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Rethinking the Philosophy of Mathematics

The Mathematical Dialogues of a Visual and Logical Thinker

By Roy D. Follendore III

January 11, 2004

 

A Logical Thinker and a Visual Thinker arrive at an ancient university to discuss their differing perceptions of the essence of the simple mathematical formula Rate (R) * Time (T) equals distance.     


Visual Thinker: Please express the algorithm and then the explicit example we are to be discussing.

Logical Thinker: Certainly: The formula is R*T=D.  A car that is traveling at velocity of 50 miles per hour will be 50 miles away in an hour. 

Visual Thinker: Unless a bridge in it's path is out, in which case the 50 miles would be the maximum distance the car could have traveled.

Logical Thinker: True, but this was not prescribed within my algorithm. What is not described does not exist.

Visual Thinker: Then your algorithm can not be faithful to a universal reality of unbounding consequence.

Logical Thinker: I maintain that my algorithm is the very essence of reality and that it is unburdened by unnecessary consequences.

Visual Thinker: Then you must be describing a different universe from the one in which we now exist at this moment, because consequence is a constant thermodynamic fact of nature, a physical fact supportable by mathematical logic.

Logical Thinker: I have simply isolated the potential consequences and reduced the issue of velocity to its basic constraints.

Visual Thinker: But what you are now saying is untrue because a potential consequence is also a basic constraint.

Logical Thinker: No, you are wrong because no other constraints exist.

Visual Thinker:  Then you must have described the situation incorrectly because the car could not possibly be traveling.

Logical Thinker: What do you mean?

Visual Thinker: You should have said: A car that was traveling at a velocity of 50 miles per hour was 50 miles away.

Logical Thinker: That of course could be true too but it would also be completely ridiculous.

Visual Thinker:  How so?

Logical Thinker: The model that I gave was intended as a means to both predict and state the distance of the car based on the velocity.

Visual Thinker: That was exactly my point.  Predictions and statements are not congruent with respect to specifications of time. You were describing the potential of the distance the car in an hour based on the potential of its velocity, which is a prediction.  Predictions always involve the potential of unknown consequences.

Logical Thinker: So in your mind my example is defining an indefinite probability even though I have intended to explicitly state all of the probable constraints.

Visual Thinker: Yes.

Logical Thinker: Then would it have made any difference in your mind if I had explicitly stated the probability of the cars velocity, time and distance as 100%?

Visual Thinker: Not if you are attempting to describe the physical universe as human beings living in coexistence with its consequences must understand it. It is only logical that placing a constraint on any constraint remains a constraint. There can be no way to avoid the consequences of limitations placed on limits with respect to probability and the fact that doing so is incongruent with the necessity of sustaining rational thought within the nondeterministic natural aspects of our universe.

Logical Thinker: Then what you are saying is that the mathematical logic of reductionism is self limiting and becomes the impediment to the tangible reality of the model that I proscribed?

Visual Thinker: Exactly. By reducing the visually dynamic concepts of distance to the relationship of velocity and time, described your algorithmic model more than your objective. You have visually disassociated your objective from realistic consequences as a fantasy.

Logical Thinker: I think I see a glimpse of your point, though to me it does not mean that the formula "rate times time equals distance" is wrong.

Visual Thinker: There is of course also a visual beauty and truth to the act of simplification.

Logical Thinker: Then you are backing off from what you have just said?

Visual Thinker: Not at all... In my mind the act of reduction is an act of velocity with potential consequences just as much as the example of the car that you gave. To me, the algorithmic expressions of mathematics themselves are simply that, essentially recursions of dynamic forms. When the form of the algorithm gets in the way, the consequences of the dynamic are lost.

Logical Thinker: But I must then return to my original argument that the consequences of my model would simply be that the car is fifty miles away after one hour.     

Visual Thinker: And I would have to argue that through your algorithm you are only representing the positive contours of the problem you intellectually circumscribing.

Logical Thinker: Then I must judge your way thinking as being too interpretive.

