Part 1

Quantum Mechanics currently faces two major problems, one mathematical and the other theoretical. The mathematical problem concerns the accretion during the 20th century of a large quantity of heuristics. The perfect example of this heuristics is renormalization. Even its inventor, Richard Feynman, called renormalization a “dippy” process¹. There are a lot of dippy processes in Quantum Mechanics, though few are as dippy as renormalization. Feynman and many of the other big names in Quantum Mechanics made a lot of mathematical messes. These messes are going to have to be cleaned up at some point in the near future.

The other problem in Quantum Mechanics is theoretical, and that is what this short paper is about. Bohr and Heisenberg made several theoretical messes in the early part of the 20th century, and these messes have been augmented and multiplied by many others in the years since then. In that time, the best commentary on this mess was probably provided by Karl Popper. But there have been two problems with Popper’s commentary. One, Popper was not a professional physicist or an insider, and so he was treated with condescension and even ridicule. That is to say, his argument was dismissed not on logical or scientific grounds, but on grounds of clubbability. Most physicists have never felt it necessary to read Popper or to reply to his critique. He has become a modern Bishop Berkeley. This is a shame since Popper came as close as anyone to solving some of the paradoxes and conundrums of Quantum Mechanics. Two, Popper was often unnecessarily complex in his arguments. He was less complex and wordy than many of his contemporaries, but even the least complex arguments in this era have been unwieldy. The 20th century followed previous centuries in preferring dense treatises. Sentences may have become shorter and commas less common, but contemporary math and physics and philosophy have made up for this with an astonishing proliferation of variables and terms.

There might be added a third problem to Popper’s critique: he combined the terminology of philosophy with the terminology of physics. Physicists who were already swamped by terms and variables were asked to learn a whole new list in order to follow his argument. But 20th century physicists, more than any physicists in history, were specialists. They were not generalists and could not be assumed to know the terminology of philosophy. They were also more arrogant: most expected the world to learn their terms but could not be bothered to learn the terms of other fields.

The rational thing to do—in order to bridge this gap—would have been to speak in the common tongue. Popper should have fled the two lingoes of science and philosophy and attempted to resolve both into a simpler, more direct language. This would have made it much more difficult for him to be misunderstood. Unfortunately, the milieu conspired against this. Popper felt he must prove his intellectual standing, and to do this one did not dare to speak a stripped-down language of common people. In all areas of academia, one proved oneself with a specialized language. Since Popper felt himself watched by both philosophers and scientists, he felt it necessary to include many of the terms and variables from both disciplines. This made his papers nearly unreadable to scientists. He had spent years studying their field, but most of them had spent no time at all studying his. They were therefore in no position to penetrate his arguments. The only way to protect themselves against this inability to comprehend was to deny that there was anything to comprehend. They simply dismissed Popper out of hand, as an amateur.

In all my arguments in all my various papers I have tried to correct Popper’s formal mistakes. That is to say, I have chosen to ignore or translate into common language all variables and terms that were not absolutely necessary to the immediate point. As might be have been predicted, this has hurt my credibility in the short term. Science speaks highly of grace and simplicity but it is not initially impressed by either one. It is impressed at first sight by up-to-date specialist language, lots of variables, and other preconceived esoterica. But I believe that in the long term it will be impossible to ignore true statements baldly stated. Popper will always remain dense, no matter what happens in or out of the field of science. He will remain permanently beyond the comprehension of most scientists. My papers, comprehensible but unpopular, will sit there patiently until the status quo tires of beating the wrong bushes. Eventually scientists will find a practical reason to fix their equations.

The basic mistake of Quantum Mechanics is a mistake of theory. To speak even more directly, it is a lack of precision in defining terms. To show what I mean, recall my lengthy discussions of the definition of the point. I have shown in great detail, and in simple language, that we must differentiate between a real point and a mathematical point [see my paper A Re-definition of the Derivative]. Throughout history we have failed to make that distinction many times and it has cost us clarity in many fields. This imprecision has lain at the foundation of the calculus since the beginning and it has since infected all physical and mathematical fields.

