The
Fusion of Reason and Experience

**Copyright © 2005, David
A. Epstein. **

All Rights Reserved.

October 18, 2005

All Rights Reserved.

October 18, 2005

Deductive
Reasoning and the Markers of Experience I was thinking about deductive reasoning and whether a pure form of it really exists. This type of reasoning has its roots in Aristotle's theory of logic, where a series of inferences would lead a set of premises (or a single premise) to a formal conclusion. It's referred to as "deductive" because the premises are more general than the conclusion. The conclusion is deduced through a process of narrowing scope, from an initial generalization to a specific instance. This is clearly illustrated by the example of the syllogism. The syllogism has the form: "All X are in Y. Q is in X. Therefore, Q is in Y". The corollary in set theory is that X is a subset of Y, and if something is in X, it must be in Y. Thus, "If all men are mortal, and John is a man, then John must be mortal". Any proof that conforms to this process of {premise => inferences => conclusion} is called "valid". The syllogism above is valid because the premises lead to a correctly inferred conclusion. It's also considered to be true because both the premises and conclusion are true. Such a valid and true proof is termed "sound". However, if any of the premises are untrue, then the proof won't be true. Such a valid and untrue proof is termed "unsound". An example of this type of proof is: "All men are happy. John is a man. Therefore John is happy." This is a perfectly valid proof because the conclusion (John is happy) follows from the premises; but it's obviously false because the first premise (All men are happy) is false, even though the conclusion might be true. In
theory, this process is strictly an internal product of logic
and a priori reasoning and is completely
independent of empirical
data, observations, experimentation, and a posteriori reasoning.
Indeed, the composition of a
valid proof is constructed using logic and is absolutely insulated from
the world of experience. But
what about the process of configuring the proof? Is this a
deductive process in practice?
After thinking about it for
a while, I said "no". Ha, it took a pure act of deductive reasoning
to conclude that there probably is
no such
thing as pure deductive reasoning! I
came to this conclusion by thinking about how the deductive thinker
conceives of her thought experiments, abstract concepts,
visualizations,
mathematical proofs, logical arguments, etc. What I was
considering is that there are pointers, markers, clues, catalysts,
situations,
and events in the real world that influence the thinker during her
deductive
pursuits. In
some ways, they are the road maps to the deductive process. These
pointers,
markers, and such, could be termed empirical seepage because they leak
into
this process. I emphasize here that it could be a subconscious leakage,
for
the thinker
might not even be consciously aware she is doing this. Some
examples of these markers and pointers include "flash insights", a
direct observation of something triggering a mental connection,
discovered patterns in a drawing, natural phenomena, or simply a
redirection in a thought process. With the infusion of such markers,
the
thinker doesn't strictly work top-down from premise to conclusion. She
probably begins from the top, but depending upon her success or
failure, might shift to the
bottom, then to the middle,
then back to the top, and so forth. This certainly would be expedited
if the thinker has a conception of the conclusion. I liken this to
trying to solve a word pyramid,
crossword
puzzle, or even taking an exam. One attempts to solve part
of it,
but might get stuck, so jumps to another part. If
the thinker is clever, she will be able to partition these real
world markers into separate classes, perhaps even forming relations
between
them. This establishes the basis of a multi-tier empirical process, the
1st
tier
being
the markers themselves, the 2nd being the classes, and maybe a 3rd if
we
consider
class relations as a separate tier. The main idea is that the proof
construction will be more effective when the thinker utilizes higher
tiers of marker classes. At this point, the thinker leaps
to the
deductive stage where the problem, idea, hypothesis, postulate, or
theorem
is
formulated. The intent is to utilize logic to draw inferences from the
premises
(moving from step to step),
and finally reach the conclusion. The proof is constructed using these
test markers, but most likely, it won't be successfully proven on the
first try.
