The Mathematics of The Mathematics of Information Technology – The Information Technology – The

Road AheadRoad Ahead

V. Kumar MurtyV. Kumar Murty

Department of MathematicsDepartment of Mathematics

University of TorontoUniversity of Toronto

A Mathematical UniverseA Mathematical Universe

The universe seems The universe seems to exhibit to exhibit mathematical mathematical properties.properties.

Pythagoras and the Pythagoras and the music of the spheres.music of the spheres.

Aryabhatta and his Aryabhatta and his algorithm inspired by algorithm inspired by astronomy.astronomy.

DescartesDescartes’’ vision vision

On November 10, 1619 On November 10, 1619 when Descartes was 23, when Descartes was 23, he had a vision in which it he had a vision in which it was was ““revealedrevealed”” to him that to him that the universe is the universe is mathematical and logical.mathematical and logical.

Since Descartes, science Since Descartes, science was gripped with the idea was gripped with the idea of a universe that can be of a universe that can be described described mathematically.mathematically.

DescartesDescartes The universe that Descartes referred to is the The universe that Descartes referred to is the ‘‘physical physical

universeuniverse’’ from which mind and thought are completely from which mind and thought are completely separated. This is the dualism of mind and matter.separated. This is the dualism of mind and matter.

DescartesDescartes’’ vision itself is neither mathematical nor vision itself is neither mathematical nor logical and so according to him stands outside the logical and so according to him stands outside the universe. universe.

Paradoxically, one of his main contributions was in Paradoxically, one of his main contributions was in taking mathematics from the physical to the level of taking mathematics from the physical to the level of ideas: he is known for his contribution of algebraizing ideas: he is known for his contribution of algebraizing geometry. A straight line is identified with the equation geometry. A straight line is identified with the equation that describes it, and so on.that describes it, and so on.

Abstraction is a common ingredient in all attempts to Abstraction is a common ingredient in all attempts to mathematically describe a phenomenon.mathematically describe a phenomenon.

A mathematical universeA mathematical universe

In what sense is the universe In what sense is the universe mathematical? mathematical?

Physical laws can be formulated Physical laws can be formulated unambiguously.unambiguously.

Mathematics gives a language for Mathematics gives a language for unambiguously representing, organizing unambiguously representing, organizing and manipulating information.and manipulating information.

We rely on it because it has a predictive We rely on it because it has a predictive power.power.

A mathematical universeA mathematical universe

Why is the universe Why is the universe mathematical? No mathematical? No one knows.one knows.

Eugene Wigner called Eugene Wigner called it it ““the unreasonable the unreasonable effectiveness of effectiveness of mathematicsmathematics””..

But it does have But it does have limitations.limitations.

Mathematics as a languageMathematics as a language

Mathematics is a Mathematics is a language which language which seems to be well seems to be well suited for describing suited for describing the physical universe.the physical universe.

Defining characteristic Defining characteristic is precision.is precision.

It shares some It shares some similarities with similarities with music.music.

The Uncertainty Principle of The Uncertainty Principle of LanguageLanguage

Breadth vs. precisionBreadth vs. precision Mathematics is not Mathematics is not

suitable for suitable for expressing certain expressing certain ideas (for example, ideas (for example, feelings)feelings)

It can capture It can capture quantifiable quantifiable phenomenaphenomena

Mathematics and ScienceMathematics and Science

The success of mathematics in the The success of mathematics in the physical sciences had a profound impact physical sciences had a profound impact on many other branches of enquiry.on many other branches of enquiry.

For a long time, no field of enquiry was For a long time, no field of enquiry was considered scientific unless it could be considered scientific unless it could be expressed mathematicallyexpressed mathematically

We therefore see many new disciplines We therefore see many new disciplines emerge as an attempt to use mathematical emerge as an attempt to use mathematical methods in novel fields.methods in novel fields.

Hobbes and GeometryHobbes and Geometry

Thomas Hobbes (1588-Thomas Hobbes (1588-1679) accidentally came 1679) accidentally came across a copy of Euclidacross a copy of Euclid’’s s elements when he was elements when he was 40.40.

He read the statement of He read the statement of ““Proposition 47Proposition 47”” (the (the theorem of Pythagoras) theorem of Pythagoras) and exclaimed aloud and exclaimed aloud ““this this is impossible!is impossible!””

Hobbes and GeometryHobbes and Geometry

However, he read the demonstration in which he However, he read the demonstration in which he was led to earlier propositions and their proofs was led to earlier propositions and their proofs until he was convinced.until he was convinced.

