Great Circle of Mysteries: Mathematics, the World, the Mind

209 Pages·2017·5.453 MB·English

by  Misha Gromov

Misha Gromov
Great Circle of Mysteries
Mathematics, the World, the Mind
Misha Gromov
Institut des Hautes Études Scientifiques
Université Paris-Saclay
Bures-sur-Yvette, France
and
Courant Institute of Mathematical Sciences
New York University
New York, USA
ISBN 978-3-319-53048-2 ISBN 978-3-319-53049-9 (eBook)
https://doi.org/10.1007/978-3-319-53049-9

Library of Congress Control Number: 2018950059

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Cover illustration: homographic sequences: projective dynamics © Benoît Kloeckner, http://perso-math.univ-
mlv.fr/users/kloeckner.benoit/posts/2012-04-23-SuitesHomographiques.html

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Contents
Introduction ............................................................. vii
Quotations and Ideas
1 Beautiful Elsewhere..................................................... 1
2 Science.................................................................. 2
3 Numbers................................................................ 4
4 Laws.................................................................... 9
5 Truth.................................................................. 26
6 Life.................................................................... 30
7 Evolution............................................................... 37
8 Brain................................................................... 58
9 Mind................................................................... 62
10 Mysteries Remain...................................................... 68
Memorandum Ergo
1 Brain, Ergo-Brain,and Mind........................................... 73
2 Ergo Project........................................................... 78
3 Formality and Universality – Meaning, Folding, and Understanding..... 80
4 Universality, Simplicity and Ergo-Brain................................. 89
5 Freedom, Curiosity, Interesting Signals, and Goal Free Learning......... 92
6 Information, Prediction, and a Bug on the Leaf......................... 94
7 Stones and Goals...................................................... 100
8 Ego, Ergo, Emotions, and Ergo-Moods................................ 102
9 Common Sense, Ergo-Ideasand Ergo-Logic............................ 104
10 Ergo in the Minds..................................................... 107
v
vi Contents
11 Language and Languages.............................................. 111
12 Meaning of Meaning................................................... 115
13 Play, Humour, and Art................................................ 120
14 Ergo in Science........................................................ 122
15 Unreasonable Men and Alternative Histories........................... 125
16 Mathematics and its Limits ........................................... 128
17 Numbers, Symmetries, and Categories................................. 132
18 Logic and the Illusion of Rigor........................................ 137
19 Infinite Inside, Finite Outside......................................... 141
20 Small, Large, Inaccessible............................................. 144
21 Probability: Particles, Symmetries, Languages......................... 149
22 Signal Flows from the World to the Brain............................. 156
23 Characteristic Features of Linguistic Signals........................... 161
24 Understanding Structures and the Structure of Understanding......... 164
25 Sixteen Rules of an Ergo-Learner...................................... 168
26 Learning to Understand Languages:
From Libraries to Dictionaries......................................... 171
27 Libraries, Strings, Annotations, and Colors............................ 174
28 Teaching and Grading................................................. 176
29 Atoms of Structures: Units, Similarities, Co-functionalities,
Reductions............................................................ 178
30 Fragmentation, Segmentation, and Formation of Units................. 184
31 Presyntactic Morphisms, Syntactic Categories, and Branched
Entropy............................................................... 186
32 Similarities and Classifications, Trees and Coordinatizations........... 191
33 Clustering, Biclustering, and Coclustering............................. 193
References............................................................. 201
Introduction
©pixelparticle/GettyImages/iStock
Geometry is one and eternal shining in the mind of God.
Johannes Kepler
God’s plan for mathematics is beyond human reach but mathematics is the only
light that can illuminate the mysteries of this world.
Whatwecallthescienceofphysicsisfilledwiththeradianceofmathematics;
the fog of ignorance around what we now call laws of nature receded in the
brilliance of this physics.
But the beam of this light has barely touched the edges of the kingdom of
life and the face of princess thought remains hidden from us in the shroudof
darkness.
Andunlessweknowthewaysofthoughtwecannotunderstandwhatmath-
ematics is.
The first part of the book – Quotations and Ideas – is sprinkled with the
ideas of those who saw sparks of light in the dark sea of the unknown. In the
secondpart– MemorandumErgo– we reflectonwhatin mathematics couldshed
light on mystery thought.
vii
Quotations and Ideas
1. Beautiful Elsewhere
©CopyrightIanMorrisonandlicensedforreuseunderthis
CreativeCommonsLicense.
