Full podcast episode: kzfaq.info/get/bejne/hdmYY5B7mLqWno0.html Lex Fridman podcast channel: kzfaq.info 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.

This is now one of my favorite interviews. Love and math. What a guy

Frenkel has to be the most interesting mathematician ever. The whole interview is tops.

He is a top tier mathematician and explains to the audience in simple, digestable but still leapful mysterious manner. Giving 'simple' examples like srqt(-1) to explain the emotional concept hiding behind just brings joy to me, such a lovely person!

Wow I not only totally get his point but simultaneously finally understood complex numbers from this. Fantastic guest, Lex

Where in this clip did Edward Frenkel debunk AI intelligence?

When going through education, consider yourself blessed if you got teachers/instructors/professors who are as passionate about their field as Frenkel is about math.

Wow! This dude is absolutely passionate about math. There was pure joy in his eyes as he spoke.

Edward Frenkel... A beautiful Mind but What a beautiful Soul this man has, It shows through his eyes wise yet childlike. Awesome Thanks you Lex great interview.

AI has functional intelligence but it is not the same as human intelligence. It's the same as birds do not fly in the same way as aircraft fly.

The title sounds awkward and wrong somehow....

When I was a kid many people explained why computers would never be as good at chess than the best humans. The explanations where similar to Edward Frenkels explanations: "There just is something that we get and computers won't get in chess / mathematics.

How mathematicians can be romantics

It might depend on how you define imagination, but then how do you categorize experts learning new moves from AlphaGo/Zero? Were those moves not imagined if they were not done before?

He asserted it. There was no evidence provided.

LLMs are not AGI. However if something exists in nature, like intelligence, it can be atificially replicated.

such a great guest! thanks for bringing him on. he's eloquent, seems enthusiastic with such affectiveness for the topic and probably the kind of person I could sit down and talk about stuff for a long time. I love his accent too! adds "flavor"

I want to read this man’s ChatGPT history. I’m sure it would be fascinating

In order to search for new mathematical concepts, a LLM would have to be grounded not only on natural language, but also on things like formal logic, like a mathematician is. Because natural language already carries some logic in it, current LLMs already can "create" new concepts.

Dear Lex, I can’t resist commenting on what Edward Frenkel says in this interview. He uses the discovery of complex numbers as an example of something that an artificial intelligence could not cum up with. I think that that example shows precisely the opposite. Let me first mention that since the late 1960s, mainly thanks to the work of David Hestenes, we have known what complex numbers are, their intuitive and simple geometrical meaning, contrary to what E. Frenkel suggests. Geometric Algebra defines the “well behaving” product in 3d, which exists in any dimension, not only in 2, 4 and 8, that Frankel says does not exist. You can look up “Geometric Algebra” yourself. I am well convinced that an AI with some “model based reasoning” would have discovered the marvellous and beautifully symmetric structure of Geometric Algebra together with the few rules for 2D that Gauss and other mathematician discovered centuries ago, when the story of the complex numbers originated. The absence of the structure of Geometric Algebra kept the simple significance of complex numbers (rotors) hidden and created the myth that Frenkel describes. In other words, an AI would not have been foolishly fascinated with the mysteriousness of the complex numbers, so incomplete and unjustified, because it would have arrived straight to the structure of Geometric Algebra, inside which complex numbers, quaternions, octions, the vector product, rotation in any dimension, … are all easily explained with a single product! Geometric Algebra impacts quantum mechanics, computer graphics, general relativity, … Complex numbers are just rotors … Have a nice weekend Lex! Well done, as always!!!!!

"Everybody knows that the dice are loaded, everybody rolls with their fingers crossed"

AI cannot come up with new ideas, but it can see patterns in a large set of data that we didn't notice. It's not an emergent property, it's an unexpected result. Given the impossibility for a human being to read and memorize said dataset, unexpected results are to be expected.

Wolfram told the story of him playing with GPT 3.5. He asked it to write a persuasive essay arguing about the bluest bear that exist. So chat gpt started with " Most people don't know this fact but blue bears do exist... They are found In the Tibetan Mountain... their color doesn't come from pigment, instead it comes from a phenomenon analogous to how butterflies produce colors ....." Near the end of the essay, he was like , " wait a minute do blue bears actually exist ? " He had to google it to make sure. Now tell me again, AI can't have imagination.

The “observed and observer” is a phrase coined by Jidu Krisnamurti, then taken by David Bohm and used in his description of quantum mechanics. I always find that fascinating.

