Is Math Objective?

-On the nature of mathematics and its claims to objectivity:

There are three primary viewpoints taken on the nature of mathematics.1

1. Platonism: Numbers and Mathematical concepts actually exist.

"Platonism is view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental."2

2. Nominalism: Mathematics is about objects in the world. 

This can most simply be explained as the most common view of mathematics — the belief that at a fundamental level the numbers reference real objects, just like we were taught in elementary school. The number 3 therefore simply represents a collection of three apples, bottles, or whatever.

3. Fictionalism: Mathematics is fundamentally false. 

This final view of mathematics is definitely bold, but it is difficult, if not impossible, to disprove. Although through this view mathematics is false, it does not dispute the fact that mathematics is useful. After all, math, which is the heart of physics, has given us much of modern technology. Fictionalism therefore draws a comparison between math and the moral principles of many religions. Just because these principles and stories may be useful and successful does not mean that they are necessarily true.

I would like to suggest that perhaps none of these views are particularly correct. Instead, a combination of these views must be considered in order to come to a conceptual understanding of the process that is mathematical thought.

In order to come to this understanding, we must first understand the relationship between math and physics. Physics is nothing more than a description of our universe. It claims no more and no less. Its purpose is to predict empirical occurrences. For example,  the basic kinematic equation $v_f=v_i+a*t$ tells us that if an object is accelerating with acceleration $a$, by accounting for its initial velocity $v_i$ we can know what that object's velocity is after a certain time $t$. This final velocity $v_f$ will likely be measured in mph, kph, or m/s. And although these assignments of numbers to objects and events are of human creation, they still give us a method to describe and predict the rate of change of the position of an object without having to do an experiment every time we would like to find velocity. This is the method of discourse for all of physics. We take a specific occurrence and describe it in order to predict future outcomes. This means that for every physics equation there was a specific phenomena which we set out to model. Without the existence of this phenomena the equation would have no meaning.

Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover. -Bertrand Russell

Having a deep knowledge of mathematics is therefore essential to the physicist because it simply gives you more tools to describe phenomena accurately and efficiently. The true mystery is uncovered when we ask the question: why is mathematics so good at describing the universe?

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning. - Eugene Wigner

Keeping that idea in mind, lets turn to nominalism. Nominalism is an easily comprehensible viewpoint on the nature of math. It references every number to a set of real or abstract objects and as such we can say that 9*5=45 because 9 sets of 5 objects gives us a total of 45 objects. Therefore, nominalism places the number line (more specifically the positive integers) as the base idea from which mathematics is drawn.

To explain where this viewpoint leads, I'm going to make up a term — the "idea set". The positive integers are an idea set because the elements of the set are completely dependent on each other. For example, the number 1 does not exist without 2 and 5 does not exist without 604. Each positive integer could not be thought of without each and every other positive integer and together they form an idea — the positive number line. This is akin to the fact that 1 degree of a circle cannot exist by itself. The degrees 0-360 are all completely necessary in order for us to create a concept of what "1 degree" is. Numbers don't exist without the number line just as degrees don't exist without circles. However, when we place the positive integers as the most basic idea set of mathematics we run into some problems. Positive integers do not necessitate negative numbers, irrational numbers, or imaginary numbers. 

So lets specifically address those problems...

To the mathematical nominalist (most people), what does 6-7 mean? Can we take 7 apples from 6 apples? And when we multiply (-6)*(-5), what does it mean to have negative 6 sets of negative 5 apples? We are unable to conceptualize how a nominalistic concept of numbers can necessitate negative numbers, yet we use them regularly and are fully able to understand their worth. We can think of negative numbers as debt; a concept similar to potential energy. Just like potential energy, which represents the energy of an object or a system due to the position in space, negative numbers represent a number's position in worth. It describes the consequences of such a position. We did not invent negative numbers out of necessity; we invented this theoretical set because of its useful properties. 

We can also explain imaginary numbers using the same train of thought. $i = \sqrt{-1}$. Clearly there is no physical representation of $\sqrt{-1}$. It is seemingly absurd to find the square root of a number less than zero. Well, consider the problem $x^2 = -1$ or equivalently $1*x*x = -1$. This question asks us, what transformation $x$, when applied twice, will allow you to transform 1 into -1? Here's where a spark of genius comes in. Well, what if we placed the one dimensional number line on a two dimensional plane. Can we then use this second dimension to transform 1 into -1 by using the same transformation twice? Absolutely; by simple rotation. $i$ is a transformation by rotation into a second dimension. 

