In my continued series investigating the philosophical
challenges underpinning science I would like
to focus on the concept of the theory.
This video is predominantly a reading of an
essay written on the subject my Michael Langan.
Michael Langan is supposedly the most intelligent
man alive, clocking in an IQ somewhere between
190 and 210.
He has his own theory of reality called the
cognitive theoretic model of the universe
for those who might be interested.
In essence it is a theory of everything.
However, this video is not so much concerned
with Michael Langan’s work in general but
particularly an exposition on theories in
general.
I will be reading the essay and interjecting
my own clarifying comments as some of the
content could benefit from further examples
and exposition.
What I hope you get out of this video is a
better understanding of the relationship between
science, mathematics, and philosophy.
So, let’s get started.
What is a “theory”?
Is a theory just a story that you can make
up about something, being as fanciful as you
like?
Or does a theory at least have to seem like
it might be true?
Even more stringently, is a theory something
that has to be rendered in terms of logical
and mathematical symbols, and described in
plain language only after the original chicken-scratches
have made the rounds in academia?
A theory is all of these things.
A theory can be good or bad, fanciful or plausible,
true or false.
The only firm requirements are that it (1)
have a subject, and (2) be stated in a language
in terms of which the subject can be coherently
described.
Where these criteria hold, the theory can
always be “formalized”, or translated
into the symbolic language of logic and mathematics.
Once formalized, the theory can be subjected
to various mathematical tests for truth and
internal consistency.
But doesn’t that essentially make “theory”
synonymous with “description”?
Yes.
A theory is just a description of something.
If we can use the logical implications of
this description to relate the components
of that something to other components in revealing
ways, then the theory is said to have “explanatory
power”.
And if we can use the logical implications
of the description to make correct predictions
about how that something behaves under various
conditions, then the theory is said to have
“predictive power”.
From a practical standpoint, in what kinds
of theories should we be interested?
Most people would agree that in order to be
interesting, a theory should be about an important
subject…a subject involving something of
use or value to us, if even on a purely abstract
level.
And most would also agree that in order to
help us extract or maximize that value, the
theory must have explanatory or predictive
power.
For now, let us call any theory meeting both
of these criteria a “serious” theory.
Those interested in serious theories include
just about everyone, from engineers and stockbrokers
to doctors, automobile mechanics and police
detectives.
Practically anyone who gives advice, solves
problems or builds things that function needs
a serious theory from which to work.
But three groups who are especially interested
in serious theories are scientists, mathematicians
and philosophers.
These are the groups which place the strictest
requirements on the theories they use and
construct.
While there are important similarities among
the kinds of theories dealt with by scientists,
mathematicians and philosophers, there are
important differences as well.
The most important differences involve the
subject matter of the theories.
Scientists like to base their theories on
experiment and observation of the real world…not
on perceptions themselves, but on what they
regard as concrete “objects of the senses”.
That is, they like their theories to be empirical.
Mathematicians, on the other hand, like their
theories to be essentially rational…to be
based on logical inference regarding abstract
mathematical objects existing in the mind,
independently of the senses.
And philosophers like to pursue broad theories
of reality aimed at relating these two kinds
of object.
Of the three kinds of theory, by far the lion’s
share of popular reportage is commanded by
theories of science.
Unfortunately, this presents a problem.
For while science owes a huge debt to philosophy
and mathematics – it can be characterized
as the child of the former and the sibling
of the latter - it does not even treat them
as its equals.
It treats its parent, philosophy, as unworthy
of consideration.
And although it tolerates and uses mathematics
at its convenience, relying on mathematical
reasoning at almost every turn, it acknowledges
the remarkable obedience of objective reality
to mathematical principles as little more
than a cosmic “lucky break”.
Science is able to enjoy its meretricious
relationship with mathematics precisely because
of its queenly dismissal of philosophy.
By refusing to consider the philosophical
relationship between the abstract and the
concrete on the supposed grounds that philosophy
is inherently impractical and unproductive,
it reserves the right to ignore that relationship
even while exploiting it in the construction
of scientific theories.
And exploit the relationship it certainly
does!
There is a scientific platitude stating that
if one cannot put a number to one's data,
then one can prove nothing at all.
But insofar as numbers are arithmetically
and algebraically related by various mathematical
structures, the platitude amounts to a thinly
veiled affirmation of the mathematical basis
of knowledge.
Although scientists like to think that everything
is open to scientific investigation, they
have a rule that explicitly allows them to
screen out certain facts.
This rule is called the scientific method.
