Thinking and Computing — Machines and Humans
Responsible Teachers: Joseph Almog, Vesa Halava, and Tero Harju
Substitution possible for courses: A6,A9, V11,T1, K1 (4 credits)
From 2.2. to 19.4. on Tuesdays 10–12 at Seminar room SH 150 (Philosophy Corridor)
Description:
We will discuss the very idea of “thinking” and “computing” as it occurs in mathematical work of human beings and machines. This involves simple thinking, as in adding 5 and 7 to make 12 or show—within a given geometry—that triangles have a certain angle-sum and all the way to considering very abstract problems—how many points are there on the real line? Is there a general systematic method (“algorithm”) for solving arithmetical problems in the form of polynomials (simple ones encountered at school)? Is Arithmetic consistent? Is there a general method for deciding whether when it “runs”, any such general method terminates?
The course aims to discuss very different approaches to the question of human (mathematical) thinking. We will contrast the “reductive” finitistic view, the one dominating contemporary thinking, according to which we are “finite minds” and according to which this itself is represented by the processing effected by certain kinds of “machines”, called Turing Machines. This we will contrast without our own view—a marginal one but with some interesting notable sponsors(see below the quote from Cantor) that we are not “finite minds’ (even if our brain is just 1200 cubic cm and 1.5 kg) because of our grasp of infinity, in a way that (1) is not reducible to finitistic formal operations and (2) in a way which is necessary as background to operation with “finite structures”.
We would like to read great classics on the matter that have not been really dissected enough and have been reported-on incorrectly. This includes on the “reductive” side the reading of: Hilbert, Turing and Saul Kripke’s latest (e.g. the “Turing Thesis” being for him a corollary of the completeness theorem!).
On the other side we will focus on Emil Post, the jewel in the crown, as well as Cantor and Woodin. Interesting non classifiable cases are Godel and Wittgenstein (themselves in dispute with one another). In the background, are the great figures of Leibniz and his project of Ars Combinatoria and in contrast, Descartes’ and his notion of mathematical understanding. We plan on having visitors speak of Frege’s Bwegriffschrift and Husserll/Godel phenomenology of mathematical thinking, as well as deeply involved key figure (surprise!) on Hilbert’s tenth problem.
The course is not accessible merely to people with “advanced” knowledge. If you are interested in computers, philosophy of mind, strategies in games and strategic thinking in general, you are most welcome. The discussion is going to be dialogical and informal.
We attach an inspiring motif from Cantor’s Grundlagen 1883.
Quite often the finitude of the human understanding is adduced as a reason why only finite numbers are thinkable…for by ‘finitude of the understanding’ is tacitly meant that the capacity of the understanding in respect to the formation of numbers is limited to finite numbers. If it should turn out, however, that the understanding in a certain sense is also able to define infinite i.e. transfinite (uber-endliche) numbers and distinguish them from one another, then either the words ‘ finite understanding’ must be given an extended meaning … or else the human understanding must also be granted the predicate ‘infinite’ in certain respects, which, in my considered opinion, is the only correct thing to do. The words ‘finite understanding’ which one hears on so many occasions are, as I believe, in no way on the mark as limited as human nature may in fact be, much of the infinite nonetheless adheres to it and I even think that if it were not in many respects infinite itself, the strong confidence and certainty regarding the existence (‘desseins’) of the absolute, about which we are all in agreement, could not be explained.