Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

Wednesday, December 25, 2013

LaTeX in blogger

you may or may not be aware that i also maintain a math & science & philosophy blog. an important reason for starting this as a separate blog on wordpress was that the mathematical typesetting of formulas used to be difficult on blogger. but recently things have changed, and now it is possible to use LaTeX (math formatting language/software) also in blogger, through the platform of mathjax, see here how to run this on blogger.

so, just because i like math formulas also aesthetically, let me write some simple formula here. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be given by:


whatever that may mean... and i'm left with the problem of deciding whether i should integrate the two blogs or not...

[update: since this post scores way too high on google search, let me include the instructions for installing mathjax:

to get mathjax to work in blogger, go to your blogger account. click through to your blog's dashboard called `overview', and then click `template' on the left-hand menu. (at this point i myself always backup first, top right side). next click "edit html". after the first <head> you see, paste:

<script src='' type='text/javascript'>  
        HTML: ["input/TeX","output/HTML-CSS"],
        TeX: { extensions: ["AMSmath.js","AMSsymbols.js"],
               equationNumbers: { autoNumber: "AMS" } },
        extensions: ["tex2jax.js"],
        jax: ["input/TeX","output/HTML-CSS"],
        tex2jax: { inlineMath: [ ['$','$'], ["\\(","\\)"] ],
                   displayMath: [ ['$$','$$'], ["\\[","\\]"] ],
                   processEscapes: true },
        "HTML-CSS": { availableFonts: ["TeX"],
                      linebreaks: { automatic: true } }

you can now use $...$ or \(...\) for inline equations, and $$...$$ or \[...\] for displaying equations centered in their own line.]

Sunday, July 31, 2011

Natural Topology (mathematics book announcement)

ocho infinito xxiii, frank waaldijk
ocho infinito xxiii (cover drawing for the book `Natural Topology', own work, 2004-2011, click on the image for an enlargement)

Dear readers, some of you may know that Wim Couwenberg and I started a project called `Natural topology' some years ago. The aim of the project is a) to explain intuitionism to classical mathematicians b) to simplify aspects of formal topology c) to show that intuitionistic topology is elegant and constructive, and its results can be easily translated to Bishop-style mathematics if one inductivizes the definitions d) to clarify in-our-eyes-important aspects of the relation between foundations of constructive mathematics and the foundations of physics.

Last January, we gave an informal talk about this project at the University of Nijmegen for a general audience. Positive and sceptical reactions together prompted me to work out the basic idea rigorously, all the time linking it to other developments. I am happy to announce that the long-promised book Natural Topology is now available online, from my website at

An abstract of the book is given below. I hope it will provide food for thought and discussion, also for the philosophy of mathematics and physics. The book is essentially self-contained and on the advanced undergraduate level (I believe), meaning that anyone with a knowledge of basic topology and some perseverance should be able to read it. For a good appreciation of the whole book, a certain knowledge of constructive mathematics probably is necessary, I suspect. But enough interesting elements should be accessible to anyone, I hope. (An interesting general-audience example concerns the line-calling decision-support system Hawk-Eye used in professional tennis.)

Of course all comments and reactions are welcome.

Kind regards,
Frank Waaldijk


We develop a simple framework called `natural topology', which can serve as a theoretical and applicable basis for dealing with real-world phenomena. Natural topology is tailored to make pointwise and pointfree notions go together naturally. As a constructive theory in BISH, it gives a classical mathematician a faithful idea of important concepts and results in intuitionism.

Natural topology is well-suited for practical and computational purposes. We give several examples relevant for applied mathematics, such as the decision-support system Hawk-Eye, and various real-number representations.

We compare classical mathematics (CLASS), intuitionism (INT), recursive mathematics (RUSS), Bishop-style mathematics (BISH), formal topology and applied mathematics, aiming to reduce the mutual differences to their essence. To do so, our mathematical foundation must be precise and simple. There are links with physics, regarding the topological character of our physical universe.

