Wittgenstein

user21820

Sep 5, 2017 15:39

@user170039: I've read about 20% of the pdf so far. I wish to start mentioning key points so that I don't forget them later. The first point is that Wittgenstein is being very imprecise in what he is said to have written (page 10). I will fault him for at least that, even if one argues that he actually had complete grasp of the incompleteness theorems. Specifically, he implies that Godel's "true" means "proved in Russell's system".

It is easily understood that Godel was working in a meta-system that already assumes the existence of a model of PA. I always emphasize that the meta-system must have such an entity, otherwise (arithmetically) "true" and "false" have no meaning at all. And that notion of arithmetic truth is definitely not coincident with "proved in Russell's system".

Anyone can, if they wish, question whether there is a model of PA in the first place. If there isn't, then all mathematics is built on uncertain foundations, because of the various reasons I've explained before: Accepting any formal system as meaningful already requires commitment to the closure of finite strings under concatenation, whose lengths are a model of PA.

But once one accepts the existence of a model of PA, then there is nothing wrong with the MS already assuming existence of such an entity, and Godel's proof goes through.

user21820

Sep 5, 2017 16:48

The second point is that my above comments also addresses Shankar's claim (page 24). He is correct that to treat meta-mathematical statements as purely absolute is an error; it is merely stated within MS, and it's entirely possible that MS is wrong (if you're platonist) or meaningless.

However, he's wrong to say "Gödel was barred by virtue of the logical grammar of mathematical proposition from claiming that he had constructed identical versions of the same mathematical proposition in two different systems." Godel didn't do such a thing, as would be clear to anyone who has basic grasp of logic and model theory.

user21820

Sep 5, 2017 17:01

And Floyd's opinion on page 38 (whether an accurate interpretation of Wittgenstein or not) is invalid. It is true that not every English sentence that looks like a factual statement can be assumed to have a truth value. I explicitly explain that issue in one of my posts:

3

A: Is Godel's modified liar an illogical statement?

Your question has two main facets. The first is that you did not grasp the way logic does not fall to the liar paradoxes. The second is that there are deeper reasons as to why we have such apparently innocuous sentences in natural language that seem to defy assimilation into formal logic systems....

But it is invalid to go from that claim to the general claim that all mathematical statements are of the same nature.

If one denies existence of a model of PA (in a physical form), then fine you can say the incompleteness theorems aren't valid (no platonic meaning), but then you need to give a convincing explanation for why RSA decryption works so well. You can find more detail of such issues in the following:

9

A: What are the arguments for and against "one true arithmetic"?

In short: The so-called definition of natural numbers as those that can be obtained from 0 by adding 1 repeatedly is circular, but there is no viable alternative, which already makes it impossible to uniquely pin down natural numbers mathematically. Worse still, there does not seem to be ontologi...

Anyway, as of now my opinion on Wittgenstein hasn't really changed; he (like many others) fail to realize the significance of the incompleteness theorems. I'll grant that it may be difficult to see it through Godel's initial work. To appreciate it, one has to fully grasp the computability aspects and the constructive nature of some of the proofs.

I'll try to state the significance very briefly: Formal systems were invented as a way of precisely delineating rules of reasoning by which we would like to ensure sound reasoning via syntactically verifiable deductive steps. But the very notion behind formal systems, namely string manipulation, requires prior philosophical commitment to meaningfulness of finite strings, and requires commitment to the theory of concatenation TC (see details in the paper linked from my post), more or less.

But no formal system that (computably) interprets TC is complete and consistent, as a result of the generalized incompleteness theorems (Godel merely proved them for a single formal system, so it may not have been obvious to casual readers how to generalize them).

To be able to state and prove this fact of incompleteness, the meta-system MS needs very weak assumptions. Definitely ACA is enough, and ACA is essentially what you get if you assume that there is a classical model of TC, plus induction.

So there are only two main defensible positions:
(1) Accept TC as classically meaningful, and hence accept the incompleteness theorems, which apply to anything that computably interprets TC, which include all humanly conceivable formal systems.
(2) Reject TC as classically meaningful, and hence reject the very assumptions underlying all humanly conceivable formal systems themselves! Unless of course you believe that all 'correct' formal systems have some cutoff string length.

Incidentally, as mentioned in my post there are self-verifying theories, which give an intriguing possibility lending support to the last part of (2). Strange, but possible.

By the way, I dislike popular accounts of the incompleteness theorems that do not make clear the dependence on the model of PA in the meta-system. Every explanation in my opinion must define clearly what is meant by (arithmetic) truth. Otherwise it's indeed misleading, and many philosophers have (again in my opinion) done injustice to logic by misusing the incompleteness theorems.

@user170039 Feel free to respond to any point. Though I'd encourage you to first understand the proof of the incompleteness theorems (at least syntactically) before attempting to judge its semantic content or possible lack of it. =)

Wildcard

Sep 6, 2017 03:47

@user21820 I'm interested.

@user21820 Slightly off-topic, but I find it fascinating that people study this sort of thing and write papers about it, interpreting it, rather than actually philosophizing.

Also, existentialcomics.com/philosopher/Ludwig_Wittgenstein

user131753

Sep 6, 2017 05:02

13 hours ago, by user21820

@user170039: I've read about 20% of the pdf so far. I wish to start mentioning key points so that I don't forget them later. The first point is that Wittgenstein is being very imprecise in what he is said to have written (page 10). I will fault him for at least that, even if one argues that he actually had complete grasp of the incompleteness theorems. Specifically, he implies that Godel's "true" means "proved in Russell's system".

user131753

I agree with what you say in the above messages.

user131753

Sep 6, 2017 05:13

I couldn't find (in page 24) what you claim Shankar to have said in the following,

user131753

12 hours ago, by user21820

However, he's wrong to say "Gödel was barred by virtue of the logical grammar of mathematical proposition from claiming that he had constructed identical versions of the same mathematical proposition in two different systems." Godel didn't do such a thing, as would be clear to anyone who has basic grasp of logic and model theory.

user131753

Sep 6, 2017 05:36

Regarding,

user131753

13 hours ago, by user21820

But it is invalid to go from that claim to the general claim that all mathematical statements are of the same nature.

user131753

Can you explain the reason for saying so?

user131753

Sep 6, 2017 05:55

Since you didn't say anything else regarding the remaining part of the paper, I will only ask you questions when you have gone through it @user21820.

♦ $Mathematics$ Logic

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Wittgenstein

♦ Logic

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♦ $Mathematics$ Logic