Essentially, the scientific method says that
every scientist’s job is to (1) observe
something in the world, (2) invent a theory
to fit the observations, (3) use the theory
to make predictions, (4) experimentally or
observationally test the predictions, (5)
modify the theory in light of any new findings,
and (6) repeat the cycle from step 3 onward.
But while this method is very effective for
gathering facts that match its underlying
assumptions, it is worthless for gathering
those that do not.
In fact, if we regard the scientific method
as a theory about the nature and acquisition
of scientific knowledge (and we can), it is
not a theory of knowledge in general.
It is only a theory of things accessible to
the senses.
Worse yet, it is a theory only of sensible
things that have two further attributes: they
are non-universal and can therefore be distinguished
from the rest of sensory reality, and they
can be seen by multiple observers who are
able to “replicate” each other’s observations
under like conditions.
Needless to say, there is no reason to assume
that these attributes are necessary even in
the sensory realm.
The first describes nothing general enough
to coincide with reality as a whole – for
example, the homogeneous medium of which reality
consists, or an abstract mathematical principle
that is everywhere true - and the second describes
nothing that is either subjective, like human
consciousness, or objective but rare and unpredictable…e.g.
ghosts, UFOs and yetis, of which jokes are
made but which may, given the number of individual
witnesses reporting them, correspond to real
phenomena.
The fact that the scientific method does not
permit the investigation of abstract mathematical
principles is especially embarrassing in light
of one of its more crucial steps: “invent
a theory to fit the observations.”
A theory happens to be a logical and/or mathematical
construct whose basic elements of description
are mathematical units and relationships.
If the scientific method were interpreted
as a blanket description of reality, which
is all too often the case, the result would
go something like this: “Reality consists
of all and only that to which we can apply
a protocol which cannot be applied to its
own (mathematical) ingredients and is therefore
unreal.”
Mandating the use of “unreality” to describe
“reality” is rather questionable in anyone’s
protocol.
What about mathematics itself?
The fact is, science is not the only walled
city in the intellectual landscape.
With equal and opposite prejudice, the mutually
exclusionary methods of mathematics and science
guarantee their continued separation despite
the (erstwhile) best efforts of philosophy.
While science hides behind the scientific
method, which effectively excludes from investigation
its own mathematical ingredients, mathematics
divides itself into “pure” and “applied”
branches and explicitly divorces the “pure”
branch from the real world.
Notice that this makes “applied” synonymous
with “impure”.
Although the field of applied mathematics
by definition contains every practical use
to which mathematics has ever been put, it
is viewed as “not quite mathematics” and
therefore beneath the consideration of any
“pure” mathematician.
In place of the scientific method, pure mathematics
relies on a principle called the axiomatic
method.
The axiomatic method begins with a small number
of self-evident statements called axioms and
a few rules of inference through which new
statements, called theorems, can be derived
from existing statements.
In a way parallel to the scientific method,
the axiomatic method says that every mathematician’s
job is to (1) conceptualize a class of mathematical
objects; (2) isolate its basic elements, its
most general and self-evident principles,
and the rules by which its truths can be derived
from those principles; (3) use those principles
and rules to derive theorems, define new objects,
and formulate new propositions about the extended
set of theorems and objects; (4) prove or
disprove those propositions; (5) where the
proposition is true, make it a theorem and
add it to the theory; and (6) repeat from
step 3 onwards.
The scientific and axiomatic methods are like
mirror images of each other, but located in
opposite domains.
Just replace “observe” with “conceptualize”
and “part of the world” with “class
of mathematical objects”, and the analogy
practically completes itself.
Little wonder, then, that scientists and mathematicians
often profess mutual respect.
However, this conceals an imbalance.
For while the activity of the mathematician
is integral to the scientific method, that
of the scientist is irrelevant to mathematics.
At least in principle, the mathematician is
more necessary to science than the scientist
is to mathematics.
As a philosopher might put it, the scientist
and the mathematician work on opposite sides
of the Cartesian divider between mental and
physical reality.
If the scientist stays on his own side of
the divider and merely accepts what the mathematician
chooses to throw across, the mathematician
does just fine.
On the other hand, if the mathematician does
not throw across what the scientist needs,
then the scientist is in trouble.
Without the mathematician’s functions and
equations from which to build scientific theories,
the scientist would be confined to little
more than taxonomy.
As far as making quantitative predictions
were concerned, he might as well be guessing
the number of jellybeans in a candy jar.
