast. We have to figure out which are the valid ones. There's a lot of junk, just random symbol strings. There are sort of coherent ones which sadly peter out or end in internal contradiction. So out they go too. It's a lot of work. We would practically have to invent a methodology to sort through them all, and find the gems. This, I claim, is a creative act which requires a mind. So in that sense at least they are INVENTED.
Here is the simplest explanation at a 5th grade level: It's like a rulebook that can't prove itself is correct. Just like how you need a teacher or a parent to check your homework, the rulebook needs something outside of itself to make sure it's right.
The line between A & B can't prove itself because it lacks perspective.
You still cannot take out Ruzzian from Lex. Sad
Bull. Mathematics is representation. Pure math is just when someone places a wall around that representation and tries to compartmentalize the representation without it's counterparts. The counterparts being the things represented.
Mathematics certainly started as REPRESENTATIVE of cows and bales of grain and whatnot. That was thousands of years ago.
It's long since left that constraint behind. It's now a formal system, reliant only on its axioms. If it happens to produce a result that can be applied usefully in the world, that's great. It often happens, sometimes in surprising settings. But it's absolutely not a requirement.
If there was a god then god could easily subvert any logic because every single part of the sublogic is held by superlogic and anything could even reverse. Trying to overthink anything divine using logic is just stupid waste of time.
If there were commonplace violations of logic - whether due to the interventions of deities or for whatever cause - we would observe them, I presume.
We don't. Therefore there AREN'T such causes. This is basic Modus Tollens.
Math is based on axioms taken for granted. Everything else is proven! It looks to me like another religion.
if it is making it with time then it is fine
Nerd
There is no "r" in Godel. There's your incompleteness, right there. Now, tell us how to pronounce "colonel". Godel followed Einstein around like a puppy. Eff Godel.
The correct spelling is Gödel. The umlaut (diacritical mark over the 'o') confers the 'r' sound; Frenkel's pronunciation was correct.
Please move the bottle of coke lol.
It's there for atmosphere
How come 90% of Lex' guests are Jews?
Actually, it's slightly over 40%, which is still, in terms of the general population, disproportionately high. That being said, I doubt that Lex Fridman is concerned about filling an imaginary quota; he wants the leading experts in their fields who are still articulate enough to carry a 3+-hour podcast. If that means they are representative of one ethnic group more than others, the question you could ask yourself (but please, not here!) is what causes that intellectual preeminence.But that would mean research instead of being a casual anti-Semite on a public platform. Think you can handle being that responsible, that inquisitive...?
Because there are a lot of Jewish people who are intelligent and interesting to talk to
@@AB-wf8ek
Fixed that for you:
Because of the Khazarian Mafia.
I am sure there are a lot of goyim as smart as and even smarter but lex is discriminating against the non-khazarians i guess. He is racist!
@@astroid-ws4py Spoken like a brainwashed follower of the thirteenth tribe narrative.
Lex seems to be drinking a lot of CocaCola... that can't be healthy
eastern european things
Godel is overrated. Glorified liar paradox.
Well, sooner or later someone eventually has to walk in and say something truly dumb. I guess you're it. Congratulations.
Physics is based on mathematics? Physics is based on observation. Mathematics helps make the observations possible. But that still doesn't mean that Physics is based on mathematics.
Physics is based on both mathematics and observation. Mathematics provides the language and tools necessary for describing and analyzing physical phenomena, while observation provides the data that informs our understanding of the physical world.
Are you looking with 1 eye? or two?
Math is everything everywhere only most of the time. lol
I thought science or the scientific method, in general is based on observation, and many things can be observed that physics does not accept or even include in their studies.
Math seems to launch into theory quite quickly in physics. For instance, 1+1 equals 2 if counting apples, but not in the practical application of life in general. Also, if every action has an equal and opposite reaction, wouldn't this create a completely inert status ?
If the body compensates for the loss of senses, by strengthening others, what sense is sacrificed or strengthened when focusing on physics or math ?
@@artstrology Mathematics is not a science. It's not based on any observation. As explained in the video it's axiom based plus logic. I'm mathematician btw...
@@martindindos9009 Awesome, thank you. I am definitely not a mathematician. It has always seemed quite foreign to me, but I do see much value in it. As a carpenter, I have always preferred an engineer to an architect.
You may want to just accept a simple truth as your axiom that would create the space in your mind and in Time itself to answer all your questions in a given moment, GOD. That, that you would come to know would be infinitely more than what mathematics and science can ever answer . But that of your humanity and its true computing potential (mathematical mind) and its being (scientific lived reality) can transcend reality itself past our limits grounded in all we are (human) and know all via our spiritual self (soul) and be at peace, meaning all your questions have been answered. If one still seeks more answers in the elemental word, ground your intentions to the betterment and liberation of Humanity and not Vanity and be one with GOD.
Yep, it could be it's our fixation on numbers that is holding us back. Maybe numbers are just a trick the devil played on an idle mind. An endless, fractal loop of recurring numbers.
When I learned about recurring numbers, in 2nd grade, I lost any faith I might have had in maths being a solvable problem. As someone intimated to above, everything can be found in 1.
I am he. He is me. We are one.
Um, no.