This second dimension, labeled the imaginary dimension, simply represents an alternate method of measuring a number(just like debt does). Therefore, when we multiply a number by $i$ we are signifying that this number is rotated 90 degrees counterclockwise using the imaginary dimension. As such, $1*i=i$. Additionally, $1*i*i$ rotates $1$ 180 degrees, transforming 1 into -1. 

Positive integers don't depend on negative numbers. Similarly, positive integers don't necessitate imaginary numbers. Here, numbers are two dimensional only because it is useful. Once you create "actual" numbers through nominalistic thought, it doesn't mean that you have the basis for all of math. Not everything we use is built from that foothold. However, through the use of ideas like negative numbers and imaginary numbers, we can now mathematically represent systems which we could not have before. Imaginary numbers are just as legitimate a concept as real numbers — a tool to describe our world.

My point here is that math, just like physics, stems from goal oriented logic.  Math doesn't create a singular set which is the basis for all sets so that the original set necessitates everything we can derive. It also does not claim to exist or not exist in physical or abstract form. That is not its purpose nor its rooting. Instead, math sets out to describe concepts and systems by using consistent sets that may or may not directly exist in observable phenomena. It holds the principle that logic derived from some useful tools can create as accurate a description of our world as direct observation can. Whether or not these tools are "real" or not isn't a real question. It means nothing. Math is not metaphysics. Don't defile it with such ambiguity. 

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Active Irresponsibility

As a aspiring Physicist, it must have been amazing to live in the 1920's. Theoretical Physics saw a barrage of new discoveries in one of the most ridiculously successful periods for the advancement of technology and the scientific community in general. Einstein's theories for relativity and the photo-electric effect were finally being accepted as scientific fact, and early quantum theory was significantly reformulated. During this time period Neils Bohr developed the Bohr model for the atom and Schrodinger created a basis of wave mechanics by formulating the wave equation. The discoveries of the 1920's gave us insight into a world we know very little about. It was a period driven by the obsessions of great individuals who used their limited intelligences in specific directions. They were focused on what they did not yet understand. 

Isaac Newton, Charles Darwin, Nicola Tesla, Albert Einstein, Richard Feynman, Sigmund Freud, Francis Crick, and Marie Curie. These individuals have one significant thing in common. Yes, they were all great innovators and scientists who have each influenced our world in unique ways. But as human beings they embodied one trait; they were decidedly passionate. Passionate about their craft. To these people, science was a means onto itself. The accomplishments of these great scientists illustrate that scientific breakthroughs have been achieved through passionate, unrestricted exploration, where short-term gain was not a prerequisite for advance. 

Here's some proof: 

In March 2013 we finally confirmed the existence of the now well known elementary particle, the Higgs boson. More importantly, by finding the Higgs boson we were able to confirm the existence of the Higgs field. This is because just as a photon is a vibration in the electromagnetic field, the Higgs boson is a vibration in the Higgs field. But why is this significant? Well it could be said that there are three fundamental ingredients to our reality: matter particles, forces and carrier particles, and the Higgs. The importance of this discovery was tremendous and it has opened new exciting territory for physicists to explore. When the LHC was created it was expected that the collider would either demonstrate or rule out the existence of the elusive Higgs boson, thereby allowing physicists to consider whether the Standard Model or its Higgsless alternatives were more likely to be correct. At the time, the U.S. was building a competitor to the LHC, the Superconducting Super Collider. American physicists were devastated when in 1993, its construction was cancelled. We could not see the practicality in it; the purpose of such a project. It was a huge loss for science, but it gives us insight into the unscientific mind that drives much of the world today.

There was a time when knowledge for the sake of knowledge was a central axiom in scientific inquiry. The quest for knowledge was driven by the obsessions of great minds who found discovery to be a worthwhile reward. Now, we require practicality. We expect our scientists to give us a spaceship to Mars, a better iPhone, or faster internet. We revere the names of Einstein, Tesla, Bill Gates, and Steve Jobs, yet we have no interest in their craft. We care more about their apparent success than the fascinating innovations which made them successful. And our worship of success and recognition has reduced research to a process which produces products that we can go buy at the mall. Curiosity isn't dead, but it's scarcely breathing.

