[FOM] choice sequences

[Markus Van Atten]

William Tait wrote:

>If alpha* is not law-like, then it is onaf: i.e. the symbol
>``alpha*'' denotes, not a single sequence, but a basic open set of
>them (which will include law-like sequences), and so does not pick
>out a lawles sequence at all.

This was not Brouwer's view. The difference is that for him,
certain unfinished (`onaf') objects are just as much individual objects.

`In intuitionist mathematics a mathematical _entity_ is not necessarily 
predeterminate, and may, _in its state of free growth_, at some time acquire a 
property which it did not possess before.' 
[`The effect of intuitionism on classical algebra of logic', 
in Collected Works I, p.552; emphasis mine]

Brouwer based mathematics on the intuition of time, and for him the fact that
all acts of the mathematician are acts in time was not contingent but an 
essential feature of mathematics. Hence the theory of the creating subject, and 
the strong counterexamples based on it. Non-lawlike sequences (a wider notion
than lawless) are individuated by the moment in time that the creating subject 
began choosing them. (That this is the case has been made explicit in my 
dissertation, see the end of this message.)

Brouwer formulated `the two acts of intuitionism', of which for present 
purposes the salient features are that the first introduces the two-ity and 
hence the natural numbers, and the second introduces both lawlike and non-
lawlike sequences. Later in life, Brouwer came to see that this part of the 
second act is really just a consequence of the first act: the two-ity
and its iteration give us a notion of sequence that is more general than that 
of an algorithmic or lawlike one. In particular, it follows that for Brouwer a 
non-lawlike sequence is ontologically no more obscure than the sequence of the 
natural numbers. Both are instances of the general concept of sequence given by 
the two-ity.

The relevant sources for the preceding paragraph are

1. Brouwer 1947, `Guidelines on intuitionistic mathematics' (p.477 in Collected 
Works)
2. Brouwer 1981, Brouwer's Cambridge Lectures on Intuitionism, D. van Dalen 
(ed.),
Cambridge: Cambridge University Press

The issues are further discussed in my dissertation from 1998, Phenomenology of 
choice sequences. To those interested I could send a PDF file of a slightly 
revised version.

Best wishes,
mark.

-- 
Institute of Philosophy, KU Leuven
Kardinaal Mercierplein 2, B-3000 Leuven, Belgium