Introduction
Commutative
operators
Demarcation
between science and metaphysics
Endothermic and exothermic reactions
Foundationalism and coherentism
Fuzzy ('indeterminate') logic
Implication
Induction
Inertia
Infinitesimals
Interchangeability
Intervals
Inverses of functions
Kantian categories
Mendelian factors
Meta- and para-studies
Newton's first law of motion
Newton's third law of motion
Pacifism
Particle in a box
Perspective
Platonic objects
Polish (prefix) notation
Referents (ambiguity of, in similes)
Regions
Schopenhauer's pessimism,
Selection: natural and artificial
SI units
Thermodynamic
systems as partitions
Introduction
No matter how extensive this page became, it could present only a small selection
of the possible interpretations of {theme} theory.
Established concepts from very varied fields are interpreted (very concisely)
in terms of my {theme} theory. There's next to no explanation of the theory
here, or of the notation I use. For this exposition, see in particular {theme}
theory. Without some study at least of {theme} theory and its
notation, much of the material here will be very difficult to understand.
In every case, the topics discussed are ones with a literature which is very,
very extensive, in almost all cases vast. All I do here is indicate some of
the thematic activity which underlies and often links these topics.
Commutative
operators
Non-rigorously, Ô :- (mathematical arguments) ) 
(the mathematical result).
In the real numbers, addition is commutative but not subtraction.
A group G with the binary operation + is abelian if 
g, h 
G, g + h = h + g.
One form of the Heisenberg Uncertainty Principle:
where
p is the uncertainty in the linear momentum parallel to the axis q and
q is the uncertainty in position along this axis. Dn: /
/. These observables are complementary. The term on the right hand
side of the equation is the modified Planck's Constant. Complementary observables
have non-commuting operators: the application of O1 followed by O2 has a different
result from the application of O2 followed by O1 so that Ô is significant.
The Heisenberg Uncertainty Principle has this {thematic} effect: == :- (precision
in specifying momentum and position of a particle). By 
,
== :- (any pair of non-commuting operators).
I see the need for
a generalized theory of indeterminacy which includes amongst other instances
the Heisenberg Uncertainty Principle and the instances of indeterminacy of
interest to philosophers. I regard indeterminacy as a sub-theme, {/indeterminacy}
and indicate it symbolically by Î. The {thematic} action
is 'to make indeterminate.' The most generally useful form is '... is indeterminate.'
This can be shown by the {indeterminacy} indicator, Î Î ,
usually followed immediately by expansion brackets to explain why a
statement or other entity is indeterminate or vague.
Demarcation
between science and metaphysics
The desirability of using 'demarcation,' if it should be used at all (rather
than, eg, 'boundary') advisedly, after first using ® :- //. If
(1) and (2) are conditions then the default condition (1) is one in which
Ô :- (1,2)
> (1). In speculations concerning [science] < Qn > [metaphysics]
then I use as a convention the default condition [science] // [metaphysics],
which is to say that // here is not argued but that < > has to be argued.
Obviously, there is the problem of resolvability: 
® (science-metaphysics) Qn. Karl Popper claims as a criterion of
resolvability the fasifiability of scientific claims but not metaphysical
claims. I dispute this. I think that some metaphysical claims have been falsified,
although with Î .
However, claims to scientific knowledge which have been falsified have also
been falsified with Î ,
to a far lesser degree. Claims to scientific knowledge not yet falsified or
never to be falsified are also subject to Î.
Endothermic
and exothermic reactions
The powerful division of the universe into two parts in some thermodynamic
analyses:
® : - (the universe)
(1) the reaction vessel, regarded as a thermodynamic system (2) the rest of
the universe. 
.
Giving to {direction} the interpretation 'flows into' or 'flows from' and
treating the quantity q as the subject (compare the subject of
a verb), then q is +ve in the case of an endothermic reaction, when q: (rest
of universe)
(the thermodynamic system) and q is -ve in the case of an exothermic reaction,
when q: (thermodynamic system)
(rest of the universe).
q is a path-function, but it's more convenient to analyze reactions in terms
of state-functions. Interpretation of 'independent of' in 'state functions
are independent of the way the reaction is carried out:' > < .