Those who have understood my argument concerning the point will see that the basic problem is one of mistaking the math for the reality. Applied mathematics represents physical reality, but it is not that reality itself. A mathematical point represents a physical point, but it is not that point itself. This is not simply a matter of metaphysics or semantics, since the difference between the mathematical point and the physical point is not just a matter of words or ideas. The difference between the two points can be stated in mathematically precise language. There is nothing “fuzzy” about it. A physical point has no mathematical dimensions. A mathematical point has at least one mathematical dimension (and more commonly has two or more). You can perform mathematical calculations upon a mathematical point, but you cannot perform mathematical calculations upon a physical point. That is why we create the mathematical point in the first place: so that we can do math.

The fundamental problem of Quantum Mechanics is a problem of the same sort. It is a mistaking of the math for the reality. The current theory of QM starts with the assumption that the probability wave is the reality. But the probability wave is the math. The math cannot be the reality. The math represents the reality. But it is not logically equivalent to the reality.

Heisenberg’s main fault therefore was not in his math but in the interpretation of that math. He made a simple definitional error, one of equating the math with the reality. Bohr accepted this error, it became the famous Copenhagen interpretation, and particle physics has followed it ever since. All of the biggest paradoxes in QM are caused by this error. Superposition and entanglement, for instance, are both caused by mistaking the math for the reality. Superposition was historically just an addition of wave amplitudes. In Quantum Mechanics, these waves are probability waves, and so superposition seems to imply, in some circumstances, a multiple existence. Schrodinger’s cat is both alive and dead until we open the box.

The entire problem is in assuming that the math is the reality. It is not. The math is the math, and the reality is the reality. The math in QM is statistical. The wave is a probability wave. Therefore the math can never transcend the probability. Probability math cannot fully represent reality. Even regular math cannot fully represent reality, in that the dimensions will always be incommensurate: mathematical fields cannot match physical fields due to the fact that you cannot mathematically represent (or graph) a zero-dimensional variable. But probability math represents reality even less fully, for obvious reasons. Probability math gives us only probabilities.

This used to be common sense. Mathematicians understood that probabilities were probabilities. Probabilities were imprecise, due to the very definition of the word. But scientists in the 20th century could not live with this imprecision. They were so proud of their new theory that they could not bear to admit that it was not a full expression of reality. They couldn’t live with the “gap” in knowledge. So they simply closed the gap, by main force. They just defined probability as reality. They said, in effect, “This is what we know. Our math is all we know and it is all we can know. Therefore, it is reality for us. Therefore it is reality.”

The most ironic thing historically is that this mirrors the idealism of Bishop Berkeley. Berkeley is one of the goats of mathematicians and physicists. By their standards, he made two cardinal errors. One, he was an anti-materialist. Two, he contradicted Newton. They have never been able to forgive him for either slight. I have discussed in some detail his critique of Newton elsewhere. His critique of materialism boils down to this: “Our ideas are all we know and all we can know. Therefore our ideas are our reality. The existence of material objects is therefore just a prejudice. It is unproven and unprovable.”

Berkeleys’s idealism has always been unpopular among average people, for obvious reasons. It is considered to be counter-intuitive. It is even more unpopular among scientists, again for obvious reasons. Scientists are materialists. Until the 20th century, the first assumption of science was that the physical world existed. Quantum Mechanics has overturned that assumption. What exists for the modern physicist is the mathematics. By closing the gap between probability and reality, Heisenberg made the math the reality. But math is an abstraction and therefore an idea. In this way, modern physicists are idealists. They have accepted the argument of Berkeley without realizing it.