What we know about consciousness already seems prove something fantastical lies beneath the surface of our current knowledge. Ancients seemed to have understood this much more than we do now.

The square of -1 is equal to 1. In other words, (-1) multiplied by (-1) gives us 1. However, it is not possible to take the square root of -1 in the real number system. In order to represent the square root of -1, mathematicians use the imaginary unit "i", which is defined as the square root of -1. Therefore, the square root of -1 is represented as "i" in mathematics. So the square root of -1 can be written as √(-1) = i.

You could make a good argument that a machine intelligence would more easily embrace complex numbers than humans do. After all, humans are endlessly constrained by “that’s not allowed” or “that doesn’t make sense.” These are basically emotions. A program that can self improve would already have the quality of “there are some things that are true but that I don’t already know and understand.”

I remember being taught imaginary and complex numbers, and I remember hearing my brain say, "That's it, I'm out of here". That was the point at which me'n'maths bifurcated. But I often reflected on the first maniac to pursue imaginary and complex numbers; what sort of lunatic does that? Now I know he was the same fella that invented the double cv joint - weird.

For GAI to understand that sqrt(-1) could have real value its matter of learning to do design of experiments which means it will eventually construct it.

I think that the creative spark that Frenkel is describing is what philosophers call abductive inference, as opposed to deductive and inductive inference.

It should say "Mathematician Imagines He Debunks AI Intelligence"

Regarding the complex number point, you can just explicitly ask the “agi” to probe things in mathematics that has historically been seen as impossible or unintuitive. Seems a very simple “fix” for a advanced llm (with mathematical reasoning) to discover complex numbers etc.

In my opinion this argument starts off with the assumption that the LLM can't deduce the new way of thinking, Simply by means of the historical data of said mathematician that pondered sqrt(-17). It can simply deduce from even only that 1 instance that divergent "thinking" needs to be done. If, how am I wrong?

To summarize, LLMs predict the most common answers to a particular input. Solving complex problems requires imagination and predictions that go AGAINST the grain and expected future. LLMs have to keep predicting future based on the past.

There's nothing stopping an AI from adopting the hypothesis that the square root of a negative exists and seeing what happens.

Do we have any use for fractional dimensions or even complex ones and not just integral ?

It's so embarrassing at the moment, when so many super bright people shows their simplistic 'human exceptionalist' worldview. People that say: 'oh no, AI can't do this like we can', is deniers bc it clashes with their worldview that we are something special - which it turns out - we are not. I look forward to hear them again when they have moved on from their old standpoints.

AI doesn't need to be exactly like human to have intelligence or to replace human.

Terrible title @lex he’s not talking about “AI” generally but about LLMs specifically.

I mean, LLNs aren't going to be the models that advance mathematics, but even still you could try to program an neural network to 'play', e.g. allow ignoring certain rules I'm order to solve problems. Cool episode though.

I hoped he would talk about AI more but he's right - language models are mostly not built on their own experience (mostly, it is supervised learning although there is some reinforcement learning in newer models). They act more like function interpolaters which can produce impressive results in the right context. Getting them to extrapolate anything and they can produce complete nonsense. They don't understand concepts deeply, there simply very very good mimics of the training data they have seen.

A sufficiently smart model can extract deeper meaning from less evidence. Who is to say new mathematics is not already hidden in the relationships found in the existing training data? The fact that the canon implies that something is not possible would not necessarily detain a LLM, because the it is not explicitly trained to respect the rules of Mathematics or take them with any special regard. There’s actually nothing blocking it from going beyond, from using obscure references or just stumble into a new way of solving a problem, thus creativity needs to be considered using non-anthropomorphic lens in this case.

Best clip ever. Wow. Raw and true and vulnerable. Great job catching this moment. My goodness

It's all nice and well explained but how does it relate to AI?

He's factually wrong; imaginary numbers are not required for quantum theory, we simply use them because equations are nice and easy. Rumblings of a man who doesn't understand neural nets, although none of us do

as they say.. when u know the rules, u can break the rules

A mathematician is one who knows how to find analogies between theorems, a better one who sees analogies between proofs, a still better one who sees analogies between theories, and one can imagine one who sees analogies between analogies. Stefan Banach, polish mathematician, one of the greatest ever lived...Feynman/Frenkel type, so you all would love him. Fantastic interview ))))

How was AI intelligence debunked? Did I miss something?

What seems like a jump to us might be obvious to AI

It's not artificial intellence that scares me. It's artificial stupidity...

