Dear Ongaro Nyang'
Although Israel is no longer so young, it was young, (even
younger than most other countries,) not so long ago, and
it is a small and rather isolated place, with some difficulties.
So some lessons from Israeli mathematics, especially in its early days
may be relevant.
A) Immigration
1. Immigration
Israeli mathematics was initially based on and had benefited all the
times from immigration of mathematicians to Israel. The ability to
absorb immigration, in general, and to attract and absorb immigrating scientists, in particular,
is crucial.(Keeping the relations with mathematicians who immigrated from
Israel is also an important issue.)
B) Financial matters.
Investing resources is, of course, crucial. There was a large public investment
in universities in Israel's first years even when the country itself
was in rather bad economic shape. Overall, theoretical academic
subjects like mathematics are "cheaper." Keeping the right balances
regarding policies for spending the money is very important.
Let me mention two items.
2. Sabbatical/Travel money
1.1 Good Sabbatical opportunities: Israeli mathematicians (and scientists
in general) had relatively good sabbatical terms which make it possible
for them to get (from the Israeli institution) a reasonable European/US
salary while in sabbatical abroad. This was especially effective when the
Israeli salaries were very low compared to salaries abroad. The academic
system is built on 15% or so of the faculty being on sabbatical at
any time. (Often people spend additional time abroad on leave.)
2.2 In addition, Israeli scientist (with academic university positions) and graduate students (who works as T.A.'s) have funds for short-term
travels: (Fixed amounts depending on the academic rank per year) This
enable participation in conferences and joint research.
Both the sabbatical and travel money apply to all people with academic
positions (Sabbatical only to lecturer/asst professor position and above)
and are essentially automatic. (Minimal amount of bureaucracy, no
committees to judge qualification and to decide on amounts, no requirement
to be an invited speaker in a conference, etc. etc..)
3. Salaries
Keeping the right balances when it comes to salaries is also important.
Low salary gives incentives to leave but very high salaries (compared to
average salaries in the country) are morally problematic and may
give incentive for corrupting the hiring and promoting system.
The salary system in Israel is based on essentially equal salary for equal
academic rank and (overall) there are no substantial salary awards for
academic excellence beside the academic rank. (There is some (modest)
awards for people getting external grants and larger but still rather small
for people serving in administrative positions.)
I think that not having overly differential salaries and not
being "in the game" of offers and counteroffers is actually beneficial.
C) Activities
4. Activities for national math society
There are, since early times,
regular annual meeting of the Israeli Mathematics Union and some
other local activities.
5. National mathematical journal
There was a substantial effort, again from early times,
to create and maintain an Israeli journal of
mathematics. (In the 40th there was even a professional research level
journal in Hebrew for a few years.)
6. Conferences and visitors.
There was, again from early times, some resources devoted to conferences
and visitors. Carefully administered and with attention to
the added value for the local people this can be very fruitful.
Arranging visits of top people in mathematics for lecture series and visits
can also be useful. It seems that it is a good policy (when the country is not
rich) not to over-pay for such visitors (among other reasons, because this set standards which push the cost of visitors in general too high.)
Of course, warm hospitality is priceless.
D) Content
7. Maintaining a sense of tradition.
Basing activities on areas with long tradition of success which are
identified with the country's mathematical strength can be a successful
and well excepted by the whole mathematical community.
8. Self-breeding can work
The success of Israeli mathematical departments was largely based on
successful self-breeding, namely absorbing as faculty member people who
graduated at the department.
9. Self-confidence, Tolerance for "sporadic"(or non-main stream) areas
(and tolerance in general)
This seems to me a strength of Israeli mathematics and looks (to me) a
good attitude especially for a peripheral and somewhat isolated place.
Tolerance is important especially since mathematical quality is rather
high dimensional (some dimensions being
importance/depth/visibility/applicability/usefulness.)
(There is also
complete tolerance and essentially indifference in the context of
mathematical life towards matters of politics, attitude towards religion,
etc, issues that Israel is very torn apart about.)
10. Patience, realistic goals, unrealistic dreams
Building a good mathematical activity takes time, and there are
ups and downs as well.
E) Outreach
11. Issues concerning high school mathematics
There is some efforts to promote interest in mathematics among gifted high
school students: special mathematical journal (in Hebrew), some "clubs"
and "summer camps" math Olympiads etc. I think this had some factor in
promoting math among young people. Usually, what it takes is some
mathematician in academics which is devoted to this issue and some (not
large) budget. Giving an incentive for such an activity and such a
mathematician may be a good idea. Popular Math books and text books in
Hebrew had substantial influence. A. Frankel (the set theorist)
wrote wonderful five-volume Hebrew books introducing
mathematics (It was called "An Introduction to Mathematics") in the 40s/50s.
(More recently, the Hebrew edition of Singh's
book on Wiles proof of FLT increased popularity of mathematics.)
F) Relations with other areas
12. Relation with CS, physics and other disciplines
In Israel CS department were largely built out of math departments (not
electrical engineering) and there are still strong academic ties between
these communities. Connection with CS seems valuable. Of course, relations with physics are very important (not so strong in Israel). Relations between math and economics seems strong in Israel.
(In Jerusalem there is an interdisciplinary "center for
rationality" which involves people from math/economics and some from
statistics/psychology/philosophy/law/biology.)