Foundationalism
and coherentism
For foundationalists, foundational facts are epistemologically prior to
non-foundational facts. [foundational facts] / (partitioning)
/ [non-foundational facts]. Ô :- (foundational facts, non-foundational
facts]
(foundational facts) > (non-foundational facts): (foundational facts) have
{prior-ordering}. Non-foundational facts have {dependence} on foundational
facts: (non-foundational facts) 
(foundational facts).
Foundationalists have often granted epistemological exemption to
the foundations of knowledge. == :- (error 
doubt
refutation) so that the foundations of knowledge have infallibility, indubitability
and incorrigibility. So, for many empiricists, == (+ error, doubt, refutation)
:- (immediate sensory experience). Berkeley and other phenomenalists have
claimed that (physical objects)
(sensory experience): ontological dependence. For coherentists, Ô
:- (beliefs) so as to give [foundational beliefs] / partitioning / [non-foundational
beliefs].
A general theory of dependence refers to foundationalism and coherentism
and also, eg, Kant's categorical and hypothetical imperatives, where Ô:-
(imperatives) and (hypothetical imperative)
(categorical imperative).
Fuzzy
('indeterminate') logic
'Indeterminate logic' is the term I use for the established 'fuzzy logic.'
'Indeterminate set' is the term I use for the established 'fuzzy set,' introduced
by Lotfi Zadeh. Indeterminate logic
indeterminate set i{indeterminacy}.
[indeterminate logic] < > [many-valued logic].
Zadeh: 'fuzzy logic in broad ... and narrow sense.'
Broad sense:
[ fuzzy (indeterminate) logic < synonymity > ['indeterminate' set theory
and its applications]
Narrow sense:
(many-valued logic)
('indeterminate' logic).
The proposition 'X is tall' involves ® :- (truth), which gives Î
in the construction of the sub-set (tall people) 
(people) - but not, in Mendelian genetics, in the construction of the sub-set
(tall plants belonging to the species Pisum sativum)
(plants belonging to the species Pisum sativum).
Implication
When statement 1 implies statement 2, then the truth of statement 1 ensures
the truth of statement 2. Statement 1 'leads to' or 'directs to' statement
2. This is an instance of
(1)
(2). ® :- (
)
/implication/ or {resolution} applied to {direction} gives ['directs
to'] /implication/.
Induction
[induction] < > [serial ordering: the natural number following a given
natural number]
and [element of a serial ordering] < > [datum element]
® :- Ô
/serial ordering/
[Inductive successors taking the form n, n, n, ...] ( ) [natural number successors,
n, n + 1...]
® :- (n), to take account of vagueness in n.
[inductive successors] (certainty, lack of certainty) [natural number successors]
Inertia
[Continuance of a process] < > [].
'Inertia' :- Physics, metre (in my approach), Humean philosophy.
.
From my page metre: ''Metrical
inertia' is the counterpart of inertia in Physics, which, omitting any detail,
is the property which causes a mass to resist changes - so, a body in motion
tends to continue in motion.'
From Donald L M Baxter, 'Identity and Continued Existence' in 'The Blackwell
Guide to Hume's Treatise,' (Page 119), of an experience when we 'assume more
regularity than we observe:' 'This is not mere causal reasoning, which is
constrained to observed regularity. It is rather an inertia of the mind in
continuing a way of thinking once begun ... Hume earlier discussed a precedent
for this mental inertia when explaining how we come to the fiction of perfect
equality in geometry.'
Infinitesimals
The mathematical notion that a continuum can be divided without limit I interpret
as ® :- the
continuum ~
‚. ('not possible to complete.') dy/dx as showing the limit of the ratio
y/x
as x tends to
zero. I interpret again in terms of {completion}. Where the process concerned
is the giving of the smallest non-zero values to x
and y then ‚:-
x 
dx and similarly for y.
Abraham Robinson's 'nonstandard analysis' translates every statement of analysis
which involves limits into the language of infinitesimals.
Interchangeability
Using as examples of possible interchangeabilty points and lines in projective
geometry or existential and universal quantifiers in predicate calculus, trivially,
[name: 'line'] ( ) [name: 'point'] and [name: 'existential quantifier'] (
) [name: 'universal quantifier'] but [application-sphere of name: 'line']
) ( [application-sphere of name: 'point'] and [application-sphere of name:
'existential quantifier'] ) ( [application-sphere of name: 'universal quantifier.']
Intervals
Rhyme-interval, musical
interval and mathematical interval i{distance}.
