Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century:...
Transcript of Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century:...
![Page 1: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/1.jpg)
Mathematics of 21st Century: A Personal View
SCSS 2013, RISC, July 6, 2013
1
Bruno Buchberger
![Page 2: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/2.jpg)
2
Mathematics of …, 19th, 20th Century Mathematics of 21st Century Demo
![Page 3: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/3.jpg)
3
Mathematics of …, 19th, 20th Century Mathematics of 21st Century Demo
![Page 4: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/4.jpg)
“Ancient” Mathematics: “see the truth”
• observe (“mathematical”) objects in reality
• and “see” a (general) truth
4
![Page 5: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/5.jpg)
No clear distinction between “seeing” = observing and “seeing” = thinking
5
![Page 6: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/6.jpg)
If the situation is so and so … then one “sees” that …
6
![Page 7: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/7.jpg)
“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
![Page 8: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/8.jpg)
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
![Page 9: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/9.jpg)
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
![Page 10: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/10.jpg)
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
![Page 11: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/11.jpg)
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
![Page 12: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/12.jpg)
A common misunderstanding:
Computational methods are just a „silly“ iteration of simple steps.
The truth: Symbolic algorithms need „deeper“ mathematics than „pure mathematics“.
12
![Page 13: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/13.jpg)
13
Wolfgang Gröbner (1899 – 1980)
“lived” in 19th and 20th century math, in “numerics” and “symbolics”
![Page 14: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/14.jpg)
14
Mathematics of …, 19th, 20th Century Mathematics of 21st Century Demo
![Page 15: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/15.jpg)
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
![Page 16: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/16.jpg)
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
![Page 17: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/17.jpg)
17
observe thinking
think about thinking
MATH REFLEXION
Thinking (reasoning, proving, ...) about observed laws, ...
automated reasoning
![Page 18: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/18.jpg)
18
MATH REFLEXION is natural math i s reflexion
![Page 19: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/19.jpg)
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
![Page 20: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/20.jpg)
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
![Page 21: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/21.jpg)
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
![Page 22: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/22.jpg)
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
![Page 23: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/23.jpg)
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
![Page 24: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/24.jpg)
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
![Page 25: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/25.jpg)
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
![Page 26: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/26.jpg)
Yes, raw plus raw may result in refined:
26
+
![Page 27: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/27.jpg)
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
![Page 28: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/28.jpg)
Historically,
this hierarchy went through a couple of amazing rounds
full of emotion, effort, surprises, and excitement.
(With increasing speed.)
28
![Page 29: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/29.jpg)
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
![Page 30: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/30.jpg)
30
Mathematics of …, 19th, 20th Century Mathematics of 21st Century Demo
![Page 31: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/31.jpg)
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
![Page 32: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/32.jpg)
Demo: see file Buchberger-Future-Math-Demo-Lazy-Thinking.nb
32
![Page 33: Mathematics of 21 Century: A Personal View...2013/07/06 · Symbolic computation in 21st Century: One „floor“ higher. Symbolics 1st floor: Invent new mathematics for symbolic](https://reader034.fdocuments.in/reader034/viewer/2022052008/601da3106dffac27f42d3b48/html5/thumbnails/33.jpg)
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