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I've been pondering on the nature of mathematics, after the various
discussions that have recently been going on here on Tim's list... And it
strikes me that mathematics is a language of interactivity, or :"interactive
causality". As such, it has context built right into it, which cannot be
expunged as my father observed in his various writings. No wonder he found it so
useful for illustrating various aspects of complexity!
Any equation is describing the interactions of numbers with each
other and themselves, via specified modes which are one type of constraints. But
those constraints themselves are context dependent and can mean something
different when interacting with other constraints (the sequence of events in an
equation can be changed by a pair of parentheses, for example) which totally
impacts the result. Thus, any mathematical equation can be characterized as
a description of interactive causality and the "answer" is an interactive result
or effect.
Mathematics as a modeling tool for describing nature can
be extremely useful, partly because the interactive causality in
mathematics is an echo of the interactive causality in the universe. However,
where and when various modes of interaction are applied will totally affect the
results and this is as true within mathematics as it is about applying
mathematics as a modeling tool to natural systems. In other words, there are
times when it's appropriate and times when it isn't, and if used badly or
used when it's not appropriate, its use will change the results we are
trying to learn or predict. This is where modes of modeling and modes of
measurement can add artificatual information that deforms the results of
analysis.
Generally speaking, in natural systems where context is radically
important, mathematics may only be useful for illustrating where,
when, and/or why it should NOT be applied as a modeling tool. Subtle additions
to context in the natural system can have radical effects on the outcome. So, if
those aspects of context are not included in the mathematical analysis, clearly
the predictions generated by the model will be way off. Yet mathematics is used
all the time to generate working models of ecosystems, meteorology, human
physiology, medical applications, and various agribusiness decisions to name a
few. We can prove mathematically that this is a bad idea and yet such proof
is ignored. I wonder why that is?
How important is context in natural systems? Well, in mathematics;
one plus one equals 2. I've seen many applications of mathematics to population
statistics where some huge number is generated as being the potential number of
offspring of some species of organism. So, in mice, for example, one plus one
may equal X thousand. There is enormous contextual information included in an
application like this, and this could be very wrong significant for the result.
What if we have two related mice which have genetic recessives? What's our total
number likely to be, then? What if we have two mice of the same gender?
Mice cannot reproduce that way. What if a reproducing pair are living in some
area that has built in environmental hazards of one or more sorts? What if
we're talking about bacteria instead of mice, and this specific species of
bacteria has more modes of reproduction than we know about?
The point is that mathematics is, itself, extremely
context-dependent as an interactive system in its own right. So, to use it
to analyze and generate predictions about some natural complex system where
context is even more crucial to outcome-- and make decisions based on the
analysis (about allowable exposure to various chemical byproducts of industry,
for example)... is dangerous and foolhardy if the contexts are not handled very,
very carefully. Can we even count the number of ways it can go
wrong?
I've mentioned on the list before that my father wrote a paper
based on "the tragedy of the common", where he showed, mathematically, that what
is optimal (in terms of strategy with the common and one's herd of
sheep) using a short time horizon turns out to be very sub-optimal if the
time horizon is lengthened. So the decisions made based on a short-term view are
very different from the decisions that would be made if the view of time were
lengthened. I should think it would be possible to show, mathematically, how
small changes in context can radically alter values and plug the results into a
similar gradient like the one involving time. Time is also a context, so this
kind of analysis can be extremely useful in a cautionary way. Any takers on the
list?
Judith
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