In summary, when it comes to mathematical life in a small somewhat isolated and
at times a bit troubled place, it seems that it is valuable to make the
right balances in the local mathematical community between competition and
solidarity, to practice a lot of patience and tolerance, to be open to new people and new directions, and to be careful about incentives.
Like in Brazil the efforts of few pivotals mathematicians was very crucial.
As usual, luck is useful too. Good luck.
I was not going to write anything, as I am a latecomer to this masterful troll question and not many are likely going to scroll all the way down, but Paul Taylor's call for Proof mining and Realizability (or Realisability as the Queen would write it) was irresistible.
Nobody asks whether numbers are just a ritual, or at least not very many mathematicians do. Even the most anti-scientific philosopher can be silenced with ease by a suitable application of rituals and theories of social truth to the number that is written on his paycheck. At that point the hard reality of numbers kicks in with all its might, may it be Platonic, Realistic, or just Mathematical.
So what makes numbers so different from proofs that mathematicians will fight a meta-war just for the right to attack the heretical idea that mathematics could exist without rigor, but they would have long abandoned this question as irrelevant if it asked instead "are numbers just a ritual that most mathematicians wish to get rid of"? We may search for an answer in the fields of sociology and philosophy, and by doing so we shall learn important and sad facts about the way mathematical community operates in a world driven by profit, but as mathematicians we shall never find a truly satisfactory answer there. Isn't philosophy the art of never finding the answers?
Instead, as mathematicians we can and should turn inwards. How are numbers different from proofs? The answer is this: proofs are irrelevant but numbers are not. This is at the same time a joke and a very serious observation about mathematics. I tell my students that proofs serve two purposes:
- They convince people (including ourselves) that statements are true.
- They convey intuitions, ideas and techniques.
Both are important, and we have had some very nice quotes about this fact in other answers. Now ask the same question about numbers. What role do numbers play in mathematics? You might hear something like "they are what mathematics is (also) about" or "That's what mathematicians study", etc. Notice the difference? Proofs are for people but numbers are for mathematics. We admit numbers into mathematical universe as first-class citizen but we do not take seriously the idea that proofs themselves are also mathematical objects. We ignore proofs as mathematical objects. Proofs are irrelevant.
Of course you will say that logic takes proofs very seriously indeed. Yes, it does, but in a very limited way:
- It mostly ignores the fact that we use proofs to convey ideas and focuses just on how proofs convey truth. Such practice not only hinders progress in logic, but is also actively harmful because it discourages mathematization of about 50% of mathematical activity. If you do not believe me try getting funding on research in "mathematical beauty".
- It considers proofs as syntactic objects. This puts logic where analysis used to be when mathematicians thought of functions as symbolic expressions, probably sometime before the 19th century.
- It is largely practiced in isolation from "normal" mathematics, by which it is doubly handicapped, once for passing over the rest of mathematics and once for passing over the rest of mathematicians.
- Consequently even very basic questions, such as "when are two proofs equal" puzzle many logicians. This is a ridiculous state of affairs.
But these are rather minor technical deficiencies. The real problem is that mainstream mathematicians are mostly unaware of the fact that proofs can and should be first-class mathematical objects. I can anticipate the response: proofs are in the domain of logic, they should be studied by logicians, but normal mathematicians cannot gain much by doing proof theory. I agree, normal mathematicians cannot gain much by doing traditional proof theory. But did you know that proofs and computation are intimately connected, and that every time you prove something you have also written a program, and vice versa? That proofs have a homotopy-theoretic interpretation that has been discovered only recently? That proofs can be "mined" for additional, hidden mathematical gems? This is the stuff of new proof theory, which also goes under names such as Realizability, Type theory, and Proof mining.
Imagine what will happen with mathematics if logic gets boosted by the machinery of algebra and homotopy theory, if the full potential of "proofs as computations" is used in practice on modern computers, if completely new and fresh ways of looking at the nature of proof are explored by the brightest mathematicians who have vast experience outside the field of logic? This will necessarily represent a major shift in how mathematics is done and what it can accomplish.
Because mathematicians have not reached the level of reflection which would allow them to accept proof relevant mathematics they seek security in the mathematically and socially inadequate dogma that a proof can only be a finite syntactic entity. This makes us feeble and weak and unable to argue intelligently with a well-versed sociologist who can wield the weapons of social theories, anthropology and experimental psychology.
So the best answer to the question "is rigor just a ritual" is to study rigor as a mathematical concept, to quantify it, to abstract it, and to turn it into something new, flexible and beautiful. Then we will laugh at our old fears, wonder how we ever could have thought that rigor is absolute, and we will become the teachers of our critics.
Best Answer
In a letter to Frobenius, Dedekind made the following curious observation: if we see the multiplication table of a finite group $G$ as a matrix (considering each element of the group as an abstract variable) and take the determinant, then the resulting polynomial factors into a product of $c$ distinct irreducible polynomials, each with multiplicity equal to its degree, where $c$ is the number of conjugacy classes of $G$. This is now known as Frobenius determinant theorem, and it is what led Frobenius to develop the whole representation theory of finite groups (https://en.wikipedia.org/wiki/Frobenius_determinant_theorem).