sundarramchand
03-08-2011, 01:05 AM
MATHEMATICS , A CREATIVE ART
(Note : My philosophies have changed in that though i still think the pursuit of Mathematics is an art in addition to being a science, i now believe that mathematical truths are not just creations of the human mind and are fundamental truths which are as real as the laws of physics etc and that these are discovered by the human mind. The ideal would be a synthesis of the two.)
Mathematics has always been thought of as a cold and austere science. The purpose of this essay is to show that Mathematics is not just about dry facts , tautologies and proofs but is also a growing and organic discipline. This essay also tries to show that Mathematics is an art in addition to being a science.
I have tried to touch upon the following aspects to show that Mathematics is a creative art.
- The relation between Mathematics and logic and the role of constructive activity and intuition in Mathematics.
- Mathematics and unexpected relationships
- Mathematics and the spirit of fun
- Applications of Mathematics
- New developments in Mathematics and some novel applications
RELATION BETWEEN MATHEMATICS AND LOGIC AND THE ROLE OF CONSTRUCTIVE ACTIVITY AND INTUITION IN MATHEMATICS.
Many people (including people who are good in solving Mathematical problems) aver that all mathematical activity is discovery of existing tautologies through deductive logic, i.e. discovery and proof of existing facts through deductive logic. In other words, they state that mathematics is just a language.
If such were the case, mathematics would have been struck at the Euclidean level of theorems and proofs on the one hand and basic arithmetic on the other. But the fact of creation of such constructs as sequences, infinite sequences, the operation of limits , divergence and convergence which led to the discovery of analysis and calculus refutes the above theories of mathematics as just a language for communication and reveals the facet of mathematics as a mode of thought.
New classes are forever being constructed in mathematics and all classes being widened through abstraction.
One Prime example is the generalization of the concept of dimension to non integral (or fractional ) and other values
Another example is the creation by weistrass of a function which was continuous everywhere but not differentiable anywhere.
Yet another example is the extension of differentiation and integration to fractional orders (say instead of 1st derivative, the 1/2th derivative etc)
nowadays when we say A + B = C, we could be referring to either addition of 2 numbers (whether natural, real or complex) or it could mean the addition of 2 vectors or 2 matrices or it could be referring to 2 elements of a group and the operation may not necessarily denote addition in the normal sense.
Categories are not predefined for all time. They are fluid in the sense that their meaning (or semantics) in what they symbolize or stand for or what they are symbolized by changes over a period of time
This shows that the same elements of the language may symbolize different things according to the context and the mapping between the real world and mathematics
At the turn of this century, there was a project (in which great mathematicians like Hilbert and Bertrand Russell participated) to unify mathematics and logic and breakdown the body of mathematics into a few building blocks of axioms from which the rest of mathematics can be inferred by shuffling the building blocks using the rules of logic.
But this project failed due to :
1) A paradox involving self-referential sets which was a variation on the old paradox embodied in the statement “I am a liar”.
2) The proof by Godel that any consistent system (even the simplest one i.e. natural numbers) cannot logically prove its own consistency and that such a system would have at least one true statement which cannot be proved using the axioms of that system.
The proof itself involved a burst of creative insight in which Godel managed to represent statements about natural numbers through natural numbers themselves and managed to construct a true statement that is self-referential.
These developments proved a nail in the coffin of the premise that mathematics could be reduced to logic.
The other development in the last century and at the turn of the current century was the discovery that if we change/replace the parallel line postulate(which had to be made an axiom because it cannot be proved through other axioms) of Euclid (Through a given point one and only one line can be drawn parallel to a given line) , other consistent geometries (For e.g. hyperbolic , Reimannian , elliptic).
All this goes to show that the predominant feature of modern mathematics is the creation of new constructs (especially when defining axioms ) . Mathematicians have gone on to extend Euclidean geometry to multiple dimensions and also defined spaces with different dimensions, metrics (the distance between two points as a function of the coordinates of the two points) and topologies which significantly determine the features of that space.
Some of the examples are :
- The 4-dimensional Minkowskian space involving space and time used in special relativity
- The 4-dimensional curved space used in general relativity
- The space of signals and the space of functions (generalized to Hilbert space and used in Quantum mechanics and spectral theory)
MATHEMATICS AND THE UNEXPECTED (METAPHOR IN MATHEMATICS)
Paul Dirac, the famous scientist once said that equations , in addition to being true should also be beautiful. The criteria for beauty could be the wealth of relationships that are generated between categories unrelated till then.
