Tuesday, July 21, 2015

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.


21 comments:

Anonymous said...

This post was terrible. Just a case of someone attempting to make a coherent point without understanding any of what he's talking about. You should probably go read what 'completeness' and 'consistent' means in the context of logical systems. Once you understand this, you will see that Gödel was in fact correct. You can't have a system that's both consistent AND complete.

Nathan Coppedge said...

In the context of coherency I'm willing to make what mathematicians would call 'additional irrational compromises...'

I think acceptable premises can still solve the problem if applied at a systematic level. Or at least, some systems are better than others, which ought to still be true.

Please make additional comments if you feel impelled. I will probably post them even if they are critical. I get so few comments on my blog...

Anonymous said...

What set of axioms do you propose?

Anonymous said...

You clearly haven't the foggiest idea of what mathematical incompleteness is. You're just embarrassing yourself. If I'm wrong, why don't you simply state what you think mathematical incompleteness is. Once you do that correctly you'll see it has nothing at all to do with physics, which has not been axiomatized.

Nathan Coppedge said...

Thank you for your comment.

Nathan Coppedge said...

My simple-minded principle was to somehow condense mathematics by relying on physics. If math is hypothetically more general than physics, relying on physics for axioms of mathematics should be cogent of all viable mathematical theories, that is, if they are empirical in any way. However, I am no expert physics, so the writing is intended as a well-meaning work of philosophy of science. I'm sorry I offended anyone. I am not more than an undergraduate.

Nathan Coppedge said...

I may add, I'm a philosophy major, so they might not be the right axioms from a mathematical standpoint, if I concern myself with how to translate between math and physics.

Nathan Coppedge said...

The axioms for translating between general math and specific physics would look something like Aristotle's 10 Categories.

Nathan Coppedge said...

What is it, what does it do, etc. Obviously some of this may be basic to science.

Nathan Coppedge said...

There might be a unified formula for each of Aristotle's categories.

Nathan Coppedge said...

https://hypercubics.blogspot.com/2012/11/working-method-9-macro-and-micro.html?m=1

Anonymous said...

Ok, so can you define mathematical completeness and/or incompleteness? It's a one-sentence definition. By the way the reason you are getting responses today is because someone linked you to the Reddit group Badmathematics. https://www.reddit.com/r/badmathematics/comments/541w4t/logical_solutions_to_mathematical_incompleteness/

Ryan Simonelli said...

Just add a bit of explanation: Completeness is a property of formal axiomatic systems. If you're a philosophy major, you're likely familiar with propositional logic. Propositional logic is a formal axiomatic system. That is, it has set of basic axioms, and a set of rules for proving things from these axioms. The standard formalization has only 3 axioms and 1 inference rule, modus ponens. In an axiomatic system, you're also have a way of interpreting the symbols of language such that you can say what's true. In propositional logic, you can do this with truth-tables.

In a formal axiomatic system, there are two metamathematical properties that you might care about: soundess and completness. If a system is sound, then you're not going to be able to prove anything that isn't true. If a system is complete, then, for anything that's true, you're going to be able to prove it. Propositional logic is both sound and complete (you'll prove this in an intermediate logic course).

What Godel showed is that any formal axiomatic system that is sufficiently powerful to express basic arithmetic, there are going to formula which are true given the axioms, but which cannot be proven in that system. That's the incompleteness theorem. At the time, there were projects which attempted to formalize mathematics into an axiomatic system like propositional logic (this was the project that Russell and Whitehead attempted in the Principia). Godel showed that such projects cannot work.

The anonymous commenter is saying that none of this has anything to do with physics. The reason is that no one has tried to axiomatize physics in the way that Russell and Whitehead tried to axiomatize arithmetic, so it's not clear how Godel's results could be applicable.

Hope that helps explain some things.

Nathan Coppedge said...

I was not being conventional or mathematical, is maybe what you mean.

Nathan Coppedge said...