The greatest difference between Heisenberg and Berkeley is that Heisenberg’s argument directly concerns math. Math is the idea that seeds the idealism. But this makes Heisenberg’s idealism quite easy to disprove. To disprove Heisenberg’s idealism, all I have to do is define the gap between his math and reality using his math. I have already done this. I have shown that the gap between a mathematical point and a physical point is not just a prejudice. This gap can be defined in precise mathematical terms. A physical point has zero dimensions. A mathematical point has one or more dimensions. These two definitions are not metaphysical prejudices. They are mathematical statements with real content. To say it another way: the “field” of reality is always at least one dimension removed from any mathematics. It must be by all the rules of logic and by the definition of “math” of “field” and of “number”. This means that the gap between math and reality cannot be closed [to see a more formal disproof of the Copenhagen interpretation, go to the Appendix below].

It does not matter what existential status you give to “math” or to “reality”. It does not matter whether you believe that one, both, or neither exists, by any meaning of the word exist. The only thing that is important is that math and reality are not and cannot logically be equivalent. You cannot close the gap. You cannot say that math is reality. If you do so, you are making a logical and mathematical error. You are being inconsistent, since you are saying that mathematics is your primary operational tool or term, and then you are jettisoning a logical finding of that mathematics to suit yourself.

In this way mathematical idealism is the prejudice. Heisenberg makes mathematics primary in his definition of reality, and then he proceeds to close the gap between mathematics and reality to suit his own desire. But his own mathematics defines that gap. To close the gap, he must ignore his own mathematics. In doing so, he has killed his own god, slaughtered his own logic. You cannot accept mathematics to get you from point A to point B, and then ignore that same mathematics to get you from point B to point C. That is what Heisenberg has done, and that is what Quantum Mechanics has done.

You will say that QM mathematics is not traditional mathematics, and that therefore my argument fails. But QM mathematics is derived from traditional mathematics. QM has not supplanted calculus and linear and vector algebra and so on. The foundations of math have not changed. My definition of the gap, as a necessary separation of dimensionality, must affect QM just as much as it affects all other math and science. Besides, it is quite easy to show mathematically that probability math creates a wider gap than calculus and linear algebra and so on, not a smaller gap. That is all that it is necessary for me to show. I don’t even have to show it, I just have to remind the reader that it is already accepted by everyone, including those in QM, QED, QCD and string theory. No one in the history of the world has ever argued that probability math is more precise than addition or subtraction. But the only way to counter my argument would be to suggest that probability math somehow closes the gap simply by being probability math. The definition of “probability”, by itself, dictates against this.

Part 2

This section addresses Berkeley’s idealism and therefore it may not be of interest to everyone. It may be seen by some to be a piece of philosophy corrupting a science paper, or as a piece of dusty history blown in by an ill wind. Those who think this way are encouraged to skip down to the conclusion, or to quit now, since by their definition of science, I have finished my scientific argument [except for the Appendix]. But I think others will recognize that since I have just tied QM to idealism, it might be of some interest, historical or otherwise, to see how my new argument affects old, (for some) still standing, arguments. For of course the next question is, can I use my new arguments to counter Berkeley’s idealism as well? I can, since my argument above gives me the method. Some will have thought after reading my long paper on the calculus that I was a supporter or apologist for Berkeley. I am not. I find some of his critiques of Newton to be cutting, but his idealism does not appeal to me at all.

Just as with Heisenberg’s idealism, I do not have to describe reality in order to disprove Berkeley’s idealism. I don’t have to give it definite parameters, or discuss any of its characteristics. All I have to do is prove a necessary inequivalence between his two categories, idea and reality. Berkeley, like Heisenberg, tries to close the gap between idea and reality. He says that they are logically indistinguishable, which means that idea is reality. Above, I have shown that math and reality are logically distinguishable—the distinguishing characteristic being the field dimensionality. Well, it turns out that Berkeley’s idea and reality are distinguished in exactly the same way. Field dimensionality. But instead of mathematical dimensions, we substitute levels of abstraction. Just as a mathematical term or variable is at least one level of abstraction away from the reality it represents, any idea must be at least one level of abstraction away from the thing it represents.