Visual Thinker: And I must do the same for you way of thinking; and on that we may of course agree.*

 

*Note: We can see that the Logical Thinker is attempting to interpret reality into a timeless predictive model which is reduced without the variability of nondeterministic constraints.  The Visual Thinker has a mind that is intuitively repulsed by the limitations of this functional algorithmic argument because doing so is instantly detected and recognized for what it is, an unattainable constraint. Mathematical congruence is therefore essentially rationally incongruent to the Visual Thinker and is therefore false and cognitively rejected as unmeaningful.  On the other hand, the Logical Thinker either has the ability to overlook this incongruence or the inability to see the incongruence, depending on the perspective. Perhaps more to the point is the overall issue that the very act of even being able to recognize and intelligently discuss the implications of this issue requires a certain degree of visual thinking, which is probably the reason why it has been so overlooked for so long in institutions of mathematical academia.   

 

The next day the Logical and Visual thinkers return to their discussion.  

Logical Thinker: I find that I am disturbed by the idea that your way of thinking is so philosophically different from mine.

Visual Thinker: What is it that disturbs you most?  Is it the fact that two philosophies coexist or that they should not coexist.

Logical Thinker: See...Your questions are part of the philosophical problem. Within mathematical logic exclusion is an absolute statement of existence. On should not be able to simultaneously exclude what is included or include what is excluded. 

Visual Thinker: Only in the fantasy of mathematical logic.

Logical Thinker: Do not provoke me by calling my discipline a fantasy.

Visual Thinker: I did not intend to do so.  I was simply stating a obvious logical philosophical conclusion, particularly if mathematical logic is to be considered absolute.

Logical Thinker: Then you somehow agree that inclusionary and exclusionary statements must not simultaneously coexist.  

Visual Thinker: Not at all.  However I do agree that the ability to construct a perfectly sound philosophical argument in one respect weakens the potential of that argument in another.   

Logical Thinker: Then it is really the degree rather than the specification of mathematical logic that you disagree with?

Visual Thinker: No. My thinking about logic is not explicit, it is implicit.  No statement in mathematics can be absolute outside of the fantasy universe within which it must conform. 

Logical Thinker: There you go again.  I must take offence at the use of the word "fantasy" with respect to the discipline of mathematics.

Visual Thinker: Then in the interest of our communication I must retract the word or redefine it.  It is the relationship between the image that we know of reality that surrounds us and the construct of axioms and postulates that have been developed over time that leads me to the offending term. 

Logical Thinker: Then the "logical model" of the universe would seem to be more acceptable to us both. Is it therefore the rigidity of the logical model of mathematics that you object?

Visual Thinker: Yes. That is certainly a part of it.  But it is really the translation from the flexibility of the physical universe to the explicit rigidity of mathematical logic that weakens mathematical philosophy in my way of thinking.

Logical Thinker: But you are ignoring the fact that this same foundation of "rigid" logic also serves us by allowing a means of repeatable understanding, even with regard to such issues as chaotic dynamics.

Visual Thinker: I agree, but at what cost? The cost of logical rigidity is the loss of implicitness.  For whatever justification, the fact remains that at the informational level, when complexity is restrained, knowledge is lost.  When information is lost, knowledge is lost and the potential for error is induced.

Logical Thinker: Then you are implying that the apriority acceptance of a logical resolution within mathematics involves conformity and that changes the potential for error to solutions to problems being considered?

Visual Thinker: I am actually going one step further than that.  I am also relating the idea that by completely accepting the premise of mathematics in advance, we change our perceptions of our rational universe to fit our perceptions of rigid logic.

Logical Thinker: Because in our modern world it benefits us.  Then the problem is that human beings are acting irrational.

Visual Thinker: Yes.  But more than that, the real underlying problem is that mathematical logic is fundamentally irrational.

Logical Thinker: I assume that you are not talking about rational and irrational numbers?

Visual Thinker: Of course not.  I am speaking about the concrete idea that the rational act of rigidly defining explicit constraints through axioms and postulates places limits on the kind of solutions that are possible.

Logical Thinker: Sounds a lot like the work of Kurt Gödel, who was of course the famous mathematician who created the formal proof that mathematics cannot solve all problems.  