Some examples are :
- The discovery of links between Knot theory and mathematical physics which led to a new class of invariants to distinguish knots
- The discovery of links between the class of modular forms and the class of elliptic curves which figured prominently in the proof of Fermat’s Last theorem
- The relationship between complex numbers and vectors
- Eulers formula : e^i = cos() + isin() where i refers to sqrt(-1)
- The equations and theorems in the notebook of Srinivas Ramunajan
- The relation between the zeroes of Reimann’s Zeta function and the number of prime numbers less than a given number
- The expression of Phi through an infinite series (Gregory’s series) involving the expansion of an integrand
MATHEMATICS AND THE SPIRIT OF FUN
Creativity involves play. In mathematics too, the attempt to solve certain puzzles have led to the blossoming of certain fruitful areas of mathematics
Some examples :
The attempt to solve problems like the Konisberg bridge problem, the travelling salesman problem, the knights tour of a chessboard etc have led to the growth of the discipline of graph theory.
The attempt to solve Zeno’s Paradox of Achilles and the tortoise (i.e Achilles cannot catch up with the tortoise since first he has to cover half the distance and to cover that distance, he has to cover half of that distance and so on ad infinitum) led to the concepts of convergence of infinite series
The attempt to construct a square equal in area to a circle (with a ruler and compass only) led in the last century to the proof of transcendence of Phi
The attempt by Mathematicians to solve the Rubik’s cube (and other permutation puzzles) led to insights in group theory
APPLICATIONS IN MATHEMATICS
Like any creative art, mathematics has spawned many interesting applications (‘The unreasonable effectiveness of mathematics’) . The theory of representation of groups is useful in determining energy levels of compounds etc and their degeneracy by studying their symmetry . Representation theory of groups has had a huge success in classification (and prediction ) of elementary particles in various families.
Knot theory has found applications in the study of coiling of DNA
This is just a very small sample.
NEW DEVELOPMENTS IN MATHEMATICS AND SOME NOVEL APPLICATIONS
Catastrophe theory developed by Rene Thom studies structurally stable qualitative changes (change in number and type of equilibrium points) due to a change in parameters and it is said that it can be used to study various systems (after refinement) not only in the physical sciences but also in the biological and social sciences.
Chaos theory synthesizes the paradigms of ‘order’ and ‘disorder’ by showing that apparently random processes may be governed by deterministic albeit chaotic laws. It also shows that it is possible for a system to stay bounded within a region of state space and yet be unpredictable in the sense that it is not possible to predict where a trajectory will end up. Concepts like strange attractors , ergodicity (all parts of state space being accessible by a trajectory ) find a place in chaos theory. This theory finds applications in statistical physics (especially in far from equilibrium processes) , turbulence etc.
A related concept is that of fractals . These retain structure at all scales. There is a measure called dimension which characterizes the relationship between length of an object and the scale at which the length is measured. For fractal objects / sets which are usually very irregular, these measure is usually fractional and characterizes a particular fractal. They are self-similar either statistically or deterministically . many of them like the Koch Curve , Sieperenski’s Gasket etc are constructed by recursively applying a series of transformations to an initial object. Others like the Julia set are constructed by evaluating a complex function (for e.g. z^2 + c where z and c are complex numbers and the initial z value at a particular pixel denotes the coordinate of that pixel) iteratively at all pixels (at a certain resolution) and coloring each pixel based on the number of iterations required for the magnitude to grow beyond a certain value. The c is constant across the image. A set called the Manderbolt set is also associated with the Julia set. It is the set of all c’s for which the Julia sets are not disconnected.
Both of these involve developments in conformal mapping in the complex plane and many other subtle mathematical ideas.
In addition to signifying that mathematics is a creative and beautiful art, fractals have already been used to create beautiful images through computer graphics and also in certain cases , Fractal music.
In fact some people have hypothesized that there is a relation between fractal structure and our sense of beauty in that such structures which appeal to us seem to be delicately balanced on the edge of chaos , i.e. between boring regularity and meaningless irregularity.
The other applications of mathematics in art are tilings of a plane with a set of geometric figures (For e.g Escher’s drawings). A recent development is the discovery of a set of tiles which tile the plane only non-periodically (the Penrose Tiles) but surprisingly there is a lot of order in the sense that any given configuration is repeated somewhere.
CONCLUSION
In many cases, the misconception of people about the nature of mathematics arises because the spirit of mathematics is never conveyed due to the paucity of educative and entertaining literature on the subject. Also the mathematical curriculum throughout the world stresses, in the most part , on the historical aspects of the subject (most of which could be well over 150 years old) and does not give the feel of mathematics as a developing subject. Besides, in most cases , there is no attempt to communicate at least a sense of what compelled a mathematician to solve a particular problem in a certain fashion or what prompted him / her to propound a particular hypothesis.
I do not think it is a vain hope that in the future, many more people (and not just the professional mathematicians) would be able to view mathematics as a means of creative self-expression and not just as a necessary evil or a drudge.
I hope this essay will convince people that Mathematics is, in the deepest sense, a creation of the human mind and which reflects the structures and patterns that the human mind creates and tries to impose on the external universe.
