Terry Tao, the greatest mathematician in the world

[MiloRambaldi]

one only needs to look at the different fields of mathematics Tao has touched to see how intelligent he is... people here saying 'he isn't as smart as Perelman' are relegating themselves to old women gossiping on a Sunday afternoon at a gathering.. quite pathetic.... that is not the point of mathematical discussion...

...

the simpletons working in finance. ...

I guess you must work in finance?

Look at the title of the thread. It invited debate on who is the best mathematician. IMHO, it the best result that counts and my impression is that none of Tao's (obviously great) achievements has equaled Perelman's achievement, no matter how many fields he "touched".

 

[Daniel Duffy]

one only needs to look at the different fields of mathematics Tao has touched to see how intelligent he is... people here saying 'he isn't as smart as Perelman' are relegating themselves to old women gossiping on a Sunday afternoon at a gathering.. quite pathetic.... that is not the point of mathematical discussion...

i think people hate math because they don't understand it. if i don't understand something, then of course i won't like it... the hatred comes after many years of not understanding it.. even people who go to graduate school don't understand math, let alone the simpletons working in finance. i would say that you're on your way to understanding math when you feel that you are completely useless at it and have to resort to going to older books, a la Bourbaki, Lusin, etc...

Bourbaki has ruined mathematics.

Q what is 2+3?
A the same as 3+2
Q why?
A it is commutative

 

[Daniel Duffy]

Here about JL Lions and the PDE movement in France.

SIAM: Obituaries: Jacques-Louis Lions

At a time when the French mathematical school was almost exclusively engaged in the development of the Bourbaki program, Lions---virtually alone in France---dreamed of an important future for mathematics in these new directions; he threw himself into this new work, while still continuing to produce high-level theoretical work on PDEs.

 

[vertigo]

I guess you must work in finance?

Look at the title of the thread. It invited debate on who is the best mathematician. IMHO, it the best result that counts and my impression is that none of Tao's (obviously great) achievements has equaled Perelman's achievement, no matter how many fields he "touched".

that is such a stupid question it doesn't even merit acknowledgement, let alone a response.

i won't even go into your 'method' of classify the best mathematician. if you want a serious discussion about who is the 'best' mathematician, you should take a deep look into the roots of mathematics. for example, take a look at the history behind perelman's proof.. others were involved..

as humans we are always drawn into the framework of classifying everything and everyone - why do you have to ask the question "who is the best mathematician?" in the first place? it is a very stupid question, you are asking a question, subjective in nature, about a field that is objective in nature..

Bourbaki has ruined mathematics.

Q what is 2+3?
A the same as 3+2
Q why?
A it is commutative

blow through a dusty surface and your eyes will breathe. ever heard that quote? bourbaki's books read like fraudulent arrogant shit at first. so you stop reading and forget the pain of reading such crap. ok....

usually one takes a look at the Bourbaki books when they are in the 'exploring' stage.. you move past this stage when you finish graduate school. you then finish a PhD then you work for a hedge fund (or whatever) and eventually, you feel embarrassed at yourself. you have been working with morons, simpletons, people who add nothing to society. they are not artists. they do not practise mathematics in any way, shape or form. worst of all, they cannot even do their jobs. a mathematician is an artist, not a prostitute.

so you look back at your books, at first the sight of topics such as measure theory and group theory excite you - you have found structure again! but now you are a veteran and years of experience working with bullshit consultants have taught you not to eat up words thrown at you... so you read modern mathematics books and scientific articles.. and you start asking yourself some deep questions that these books just don't explore or answer... you ask yourself why the theory of integration is so poorly explained, why the axiom of choice does not make sense, why notation is so confusing... etc.... you read bourbaki again, and the answers begin to come to you. bourbaki has no bullshit. that does not mean that it is perfect. for example, the notation "integral f(x) mu (dx)" is awful, it shows no understanding of what a measure is, instead you should be writing "mu(f)"

 

[Daniel Duffy]

"so you look back at your books, at first the sight of topics such as measure theory and group theory excite you - you have found structure again"

Never liked group theory and measure because of lack of applications.