Visual Thinker: I suppose it does.  But I am also saying that the kinds of problems that we model using mathematics can result in invalid solutions which we may tend to accept for no other reason than because the explicit mathematical solution may be mathematically correct.

Logical Thinker: So what is wrong with that?

Visual Thinker: Our physical universe does not work the same way as the mathematical universe that we have constructed through logic. What is wrong is the rational reality of mathematics.  When translated to physical terms, ALL numbers must be considered as approximations.   

Logical Thinker: Wait a second! When we define a unit value within the physical universe, it is that value simply because we say it is and that of course conforms to mathematical logic. 

Visual Thinker: What you are saying is true only in the sense of mathematical logic. There is another truth and it is that alls statements made by man are merely informational pointers. To state that a thing is something is not to say that it is not also something else or that the thing is not constantly changing.  

Logical Thinker: But that is where we definitely disagree.  It is completely irrational in my mind to state that something is and is not, exists and does not exist.  What are you getting at?   

Visual Thinker: Simply that it means that unlike the mathematical universe, within our physical universe the reality is that the whole is ALWAYS more or less than the sum of its parts.  In fact, the probably of an explicit numerical value of a thing being what we say it is, is infinitely low.  There has never been a person who is six foot tall.  In that respect one might be tempted to say that no one has ever really worked an hour in their life. Ultimately it comes down to the fact that the physical manifestation of the number line changes because the dimensions of our physical universe change.

Logical Thinker: I am sorry but I find your observations not only irrational, they are also impractical and useless.  Even assuming that what you say is correct, to a large extent the objective of mathematics rests on the essence of its practicality.  The physical manifestation of a number in our universe immaterial if we are able to use mathematical logic for practical means.  Man negotiates agreements as to the translation of numerical value.

Visual Thinker: Exactly my point.  The rigidity of numerical logic is different from that of rational logic within our physical universe, which must be negotiated.  It is the nature of logic that some degree of tolerance must exist even to negotiate absolute rigidity of specification. 

Logical Thinker: True.  Within mathematics there certainly must be agreement to specify correct logic. I can certainly accept the core idea that the essence of rationality for mathematical logic must therefore exist within the function rather than the explicit value.

Visual Thinker: Then you must also accept that fundamentally different functions may express and represent the same aspects of intellectual integrity, even if one may appear to be more rigorous than another.

Logical Thinker: It is possible, though I suspect that the degree to which I might place my faith on the importance of rigor would be quite different than yours.  Certainly within my frame of thinking, the consequences of the function two plus two equaling more or less than four are enormous.        

Visual Thinker:  An excellent example. And yet in the totality the mathematical framework of the function you have just described is both more and less than four.  Limits to functions coexist.

Logical Thinker: How so?

Visual Thinker: For you, the first order product of two plus two is obvious, but visually the first order implication of two is the relationship of two ones, which is one. That means that the first order "implication" of both twos is one and two. But remaining in a visual sense, when the first order product of two plus two is expressed as four, its first order implication is also the potential relationship of one, two, three, four, implying eighteen to infinity and its relationship of course to zero.

Logical Thinker: Wow what a hodge-podge of undisciplined logic.  For me the function two plus two is self-contained. it is automatic. It is a balanced function to which there are limits.

Visual Thinker: Do not underestimate the true nature of discipline because there are many ways to achieve it. In my way of thinking, the limits you formally describe limit solutions and perhaps more importantly they do not describe the actual perception of the universe. The function two plus two does equal four, but it also equals many other potential relationships that can be visualized.

Logical Thinker: Regardless, it sounds confusing.

Visual Thinker: From my perspective it certainly can be but my mind tends to work that way. The way that I attempt to think about these concepts is both holistic and intuitive.  The advantage to my approach is the fact that it allows me to think outside of the proscription of standard mathematics so that I can inductively relate a variety of potentially critical assumptions simultaneously. From my perspective that makes it far easier to see the general nature of functions that have the same purpose. The disadvantages of course are that because of the mechanics of the way that I tend to think, I find it too difficult to manage explicit complex rules governing mathematical calculation and I have observed that mathematicians tend to choose problems that can be solved.