He fell in love with geometry and the axiomatic He fell in love with geometry and the axiomatic method.method.

So impressed was he with the idea that a So impressed was he with the idea that a statement which was not obvious could be statement which was not obvious could be proved by systematic and logical reasoning that proved by systematic and logical reasoning that he wondered whether all thought could be he wondered whether all thought could be formulated axiomatically.formulated axiomatically.

An axiomatic approach to An axiomatic approach to societysociety

He tried to apply this to the organization of He tried to apply this to the organization of society. He conceived of it as an artificial society. He conceived of it as an artificial being (Leviathan) composed of parts being (Leviathan) composed of parts (individuals).(individuals).

He attempted to make government and He attempted to make government and social institutions an object of rational social institutions an object of rational analysis and politics a science.analysis and politics a science.

HobbesHobbes

Perhaps he would have done a better job if he Perhaps he would have done a better job if he had had a deeper understanding of had had a deeper understanding of mathematics.mathematics.

Hobbes spent considerable effort in trying to Hobbes spent considerable effort in trying to square the circle and double the cube.square the circle and double the cube.

He also entered into a controversial He also entered into a controversial ““debatedebate”” with Wallis which began with mathematics and with Wallis which began with mathematics and spread to theological and personal questions.spread to theological and personal questions.

Hobbes wrote Hobbes wrote ““Marks of the Absurd Geometry, Marks of the Absurd Geometry, Rural Language, Scottish Church Politics, and Rural Language, Scottish Church Politics, and Barbarisms of John Wallis, Professor of Barbarisms of John Wallis, Professor of Geometry and Doctor of DivinityGeometry and Doctor of Divinity””

Montesquieu and Social LawsMontesquieu and Social Laws

His firm belief that His firm belief that everything was governed everything was governed by laws was greatly by laws was greatly influenced by Descartes.influenced by Descartes.

Montesquieu tried to Montesquieu tried to understand social facts understand social facts as objects of science as objects of science subject to laws.subject to laws.

These laws are not These laws are not created but created but ““God-givenGod-given””, , in other words, axioms.in other words, axioms.

Montesquieu and SociologyMontesquieu and Sociology

He formulated the concept of He formulated the concept of ““social typesocial typess”” and studied them by comparing different and studied them by comparing different societies. In some sense, this was the societies. In some sense, this was the beginning of the field of sociology.beginning of the field of sociology.

Montesquieu and governmentMontesquieu and government

He formulated the concept of a three-body He formulated the concept of a three-body government (executive, parliament and government (executive, parliament and judiciary) and a judiciary) and a ““separation of powersseparation of powers”” between the bodies.between the bodies.

Though he seemed to prefer a democratic Though he seemed to prefer a democratic government, he did not feel that all were government, he did not feel that all were equal. He advocated slavery and had equal. He advocated slavery and had doubts about the abilities of women. doubts about the abilities of women.

Mathematics and EthicsMathematics and Ethics

Baruch Spinoza tried Baruch Spinoza tried to establish an ethical to establish an ethical system through a system through a deductive method deductive method modeled on Euclidean modeled on Euclidean geometry.geometry.

““Ethics demonstrated Ethics demonstrated in geometric orderin geometric order””

Anything that cannot Anything that cannot be captured be captured mathematically is mathematically is illusion.illusion.

The Tool of AbstractionThe Tool of Abstraction

Each of Hobbes, Montesquieu and Each of Hobbes, Montesquieu and Spinoza attempted to formulate abstract Spinoza attempted to formulate abstract concepts that modelled the reality they concepts that modelled the reality they were trying to describe.were trying to describe.

The abstract concepts could then be The abstract concepts could then be subjected to analysis which they felt had subjected to analysis which they felt had mathematical precision.mathematical precision.

Mathematics and information Mathematics and information technologytechnology

The representation of The representation of informationinformation

The manipulation of The manipulation of information to accomplish information to accomplish a predetermined functiona predetermined function

The protection of The protection of informationinformation

Mathematics can be used Mathematics can be used in all of these aspectsin all of these aspects

Google and Linear AlgebraGoogle and Linear Algebra

Google set itself apart Google set itself apart from other search from other search engines by its ability engines by its ability to quantify to quantify ““relevancrelevancee””..

Google and Linear AlgebraGoogle and Linear Algebra

Suppose we have a connected directed Suppose we have a connected directed graph of n nodes.graph of n nodes.

We want to attach a We want to attach a ““relevance factorrelevance factor”” to to each node and use it to order the nodes.each node and use it to order the nodes.

