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8/18/2019 Bühlmann - From Linear Algebra to Algebraic Invariances
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in the light of the Master Argucomputability
PhD Kolloquium Wvera bühlmann, Dec
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Before the study of invariances, thestudy of linear equations was usuallysubsumed under that of determinants
transformations carried out on the variables („constants“) within algebraic forms
> not the solution of a system of
the solv ability of equations and th
from linear algebra to algebraic invariances propremaa systtrans
Quantics
In linalgebra,
determinana vaassociated w
a squmatrix . It cbe compu
from entries of
mat
a linear equation is an equation
in which each term is either aconstant or the product of aconstant and a single variable
(not raised to any power).
Linear Algebra studiessystems of such equations. No consideration was given to systems in wh
number of equations differed from the numbeunknowns . This changed with the interest in
2
http://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Variable_(mathematics)http://en.wikipedia.org/wiki/Constant_termhttp://en.wikipedia.org/wiki/Constant_termhttp://en.wikipedia.org/wiki/Term_(mathematics)http://en.wikipedia.org/wiki/Term_(mathematics)http://en.wikipedia.org/wiki/Square_matrix#Square_matriceshttp://en.wikipedia.org/wiki/Square_matrix#Square_matriceshttp://en.wikipedia.org/wiki/Square_matrix#Square_matriceshttp://en.wikipedia.org/wiki/Square_matrix#Square_matriceshttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Linear_algebra
8/18/2019 Bühlmann - From Linear Algebra to Algebraic Invariances
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what aboutspace and
geometry in
linear algebra ?
field - rational domain within the compelx numbers satifying certain con
vector - establishes the operability over a field, n-dimensionality, in ideal vectorspaces are taken as subsets in rings.
ring - establishes unique factorization, modularity, of a field
algebra - hypercomplex number system (a ring and a vector space over a
key terms in linearalgebra
they are dealing withinvariances - properties thatremain stable undertransformations operating onthe variables or coefficients
within algebraic forms .
> what is the nature of these invaProperties of what are we dealingThis is the main disconcertion as
with invariances from a non-techpoint of view.
3
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linear algebra (forms of homogenous transformations)- invariance and covariance, vector spaces and
matrices. no systematization, a cookbook.
Interest in the solvability of structures in abstractionfrom any specific „ground“ - through the conception ofthe ground in ideality, rings, fields, modules, groups,etc.
Interested in providing the conditions for solvability.
The ground is symbolic and engendered (not assuminga given „nature“ of numbers)
treats valences in physics and chemistry, according toprinciples of saturation (from the Latin word saturare ,meaning to fill ) - a notion of fullfillment based onreaching a maximum capacity.
algebra
quantics
abstract algebra
symbolic algebra
symbolico-physical
> Doping technologies.
ha nd l e t hi s w i t h s p e c i a l c a r e - t he dist inc t ions ar e assor t edow n r easoning ! no st andar d list of dist inc t ions!
4
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1 General Features of Invariance2 algebraic invariance3 the synthesis by transformation groups
4 codification of geometry by invariance5 intrinsic spatial invariance
E.T. Bell, in The Development of Math
chapter on Invarian"Invarian
changele
midst of
permane
world of
persistenfiguratio
remain t
despite t
stress of
hosts of c
transform
(E.T. BEL
5
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paragraph1 General Features of Invarian
along which path?
Quantics - the study of algebraic form
Abstract and Symbolic Algebra - an undiscovered continent
Relativism - the hypotheses at the foundation of geometry
Systematization - through seeking a unification of geometries
Physics - general theory of relativity (Conservation Laws)
invariant theory as aaddition to mathem
thought ?
Invariance as newunifying principle ,abstraction from thebook listing of calcuin the algebra of form
Riemann Space - a cimagination, againsabsolutist imaginati
encircling whic
George Boole
Bernhard Riemann
Albert Einstein
Felix Klein
6
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paragraph 2 Algebraic Invariance
Determinants - evolution of the symbolic me
from the Calculus of Quantics to the Algebra of Qua
from invariance as a property of quantities to invariance as the porperty of gro
settling in the undiscovered continent - devices of calculation bring about „new Provin
foundations of math
algebraic geometry (encoding of space and spac
chemico-algebraic theory (theory of chemical valence e.g. be
-> quantics was eventually absorbed into quantum
Problems of Quantics – is there a fundamental system, and a finite set of independent irreduci
for quantics YES, but for quantum theory cf the Hilbert - Brouwer/Weyl controversy . Intuitionism.7
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Bell speaks of an „army of algebraists and
projective geometers“ who storm into „the fertile
territory “ of abstract symbolic algebra. Those
who applied the symbolic method were
adventurers stigmatized as „illegitimate Kings“
striving for „profit“ in the domains of their
settlement: Bell writes how they were „recruting
masses of young mathematicians who mistake
the kingdom of quantics for the democracy of
mathematics“.
settling in theundiscovered
continent(Algebra) was
interpretedpolitically before
breaking abattlefield
betweenideological claims
for supremacy.