Logical Thinker:  Mathematicians do work simultaneously on problems that may or may not be solved.  The so-called theory of everything otherwise known as string theory reached the same solution through five independent and different conceptually mathematical models. Dr. Edward Witten proposed through his M-theory that all five approaches are part of the same concept.  You are therefore in error if you are implying that mathematical logic prevents one from operating simultaneously on several perspectives. 

Visual Thinker: I did not say that and nor did I mean to imply that is the case.  What I meant to say was that mathematical logic guides one to certain kinds of conclusions and is therefore self limiting in the kinds of approaches to solutions that can easily be confronted. Rigor and insight are not mutually exclusive, but neither are they the same thing.  Mathematical functions go a step further than understanding problems by modeling them through logic. In practical terms, many relatively simple ideas that can be easily visualized become quite difficult to mathematically model.  Simple ideas are difficult to arrive at through a mathematical model.             

Logical Thinker: Perhaps the fundamental approach to logic that you are ultimately describing is similar to the way that Leonardo Pisano Fibonacci described number theory. 

Visual Thinker: Maybe so, Fibonacci was ultimately mathematically rigorous in his visualization of patterns. The underlying justification for my approach is that without the vision there could be no rigor.  To me that means that a combination of differently oriented methodologies can be a better approach in research.   

Logical Thinker: I see.  But Fibonacci was dealing with numbers. 

Visual Thinker: But he was fundamentally dealing with patterns of numbers. At some point the instinctual cognitive aspect of logic may be inseparable from mathematical logic.  At other points along the path it is.  The point is that through logic we must tell mathematical algorithms what it can and cannot produce and then we must begin to interpret.  In order to commit to intellectual progress we must begin to infer.  Creative inference is fundamentally a cognitive problem, not a logical or algorithmic one.

Logical Thinker: But the fact remains that the kind of vision that you are describing does not have the clarity or the elegance that mathematical logic can produce through algorithms.

Visual Thinker:  Let us agree that there can no doubt that mathematics is capable of producing elegant products, and far more often than just often of if reductionism is to be the primary definition of elegant. The issue of clarity is a completely different matter.  Einstein's famous equation consists of just three variables expressing the relationships between energy, mass and the velocity of light.  

Logical Thinker: That is no fault of mathematics.  It is the scope of the subject.    

Visual Thinker: More to the point, on one hand it is an easy functional relationship to understand.  On the other hand, it is also extraordinarily difficult to conceptually embrace without a visual model.  My point is that reductionism does not in and of itself produce clarity and it may easily in fact decrease it. Once we step into the threshold of describing logic as a form of elegance and clarity we are beginning to express mathematics in terms of esthetics and the uncertainty of perception.

Logical Thinker: You have a point, though I don't know that I would it exactly that way.  Since we are now openly discussing the underlying aspects of critical thinking concerning mathematical logic, I must ask the more general question of what it is that mathematicians can do to improve our discipline.  

Visual Thinker: I believe that is appropriate in every field of endeavor for the cost of not doing so is enormous.  The first and most important change to consider is the very nature of mathematics as we have come to understand it.  

Logical Thinker:  What exactly do you mean by this?

Visual Thinker: The esthetics of mathematics is of critical importance to its progress but the idea that it is absolute criteria of excellent mathematics is incorrect.  A greater appreciation for messy math is necessary for many of the same reasons that artists like Picasso and Jackson Pollock have been so important to the fine arts. The act of doing mathematics exists within the expression of logic and the footprint of that act is the functional product. There is the craft and there is the art, which are not necessarily the same things.

Logical Thinker: Then you are suggesting that we abandon the notion that the greatest importance of mathematics is functional.

Visual Thinker: I did not say that.  I am simply asserting the point that the importance of a crafted product is not only what it does but I am also suggesting where all of this takes us and what it makes of us.  Within the field of mathematics there is not nearly enough credit given to those who deliver opportunities.  This is also the result of the social aspect of the discipline that has become and always was a part of mathematics.

