Mathematics

We usually think of basic mathematics such as introductory calculus to be fairly solid. However, recent research by UD authors shows that calculus notation needs a revision.
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In the demonstration below, a bit of acetone has been put on the corner of the brick to get the process started: This demonstrates the remarkable effects of inherent, embedded, intelligible structural, quantitative properties of fluorine and other elements and molecules. With lesser materials, we can see similar, even more spectacular effects: Notice, the table of standard electrode potentials of selected ions: A world that exhibits lawlike, reliable properties that are structural and/or quantitative shows how such properties are integrated into the fabric or architecture of being. END

For some weeks now, an underlying persistent debate on the reality of numbers has emerged in several discussion threads at UD. In part, it has been cast in terms of nominalism vs platonic realism; the latter being the effective view of most working mathematicians. Obviously, this is a first principles issue and is worth focussed discussion. Now, No. 14 in this series, on objectivity of aesthetics principles as canons of beauty, begins by pointing to an underlying challenge: We live in a Kant-haunted age, where the “ugly gulch” between our inner world of appearances and judgements and the world of things in themselves is often seen as unbridgeable. Of course, there are many other streams of thought that lead to Read More ›

He offers some here: When I teach a course, I too like to sell the sizzle at the beginning of each lecture. For a graduate course in information theory I teach, the students are told that they will learn why their cell phones use recently discovered coding that pushes the boundaries of what is mathematically possible in communication speed. I also tell them that we will prove that some things exist that we can also prove are unknowable. And there are numbers that a computer can’t compute. There also exists a single number, Chaitin’s number, that we know lies between zero and one. If we knew Chaitin’s number to finite precision, we could prove or disprove numerous open problems in Read More ›

There is a creeping scientism in our intellectual climate. We have been led to think that Science is the gold standard of reliable, substantial knowledge and that institutional science and its leaders are the curators of knowledge. This is of course deeply connected to the wider domination of evolutionary materialistic scientism, which compounds the above with the notion that the stuff studied by the physical and chemical sciences is effectively the limit of credibly, reliably knowable reality. Where, let us note that scientism is a part of the defining cluster of naturalism, in both its metaphysical and “methodological” guises. We can readily see that in that ever so humble source, Wikipedia, speaking confidently and comfortably on its own philosophical bent: Read More ›

In recent weeks, we have seen repeated attempts to suggest that Mathematics is essentially a mind game we make up as an aspect of culture. There has been a very strong resistance to the idea that there are intelligible manifestations of structure and quantity embedded in the fabric of the world (and indeed in that of any possible world). And when test cases have been put on the table, they have been consistently brushed aside as cases where our mathematical modelling has been applied; that is it’s all in our heads. So, it is appropriate to put on the table a test case that is quite literally in our heads, hearing and particularly how the cochlea works. Video: We see Read More ›

The Fourier series is a powerful technique that can be used to break down any repeating waveform into sinusoidal components, based on integer number harmonics of a fundamental frequency: Video: This is already amazing, that by summing up harmonically related sinusoids (with suitable amplitudes and lagging) we can analyse any repeating waveform as a sum of components. This then extends to any non-repeating pulse, once we go to an integral, which brings in the idea of a continuous spectrum where some wave “energy” is found at every particular frequency in a band. However, something subtler lurks: As the illustration based on clips from the video shows, a sinusoid can be seen as the projection of a rotating vector (= a Read More ›