//

What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? "Oh, curse this maths" - concludes the student (who, possibly, becomes the Minister for Science in the future).

We get a totally different situation if we start off not with the group but with the concept of a transformation (a one-to-one mapping of a set onto itself) as it was historically. A collection of transformations of a set is called a group if along with any two transformations it contains the result of their consecutive application and an inverse transformation along with every transformation.
This is all the definition there is. The so-called "axioms" are in fact just (obvious) properties of groups of transformations. What axiomatisators call "abstract groups" are just groups of transformations of various sets considered up to isomorphisms (which are one-to-one mappings preserving the operations). As Cayley proved, there are no "more abstract" groups in the world. So why do the algebraists keep on tormenting students with the abstract definition?
By the way, in the 1960s I taught group theory to Moscow schoolchildren. Avoiding all the axiomatics and staying as close as possible to physics, in half a year I got to the Abel theorem on the unsolvability of a general equation of degree five in radicals (having on the way taught the pupils complex numbers, Riemann surfaces, fundamental groups and monodromy groups of algebraic functions). This course was later published by one of the audience, V. Alekseev, as the book The Abel theorem in problems.
What is a smooth manifold? In a recent American book I read that Poincaré was not acquainted with this (introduced by himself) notion and that the "modern" definition was only given by Veblen in the late 1920s: a manifold is a topological space which satisfies a long series of axioms.
For what sins must students try and find their way through all these twists and turns? Actually, in Poincaré's Analysis Situs there is an absolutely clear definition of a smooth manifold which is much more useful than the "abstract" one.

 

[MiloRambaldi]

I guess you must work in finance?

that is such a stupid question it doesn't even merit acknowledgement, let alone a response.

It is? QuantNet is a quantitative finance forum.

Because finance is so far beneath you?

 

[Polter]

Never liked group theory and measure because of lack of applications.

I think it can be pretty interesting:
Running Probabilistic Programs Backwards | Lambda the Ultimate

Paper: [1412.4053] Running Probabilistic Programs Backwards
Slides: http://www.cs.umd.edu/~ntoronto/papers/toronto-2015esop-slides.pdf
Code: ntoronto/drbayes · GitHub

"Many probabilistic programming languages allow programs to be run under constraints in order to carry out Bayesian inference. Running programs under constraints could enable other uses such as rare event simulation and probabilistic verification---except that all such probabilistic languages are necessarily limited because they are defined or implemented in terms of an impoverished theory of probability. Measure-theoretic probability provides a more general foundation, but its generality makes finding computational content difficult.

We develop a measure-theoretic semantics for a first-order probabilistic language with recursion, which interprets programs as functions that compute preimages. Preimage functions are generally uncomputable, so we derive an abstract semantics. We implement the abstract semantics and use the implementation to carry out Bayesian inference, stochastic ray tracing (a rare event simulation), and probabilistic verification of floating-point error bounds."

"We show that measure-theoretic probability can be made computational by
1. Using measure-theoretic probability to define a compositional, denotational
semantics that gives a valid denotation to every program.
2. Deriving an abstract semantics, which allows computing answers to questions
about probabilistic programs to arbitrary accuracy.
3. Implementing the abstract semantics and efficiently solving problems."​

 

[vertigo]

I guess you must work in finance?



It is? QuantNet is a quantitative finance forum.