Logical Thinker: Why do you believe this is so? 

Visual Thinker: From my perspective mathematicians do not seem to readily deal with such issues because they see themselves as cloaked in the mysterious functional value of the work that they do.  Mathematics is so vast that it is easy to become a specialist.  The field is strewn with the debris of countless unknown mathematicians who reached localized pinnacles because they were a challenge and abandoned them.

Logical Thinker: You are making my discipline seem haphazard.  I would argue that some mathematical problems are easier than others to contribute to, while the constant advantages of the solutions that are possible are not well understood until they are associated with definable variables.

Visual Thinker: And I would argue that many mathematical problems are more readily sought after than others because they are easier than others to contribute to, while the advantages of the solutions that are possible are not well understood until they are associated with variables that are appreciated.            

Logical Thinker: In what ways?

Visual Thinker: Many of the most fundamental questions of mathematics remain without a rational foundation of understanding.

Logical Thinker: Can you give me an example?

Visual Thinker: Off the top of my head, I can give you several examples, particularly those questions with respect to mathematical philosophy.  The first of these fundamental questions deal with the origins of mathematics as logic.  From where do numbers arise? Why does the creation of dimensions occur in relation to the number line? What cognitive complexities arise between the definable nature of symbols and variables? Where does probability begin and how should it be expressed within the foundations of mathematical philosophy? 

Logical Thinker: These kinds of questions do not deal with the function of mathematics.  It is not necessary to understand everything in order to function appropriately. 

Visual Thinker: These kinds of questions can not be answered through mathematical logic. They must be arrived at through reasoning and therefore represent reasoning through inconsistencies and visual perspective. Mathematical formulas do not explain because they can't, they imply explanations. Logic can not explain itself without error. It is therefore a subjective discipline. 

Logical Thinker: But in fact many of these questions have been answered for thousands of years.  Most mathematicians feel that the discipline of mathematics is past that. Why should we mathematicians put forth the massive effort to ask these things again when we already know the answers?      

Visual Thinker: When you begin to analyze the totality of the issues, there are at least two answers to this question you are asking.

The first reason is because the truth is that we really do not know the answers to these kinds of questions. Mathematics operates on postulations of philosophical truth.  

The second reason is because these are modern problems that must be reinterpreted and resolved with respect to the knowledge we now know to be true.   Reason tells us that weaknesses in a theoretical foundation give rise to ambiguous errors in the pathology of solutions that arise. Some of these may corrupt the calculated result; others may result in systemic corruption. 

But there are also other larger questions that need to be asked in relation to the answers that we will gain.  How should we optimize the probability of reaching optimal mathematical solutions? How do we recognize the similarities and differences with respect to the logic different mathematical algorithms and functions? How do we, as well as how should we deal with the relationships between logic and illogic, rationality and irrationality within mathematics? 

Logical Thinker: Obviously "How do we" and "How should we" are often very different questions of logic.  Most mathematicians already deal with such issues through "how can we."  Aren’t the problems of modern mathematical logic difficult enough? Hasn't rapid progress in mathematics been good enough? Once again, my question stands.  Why should we mathematicians put forth the extra effort? 

Visual Thinker: Because progress in understanding mathematics is more of the matter of a question than a statement.  The most fundamental questions cannot rise to the surface if procedures are expressed as statements. But at an even deeper level, because the ultimate question that we must ask of ourselves about mathematics goes to the very heart of the nature of thought.  What is what is the optimum cognitive process for considering different kinds of mathematical logic? Are human minds the true engines of all forms of logic.

Logical Thinker:  Why do you believe that these questions are so important? 

Visual Thinker: Because modern problems are becoming more cataclysmic, presenting humanity with the fact that it is not nearly enough for us to solve equations.  Humanity must also absorb and understand them.  We must discover solutions that allow us to keep up or else allow mathematics to continue on the path of alienation. It stands to reason that we must therefore first ask the kinds of questions that would allow us to resolve the nature of mathematics.

Logical Thinker:  ...And by doing this you are saying that we can maintain a state of parity with mathematics? 