From this, one might be tempted to theorize
that the axiomatic method does not suffer
from the same kind of inadequacy as does the
scientific method…that it, and it alone,
is sufficient to discover all of the abstract
truths rightfully claimed as “mathematical”.
But alas, that would be too convenient.
In 1931, an Austrian mathematical logician
named Kurt Gödel proved that there are true
mathematical statements that cannot be proven
by means of the axiomatic method.
Such statements are called “undecidable”.
Gödel’s finding rocked the intellectual
world to such an extent that even today, mathematicians,
scientists and philosophers alike are struggling
to figure out how best to weave the loose
thread of undecidability into the seamless
fabric of reality.
To demonstrate the existence of undecidability,
Gödel used a simple trick called self-reference.
Consider the statement “this sentence is
false.”
It is easy to dress this statement up as a
logical formula.
Aside from being true or false, what else
could such a formula say about itself?
Could it pronounce itself, say, unprovable?
Let’s try it: "This formula is unprovable".
If the given formula is in fact unprovable,
then it is true and therefore a theorem.
Unfortunately, the axiomatic method cannot
recognize it as such without a proof.
On the other hand, suppose it is provable.
Then it is self-apparently false (because
its provability belies what it says of itself)
and yet true (because provable without respect
to content)!
It seems that we still have the makings of
a paradox…a statement that is "unprovably
provable" and therefore absurd.
But what if we now introduce a distinction
between levels of proof?
For example, what if we define a metalanguage
as a language used to talk about, analyze
or prove things regarding statements in a
lower-level object language, and call the
base level of Gödel’s formula the "object"
level and the higher (proof) level the "metalanguage"
level?
Now we have one of two things: a statement
that can be metalinguistically proven to be
linguistically unprovable, and thus recognized
as a theorem conveying valuable information
about the limitations of the object language,
or a statement that cannot be metalinguistically
proven to be linguistically unprovable, which,
though uninformative, is at least no paradox.
Voilà: self-reference without paradox!
It turns out that "this formula is unprovable"
can be translated into a generic example of
an undecidable mathematical truth.
Because the associated reasoning involves
a metalanguage of mathematics, it is called
“metamathematical”.
It would be bad enough if undecidability were
the only thing inaccessible to the scientific
and axiomatic methods together.
But the problem does not end there.
As we noted above, mathematical truth is only
one of the things that the scientific method
cannot touch.
The others include not only rare and unpredictable
phenomena that cannot be easily captured by
microscopes, telescopes and other scientific
instruments, but things that are too large
or too small to be captured, like the whole
universe and the tiniest of subatomic particles;
things that are “too universal” and therefore
indiscernable, like the homogeneous medium
of which reality consists; and things that
are “too subjective”, like human consciousness,
human emotions, and so-called “pure qualities”
or qualia.
Because mathematics has thus far offered no
means of compensating for these scientific
blind spots, they continue to mark holes in
our picture of scientific and mathematical
reality.
But mathematics has its own problems.
Whereas science suffers from the problems
just described – those of indiscernability
and induction, nonreplicability and subjectivity
- mathematics suffers from undecidability.
It therefore seems natural to ask whether
there might be any other inherent weaknesses
in the combined methodology of math and science.
There are indeed.
Known as the Lowenheim-Skolem theorem and
the Duhem-Quine thesis, they are the respective
stock-in-trade of disciplines called model
theory and the philosophy of science (like
any parent, philosophy always gets the last
word).
These weaknesses have to do with ambiguity…with
the difficulty of telling whether a given
theory applies to one thing or another, or
whether one theory is “truer” than another
with respect to what both theories purport
to describe.
Now, what Langan is saying in the last paragraph
is that there are significant problem in science
relating to the ability to choose between
two seemingly valid theories.
For example, if two theories seem to perfectly
explain any set of observations but are radically
different, how do you decide which one is
correct?
In a concrete case, let us say that a homosexual
man is fired by his employer; two theories
explaining this event can be as follows; either
the employer hates gays, or the gay man was
a terrible employee.
Both theories have equal explanatory force
of the phenomenon.
Though this example is trivial, it is an easy
way to frame the sort of challenges that appear
in science.
Now, let us resume the essay.
But before giving an account of Lowenheim-Skolem
and Duhem-Quine, we need a brief introduction
to model theory.
Model theory is part of the logic of “formalized
theories”, a branch of mathematics dealing
rather self-referentially with the structure
and interpretation of theories that have been
couched in the symbolic notation of mathematical
logic…that is, in the kind of mind-numbing
chicken-scratches that everyone but a mathematician
loves to hate.
