This website is about the history of math and lists famous mathematicians. The ones from the 20th century are biased towards logic, but you should go there yourself and see who you think they left out.

There entry on Paul Cohen is... odd. Its here. I quote from it:

-------------------------------------

His findings were as revolutionary as Gödel’s own. Since that time, mathematicians have built up two different mathematical worlds, one in which the continuum hypothesis applies and one in which it does not, and modern mathematical proofs must insert a statement declaring whether or not the result depends on the continuum hypothesis.

-------------------------------------

When was the last time you had to put into the premise of a theorem CH or NOT-CH?

I did once here in an expository work about a problem in combinatorics that was ind of set theory since it was equivalent to CH. Before you get too excited it was a problem in infinite combinatorics having to do with coloring the reals.

I suspect that at least 99% of my readers have never had to insert a note in a paper about if they were assuming CH or NOT-CH. If you have I'd love to hear about it in the comments. And I suspect

you are a set theorist.

Paul Cohen's work was very important--- here we have an open problem in math that will always be open (thats one interpretation). And there will sometimes be other problems that are ind of ZFC or equiv to CH or something like that. But it does not affect the typical mathematician working on a typical problem.

I have a sense that its bad to exaggerate like this. One reason would be that if the reader finds out

the truth he or she will be disillusioned. But somehow, that doesn't seem to apply here. So I leave it to the reader to comment: Is it bad to exaggerate Paul Cohen's (or anyone's) accomplishments? And if so,

then why?

# Computational Complexity

Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

## Monday, October 24, 2016

## Thursday, October 20, 2016

### Alternate Histories

Imagine if we had a machine that let us change some earlier moment in history and see what developed. We wouldn't actually change history--that would lead to paradoxes and other disasters, but we could see what would develop. Suppose Archduke Ferdinand was never assassinated. How would that have changed 20th century history and beyond?

The same could be said for an academic field. Science flows like a story, with building results from other results, "standing on the shoulders of giants" so to say. But not all theorems are dependent on earlier ones and suppose that things happened in a different order. How would that have changed our field?

Points to ponder:

Suppose Gauss was alive today instead of two centuries ago. Would he still be as famous? Would there be a big hole in mathematics that would have been left for the current Gauss to solve?

Suppose Fermat's last theorem was still a conjecture. Would there be more budding mathematicians inspired by the wonders of that famous open problem like my generation was? Would the Clay Mathematics Institute still have produced a list of those millennial problems, including P v NP?

Suppose we knew Primes in P back in 1975. Would randomized algorithms and subsequent derandomization techniques have happened without its prime example? Same for Undirected connectivity in log space. These both are small cheats as the AKS and Reingold proofs are at their core derandomization arguments and may not have happened if we didn't think about randomized algorithms.

What if someone settled P vs NP right after Cook? Would it have stopped most of the research in computational complexity theory? Would it depend on whether P = NP was answered positively or negatively?

What if you were never born? Ultimately that would be the only true measure of your influence in the world. What if your research somehow prevented other great theorems from happening? Would you even want to see the results of that experiment?

The same could be said for an academic field. Science flows like a story, with building results from other results, "standing on the shoulders of giants" so to say. But not all theorems are dependent on earlier ones and suppose that things happened in a different order. How would that have changed our field?

Points to ponder:

Suppose Gauss was alive today instead of two centuries ago. Would he still be as famous? Would there be a big hole in mathematics that would have been left for the current Gauss to solve?

Suppose Fermat's last theorem was still a conjecture. Would there be more budding mathematicians inspired by the wonders of that famous open problem like my generation was? Would the Clay Mathematics Institute still have produced a list of those millennial problems, including P v NP?

Suppose we knew Primes in P back in 1975. Would randomized algorithms and subsequent derandomization techniques have happened without its prime example? Same for Undirected connectivity in log space. These both are small cheats as the AKS and Reingold proofs are at their core derandomization arguments and may not have happened if we didn't think about randomized algorithms.

What if someone settled P vs NP right after Cook? Would it have stopped most of the research in computational complexity theory? Would it depend on whether P = NP was answered positively or negatively?

