Gödel’s Incompleteness Theorem: Ontological Mathematics vs. Science

Most people think that Gödel’s Incompleteness Theorem means something about the final answer to everything being something that can never be attained, only more and more closely approximated, because any system of logic and its axioms will always be incomplete and contain inconsistent statements inevaluable as either absolutely true or false.  Similarly, Stephen Hawking wrote in the ‘Brief History of Time’ that it seemed as though science would only ever asymptote towards a final theory of everything, but never completely get there.

To properly understand Gödel, it is helpful to understand his philosophical enemies, one being Bertrand Russell.  Russell wished to explain mathematics and numbers as having originated, and originating in, a more fundamental set of axioms that determined the truth or falsity of mathematical statements.  Basically, he wished to prove that mathematics was a creation of the human mind, that it was a convention of logic among humans.

What Gödel showed, in the Incompleteness Theorem, was that that attempt was itself always going to be inconsistent and incomplete, and hence could never be absolutely proven, and hence, you couldn’t explain the field of mathematics with such an attempt.  What Gödel showed was that Russell’s idea of axiomatizing mathematics was, basically, simply illogical.  The attempt to axiomatize mathematics would always fail, due to this logical phenomenon of “incompleteness”, or inconsistent statements that can’t be evaluated for absolute meaning given any field of axioms. For example: “This sentence is a lie.”  It meant that mathematics cannot be defined in terms of a set of axioms created by humans, and hence, mathematics was its own complete system independent of the human ability to originate.

But then the question is, what is mathematics, and where does it originate?  If mathematics is outside of the ability to axiomatize, then what is it?  Where does it originate?  What quality or class of logic does mathematics actually belong to?

The reason this works out this way is because mathematics is a class of logic called ‘necessary tautology’. One doesn’t need to axiomatize a tautology: 1 + 1 = 1 + 1, or 1 = 1, or 0 = 0, etc., is independent of requiring axiomatization for us to agree upon.  It is simply a necessary, and eternal, truth, not something we merely agree upon because of some established convention of axioms.  You don’t need to axiomatize a tautology in order to explain it, and hence, the attempt to axiomatize a tautology in order to explain it is an illogical operation, and hence, it fails.  A necessary tautology is simply accepted as an eternal truth of reason.

The equal sign at the heart of every single mathematical equation indicates a necessary tautology, an eternal truth.  When we write 1 + 1 = 2, we are fundamentally actually simply writing 1 + 1 = 1 + 1, and we use the character that looks like this symbol, “2”, to short-form represent 1 + 1, for convenience.  We would never write 1 + 1 = 1 + 1 + 1, because that equality statement is illogical in the absolute; whereas writing 1 + 1 = 1 + 1 (= 2)  is logical in the absolute.

One question that might be asked, is where does the concept of “1” originate?  Isn’t the concept of “1” an axiom that we must agree upon?

The number “1” is the concept of a unit, of a single, objective, identifiable, and sovereign (meaning not blended with anything else), entity.  This concept of an objective unity has a sufficient reason for its existence in the rational deduction of the monad.  The monad is the indivisible basis of existence and is arrived at relatively effortlessly.  We ask, what is the basis of existence?  What are its properties?  The basis of existence must be that which is indivisible, for if the basis was divisible, then the pieces from division would be sub-components of the larger, and thus more fundamental.  Thus, the basis of existence must be indivisible.  Additionally, the basis of existence must be uncreated, must be a self-evident eternal cause, because if the thing we thought was the basis was created, we would then need to ask what created its creator since the creator would be more fundamental than the created.  Hence, the basis of existence must be eternal and uncreated.

And so, that which is indivisible and uncreated is a sovereign unity by definition, called a monad, and its existence is sufficiently rationally deduced without requiring any assumptions.  The comprehension itself of what this entity actually ontologically is, is found in the mathematical zero.  Zero is indivisible, for it is the infinitely small, and can’t be divided into parts; and zero requires no effort to create since it comprises nothing, and hence there is nothing to prevent its existence since that which requires nothing to exist cannot be prevented from existing.  The monad, the ontological zero, exists effortlessly, hence it exists.  Likewise, there is nothing to stop an infinite number of instances of these entities from existing since if the conditions are such that one such entity can exist effortlessly, then any number of them can.  Something must exist rather than nothing because nothing cannot have the quality of existing, and so, something must exist, but this thing that must exist has the necessary properties we’ve already deduced.

