Introduction

Difficulties

Engineering and Physical Science degree programmes insist on a reasonable Mathematical background. This is not true for Computer Science programmes and the graduate here needs this type of background even more than an Engineer - consider, as an example, the situation where many Computer Science graduates are thrown into work which involves Field Theory.

"Programming" has been replaced by "Software Engineering" with some emphasis on the "engineering" aspect. Software engineers need a broad Mathematical Education. Computer Science graduates have no Mathematical equipment in which to analyze what they are trying to do.

Hoare, a well-known Computer Scientist, in his inaugural address made the following assertions:

• Computer programs are Mathematical expressions
• A programming language is a Mathematical theory
• Programming is a Mathematical activity.

There are great difficulties in attracting students to Mathematics degrees, especially from those who come into Arts or Science degree programmes with no specific subject in mind ("undenominated" programmes). We have to compete with other subjects which are perceived to be more vocational, are also perceived as easier options and haven't the high fallout rate that honours Mathematics has.

Students of Mathematics lack motivation, have inadequate preparation and this contributes to their subsequent subject choices.

On the other hand it is recognized that Mathematicians, when they do solve problems, are unable to communicate their solution to others.

An integrated Mathematics and Computer Science degree will go some way to rectifying many of these problems.

Common interests?

To quote a person working in the industry: "In the software engineering process, the use of Mathematical ideas requires more resources at the initial stages but the total resources are less and an enhanced product ensues."

To be more specific, the following Mathematical ideas should play an important part in the education of a software engineer:

• Formal methods
• Modern logics
• Parallel processing
• Modeling, forecasting
• Complexity
• Computability
• Encryption and coding
• Statistics and probability (Networking is a probabilistic entity).
• Neural networks

Benefits and advantages

Benefits of an integrated degree

• Volatile and exciting area.
• Focused programme.
• It still embodies essence of Mathematics when carefully thought out.
• Facilitates development of PSQs (personal skills and qualities) through group and individual projects.
• Vocational.
• Accessibility to potentially good students.
• Prepares potential school teachers in both Computer Science and in Mathematics.
• It is a laboratory based programme and can lead to better funding.
• Modern applicable area.
• Employment prospects for students are greatly increased.
• Accessibility to funding agencies and politicians.
• It leads to active applications and involvement by the student who is encouraged to build his/her own system with great satisfaction.
• The emphasis is on abstraction and proof.
• It is not the "death of proof" but a deeper understanding of what "proof" is will ensue from experiment.
• Students can see harder continuous Mathematics in a new light.
• There is less emphasis on graduates as technicians and more emphasis on Science.
• The image of Mathematics and of Computer Science is enormously improved.

Employers' perspective

• Engineering houses, Computer Manufacturers, Research Laboratories, Scientific users (such as the Meteorological Office)
• Software Houses
• Commercial users. These could be subclassified as:
• Utilities
• Manufacturers
• Finance institutions
• Retail outlets

Those concerned with rigorous development also include the military and data-security teams within commerce and finance.

These classifications are on an International basis and Ireland has its own particular problems as a small open economy, with manufacturing but little research and development. Our employers also have the narrow view that graduates should be trained for a specific job but overlook how things are changing rapidly and that a well-trained mind with the ability to think and solve problems as and when they arise is what is needed.

Where do the graduates go?

The British Sunday Times on 8/8/91 reports "Britain faces an acute shortage of Mathematicians". If this is so, where does this leave us? Many of our good Mathematics students go into Engineering and the school culture is such that those good at Mathematics are encouraged to apply there. Would those good at English be encouraged to do a degree in, say, Accountancy?

Mathematics is not looked on as a career in itself and nowadays additionally some of the best Mathematicians are attracted into Business and Commerce. If they are good at Mathematics and must pursue a career in business, why isn't it pointed out that a degree in Mathematics and Economics is a much better preparation?(1)

In the UK, 26% of Mathematics graduates(2) go into the Computer Industry and 52% into Finance. Of the Mathematics graduates in Ireland that did not go on to further study, 82% (1991 figures) went into financial work and computing. (3) Graduates with an integrated degree would be much better prepared for careers in these areas.