What if you were never born? Ultimately that would be the only true measure of your influence in the world. What if your research somehow prevented other great theorems from happening? Would you even want to see the results of that experiment?

## Tuesday, October 18, 2016

### This university does not discriminate based on....

I recently came across the following (I delete the name of the school)

and also add my own comments in caps as they relate to UMCP hiring

of professors.

X-University, located in YZ, in hiring professors

does not discriminate on the basis of

race

color . COULD AN HBCU DISCRIMINATE AGAINST WHITE PROFESSORS?

religious HAS NEVER COME UP. COULD A RELIGIOUS SCHOOL DISCRIMINATE ON THIS BASIS?

creed I LOOKED UP HOW IT DIFFERS FROM RELIGIONS- CREED COULD BE ANY SET OF BELIEFS. WHAT IF ONE OF CANDIDATES BELIEVES ARE REPREHENSIBLE BUT WOULD NOT INTERFERE WITH THEIR JOB? JUST ASKING.

age WHAT IF YOU WANT SOMEONE THERE LONG-TERM?

and also add my own comments in caps as they relate to UMCP hiring

of professors.

X-University, located in YZ, in hiring professors

does not discriminate on the basis of

race

color . COULD AN HBCU DISCRIMINATE AGAINST WHITE PROFESSORS?

religious HAS NEVER COME UP. COULD A RELIGIOUS SCHOOL DISCRIMINATE ON THIS BASIS?

creed I LOOKED UP HOW IT DIFFERS FROM RELIGIONS- CREED COULD BE ANY SET OF BELIEFS. WHAT IF ONE OF CANDIDATES BELIEVES ARE REPREHENSIBLE BUT WOULD NOT INTERFERE WITH THEIR JOB? JUST ASKING.

age WHAT IF YOU WANT SOMEONE THERE LONG-TERM?

gender. COULD A WOMEN'S (MEN'S) SCHOOL DISCRIMINATE AGAINST MEN (WOMEN)?

gender identity or expression I THINK THIS IS A NEW ONE TO

COVER TRANS AND SIMILAR THINGS. ITS NOT SEXUAL ORIENTATION WHICH IS BELOW.

national origin HAS NEVER COME UP. EVEN SO, I WOULD NEVER HIRE A WISIAN. See here for why.

marital status WE CAN"T EVEN ASK IF THEY HAVE A SPOUSE WHICH MAY BE HARD FOR TELLING THEM SOME SELLING POINTS OF THE AREA- GOOD FOR MARRIED PEOPLE? GOOD FOR SINGLE PEOPLE? COULD A RELIGIOUS SCHOOL HOLD AGAINST THEM THAT YOU WERE LIVING WITH SOMEONE BUT NOT MARRIED TO THEM?

ancestry I CAN"T IMAGINE THIS COMING UP.

present or past history of mental disorder I"M SURPRISED ABOUT THIS ONE SINCE I WOULD THINK HAVING A PRESENT MENTAL DISORDER IS RELEVANT TO THE JOB. THEN AGAIN, GODEL AND CANTOR PROB HAD SOME MENTAL DISORDERS.

learning disability HAS NEVER COME UP. I HAVE NEVER EVEN SEEN A GRAD STUDENT WITH A LEARNING DISABILITY SO IT MIGHT NOT COME UP FOR A WHILE.

physical disability HAS NEVER COME UP. I CAN SEE IT COMING UP.

political belief HAS NEVER COME UP. I WONDER IF IT COMES UP MORE IN A POLYSCI OR HISTORY DEPT.

veteran status HAS NEVER COME UP. COULD A MILITARY SCHOOL PREFER VETERANS?

sexual orientation HAS NEVER COME UP. BUT COULD.

genetic information THIS MAY BE RELEVANT IN THE FUTURE. PERHAPS THE NEAR FUTURE. SCARY?

non-position-related criminal record. WHICH CRIMES ARE POSITION RELATED? THIS COULD BE A LEGAL QUAGMIRE . MURDERING SOMEONE IN A BAR FIGHT IS NOT POSITION RELATED BUT MURDERING A STUDENT WHO COMPLAINED ABOUT A GRADE IS. OTHER CASES MAY NOT BE AS CLEAR CUT.