Musical interval: [pitch of note 1] < interval > [pitch of note 2].
The interval is named by counting the diatonic degrees between note 1 and
note 2, eg C - G is a fifth, {distance} < C - E, a third.
Mathematical intervals: the set which contains all the real numbers of points
between two real numbers or points. Notated (open interval) as / {x: a <
x < b} /.
Inverses
of functions
Interpreted as ». If A, B are sets and f: A /
/
B is a mathematical
function, then an / inverse /, » for f is a function g: B /
/ A such that
g ° f = idA and f ° g
= idB.
So,
a A
and
b B,
g (f (a) ) = a and f (g (b) ) = b
/Composition/ of functions a and b: Ô :- ( f ° g ).
The existence of an / inverse / ( interpreted: {/ reversal}) is one of the
axioms of a Group: for each element g
G there's an element h G
such that g ° h = h ° g = e. The element h is an / inverse / of g.
A group (G, ° ) is Abelian or commutative if Ô (a ° b) = Ô
(b ° a )
a, b G.
The / identity function / on A (written idA )
:- ( A ).
Kantian
categories
From Kant's 'Critique of Pure Reason,' I, Transcendental doctrine of elements,
Second Part, Division 1, Book I, Chapter I, third section: On the pure concepts
of the understanding or categories, translated by Paul Guyer and Allen W.
Wood.
Any view of Kant's
'Critique of Pure Reason' as an impregnable edifice (rather than a magnificent
one) would be mistaken, of course. Kant's list of twelve categories in four
groups has been widely criticized. I maintain that his list is confused, in
fact chaotic.
My {themes} are
Kantian to the extent that the human mind uses and must use {themes} if it
is to make sense of the world, the inner world as well as the outer world.
The {themes} are fundamental and are distinguished from spheres, which include
their application-spheres. ® :- (application-spheres)
(contingent
non-contingent spheres). I don't give a criticism in detail of Kant's scheme
here, since my {theme} theory
provides a clear and systematic approach to reality which eliminates the confusions
in Kant's Table of Categories.
The inclusion
of 'reality' with 'negation' and 'limitation' in Kant's group 'Of Quality'
is particularly confused. Reality is the most general sphere. Particular spheres
(in my terminology) belong to 'reality,' including the application-spheres.
[Kantian 'reality'] < > [Kantian 'totality']. For some reason, these
are placed in separate groups in Kant's scheme.
The application
of {restriction} 'tidies up' some of Kant's Table. == :- (totality)
(plurality). [{restriction}] < > [Kantian 'limitation'} is obvious.
Kant's 'causality'
and 'dependence' should be subject to ®, since causality belongs only
to the contingent sphere and dependence (my
) is at a higher, and different, level of generality, including the non-contingent
as well as the contingent. The Kantian possibility and impossibility are very
different in their application to the contingent and the non-contingent spheres.
Mendelian
factors
are instances of generalized factorization, which has as other instances
the Factors involved in human diet (these are not only the factors needed
for a balanced diet), the factors involved in a moral choice - and innumerable
other examples.
// :- (Mendelian factors) easily, eg, long or short pea plants, red or white
flower colour in pea plants. These are sharply distinguished.
Meta-
and para-studies
meta- is from the Greek
which
means (with the accusative, with reference to sequence or succession) 'after'
or 'next to,' and para- is from the Greek 
which means (with the accusative) 'to the side of,' beside.'
The established use of 'meta-' has reference, in my terminology, to {ordering}.
So, a metalanguage 'comes after' a language and has {post-ordering}: a language
used to describe another language, the object language. In my terminology,
the object language is the sphere of application of the metalanguage, metalanguage
:- (object language.)
Metamathematics is the study of such aspects of mathematics as consistency
and reliability. Tarski included questions of {completion} (in my terminology)
and axiomatizability. Tarski referred to the 'methodology of the deductive
sciences.)
Metaphysics has an application-sphere with less {restriction}. Metaphysics
is much broader in scope than physics - it has less {restriction} - and goes
far beyond examining the methodology of physics. Its scope includes non-physical
entities. But {ordering} is as applicable to metaphysics as to metalanguage
and metamathematics. Metaphysics, for example, can be regarded as more fundamental
than physics.
The term para-study isn't established. I use it for studies which aren't
given {ordering} but which are 'alongside.'
