Eine seltene, extrem lange Interview mit Terence Tao: Mathematik, KI und Ratschläge für junge Menschen

By Lu Yu Buyuan from Aofeitai
QbitAI | Official Account QbitAI

Terence Tao rarely gave an extremely long interview, sharing his latest insights on mathematics, AI, education, and human intelligence.

As a Fields Medalist, Terence Tao has always been regarded as one of the greatest mathematicians of our time. This conversation with Lex Fridman, a podcast guru with an MIT technical background, is his first interview by a non - academic institution lasting over 3 hours in recent years. The content covers many hardcore topics such as cutting - edge mathematics, formal verification of AI, and research methodology.

He not only shared professional viewpoints related to mathematics and physics but also, against the backdrop of the rapid development of current AI technology, pondered over popular topics such as basic education and AI applications...

Terence Tao came up with many golden sayings, for example:

The distance between AI and the Fields Medal is just one graduate student.

The human community in the plural sense will create the most top - notch super - intelligent agents, and is more likely to achieve breakthroughs in the field of mathematics than a single mathematician.

The key to mathematics lies not only in finding an effective technical path but also in eliminating wrong answers among dozens of potentially applicable methods.

Science is usually an interaction among three things: the real world, our observations of the real world, and the models of how we think the world works.

In terms of the way of understanding and perceiving the world, mathematics starts from axioms and focuses on models, while physics is driven by conclusions and pays attention to collecting results.

With the assistance of AI, mathematics will have more experiments in the future, not just theories.

The beauty of mathematics lies in that you can change the rules as you like, which is something that cannot be done in any other field.

Solving difficult problems is like a Hong Kong action movie, defeat them one by one to achieve great success.

Terence Tao believes that AI is reshaping the human scientific paradigm. In the ultimate questions of mathematics and physics, AI will become an important partner for humans to explore these paradigms, but it cannot replace human intuition and creativity.

Regarding mathematics, he also talked about the following world - class mathematical problems in an accessible way:

The Kakeya Conjecture

The Navier - Stokes Regularity Problem

Szemerédi's Theorem

The Theory of Everything

General Relativity

The Poincaré Conjecture

The Twin Prime Conjecture

The Collatz Conjecture

The Riemann Hypothesis

Fermat's Last Theorem

Well, yes, this is a high - intelligence, high - density, and high - intensity conversation. If you are also ready for a "brain scream", let's take a look at the full text of the interview we have organized~

The Full Text of the Interview with Terence Tao

Undergraduate Education Only Needs to Solve 10% of the Problems

Q: What was the first truly difficult research - level mathematical problem you encountered?

Terence Tao: In undergraduate education, you will learn about those truly difficult, even seemingly unsolvable problems, such as the Riemann Hypothesis and the Twin Prime Conjecture. But in fact, this is not a big deal. What's really interesting is that existing techniques can do about 90% of the work, and you only need to solve the remaining 10%.

During my Ph.D., the Kakeya problem caught my attention. In fact, this problem had just been solved, and it was also a problem that I invested a lot in during my early research. Historically, it originated from a small puzzle proposed by the Japanese mathematician Soji Kakeya around 1918.

This puzzle is as follows: Suppose you have a needle on a plane, or imagine making a U - turn while driving on the road. You want to turn the needle around in the smallest possible space.

But the movement of this needle is infinite, so it can be regarded as a unit pointer rotating around a center. This requires a disk with an area of π/4.

Or you can make a three - point turn, which actually only requires an area of π/8. So, it is more efficient than rotation. Therefore, for a while, people thought this was the most effective way to turn the pointer around.

But Besicovitch proved that, in fact, you can turn the needle around with an arbitrarily small area, such as 0.001. So, there can be various fancy back - and - forth turning operations to achieve this, so that the needle can pass through every intermediate direction during the turning process.

This occurs in a two - dimensional plane. We understand everything in two dimensions. Then the next question is, what happens in three - dimensional space?