## Thursday, October 13, 2016

### 2016 Fall Jobs Post

The weather cools down, the leaves change color and you start thinking about what you plan to do after you graduate. As a public service every year about this time, we offer up links and advice for the job search for PhDs in theoretical computer science.

For computer science faculty positions best to look at the ads from the CRA and the ACM. For theoretical computer science specific postdoc and faculty positions check out TCS Jobs and Theory Announcements. If you have jobs to announce, please post to the above and/or feel free to leave a comment on this post.

It never hurts to check out the webpages of departments or to contact people to see if positions are available. Even if theory is not listed as a specific hiring priority you may want to apply anyway since some departments may hire theorists when other opportunities to hire dry up. Think global--there are growing theory groups around the world.

I expect hiring this year to be similar to recent years. Most departments looking to dramatically expand in computer science but with a preference for the more applied areas that the students and the companies that will hire them desire. You can make yourself more valuable by showing a willingness to participate and teach beyond core theory. Machine learning, data science and information security are areas of great need where theorists can play a large role.

A bad job talk can sink your job prospects. Know your audience and particularly for faculty positions create a talk that can express your results and importance to those outside of theory. Pictures help, complex formulas and heavy text work against you. Practice your talk in front of non-theory students and faculty. You will greatly increase your chances if you can sell yourself as a valuable colleague, not just a smart theorist who will prove great things in isolation.

Good luck out there and I look forward to seeing your names on the Spring 2017 jobs post.

For computer science faculty positions best to look at the ads from the CRA and the ACM. For theoretical computer science specific postdoc and faculty positions check out TCS Jobs and Theory Announcements. If you have jobs to announce, please post to the above and/or feel free to leave a comment on this post.

It never hurts to check out the webpages of departments or to contact people to see if positions are available. Even if theory is not listed as a specific hiring priority you may want to apply anyway since some departments may hire theorists when other opportunities to hire dry up. Think global--there are growing theory groups around the world.

A bad job talk can sink your job prospects. Know your audience and particularly for faculty positions create a talk that can express your results and importance to those outside of theory. Pictures help, complex formulas and heavy text work against you. Practice your talk in front of non-theory students and faculty. You will greatly increase your chances if you can sell yourself as a valuable colleague, not just a smart theorist who will prove great things in isolation.

Good luck out there and I look forward to seeing your names on the Spring 2017 jobs post.

## Tuesday, October 11, 2016

### Ideal courses for a comp sci dept/I'm glad we don't do it

I once heard it said:

In our data structures course we read Knuth and ignore the proofs

In our algorithms course we read Knuth and ignore the code.

And indeed, there are many topics where the theory-part is in one course and the programming is in another.

With that in mind, here is a different way to offer courses (some depts already do some of this).

1) A course in Algorithms AND Data Structures. Actually I have seen this title on courses but its usually just a theory course. I mean you REALLY PROOF and CODE. Might need to be a year long.

2) Crypto AND Security. You learn crypto and security together and do both. If you only did the crypto relevant to security it might be a semester long and in fact it might already be being done. But I am thinking of really combining these courses--- code and prove. Might be a year long.

3) Formal lang Theory and Practice of Compilers. You do DFA, NDFA, CFG, but also do compiler design. If you also want to do P, NP, and decidability (spell check thinks that decidability is not a word!) then might not quite connect up with compilers, then again in might with theorems like: CFG equivalence is undecidable. Might be a year long.

4) Machine learning AND Prob/stat.

PROS: Theory and Practice would be more united.

CONS: Having year-long courses is hard for the students scheduling their courses. Would having just one semester of any of the above courses make sense?

CONS: Harder to find someone to teach these courses. I'll confess that I prefer to teach formal lang theory without any programming in it.

CAVEAT: I think theorists coming out now know more of the practical counterpart of their area than I did and perhaps than people of my generation did.

CAVEAT: A much less radical thing to do is to put more security in to crypto, more about compilers into Formal Lang Theory, etc. But thats a bit odd now since we DO have a course in security, and a course in compilers. Even so, seems to be a good idea and this I know many schools are doing.