We’ll see further on what the properties of this “nothing that is something” of the ontological zero actually are, and that this seeming contradiction can be totally resolved within the concept of the zero and mathematics, but for now, we can accept that the concept of zero, and unity, have both been rationally deduced with sufficient reason.  Hence, the concept of the number “1”, the unity, is a logical deduction with sufficient reason, and is not just an axiom that we’ve agreed upon.  As the mathematical equality is a simple statement of tautology, then 0 = 0, 1 + 0 = 1 + 0 = 1, 1 + 1 = 1 + 1 = 2 where “2” is the character we use to short-form the concept of having one unity and another unity, and these are simple statements of absolute logic that can never contradict themselves and can never be wrong.  It would always be illogical and wrong to write 1 + 1 = 1 + 1 + 1, whereas it is always correct that 1 + 1 = 2.  Mathematics does not originate from a set of axioms we’ve created, but rather, it exists for sufficient logical reasons which are eternally and effortlessly true.

A direct corollary of all these facts is that, at the basis of existence, reality is a mathematical substance.  It’s quite simple really – there’s no other way for reality to behave.  Can one monad and another monad make three units of sovereign monads?  That’s a complete contradiction in terms.  How else can reality behave than that two monads make up two monads?  It’s the only possibility.  Existence is composed of an infinite number of monads, of “ontological zeros”, and the only way these entities can relate to each other is as they are to each other, and each are one unit monad, or two unit monads together, or three unit monads together, etc.  The basis of existence is a substance of monads, but this substance is effectively synonymous with mathematics because there is no other possibility than that the monads relate to each other as they are, and they are each sovereign units.  So we can equivalently say that the basis of reality is a monadic mathematical substance, or a mathematical substance of monads, etc.  This is where the term “Ontological Mathematics” originates as a philosophy because it specifies that mathematics is ontology, and ontology is the study of that which exists.

Now, what does physics and science study at the fundamental level?  It studies the behaviour of energy.  Energy is simply called an “abstract idea” in science, and so it isn’t actually explained at all by science.

The most important law in physics is the law of conservation of energy.  This is accepted as a law, but it’s origin isn’t actually known or explained by science…it is simply accepted because it has been so well-confirmed.  Well, if energy is the fundamental thing about reality that science studies, and we now know that the basis of reality must be a mathematical substance, then we can ontologically identify the phenomenon of energy as a noumenal mathematical substance.  Then, the law of conservation of energy automatically follows since mathematics is necessary tautology, and such tautology is necessarily intrinsically conservative. 1 + 1 = 1 + 1 is conservative.  It is never 1 + 1 = 1 + 1 + 1, as this is illogical and a contradiction.  In ontological math and any math, 1 + 1 = 1 + 1 and this is necessary eternal truth, a necessary tautology, and conservative.  Hence, the origin of the Law of Conservation of Energy is found in that what we experience and study in science as energy, is in fact a mathematical substance.  Energy is a mathematical substance and mathematics is necessarily intrinsically conservative.  What energy actually is, is mathematical substance.  Hence, why we use mathematics and have even discovered new mathematics and extended our understanding of mathematics, when studying energy in science.

Mathematics is numbers.  Numbers can be positive and negative, real and imaginary, where imaginary simply means the square root of a negative number.  The term “imaginary” is actually a really poor choice of word for something which is totally rational and understandable because it sometimes makes people think that it refers to something fanciful; however, the word “imaginary” does have some meaning in that it requires your mental imagination to comprehend it, but it is not a fanciful type of imagination, but rather the imagination of a rational concept in your mind’s eye.  There are relatively simple deductive methods for proving the existence of the real & imaginary complex number plane, and the development of understanding the complex numbers is an interesting and intellectually provocative history.  See “First Geometric Interpretation of Negative and Complex Numbers” (http://www.cut-the-knot.org/arithmetic/algebra/JohnWallis.shtml), and “Fundamental Theorem of Algebra” (http://www.cut-the-knot.org/do_you_know/fundamental2.shtml) for very brief summaries.

A mathematical substance will necessarily comprise all of these possible numbers, because there’s no reason why some numbers should be privileged over any others.  If a mathematical substance arbitrarily did only comprise a subset of the numbers, then it wouldn’t be complete and consistent and hence wouldn’t be logical and wouldn’t be stable.  Only the entire plane of complex numbers comprising both real and imaginary and positive and negative numbers is completely logically closed and consistent.