Of course, we shouldn't define the scientific mind as void of practicality. Take, for example, Nicola Tesla. While Newton and Einstein focused heavily on the theoretical and abstract aspects of Physics, Tesla had a way of making his imagination tangible. Just as much as Tesla was an engineer, he was also a scientist. He was THE "mad scientist" and many of the tangible benefits of his ideas were realized immediately (although not publicly). The man was essentially Tony Stark. So yes, I would like us to question the notion that great science and innovation demands utility. However, I do not imply that practicality is inherently foolish or that a great mind cannot dream of tangible discoveries. Teleportation, warp drives, quantum computing, macroscopic quantum tunneling - these are the dreams of both the physicist and the layman. Instead, I propose that the mundane, auditing culture of today needs to completely redefine the word "useful" when we speak on scientific matters. 

"The way I think of what we are doing is, we are exploring, we are trying to find out as much as we can about the world...whatever way it comes out it’s nature, it’s there, and she’s going to come out the way she is. And therefore when we go to investigate we shouldn't pre-decide what it is we are trying to do except to find out more about it."  -Richard Feynman

The theorists who proposed the Higgs boson spent their entire lives creating a theory which the LHC may have found false. It's how science works and as a theorist you have to learn to deal with that.

"If you don't risk your life, you can't create a future. Right?" -Monkey D. Luffy

By definition, you cannot anticipate the results of potential innovations. We must mend the boundary between the acquisition of knowledge and the application of it. We need active irresponsibility. If history has shown us anything, it's that in order to truly change the world we must be obsessed with something other than ourselves.

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The Apparent Irreversibility of Time Pt. 2: Is Entropy Essential to Time?

Why does the process of time appear so irreversible? This question leads us to the second law of thermodynamics. It states that the entropy of an isolated system always increases or equivalently, that heat always flows from higher to lower temperatures. In order to connect the concepts of time and entropy, we must first dissect this law and define entropy. 

The increase of entropy in a system over time suggests that the number of ways a system can be arranged increases with time. Imagine a system of 20 rubber balls and 20 plastic containers in which rubber balls go into plastic containers. In the system's most orderly state (least uniform), all 20 balls exist in 1 of the 20 containers. Therefore, there would be 20 ways in which the system could by arranged. 20 balls could exist in container 1, or container 2, or container 3, etc. However, in its most uniform state, where each rubber ball can exist in any of the 20 containers, there are 20! (2.43*10^18) possible arrangements for the system. Therefore, when we say that the entropy of every isolated system always increases, we are saying that as time progresses the universe tends towards uniformity and disorder.

In the example which I used in my previous post, gas held in a bottle is released into a room. This is a perfect example of an increase in entropy. The molecules do not dare permanently restrict themselves to the confines of the bottle. The system must always proceed towards uniformity. Of course, in principle it is not impossible for the system to occur in the opposite direction. But in practice, such an occurrence wouldn't happen in a million years. The idea that the second law of thermodynamics is a law of disorder was developed by Ludwig Boltzmann. His ideas were questioned because of the reversibility of Newton's equations. Why couldn't the velocities of the molecules just be reversed and the entropy of the system decrease? To such a question Boltzmann sarcastically replied, "Go ahead and reverse them." 

This bottle of gas scenario can be represented by the following two pictures:

To sum up everything I have been saying so far, the only reason why we know the right picture occurs after the left picture is because of the second law of thermodynamics.

Of course, no definition of entropy would be complete without a mathematical representation. One might think that entropy, which we shall denote as S, should be proportional to the N number of ways a system can be distributed. This would be represented as S = kN, where k is the proportionality constant. In a combination of two isolated systems(microstates), this would mean that the combination(macrostate), S12, would equal kN1 * kN2 =  S1 * S2. However, in order for this concept of entropy to work, the entropy value must be innate to each isolated system(microstate). Therefore, in order to calculate the entropy of a combination of two systems, we must simply add S1 and S2. S cannot be directly proportional to N. Instead, S1 = k*ln(N1). In a combination of two isolated systems, S12 = k * ln(N1 * N2) = k * ln(N1) + k * ln(N2) = S1 + S2. This leads us to the formal mathematical expression for entropy in statistical mechanics:

where kB is the boltzmann constant, and P is the probability that the system is in it's ith microstate. 