Since any worthwhile theory can be formalized,
model theory is a sine qua non of meaningful
theorization.
Let’s make this short and punchy.
We start with propositional logic, which consists
of nothing but tautological, always-true relationships
among sentences represented by single variables.
Then we move to predicate logic, which considers
the content of these sentential variables…what
the sentences actually say.
In general, these sentences use symbols called
quantifiers to assign attributes to variables
semantically representing mathematical or
real-world objects.
Such assignments are called “predicates”.
Next, we consider theories, which are complex
predicates that break down into systems of
related predicates; the universes of theories,
which are the mathematical or real-world systems
described by the theories; and the descriptive
correspondences themselves, which are called
interpretations.
A model of a theory is any interpretation
under which all of the theory’s statements
are true.
If we refer to a theory as an object language
and to its referent as an object universe,
the intervening model can only be described
and validated in a metalanguage of the language-universe
complex.
Though formulated in the mathematical and
scientific realms respectively, Lowenheim-Skolem
and Duhem-Quine can be thought of as opposite
sides of the same model-theoretic coin.
Lowenheim-Skolem says that a theory cannot
in general distinguish between two different
models; for example, any true theory about
the numeric relationship of points on a continuous
line segment can also be interpreted as a
theory of the integers (counting numbers).
On the other hand, Duhem-Quine says that two
theories cannot in general be distinguished
on the basis of any observation statement
regarding the universe.
So once again, if we go back to our example
of the homosexual man who was fired, Langan
is saying that you cannot look at the phenomenon
that the homexual man was fired to decide
whether or not it is the case that the employer
hates gays or the gay man was a terrible employee.
Though this may seem obvious, it will often
clash with people’s assumption that you
could simply tell which theory is correct
by observing the evidence.
This cannot be done as the observable evidence
fits both theories perfectly.
Now back to the essay.
Just to get a rudimentary feel for the subject,
let’s take a closer look at the Duhem-Quine
Thesis.
Observation statements, the raw data of science,
are statements that can be proven true or
false by observation or experiment.
But observation is not independent of theory;
an observation is always interpreted in some
theoretical context.
So an experiment in physics is not merely
an observation, but the interpretation of
an observation.
This leads to the Duhem Thesis, which states
that scientific observations and experiments
cannot invalidate isolated hypotheses, but
only whole sets of theoretical statements
at once.
This is because a theory T composed of various
laws {Li}, i=1,2,3,… almost never entails
an observation statement except in conjunction
with various auxiliary hypotheses {Aj}, j=1,2,3,…
. Thus, an observation statement at most disproves
the complex {Li+Aj}.
 Page 8
To take a well-known historical example, let
T = {L1,L2,L3} be Newton’s three laws of
motion, and suppose that these laws seem to
entail the observable consequence that the
orbit of the planet Uranus is O.
But in fact, Newton’s laws alone do not
determine the orbit of Uranus.
We must also consider things like the presence
or absence of other forces, other nearby bodies
that might exert appreciable gravitational
influence on Uranus, and so on.
Accordingly, determining the orbit of Uranus
requires auxiliary hypotheses like A1 = “only
gravitational forces act on the planets”,
A2 = “the total number of solar planets,
including Uranus, is 7,” et cetera.
So if the orbit in question is found to differ
from the predicted value O, then instead of
simply invalidating the theory T of Newtonian
mechanics, this observation invalidates the
entire complex of laws and auxiliary hypotheses
{L1,L2,L3;A1,A2,…}.
It would follow that at least one element
of this complex is false, but which one?
Is there any 100% sure way to decide?
As it turned out, the weak link in this example
was the hypothesis A2 = “the total number
of solar planets, including Uranus, is 7”.
In fact, there turned out to be an additional
large planet, Neptune, which was subsequently
sought and located precisely because this
hypothesis (A2) seemed open to doubt.
But unfortunately, there is no general rule
for making such decisions.
Suppose we have two theories T1 and T2 that
predict observations O and not-O respectively.
Then an experiment is crucial with respect
to T1 and T2 if it generates exactly one of
the two observation statements O or not-O.
Duhem’s arguments show that in general,
one cannot count on finding such an experiment
or observation.
In place of crucial observations, Duhem cites
le bon sens (good sense), a non-logical faculty
by means of which scientists supposedly decide
such issues.
Regarding the nature of this faculty, there
is in principle nothing that rules out personal
taste and cultural bias.
