# Popper on demarcation and the power of epicycles Suppose you had a standard Fourier series representing an arbitrary function. Suppose now this function is the periodic movement of some celestial body.
As the following video explains, what the Fourier coefficients stand for are the usual Ptolemaic epicycles.

A paper by Hanson, “Mathematical Power of Epicyclical Astronomy” shows that provided a sufficient number of those frequencies are being employed, a summable function can always be approximated as close as needed.

What becomes now of Popper demarcation problem (“Conjectures and Refutations”,2nd ed.,Ch. 11, “The Demarcation between Science and Metaphysics”)? It is not evident to me that any falsificationist strategy can distinguish this Fourier based approximation algorithm – call it Ptolemaic system – from a Copernican algorithmic strategy. Since Copernicus as well as Aristarchus were right of course, how are we going to proceed? It is a vacuous theory for sure, but is there a metric for how vacuous a theory can be? Popper himself , (“Conjectures and Refutations”,2nd ed. , Ch. 10:xxi) says that “Ptolemy’s system was not refuted when Copernicus produced his”.

Another example is Lagrangian polynomial: for, (wiki) given a set of $k+ 1$ distinct data points $(x_0, y_0),\ldots,(x_j, y_j),\ldots,(x_k, y_k)$, the interpolating polynomial in the Lagrange form is the linear combination $L(x) := \sum\limits_{j=0}^{k} y_j \ell_j(x)$

of Lagrange basis polynomials $\ell_j(x) := \prod\limits_{\begin{smallmatrix}0\le m\le k\\ m\neq j\end{smallmatrix}} \frac{x-x_m}{x_j-x_m} = \frac{(x-x_0)}{(x_j-x_0)} \cdots \frac{(x-x_{j-1})}{(x_j-x_{j-1})} \frac{(x-x_{j+1})}{(x_j-x_{j+1})} \cdots \frac{(x-x_k)}{(x_j-x_k)}$,

where $0\le j\le k$.

That does not capture the Data Generating Process any more than projecting the points into a plane – as a linear regression would do.

How are we then to demarcate (Popper) in this case? Leibnitz had some ideas, later exposed by Weyl (“The Open World”, 1931) and subsequently picked up and generalized by Chaitin and Kolmogorov. You want to base your demarcation strategy on the parsimony of the identified model, à la Ockham. We are then simply begging the question: is there a good metric for parsimony? More on that later on.

Categories: Episteme

### 5 replies

1. Appreciating the persistence you put into your website and in depth information you provide. It’s good to come across a blog every once in a while that isn’t the same unwanted rehashed information. Fantastic read! I’ve bookmarked your site and I’m adding your RSS feeds to my Google account.

Like

2. Hello! I’m at work surfing around your blog from my new iphone! Just wanted to say I love reading through your blog and look forward to all your posts! Carry on the outstanding work!

Like

3. Attractive section of content. I just stumbled
upon your blog and in accession capital to assert that I acquire in fact enjoyed
account your blog posts. Any way I’ll be subscribing to your feeds and even I achievement you access consistently rapidly.

Like

4. Hi there It’s nearly impossible to find well-informed people about this topic, however, you seem like you know what you’re talking about! thank you

Like

5. Your style is unique compared to other people I’ve read stuff from. Many thanks for posting when you have the opportunity, Guess I will just book mark this blog.

Like