We might say that the relevance of a node We might say that the relevance of a node is increased by the number of other nodes is increased by the number of other nodes that link to it.that link to it.

We might weight nodes by the number of We might weight nodes by the number of outgoing links.outgoing links.

Google and Linear AlgebraGoogle and Linear Algebra

Suppose node Suppose node kk is given non-negative weight is given non-negative weight xxk k

which we are trying to define.which we are trying to define.

Consider the matrix Consider the matrix A = (aA = (aijij)) where where aaijij= 1/n= 1/njj if if

node node j j connects to node connects to node ii and and nnjj is the number of is the number of

outgoing edges from node outgoing edges from node jj.. Then we have to solve the matrix equation Then we have to solve the matrix equation Ax = Ax =

x x where where xx is the column vector is the column vector (x(x11, … x, … xnn))TT..

The columns of the matrix add to one, so it will The columns of the matrix add to one, so it will always have 1 as an eigenvalue.always have 1 as an eigenvalue.

Google and Linear AlgebraGoogle and Linear Algebra

The graph on the The graph on the right gives the right gives the matrix matrix 1

2

4

30 0 1 1/2

1/3 0 0 0

1/3 1/2 0 1/2

1/3 1/2 0 0

Google and Linear AlgebraGoogle and Linear Algebra

This matrix has a This matrix has a unique eigenvector for unique eigenvector for the eigenvalue 1, the eigenvalue 1, namely (12,4,9,6)namely (12,4,9,6)TT..

This gives node 1 the This gives node 1 the highest ranking.highest ranking.

Need to modify this if Need to modify this if the graph is not the graph is not connected or if the connected or if the eigenspace is of eigenspace is of dimension > 1.dimension > 1.

1

2

4

3

CDs and Polynomials over finite CDs and Polynomials over finite fieldsfields

Compact discs and many storage Compact discs and many storage mechanisms (DVD, Raid, etc) encode mechanisms (DVD, Raid, etc) encode information using polynomials over finite information using polynomials over finite fields.fields.

Let F be a finite field, say of cardinality q.Let F be a finite field, say of cardinality q. Order the nonzero elements as xOrder the nonzero elements as x11,…,x,…,xq-1q-1.. The code words are the (q-1)-tuples {(f(xThe code words are the (q-1)-tuples {(f(x11),),

…f(x…f(xq-1q-1)): f a polynomial over F of degree < )): f a polynomial over F of degree < k}k}

CDs and Polynomials over finite CDs and Polynomials over finite fieldsfields

A given text of k-1 elements of F to be A given text of k-1 elements of F to be encoded are viewed as the values of a encoded are viewed as the values of a polynomial f of degree < k with coefficients polynomial f of degree < k with coefficients in F.in F.

The k-1 symbols are encoded as (f(xThe k-1 symbols are encoded as (f(x11), ), …, f(x…, f(xq-1q-1)).)).

In practice, commonly used values are In practice, commonly used values are n=255 = 2n=255 = 288-1, and k=223 and it can correct -1, and k=223 and it can correct (n-k)/2 = 16 errors.(n-k)/2 = 16 errors.

CDs and Polynomials over finite CDs and Polynomials over finite fieldsfields

Reed-Solomon codes were invented in Reed-Solomon codes were invented in 1960 but were applied to CDs in 1982.1960 but were applied to CDs in 1982.

They have been generalized in many They have been generalized in many ways, including algebraic geometry codes ways, including algebraic geometry codes in which the polynomials are replaced by in which the polynomials are replaced by functions on a curve over F.functions on a curve over F.

Roomba and computational Roomba and computational geometrygeometry

Robotic vacuum Robotic vacuum cleaner.cleaner.

Navigation is through Navigation is through computational computational geometry.geometry.

Examples of Examples of problems that need to problems that need to be solved: given n be solved: given n points, find the pair points, find the pair that has the shortest that has the shortest distance. distance.

Information SecurityInformation Security

Establishing a shared secret through an Establishing a shared secret through an insecure medium.insecure medium.

Security is measured through the Security is measured through the computational difficulty of solving certain computational difficulty of solving certain mathematical problems.mathematical problems.

Examples are factoring integers, Examples are factoring integers, computing a discrete logarithm in a finite computing a discrete logarithm in a finite cyclic group, finding the shortest nonzero cyclic group, finding the shortest nonzero vector in a lattice, etc.vector in a lattice, etc.

Mathematics and Life Sciences: Mathematics and Life Sciences: the new frontierthe new frontier

Relationship between mathematics and life Relationship between mathematics and life sciences in the 21sciences in the 21stst century may be similar to century may be similar to that between mathematics and physics in the that between mathematics and physics in the 2020thth century. century.