Berkeley’s trick is in not precisely defining “idea” to start with. An idea is a representation of another thing. An idea cannot be the cause of itself. It also cannot represent itself. An idea is an abstraction that always implies a generator. An ungenerated idea is a contradiction in terms, since an ungenerated idea would have no content. The content of the idea must come from outside the idea.

Berkeley might say that math is just an idea that represents another idea, the first idea being the math, and the second idea being what we call reality. But even if we accept that, we must notice that he has still assumed representation and separation. An idea is a thing that represents another thing, and those two things are separate. The math is not the reality. The first idea is not equivalent to the second idea. The first idea represents the second idea, and is an abstraction of it. Likewise, the second idea—if it is an idea—must represent something else. If it represents this other thing, then it is separate from this other thing.

Using this logic alone, one can show that idea and reality cannot be equivalent. By definition, there must be a separation between an idea and what it represents. In this way I can always stay one step ahead of Berkeley. Every time he calls my reality an idea, I can demand that his idea have a generator. I do not even have to say what that generator is. I can be as nebulous as I like; it will not affect my logic. Berkeley will say, “OK, show me what is causing my idea.” But I don’t need to do this. I don’t need to show a specific cause or any characteristics of that cause, I only need to prove the logical and necessary existence of a cause. An idea is a representation. A representation must represent something. I say to Berkeley, “That thing that your last idea in a line of ideas represents, I call reality.” I can define it as loosely as that and still win.

His only possible defense is to deny that an idea is a representation. He will say, “Your definition presupposes what you want to prove. Defining an idea as a representation requires an infinite line of causes, and you simply call the limit of this series reality. But I deny that an idea is a representation. I claim that an idea may have no cause or generator. It may arise spontaneously, from nothing. How do you answer that?”

I answer that the burden is now on him to show us an idea that represents nothing, that is ungenerated. Logically, I was not required to describe the characteristics of ultimate reality. All I had to do was prove that all ideas as representations necessarily implied separation between what was representing and what was represented. This separation kept him from claiming equivalence between idea and reality, however they were defined. But if he denies that ideas are representations, then this is no longer a logical claim, it is an existential claim. He is claiming that ungenerated ideas exist; therefore he must show us at least one.

Look at it this way: I can easily give an example of an idea that represents something else. This one, for starters: “I have an idea of a book in my head. It is an idea of a book that is now lying on my nightstand in my bedroom.” Berkeley will say that the book in the bedroom is just another idea in my head, but that is not to the point. The point is that there is a separation of ideas. The two ideas are not equivalent, as I can prove by walking into my bedroom and finding the imagined book. Now I have a line of three ideas, all with similar content: 1) the idea of the book, 2) the idea that the first idea corresponded to a book in my room, 3) the idea that I went into my room and found that same book. You see that even if I state it in Berkeley’s idealist terminology, we have representation and separation. We do not have threes ideas that arose spontaneously out of nothing. We have a line of ideas, each idea tied to the next.

But Berkeley cannot give an example of an ungenerated idea. Any idea he presents us with, we can easily find generators for. We can usually tie it to not one, but a multitude of previous ideas. Even so-called innate ideas are not ungenerated; they do not arise spontaneously. On the contrary, innate ideas are simply the memory of the species. Animals know all the things they do because their cells have found some way to codify the knowledge of the species. If this is true then this codification is just memory. Memory is not made up of ungenerated ideas, it is made up of generated ideas. Memory is the complex storing of experience.

Berkeley might argue that we cannot prove beyond all doubt that any one of these previous ideas caused the given idea, and this is true. The mind or cell does not keep a record of all its workings; or, if it does, this record is not yet available to us. But we don’t need a record. All we need is a bit of common sense. The logical reply to Berkeley is this: “If you admit that there are thousands of sensations or ideas that could have caused a given idea, why would you need to postulate that the idea was generated spontaneously? That is like finding a bloody madman in a butcher’s shop with a dead body and a thousand knives. You notice that the body has its throat slit. Do you a) postulate that the murder was done with one of the thousand knives? or do you b) postulate that the man died of natural causes and that the throat spontaneously opened of its own accord?”

