KARL JASPERS FORUM
( 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. 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. 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. 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. I agree very strongly especially with your second paragraph.
Re. 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. 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.
Professor, Department of Mathematics and Statistics
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e-mail: wpbyers (at) mathstat.concordia.ca
phone: 514-848-2424 Ext. 3243