We can ask whether the field of all numbers and how they behave is governed by any general equation…numbers are all math right?  So, maybe there’s some math equation that describes how all possible numbers relate to one another and how they behave relative to each other.  This is a pure problem of mathematics and in fact was solved by the mathematician Leonhard Euler, and the equation is called Euler’s Formula…although Euler didn’t realize its importance at the time.  Occam’s Razor would imply that a mathematical substance should follow the simplest, single, and complete, all-encompassing rule for the behaviour of all possible numbers, and that rule is Euler’s Equation, which describes how all possible numbers relate to one another and how they behave relative to each other.  The mathematical substance which is existence is actually Euler’s formula, ontologically, as an equation you can actually write down:

eix = cos(x) + i*sin(x)

And so, Euler’s formula is indeed already found at the heart of quantum mechanics, which describes the behaviour of matter and energy, and it also derives relativity theory, which describes the space-time which matter and energy inhabits, and thus it is found at the core theories of science describing all of phenomenal reality.  We have both a rational proof and empirical evidence that the basis of reality is a mathematical substance we call Ontological Mathematics, following what we now call the God Equation, which is Euler’s Formula.  Ontological Mathematics is the noumenon – the thing behind the scenes that makes it all work, the thing behind the appearances.  Sensory experience and empiricism is the phenomenal, subjective experience, of that objective noumenon.  The color red is really just an objective specific numerical frequency of an energy wave which itself has no color at all, whereas our subjective experience of that frequency of energy wave is perceived as what we call the color red.

Euler’s equation explains this “nothing that is something” of existence via one of its corollaries.  If we set x = π, then the equation becomes e + 1 = 0.  This is a direct mathematical demonstration that something, the complicated expression on the left hand side, is equal to nothing, or zero, on the right hand side.  But there’s actually more than that too.  Euler’s equation itself shows that all numbers produce waves, sines and cosines, centered on zero.  All numbers are centered around zero and thus balance out, or reduce, to net zero; there’s all sorts of stuff happening in positive and negative real and imaginary numbers, and all manner of wave activity and vibration, but all of this activity balances out at zero.  Hence, the something of existence is actually nothing and thus can exist and have activity without effort, and all of the logic is necessarily consistent with Euler’s Formula and the substance of reality.  Here is a graphical representation which shows how Euler’s equation is centered on and thus balances out to zero:

3D Euler's_Formula_plot

An existing problem in science is the incompatibility between relativity and quantum mechanics.  The reason is because science doesn’t use Euler’s equation for deriving and explaining relativity theory – it uses math which is logically different than the Euler technique.  So, if science uses Euler’s equation in quantum mechanics, but it doesn’t use it in relativity, then what automatically gets created is a mathematical incompatibility between those two theories.  Basically there is a “substance incompatibility” between the two theories.  When relativity is re-cast in the proper substance of Ontological Mathematics, then the incompatibility disappears and hence the theories will become reconcilable.  Unfortunately no one in the public sphere is working on this yet, but the basic outline for what needs to be done is provided in the God Series (http://www.amazon.com/Mike-Hockney/e/B004KHR7DC) books.

This concept of substance incompatibility was brought up as a segue to something much more provocative.  Let us consider life vs. death, and mind vs. nonmind, in the context of the basis of reality, the substance of mathematics.  If the mathematical substance was dead and without mind, then what sufficient reason would there be for it to spontaneously pretend, to simulate, to produce the appearance and the actual experience of life and mind?  If something is dead and without mind then it is a category error to say that it has the experience of life and mind.  That would be a basic contradiction.  The only thing which isn’t contradictory is to say that dead things without mind are dead and don’t experience mind, whereas things with life and with mind therefore experience life and mind.  What could be more simple and logically obvious?  There’s a basic substance incompatibility between death and life, between non-mind and mind.  Death can’t pretend to be alive and nonmind (especially!) can’t pretend to think and to choose what it wants to do.  The only thing that can think is mind and the only thing that can be alive is life.

So what does this say about the mathematical substance which makes up reality?  If life and mind exist in this reality, and they do, and reality is fundamentally a mathematical substance, then life and mind has to be a property of the basis of reality.  Since life and death are incompatible substances, then the noumenon of mathematical substance, the basis of reality itself, must be alive and minded, not dead and nonminded.  The mathematical substance which is the basis of reality must be the source of mind and life, because there is no reason for death and non-mind to magically and purposelessly pretend to be alive and minded.  This isn’t new-age “woo-woo”, it really is just simple logical rational facts. In fact, the “woo-woo” position would be to say that dead and nonminded matter magically and inexplicably pretends to be alive and to have mind, to have feelings, and to arrange itself into subunits which create the drive to find meaning and purpose!  There is nothing more silly and illogical than that.  Only life and mind can have the drive to find meaning and purpose, to experience, feel, think, evolve…and have life and have mind!

Finally, as the basis of reality is the mathematical unit called the monad, and this monad follows Euler’s equation which contains all sorts of activity and vibration centered around zero, and this activity must be alive and minded not dead and nonminded, then we have in fact identified the individual soul.  The mathematical substance of reality is actually a substance of alive souls, souls which necessarily, individually, subjectively experience their mutual shared objective existence.

And as souls being monads, then they are uncreated, and are hence eternal, and hence indestructible and immortal.  These are the basic properties of what we would normally define as Divine.