Suppose the Hubble Space Telescope is a tube in space. If you want to observe every star in the universe, you need to rotate the telescope to point in every direction.

Make an unrealistic assumption: Suppose space is very precious, although it is not actually the case. You need to occupy the smallest possible volume to rotate your needle, that is, the telescope. Then how small a volume is needed to do this? You can try to modify the basic structure so that the telescope has zero thickness, so that you can use an arbitrarily small volume.

But the problem is, if your telescope does not have zero thickness but a very thin thickness delta, then as a function of delta, what is the minimum volume required to see all directions? And as delta becomes smaller and the needle becomes thinner, the volume will also decrease, but how fast does it decrease? The conjecture at that time was that it decreased very, very slowly, like a logarithmic decrease, and this was also confirmed after a lot of work.

What's interesting about this problem is that it turns out to have amazing connections with many problems in partial differential equations, number theory, geometry, topology, and combinatorics.

For example, in wave propagation, when you splash some water around and create water waves, they will spread in all directions. But waves also exhibit both particle - like and wave - like behavior, resulting in a so - called "wave packet", which is very localized in space and moves in a certain direction over time. If you plot it in space and time, it will show a tubular region.

So, it is possible that an initially very dispersed wave will all focus on one point at a later time. Just like throwing a pebble into a pond, the ripples will spread out. But if you reverse the time of this scene, and knowing that the wave equation is time - reversible, you can imagine that the ripples will converge to a point, and then there will be a huge splash, and perhaps even a singularity will occur.

Geometrically, the same is true for light rays. If this wave represents light, and the wave is regarded as a superposition of photons, and the photons all propagate along light rays and eventually focus on one point.

So, you can focus a very dispersed wave into a very concentrated wave at a certain point in space and time, but then it will disperse and separate again.

But if the conjecture has a negative solution, it means there is a very efficient way to pack tubes pointing in different directions into a very narrow small - volume region.

Then you will also be able to create some waves that start out very dispersed, but they will not only concentrate at one point but also have a large number of concentrated points in space - time. You will create what is called an explosion, and the amplitude of the waves will become so large that the physical laws they rely on are no longer the wave equation but something more complex and non - linear.

So, in mathematics and physics, we are very concerned about whether certain wave equations are stable and whether they can produce singularities.

Create an Explosion by Changing the Laws of Physics

Q: Could you talk about the Navier - Stokes problem?

Terence Tao: There is a famous unsolved problem called the Navier - Stokes regularity problem, which controls the flow of incompressible fluids such as water.

The problem is, if you start with a smooth velocity field of water flow, will it concentrate to such an extent that the velocity becomes infinite from a certain point, and that point is a singularity. Of course, we don't see this in real life.

In fact, in recent years, there has been a consensus that for some very special initial configurations, such as water, a singularity can form, but it has not been truly proven yet.

Among the seven Millennium Prize Problems of the Clay Mathematics Institute, solving one of them can earn a one - million - dollar prize, and this is one of them. Only the Poincaré Conjecture has been solved among them.

So, although the Kakeya Conjecture is not directly related to the Navier - Stokes problem, understanding the Kakeya Conjecture helps us understand problems such as wave concentration, which may indirectly help us better understand the Navier - Stokes problem.

This is where mathematicians differ from almost everyone else. If something holds in 99.99% of cases, for most things, that's enough. But mathematicians care about whether it holds in 100% of cases. Most of the time, water doesn't explode, but maybe you can design a very special initial state to make it explode.

The Clay Prize problem involves the so - called incompressible Navier - Stokes equations, which control fluids like water. There are also the so - called compressible Navier - Stokes equations, which govern fluids like air.

In weather forecasting, there are a lot of fluid dynamics calculations. A large amount of data needs to be collected to initialize the Navier - Stokes equations and even solve them as much as possible, so it is very important in real life.

Q: Why is it so difficult to prove the general conclusion of the equation?