We do not need to imagine that ideas arise spontaneously, since it is fabulously easy to point to generators. From the moment a baby opens its eyes, its entire existence is a generating of ideas. Undoubtedly, the baby’s mind provides it with many tools to collate, combine, categorize, and reduce all these generated ideas. But even if we decide to call some or all of these tools “innate ideas” we cannot reach Berkeley’s idealism. Berkeley’s theory requires that all ideas be innate ideas. If any idea were generated from the outside, then that would entail there being an outside, which would entail an external reality, which Berkeley denies.

Now that I have critiqued the historical basis of idealism, I think it is worth pointing out that Berkeley himself never argued as far in his own defense as I have argued for him. He never argued that ideas were innate ideas or that ideas were not representations. In The Principles of Human Knowledge he accepts that an idea must be “perceived.” By perception he means, “imprinted upon the mind.” He also accepts that some ideas are imprinted directly on the senses. He believes in senses. To me all this implies representation. Furthermore it implies both a perceiver and a thing perceived. Berkeley explicitly accepts the existence of the perceiver, which he calls “mind, spirit, soul, or myself.” But he does not accept the material existence of the thing perceived, except when it is being perceived (in which case it is equivalent to the idea of it). His argument is this: we can list or imagine no characteristics of a thing beyond our sensible knowledge of it. Its existence is the sum of its sensible characteristics. Therefore, when it is not being sensed, it does not exist.

You can immediately see that Berkeley’s actual argument falls to my Heisenberg critique far faster than I have so far admitted. I do not need to address the problem of innate ideas or ungenerated ideas. I can limit my critique to my initial logical argument, and that argument is precisely the one I used against Heisenberg’s idealism. That logical argument is that perceptions must have a cause outside the perception itself. The cause of the perception is the external object. The existence of the external object is defined by its ability to cause the perception. The gap between the external object and the perception cannot be closed, since there is a logically necessary separation between the two. They are at different levels of causation and abstraction and cannot be equivalent.

According to this argument it is beside the point whether the external object has characteristics beyond those perceived or not. Berkeley’s argument is meaningless, since I can easily accept his assertion that nothing exists in the perceived object except qualities capable of being perceived by the five senses, and still falsify his idealism. I can do this because I do not define existence by “objective” qualities—or by essential characteristics that transcend or underlie the directly perceived characteristics. I define existence as the ability to cause perceptions in a perceiver, and that is all.

Interestingly, John Ruskin made a similar argument in 1856. He put it this way, “[The color] blue does not mean the sensation caused by a gentian [flower] on the human eye; but it means the power of producing that sensation; and this power is always there, in the thing, whether we are there to experience it or not, and would remain there though there were not left a man on the face of the earth.”²

Berkeley denies material existence because it has no characteristics of its own. Ruskin and I both bypass this by defining material existence not by a set of exclusive and objective characteristics, but by the power to cause perceptions. In this way we might agree that the existence of the object was in some sense the sum of the possible perceptions, but we would not agree that the object was the perceptions. Just as with Heisenberg, there is a necessary logical and mathematical gap between the perception and the cause of the perception. This gap cannot be closed. Closing the gap is a logical contradiction since it implies that the perception and the cause of the perception are equivalent. They cannot be equivalent, and this is guaranteed by definition, including the definition of “perception” and the definition of “representation.” The perception and the cause of the perception are axiomatically separated, just as “math” and “reality” are axiomatically separated.

