⚠ It’s A Trap ⚠

[Jon Awbrey]

Cf: ⚠ It’s A Trap ⚠
https://inquiryintoinquiry.com/2013/05/18/%e2%9a%a0-its-a-trap-%e2%9a%a0/

Re: Kenneth W. Regan • Graduate Student Traps
https://rjlipton.wpcomstaging.com/2013/05/15/graduate-student-traps

So it begins ...

The most common mathematical trap I run across has to do with
Triadic Relation Irreducibility, as noted and treated by the
polymath C.S. Peirce.

This trap lies in the mistaken belief that every
3-place (triadic or ternary) relation can be analyzed
purely in terms of 2-place (dyadic or binary) relations —
“purely” here meaning without resorting to any 3-place
relations in the process.

A notable thinker who not only fell but led many others
into this trap is none other than René Descartes, whose
problematic maxim I noted in the following post.

• Château Descartes
https://inquiryintoinquiry.com/2013/02/14/chateau-descartes/

As mathematical traps go, this one is hydra-headed.

I don’t know if it’s possible to put a prior restraint on the
varieties of relational reduction that might be considered,
but usually we are talking about either one of two types
of reducibility.

• Compositional Reducibility.
All triadic relations are irreducible under relational composition,
since the composition of two dyadic relations is a dyadic relation,
by the definition of relational composition.

• Projective Reducibility.
Consider the projections of a triadic relation L ⊆ X × Y × Z
on the 3 coordinate planes X × Y, X × Z, Y × Z and ask whether
these dyadic relations uniquely determine L. If so, we say L is
“projectively reducible”, otherwise it is “projectively irreducible”.

Et Sic Deinceps …
=================

More Discussion of Relation Reduction
• https://oeis.org/wiki/Relation_reduction
• https://planetmath.org/RelationReduction

Previous Posts on Triadic Relation Irreducibility
• https://inquiryintoinquiry.com/2013/01/17/triadic-relation-irreducibility-1/
• https://inquiryintoinquiry.com/2013/01/18/triadic-relation-irreducibility-2/

Regards,

Jon