_@Lex Fridman_ -- Edward Frenkel is brilliant and I appreciate his insights and his humility very much. Thanks for posting this video clip. For another, more down-to-earth, perspective on complex math, you should interview an engineer. You know, the people who apply the crazy things mathematicians dream up. A good electro-mechanical engineer can easily provide tons of real-world examples where the "imaginary" number system is essential to describe the day-to-day reality we observe. For example, audio engineers would know nothing about phase without complex math. They would have no idea how two seemingly identical sound waves (identical magnitudes) can completely cancel (when they are 180° out of phase). And it goes even deeper because complex math is at the center of the Heisenberg uncertainty principle. In audio we can know everything about the magnitude of sound. But if we do, we'll know nothing about when in time the sound occurred. On the other hand, we can know everything about the time when a sound occurred, but we'll know nothing about its magnitude. Both cannot be fully known at the same time, creating the uncertainty. This is why advanced audio measurements systems must trade the magnitude-frequency domain for the time domain, depending on the job requirement. And it illustrates how complex math affects the macro world -- not just the micro world of quantum mechanics. Then along came a clever guy (Richard Heyser 1931-1987) who discovered that you could map mathematically into an abstract dimension via a Hilbert transform and operate simultaneously on both the magnitude and phase of sound, then map back to our reality with the result. The technology this birthed is Time Delay Spectrometry or TDS. Heyser applied this same "trick" to medical MRI (magnetic resonance imaging) systems to greatly increase their resolution. This just touches the surface of the amazing way complex math weaves throughout our world. Another great example is kinetic vs potential energy. Kinetic energy requires the "real" numbers and potential energy requires the "imaginary" numbers. It bugs me no end that we are stuck with these awful names for these two essential number systems. I wish we could do away with the "real" and "imaginary" labels and call them something else.

the moment we create an AI machine without any concrete goals or set objectives/tasks but yet it still goes on to do something (even something as simple as moving around (if it has a robotic body)) is the moment we know it's self-aware and conscious. IDK maybe I'm dumb but that's the way I imagine we would know that it's the real deal

From about 300BCE to until the early 19th century, humanity made a pretty basic assumption that two parallel lines would never intersect. Euclid. It was taken as law. Yet, it's pretty clear to anyone that studies a globe, that parallel lines can indeed intersect, they will at the poles. It's not an abstraction that difficult to come to terms with, you don't need to be Einstein to grasp it. Yet all of humanity missed it, even when they were actively looking for it, for a very long time. Sometimes even things that are staring at us in plain sight, elude us, because we fall into dogmatic beliefs about what we take as law.

And it only took 200,000 years to find those imaginary numbers.

Chat GPT (I don’t care which version) is an absolute train wreck when it comes to higher math

I like the use of the phrase "imaginary parts" this is very similar to how people have their "real parts" or their bodily parts, and then they have their "imaginary parts" which would be the mind, thoughts values, feelings, goals etc. Even numbers have this interior terrain

He didn't really debunk it, he explained why he thinks it is not intelligent. And he wasn't even ralking about AI in general but about Large Language models. People often forget that AI is a ridiculously large subject and LLMs (and more specifically transformers) are just one way to go about it.

The devices we employ to build and run ”AI" models would not exist in the absence of the mathematics which the models themselves are unlikely to be capable of even conceiving. It seems to me that a (ironic) parallel could also exist with regard to humans attempting to decipher consciousness?

Nice title.... Ai intelligence Artificial intelligence intelligence

While current AI cannot imagine or conceive of the square root of minus 1, does he believe it won’t be able to in the future. Current AI can be considered as infants and just like human infants might not be able to imagine the square root of minus 1, adult humans and likely adult AI might.

A complete and open minded person isn’t one thing, intuition is important.

This is plainly speaking inbox, the yellow brick road of progress.

AI: “how many bullets does not take to kill humans? How long can humans survive poisoning the air, food and water?”

Mr. Einstein solved his mysteries on a solid couch, while lucid daydreaming.

I invented a language with my farts. Let's see AI do that!

AI doesn't need sqrt(-1). It just needs bits or qubits. It doesn't have to develop math for humans to understand.

