Thursday, March 17, 2005

Indeterminate objects and contemporary platonism in maths

This evening, having dinner with Nenad and Ira at the Kerepesi dorm he asked me the following question [having in mind that I sympathize to Jim Brown's mathematical platonism]: if we take Jim's example [an actually proven theorem in number theory] that

illustrated by a simple diagram where every [natural] number is substituted by a square like in

then it seems that it is not necessary to imagine a determinate number of squares, i.e. a deteminated value of n, since the theorem holds for all n [defined over Z]. Then, if following Jim, we agree that we have a case with a platonic kind of grasp for the truth of the theorem by virtue of the diagram alone it is natural to ask whether we actually grasp an indeterminate abstract object or we indeterminately grasp a [determinate] abstract object? Nenad believes that this question presents us with a problem about the platonist. Intuitively, I do not have an immediate answer about whether there are indeterminate abstract objects in the platonic realm or only determinate ones and also I am not sure that this is specific problem for platonism alone. Nevertheless the issue seems prima facie interesting. One of the options is the traditional answer that there are only determinate abstract objects and sometimes we only manage to get an indeterminate grasp of them. Another option is that all [possible?] sorts of abstract objects are hosted in the realm, that would mean determinate and indeterminate and for a particular grasp there is a paricular object that corresponds. Anyway, questions like grasp of infinite structures and the like does not make the situation easier.

M.C. Escher - The Official Website

When browsing Sid Smith's web page Platonic Realms [] I bumped on a link to the official web site, devoted on the life and the work of Escher, finally. In the subsection gallery you would find probably most of his work, available for download. As a separate collection are given the popular variations on the symmetry theme. Argubly being the most popular artist among philosophers and especially among philosophers of science [and in particular - space-time freaks] Escher's work reproduction could be found in a lot of books and articles, illustrating space properties or simply attempts to visualize non-Euclidean space. Personally, I disagree with the last interpretation being taken for successful for reasons best exposed by J.R. Lukas in his Euclides Ab Omni Naevo Vindicatus. Nevertheless the mind seems to struggle endlessly to cope with Escher's visions using nothing but its own [in my oppinion - Euclidean] limitations.

Tuesday, March 15, 2005


Taking into account the physical limitations as the uncertainty principle, causal decoherence, speed of light limit, etc? Naturally, the real experiment is considered to be the most powerful weapon of the scientist and yet there are substantial parts of the physical reality where real experiments are impossible to conduct, whether because of physical limitations or because of lack of money. What should be the right attitude towards the real experiments and their actual role in science? The growth of technology would undoubtedly reveal new ways of coping with the old problems and would make possible to perform some experiments which were unthinkable in the past. Yet, a lot would remain just physically impossible. Should then scientists start relying more on alternatives [like thought experiments, mathematical reasoning, etc.] or just sit and wait for the laws of nature to change?


In Dynamics of Reason, when he speaks about the second layer of principles in his model of scientific knowledge, Friedman admits that they could change in response to "empirical pressure". Since there is no explicit defense of the a priori character of these principles when viewed as revisable in conditions of empirical pressure I wonder what would actually stop the empiricist of arguing that in this case the principles are empirically revisable, i.e. their a priori status, no matter how "relative", would lose most of its power.


Since the so called "popular science" trend has something like an intuitive appeal to quite a lot of people where is the line between the superficial and the professional? I was just wandering, some professional physicists make a whole bunch of money by writing pop-scince books and the usual accusation is that the innocent laymen get all sorts of wrong impressions about the subject. How professional then all these books should be, facing the fact that getting closer to the "real stuff" would repulse most potential readers because of the incomprehensible language.