Information Technology

• Software Engineering or Knowledge-based Information systems
• AI (Artificial Intelligence)
• VLSI (Very Large Scale Integration)
• Communications
• Human Interface

Computer Algebra Systems(4)

Computer Algebra systems are likely to become as useful to Scientists and Engineers as word-processors and databases have become to all.

Cohen remarks: "A mediocre Mathematician with a computer might be able to simulate the creative powers of a top notch Mathematician with pen and paper". How much more could a top-notch Mathematician produce?

Packages like MAPLE, MATHEMATICA, REDUCE, CAYLEY (to be replaced by MAGMA soon), GAP, AXIOM should be, in fact must be, integrated within our courses. From our experience it is much easier to integrate these within an integrated Mathematics and Computing degree, where an element of laboratory and experimental work already exists.

Generally speaking a package like MAPLE, REDUCE or MATHEMATICA would go well with Analysis-type courses or Applied/Mathematical Physics courses, and one of CAYLEY (soon to be MAGMA), GAP or maybe AXIOM (5) should be integrated within abstract Algebra courses such as Group Theory, Field Theory, Coding Theory or even Homological Algebra. Also MATRIX, for Linear Algebra and Linear Programming, is an excellent CAL (Computer Assisted Learning) package which goes down well with students. In addition, MACTUTOR, for the Apple Mac, is an excellent all-purpose CAL package.

Use of Computer Algebra does however tend to increase the workload on the instructor and many Mathematics Departments and Colleges are as yet unwilling to recognise this fact and indeed do not recognise the importance of considering Mathematics as a laboratory-based subject.

Leibnitz stated: "...it is unworthy of excellent men to lose many hours like slaves in the labour of calculation, which could safely be relegated to someone else if the machine were used."

Syllabus

There can be core courses and options to suit individual tastes. It is important to note however that this an integrated programme and that although some of the Mathematics and Computer Science courses are independent, the programme should be drawn up by reference to common areas of interest and dependency. It is not a matter of simply combining the subjects as in a traditional two-subject degrees.

To spell out a full syllabus would be pointless here but some suggestions on related areas that could be included are given below - fundamental courses are in addition to these.

Within Mathematics, the importance of Discrete Mathematics to Computer Science is fundamental and has now been recognized as such but this is only part of the picture. Discrete Mathematics should include algorithms, recursive function theory, Boolean algebra, logic and circuit design. Analysis courses are also an important element and must be compulsory in the early years. A good idea which works well at UCG is to have courses on Metric Spaces and Fractal Geometry. Other ideas to think about: Have a course in Number Theory and Cryptography; include Coding Theory with Field Theory, and semigroups and machines with a Group Theory course. Category theory: "Categories themselves are the models of an essentially algebraic theory and nearly all derived concepts are finitary and algorithmic in nature" (John Gray in Computational Category Theory). This is all good Mathematics.

Numerical Analysis cuts across both areas and could be included either within the Mathematics core or within the Computer Science core areas.

In Computer Science, courses on Programming, Operating Systems, Networking and Communications, Databases, Architecture, Modeling, Algorithms, Computability, Complexity, Graphics, Parallel Processing, Artificial Intelligence and logical aspects of Computing, would appear to be fundamental. Consideration should also be given to courses on Automated Reasoning and Neural Networks which would fit in well with the Mathematics. In general, the Computer Science element in a joint programme should be directed more to software considerations .

Everyone has their own favourite language but at the risk of upsetting some persons I would suggest that C and C++ are the most useful (and Mathematically oriented) languages now used in Science and Industry. This also fits in well with Operating Systems as many of these were originally written in C.

Conclusion

Both subjects have much to learn from one another. The debate will continue!

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Mathematics Department, Maths Physics Department, IT Department, Physics Department.
National University of Ireland, Galway, University Road, Galway, Ireland.
Phone: +353 (0)91 750442 (direct line) , +353 (0)91 524411 x2332 (via switchboard)
Fax No.: +353 (0)91 750542, E-mail: John.Burns@nuigalway.ie
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