Does it follow from Gödel's theorem that this world cannot be fully described by math?

Question

What are the flaws in the following reasoning?

By Gödel's theorem, for any non-contradictory formal system $F$, at least as complex as ordinary arithmetic, there exists a true statement $G(F)$ (or statements $G_i(F)$) that cannot be derived from this formal system.

If we extend it to math in general (math is a formal system, that certainly includes ordinary arithmetic; we'll mark the whole math with $M$ here): in math ($M$) there exist true statements $G_i(M)$ that cannot be derived by any formal rules that currently exist in math. In other words, math in general cannot contain within itself an algorithmic procedure that will discover all possible true statements in math ($G_i(M)$).

Which basically means that there are true statements in math that need an external explorer to discover and prove them. We, people, are actually such explorers and we have the capability of finding and proving some of $G_i(M)$. (Well, it can be shown by using Gödel's theorem itself, but I'll omit this proof for brievity).

And finally, the above means that the world we live in cannot be described by math in all it's entirety. Because, if it could, our human mind (that is certainly a part of this world) would be also fully described by math, which would include an algorithmic description of our capability to find and prove $G_i(M)$, which is a contradiction.

UPDATE 1:

Second take on the above reasoning (trying to be a bit more formal):

1. By Gödel's theorem, in any non-contradictory formal system $F$, at least as complex as ordinary arithmetic, there exists a true statement $G(F)$ (or statements $G_i(F)$) that can be formulated in terms of $F$ but cannot be proven with $F$.

2. All current math (marked with $M$) is a formal system, to which Gödel's theorem is applicable. Hence there exist $G(M)$ (or multiple $G_i(M)$) that can be formulated with $M$, but cannot be proven by $M$.

3. Also, $M$ does not contain within itself an algorithmic procedure to even suggest $G(M)$, because the existence of this algorithmic procedure within $M$ would actually be a proof of $G(M)$. (More rigorous proof is required here, I guess.)

4. Humans (mathematicians) can certainly suggest some $G_i(F)$ and prove them afterwards using rules/approaches additional to $F$, thus forming a new formal system $F'$, where those $G_i(F)$ are proven to be true. (I need to provide some good examples here). Same holds for $M$.

5. Finally, currently known math $M$ cannot describe the world in all it's entirety, because it at least cannot describe our human capability to suggest $G_i(M)$. After we prove $G_i(M)$, we come up with new math $M'$. But $M'$ now contains $G_i(M')$ by Gödel's theorem, and we come back to step 3. This can go on infinitely many times, and neither $M$, nor $M'$, nor $M''$ etc. would be able to fully describe the world and, in particular, our mathematical thinking and the way we do math.

So, what are the flaws/mistakes in the above reasoning? We already have some good answers here, but maybe some additions/corrections?

Answer 1

The major flaw is in your presumption that humans can find and prove statements that are true but not provable within math. Think about what exactly you mean by "math": you presumably mean something like "all math people do". In other words, $M$ would be a set of axioms so that everything mathematicians have ever proven follows from $M$. If that's so, what's your evidence that a human can prove something not proven by $M$? By definition, that has never happened before! And if they could, what would such a proof look like? It would have to invoke axioms or steps of reasoning that aren't in $M$ - but that means it would be assuming something that we don't actually know is true. So how could we believe that "proof"?

Answer 2

Godel's theorem is about axiomatic formal systems. Reality is not axiomatic; no humanly postulated premises are necessary for its existence, completeness, or self-consistency. So what is is part of a non-axiomatic system, raising the hope that it can be described fully in the context of its own system (whatever that might be). Math, which is an axiomatic system, can describe a larger universe of entities than exist in physical reality (for example, there is no physical correlate to Fermat's Last Theorem). Some of those entities apparently include truths that are not provable within given axiomatic formal systems. All of that neither establishes nor disproves that every actual thing can be represented by math. Whether or not the world is fully describable by math might be, but is not necessarily, commented on by Godel's theorem.

Answer 3

You have to be careful with what you mean by "true". In Gödel's incompleteness theorem for instance, "truth" refers to truth in a specific model of arithmetic ($\mathbb{N}$, to be specific- the so-called standard model of arithmetic).

By contrast, Gödel's completeness theorem that anything which is true in any model (understand "any world" if you don't know any model theory) is provable: in other words, what is true is provable. Of course this doesn't contradict the version of the incompleteness theorem you stated.