Is there a biological counterpart to physical Is there a biological counterpart to physical force?force?

Functional and non-local interactions.Functional and non-local interactions. Need for a language to express concepts such Need for a language to express concepts such

as as ““self-organizationself-organization”” and and ““emergent propertiesemergent properties””..

Information in Living OrganismsInformation in Living Organisms

Information technology of the past has been the Information technology of the past has been the reading and manipulation of information in the reading and manipulation of information in the physical universe.physical universe.

Information technology of the present is largely Information technology of the present is largely the representation and manipulation of the representation and manipulation of information that we generate.information that we generate.

The information technology yet to come is in the The information technology yet to come is in the reading and manipulating of information in living reading and manipulating of information in living systems.systems.

Organisms differ from inanimate matter in that Organisms differ from inanimate matter in that they possess coded information.they possess coded information.

Mathematics in the Life SciencesMathematics in the Life Sciences

Mathematical neuroscience tries to model Mathematical neuroscience tries to model neuronal activity.neuronal activity.

Main problem of neuroscience: how does Main problem of neuroscience: how does the nervous system process information?the nervous system process information?

Population genetics: how genetic Population genetics: how genetic mutations and selection are propagated in mutations and selection are propagated in a population.a population.

Epidemiology: dynamics of diseases.Epidemiology: dynamics of diseases.

Systems BiologySystems Biology

Information is not only in Information is not only in the nucleus of the cell, the nucleus of the cell, but in the entire cell.but in the entire cell.

A view of the cell as a A view of the cell as a system consisting of system consisting of functional components.functional components.

The aim is to predict and The aim is to predict and control the behaviour of control the behaviour of the system.the system.

Synthetic BiologySynthetic Biology

Natural successor is Natural successor is ““synthetic biologysynthetic biology”” in in which the system is which the system is ““engineeredengineered””..

Intelligence and ConsciousnessIntelligence and Consciousness

A more difficult array of questions are A more difficult array of questions are centered on intelligence: what is it? centered on intelligence: what is it?

Problems of cognition: how do we find Problems of cognition: how do we find meaning? What is meaning?meaning? What is meaning?

What is consciousness?What is consciousness?

Mathematics and the Human Mathematics and the Human BrainBrain

Von Neumann Von Neumann compared the human compared the human brain to a digital brain to a digital computer and found computer and found that the superior that the superior power of the brain power of the brain comes from massive comes from massive parallelism.parallelism.

The Nature of Mathematical The Nature of Mathematical CalculationCalculation

In numerical calculations that involve In numerical calculations that involve approximations or approximations or ““errorserrors””, repeated , repeated calculations can compound the calculations can compound the ““errorserrors”” to to the point of rendering the calculation the point of rendering the calculation meaningless.meaningless.

““ArithmeticalArithmetical”” or or ““logicallogical”” depth refers to depth refers to the number of serial arithmetic operations the number of serial arithmetic operations that have to be performed in a calculation.that have to be performed in a calculation.

The Language of LifeThe Language of Life

Biological calculation Biological calculation seems to be more seems to be more ““horizontalhorizontal””..

This This ““horizontalhorizontal”” nature of nature of biological calculation biological calculation eliminates the problem of eliminates the problem of compounding errors.compounding errors.

The language of the The language of the nervous system seems to nervous system seems to have less have less ““arithmetical arithmetical depthdepth”” than we are used than we are used to in mathematics.to in mathematics.

The Language of LifeThe Language of Life The nervous system The nervous system

seems to use a radically seems to use a radically different method of different method of notation which is notation which is stochastic: not the stochastic: not the positional system.positional system.

Meaning is conveyed by Meaning is conveyed by statistical properties of statistical properties of the message. the message.

Meaning is also conveyed Meaning is also conveyed by statistical properties of by statistical properties of different messages different messages transmitted transmitted simultaneously.simultaneously.

Implications for mathematicsImplications for mathematics

The mathematics we know may in fact be The mathematics we know may in fact be a a ““secondarysecondary”” language language ““derivedderived”” from the from the ““primaryprimary”” language of the nervous system. language of the nervous system.

Mathematics and the human Mathematics and the human brainbrain

DARPA recently asked for proposals to DARPA recently asked for proposals to build a mathematical model of the human build a mathematical model of the human brain.brain.

That effort will radically alter the way we That effort will radically alter the way we construct digital computers.construct digital computers.

It may also radically alter our views of It may also radically alter our views of intelligence and consciousness and our intelligence and consciousness and our humanity.humanity.