Are the realms of abstraction „terrundiscovered continent“ t
Are adventurers in abstraction supposed to be
Renaissance, obliged to sail in the name of the Queen > then the State turns into a monarch of the multitude, capable
L. conquirere for "to search for,procure by effort, win"
from com- for "together, together with, in combination,"
and quaerere "to seek, acquire"
conto susubmundanomaymat
8
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„Cayley's numerous successes, quikly followed by those o
Sylvester, unleashed one of the most ruthless campaignscalculation in mathematical history . [...] Such misdirectenot peculiar to the algebra of quantics in mathematics siaccompanying theory of groups, for example, especially groups, there was a similar panic. Once the means for raisupplies of a certain crop are available , it would seem to bcaution to keep on producing it till the storehouses burstcourse, the crop is to be consumed by somebody. There hconsumers for the calculations mentioned, and none foreasily digested. Nevertheless, the campaign of calculatioapparently, of mere calculation did at least hint at undiscin algebra, geometry, and analysis that were to retain thedecades after the modern higher algebra of the 1870's hato the dustier classics.“
(E.T.Bell p. 429/30)
Enlightenment – political seculariz
The Algebraists were accused of
totalitarian calulation: as
campaigning torecruit
mathematiciansfor theory with no
application
(useless)
science detached from metaphysics/theology
> fear of the potential applications that werBell‘s stance: criticizes the filling of the „storages“ with „intellectual nourishment“ that 9
b l f h L f h E l d d Middl
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Background to the Hilbert – Brouwer/Weyl controversy
Intuitionism
Gordan in 1868 proved the existence of finite fundamensystems of invariants and covariants for any binary quaand in 1870 did the same for systems of such forms. Hismethod was constructive, he demonstrated a procedur which can generate all the instances.
Hilbert extended the finite theorem significantly by givipurely formal proof without procedure to actually geneall instances. On purely logical grounds – by applying thLaw of the Excluded Middle to Cantor‘s infinite sets.
"This is nomathematis theology
„For this memorable victory overthe barbaric hordes of algebraicformulas, Gordan was crowned
"King of Invariants" [...] Heoccupied the throne exactly twenty-
two years, until Hilbert, a merestripling of twenty-eight, in 1890
snatched the crown from Gordan'sageing head and rammed it firmly
down on his own.“ (E.T.Bell)
Jan Brouwer and Herman Weyl (both students of Hilbercritizized the formalist (logical) method and instist ininfinitary methods when dealing with infinities.
Aristotle‘s privation-dynamicsis a prototype for infinitary method; the same
for Booles‘ Algebra of logics , and for Dedekind‘sand Noether‘s conceptual approaches .
contradictories for theaccidental properties,Contradiction for theessential properties.
non-absoluteness of the Law of the Excluded Middle
10
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paragraph3 The synthesis by transformatioStructural theory – inversion of problem-solving approaches: not is there a solu
problem, but which operations are sufficient and necessary, and wh
objects must be invented, to provide a solution for a problem of a
elimination through composition series (Liand algebraic characterization of the nature of irrationals (fields, domains, etc) dire
looking for integrability – The primary objectives (of algebraic structural theory) are
can be done rather than to do it , and to give criteria for wha
Distinguish between reasonable formulatio of problems and not reasonable formulation of problem
physical and chemical valences (doping) – construct manifestations of invarian(partial) equations whose conditions the abstract invar
Organic and non-organic ch
logistics, geodesics, analytical mechanics – application of invariants to kinem
(describe Euclidean Space kinematically (not static!) by working with differ
Lie secured linearization for differential equation like Newton did for infi
operat11
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paragraph4 The codification of geometry
Codification - from
Paradigms of langauge, informati
from kinematic space to quantum space - from equivalence transformations between group
commensurable by schemata (quantified), to transformations within groups that are iden
and need to be quantized
search for a unified field theory - abstracting from Einstein‘s gravitMaxwell‘s electromagnetic field – attempts developed by Weyl, leading to quan
point-set geometries – geodesics, Levi Civita parallel displacement (the sum of the angle
the earth is more than 180°), generalization to the geometry of path, eventually: break
connections (Dirac‘s Algebra of Q
12
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„A geometry is defined as the system of definitions
and theorems invariant under a given group of
transformations.
