It is axiomatic in mathematics research that all steps of an argument or proof are open to scrutiny. However, a proof based even in part on commercial software is hard to assess, because the source code---and sometimes even the algorithm used---may not be made available. There is the further problem that a reader of the proof may not be able to verify the author's work unless the reader has access to the same software.
For this reason open-source software systems have always enjoyed some use by
mathematicians, but only recently have systems of sufficient power and depth become available which can compete with---and in some cases even surpass---commercial systems.
Most mathematicians and mathematics educators seem to gravitate to commercial systems partly because such systems are better marketed, but also in the view that they may enjoy some level of support. But this comes at the cost of initial purchase,
plus annual licensing fees. The current state of tertiary funding in Australia means
that for all but the very top tier of universities, the expense of such systems is
harder to justify.
For educators, a problem is making the system available to students: it is known that
students get the most use from a system when they have unrestricted access to it: at home as well as at their institution. Again, the use of an open-source system makes it trivial to provide access.
This talk aims to introduce several very powerful and mature systems: the computer
algebra systems Sage, Maxima and Axiom; the numerical systems Octave and Scilab; and the assessment system WeBWorK (or as many of those as time permits). We will briefly describe these systems: their history, current status, usage, and comparison with commercial systems. We will also indicate ways in which anybody can be involved in their development. The presenter will describe his own experiences in using these software systems, and his students' attitudes to them.
Depending on audience interests and expertise, the talk might include looking at a
couple of applications: geometry and Gr\"obner bases, derivation of Runge-Kutta
explicit formulas, cryptography involving elliptic curves and finite fields, or
digital image processing.
The talk will not assume any particular mathematics beyond undergraduate material or material with which the audience is comfortable, and will be as polemical as the