Any natural space is isomorphic to a quotient space of Baire space, which therefore is universal. We develop an elegant and concise `genetic induction' scheme, and prove its equivalence on natural spaces to a formal-topological induction style. The inductive Heine-Borel property holds for `compact' or `fanlike' natural subspaces, including the real interval [α,β]. Inductive morphisms preserve this Heine-Borel property. This partly solves the continuous-function problem for BISH, yet pointwise problems persist in the absence of Brouwer's Thesis.

By inductivizing the definitions, a direct correspondence with INT is obtained which allows for a translation of many intuitionistic results into BISH. We thus prove a constructive star-finitary metrization theorem which parallels the classical metrization theorem for strongly paracompact spaces. We also obtain non-metrizable Silva spaces, in infinite-dimensional topology. Natural topology gives a solid basis, we think, for further constructive study of topological lattice theory, algebraic topology and infinite-dimensional topology.

The final section reconsiders the question of which mathematics to choose for physics. Compactness issues also play a role here, since the question `can Nature produce a non-recursive sequence?' finds a negative answer in CTphys. CTphys, if true, would seem at first glance to point to RUSS as the mathematics of choice for physics. To discuss this issue, we wax more philosophical. We also present a simple model of INT in RUSS, in the two-player game LIfE (Limited Information for Earthlings).

Monday, January 24, 2011

self-fulfilling post: more posts in january 2011 than in the entire year 2010 (self reference intermezzo)

title says it all. this post is self-fulfilling ;-) since it is the 20th post of january, and in 2010 there were only 19 posts on this blog. just to show you how empty self-reference can be...although in the foundations of mathematics, self-reference is a profound tool. kurt gödel's incompleteness theorems hinge on the possibility of encoding mathematical statements about our number system in numbers. [the whole numbers 0,1,2,...with addition and multiplication]. combine this with the fact that formal derivations in the number system can also be coded as calculations on numbers, and with quite some work one gets a statement Q about numbers which talks about itself...namely Q, when decoded, reads:

the statement Q cannot be formally derived in the formal number system.

suppose Q can be formally derived...then -assuming the number system is consistent- this means that Q is true, but then Q cannot be formally derived! contradiction. since the assumption that Q can be formally derived leads to contradiction, we conclude that Q cannot be formally derived. this means that Q is true!

wow, you say, so what. but this is one of the most profound insights in the limited power of formal human reasoning that i have ever come across. roughly speaking, it means that no matter how hard we try to formalize our reasoning, if the formal system is strong enough (and consistent) then we will always come across statements which are obviously true but which cannot be derived in our formal system hence the name `incompleteness theorem'. the second incompleteness theorem states that especially the consistency of the system cannot be formally derived within the system.

gödel's incompleteness theorems were a shattering blow to the program of david hilbert, who wanted to formalize all of mathematics. the dutch mathematician l.e.j. (jan) brouwer had already predicted in his phd-thesis (1907) that this would be impossible, on mathematical-philosophical grounds. but gödel gave a sharp mathematical proof, in 1931.


for me, there is some relevant personal history here...since you could say that both my mathematical career and my artistic career were fueled by the absolutely marvelous book `gödel escher bach - an eternal golden braid' (written by douglas hofstadter, and winner of the pulitzer prize for non-fiction in 1980 - when i was fifteen).

gödel escher bach, douglas hofstadter

few other books have so sparked my interest in art, neuroscience and mathematics as this book, and it is wonderful that there are people like douglas hofstadter devoting time and energy to translate difficult concepts from mathematics and natural science to a more general audience.

and yes i will come back to maurits escher once more in later posts...(did i mention somewhere how much i love bach's music? but i know very little about music, so i won't write about it on this blog i think).

in art, also self-reference can play various important roles. one obvious role is that of the self-portrait...(see some recent previous posts for digital self-portraits) and i will come back to that also, after i finish the thread on nuclear energy and art.


this post was partly sparked by my lunch today with paul, staunch supporter of this blog and its author, who repeated his earlier remark that i should not forget to combine my mathematical background with my artistic endeavour from time to time.

so thank you paul!