Two groups of transformations are to be considered in
connection with any geometrical relation: a group by
means of which the relation may be defined ; a group
under which the relation is left invariant . The more
restricted the group, the more figures will be distinctrelatively to it, and the more theorems will appear in
the geometry.
The extreme case is the group corresponding to the
identity [the transformation leaving everything
considered invariant], the geometry of which
is too large to be of consequence .“
codification of geometry
The Erlangen Program 1872 (Felix Klein 1872 )
note the indefinite article –a geometry, like a language
If at least one postulate of a mathematical (hypothsystem be suppressed , the sfrom the modified set of porestricted than the original modified system is more fumore general than the origi
group - structural solution space (can be expresse
all the properties of a spaceconsidered if an identity re
instead of equivalence rela13
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In the calculation of group characters, analytical considerations offeran alternative to finite summations of subsets, which are beingreplaced by integrations over the group manifold , the invariant
element of volume of a compact group having been suitably defined.
Klein‘s'Erlanger Programm' of 1872 for the codification of geometry as it existed at the time, essentially reduced geometry to the study of
equivalence under certain groups of transformations. In the modernizedpresentation of Klein's project (e.g. in Hilbert axiomatization of
mathematics), groups appear as groups of automorphisms and ofpreferred coordinate systems of the base space . The automorphisms leave
invariant the characteristic relations of the particular geometryconcerned. These automorphisms induce transformations on
the vector subspaces of the base space .
the characterization of groups throughinvariant integrals (not summation of subspaces)
(and Felix Klein) the definition of sets by code, working withrepresenting groups by summation of subspaces within coordinate systems
Weyl
Klein/Hilb
this needs a re
of groups as obassumes coordit cannot deal space, only wirepresentationtherein.
is purely oper works directlyspaces (non-c
spaces, manif14
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This brought to attention precisely those
Riemannian geometries about which the
Erlanger Programm said nothing, namely
those whose group is the identity. In such
spaces there is essentially only one figure,
namely the space structure as a whole.
„This [Klein's] point of view was the dominant one for
the first half century after it was enunciated. It was a
helpful guide in actual study and research. Geometers
felt that it was a correct general formulation of what
they were trying to do. For they were all thinking of
space as a locus in which figures were moved about
and compared [as was implied earlier in connection
with Lie's revision of Helmholtz' kinematic geometry].
The nature of this mobility was what distinguished
between geometries.“ (E.T.Bell)
space as alocus in whfigures we
moved aboand comp
With the advent of Relativity wbecame conscious that space
need not be looked at only as
"locus in which," but that it
may have a structure, a field-
theory , of its own. Einstein‘s Relativity Th> space-time.
„nature of this mobility“ > K
classical mechanics that des
bodies (objects) and systems
without consideration of th
No unified field theory for the
space of a geometry: Einstein‘s field of Gravitation andMaxwell‘s field Electromagnetism
15
http://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Classical_mechanics
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"A space is a set of objects with a definitesystem of properties, called the structure of
the space." (cited in Bell p. 448)
„A geometry is defined as the
system of definitions and theorems
invariant under a given group of
transformations“.
With relativity theory – matter itself baspect of spacetime . The measuremencurvature varies from point to point icorresponding to the amount of mat
In the absence of matter, spacetime i
Space is not a locus in which objects are placed and subjects act
(governed by whom/whatever).
metricsspace
matter
> this is what is so silly aboPhilosophy Movement! (O
tecthey are dealing withquantification, not with quphilosophy were game desi
16
C di h
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A „wor
signs ovsigns a
com
(sy
{a, b, c .... x, y, z}
{0, 1, 2,.....8, 9} decimales alphabet
alphabet in letters
involves double discretization of amplitud
1) sampling (Abtastung)
2) quantization (Quantelung)
change of an alpe.g. analog/digital con
Coding – quantification vs quantization
formal Language vs Quantutransformation groups as alphabets
{0, 1} binary alphabet
... ...