This leads to an
interactive
phase in the process. The thinker will reformulate the theory such that
it will
encompass or pass through the markers in a meaningful way. This
reformulation
might be a
major
overhaul or it could be a series of simple tweaks. Perhaps the markers
will be
slightly repositioned to fit the model. This is where approximation
theory
could enter the
picture to explain how the process leads to a convergence towards the
valid conclusion. Objectivity, Logic, and Superposition of minds In the study of ontology, pioneered by the early Greek philosophers, all objects are considered real in that they truly exist as hardware in nature. This forms the basic of what is arguably objective reality; the "being" of something is tangible, observable, and verifiable by other "beings" existing in reality. In his development of formal logic, Aristotle used ontology to derive all types of categorial relations between objects and to form relevant classifications. The existence of an objective framework encapsulates these objects and their relationships. This "out there" reality was air tight until the advent of epistomology, the study of knowledge. Hume and Kant opened the door to the great interface, filter, and interpretor between man and the external world: the human mind. And the mind turns out to be an ultra-complex entity that could very well be playing tricks on us. We could seriously consider the proposition that logic is the projective symbolic interface of the human mind. What would make it "objective" are the shared projective mechanisms we all possess. The basis of this objectivity are the inner workings of the brain and its relationship to the human mind. In other words, to be human is to have the facilities to invoke a built-in logic mechanism. In its incubation, the existence of logic is subjective, for it namely exists in the human mind; however, the superposition of these minds forms what "we" perceive to be the objective world and logic is necessarily a part of it. The coherency of this superposition is proportional to the best-fit methods of overlapping mentalism, which requires the projective mechanisms just mentioned to assemble the clearest jigsaw puzzle with imperfectly cut pieces. The usage of logic could then be considered objective because the symbols and the rules for representing and manipulating objects with these symbols are shared amongst humans. It's a rationally accepted convention that's been programmed into collective existence. What I'm suggesting is that logic is not an "out there" reality, but more like seeds planted in human minds that have flourished more in some minds than others. It should be seen more as a dynamic and evolutionary development, a product of our intelligence that has congealed over time, rather than as a static and permanent apriori reality. Mathematics, Logic and Experience One
can not simple equate reality with experience because logic is included
in reality, but logic is fundamentally removed from the world of
experience.
Experience is
not an input into mathematical proofs (at least not in Pure
Mathematics), but logic and math are intimately related. Mathematics
uses logic, reason, deductive and (mathematically) inductive processes.
Mathematicians don't send out questionaires to people asking them about
their
personal experiences, their opinions, etc. They reference the above
mentioned
tools and the historical body of mathematical knowledge. It's certainly
true that Applied Mathematics
uses real world examples, problems and constructs. Equations are
modeled upon
these things, but even here, the rigors of the proof are confined to
non-experiential, non-empirical processes. To
help us visualize this situation, let's suppose there are three
entities: logic, math, and
experience. They're not mutually exclusive, but they are
distinct, recognizable entities. None of the three can be equated with
the
others; otherwise, we could eliminate at least one from the system. My
point is
that the the three are needed in a data flow diagram, finite or minimal
state machine,
minimal
spanning tree, entity relationship chart, or any other construct one
wishes to employ. Now,
there
are data flows between the three entities. I would argue that these
flows
can even
represent their overlapping regions. It depends upon whether the system
is
represented structurally or by event-driven prototypes. The flow from logic to experience is pretty strong, not quite as strong as those between logic and math, but stronger than the flow from experience to pure math. Logic is used to explain phenomena that occurs in reality. It also can help us to relate, order, filter, explain, and even forecast events that happen in the real world. However, the weakest of the flows is from experience to logic. Notice here that I recognize there’s a data flow; but I'm claiming that it's weak and sporatic. Logic and mathematics are like apples and oranges. Both are fruit, but they're different fruit. Furthermore, there are countless examples of abstract systems being applied to the real world, but we can't equate these systems with the world of experience. For example, there are branches of mathematics like group & set theory that are applied in Computer Science, and Lie Algebra describes symmetries found in quantum mechanics. There are countless other examples to consider; but each would sufficiently demonstrate that the abstract and real worlds are distinct and separate. Provability and Empiricism Applying logic to the visible universe (re: "real world") is very problematic. Ideally, we would like nothing better than to keep the problem relegated to the realms of deductive reasoning, mathematics and logic; however, scientific experiments can't be conducted without direct observation of physical phenomena. This is an integral part of the scientific method, which in fact is the marriage of reason and observation. Unfortunately, observation depends upon the senses and our mind, which experience has taught us