Visual Thinker: Not exactly.  Humanity constantly increases its technical ability to calculate more complex algorithms while individually we fall far behind in our ability to understand what it is that we are calculating.  We know how to calculate.  We don't understand how to use the calculations we produce. It is the analysis of mathematics rather than the calculation of mathematics that have become so difficult.  As mathematicians we are a primitive lot, where the possibility of tools solutions surround us, but we just can't recognize them because we don't know that we must reach out and make those solutions a part of us. 

Logical Thinker: What you appear to be saying seems to be a condemnation of the state of mathematics.

Visual Thinker: I suppose you could take it that way.  On the other hand you could also take it as an observation that the potential products of mathematics are about to accelerate the potential of humanity beyond our present imagination.  

Logical Thinker: You are predicting an explosion of mathematical productivity?

Visual Thinker: I am saying that if you consider humanity's ability to calculate in terms of a reserve of potential progress, the spark that ignites that reserve could accelerate the progress of humanity in the direction of our choosing.  By analogy we only have to look at the history of the space shuttle program to understand the implications of what I am saying.  When you think about it, the fundamental difference between a rocket and a bomb is that the rocket is designed to have a directed output that quickly transports cargo without damaging it.

Logical Thinker: You are using the analogy to differentiate the differences between the potential destructive and constructive nature of mathematics  

Visual Thinker: I am also using the analogy to state that it is not enough that an expert understands that there will be the potential of a tremendous burst of energy from a reserve. To be constructive that burst of energy must be directed in context with the overall systemic objectives. To put it a different way, there are limited constructive uses for a nuclear bomb but there there are an enormous number of uses for directed power that comes from a process of controlled nuclear fission.  I am saying that the understanding that is derived for the control of the products of mathematics determines its potential benefits to humanity and we are barely scratching the surface on that kind of mathematical understanding?

Logical Thinker: Others in the field of mathematics might obviously disagree.  But why do you feel that that is that humanity is scratching the surface?

Visual Thinker: Once again, I think that the underlying reason is because we don't seem to be able to understand the complete underlying nature of mathematics.  Part of that is probably because of the way that mathematics has been institutionalized.  Educational institutions tend to try to manufacture mathematicians as though they are interchangeable parts. The part that I am talking about most is the fact that the nature of mathematics as a discipline is an artifact of the necessity vs. opportunity.  Institutions have never actually defined mathematical curriculum that fits the kinds of minds that can and should deal with all of the discarded opportunities that go with mathematical unification.      

Logical Thinker: Then what you are implying is that mathematics should not be unified?  If so that would go against the potential to manage the very productive outcomes that you have just been so adamant. To me it sounds like a contradiction in your philosophy.

Visual Thinker: I don't think it is when you look at the entirety.   Along the path of intellectual achievement humanity ultimately chooses both process as well as the symbols to achieve effective and efficient ends.  Mathematics is made up of those processes and symbols.  But what changes along the path is the nature as well as the scope, and degree of intellectual achievement.  Humanity simply did not need certain forms of mathematics until we needed them.  The modern problem is really that we would not know that we need them. The implication is the tremendous waste of opportunity to catch catastrophic failures that would otherwise sustain humanity as well as it is an implication that we are wasting our opportunities to advance humanity.

Logical Thinker: What kinds of catastrophes and advances are you referring to? 

Visual Thinker: This is a massive multidimensional set of problems.  There are social and cultural catastrophes as well as those that are technical and have an economic basis.  These kinds of problems are tricky because they are often recursive as well as interconnected and that becomes a computational research issue. We still do not know how to do that kind of statistical work. What we tend to do instead is use the lowest conventional statistical approach arrived at by calculating the average and mean.  Statistics can quickly devolve to become pointless and wasted efforts.    

Logical Thinker: How does your way of thinking differ from my existing approaches?

Visual Thinker: Organizations of people like to throw hardware at problems like these, probably because they promise instant gratification with predictable results. Many of the most difficult problems that humanity must and will face, involve the problem of how we should best approach the logic of mathematics with respect to the organization, not the computational problem itself.  We do not have good tools and standardization methodologies for assuring the potential best solution to such problems. 

