KARL JASPERS FORUM

TA112 (Müller)

Commentary
1

( MATHEMATICAL THINKING )

by William Byers

6 February 2009, posted 14 February 2009

Thank you for your kind words about my book. Here are some comments:

Re.[18] I don't know that I said that Platonic
means mind-independent. Platonic as it is used in math remains that the
structures of math exist in some objective way. The best statement on
this is by Gödel. Thus a mathematical result can be both true and unprovable as he showed.

Re.[22] I don't know that I agree that numbers
start out as tools. The nature of number
is very subtle and I am writing about this now. One has to go back to the
Greeks, the Pythagoreans to see that the original notion of number was much
more general than ours. We think of number as purely quantitative, that
number comes from counting or measuring but, to me, the Greeks understood number also in a way that one could call
qualitative. The world is ordered by number. Number is the
fundamental principle through which order comes into the world. The
Jungians call number an archetype and this for me has echoes of the qualitative
meaning of number for the Greeks. By the way my book by a Jungian analyst
who seemed to like my definition of archetype as something that has standing in
both the so-called objective world and also the world of mind. So don't
underestimate the depth and complexity of number.
I note that one and two were not numbers of these Greeks, numbers began with
three. So what are one and two. They are
Gestalts, primary aspects of reality. I discuss one and two in the book
but superficially. Remember Plotinus and the One. You could say
that every whole integer number gives an insight into the deep structure of the
One. But I must stop here or I will write another book.

Re.[23] I don't agree that "Byers shows that
mathematical formalisms ...." Ideas, perhaps, arise out of such problematic
situations. Formalisms and algorithm serve the psychological function of
helping us escape from ambiguity which is a given of the human situation.
Ambiguity deconstructs the objectified MIR and ambiguity is inevitably round
because MIR is not identical to reality. The fundamental nature of the
world is not that of logical consistency; it is that of ambiguity, one aspect
of which is the illusion of absolute non-ambiguity.

Re.[24] I agree very strongly especially with
your second paragraph.

Re.[26] In the Feyerabend
comment, I would say islands of rationality in a sea of non-rationality.
I don't remember with the Lakatos
statement. Did it come from my book?

Re.[27] There is a question here. Are all
ideas in individual mathematician's minds? According to Popper and people
like Hersh there is an objectivity
to the results of math but this objectivity lies in a social and cultural
realm. Once things are given there commonly
understood meaning then questions can be asked
and answers given that are not dependent on any particular person. They
are objective but it is a relative objectivity not the Platonic absolute
objectivity.

----------------------------------------------------------

W. Byers

Professor, Department of Mathematics and Statistics

Concordia University

1455 De Maisonneuve Blvd. W.

Montreal, Quebec H3G 1M8

Canada

e-mail: wpbyers (at) mathstat.concordia.ca

phone: 514-848-2424 Ext. 3243

fax: 514-848-4511