So, the entropy of the universe is always increasing. As a result, we are able to differentiate between the past and the future. The world tends towards disorder and uniformity and time is irreversible because of this. But we still have not answered WHY the universe must become more uniform and why heat always flows from higher to lower temperature. If we go back in time the entropy of the universe should be lower than it is now. Consequently, it can be assumed that the universe began with a low entropy. Why is this? Maybe, increasing entropy is a result of the expansion of the universe(cosmic inflation). Or maybe it's the other way around. 

I'll leave you with an interesting proposal. Space has no innate directionality. There is no up, down, or sideways. It also has no discernible center. However, in the presence of a large object such as the earth, we are able to create a perceived directionality. With the Earth as center, it becomes the authority on what we call north, south, east, and west. Perhaps there is a large event which has created a perceivable directionality in time; the big bang. 

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The Apparent Irreversibility of Time Pt. 1: The Problem

Initially, the directionality of time does not appear to be a problem. The past becomes the present and the present becomes the future and we ignore the intricacies of its definition. After all, time is so intrinsic to the universe and our daily lives. Time moves forward. Without that direction we would find it difficult to even call time, time. But, what is time if it lacks a past or a future? What if there is no distinction between the processes of past to future and future to past; is it still "time"?

As you view the motion of this pendulum can you decide whether this video is playing forward in time or in reverse? The problem with the arrow of time is its apparent reversibility. That is to say, if this system is reversible then it can proceed in either the forward or reverse temporal directions without breaking physical law. Lets take a look at another simple system; a planet in orbit.

By reversing this video we would find that at every corresponding position the velocity of the planet would also be reversed (-v) and the acceleration would remain constant. This completely agrees with physical law and in neither clip can we discern whether we pressed play or rewind. Thus, Newton's equation for the force of the gravity (F = GMm/r^2) between these 2 planetary bodies is time reversible and in fact this entire system works just as well in reverse as it does in the forward direction. Actually, all of Newton's laws and equations happen to be time reversible. 

Whether or not these simple systems are time reversible, we are still able to maintain our previous perception of time. There are many events in our daily lives that are apparently irreversible. For example, here's a glass breaking.

And here's it breaking in reverse...

Surely this situation is different since glass does not put itself back together in such a manner by the laws of physics. In a complicated system such as this our original perceptions are justified and we find that there must be some difference between the past and the future. It is a seemingly irreversible event. Now consider another complicated system where gas, which is held in a bottle, is released into a room. As time goes by, the gas exits the bottle and evenly distributes itself throughout the room. Again, it is seemingly impossible for this situation to reverse itself and for all the dispersed molecules of gas to gather and enter the bottle once again. However, this event is dictated entirely by Newton's laws, which are time reversible. How can a situation described by time reversible equations produce such apparently irreversible results? One might even argue that perhaps if we were able to obtain data on each molecule of gas, including its position, velocity, and the results of their elastic collisions, that the entire system could be described in time reversible simple states. 

To determine the validity of this claim let us consider a third event. You stand at the edge of a pool. The water is completely still. You take your finger and touch the water, creating a series of circular ripples emanating outwards. Laws determining the propagation of waves are time reversible and therefore, at a fundamental level this event should be completely reversible. This means it is possible to design boundary conditions at the edges of the pool such that you create waves emanating inward to a single point. Additionally, at the moment those waves converged it would be necessary for your finger to come into contact with the water to absorb the momentum. For all practical purposes, this is impossible. Therefore, in principle these three events are not irreversible. Rather, the chances that the reverse of these situations may occur(in reality) by accident or design are extremely low. Why is this?

The thing is...ALL the fundamental laws of Physics are time reversible. This includes Newton's laws, the laws of electromagnetism, and even the wave equation of quantum mechanics. How can we reconcile this fact with our macroscopic observations and interpretations of time and reality? Why does the process of time and the processes of the universe appear so irreversible? I will attempt to answer this question in my next post...

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Short Introduction to Special Relativity

The key to special relativity, as suggested by its name, is reference frames. The picture below shows an observer standing in the back of a pick-up truck. The pick-up truck is moving with a velocity v in reference to a second observer who is stationary on the ground. The first observer throws a ball straight up into the air and then catches it. To this first observer it appears as if the ball goes up and down without changing its horizontal coordinate. However, as the truck moves by the second observer, she observes the the path of the ball forming a parabola when the ball is thrown into the air. What Einstein purposed, in short, is that neither observation of the ball is the "true" observation. Because there is no reference frame with more accuracy or consistency that any other reference frame we can begin to grasp Einsteins two postulates that form his basis for special relativity. 

