Mathematics of 21st Century: A Personal View

SCSS 2013, RISC, July 6, 2013

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Bruno Buchberger

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

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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

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

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

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“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

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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)

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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.

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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.

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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, …)

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A common misunderstanding:

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

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

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Wolfgang Gröbner (1899 – 1980)

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

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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.

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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.

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observe thinking

think about thinking

MATH REFLEXION

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

automated reasoning

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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.

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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”.)

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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.

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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).

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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).

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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.

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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?

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Yes, raw plus raw may result in refined:

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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.

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Historically,

this hierarchy went through a couple of amazing rounds

full of emotion, effort, surprises, and excitement.

(With increasing speed.)

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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,)

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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.

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

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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.

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