Hi Luke,
Many thanks for your comments. I'm guessing the position you describe is the kind of perspective on maths that many hold. Its useful in the sense that it makes sense out of some of the use of mathematics at school and also as emphasising a role that maths might play in clarifying complex rhetorical claims.
I suggest, however, educators need to take people past the idea that maths is primarily logic to better envisage the other more powerful roles of maths in creative professional work.
What you described is what has elsewhere been called the dialectic approach to mathematics.
Practical mathematics is rather different and usually more complex and flexible in its nuances and its use in argument and reasoning. A brief introduction using some straightforward maths ideas is found in
www.math.uoc.gr/~ictm2/Proceedings/invSiu.pdf comparing dielectic maths to algorithmic maths.
Maths has many rhetorical styles over and beyond the limited picture of logical reasoning, see for example, Ernest at http://link.springer.com/article/10.1023%2FA%3A1003577024357#page-2 but again this is at school level rather than practical use of graduate and post-graduate maths in professional creative work.
Perhaps most significant and not addressed at all really in the limited idea of seeing logic as representative of maths, is the power of abstraction used in maths and its role in creating models of situations and their behaviour, and then, at a higher level, creating models of the behaviours of the factors that shape the models. This can be done in ways that we can obtain an understanding of characteristic behaviours of classes of situations that are hard to obtain from conventionally thinking about the situations themselves.
An example of this kind of meta-abstraction in another realm is how we might start to understand that individuals educated under particular systems will tend to exhibit visually-based fixation in their creative idea generation (see, for example, Gero https://docs.google.com/document/preview?hgd=1&id=1QbFQzvbBbV9U4W5dIk75ty6B4mJHQR1q0CPkkamzjfc
At the lowest level we can observe designing happening. Abstraction from that shows particular characteristics. Abstraciton of those charactersitics of the behaviour of designing indicates a property that has been called fixation. Abstraction fo the factors that shape fixation can at a higher level fo abstraction start to be grounded in various nature or nurture factors such as education. Abstraction of those. . . and so on.
Possibly understanding the role of these other areas of maths gives a different understanding of the potential in creative work from the idea that maths is primarily a logic tool?
Best wishes,
Terry
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Dr Terence Love
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-----Original Message-----
From: [log in to unmask] [mailto:[log in to unmask]] On Behalf Of Luke Feast
Sent: Wednesday, 23 April 2014 8:37 AM
To: PhD-Design - This list is for discussion of PhD studies and related research in Design
Subject: Re: Knowledge
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.
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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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