1. Object Oriented Programming & Mathematics
The Beauty of Implementing Abstract Structures as
Abstract Structures.
Marc Conrad, University of Luton.
Marc Conrad, University of Luton

2. Once upon a time... Mathematics before the 20th century 17.11.2018
Marc Conrad, University of Luton

3. there was two branches of Mathematics
axiomatic
Mathematics before the 20th century
algorithmic
Marc Conrad, University of Luton

4. Axiomatic Hilbert, N. Bourbaki, etc. axiomatic
Mathematics before the 20th century
algorithmic
Marc Conrad, University of Luton

5. Algorithmic axiomatic Mathematics before the 20th century algorithmic
Turing,
Church,
etc.
Marc Conrad, University of Luton

6. Pure Mathematics & Computer Science
axiomatic
"pure" mathematics
Mathematics before the 20th century
algorithmic
computer science
Marc Conrad, University of Luton

7. With links in between. axiomatic "pure" mathematics
Mathematics before the 20th century
computer
algebra etc.
algorithmic
computer science
Marc Conrad, University of Luton

8. However, some topics of Computer Science seemed to be unrelated to mathematics...
axiomatic
"pure" mathematics
Mathematics before the 20th century
algorithmic
computer science
software design
operating systems
Marc Conrad, University of Luton

9. as e.g. object oriented programming.
axiomatic
"pure" mathematics
Mathematics before the 20th century
algorithmic
object oriented programming is a technique to solve the "software crisis". It evolved in a context completely unrelated to mathematics.
computer science
e.g. object oriented programming
Marc Conrad, University of Luton

10. But object oriented programming is closer to "axiomatic" mathematics than it appeared in the first place.
object oriented programming...
... allows to implement abstract structures in an "axiomatic" way.
"pure" mathematics
axiomatic
Mathematics before the 20th century
algorithmic
object oriented programming
computer science
Marc Conrad, University of Luton

11. Example: A ring (abstract).
We cannot implement:
addition
negation
multiplication
inversion
"zero"
"one"
check if zero
We can implement:
subtraction (because of addition and negation)
exponentiation a
embedding of Z, Q.
check for equality
polynomials over this ring
etc. n
Marc Conrad, University of Luton

12. Marc Conrad, University of Luton
Example: A ring.
We cannot implement:
addition
negation
multiplication
inversion
"zero"
"one"
check if zero
We can implement:
subtraction (because of addition and negation)
exponantiation a
embedding of Z, Q.
check for equality
polynomials over this ring
etc.
Object Oriented programming allows having objects which do not implement everything! (Concept of overriding abstract methods)
The good news is: n
Marc Conrad, University of Luton

13. Marc Conrad, University of Luton
A "UML" approach to a ring.
Ring Z Q
Polynomial
Ring
The child classes implement (override) the missing functionality of the parent class.
Marc Conrad, University of Luton

14. Marc Conrad, University of Luton
A "UML" approach to a ring.
Ring Z Q
Polynomial
Ring
But things are more complicated, a polynomial is defined over a ring. It both inherits and aggregates a ring.
Marc Conrad, University of Luton

15. Marc Conrad, University of Luton
A "UML" approach to a ring.
Leads to multivariate polynomials by implementing univariate polynomials!
Ring Z Q
Polynomial
Ring
But things are more complicated, a polynomial is defined over a ring. It both inherits and aggregates a ring.
Marc Conrad, University of Luton

16. Marc Conrad, University of Luton
A "UML" approach to a ring.
Ring
Element
Ring Z Q
Polynomial
Ring
And in order to perform computations we also need a class for the elements of a ring.
Marc Conrad, University of Luton

17. Marc Conrad, University of Luton
The practical side.
In order to get experience with the idea, an experimental implementation in Java classes has been developed.
Java is highly object oriented but not very common in mathematical context.
Results can be viewed at
The name of the Java package is consequently:
com.perisic.ring
Marc Conrad, University of Luton

18. Results, Remarks, Conclusions
It is possible to work with abstract structures!
E.g. Modular Ring R/p(x), where R is abstract.
E.g. Quotient Field Quot(R), where R is abstract.
Same amount of work as implementing Z/mZ or Q.
Multivariate polynomials can be used although only univariate polynomials have been implemented (over R).
Complex structures can easily be derived as child classes:
Cyclotomic fields, complex numbers, rational function fields, ...
Concepts for automatic mapping from one ring to another.
Marc Conrad, University of Luton

19. Results, Remarks, Conclusions
It is astonishing simple to implement complex mathematical structures in an object oriented environment "from scratch".
You are invited to experiment, contribute, or share experiences. The package com.perisic.ring is available and documented at
Caveat: There are drawbacks: performance, decisions on how to organise classes, implementing specialised algorithms (primality testing, factoring, …)
Marc Conrad, University of Luton

20. Results, Remarks, Conclusions
The experiments with the Java package com.ring.perisic show that object oriented programming deserves a closer look in the context of mathematics:
The mechanism of overriding and dynamic binding allows protoyping of abstract mathematical structures.
Object oriented programming should be a main feature in CAS (as user defined functions a couple of years ago).
Mathematical software should use object oriented terminology instead of "reinventing the wheel".
Disseminate object oriented concepts to the mathematical community.
Marc Conrad, University of Luton