The Mathematics of Information Technology – The Road Ahead V. Kumar Murty Department of Mathematics University of Toronto

A Mathematical Universe  The universe seems to exhibit mathematical properties.  Pythagoras and the music of the spheres.  Aryabhatta and his algorithm inspired by astronomy.

Descartes’ vision  On November 10, 1619 when Descartes was 23, he had a vision in which it was “revealed” to him that the universe is mathematical and logical.  Since Descartes, science was gripped with the idea of a universe that can be described mathematically.

Descartes  The universe that Descartes referred to is the ‘physical universe’ from which mind and thought are completely separated. This is the dualism of mind and matter.  Descartes’ vision itself is neither mathematical nor logical and so according to him stands outside the universe.  Paradoxically, one of his main contributions was in taking mathematics from the physical to the level of ideas: he is known for his contribution of algebraizing geometry. A straight line is identified with the equation that describes it, and so on.  Abstraction is a common ingredient in all attempts to mathematically describe a phenomenon.

A mathematical universe  In what sense is the universe mathematical?  Physical laws can be formulated unambiguously.  Mathematics gives a language for unambiguously representing, organizing and manipulating information.  We rely on it because it has a predictive power.

A mathematical universe  Why is the universe mathematical? No one knows.  Eugene Wigner called it “the unreasonable effectiveness of mathematics”.  But it does have limitations.

Mathematics as a language  Mathematics is a language which seems to be well suited for describing the physical universe.  Defining characteristic is precision.  It shares some similarities with music.

The Uncertainty Principle of Language  Breadth vs. precision  Mathematics is not suitable for expressing certain ideas (for example, feelings)  It can capture quantifiable phenomena

Mathematics and Science  The success of mathematics in the physical sciences had a profound impact on many other branches of enquiry.  For a long time, no field of enquiry was considered scientific unless it could be expressed mathematically  We therefore see many new disciplines emerge as an attempt to use mathematical methods in novel fields.

Hobbes and Geometry  Thomas Hobbes ( ) accidentally came across a copy of Euclid’s elements when he was 40.  He read the statement of “Proposition 47” (the theorem of Pythagoras) and exclaimed aloud “this is impossible!”

Hobbes and Geometry  However, he read the demonstration in which he was led to earlier propositions and their proofs until he was convinced.  He fell in love with geometry and the axiomatic method.  So impressed was he with the idea that a statement which was not obvious could be proved by systematic and logical reasoning that he wondered whether all thought could be formulated axiomatically.

An axiomatic approach to society  He tried to apply this to the organization of society. He conceived of it as an artificial being (Leviathan) composed of parts (individuals).  He attempted to make government and social institutions an object of rational analysis and politics a science.

Hobbes  Perhaps he would have done a better job if he had had a deeper understanding of mathematics.  Hobbes spent considerable effort in trying to square the circle and double the cube.  He also entered into a controversial “debate” with Wallis which began with mathematics and spread to theological and personal questions.  Hobbes wrote “Marks of the Absurd Geometry, Rural Language, Scottish Church Politics, and Barbarisms of John Wallis, Professor of Geometry and Doctor of Divinity”

Montesquieu and Social Laws  His firm belief that everything was governed by laws was greatly influenced by Descartes.  Montesquieu tried to understand social facts as objects of science subject to laws.  These laws are not created but “God-given”, in other words, axioms.

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

Montesquieu and government  He formulated the concept of a three-body government (executive, parliament and judiciary) and a “separation of powers” between the bodies.  Though he seemed to prefer a democratic government, he did not feel that all were equal. He advocated slavery and had doubts about the abilities of women.

Mathematics and Ethics  Baruch Spinoza tried to establish an ethical system through a deductive method modeled on Euclidean geometry.  “Ethics demonstrated in geometric order”  Anything that cannot be captured mathematically is illusion.

The Tool of Abstraction  Each of Hobbes, Montesquieu and Spinoza attempted to formulate abstract concepts that modelled the reality they were trying to describe.  The abstract concepts could then be subjected to analysis which they felt had mathematical precision.

Mathematics and information technology  The representation of information  The manipulation of information to accomplish a predetermined function  The protection of information  Mathematics can be used in all of these aspects

Google and Linear Algebra  Google set itself apart from other search engines by its ability to quantify “relevance”.

Google and Linear Algebra  Suppose we have a connected directed graph of n nodes.  We want to attach a “relevance factor” to each node and use it to order the nodes.  We might say that the relevance of a node is increased by the number of other nodes that link to it.  We might weight nodes by the number of outgoing links.

