




Recent scholarship and research. A review of the book Everywhen: Australia and the language of deep history appeared in The Journal of the Royal Australian Historical Society, December 2023.
***The paper Equal Representation will appear in the
Australian Mathematics Education Journal. This is a highschool level mathematical discussion about equal representation of groups in institutions in society, and how this is related to fractions.
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A paper on orbit points and sums in the binary dynamical system will appear in the journal Real Analysis Exchange. The idea for this research arose from some work done by G. H. Hardy and J. E. Littlewood in 1930.





Mathematics, history and philosophy. The article Infinitesimal Knowledges appeared in Axiomathes (2022). In the seventeenth century, Italian mathematicians were using new methods derived from the notion of
indivisibles used in ancient Greece. This often led to paradoxes that undermined the certainty that had previously characterised mathematical knowledge. This article discusses aspects of this with reference to the book Infinitesimal by Amir Alexander, and also considers related issues of language, history, pedagogy and knowledge.





The two universities. "The real University, he said, has no specific location. It owns no property, pays no salaries and receives no material dues. The real University is a state of mind. It is that great heritage of rational thought that has been brought down to us through the centuries and which does not exist at a specific location. It's a state of mind which is regenerated throughout the centuries by a body of people who traditionally carry the title of professor, but even that title is not part of the University. The real University is nothing less than the body of reason itself." Read more by Robert M. Pirsig on The Two Universities.





Applications of school mathematics. "In a democratic society, there are many substantial social and scientific issues raised or influenced by public opinion, so it is important that citizens can critically examine those issues by using and interpreting mathematical perspectives"  The Australian Curriculum Assessment and Reporting Authority. There are applications to the funding of government and private schools, resource allocation, banking investments and the role of quadratic functions in balancing contrary effects. Typically, policy decisions require a 'balance' between desirable and undesirable effects, Read about applications of school mathematics. Also, read about pentagons and the number $(1+\sqrt{5})/2$.





Dynamical systems. Randomness and Recurrence in Dynamical Systems: a real analysis approach, with a foreword by
Kenneth A. Ross was published by The Mathematical Association of America in 2010. Many results in real analysis and probability theory make use of measure theory, and this can act as a barrier for learning about phenomena of randomness and recurrence. An aim of this work is to avoid the need for prior knowledge of measure theory, and to consider randomness and recurrence through the unifying concept of a dynamical system. Read reviews of this book by
Svitlana Rogovchenko and James A. Walsh. You may read more about this book and dynamical systems.





Fourier analysis research. We consider only $2\pi$periodic functions on the real line, square integrable over $[0,2\pi)$. A function $g$ is called a difference if there is a real number $b$ and a function $h$ such that $g(x)=h(x)h(b+x)$ for almost all real $x$. In 1972, Gary H. Meisters and Wolfgang Schmidt proved the following statement about a function $f$:
$\int_0^{2\pi}f(t)\,dt=0$
if and only if there are three difference functions $g_1,g_2.g_3$ such that $f=g_1+g_2+g_3$.
Building on this work, Difference Spaces and Invariant Linear Forms identifies a new phenomenon in Fourier and harmonic analysis and develops an associated theory of difference spaces. Read more about this and about further Fourier analysis research.



Felix, qui potest rerum
cognoscere causas  Virgil
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"All truly liberating acts are genuinely altruistic and involve freely accepted renouncement of some freedoms"  Igor Kluvánek.
Read about the mathematics and ideas of Igor Kluvánek, and some more of his quotations.
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"If one stops oneself from thinking about all this, one makes oneself an accomplice of what is happening"  Simone Weil.
Read a review of Robert Zaretzsky's book on Simone Weil.
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"Integrity requires us to view things in their organic reality, not their mechanical reality." Read more about integrity in teaching and learning.
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"...... postmodern thought, for all its 'radical' and 'subversive' pretensions, is merely an application of the ideas of market forces ... to the world of the intellect. Of course, both [postmodernism and market economics] have the same effect of trivialising the serious life of the mind."
Read more about changing intellectual culture.
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Read a review of Symbols of Australia, Melissa Harper
and Richard White (eds.), NewSouth Press, published in The Journal of
the Royal Australian Historical Society 2022.
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Read a review of Everywhen: Australia and the language of deep history, Ann McGrath, Laura Rademaker and Jakelin Troy (eds),
NewSouth Press, published in The Journal of
the Royal Australian Historical Society 2023.
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Read a review of Richard Dawkins' book The God Delusion.
Site updated 6th October, 2024