Logical Thinker: Humanity's mathematical progress is both the result of, as well as the basis of, our world’s current level of technological growth.  

Visual Thinker: That growth is defined by our success, especially with respect to these kinds of issues that we have been discussing.  It is inevitable that civilization will find such a problem that will spell disaster for humanity if we do not effectively and efficiently solve them within an increasingly indefinable time period.  Many of these problems aggregate and compound with respect to each other so that some may actually have come into existence because we have not created the necessary logical tools of inclusion for our social intellect.

 http://www.cut-the-knot.org/hall.shtml
 

Logical Thinker: I have studied the problem as you asked.  The logic of probability simply outweighs the typical "natural" perspective. I see no true mathematical paradox.

Visual Thinker: My reason for asking you to review this particular problem is because, particularly within the field of security and cryptography, this is in fact a most important and unusual mathematical/psychological paradox. 

Logical Thinker: Psychological is not mathematical. 

Visual Thinker: Oh but I argue that it is. This issue is the thread of a fundamental paradox because it goes to the root of most of the security and engineering problems humanity must face. If security is to be rational and practical then people must be able to fit the idea of probability into their daily frame of reference.  More than that, It is the fundamental function of mathematics to assume that perfection exists; that probability exists in the physical universe and can be calculated prior to the result. Another part of the mathematical philosophy is that of neutrality of the mathematical perspective; that this expectation of what I call a neutral "third eye" of mathematics is unbiased.  

Logical Thinker: What is this "third eye"?

Visual Thinker: The idea that perspective plays a role in the existence of mathematics.  To understand that we have to go back to basics. There is the perspective of, say, the number "1".  Whose perspective is that?

Logical Thinker: It is yours.  You just said it.

Visual Thinker: Actually there are many perspectives.  There is my concept of the number "1". There is your concept of the number "1". There is also "our" concept of the number "1".  Our concept is the differences and the similarities of our individual understanding.  But there is also a third eye involved.  The eye that I am referring to is the optimal or absolute view. This is the view that represents the absolute abstraction of the number "1".  I sometimes think of it as "Gods" view because it allows us to simultaneously see many kinds of perspectives.  By assuming omnipotent simultaneous viewpoints we achieve mathematical empathy; we solve problems that we could not otherwise solve.

Logical Thinker: You are saying that this is involved in the Monty Hall problem? 

Visual Thinker: Yes. The underlying problem within the Monty Hall paradox exists the underlying fact that within the specification there are simultaneous mathematical perspectives taking place. The communication of information must be interpreted. When and if the contestant allows their guess be known then Mr. Hall is given information that could be used to bias the decision of the outcome of the problem.  If Monty Hall also knows the location of the prize in advance then the act of opening a door with a goat could represent a neutral, a benevolent or a malevolent transaction of information. In other words, Monty could attempt to bias the contestant to either switch or stay put with their decision. It is not enough to specify what Monty knows; we must also assume Monty's intention.

 

Logical Thinker: Within the specifications have reduced the problem.  Within mathematics it is important that we specify our assumptions as statements, even if they are probabilities. These are acts that have taken place.  Monty's intentions have nothing more to do with it.  

 

Visual Thinker: Oh but it does.  The reason why has its roots in the nature of the prisoners dilemma.  As I am sure you remember, the perception of the prisoners and the interrogator becomes critical to the probability of the outcome. Ignorance of this area of mathematics is also part of the dynamic of the expectations of the outcome. The mindset of participating actors who are not godly observers of the physical universe plays a critical part in the potential distribution of probability.  Knowledge of human perception on the part of Mr. Hall and the contestant plays a part in the probability of the outcome.

Logical Thinker: You are saying that the the past affects the future, even after the action has occurred and the mathematical function has been specified?

Visual Thinker:   Yes. This is an essential perspective of the third eye that we were discussing.  The failure of mathematical formula is the inability to express intent as something other than a "neutral" perspective; a neutrality that may not have truly existed.  This implies that perhaps our "Gods" view is not really so godly after all.  