{theme} theory is applied to language, mathematics, metaphysics and innumerable
other areas. These are regarded as un-ordered for the purposes of this examination.
They are application-spheres for {theme} theory.
Newton's
first law of motion
'Every body continues in its state of rest or uniform motion in a straight
line, unless compelled to change its state by an external force.'
'Every body:' ==
: - (°
P) where 'bodies'
are P.
® :- state
rest
motion.
® : - motion
uniform
non-uniform motion
rectilinear curvilinear motion.
the state, the agent of {modification} being an external force.
The law is subsumed
under generalized thematic action whilst retaining all its empirical testability
and falsifiability.
Newton's
third law of motion
Action and reaction are equal and opposite.
Force is a vector quantity.
(( vector quantities, eg force )) :- (magnitude
direction).
bodies,
pairs of bodies, eg P and Q,
Ô (magnitude of forces, force of P on Q and Q on P.)
bodies,
pairs of bodies, eg P and Q, if
:- (force of P on Q) is
(1) then
:- (force of Q on P) is » :- (1).
Pacifism
The claim - unjustified,
I think - that
® :- (reasons for waging war) to establish moral justification for engaging
in war. The anti-pacifist claim, which I support, is that ® :- (reasons
for waging war)
(offensive war, defensive war), (unjustifiable war, justifiable war), (war
not likely to give the consequentialist advantage of reducing suffering, war
likely to give the consequentialist advantage of reducing suffering) ...
Particle
in a box
== :- (space) so that
there are boundary conditions.
== :- (wavefunctions, found by solving the Schrödinger equation for the
system) so that only certain wavenfunctions are acceptable.
== :- (observables) so that only discrete, not continuous, values are observable.
The energy of the particle is quantized.
Perspective
Linear and aerial perspective involve {distance} into the picture plane,
not the illusion of distance into the picture plane. I claim that
Musical tonality is
linked with perspective in art - both are instances of {distance}:
[musical tonality] < i({distance}) > [perspective in art].
Platonic
objects
include circles, the geometrical objects which are 'approximated-to' by
the drawn objects used to illustrate the properties of these Platonic geometrical
objects. An illustration of, in my terminology, 'the principle of the primacy
of perfection.' Ô
:- (objects)
(perfect objects) > (real-world objects) and == : - (perfect objects)
(real-world objects). {distance} expresses the degree of approximation of
real-world objects to perfect objects.
Polish
(prefix) notation
Instead of writing 'this notation places the operators before their arguments'
or 'this notation places the operators in front of the schemata over which
they are ranging' or 'constants precede the variables they govern' I use consistent
(common) 'application-sphere.' I retain 'operator: ' the operator has an application
sphere. Ô:- (operators, application-spheres of operators).
[Established infix notation, eg p
q (+
'inclusive'] ( Ô of operator-application-sphere of operator)
[Polish (prefix) notation) Apq] and
[Established infix notation, eg p
q (+
'exclusive')] ( Ô of operator-application-sphere of operator)
[Polish (prefix) notation Jpq]
Reverse (postfix) notation, » :- ( Ô operator-application-sphere
of operator). So, » (Apq)
(pqA).
Referents,
ambiguity of in similes
Seamus Heaney, 'Blackberry-Picking,' 'At first, just one, [blackberry] a
glossy purple clot / Among others, red, green, hard as a knot.'
[ primary subject
] <
>
[ secondary subject S1, knot
in string/rope
secondary subject 2, knot in wood]
Regions
See also Region poetry and zoning
in the page 'Glossary of literary linkage terms' and Regional
differences in the page 'Web design.'
The concept of region has a high degree of generality and,
® :- (this generalized concept of region)
(these regions of region poetry
these regions of Web design
.) Obviously, this ((survey)) is far from complete. A fuller ((survey)) would
include geographical regions of a country, region in Heidegger's 'Sein und
Zeit' (German 'Gegend') the present belonging to a different region from the
future, 'regions of the mind,' regions in scientific knowledge and 'tonal
regions' in music. 'any large, indefinite, and continuous part of a surface
or space' but in my interpretation, ['region'] < > [{distance}] and
{distance} i(spatial and non-spatial distance). /Geometrical distance/ i-spatial
distance, contrary to very many accepted conceptions.