                           

1. The laws of physics must be the same in all inertial reference frames.

2. The speed of light in a vacuum(300,000,000 meters per second)has the same value regardless of the velocity of the observer or the velocity of the source emitting the light.

The second postulate is really implied by the first*

When you are moving in a truck at velocity v and you throw a ball, at velocity v', in the same direction as the velocity of the truck, it makes sense that the total speed of the ball with respect to a stationary observer would be v + v'. In classical mechanics, any projectile fired from a non-stationary object, adds to the velocity of that non-stationary object. So since light travels at a velocity c it would appear that when emitted from a moving reference frame with speed v, the total speed of that light with respect to a stationary reference frame would be v+c. Light, however, is a fundamental property of the laws of physics. According to Einstein's two postulates the speed of light must be the same in each reference frame. Regardless of if the observer is moving at a velocity close to that of light, say 0.95c, the observer should still measure the speed of light at 300,000,000 m/s. Regardless of if he thinks he is moving or stationary. 

This seems quite counter-intuitive. To understand this concept we must first think about how velocity is measured. Meters per second is an amount of space(meters) covered in some period of time(seconds). In order to keep the speed of light the same when changing inertial reference frames we need to subtlety edit this definition. This brings us to the topic of time dilation. Einsteins theories remove some of the separation we intuitively place on space and time. We need to alter our perceptions of space and time in order to conceptualize the effects of special relativity. Every time we change inertial reference frames(move) we affect not only space but time also. Our perception of time changes as our velocity changes. This is not illusion. It is a principle of time itself. The reason why light appears to move at the same velocity with respect to an observer regardless of if that observer is also moving is because as that observer moves, their perception of time changes. Time is slower in a moving reference frame than in a stationary one. However, to the observer, time always appears to "pass" the observer, in each reference frame, at the same speed. In other words, though less time occurs in a moving reference frame in comparison to a stationary reference frame, no observer in any reference frame thinks they're in fast or slow motion. It is interesting to explain the effects of such a proposition. The equation for this time dilation is  

                                     

This gives us the amount of time passed in a moving reference frame in comparison to the amount of time passed in a stationary one. For example, if 0.5 seconds passes in stationary time(t) then the amount of time passed in a reference frame with a velocity of 200,000,000 meters per second is 0.5(sqrt(1-((2*10^8)/(3*10^8)))) or about 0.29 seconds. If apparent time is shorter and therefore occurring slower, it follows that apparent distance must be shorter. Another way of explaining this is that to keep the consistency over reference frames, as the value of seconds decreases, meters must also decrease. In order to maintain the same apparent velocity the ratio m/s must remain constant.  This is called length contraction.  It is described by the equation 

           

Where L = stationary length and L' = contracted length

I'll give one last example on this post regarding the effects of time dilation. This involves an experiment involving a set of twins. Both twins are 20 years old. One of the twins happens to be an astronaut and sets out on a journey to a planet 20 light years away. His ship is capable of reaching a speed of 0.95c. When he arrives at the planet he immediately turns around and comes back to Earth at the same speed.  To the twin that stayed stationary 42 years have passed. He is now 62 years old. According to the equation for time dilation, when the traveling twin returns he would only be 33 years old. To him, a mere 13 years would have passed. This is an famous example of time dilation used in the twin paradox. (http://en.wikipedia.org/wiki/Twin_paradox). 