Logical Thinker: I hate it when you do that. You specify something and then flip.  You are being too inconsistent. Try boiling down what you are saying into something that is logical. 

 
Visual Thinker: The "neutral" mathematical assumption taking place is that a zero sum game exists and that the relationship and actions taking place between the contestant and Monty Hall has been adversarial. What is so interesting about this from a communications behaviorists perspective is that such a posture illuminates the phenomena that the concept neutrality within mathematics isn't psychologically neutral.  It calls into question the whole idea of mathematical neutrality and in particular its utility with respect to the properties of translating probability into physical reality. Part of the paradox for me includes the fact that that in a world of intent, useful mathematical logic exists at all.

Logical Thinker: You are implying that mathematics be adapted to include intent.  We can easily accommodate  that through the use of probability.  

Visual Thinker: That is just part of it. There is also the probability that is undetectable and the probability that is unspecifiable.  How do we define that probability?

Logical Thinker: We can't.

Visual Thinker: A truly "neutral" observer could just as well have assumed that the objective of the game Monty Hall was playing was to "deliver" prizes in an interesting way that sells soap rather than "limiting" prizes that sell soap. This makes a fundamental difference in motivation that leads to the assumption of a mathematical solution. The act of presenting a door with a goat is in effect a new opportunity of the contestant to change a decision and the motivation for that could have been centered on the fact that not enough prizes were given away or the fact that a delay was needed because a commercial was about to interrupt the outcome.

Logical Thinker: Then you are saying that two solutions bifurcate from the same conditions.  A probability exists that a channel of cooperation between Monty and the contestant exists which is unspecifiable. I go back to my point that this was not stated.

Visual Thinker: But it was in fact rationally implied by the complexity of the situation.  Human beings interact and communicate both by the significance of what is stated and what is not stated.  Mathematics in its current state only operates through what is stated and not even necessarily what can otherwise be stated. For every significant problem there are dimensions that can not be expressed. For me it is as though mathematics are shadows that are ultimately less important than the qualities of the surface that that they reveal.

Logical Thinker: Let me get this straight. I look at a formula and see a set of interactions that relate to a solution.  I look to reducing the functions as part of what becomes the most meaningful solution for me. Simplification is practical. But that does not seem to necessarily be the way that you perceive the implications of mathematical solutions.  

Visual Thinker: You are correct.  Complexity must exist in order to model reality.  Reduction of complexity is the reduction of reality. Since it is impossible to calculate actual reality we must constantly be on our guard that the conclusions we are reaching through the use of mathematics is a matter of our will and not simply the product of an arbitrary mathematical path. 

Logical Thinker: Then you are saying that that this so called "third eye"  is a manifestation of the need to express mathematical logic in terms of a universal reality.

 
Visual Thinker: Actually what I am saying is that no matter how we may perceive it to be otherwise, the minds eye is not neutral and that means that the "third eye," our perspective of "God's eye" also can not be. "God's eye" is implied. The whole point of probability seems to be the fact that within our physical universe the potential of different outcomes coexist.  In the Monty Hall paradox, whether there is a mathematical probability of 2/3 for switching or the subjective probability is 50/50 is immaterial.  These statistical differences are manifestations of expectations associated with setting up the problem. What is important is the probability with respect to the dynamic condition of reality.

Logical Thinker: Probability becomes a matter of human perception.    

Visual Thinker: Exactly; for the visual thinker when a one in a million lottery ticket number is guessed, the idea that a one in one perspective of probability must coexist with the one in a million is not a problem. The probability of one in a million is not displaced by a probability of one in one. You glimpse at the fleeting simultaneous possibilities of the whole and return to the specific logic.  I glimpse at the specifics of logic and return to possibilities of whole. This is the reason why what may be a simple formula for you becomes so difficult for me.  Within my mind there are too many potential intervening states for me to ignore. This is a fundamental difference between visual and logical thinking.

Logical Thinker: This has been enlightening.  I must leave to consider these things. Thank you.

Visual Thinker: Thank you for the opportunity to share my views.  

 

 

 

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