can be unreliable. Let us consider an example of something "we know to be true" but we really can't prove: the fact that the earth revolves around the sun? Maybe our senses are playing tricks upon us. Or perhaps the equations of motion governing planetary movements are falsely being modeled upon an incorrect reference frame, and hence what is viewed as planetary movement of Earth is in fact movement of the Sun. Now, most people would ridicule these hypothetical examples. Everyone knows the Earth rotates around the Sun, right? But this heliocentric theory must be demonstrated empirically, and empiricism inherently requires the use of our senses to validate the theory. However, the limitations or misperceptions of our senses can mislead us; hence, the theory can be falsified. This is one of the quandaries of theories that fall into this category of knowledge: it can't be strictly proven, yet can be falsified. It's susceptible to being invalidated in two ways. (Pure) Mathematical knowledge, on the other hand, does not rely upon empiricism. This body of knowledge is not validated by the use of our senses. It is validated by logic and reason. True, the mind often deceives us as well, and we could present an argument that mathematical theories, even those that are proven, can in reality be at least partially false bodies of knowledge. Remember what Einstein said about mathematics and certainty: "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality". We can't say for certain that mathematical proofs are true just as we can't say that observational demonstrations (what I'm calling the process to validate empirical-based theories like heliocentrism) are certainly true. What can be said, however, is that mathematics does not depend upon empiricism while physics & science are dependent upon it. The Experience of Provability Let
us now turn to the subject of the socialization process of accepting
proofs as a source of
truth. This is definitely distinct from the internal validation of
those proofs. Yes, the public
will
examine the proofs, but whether or not they are examined, or even if
the authors formally
present their conclusions to a avidly listening audience, it will not
say anything about the
validity of
the proofs. It is independent of what people think; yet there can be no
doubt that
the
presentation of a proof, corroboration by independent mathematicians,
refinements, exchange of ideas among academicians, or even refutations
of the
proof can trigger a variety of other proofs, developments, and
investigations.
These experiences are valuable inputs into the overall endeavor of the
provability enterprise.
Sometimes, but
not too often, they even lead to refinements in the way theorems are
proved.
But these refinements will not lead to a refutation of a theorem
previously
shown to be valid. Both
scientific and mathematical statements can be proven to be true or
false. The falsification criterion for determining scientific validity,
the crux of Popper's
theory, is invoked to differentiate between what can and cannot be
proven. For example, the statement that "All people live in Asia" can
be proven false through a scientific investigation of where people
actually live. This is not to say this is necessarily a false
statement, but merely that it is inherently possible to use the
scientific method to show that it is false. A converse argument can be
established that all scientific statements are capable of being shown
to be true. "All people must live on Earth", for example can be shown
to be true. There might be some point in the future where some people
might live in space, and hence the statement will be disproven; but
that does not detract from the criterion that is being applied to
determine its truth or falsehood. As
already noted, science and mathematics differ in their methods of
provability. In a nutshell, science embraces the observation of the
real world, while pure mathematics is insulated from it. As an
extension of this fundamental difference, scientific proofs are
dependent upon the introduction of observable, real-world facts, while
mathematical ones are based upon logical statements, inferences, and
deductions. While both types can be socialized in one respect or the
other, it is the scientific ones that provide the greatest variability
and uncertainty in stating their conclusions. In great measure, this is
due to another type of falsification that can't be introduced into
mathematical theorems, but most certainly can be injected into
scientific investigations: the introduction of fake evidence. If there
is a
scientific claim that ghosts are real, and it is based upon the
observation of phantom objects, the positivist scientific method can be
utilized to prove or disprove its veracity. The scientist will use
measuring instruments, optical equipment, listening devices, and even
her direct senses to determine if the hypothesis is true. The fact that
"ghosts" can be holographically projected into a dark area is an idea
that the scientist will consider, and hence search out, but such
holographic technology might be "out of view" of the scientist's
experimental radius. Indeed, the tangibility of those "ghosts" might be
so craftily constructed from the utilization of such technology, that
the scientist might reach out and feel them directly, and conclude the
ghosts in fact are real. Of course the best scientists will be more
clever in their approaches, and most likely will disprove the
hypothesis. But it is the inherent nature of scientific investigation
that what is sought to be proven or disproven can be falsified through
various means, i.e. faked one way or the other. If it wasn't possible
to
fake the evidence, or generally falsify the observables along these
lines, then the hypothesis would be self-evident and there would be no
reason for conducting any scientific experiments. Creative Thinking and Mathematics In
addition to utilizing pure logic, mathematicians
use creative thinking, visualization, thought experiments, and even