This man almost convinced me that LLMs emulate rather imitate human behavior based on all the existing knowledge but not reproduce the vast imaginative capabilities exercised by humans. ChatGPT would've responded that Sqrt of a a negative number doesn't exist. whereas a human has gone beynd what already exists and made new contributions

So, Bilbo Baggins was a mathematician too! (Huge respect for the guest)

In my opinion, square root of negative one is just two number lines or axises that can be oriented in any way. If it is on the same numberline, say x, then it should be one. I think its a fake problem since weve favord the x, y, and z which are arbitrarily at 90°s from eachother. Spherical coordinates are the next step. Negative numbers are literally whole numbers if you just translate their values into the real plane. Always have a camera or measuring device to measure your original data called the origin.

And why exactly shouldn't a machine be able to do such discoveries?

Brilliant discussion.

I understand none of this but am super relieved to see a brilliant person confirm my hope and feeling that AI cannot ever be sentient

At the core of his statements is the fact that humans use inferential reasoning not just the compilation of data. There's the learning of facts, even the curation and organization of facts, but the leaps come from our DRAM. Not a Mathematician, but a very enjoyable video. Kudos for the Einstein references !

Great interview indeed! Here's an example of how far we still have to go...Just take a look at this simple instruction that the poor bot fumbled up this evening as I enjoyed a cup of coffee and a few giggles. User: aAbcde9fz list every 2nd character in the above string please. ChatGPT: Sure, here are the every second characters in the string "aAbcde9fz": "Abd9z" User: why did you include 'b' ? ChatGPT: I apologize for the confusion. The correct every second character in the string "aAbcde9fz" is: "Abe9z" User: No. The correct answer is actually: Acef ChatGPT: I apologize for my mistake earlier. You are correct that the correct every second character in the string "aAbcde9fz" is: "Acef" //Unless the bot, being a computer takes it for granted that the first character of any given string is always '0' and then 1 2 3 etc ? But even so, its first answer is still wrong. Or am I missing something?

ChatGPT's answer to could you have invented complex numbers: "The potential of an AI to conceive of a new concept like complex numbers depends on its architecture and training. If an AI is designed with the ability to learn, reason, and make creative connections, it may be able to come up with new concepts that were previously unknown or unexplored. However, this would require a rich and varied set of input data, as well as the possibility for the AI to engage in hypothesis generation, conjecturing, and testing. That said, since AI is typically a product of human knowledge and ingenuity, it is important to consider that the creative spark underlying the development of something as novel as complex numbers is still driven by human creators and programmers. In conclusion, it's theoretically possible for an AI to come up with a concept like complex numbers if they hadn't been invented before, but it would require a suitable architecture and a deep, diverse set of input data to facilitate that level of creativity."

Its almost like complex numbers exist in a higher dimention and we know they exist but dont know what they are. Yet they are directly linked to us and some how calculate our physical world in the mandbrot set.

Lex should have Swami Sarvapriyananda on for a discussion on consciousness and vedanta.

Math being rational is a subset of reality which includes rational and irrational. For example, emotions are very important features of our consciousness and they are irrational. What's remarkable about the irrational/emotional aspects of consciousness is how creativity comes from our emotional aspect. It's the irrational that leaps from what we know to entirely new possibilities. Conversely, the intellect relies on logic which plods along from what we know inching toward a slightly different idea. That's why revolutionary new ideas first appear using our irrational emotions. However, since irrational emotions lie outside the limits of rational math and logic computers cannot explore emotions or use those irrational features of consciousness to leap to entirely new perspectives, solutions, etc. “If I create from the heart, nearly everything works; if from the head, almost nothing.” -Marc Chagall (1887-1985), artist

The title does not correspond with the chat. The mathematician does not debunk anything. He doubts it can be and points that human mind does not operate in the same way. Clickbait like this is detrimental. Stop doing it.

I have a degree in math and computer science, and I know quite a bit about ML and AI. I agree with 100% of what he says. It will be exciting when there is a new AI technology that can think abstractly and generate new knowledge, but we don't have that yet!

As a highly creative designer-craftsman I was fortunate to work in Silicon Valley for some of the best and brightest, and richest, people in the world. I listened to their "theories on their dreams" and brought it to fruition. I also witnessed their private lives, and decisions they had made about their dream.

quaternion multiplication does not commute

Later: AI debunks mathematicians intelligence.

Maybe this guy is missing the point: LLM would say, sqrt(-1), why not? I imagine chatpgpt saying: sqrt(-1) is not a usual integer, however if we accept it ..."

