Alekandr Lokshin has a book Free Will and Mathematics (in Russian)
and he has started a discussion on the everything list with an abstract of the book in English
Alekandr Lokshin, 27.05.2012 20:51:
Alexander A. Lokshin
FREE WILL AND MATHEMATICS
Moscow, MAKS-Press, 2012 , 40 pages (abstract)
The general idea of the booklet is as follows. All main mathematical notions ( such as infinity, variable, integer number) implicitly depend on the notion of free will. Therefore a scientist employing mathematics when modeling nature cannot deny the existence of the free will. (Unfortunately, Stephen Hawking made this incorrect conclusion in “The Grand Design”).
To make the general idea more clear, suppose we are proving the well- known formula S = ah/2 for the area of a triangle. Our proof will necessarily begin as follows:
“Let us consider AN ARBITRARY triangle…” Here we obviously apply the operator of the free will choice which cannot be replaced by the random choice. In fact, let us imagine that our proof begins in such a way : “Let us consider A RANDOMLY SELECTED triangle…” Surely, such a beginning will not lead us to the desired proof. The formula obtained for a randomly selected triangle is not necessarily valid for all triangles!
On the other hand when proving the formula S=ab/2, obviously, it is impossible to consider all the triangles simultaneously. Thus the operator of the free will choice must be used inevitably. More widely, let us consider a variable x which is running about a sphere of radius 1. Let us pose a question: what does x denote?
a) x does not denote an object,
b) x does not denote a multitude,
c) x does not denote a physical process.
In my opinion, x denotes the free will choice which the reader of the mathematical text must do. So, the notion of a variable inevitably is based on the notion of the free will.
Moreover, considerations adduced in the book show that the notion of an integer number is based on the notion of the free will as well. In fact, when constructing integers by means of one-to-one correspondence we implicitly assume that the mentioned correspondence is realized by means of continuous lines connecting pairs of objects. But to ensure continuity of lines above-mentioned one must consider an arbitrary point on such a line!
A new approach to the Alan Turing problem (how to distinguish a person from an android) is also proposed ; this approach is based on the idea that an android cannot generate the notion of an arbitrary object.
Evgenii Rudnyi 07.06.2012 13:59:
I have started reading Collingwood’s An Essay on Metaphysics and I see one definition that seems to be pertinent to this discussion.
p. 27 “Def. 4. To assume it to suppose by an act of free choice.
A person who ‘makes an assumption’ is making a supposition about which he is aware that he might if he chose make not that but another. All assumptions are suppositions, but all suppositions are not assumptions: for some are made altogether unawares, and others, though the persons what make them may be conscious of making them, are made without any consciousness of the possibility, if it is a possibility, that others might have been made instead. When correctly used, the word ‘assumption’ is always used with this implication of free choice, as when it is said ‘let us assume x = 10’.”
What about that? It looks to be not far away from what Aleksandr Lokshin has suggested.
Discussion on Facebook
10 June 2012:
I am afraid you have missed the point. This was not about statements proved by mathematics, but rather how mathematicians prove the statements. For example, an assumption requires a free choice, otherwise it is not an assumption. Let us take a point and a line. It is possible to assume that there is only one non-intersecting line through a point in respect to the original line, or infinite number of non-intersecting lines, or even zero non-intersecting lines. All three choices are valid. What forces a mathematician to take one particular assumption?