John 10:34

Jesus answered them, “Is it not written in your Law, ‘I have said you are “gods”‘?

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8 Responses to Gödel’s Incompleteness Theorem: Ontological Mathematics vs. Science

  1. Bill Waters says:

    I must admit to having a kind of epiphany when reading “Why Math Must Replace Science” and the related post above, since it is difficult for me to disagree with the ‘necessary tautology’ that seems to be its basis. I have for years felt that the idea of a Necessary Tautolog was not much different than a Personal Conviction or an item of Faith. Perhaps I might be just as easily convinced that there are no satisfactory symbolic representations for human experience except for thier usefulness in representational descriptions of it….is there any help for such a skeptic?

  2. Both rational logical deductive proof and empirical evidence demonstrate that reality is a mathematical substance. Where else do you have left to go? What else do you need? What else do you want? To keep being skeptical at this point reduces, in fact, to a *faith* that you can get away with refusing to acknowledge reason and any possibility for absolute truth. If all this hasn’t amounted to a sufficient proof of its subject matter…then reason isn’t in fact what someone is looking for to convince them – they’re looking for something else aside from reason…probably something to satisfy their feelings or something.

  3. The bible quote at the end is put there for Christians, to help them think about whether a faith that doesn’t really explain anything at all, or reason which does actually explain everything, is a better way to understand your soul and your existence. If your soul ins’t created, and it isn’t, then who’s this fellow claiming that he owns you, who threatens you with eternal punishment, and who’s books have created untold bloodshed for thousands of years?

    What really needs to be there at the end is a discussion of the other divine attribute of omnipotent power, and hence Will to Power and Becoming, the Dialectic, etc etc. But this was also a good place to tie it off and stop from getting too far away from the title topic…the essential points had been made.

  4. Bill Waters says:

    I have a great admiration for the notion that we humans could clean up our act by adopting a rational basis for our behavior, but there is a kind of madness in human behavior that is worshipped even more than any of its deities and this is irrationality it self. It seems to those who indulge in it (all of us to some extent) an inexpensive addiction to at least temporarily satisfy all desires.

  5. Indeed…more than anything else, humans worship the abject irrational; everything else follows from that.

  6. paul says:

    Hi joe…great post,the links and diagram you added are very helpful.


  7. For someone who complains about sophistry, you have done an amazing job here of engaging in it yourself. Though the deception is to yourself first. What is everything you have just written if not abstract? There is nothing ontological in this. There is nothing real, nothing physical which you can test this on. Zero is not something. Infinity is definable. If everything is zero, then how does 0+0=1? How does nothing become something? Answer: It doesn’t. It happens only in your mind. Only with tricks of mathematics, with no grounding in the world you experience. All you know, all you can ever know, is that you are finite. You are born and you die. The space and time your life occupies, thus become finite around you. You may understand everything you experience by logic and deduction and ontological math, but only if you also frame what you experience in finite terms. But (so far at least) outside of anything you quantify, there is more to explore. It doesn’t matter how big, how small, how cold, how hot, how long, short, wide, deep, etc, etc. The only fundamental was your last attempt to define fundamental. Beyond that there is more to discover.
    The search for THE ANSWER is itself eternal. The greatest pride of the great mind is thinking you know it. You say zero. It could be 42.
    What’s the quote again? “A beautiful hypothesis destroyed by an ugly fact”

  8. “nothing physical which you can test this on”

    There is truth beyond empiricism. But you have to be a rationalist to be able to understand that.

    “Zero is not something”

    Only according to the paradigm of materialism.

    “How does nothing become something?”

    It doesn’t. Nothing IS something.

    “Only with tricks of mathematics”

    1 + 1 = 1 + 1 = 2 is not a trick, but 100% truth.

    “with no grounding in the world you experience”

    The world we experience behaves mathematically.

    “All you know, all you can ever know, is that you are finite. You are born and you die. The space and time your life occupies, thus become finite around you.”

    Only according to the illogical paradigm of Scientific Materialism.

    “The only fundamental was your last attempt to define fundamental.”

    Infinite regression is the least rational answer there can possibly be.

    “Beyond that there is more to discover.”

    Prove that there is no fundamental. If there is no fundamental then there is no answer at all. You seek a state of no answer, and seek only experience with no founding. You can still have experience with a fundamental founding.

    “The search for THE ANSWER is itself eternal.”

    Claim. Prove it. Prove it experimentally at that, since you’re an empiricist.

    “The greatest pride of the great mind is thinking you know it. You say zero. It could be 42.”

    Claim. Prove that it could also be 42.

    “What’s the quote again? “A beautiful hypothesis destroyed by an ugly fact””

    Try again. But before you try again, please read all of the God Series books by Mike Hockney (28 books currently), so that you have sufficient knowledge of what you’re trying to take on.

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