To put this one final way: Berkeley admits that ideas may be imprinted on the mind. Well, “imprint” is an active verb. It requires a subject. An idea must be imprinted by a something. An idea cannot be imprinted by a nothing. The something that does the imprinting is the thing that is external to the mind. This thing I call an object. Its power to imprint ideas on a mind I call existence external to the mind. In this way, my argument is no different than Berkeley’s unstated argument for the existence of the mind. He says that an idea must be imprinted somewhere. Imprinting requires a place of imprint. You cannot imprint nowhere. You must imprint somewhere. The somewhere of imprinting, he calls mind or myself. Its ability to receive an imprint defines its existence. Logically, how can he accept one argument and not the other? How can he accept a perceiving subject and not accept an object of perception? The answer is, he cannot do so and remain consistent.

In the name of thoroughness I feel I must address one last argument of Berkeley. I have said that he does not address the subject of innate ideas; but he does address the subject of ideas not directly generated by sensations. This class of ideas he calls dreams. This argument has historically been considered one of Berkeley’s most fascinating, though I cannot say precisely why. He says that if we can show definite instances of ideas being created without the imprint of “material objects”, then we can imagine that all instances of imprinting are achieved in a similar manner. A materialist can provide no proof against this, he says. A materialist may not be able to provide any material proof, but he can very easily provide logical proof. It is clear that the ideational content of dreams is made up from recombination and selection from memory. The content of dreams is usually quite easy to trace by the dreamer himself, without recourse to hypnosis or other outside help. In the cases where it is more difficult to trace, one may imagine that the memories are old, buried, unconscious, or perhaps that they are memories of the species, stored in ways unknown to us. We do not need to hypothesize that dreams come from nowhere, or are created from whole cloth by the dreamer. Even Berkeley admits this. He never argues that ideas are uncaused or untraceable. He simply believes in another cause than a world of objects (see below).

Berkeley’s dream hypothesis became the famous brain-in-a-tank hypothesis, where philosophers proposed that all experience might be fake—created chemically by evil scientists in a lab. Both hypotheses do show the difficulty in differentiating between dream, evil experiment, and life. But they don’t really put into doubt the real world. The reason they don’t is that the content of dreams and inflicted hallucinations both have to come from somewhere. Those evil scientists must inflict us with specific ideas. We can say that the brain in the tank gets its ideas from the evil scientists. But where do the evil scientists get their ideas? Say there is a chair in the hallucination. If the evil scientists never experienced a real chair, where did they get the idea from? Are their brains also being manipulated? You can see that Berkeley and the idealists only put off the question one more step, but they never give a meaningful answer to anything. The content of dreams and inflicted hallucinations requires causation just as much as the content of normal waking experience. Fleeing into “life as a dream” does not address this content.

The greatest insight into idealism may be found, in my opinion, in discovering precisely why Berkeley wanted to deny materialism. What was his starting point? His starting point was this: “Only spirit exists”. This was his first postulate or axiom, the idea that stood without proof or much argument of any kind. This is why he did not find it necessary to prove the existence of mind. For him it was the first given. Mind, as a sensing, willing agent, was part of spirit. Chairs and tables could not be admitted to exist in the same way, since they were not spirit. They were not ensouled. These things existed only as ideas imprinted on spirit. Strangely enough, Berkeley did believe that this imprinting had a cause. Ideas did not arise spontaneously and they were not created by myself. In proof of this he offered many sensations that were beyond his control—things like the weather and the movement of other people. The cause of all sensations was, for him, God. This made it possible for all content, existence, and action, to be given to spirit.

You can see that this is not so far from the evil scientist explanation, except that God is not evil and he is not human in any sense. But life is still explained by Berkeley as a sort of manipulation by a higher power. The problem, of course, is that Berkeley’s God is open to the same questions as the evil scientists. Where does God get these ideas he us supplies us with? Shouldn’t he have to create and experience a sun before he can give us the idea of it?