quantization is wher
model of language-in
life streams and SOM
breeding clusters of d
be interpreted and fo
17
in codification theory:
i i i i l i i
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paragraph5 intrinsic spatial invariance
topology - can it be interpreted as „the royal road „the devil of abstract algebra and the angel of topology“ (Herman Weyl): one striving for articulation and differentiatio
properties of a space - topology works with its own categories of how neighborhood, region, bounary, etc. These allow to predicate the properties of a
topological space - topology constructs its spaces according to the transformation gri.e. by the qualitative properties of space (independent of size, location, and shape): continous, homeomorphic (finite and con
knots – structural „objects“ (knots) in non-coordinated space (networks) can be a
aspectual space only (their appearances). They have relative dimensionality (topological invariances, c Analysis allows to enumerate and characterize all possible knots (Euler‘s bridge problem) relative t
topology illustrates properties of functions – appearances are constituted by their qualitat
topology in analysis - inversion of Descartes Accidential properties (dimensions in topology) for Descartes were series , starting from an absolute point. Construction had to build
Now we can postulate properties as invariants of groups of complex bodies, and then construct the spaces accordingly. In complex
imaginary numbers in the real numbers (Dedekind Cut). The r18
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an Aristotelian mindset
and contemporarymathematics ?recap the dynamics of privation
Everything is subject to privation in the events that can happen tothem, they gain their individuality thereby.
The space of quality is full – but never exhaustible!
He ascribed privationCan we ascribe it to
19
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„This garment, for example, may be cut in two and yet wibe cut in two, but will wear out first. In the same way, it mnot be cut, for it could not wear out first were it not poss
for it not to be cut in two. This holds for all other events a which are mentioned as having the same kind of potentia
Is this real or virtual ?
Aristotle‘s exampleof probabilistic
truth value based
on opinion:
Aristotle‘s future contingents, as in the example below
theory of the potentiality for contrar
A constradiction, if it concerns the accidoperationally and can be harvested in thin ethics .
abstracaccomodiversitbrings r
20
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to have a potentiality formeta-property for Aristo
For Aristotle, privation is if a thing is
hindered in fullfilling its potential.
To what is this kind of potentiality a proper potential (to whom or what does itbelong)? What could possible „fullfill“ it?Or is it not subject to privation at all?
Privation is what many (modern) theories hold asconstitutive for „the human“, and for ethics. The idea of
socalled Mangelontologien (ontologies of lack) is thatonly through affirming privation – the being hindered
through communality in fullfilling exhaustively ones
potential – can we live together socially.
excursion: privationthis is the kind of pro
abstract entities likeconstitution, plan, a generali
a form or a schema, etcpotentiality an abstract en
has, the higher its va
Abstraction allows to conse
myabstraction
the degree of(freedom) w
mastering of a govern its s
Not MANGELONTOLOGIE:
abstraction createsan excess of potential
properties for itssubjects. It
introduces aneconomy principle 21
b i i
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being-in-actbeing-in-potency
composed and divided yes, but
essential
accidential/probable
(the properties belong to a subje
no!
(the properties belong to an eve
no contradictions allowed in the essential
contradictions are operationalized within the dynamics of privation
as contradictories the can be
conjuncted and disjuncted
Contingency in the universe due todynamics of privation.
This is the Reality-Principle of A
Aristotle economized necessitarianist prede
for Aristotle,Truth had a Nature! „lending articulate voice to that with inarticulate eloquence“
A R T I C U L A T I O N
can algebraic invarianceplay the role of Aristotle‘
essential today?
22
i i l f d
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[For Aristotle:]
„Accidental beings are not necessary butindeterminate, and their causes are unordered andinfinite.“
The principle of correspondence (at work in
language) transmits the properties of the things andtheir causes to the statements about them.
principle of correspondence And the Nature of Truth (within Realist Philosophy in Aristotle)
> correspondence is not representation but articul
how to te without coerci
for accidentials
the idea of a natural flow we ca
topologby estab
co
23
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being is not absolute for Aristotle
being-in potency can never be fully actualized.motioninfinity voidfullness
Reality actualizes from „linking up“ being(essences) and being-in-potency (accidencaught up in the dynamics of contrarity wdriven by potentiality, privation).
but numerouquantificationquantization because of the simultanousexistence of contraries
L. absolutufree, make "without renot relative
Aristotle‘s Reality is
analytical, ofdifferential make-up
which allows forgeneration and decay,
transformation,becoming
24
th t hi h i ti l t ElAristotle‘s
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that which expresses inarticulate Eleganc
that which expresses the appearancesTarski‘sNature
Aristotle sNature
cannot be exhausted by language, the poetic principle of th
can be determined by formal language, the semantic principle of ththe domestication of Language by Algebra
the domestication of Nature by Language
What happens to theinarticulate elegance
of that whichappears without
being forced into
expression?
How can Algebra
domesticate Natuand not only coer form and consequthrough controlli(de dicto)?
25
the beauty
8/18/2019 Bühlmann - From Linear Algebra to Algebraic Invariances
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the beauty equation doe
appear if we solution, but from
promise of integradifferences wit
conif we can see inarticulate
promise in it! 26