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Why I Don't Quit Physics

Recently I watched a short video (http://www.youtube.com/watch?v=NK0Y9j_CGgM) from a talk with Michio Kaku, the Henry Semat Professor of Theoretical Physics at the City College of New York. He speaks about the decline of scientific minds within the United States. According to Kaku 50% of PHD candidates in the United States are foreign born. So why is this? If I were to show that statistic to every single person in America I would guess that the vast majority of people would agree with the statement that America is in fact getting stupider. And yes, the vast majority of America saying that the vast majority of America is stupid is obviously ironic. But I think, personally, that this statistic can be attributed to a much bigger problem. Or what seems to me to be a problem. During high-school I adopted the idea that the only real purpose of going to school is to get a job and make money. This is why I came to college. I believed that without college I was doomed to an unsuccessful and thus unhappy future. However, this was not my opinion in my early childhood. I enjoyed learning rather than thought of it as a necessary evil. I think this problem, which affects a large amount of the American population, stems from exactly what makes America, America. Capitalism. I'm not addressing capitalism as a economic system but instead as a mind-state. We have adopted the ideas of capitalism as a mindset and even a moral system. It should be obvious why thinking that school is only good for getting a job could be detrimental to science. Ridding ourselves of this capitalistic mindset however, is impossible. It is difficult to convey just how deeply it invades every part of our thought processes. Things which we fully consider to be simply human traits are rather only consequential ideas of a capitalistic mindset. When I came to college I originally decided on a career in Engineering. I chose this not because I enjoyed engineering but instead because I believed I had a talent for math and science and that engineering was the most sure way to take those talents and make them into a stable career. This may seem like a slightly valid reason. But you should be able to see how this could only be considered a valid reason if you have capitalistic bias'. I am now a Physics major. Not because I know that I have a sure and specific future career ahead of me. I am a Physics major because I have decided I want to know what I want to know. School is for gaining knowledge is it not? So yes. America is losing its ability to create scientific minds. It is disgusting that an inquisitive mind, like my own, would have difficulty in deciding to allow itself to be inquisitive simply because of a definition of success which only caters to a capitalistic mind-state. Even now I struggle. When I arrive at topics in my Physics textbook which confuse me I sometimes think, maybe this isn't for me. It would be so much easier to study a subject where the material you study doesn't takes longer to comprehend than to actually learn. And yet, one simple thought allows me to keep going. I imagine myself as a 40 year old, reading a popular science book on Physics. The book in this dream is a book regarding the formation of black holes. In speaking of how black holes form it makes one simple statement. "The math behind these ideas suggests that"...etc. I imagine myself reading this and not knowing what the math suggests and also knowing that if I were to find the math which suggests this proposition that I would not be able to comprehend it. That thought is terrifying to me. I would feel as if I have wasted my entire youth. I cannot stand the thought of being in my 40's and not knowing answers or at least theories to the questions which now keep me up at night. That thought is terrifying enough to allow me to keep doing what I am doing. I simply want to know more about the world. So when the statement is made that America is falling behind in Math and Science it doesn't surprise me one bit. I know from personal experience how little the majority of America caters to and respects the scientific mind. Too often I hear students say "Why do I need to learn this? This isn't going to be useful to me in life. I don't care about this subject. How is this going to useful in my career? Who really cares?" Genuine curiosity about the world and the desire for knowledge are very rare traits. What I can do is surround myself with the people who have developed their own scientific minds. Inquisitiveness can't possibly a bad thing.

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Reference Frames

Everything needs an initial reference frame. Every idea/principle. It’s not that everything should be given an initial reference frame. It’s that nothing can be thought of without one. Again, not that a given idea doesn't exist without a reference frame, but that without the initial reference frame the idea simply couldn't be thought of. It couldn't be an idea or be imagined. Everything that enters the mind should have reference frame. But reference frames can be misinterpreted which allows for some literally unthinkable ideas. This happens when ideas are built off “ideas” beyond the actual initial reference frame. An idea of an “idea”. This latter idea often times looks to be rational and may fit very well with the order of things. The initial “idea” however, has not been considered. But once considered it can become evident that the initial idea is actually unimaginable  It cannot be thought of. It’s an “idea” or reference frame which has been misinterpreted in order to come to later conclusions. Misinterpreted in the sense that it should not have been interpreted to begin with. This might seem impossible since the brain cannot think of things which it cannot think of, clearly. But if the later conclusions are appealing enough intellectually or emotionally there can be a misinterpretation of unknown reference frames. The known becomes intertwined with the unknown and eventually the entire idea becomes seemingly plausible. These misinterpretations are usually difficult to recognize especially within widespread ideas. Whether the “idea” is plausible or not is irrelevant since it is not an actual idea. The correct conclusion once realizing the nature of the nonsense of the “idea” is to conclude that all proceeding ideas are also nonsensical. Again, not that they don’t have substance or that they don’t exist. That wouldn't mean anything. Anything can exist. Only a few things can be thought of, in proportion to that which can exist. But yeah, these “ideas” cannot be thought of as ideas. When I encounter these ideas I tend to ask the question, “What does that mean?” The answer is never satisfactory because of either blindness to this entire concept or misinterpretation of my question. Need a better approach.  

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