Terence Tao: An example is Maxwell's demon. Maxwell's demon is a concept in thermodynamics. Suppose you have a box containing both oxygen and nitrogen without a partition between them, and they should remain mixed.

And at this time, there may be a microscopic demon called Maxwell's demon that causes each time an oxygen and a nitrogen atom collide, they will bounce in a certain way, so that all the oxygen floats to one side and the nitrogen flows to the other side.

This is a statistically impossible configuration, but it often occurs in mathematics.

For example, the digits of pi, 3.14159... These digits seem to have no pattern, but in the long run, there are as many 1s, 2s, and 3s as 4s, 5s, and 6s. In theory, it should not favor a certain digit, but maybe there is also a demon in pi. When you calculate more digits, it will favor a certain digit, but our current technology cannot prove it.

So, back to the Navier - Stokes problem, the fluid has a certain amount of energy, and because the fluid is in motion, the energy is transferred around. At the same time, water has viscosity. If the energy is dispersed in many different positions, the natural viscosity of the fluid will cause the energy to decay and tend to zero.

This often happens when we actually do experiments with water. For example, the turbulence or waves that occur when splashing water will eventually stabilize. The lower the amplitude and the smaller the velocity, the calmer it becomes.

But maybe there is a certain demon that constantly pushes the energy of the fluid to smaller and smaller scales, moving faster and faster. At a faster speed, the viscosity has less influence.

So, it may create a so - called self - similar blow - up scenario, where the energy of the fluid starts from a large scale and is all transferred to a smaller region in the fluid, and then moves to an even smaller region at a faster speed, and so on.

Each time may take half the time of the previous time, and then all the energy can converge to a point in a finite time, that is, a finite - time blow - up.

So, if you have a large water vortex, it tends to split into smaller vortices, but it won't transfer all the energy from a large vortex to a certain small vortex. Instead, it will be dispersed and transferred to three or four small vortices, and the small vortices will each split into three or four even smaller vortices. The energy is dispersed to the extent that viscosity can control everything.

But if you can concentrate all the energy in a certain way and do it fast enough so that the viscous effect doesn't have enough time to calm everything down, then an explosion will occur.

So, some papers claim that by only considering energy conservation and carefully using viscosity, you can control the Navier - Stokes equations and even many similar types of equations.

So, in the past, people have tried many times to obtain the global regularity of the Navier - Stokes equations, which is the opposite of finite - time explosion, and the velocity needs to remain smooth. But all attempts have failed, and there is always some irreparable sign error or other error.

So, what I'm interested in is why we can't disprove finite - time blow - up. If I can average the Navier - Stokes equations of motion, then I can turn off some ways of water interaction and only keep the parts I need.

Especially if there is a fluid that can transfer energy from a large eddy to a small eddy, I will close the channels that transfer energy to other vortices and only guide it into this small vortex, while still maintaining the law of energy conservation.

I hope to create an explosion by changing the laws of physics, which is something that mathematicians can do.

But there are also mathematical obstacles. So, the work I did is that when I turn off some parts of the equation, its non - linearity decreases, and it becomes more regular and less likely to explode. But I also found that by turning off a carefully designed set of interactions, I can force all the energy to explode in a finite time.

This means that if you want to prove the global regularity of the Navier - Stokes equations, you must use some characteristics that the real equation has, but my artificial equation does not satisfy. Therefore, some methods can be excluded.

So, the key to mathematics is that it is not just about finding an effective technique and applying it, but you need to avoid those ineffective techniques. For really difficult problems, there are usually dozens of methods that you may think are applicable, but only after accumulating a lot of experience can you realize that these methods simply don't work.

So, finding counter - examples for neighboring problems can rule them out, thus saving time and not wasting energy on things that are impossible to succeed.

The technique I used is based on super - criticality. In partial differential equations, the equations are like different forces tug - of - warring. In the Navier - Stokes equations, there are two competing terms: the dissipation term and the transport term. If the dissipation term from viscosity dominates, then you will get regularity. If the transport term dominates, it will be an unpredictable non - linear situation.