Because finance is so far beneath you?

in this modern age using the brain is optional.. nevertheless i would have hoped you would use it sometimes. i wasn't referring to that question, i was referring to the question of who is the best mathematician.

like a lot of people who don't use their brain, it is you who is the fool.. and the moron who does not understand anything. often it is the people who criticise XXXwhom know the most about XXX. if you want to know how something works, learn how to break it. ever heard that? by saying that finance is beneath me, you are projecting me to be arrogant. no, it is you who is stupid and is arrogant, because you do not use your brain. first, use your brain and think, then post... instead of having a discussion on the correct question "what is the best mathematical result?" we are now discussing what is beneath us... you know what this reminds me of? sunday afternoon tennis club sessions with soccer moms who complain about their peers and 'speculate' about things that they will never understand.. it is pathetic. this is destroying mathematics, not helping it.

i have worked in finance for many many years and in other industries such as insurance and academia. yes, i call academia an industry because researchers, in this age, are prostitutes. in mathematics it is less embarrassing as opposed to psychology, statistics, economics, etc... but still, there is something wrong with humanity in the modern age, it is a crisis, a farce. when you consider mathematicians like Abel and Galois, whom died quite early and had shit lives, they still knew more than what a modern day PhD student will ever know. that is not because mathematics is so much weaker now, that is completely wrong, it is because how we approach mathematics is crap. for example, there should be no concept of 'tenure'. it is like saying 'i have been a slave long enough, so i deserve respect now! i have tenure!!!' no... you should never have been a slave in the first place.

when i am working i don't show off about mathematics or complain about the simpletons, i focus on generating revenue for the business, working with peers, etc.. that is the attitude that one needs to have in business. simpletons (people who work in finance) do not understand mathematics. in fact, simpletons do not understand finance. whenever i hear of someone being an 'expert' in finance, i am always reminded by a very famous book on corporate finance, now in its 8th edition. in the 3rd edition, the efficient markets hypothesis was treated as an obvious truth, and now in its 8th edition it is treated with extreme skepticism. the point here is that the authors whom wrote that book knew everything about finance... and even they got it wrong. so let's not treat finance as anything more than a simple business with extremely simple people.

despite having such a 'business oriented' and 'practical' attitude, i can tell you that finance is a place for simpletons. it is full of morons. and who are these morons? risk managers, directors, traders, quants, consultants, sales, brokers, etc, all of them. they produce nothing. even if they do, when there is a massive loss they get fired and their results, like them, follow the garbage can. they are the human embedding of Ebola - they spread their stupidity to others, without any thought of what they are doing... as a general heuristic: the more senior someone is the more naive minded and useless they are. another heuristic: the less someone is involved with the direct actions in the market, the less they know. that is why risk managers know nothing about risk

i am reminded of the scene in fight club where the author has to suffer desolation and hit rock bottom before he realises the state of reality. that is what working in finance is like to me - you need to hit rock bottom and realise that the people you are working with, are essentially slaves, if not prostitutes. they are... nothing.

my views are not unique... if you don't believe me, try working in any financial institution. ok... let's say you are the special angel that is blind to reality and you say to me 'i work in job XXX at firm YYY its not like that'... my suggestion: read well stablished finance books by respected authors and veterans (the black swan, physicists on wall street, derman life as a quant, mandelbrot, etc..) for verification that finance is full of morons. then learn to open your eyes...

"so you look back at your books, at first the sight of topics such as measure theory and group theory excite you - you have found structure again"

Never liked group theory and measure because of lack of applications.

//

What is a group? Algebraists teach that this is supposedly a set with two operations that satisfy a load of easily-forgettable axioms. This definition provokes a natural protest: why would any sensible person need such pairs of operations? "Oh, curse this maths" - concludes the student (who, possibly, becomes the Minister for Science in the future).

