[FOM] mathematical understanding
steven at semeiosis.com
Sun Feb 6 22:44:35 EST 2005
Regien Stomphorst wrote ..
> I am interested in 'Mathematical Understanding'. I would like to compare
> Mathematical Understanding with Scientific Understanding.
>From the point of view of semeiotics of it is not clear exactly what you mean by
"understanding", and, in particular, it is not clear what you mean by "scientific understanding."
Scientific understanding, as I think you mean it, is captured in a set of domain
specific formal languages that might generally be called mathematical languages.
These are languages that in their experience and concurrence provide an
embodied "understanding" of the facts of the world.
Hence, I will argue, all scientific understanding is founded on mathematical
understanding, mathematics is the language of science.
So, in comparing the scientific method of observation, experiment, and reason
with the mathematical methods of prediction from inductive and deductive
structures; axioms are observations held to be obvious facts of the world,
experiment is the search for new axioms, and reason is analysis by which we
derive new insight.
> The way I see the distinction between scientific understanding and
> mathematical understanding is as follows:
> Mathematics seeks to prove new theorems within its own domain. Propositions
> are proved within the mathematical (axiomatic) framework. Physical theories
> need experimental verification. Any physical theory described in
> mathematical terms needs a direct relation to experimental conditions.
> mathematician will develop a familiarity with the behaviour of the equations
> (s)he deals with. This familiarity is mathematical in nature. To explain
> phenomena, physical theories are used. These theories can be formulated
> abstract mathematical equations. To understand the relation between the
> theories and phenomena mathematical understanding is not enough. It should
> be supplemented with scientific understanding.
> More specifically, I would like to know when do mathematicians understand
> their subject matter? Do they agree on when something is understood and
> on what is understood?
Proofs are the marks of concurrence.
Meta-mathematics - from the point of view of a semeiotician - is a science that
treats mathematical language as a fact of the world. The understanding derived
is no different in nature than any other scientific understanding. It is
tempting to measure all science by its level of rigor.
Pure mathematics and its refinement is the evolution of language that describes
certain general facts of the world - e.g., common properties of geometry. No
language is without subject matter. A computer might only manipulate symbols
and I am told that some mathematicians can act this way (Godel like) - at this
point the subject matter has become the mechanism.
It often requires some analysis - as is apparent from Godel's contributions - to
see the broader implications of that discovery and apply it to other subject
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