Google and Linear Algebra  Suppose node k is given non-negative weight x k which we are trying to define.  Consider the matrix A = (a ij ) where a ij = 1/n j if node j connects to node i and n j is the number of outgoing edges from node j.  Then we have to solve the matrix equation Ax = x where x is the column vector (x 1, … x n ) T.  The columns of the matrix add to one, so it will always have 1 as an eigenvalue.

Google and Linear Algebra  The graph on the right gives the matrix /2 1/3000 1/31/201/2 1/31/200

Google and Linear Algebra  This matrix has a unique eigenvector for the eigenvalue 1, namely (12,4,9,6) T.  This gives node 1 the highest ranking.  Need to modify this if the graph is not connected or if the eigenspace is of dimension >

CDs and Polynomials over finite fields  Compact discs and many storage mechanisms (DVD, Raid, etc) encode information using polynomials over finite fields.  Let F be a finite field, say of cardinality q.  Order the nonzero elements as x 1,…,x q-1.  The code words are the (q-1)-tuples {(f(x 1 ),…f(x q-1 )): f a polynomial over F of degree < k}

CDs and Polynomials over finite fields  A given text of k-1 elements of F to be encoded are viewed as the values of a polynomial f of degree < k with coefficients in F.  The k-1 symbols are encoded as (f(x 1 ), …, f(x q-1 )).  In practice, commonly used values are n=255 = , and k=223 and it can correct (n-k)/2 = 16 errors.

CDs and Polynomials over finite fields  Reed-Solomon codes were invented in 1960 but were applied to CDs in  They have been generalized in many ways, including algebraic geometry codes in which the polynomials are replaced by functions on a curve over F.

Roomba and computational geometry  Robotic vacuum cleaner.  Navigation is through computational geometry.  Examples of problems that need to be solved: given n points, find the pair that has the shortest distance.

Information Security  Establishing a shared secret through an insecure medium.  Security is measured through the computational difficulty of solving certain mathematical problems.  Examples are factoring integers, computing a discrete logarithm in a finite cyclic group, finding the shortest nonzero vector in a lattice, etc.

Mathematics and Life Sciences: the new frontier  Relationship between mathematics and life sciences in the 21 st century may be similar to that between mathematics and physics in the 20 th century.  Is there a biological counterpart to physical force?  Functional and non-local interactions.  Need for a language to express concepts such as “self-organization” and “emergent properties”.

Information in Living Organisms  Information technology of the past has been the reading and manipulation of information in the physical universe.  Information technology of the present is largely the representation and manipulation of information that we generate.  The information technology yet to come is in the reading and manipulating of information in living systems.  Organisms differ from inanimate matter in that they possess coded information.

Mathematics in the Life Sciences  Mathematical neuroscience tries to model neuronal activity.  Main problem of neuroscience: how does the nervous system process information?  Population genetics: how genetic mutations and selection are propagated in a population.  Epidemiology: dynamics of diseases.

Systems Biology  Information is not only in the nucleus of the cell, but in the entire cell.  A view of the cell as a system consisting of functional components.  The aim is to predict and control the behaviour of the system.

Synthetic Biology  Natural successor is “synthetic biology” in which the system is “engineered”.

Intelligence and Consciousness  A more difficult array of questions are centered on intelligence: what is it?  Problems of cognition: how do we find meaning? What is meaning?  What is consciousness?

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

The Nature of Mathematical Calculation  In numerical calculations that involve approximations or “errors”, repeated calculations can compound the “errors” to the point of rendering the calculation meaningless.  “Arithmetical” or “logical” depth refers to the number of serial arithmetic operations that have to be performed in a calculation.

The Language of Life  Biological calculation seems to be more “horizontal”.  This “horizontal” nature of biological calculation eliminates the problem of compounding errors.  The language of the nervous system seems to have less “arithmetical depth” than we are used to in mathematics.

The Language of Life  The nervous system seems to use a radically different method of notation which is stochastic: not the positional system.  Meaning is conveyed by statistical properties of the message.  Meaning is also conveyed by statistical properties of different messages transmitted simultaneously.

Implications for mathematics  The mathematics we know may in fact be a “secondary” language “derived” from the “primary” language of the nervous system.

Mathematics and the human brain  DARPA recently asked for proposals to build a mathematical model of the human brain.  That effort will radically alter the way we construct digital computers.  It may also radically alter our views of intelligence and consciousness and our humanity.