In our data structures course we read Knuth and ignore the proofs

In our algorithms course we read Knuth and ignore the code.

And indeed, there are many topics where the theory-part is in one course and the programming is in another.

With that in mind, here is a different way to offer courses (some depts already do some of this).

1) A course in Algorithms AND Data Structures. Actually I have seen this title on courses but its usually just a theory course. I mean you REALLY PROOF and CODE. Might need to be a year long.

2) Crypto AND Security. You learn crypto and security together and do both. If you only did the crypto relevant to security it might be a semester long and in fact it might already be being done. But I am thinking of really combining these courses--- code and prove. Might be a year long.

3) Formal lang Theory and Practice of Compilers. You do DFA, NDFA, CFG, but also do compiler design. If you also want to do P, NP, and decidability (spell check thinks that decidability is not a word!) then might not quite connect up with compilers, then again in might with theorems like: CFG equivalence is undecidable. Might be a year long.

4) Machine learning AND Prob/stat.

PROS: Theory and Practice would be more united.

CONS: Having year-long courses is hard for the students scheduling their courses. Would having just one semester of any of the above courses make sense?

CONS: Harder to find someone to teach these courses. I'll confess that I prefer to teach formal lang theory without any programming in it.

CAVEAT: I think theorists coming out now know more of the practical counterpart of their area than I did and perhaps than people of my generation did.

CAVEAT: A much less radical thing to do is to put more security in to crypto, more about compilers into Formal Lang Theory, etc. But thats a bit odd now since we DO have a course in security, and a course in compilers. Even so, seems to be a good idea and this I know many schools are doing.

## Friday, October 07, 2016

### Typecasting from Avi's 60th Celebration

Lance: Live from the Institute for Advanced Study in Princeton, New Jersey, Bill and I are doing a typecast from the Avi Wigderson 60th Birthday Celebration. Hi Bill.

Bill: Hi Lance. Last time I saw you was for the Mike Sipser 60th birthday and next for Eric Allender and Michael Saks.

L: New Jersey again. The Garden State.

B: New Jersey rocks! After all Bruce Springsteen is from here.

L: So was I.

B: You both rock! You are better at math. But I don’t want to hear you sing. I got in last night. How was the first day of the conference.

L: There’s two kinds of talks at a celebration like this. Some people who survey how Avi has shaped the field and their research in particular. Scott Aaronson gave a great talk titled “Avi’s Permanent Impact on me” on how a survey talk on the permanent problem eventually led to Scott’s work on boson sampling.

Most people though are giving talks on their latest and greatest research. At least they have some good Avi jokes to start their talks.

B: So what category did Oded Goldreich’s “Canonical depth-three Boolean circuits for multi-linear functions, Multi-linear circuits with general gates, and matrix rigidity” talk fit in.

L: Oded did say the title was a joke and he did start with an Avi story, Actually it was a Silvio and Shafi story about when to leave for the airport.

B: I liked Alex Lubotzky’s ruminations on pure math versus computer science. He went into pure math thinking computer scientists had to know how to use computers. Had he known that they didn't he may have gone into computer science. He likes that his work on expanders has “applications” to computer science.

L: How did you like the rest of his talk.

B: I don’t know--it’s still going on.

L: Once I gave a talk at a logic meeting about generic oracles. People came up to me afterwards so excited that logic had such applications in computer science.

B: I liked Noga Alon’s talk “Avi, Graphs and Communication”. There is a fixed graph G on k nodes with k players each with an n bit string. How many bits do they to communicate over the edges to determine if all the strings are identical? Basically if the graph is 2-connected you need half of the trivial kn bits.

[Intermission]

L: A day has passed and we find ourselves talking again on Friday at IAS. It’s a beautiful day and we are sitting on lawn chairs on the IAS grounds. This is how Einstein must have felt.

B: Although he wouldn’t be skipping a talk on quantum physics.We are listening in through the magic of the Internet.

I liked Noam Nisan’s talk on the complexity of pricing. Perhaps surprisingly a seller can do better bundling two independent items than using individual prices. I plan to use that example in my fair division class. I may even do a short blog post.

