history of mathematics - National STEM Centre

7 / Towards a rigorous approach
De Morgan was very keen to describe the number system in an abstract way, and to
keep numbers themselves out of an axiomatic description of the number system.
There was a major implication of this approach. All systems which obey the axioms
also obey any rules derived from the axioms. This is true of the real and complex
numbers, both of which satisfy the axioms. However, Hamilton was inventing an
extension (which he called quaternions) of complex numbers. You will see later that
quaternions do not satisfy the same rules as numbers.
Activity 11.7 Do all systems have the same axioms?
Activity 11.8 Boolean algebra
1 Check that De Morgan's view is correct at least as far as the real and complex
numbers are concerned. That is, check that the axioms as specified in Activity 11.5,
question la, for combining elements under + and x are the same for both the real
and the complex number systems.
2 Also, given any real number a, the axioms require you to be able to specify
another number b such that a + b = 0.lfa^0, the axioms also require you to be
able to specify another number c such that a x c = 1. If a is the complex number
2 + i, find b and c.
George Boole (1815-1864) had no formal mathematical education. He became a
primary school teacher and during that time taught himself more mathematics and
became interested in a debate over logic which was taking place. As a result, in
1847 he published a short work on logic which was recognised by De Morgan as
being revolutionary. Nevertheless it did not receive much recognition at the time
although it probably enabled Boole to get a position teaching at Queen's College in
Cork. Seven years later he produced the classic Investigation into the laws of
thought, which established the algebra of logic, or the algebra of sets, now known as
Boolean algebra. In other words, Boole set up the rules by which the elements in his
sets, or in his logic, might be combined.
1 a Find out as much as you can about Boole and Boolean algebra and make
notes as you do your research.
b As you do your research, decide for yourself whether the elements of Boole's
algebra satisfy the same rules of combination as the real numbers.
c Use your notes to prepare and make a presentation to your fellow students on
Boole and Boolean algebra. As part of your presentation, include your answer to
part b. Remember that your fellow students may not know anything about axioms,
so make your presentation in a way which is easy for them to understand.
William Kingdon Clifford (1845-1879) was a graduate of Trinity College,
Cambridge. He was renowned for his teaching, and for his ability to visualise
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