So, with that in mind, what do we mean by "true" ? For instance, for set theory (e.g. ZFC) it doesn't make sense to refer to the standard model of set theory, because there is no such thing. In particular, the version of Gödel's theorem you quoted makes no sense for theories that cannot be interpreted in $\mathbb{N}$ (or some "standard model") !

In particular, you cannot apply the version "there are true statements that are unprovable" of Gödel's theorem to the mathematical universe (and there's little hope in applying the theorem to the physical universe, anyway)

Answer 4

I would like to add a different perspective. One problem with the reasoning is the assumption that we need to capture reality in just one mathematical system. Note that Gödel's incompleteness theorems generate, for each "good" (i.e. sufficiently similar to ordinary arithmetic) formal system $F$, a Gödel sentence $G$ which depends on $F$. So, at best, this would rule out having one big formal system which globally represented "reality". But it does not rule out having many local systems which represented aspects of reality and which jointly exhausted it.

Answer 5

There's an unjustified assumption in there that the human mind and human thought process is a part of this world.

...our human mind (that is certainly a part of this world)...

Not to get religious on the math stack exchange, but strictly logically, you must proceed from basic assumptions only to conclusions based on your assumptions. You can't make hidden assumptions and get a universally valid conclusion.

Taking the same set of assumptions you have, it could still be possible that mathematics could be used to describe everything about the world we live in, except the human mind.

You've also assumed that the human mind's ability to resolve problems only proceeds along lines that could be algorithmically described. There are plenty of counterexamples, such as ESP, telepathy, prescience, etc., though of course due to their very nature they are usually ignored by scientists seeking solid provable evidence about the nature of the world. If you only consider the behavior of human minds that are behaving like Turing Machines, then of course it will appear to you that minds operate like Turing Machines (a sneaky sort of selection bias).

The nature of life is the key to your question, in actual fact. If you are only a Turing Machine (which is utterly unproven), then your reasoning holds. Otherwise, it doesn't.

Answer 6

Maybe the theory does not need to be complete in Gödel sense to describe the world well enough for our purposes. Maybe it just needs to be good enough.

1. It is not sure the propositions describable but undecidable in the theory will be observable in the world. If they are not observable, then why should we care about them?
2. Then we won't have any practical need to add neither them nor their negations to the theory.
3. It might be that propositions can become observable because of advances in technology and measurement equipment. Then there is nothing that stops us from expanding our theory by adding the proposition or it's negation as a theorem. That is the nice thing about mathematics never being possible to finish (in Gödel sense). We can always expand on it when we learn more about the world.

Answer 7

I've looked at the "bit more formal" version of your question. I think the problem is with point 3.

First, let me quote your first two points:

1. By Gödel's theorem, in any non-contradictory formal system $F$, at least as complex as ordinary arithmetic, there exists a true statement $G(F)$ (or statements $G_i(F)$) that can be formulated in terms of $F$ but cannot be proven with $F$.

2. All current math (marked with $M$) is a formal system, to which Gödel's theorem is applicable. Hence there exist $G(M)$ (or multiple $G_i(M)$) that can be formulated with $M$, but cannot be proven by $M$.

I'm going to assume that "all current math" means ZFC. Certainly, some people (including me!) are going to disagree with the notion that ZFC satisfactorily captures "all current math", but my objection to your argument is going to be pretty much the same regardless of exactly which system is chosen for $M$.

3. Also, $M$ does not contain within itself an algorithmic procedure to even suggest $G(M)$, because the existence of this algorithmic procedure within $M$ would actually be a proof of $G(M)$. (More rigorous proof is required here, I guess.)

This isn't true at all.

It is, in fact, possible to define ZFC inside of ZFC. If you're doing mathematics inside of ZFC, then you could define a set $z$ which represents ZFC itself.

It's also possible, inside of ZFC, to define a function $f$ which takes any formal system $F$ and returns the corresponding statement $G(F)$. The function $f$ can even be defined as a particular algorithmic procedure.

And so, now that you've defined the function $f$ and $z$, the expression $f(z)$ refers to the Gödel statement of ZFC, $G(\text{ZFC})$.

You say that the existence of $f$ "would actually be a proof of $G(M)$". As a matter of fact, $f$ does exist, but it doesn't prove $G(M)$ at all.