That scientists prefer lofty appeals to Occam’s
razor, while mathematicians employ justificative
terms like beauty and elegance, does not exclude
less savory influences.
So, what Langan has just described is that
in most cases, when two theories both seem
to fit the evidence, the ultimate tool the
scientist uses to decide which one is correct
is his feeling, intuition, or something else
completely subjective.
Though in the case of the orbit of Uranus,
there was an experiment that could decide
what the case was, Duhem argues that in most
cases when a scientist needs to decide between
two theories, there is no experiment, and
therefore the theory that feels correct is
concluded to be correct.
Now back to the essay.
So much for Duhem; now what about Quine?
The Quine thesis breaks down into two related
theses.
The first says that there is no distinction
between analytic statements (e.g. definitions)
and synthetic statements (e.g. empirical claims),
and thus that the Duhem thesis applies equally
to the so-called a priori disciplines.
To make sense of this, we need to know the
difference between analytic and synthetic
statements.
Analytic statements are supposed to be true
by their meanings alone, matters of empirical
fact notwithstanding, while synthetic statements
amount to empirical facts themselves.
Since analytic statements are necessarily
true statements of the kind found in logic
and mathematics, while synthetic statements
are contingently true statements of the kind
found in science, Quine’s first thesis posits
a kind of equivalence between mathematics
and science.
In particular, it says that epistemological
claims about the sciences should apply to
mathematics as well, and that Duhem’s thesis
should thus apply to both.
Remember, where in science a scientist would
decide which of two theories corresponding
to the same observation is correct is based
on good sense, feeling, or intuition.
The mathematician’s version of this is elegance.
Now, I am sure that you have encountered the
term “elegance” in describing scientific
theories before.
This term is also used in the discipline of
computer science.
In software development, when two pieces of
code both perform the same function, often
times what software developers use to judge
which solution is better is the concept of
elegance.
Remember, both pieces of code accomplish the
exact same goal and from an users point of
view, there is absolutely no different, yet
still one piece of code is considered more
correct than the other because it is more
elegant.
Langan writes.
Quine’s second thesis involves the concept
of reductionism.
Reductionism is the claim that statements
about some subject can be reduced to, or fully
explained in terms of, statements about some
(usually more basic) subject.
For example, to pursue chemical reductionism
with respect to the mind is to claim that
mental processes are really no more than biochemical
interactions.
Specifically, Quine breaks from Duhem in holding
that not all theoretical claims, i.e. theories,
can be reduced to observation statements.
But then empirical observations “underdetermine”
theories and cannot decide between them.
This leads to a concept known as Quine’s
holism; because no observation can reveal
which member(s) of a set of theoretical statements
should be re-evaluated, the re-evaluation
of some statements entails the re-evaluation
of all.
Quine combined his two theses as follows.
First, he noted that a reduction is essentially
an analytic statement to the effect that one
theory, e.g. a theory of mind, is defined
on another theory, e.g. a theory of chemistry.
Next, he noted that if there are no analytic
statements, then reductions are impossible.
From this, he concluded that his two theses
were essentially identical.
But although the resulting unified thesis
resembled Duhem’s, it differed in scope.
For whereas Duhem had applied his own thesis
only to physical theories, and perhaps only
to theoretical hypothesis rather than theories
with directly observable consequences, Quine
applied his version to the entirety of human
knowledge, including mathematics.
If we sweep this rather important distinction
under the rug, we get the so-called “Duhem-Quine
thesis”.
Because the Duhem-Quine thesis implies that
scientific theories are underdetermined by
physical evidence, it is sometimes called
the Underdetermination Thesis.
Specifically, it says that because the addition
of new auxiliary hypotheses, e.g. conditionals
involving “if…then” statements, would
enable each of two distinct theories on the
same scientific or mathematical topic to accommodate
any new piece of evidence, no physical observation
could ever decide between them.
The messages of Duhem-Quine and Lowenheim-Skolem
are as follows: universes do not uniquely
determine theories according to empirical
laws of scientific observation, and theories
do not uniquely determine universes according
to rational laws of mathematics.
The model-theoretic correspondence between
theories and their universes is subject to
ambiguity in both directions.
If we add this descriptive kind of ambiguity
to ambiguities of measurement, e.g. the Heisenberg
Uncertainty Principle that governs the subatomic
scale of reality, and the internal theoretical
ambiguity captured by undecidability, we see
that ambiguity is an inescapable ingredient
of our knowledge of the world.
It seems that math and science are…well,
inexact sciences.
Thanks for listening.
Go Team!