I'm not going to dig into my heart and scream about the technical (pathological) issues regarding mathematics. i respect that you took the time to explain your background and what you feel about mathematics... and also i can see that you know a lot about mathematics - teaching group theory to schoolchildren takes balls. i am sure they appreciated your effort. if i had to teach group theory, like you, i wouldn't use axioms or symbols... i would use shapes. that is how i was taught group theory. i spent more time making shapes then i did writing equations down. and poincare is one of my favourite mathematicians/philosophers..

group theory has many applications.. Burnside's lemma can be applied to the production of security identification cards, for example.. as for the lack of applications regarding measure theory.... i don't know where to begin on that...

the internet wasn't designed for what it is used for now, nor was the laser, and penicillin was found out by accident.. twitter, a social media website where you are limited to X characters seemed stupid, pointless and having no real application... until people realised. my point is, you should not hate on XXX because it doesn't have an application YYY.. penicillin being XXX didn't have an application YYY for many years... until people realised. until they used their brains and realised. my point? group theory and measure theory do have applications, but your mind is telling you "based on my previous experiences they have no application so i do not like them", so of course you will never dig into them - you will have to use your brain and realise. that is the essence of reality - to dig into something to and find the truth about it and to be as objective as possible, not to say 'i don't like it because it doesn't have applications'... how do you know it doesn't have applications?

 

[Daniel Duffy]

"'i don't like it because it doesn't have applications'... how do you know it doesn't have applications?"

Yes and no. Much of current maths does not have applications yet (but you could get lucky) because the initial spark is not based on reality. Just compare how Euler's ruminations on the Seven Bridges of Konigsberg led to the invention of graph theory and topology.

It's the disregard for applications that has changed the face of mathematics IMHO.

 

[TehRaio]

It's the disregard for applications that has changed the face of mathematics IMHO.

Same could be said of economics models, been reading a bit about "mathiness" lately. it's a bit of an intellectual disease propagating itself.

 

[Daniel Duffy]

Same could be said of economics models, been reading a bit about "mathiness" lately. it's a bit of an intellectual disease propagating itself.

Based on their track record, I find it had to disagree.

 

[tfors]

Based on their track record, I find it had to disagree.

Will be interesting to see if the currently upcoming "complex systems" approach in economics will lead to better results.

 

[Daniel Duffy]

Will be interesting to see if the currently upcoming "complex systems" approach in economics will lead to better results.

I have just read "Critical Mass" b y Philip Ball who has some interesting discussion on these issues.

 

[TehRaio]

Will be interesting to see if the currently upcoming "complex systems" approach in economics will lead to better results.

Sorry for the *terry tao discussion*, my last post on this: rare are the economists (i.e. brian arthur) who work in that domain, I've seen a significant number of physicists (i.e. Eugene Stanley) in there however. From what I gather, finance/economics scholars are not familiar with physicists work in their discipline. I think it's a shame. One can only hope a bridge is found sometimes in the future.

 

[Daniel Duffy]

I think it can be pretty interesting:
Running Probabilistic Programs Backwards | Lambda the Ultimate

Paper: [1412.4053] Running Probabilistic Programs Backwards
Slides: http://www.cs.umd.edu/~ntoronto/papers/toronto-2015esop-slides.pdf
Code: ntoronto/drbayes · GitHub

"Many probabilistic programming languages allow programs to be run under constraints in order to carry out Bayesian inference. Running programs under constraints could enable other uses such as rare event simulation and probabilistic verification---except that all such probabilistic languages are necessarily limited because they are defined or implemented in terms of an impoverished theory of probability. Measure-theoretic probability provides a more general foundation, but its generality makes finding computational content difficult.

We develop a measure-theoretic semantics for a first-order probabilistic language with recursion, which interprets programs as functions that compute preimages. Preimage functions are generally uncomputable, so we derive an abstract semantics. We implement the abstract semantics and use the implementation to carry out Bayesian inference, stochastic ray tracing (a rare event simulation), and probabilistic verification of floating-point error bounds."

"We show that measure-theoretic probability can be made computational by
1. Using measure-theoretic probability to define a compositional, denotational
semantics that gives a valid denotation to every program.
2. Deriving an abstract semantics, which allows computing answers to questions
about probabilistic programs to arbitrary accuracy.
3. Implementing the abstract semantics and efficiently solving problems."​

Looks interesting. Other languages
Probabilistic programming language - Wikipedia, the free encyclopedia