(Note : My philosophies have changed in that though i still think the pursuit of Mathematics is an art in addition to being a science, i now believe that mathematical truths are not just creations of the human mind and are fundamental truths which are as real as the laws of physics etc and that these are discovered by the human mind. The ideal would be a synthesis of the two.)
Mathematics has always been thought of as a cold and austere science. The purpose of this essay is to show that Mathematics is not just about dry facts , tautologies and proofs but is also a growing and organic discipline. This essay also tries to show that Mathematics is an art in addition to being a science.
I have tried to touch upon the following aspects to show that Mathematics is a creative art.
- The relation between Mathematics and logic and the role of constructive activity and intuition in Mathematics.
- Mathematics and unexpected relationships
- Mathematics and the spirit of fun
- Applications of Mathematics
- New developments in Mathematics and some novel applications
RELATION BETWEEN MATHEMATICS AND LOGIC AND THE ROLE OF CONSTRUCTIVE ACTIVITY AND INTUITION IN MATHEMATICS.
Many people (including people who are good in solving Mathematical problems) aver that all mathematical activity is discovery of existing tautologies through deductive logic, i.e. discovery and proof of existing facts through deductive logic. In other words, they state that mathematics is just a language.
If such were the case, mathematics would have been struck at the Euclidean level of theorems and proofs on the one hand and basic arithmetic on the other. But the fact of creation of such constructs as sequences, infinite sequences, the operation of limits , divergence and convergence which led to the discovery of analysis and calculus refutes the above theories of mathematics as just a language for communication and reveals the facet of mathematics as a mode of thought.
New classes are forever being constructed in mathematics and all classes being widened through abstraction.
One Prime example is the generalization of the concept of dimension to non integral (or fractional ) and other values
Another example is the creation by weistrass of a function which was continuous everywhere but not differentiable anywhere.
Yet another example is the extension of differentiation and integration to fractional orders (say instead of 1st derivative, the 1/2th derivative etc)
nowadays when we say A + B = C, we could be referring to either addition of 2 numbers (whether natural, real or complex) or it could mean the addition of 2 vectors or 2 matrices or it could be referring to 2 elements of a group and the operation may not necessarily denote addition in the normal sense.
Categories are not predefined for all time. They are fluid in the sense that their meaning (or semantics) in what they symbolize or stand for or what they are symbolized by changes over a period of time
This shows that the same elements of the language may symbolize different things according to the context and the mapping between the real world and mathematics
At the turn of this century, there was a project (in which great mathematicians like Hilbert and Bertrand Russell participated) to unify mathematics and logic and breakdown the body of mathematics into a few building blocks of axioms from which the rest of mathematics can be inferred by shuffling the building blocks using the rules of logic.
But this project failed due to :
1) A paradox involving self-referential sets which was a variation on the old paradox embodied in the statement “I am a liar”.
2) The proof by Godel that any consistent system (even the simplest one i.e. natural numbers) cannot logically prove its own consistency and that such a system would have at least one true statement which cannot be proved using the axioms of that system.
The proof itself involved a burst of creative insight in which Godel managed to represent statements about natural numbers through natural numbers themselves and managed to construct a true statement that is self-referential.
These developments proved a nail in the coffin of the premise that mathematics could be reduced to logic.
The other development in the last century and at the turn of the current century was the discovery that if we change/replace the parallel line postulate(which had to be made an axiom because it cannot be proved through other axioms) of Euclid (Through a given point one and only one line can be drawn parallel to a given line) , other consistent geometries (For e.g. hyperbolic , Reimannian , elliptic).
All this goes to show that the predominant feature of modern mathematics is the creation of new constructs (especially when defining axioms ) . Mathematicians have gone on to extend Euclidean geometry to multiple dimensions and also defined spaces with different dimensions, metrics (the distance between two points as a function of the coordinates of the two points) and topologies which significantly determine the features of that space.
Some of the examples are :
- The 4-dimensional Minkowskian space involving space and time used in special relativity
- The 4-dimensional curved space used in general relativity
- The space of signals and the space of functions (generalized to Hilbert space and used in Quantum mechanics and spectral theory)
MATHEMATICS AND THE UNEXPECTED (METAPHOR IN MATHEMATICS)
Paul Dirac, the famous scientist once said that equations , in addition to being true should also be beautiful. The criteria for beauty could be the wealth of relationships that are generated between categories unrelated till then.