Once we start talking of the mind and experience and intention of God, we are in the realm of fancy. For the sake of argument, let us say we accept the proposal that all things exist either as ideas or as objects (it does not matter which). Let us say that we also accept that all these things were created either by a purposeful or unpurposeful agent (it does not matter which). Let us agree to call this agent God. Now, we cannot know whether God created objects and then experienced them himself, or whether just thinking of them made them real in the sense we commonly mean. It is all intellectual quibbling. But we can apply logic even to the mind of God, in the sense that we are calling the mind of God the totality of all those created things. Let us say that we accept Berkeley’s argument, that argument being, basically, that God did not need to give physical or material being to all the myriad things he created in Genesis. When he created them, he did not put them into a material world, he simply created them in his mind. Parts of his mind he imprints upon our minds as his will would have it. Objects in the mind of God we call reality. Is there a further logical problem here, one we haven’t already addressed?

Yes, there is. Berkeley believed that only spirit could have a primary existence. Non-spiritual things like chairs and tables could have only a secondary existence, as ideas imprinted upon the minds of spirits. Finally, he told us that these ideas are caused by God. This would mean that before the imprinting took place, the content of these ideas “existed” only in the mind of God. Before I can have the idea of “sun”, God must create or have the idea of “sun”, which he shares with me at his pleasure.

Now, it is clear why Berkeley might want to downgrade the existential status of ideas in the mind of a mortal spirit like myself. The mortal spirit did not create these ideas, they are beyond his will in most ways, and they are pale shadows of the ideas in the mind of God. But Berkeley has also, by his theory, downgraded the ideas in the mind of God. It is not that he denies them materiality, which is almost beside the point in this context. It is that he denies them continuity. If objects are ideas in the mind of God, can we imagine that God forgets them in between imprintings? No, if God has created them in his mind, then they cannot possibly stop existing just because some mortal spirit looks away or closes his eyes. These objects must persist until God decides to end them or change them into another form. Berkeley is inconsistent even in his theology. The discontinuity of ideational objects in the mind of God is just as illogical (some might say heretical) as the discontinuity of physical objects in a material world.

Appendix

This is a formal disproof of the Copenhagen interpretation. I will not use math or symbols, since I am not interested in stenography but in direct and broad communication. I will therefore use simple sentences. The underlined parts are the most formal parts of the proof; non-underlined parts may be taken by purists as commentary or elucidation for non-purists.

Definition 1: A physical point, line, curve, or figure exists in the physical world. This world we call reality.
Definition 2: A mathematical point, line, curve, or figure represents the physical world. It is therefore an abstraction of the physical world.
Definition 3: A physical point has zero dimensions and may not be graphed, diagrammed, or mathematically represented in any way, including the assigning of a number to it. A mathematician cannot possibly assign a number or variable to a physical point.
Definition 4: Mathematics must be performed on a mathematical point, which point must have at least one dimension. A point diagrammed on one axis (a line) has one dimension. A point diagrammed on two axes (a Cartesian graph) has two dimensions, and so on. Assigning a number or potential number (variable) or symbol to a point automatically assigns it at least one dimension.

Deduction 1: deduced from def. 3: A point drawn on a piece of paper or on a computer screen or diagrammed upon the memory is a mathematical point, not a physical point. This is because we draw or diagram the point in order to assign it a number or variable or other symbol. If we do not assign it a number or variable or at least one dimension, then it is useless to us mathematically or as an abstraction. In that case it remains a dot on a piece of paper, which is, of course, a physical thing. Once we use it mathematically, however, its physical status is overwritten and is no longer important. Its use determines its status.
Deduction 2: deduced from defs. 3 & 4: All mathematics and all logical symbolism is at least one dimension away from the physical world it represents.

Result 1: from ded. 2: A mathematical field cannot be dimensionally equivalent to the physical field it represents. This means that mathematics cannot fully express reality. It also means that mathematics cannot be defined as reality.

Final result: Quantum Mechanics is applied mathematics. As such, it must be dimensionally inequivalent to the physical situation it represents. The fields that QM creates are not physical fields. Therefore QM cannot claim that its mathematical field is reality. This falsifies the Copenhagen interpretation.

QM is only a statistical representation of reality. Reality cannot be fully symbolized; its ultimate qualities must be deduced. That is to say, we must use reason to interpret our symbols as best as we can, avoiding contradiction.