LOGICAL SOLUTIONS TO MATHEMATICAL INCOMPLETENESS

I have been musing about the dialogues between Einstein and Niels Bohr, and also the relations between the greatness of Einstein and the greatness of Kurt Godel. My solution to this dialogue is that there are key insights which overshadow the doubts that emerged at the time. You will excuse me if I introduce some psychology in the schematizing of the thoughts of the time, as psychology is often what is warranted in solving intractable problems. In part because mathematics and physics have often been seen as related disciplines, it can be seen that there is some factuality to the influence of hormonal competitiveness with Einstein in the specific design of theories of his competitors, such as Godel’s incompleteness.

I will leave that complex idea as an initial lemma, except that there is a particular relation between logic and mathematics that may be worth discussing. This is specifically the role between proof theory (a.k.a. Einstein), and the causal relationship between mathematics and physics.

For, we would not ordinarily say that physics is incomplete----at least I wouldn’t. Instead of criticizing this point as having irrelevance out of the inherent incompleteness of physics, we can adopt a logical tool, and say that physics is semantically complete. For, after all, in logic we would not say that a theory is incomplete if it appeared (like Aristotle’s syllogisms), to provide an explanation for any type of conclusiveness we might imagine. Nor would I say that this view is naïve, since even in science the deference is to the best available theory. We would not say science is wrong because it does not know absolutely everything. Instead, we would say that it is relatively complete. And, I argue, it is the same with physics. Unless we are being very technical about what completeness means----and I think we aren’t absolutely technical here, just as the physics is not absolutely complete----then it makes sense to consider physics as though it has some degree of completeness, e.g. it is a relatively successful attempt at a complete explanation. Even if there are multiple categories of physics (relative, quantum, string theory), each category undoubtedly contributes to the completeness of physics, or the theories would be regarded as quack science.

Now that I have argued that physics is considered relatively complete, I would like to make an interesting stipulation. If physics is considered complete, and math is not considered complete, is there something that can be had here? Although math is not normally considered ‘physical’ ---- could it be that the immaterialism of math is groping with a primitive spiritual idea, instead of what it should be doing, which is accepting some perhaps unseen form of objectivity?

For, what is ‘incompleteness’ saying except that math cannot be objective? Wouldn’t objective incompleteness be an oxymoronic definition? Or would it just mean that math cannot ever be complete? Then, are we saying that math cannot ever be complete in a complete sense, or are we saying that math is itself un-objective? I think no one will make the claim that math is un-objective, and asking if math can be complete in a complete sense begs the question of whether we are in fact being relativistic. For there is no sense of math apart from the objective sense, unless it becomes a math of un-objective things. But, instead of wading deeper and deeper into the sense of math as an un-objective application----which clearly leads to un-objective conclusions----the principle that truth is the obviation of the obvious gives us several options: (1) Truth is obvious, (2) Truth is about obviating, (3) Truth isn’t obvious, and (4) Truth isn’t about obviating. Clearly I think it is the case where math is not about obviating that seems like the weak point. But have we proved for definite that math cannot obviate truth? I think regardless of the amount of education required to learn math, it certainly can! And this goes against the principle of Godel’s incompleteness.

However, to prove for definite that there is some mathematical principle based on physics that could be foundational for math requires additional reasoning. But there is nothing which says such ideas could not be foundational in some exceptional, acceptable sense.


In my own theories on logic, I have arrived at a concept of a bounded Cartesian Coordinate system defined by polar opposite word pairs. In this case, it was simply conceptualizing differently which permitted the context to be understandable. I would advocate a similar solution for math. The concept that some mathematical principles are complex, unavailable, or infinite may be limiting the cogency of mathematics. Furthermore, there may be some way in which infinity is not being conceptualized appropriately. My own solution has been that the trans-finite is a product of division rather than multiplication. Unless there is a concept of a whole, math will remain incomplete. But if infinity is a byproduct of multiplying and adding exponents, this assumes the consequence that there is no whole, and thus, that math is incomplete. But the process by which this occurs is not mathematical, instead, it is a more rudimentary logic that might be proven wrong, as I have shown. Certainly the concept that math is not whole does much to refute mathematics, if it comes last. But I think it could just as easily have come first. If it is a matter of cause and effect, and it could be either one, then it is clear that incompleteness is not 100% correct.

In logic, if there is something ambiguous, the process is to search for new and creative ways to solve the ambiguity. These tools are not as easy to apply in math, when the assumption is that it is a product of its products. However, math sometimes involves philosophy. It sometimes involves logic. Some of its assumptions could be wrong. And I think this is the most likely explanation for any form of absolute mathematical incompleteness.

As I mentioned, I have thought of some possible ways to support math by combining it with physics. What if, for example, there was some strength to physical arbitrariness? Once we assume that math is physical, as it may well be in some sense, we can then perhaps prove that since math is more arbitrary than physics, then math has greater support than physics! By this form of arbitrariness, what I mean is that it is not as directly influenced by causal laws. Or, more precisely, it applies to a wide range of phenomena without participating directly in their chain of causality. And, even if math did participate in an object’s chain of causality, this would not make math less arbitrary than the objects being determined. The objects are by definition, the most determined things about observation. Math, then, whether it depends on observations, or does not depend on observations, remains less determined than matter. And, where it is less determined, so far as it does not disappear, it has a more permanent influence.

So, at this point we have a few options. Either (1) math disappears, or (2) math has some influence. But, here is the important corollary. If math has influence, math is physical. And what is physical is not incomplete. We already know, since it is less determined than matter, that it is more influential than matter. Therefore, math is more complete than matter!


---Nathan Coppedge
July 21st, 2015
New Haven, CT


It will also be available as an academic paper at: https://www.academia.edu/14269683/Logical_Solutions_to_Mathematical_Incompleteness if you want a more cite-able format.