Dear Terry,
You asked about how others see knowledge when it is transformed in the
language of mathematics. In this post I will offer some thoughts on the
method of mathematics. Whether knowledge plays a role in mathematics is a
larger question that I am not directly concerned with in this post. Also, I
will also not offer any response to the ‘three interesting things’ about
mathematics that you state in your post. The aim of the post is elaborate
the method of reasoning in mathematics. I will be drawing on the work
of Chaim Perelman
and Lucie Olbrechts-Tyteca (1969, pp.1-14) in what follows.
Mathematics takes necessary propositions and self-evidence as the marks of
reason. It is opposed to deliberation and argumentation, which concerns the
domains of the credible, the probable, and the plausible.
Mathematics cannot be concerned with probable opinions, it must produce a
system of necessary propositions concerning which agreement is inevitable,
and every rational being must submit to it. Disagreement is a sign of
error. According to Descartes:
“Whenever two men come to opposite decisions about some matter, one of them
must certainly be wrong, and apparently there is not one of them that
knows; for if the reasoning of one was sound and clear he would be able to
lay it before the other as finally to succeed in convincing his
understanding also” (Descartes quoted in Perelman, 1969, p. 2).
The quote shows that according to Cartesian rationalism, the correct
decision is certain and to be obtained by demonstration.
The formal logic of the mathematical sciences concerns the geometrical
method of demonstration that establishes a proof that is the reduction to
the self-evident. Doubt and ambiguity must be avoided at all costs.
Inductive reasoning is less concerned with the necessity of propositions as
to their truth, that is, their conformity with the facts. Evidence in
inductive reasoning is obtained through sensible intuition. In contrast,
the formal system of mathematical logic is not related to evidence at all.
The logician is free to create the system of symbols and combinations of
symbols as he pleases. He decides what are the axioms that are considered
without proof as valid, and the rules of transformation that make it
possible to deduce, from the valid expressions, other expressions of equal
validity in the system. The only aspect that is important, and the aspect
that gives demonstration its force, is choosing axioms and rules in a way
that avoids ambiguity and doubt. It is essential, without hesitation, even
mechanically, to be able to establish whether a sequence of symbols is
valid because it is an axiom or an expression deducible from the axioms
(Perelman, 1969, p. 13).
In formal demonstration, the meaning of the expressions is irrelevant;
interpretation of the elements can be left to those that will apply it. If
the demonstration is questioned, it is sufficient for the logician to
indicate the process by which the final expression of the deductive system
was obtained. Where the first elements that became the axioms selected by
the logician come from, whether they are impersonal truths, divine
thoughts, or results of experiments for instance, is not part of the
logician’s discipline.
As soon as the logician concerns himself with the meaning the elements in
the system, or their application to particular situations, then the context
and social conditions cannot be neglected. He then enters the domain of the
controversial and of argumentation. An argument presupposes a form of
intellectual contact within a community of minds. In an argument, it is not
enough to simply speak, one must also be listened to.
Warm regards,
Luke
Perelman, C., & Olbrechts-Tyteca, L. (1971). *The new rhetoric: a treatise
on argumentation* (J. Wilkinson & P. Weaver, Trans.). Notre Dame, Indiana:
University of Notre Dame Press.
--
Luke Feast | Lecturer | Early Career Development Fellow | PhD Candidate |
Faculty of Health, Arts and Design, Swinburne University of Technology,
Melbourne, Australia | [log in to unmask] | Ph: +61 3 9214 6165 |
http://www.swinburne.edu.au/health-arts-design/
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