The concept of region includes the mathematical concept of region, but not
simply because the word 'region' is established in mathematics. The mathematical
conception of 'region' makes reference to a connected subset of two
dimensional space. 'Space' should not be interpreted as physical space here.
Again contrary to many accepted conceptions, [Euclidean space with Cartesian
coordinates] // [the 3-dimensional space of our common-sense world]. Here,
'//' as always does not imply lack of all < >.
Heidegger's conception of 'region' is not at all mine. For example, ' 'In
the region of ' means not only 'in the direction of ' but also within the
range [Umkreis] of something that lies in that direction.' (1, 3: 103. Translation
of John Macquarrie and Edward Robinson.) The view of 'direction' here has
very little in common with my own {direction}. Although Heidegger's view seems
to have great generality, this is only an appearance. He writes, ' ... Dasein
itself is 'spatial' with regard to its Being-in-the-world.' (1, 3, 104). Space
shares the concreteness of Dasein and Heidegger's ontology. Just as Heidegger's
'concrete epistemology' attempts to displace Cartesian epistemology rather
than accommodate it - find (non-spatial) 'room' for it - his understanding
of space finds no room for such mathematical conceptions as the Euclidean.
He writes, 'When we let entities within-the-world be encountered in the way
which is constitutive for Being-in-the-world, we 'give them space'. This 'giving
space', which we also call 'making room' for them, consists in freeing
the ready-to-hand for its spatiality.' (1, 3: 111.) Here, 'giving space' translates
'Raum-geben' and 'making room' translates 'Einräumen.' But the nature
of his philosophy - cramped rather than spacious - had the effect of imposing
{restriction} to a very great degree.
Heidegger's conception of region is too limited, and shares the limitations
of his philosophy. I think it important that a ((survey)) of regions should
include scientific instances. Regions i-regions of phase diagrams, which,
in the case of a pure substance, show regions of pressure and temperature
where the states solid, liquid and gas are thermodynamically stable. And i-regions
of relative negative potential and relative positive potential on the surfaces
of molecules, shown by an electrostatic potential surface.
In connection with geographical regions, 'boundary experiences' are of particular
interest to me - the extent to which, often, those within a boundary but near
to it identify more with those beyond the boundary, rather than giving weighting
to the < > with those deeper within the boundary.
Schopenhauer's pessimism
isn't supported in my thinking. Disappointment and unhappiness are == :-
(happiness). The fact that disappointment and unhappiness are often much more
in evidence than happiness doesn't modify 'the principle of the 'primacy of
perfection,' which isn't at all a principle of 'optimism despite reality.'
Selection:
natural and artificial
Natural and artificial selection are instances of selection, which is an
aspect of filtering, the non-systematic term which gives {substitution}
for the more systematic == :- (x), the term which in this case refers to genotypes.
In the case of natural selection, the filtering is bound. In the
case of artificial selection, the filtering is free.
SI
units
In diagrams, it's often convenient to use linkage-lines rather than
indicate linkage by < >. < > is at a higher level of generality
than {direction}, which can be indicated in a diagram by direction-lines,
formed by {extension} of
.
{direction} has {/derivation}, which can be shown by using dependence-lines
which are extended. {dependence} has {/derivation}. In the diagram below,
< > between some of the base units in the SI system, kg, m, s and some
derived units, acceleration, force, energy and power, are shown by means of
linkage lines. These can be replaced by derivation-lines, at a lower level
of generality. The derivation-lines show that the derived units are derived
from the base units, and show the {direction} of derivation.
Thermodynamic
systems as partitions
The need for a generalized theory of partitions. Two partitions can be interpreted
as //2 after ®. The differentia of a mathematical
partition includes the need for an 'equivalence relation' (+ ... ). At a high
level of generality, [ thermodynamic system and surroundings ] < i-generalized
partition > [ 2 mathematical partitions ] and (generalized partition) :-
(mathematical partition)
(thermodynamic system and surroundings).
Mathematically, if A is a collection of subsets of the set S, A is a partition
of S iff the union of all the subsets which belong to A is the whole of S,
the unequal members of A are disjoint and each subset belonging to A is non-empty.
Thermodynamically, in my notation, ® :- (the universe)
(the system
its surroundings). (system) // (surroundings) but there are degrees of //.
In the case of open systems, matter can be transferred through the system-surroundings
boundary. Otherwise, the system is closed.