conceptual
leaps of faith to formulate and present their theories. I've already
alluded to arguably the best example to illustrate this article of
faith: Einstein's thought experiments that were instrumental in
formulating his Theory of Relativity. In addition, there is an
antecedent to it that is most noteworthy, and that is the advent of
non-Euclidean geometry. These
constructs didn’t completely invalidate Euclidean proofs, but
limited them in scope and domain. For example, because Euclid confined
his
proofs to two dimensional surfaces, the sum of the interior angles in a
triangle always equals 180 degrees. However, in non-Euclidean geometry,
surfaces are never confined to 2-D spaces, but 3-D or higher spaces, so
the sum
never equals 180. In fact, on a hyperbolic surface, the sum can equal
270 (On a
sphere, start at the north pole, form a right angle at the pole, and
extend the
lines down to the equator).. The
creative visualization of curved space superceded the flat space
concept upon which Euclidean geometry was constructed. This allowed for
the
propagation and eventual proof of a non-sensical idea that an
infinite number of parallel lines pass through a singular point (see hyperbolic
geometry). Creative thinking played an instrumental role in the
development of these theories (* see below). Reason, Experience, and
System Theory There
can be no question that reason and experience must form a lasting
partnership because while they're not mutually dependent upon each
other, the interrelationship between the two enhances the abilities of
each partner
and the quality of life in general. Reason can dispatch to experience
the most rational choices available in any given situation,
and can order them accordingly, while experience can present reason
with choices it has made in the past, and thus give reason a narrower
range of choices to analyze in the future. The primary tools of reason,
logic and a priori knowledge, converge with the fundamental instruments
of experience, empiricism and observation, to form a binding union in
the greatest of all modern pursuits, science. The scientific
methodology gainfully employs all of these wonderful labors in the
cause of discovering the true inner workings of nature. Reason
and Experience also fit in nicely with general system theory, and
specifically with Complex Adaptive Systems (I will write another paper
on this sometime in the near future). The basic relationship is that as
reason refines the choices and their prioritization for experience to
pursue, experience provides feedback to reason that alters the range of
choices
and perhaps the objectives for reason to consider. As this process
repeats iteratively over a prolonged period of time, it will achieve a
regular state of stability in performing these designated operations,
with a greater stability being achieved with each procedural iteration.
This will occur even as errors are made, for there will either be a
self-correcting or programmed error-correcting mechanism in the system.
From this functional stability emerges a process of learning and
refinement that will improve the efficiency of the system: both reason
and experience will become sharper in their mutual decision making
process. As
experience guides reason in its pursuits, reason will no longer solely
need to
depend upon a priori knowledge to reach its conclusions, for the system
does not operate in a purely deductive process. It certainly will
imploy a priori knowledge, but a posteriori knowledge will also be incorporated in a feedback
loop process to help refine reason's investigation of available
choices. Conversely, a posteriori knowledge won't solely
depend
upon experience as its "great teacher", for a priori knowledge will be utilized
to lead to a more rationally useful set of a posteriori mandates. This article has focused upon the interconnectivity between the worlds of reason and experience. While the methods of logic and empiricism are distinct, they cross paths and perhaps cross-pollinate in a process of inquiry and formulation. I have also attempted to take a creative look at logic itself. While I'm contending that logic could have subjectivist origins, this is not to say that it is relativistic. In no way should this be viewed as embracing any Post-Modernist jibberish. I thoroughly reject such philosophical narcissism. The shared experience of logic and its applications to an object oriented world certainly would put it closer to the realm of objectivity than subjectivity, and most definitely out of the reach of any alleged solypsism. Originally Written between April 24, 2000 and May 17, 2000 Updated in October 2005. Updated "The Experience of Provability" on 10/19/10. * An great description of the interplay between creativity and mathematics is found on page 185 of the book "On Intelligence" by Jeff Hawkins (Times Books 2004): "We do however, believe we are being creative when our memory-prediction system operates at a higer level of abstraction, when it makes uncommon predictions, using uncommon analogies. For example, most people would agree that a mathematician who proves a difficult conjecture is being creative. But let's take a close look at what's involved with her mental efforts. Our mathematician stares hard at an equation and says, 'How am I going to tackle this problem?' If the answer isn't readily obvious she may rearrange the equation. By writing it down in a different fashion, she can look at the same problem from a different perspective. She stares some more. Suddenly she sees a part of the equation that looks familiar. She thinks, 'Oh, I recognize this. There's a structure to this equation that is similar to the structure of another equation I worked on several years ago.' She then makes a prediction by analogy. 'Maybe I can solve this new equation using the same techniques I used successfully on the old equation.' She is able to solve the problem by analogy to a previously learned problem. It is a creative act." |