Round us, near us, in depth and height,
Soft as darkness and keen as light.
Algernon Swinburne, Loch Torridon
“Math´ematiques, un d´epaysement soudain,” was an exhibition organized in 2011
by Fondation Cartier pour l’art contemporain in Paris. It featured, among other
things, The Library of Mysteries:
the mystery of Physical Laws
the mystery of Life
the mystery of the Mind
the mystery of Mathematics
These were presented as quotes from writings of great scientists in a film
made by David Lynch – an artist’s visualizations of the ideas of Time, Space,
Matter, Life, Mind, Knowledge, and Mathematics.
© Springer International Publishing AG, part of Springer Nature 2018 1
M. Gromov, Great Circle of Mysteries, https://doi.org/10.1007/978-3-319-53049-9_1
2 Quotations and Ideas
Michel Cass´e and Herv´e Chand´es have persuaded me to try to do a Naive
Mathematician’s version of what David Lynch has done – to project a vision of
theseideastotheinvisiblescreeninourmind,animageilluminatedbytheeternal
shining of mathematics rather than by the reflection of the beauty of the goddess
of arts.
I knew it wouldbe impossible but I triedanyway.Much of what I wrote was
doneindiscussionswithGiancarloLucchiniandsomeofmyEnglishwascorrected
by Bronwyn Mahoney. Below is a modified version of what came out of it.
2. Science
Nothing exists except atoms in the void;
everything else is opinion.
Democritus of Abdera (?), 460–370 BCE
Allmenbynaturedesireknowledge.Thinkingisthetalkingofthesoulwithitself.
... knowledgeistheonemotiveattractingandsupportinginvestigators... always
flying before them ... their sole torment and their sole happiness. To perceive is
to suffer.
Common sense is the collection of prejudices acquired by age eighteen.
... Science... [is]asking,notwhetherathingisgoodorbad... butofwhat
kind it is?
Science is the belief in the ignorance of experts. What we know already ...
often prevents us from learning. Our freedom to doubt was born out of a struggle
against authority in the early days of science.
Science is no more a collection of facts than a heap of stones is a house. A
fact is valuable only for the idea attached to it, or for the proof that it furnishes.
He who does not know what he is looking for will not understand what he finds.
The investigator should have a robust faith – and yet not believe.
Science increases our power in proportion as it lowers our pride. Ignorance
more frequently begets confidence than does knowledge. Whoever undertakes to
set himself up as a judge of Truth and Knowledge is shipwrecked by the laughter
of the gods.
Thegodsarefondofajoke–theuniverseisnotonlyqueererthanwesuppose,
butqueererthanwecansuppose.Andthemostincomprehensiblethingaboutthe
world is that it is comprehensible.
Thereisgeometryinthehummingofthestrings,thereismusicinthespacing
of the spheres. A hidden beauty is stronger that an obvious one. It is godlike ever
to think on something beautiful and on something new.
Themostbeautifulthingwecanexperienceisthemysterious.Itisthesource
of ... art and ... science – branches of the same tree. He to whom this emotion
is a stranger ... is as good as dead: His eyes are closed.
2. Science 3
When it comes to atoms, language can be used only as in poetry. Poetry is
nearerto vital truth than history.Knowledgeis limited. Imaginationencirclesthe
world.
But put off your imagination, as you put off your overcoat, when you enter
the laboratory. Put it on again when you leave.
The objective reality of things will be hidden from us forever; we can only
know relations. Everythingwe call real is made of things that cannot be regarded
as real. The internal harmony of the world is the only true objective reality.
It is not nature that imposes time and space upon us, it is we who impose
them upon nature because we find them convenient. ... the distinction between
past, present, and future is a most stubbornly persistent illusion.
A human being is a part of a whole, called by us “universe” – a part ...
restricting us to our personal desires and to affection for a few persons nearest to
us. Our task must be to free ourselves from this prison by widening our circle of
compassionto embrace all living creatures and the whole of nature in its beauty.
Pythagoras 1 Charles Darwin Niels Bohr
Heraclitus 1 Claude Bernard Albert Einstein
Plato 1 James Clerk Maxwell John Haldane
Aristotle 1 Henri Poincar´e Richard Feynman
What these people think and how they write enlightens your mind and ele-
vatesyourspiritbutthesethoughtshavelittlelifeoftheirown.Theydonotgrow,
they do not transform, they do not shoot new green sprouts – they luminesce as
crystalsoffieryflowersfrozenineternity.Theyarenotquitewhatmathematicians
call ideas, they are halfway between ideas and opinions.1
Greatscientific ideas aredifferent– they arealive,they ignite yoursoulwith
delight, they invite you to fight and to contradict them. Unlock your spirit from
the cage of the mundane, let your imagination run free, start playing with such
ideasasalittlepuppywithitstoys–andyoufindyourselfintheworldofBeautiful
Elsewhere – that is called Mathematics.
1AnopinionaboutXisafunction,sayOPX =OPX(p),thatassignsyes(agree)orno(disagree)
to a person p who is dying to saywhat he/she thinks about X; e.g., about the existence (non-
existence) ofvacuum. Democritus didnotcareabout specificyes/no-values ofOPX(p), except
forpbeing hisbest friend,maybe. Butthe philosopher wouldlovetosee acorrelationbetween
OPX(p)forX =“vacuum exists”andthedistancefromthehouseofptoAbdera.