Some examples are :
- The discovery of links between Knot theory and mathematical physics which led to a new class of invariants to distinguish knots
- The discovery of links between the class of modular forms and the class of elliptic curves which figured prominently in the proof of Fermat’s Last theorem
- The relationship between complex numbers and vectors
- Eulers formula : e^i = cos() + isin() where i refers to sqrt(-1)
- The equations and theorems in the notebook of Srinivas Ramunajan
- The relation between the zeroes of Reimann’s Zeta function and the number of prime numbers less than a given number
- The expression of Phi through an infinite series (Gregory’s series) involving the expansion of an integrand
MATHEMATICS AND THE SPIRIT OF FUN
Creativity involves play. In mathematics too, the attempt to solve certain puzzles have led to the blossoming of certain fruitful areas of mathematics
Some examples :
The attempt to solve problems like the Konisberg bridge problem, the travelling salesman problem, the knights tour of a chessboard etc have led to the growth of the discipline of graph theory.
The attempt to solve Zeno’s Paradox of Achilles and the tortoise (i.e Achilles cannot catch up with the tortoise since first he has to cover half the distance and to cover that distance, he has to cover half of that distance and so on ad infinitum) led to the concepts of convergence of infinite series
The attempt to construct a square equal in area to a circle (with a ruler and compass only) led in the last century to the proof of transcendence of Phi
The attempt by Mathematicians to solve the Rubik’s cube (and other permutation puzzles) led to insights in group theory
APPLICATIONS IN MATHEMATICS
Like any creative art, mathematics has spawned many interesting applications (‘The unreasonable effectiveness of mathematics’) . The theory of representation of groups is useful in determining energy levels of compounds etc and their degeneracy by studying their symmetry . Representation theory of groups has had a huge success in classification (and prediction ) of elementary particles in various families.
Knot theory has found applications in the study of coiling of DNA
This is just a very small sample.
NEW DEVELOPMENTS IN MATHEMATICS AND SOME NOVEL APPLICATIONS
Catastrophe theory developed by Rene Thom studies structurally stable qualitative changes (change in number and type of equilibrium points) due to a change in parameters and it is said that it can be used to study various systems (after refinement) not only in the physical sciences but also in the biological and social sciences.
Chaos theory synthesizes the paradigms of ‘order’ and ‘disorder’ by showing that apparently random processes may be governed by deterministic albeit chaotic laws. It also shows that it is possible for a system to stay bounded within a region of state space and yet be unpredictable in the sense that it is not possible to predict where a trajectory will end up. Concepts like strange attractors , ergodicity (all parts of state space being accessible by a trajectory ) find a place in chaos theory. This theory finds applications in statistical physics (especially in far from equilibrium processes) , turbulence etc.
A related concept is that of fractals . These retain structure at all scales. There is a measure called dimension which characterizes the relationship between length of an object and the scale at which the length is measured. For fractal objects / sets which are usually very irregular, these measure is usually fractional and characterizes a particular fractal. They are self-similar either statistically or deterministically . many of them like the Koch Curve , Sieperenski’s Gasket etc are constructed by recursively applying a series of transformations to an initial object. Others like the Julia set are constructed by evaluating a complex function (for e.g. z^2 + c where z and c are complex numbers and the initial z value at a particular pixel denotes the coordinate of that pixel) iteratively at all pixels (at a certain resolution) and coloring each pixel based on the number of iterations required for the magnitude to grow beyond a certain value. The c is constant across the image. A set called the Manderbolt set is also associated with the Julia set. It is the set of all c’s for which the Julia sets are not disconnected.
Both of these involve developments in conformal mapping in the complex plane and many other subtle mathematical ideas.
In addition to signifying that mathematics is a creative and beautiful art, fractals have already been used to create beautiful images through computer graphics and also in certain cases , Fractal music.
In fact some people have hypothesized that there is a relation between fractal structure and our sense of beauty in that such structures which appeal to us seem to be delicately balanced on the edge of chaos , i.e. between boring regularity and meaningless irregularity.
The other applications of mathematics in art are tilings of a plane with a set of geometric figures (For e.g Escher’s drawings). A recent development is the discovery of a set of tiles which tile the plane only non-periodically (the Penrose Tiles) but surprisingly there is a lot of order in the sense that any given configuration is repeated somewhere.
CONCLUSION
In many cases, the misconception of people about the nature of mathematics arises because the spirit of mathematics is never conveyed due to the paucity of educative and entertaining literature on the subject. Also the mathematical curriculum throughout the world stresses, in the most part , on the historical aspects of the subject (most of which could be well over 150 years old) and does not give the feel of mathematics as a developing subject. Besides, in most cases , there is no attempt to communicate at least a sense of what compelled a mathematician to solve a particular problem in a certain fashion or what prompted him / her to propound a particular hypothesis.
I do not think it is a vain hope that in the future, many more people (and not just the professional mathematicians) would be able to view mathematics as a means of creative self-expression and not just as a necessary evil or a drudge.
I hope this essay will convince people that Mathematics is, in the deepest sense, a creation of the human mind and which reflects the structures and patterns that the human mind creates and tries to impose on the external universe.