When I speak of soundness, I speak of soundness of truth, not just axioms. That's why I find many attempts with propositional logic embarrassing. They are not formally coherent about absolute truth. Aristotelian say that well of course one proposition could contradict another, it's called a paradox. They don't believe paradoxes are resolvable in the same way. In my system the most ultimate paradoxes concern pure opposites, and pure opposites must be compared indirectly or they create pure contradiction. I have proposed a general solution to all incoherent paradoxes that seems to work, and coherent paradoxes reduce to coherent deductions that I have formulated. However, my results have not been accepted. Scholars seem to ASSUME that it reduces to conventional truth tables, or worse, that coherent (categorical) deduction is the same thing as syllogisms or imperatives. But syllogisms can be contradicted no matter what, and imperatives only apply to morality.

Nathan Coppedge said...

The advantage of my system is that it is absolutely coherent within ita assumptions. The exceptions are countable: 1. Naive realism, 2. Irrationality, 3. Paradoxicality, 4. Incoherency, and perhaps 5. Non-formalism, if different from the above.

Anonymous said...

So you have no idea what mathematical completeness is, even after Ryan Simonelli graciously provided you a detailed explanation. So why don't you take the time to learn what it means?

Ryan Simonelli said...

Gosh, you're all over the place, dude. You seriously need to slow down if you want your thoughts to be comprehended. Here are my comments from what I could parse:

First, what do you mean that you " find many attempts with propositional logic embarrassing"? Attempts at what? There are a lot of different logics that are good for different things. Classical propositional logic is a very simple formal system. It works for modeling some simple mathematical reasoning, but, it's actually very bad at modelling natural language reasoning if you try to employ it to model all the inferences that we make in natural language.

Consider this example: In classical logic, the conditional is such that, if an inference is good and you add additional premises to the antecedent, it will still be good inference. For instance, if the conditional "If a number is divisible by 2, it is even" is good, then the conditional "If a number is divisible by 2 and divisible by 4, then it is even" is also good. It turns out however, that a conditional of this sort doesn't equate very well to the sort of conditional we normally employ in reasoning about things in the world. There are good inferences in natural language, such as "If you strike a match, it will light," such that, if you add an additional premise to the antecedent, it won't be a good inference. So, while "if you strike a match, it will light" is a good inference, "If you strike a match and you're in a chamber without oxygen, it will light" is not a good inference. Thus, non-monotonic logics might best capture the norms of everyday empirical reasoning, and lots of philosophers who want to model this sort of reasoning employ non-monotonic logics.

Also, scholars don't "ASSUME" that everything reduces to conventional truth tables. What is the case is that you can do the semantics for classical propositional logic, a very simple formal system, in terms of truth tables. But there are other formal systems (like first order logic) that you don't do the semantics for in terms of truth tables.

Second, a contradiction is not the same as a paradox. Paradoxes usually involve ending up at a contradiction by a line of reasoning that seems sound, so the two notions are related, but they're not the same thing.

Third, I've tried to understand your own "system of logic," but I can't make any sense of it. The problem is that you're using mathematical terms, but you're not using them in the way that mathematicians use them (like "incompleteness"), so, if you are, in fact, saying anything coherent, it's impossible to tell what that is, because it's impossible to tell what your words mean. If you want to define your own terms, fine, but be explicit about what you're talking about.

You say you have a philosophy degree, or are working towards one? Didn't you learn how to reason cogently in your studies? You might have the best ideas in the world, for all I know, but if you can't express them in a clear manner, no one will be able to discern them from absolute rubbish, and, to be honest, I can't discern what you're writing from nonsense. I don't doubt that it makes sense to you, but, if you want to be taken seriously from an academic standpoint, you need to learn how to convey your ideas clearly.

Nathan Coppedge said...

It is bad at modelling absolute truth, which is what was meant by truth in any sense originally. Plato placed science itself in the second, third, and fourth out of five or six divided lines because he felt truth was something more important than speculation. In my mind, the coherent system I have found (just search for categorical deduction on Quora.com) realizes the aim of describing truth's absoluteness.

Nathan Coppedge said...

Simonelli, thank you for taking the time to respond. I think I understand you better now. Math is true, but unproven. But in philosophy, things are only true that ARE proven, so from a philosophical standpoint it looks like Fidel, Tarski, and Russell kicked the ground out of mathematics. Mathematicians have already accepted the premises, that's what I understand. But that's the problem with any argument. But they are a very small number of premises, as you said, so it seems to amount to a coherence problem, which was what I was originally addressing (and usually not in relation to physics).

Nathan Coppedge said...

I meant Godel, not Fidel. Damned spellcheck.