L: I can’t wait. Madhu gave a great talk on low-degree polynomials and how error-correcting have gone from obscurity in theoretical computer science in the early 90’s to a standard tool just a few years later. And we have Madhu (inspired by Avi) to thank for that.

B: Eyal Wigderson, son of you know who, is a grad student studying neuroscience. Talked on “Brains are Better Computers than Computers” which I found flattering.

L: He was talking about rats, not your brain Bill.

B: Should I feel complemented or insulted?

L: Yes.

B: It’s kind of nice to have a birthday person’s child talk a conference like this. In TCS there are several including Paul Valiant who talked at Les Valiant’s 60th, Tal Rabin talked at Michael Rabin’s 80th, father and son Edmonds are here, and Molly Fortnow will surely talk at Lance Fortnow’s 60th. I remember when she interviewed us.

L: Let’s leave it at that.

B: Silvio Micali “knighted” Avi and used Byzantine agreement to improve bitcoin-like chains. I was enlightened to learn bitcoin is so expensive only five companies do it regularly.

L: When people claim to solve P = NP I asked them to mine a few bitcoins and get back me. I have yet to see any bitcoins.

B: I asked them to find Ramsey of 5. I’m still waiting.

L: On that note time to say goodbye until we meet again.

B: Allender/Saks New Jersey again! Take us out Lance.

L: In a complex world, best to keep it simple, and

B/L: HAPPY BIRTHDAY AVI!

## Tuesday, October 04, 2016

### A Second Order Statement true in (R,+) but not (Q,+)

In my last post I asked

Is there a first order statement true in (R,+) but false in (Q,+)

Is there a second order statement true in (R,+) but false in (Q,+)

**First order: No.**

One reader said it followed from the Lowenheim-Skolem Theorem. I can believe this but don't see how.

One reader said the theory of (Q,+) is complete. True, but would like a reference.

I would prove it by a Duplicator-Spoiler argument.

**Second order: Yes**

Some readers thought I had < in my language. I do not so I think that examples using < do not work- unless there is someway to express < in my language.

Some readers thought that second order meant you could quantify over functions. My intent was just to be able to quantify over sets.

Some had the right answers. The one I had in mind was

*There exists A,B such that A,B are subgroups with at least two elements that only intersect at 0.*

*Here is a writeup.*

## Thursday, September 29, 2016

### Give a second order statement true in (R,+) but false in (Q,+) or show there isn't one

Here is a logic question I will ask today and answer next week. Feel free to leave comments with

the answer- you may come up with a different proof than me and that would be great!

Our lang will have the usual logic symbols, quantification over the domain, quantification over subsets of the domain (so second order) the = sign, and the symbol +

Examples of sentences:

(∀ x)(∀ y)[ x+y=y+x]

true over Q,R,N. False in S_n for n\ge 4 (group of perms of n elements)

(∃ x)(∀ y)[ x+y=y]

true in Q, R by taking 0. not true in {1,2,3,...}

Lets assume it is true and call the x 0

(∀ x)(∃ y)[x+y=0]

True in Q, R, Z, not true in N.

QUESTION ONE: Is there any sentence in the first order theory that is TRUE over (Q,+) but

FALSE over (R,+)?

QUESTION TWO: Is there any sentence in the second order theory that is TRUE over (Q,+)

but false over (R,+)?

the answer- you may come up with a different proof than me and that would be great!

Our lang will have the usual logic symbols, quantification over the domain, quantification over subsets of the domain (so second order) the = sign, and the symbol +

Examples of sentences:

(∀ x)(∀ y)[ x+y=y+x]

true over Q,R,N. False in S_n for n\ge 4 (group of perms of n elements)

(∃ x)(∀ y)[ x+y=y]

true in Q, R by taking 0. not true in {1,2,3,...}

Lets assume it is true and call the x 0

(∀ x)(∃ y)[x+y=0]

True in Q, R, Z, not true in N.

QUESTION ONE: Is there any sentence in the first order theory that is TRUE over (Q,+) but

FALSE over (R,+)?

QUESTION TWO: Is there any sentence in the second order theory that is TRUE over (Q,+)

but false over (R,+)?

Subscribe to:
Posts (Atom)