I disagree with the example of imagining the square root of -1 for a rather concrete reason. Neural networks have features that are fundamentally made up of real parameters, and these features can achieve extremely high levels of abstraction - for example a feature representing whether a picture has a cat in it! Even that should be a strong clue they can come up with other sorts of abstraction, like the square root of minus one. There is one much simpler type of feature which is relevant to the claim. Topologically, a real-valued feature has no loop - if you keep increasing or decreasing it, you never see the same values again. But from a single such feature it is possible to generate two new features using sine and cosine that are related by the familiar sin^2 + cos^2 = 1 rule. This effectively maps the line of a single feature to a circle in the complex plane by the transformation x -> e^ikx. The two transformed features are effectively a single new feature with different topology. More generally two features always have the capability of being used so that they represent complex numbers, and where complex features are useful to a model they can emerge naturally. So it is safe to say that not only can general neural networks come up with the notion of a square root of minus one, they can do this sort of thing quietly in the background where it turns out to be useful to a model. And if they can do it quietly, it is certainly reasonable to believe they could talk about it if they had a large language model as well!

I respect a mathematicians perspective, but I sort of look at it in the same way I would look at a physicist describing the human body. If you break us down to our atomic structure; mostly just hydrogen, carbon, nitrogen and oxygen; it's hard to imagine how it can all come together to create consciousness. AI, although composed of simple 1s and 0s, could create emergent intelligence at a higher level when those 1s and 0s are combined in certain ways. What we think of as intelligence may simply be a boring and predictable outcome, but we just can't comprehend all the variables so it looks unique and special. We already see AI "hallucinating" beyond a machine's traditional cold and strict boundaries. Also, programming frequently injects randomness into the process to encourage unique outcomes amongst the array of potential solutions. I would say these things are demonstrations of a type of imagination, allowing for new ideas to emerge from unlikely places. We see glimpses of what can be described as "common sense" within the latest LLMs that may be an indicator how future AI will be able to self-assess failure and make new attempts to remedy the situation. Hell, AlphaFold's very purpose is to discover new protein structures, which is a practical example of AI thinking of unique ideas beyond just a brute-force approach.

Irrational numbers and complex numbers are not the same thing. Why are irrational numbers irrational? There lies the great divide of mathematics and reality

This guy is truly awesome, great interview Lex!

So in the 4th dimension, you can't multiply by 3, 5, or 7. I would want to look into the significance of that.

A ubiquitous presence does not need to be intelligent in order to be dangerous. Think of termites. The risk is not that computers will of necessity outsmart us, the risk is we will hand them the steering wheel.

What about searching for numbers structures in the 16 dimension? I got curious as he said only powers of 2 made sense and 16 is a power of 2

What if imaginary numbers dont exist and we are just dumb for inputting a negative under the square root? Do imaginary numbers appears spontaneously in regular math equations?

The tool is phenomenology. The philosophical method for describing subjective experience has already been developed.

This is a profound intellectual discussion about mysticism, very rare. That said, perhaps it's not a matter of imagination to come up with ingenious mathematical constructs like the squareroot of -1, but merely the ability to challenge one's own assumptions. If or when AI reaches that pinnacle, we would be making profound scientific discoveries that could let us bend spacetime and travel to furthest parts of the universe.

So elegant in conversing such a complex and mysterious topic.

Prof. Frenkel stops at octonions, but I’ve read that numbers of dimension 2 to the power of 4, or 16, called sedenions, have also been defined and studied, and have very curious properties. Or rather, they lack properties that are fundamental to real or complex numbers, like associativity and commutativity. They also allow division by zero, where multiplying two non-zero sedenions can give zero as an answer!! I don’t know that they have found any practical applications though.

I think AI would be capable of coming up with sqrt(-1), because it doesn't require imagination, it just requires generaliization of algebraic rules. An AI trained on math would have seen the concept of generalization already numerous times and be able to apply it to new fields it hasn't seen yet.

As a set of instructions, ai will be able to sample every subset of any sample.

Yeah, sorry, not buying it. He says there are tools for analysing things like consciousness, then fails to enumerate or describe any of them. What he did do, was to point out the mysterious intuitive leaps (such as sqrt(-1)) that human intelligence makes. Meanwhile, many of the people involved with advanced AI systems like GPT, are all worried about the way that they don't have good explanations for the "emergent behaviours" of their AI systems. They're learning to do things that are not explicitly taught. How are these not the same phenomena?

Love this podcast, but why the clickbait titles? What does it even mean to "debunk AI"? AI stands for artificial intelligence, why would you write "AI intelligence"? I've always admired Lex for his integrity, so why is this title so disingenuous?