The Fusion of Reason and Experience
Copyright © 2005, David A. Epstein.
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.

One example that comes to mind is Einstein's Special Theory of Relativity. This was not formulated through a pure deductive process, although most of it certainly was deductive. He started out with some ingenious thought and visualization experiments. He used real world examples: moving trains, falling elevators, and light interacting with these bodies. These are the markers I was talking about. Of course he had the trains and elevators sometimes moving at incredibly fast speeds, but nevertheless, the foundation of these experiences was rooted in the real world. After this, he utilized the mathematical ideas & equations of Maxwell, Riemann, Mach, Lorenz, and others.

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.

Now, I'm proposing a theory where logic is the symbolic interface of the mind. In a sense, it would be the evolutionary product of the mind, and hence would have a subjectivist origin (the 'objectivity' would result from the exchange of ideas, symbol manipulations, shared interface of this logic). The world of experience is also, in part anyway, a product of the mind or our entire being. So in this theory, logic and experience, more or less, have a common origin. Furthermore, logic is used in the world of experience to help us make better decisions in life. Conversely, experience helps spur on logicians and mathematicians to formulate, evaluate, refine, and/or complete their works. Experience can even influence the development of logic itself. However, when proving something, at that point, logic is frozen in time, crystallized, or quickened. It's divorced from the world of experience.

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.

Logic and math certainly have a bidirectional data flow relationship, since logic is used in math and mathematical equations appear in logic (particularly symbolic logic); they are strongly related. Think of them as gushing streams of water from a fire hydrant. Experience and math have weaker bidirectional data flows, maybe comparable to the water spraying from a home sprinkler. Experience helps us to model systems, derive equations, or expedite a proof, but it doesn't contribute to the internal validation of a proof, whereas, logic makes a vital contribution. Now, if it's applied math were considering, then the data flow between math and experience is obviously stronger in the here and now, but we know that the pure mathematical ideas of today can become the applied math realities of tomorrow. So this model becomes more complex when we made a distinction between pure and applied math.

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.

There is no question that there must be a separation between the processes of external socialization and the internal validation of proofs. Social acceptance is an evolutionary process (hopefully evolving to some stable point) while the validation of proofs is an archeological one (new discoveries can be dug up, but they were there to be discovered). Admittedly, this analogy has flaws, for it is egregious to strictly think of new deductive processes, syllogisms, etc, to be non-evolving fossils. But I think this separation is a good starting point; procedurally, it's a potent first step. After this, we can commence with the creative mixing of the elements. For example, color separation is a prerequisite for painting, making textile dyes, computer graphics (i.e.for rasterization).

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.

For about two thousand years, Euclidean geometry was the most established branch of mathematics. Such concepts as the Parallel Postulate and the Pythagorean Theorem were carved in stone. Yet in the 19th century, there were revolutionary changes as flat space assumptions were abandoned.  The non-Euclidean geometries of Lobachevsky, Bolyai, Gauss, Riemann, and Minkowski were developed during this period. They were precedents for Einstein’s General Theory of Relativity. In other words, the theoretical constructs of these new geometries, such as non-Euclidean curved space and the metric tensor, provided the basis for facets of Einstein’s theory. It was the rational and creative minds of these individuals, and not new physical observations or empirical evidence, that challenged the ancient wisdom. They were able to visualize the theorems or their manifestations and independent observations confirmed, more or less, what was conceived in their minds.

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.

Conclusion

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."