mathematics of 21 century: a personal view...2013/07/06  · symbolic computation in 21st century:...

Mathematics of 21st Century: A Personal View

SCSS 2013, RISC, July 6, 2013

1

Bruno Buchberger

2

Mathematics of …, 19th, 20th Century Mathematics of 21st Century Demo

3

Mathematics of …, 19th, 20th Century Mathematics of 21st Century Demo

“Ancient” Mathematics: “see the truth”

•  observe (“mathematical”) objects in reality

•  and “see” a (general) truth

4

No clear distinction between “seeing” = observing and “seeing” = thinking

5

If the situation is so and so … then one “sees” that …

6

“Modern” Mathematics: “prove new truth from seen truth” Clear distinction between

“seeing” = observing and “seeing” = thinking (reasoning, … proving)

(see) (a+b) (a+b) = c.c + 4.(a.b)/2 = (think) a.a + 2.a.b + b.b

7

Mathematics of … 19th Century strong intutions about important (real-world / physics relevant) results, comprehensive view of all areas of math (geo, algebra, analysis, number theory, …), still, an amazing source of deep results (e.g. foundation of Risch’ algorithm)

8

Mathematics of 20th Century

•  mathematical logic: meta-mathematics

•  “Bourbakism”: all from “zero”

•  the universal computer (executing “algorithms”)

The three aspects are deeply connected. However, pursued in three different communities.

9

Computational mathematics was “only” concerned with “approximate” problems using “approximate” numbers: x^2 + b x + c = 0, x = ? in case: b = 3, c = 5: x = -1.5… ± 1.65831… i Around 1950: ‘’Symbolic Computation ’’: x = 1/2 (-b ± Sqrt (b^2 - 4 c )) if you want, plug in b = 3, c = 5.

10

How far can “symbolic computation” go? Answer: Very far ! But may be very difficult. Why difficult? 1950 – now: Symbolic algorithms for traditional mathematical problems (e.g. general non-linear systems, symbolic integration, …)

11

A common misunderstanding:

Computational methods are just a „silly“ iteration of simple steps.

The truth: Symbolic algorithms need „deeper“ mathematics than „pure mathematics“.

12

13

Wolfgang Gröbner (1899 – 1980)

“lived” in 19th and 20th century math, in “numerics” and “symbolics”

14

Mathematics of …, 19th, 20th Century Mathematics of 21st Century Demo

Symbolic computation in 21st Century:

One „floor“ higher.

Symbolics 1st floor: Invent new mathematics for symbolic algorithms for traditional math problems.

Symbolic 2nd floor: Invent algorithms for inventing and proving new mathematics.

15

Mathematics of 21st Century: Invent and implement algorithms (software, systems, ...) for „mathematical knowledge management“:

–  for inventing definitions –  for inventing and proving theorems –  for inventing problems –  for inventing and proving algorithms –  for building up and managing mathematical

theories in a structured way.

16

17

observe thinking

think about thinking

MATH REFLEXION

Thinking (reasoning, proving, ...) about observed laws, ...

automated reasoning

18

MATH REFLEXION is natural math i s reflexion

Mathematics: Given a mathematical problem, whose solution, for each instance, at a given historical moment, needs human intelligence, one strives at inventing, again by human intelligence, a general algorithm, that solves the problem in all infinitely many instances.

19

In other words, it is the essence of mathematics to think once deeply (by “natural” intelligence) on a problem and its context in order to replace thinking infinitely many times for solving each of the infinitely many problem instances individually by non-intelligent computation. (In other words: Generate “articifial non-intelligence” by “natural intelligence”.)

20

Example:

At some stage, finding a shortest path in a graph needed an individual consideration for each graph. At a next stage, somebody (Dijkstra) managed (and proved) one algorithm for finding a shortest path for all graphs.

21

Example:

At some stage, solving a Sudoku puzzle needed an individual consideration for each puzzle. At a next stage, somebody (BB, …) managed (and proved) one algorithm for finding a solution for all Sudoku puzzles (e.g. by the Gröbner bases method).

22

Example:

At some stage, finding the integral (in “closed form”) of a function (expression) needed an individual consideration for each function. At a next stage, somebody (Risch) managed (and proved) one algorithm for finding a solution for all functions (out of a particular class).

23

Mathematics is automation of mathematical invention on some level

by a mathematical invention on a higher level.

In other words, it is the goal of mathematics

to trivialize itself.

24

For making math easy (“non-intelligent”)

on one level one must do more difficult (“intelligent”) math

on a higher level.

Discussion: Is this always true?

Can general invention on higher level be easier than the individual inventions on the lower level?

25

Yes, raw plus raw may result in refined:

26

+

The principle of mathematics can be iterated.

Mathematics is a hierarchy of “intelligent” inventions:

One invention on a higher level avoids infinitely many inventions on a lower level.

27

Historically,

this hierarchy went through a couple of amazing rounds

full of emotion, effort, surprises, and excitement.

(With increasing speed.)

28

There is no upper bound to the rounds of automating mathematical invention.

(Gödel’s Incompleteness Theorem

can be “felt” by all who attempt to add a next round,)

29

30

Mathematics of …, 19th, 20th Century Mathematics of 21st Century Demo

My personal contribution: •  to the „first floor“ (1965 -...): Gröbner bases

•  to the „second floor“ (1995 - ...): Theorema

In this talk, I will give a demo about the „second floor“:

An algorithmic method (called „Lazy Thinking“) for inventing and proving algorithms from problem specifications.

31

Demo: see file Buchberger-Future-Math-Demo-Lazy-Thinking.nb

32

Conclusions

Style of “working mathematicians” will change. Style of math “journals” as archives will change. Style of math “journals” as math quality control will change.

33