Gödel's incompleteness theorems & PHILOSOPHY

Statement one seems to be true. However, all mathematics is based on 'assumptions', usually called 'axioms' or 'postulates'. One such is 'a straight line is the shortest distance between two points'. That is a definition of a straight line, not a finding, conclusion or deduction.

Assumptions are not proven, usually. (NOT the same thing as 'demonstrated'.) That is why they are 'assumed'. Usually, they are what seems to be observed in nature (so to speak), and fall under 'common sense' or 'conventional wisdom'.

Statement two is a bit fuzzier to me. I'll work on it a bit further.

AionPhanes said:

What, if any, light can Godel's Incompleteness theorems shed on philosophy? Specifically we could discuss epistemology or even ontology? Other areas welcome too. I might not be able to pontificate like I normally enjoy doing on this specific subject and I'm putting my trust in the philosophy geeks and masterminds here to help me out. Math isn't my thing.

For those who don't know Godel's two famous mathematical theorems they can be explained as follows:

1. Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistant and complete.In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,but not provable in the theory (Kleene 1967, p. 250)​

and

2. For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.​

Now, one might think this could lead to nihilism but interestingly Godel was a strong Platonist of sorts. According to the philosophy professor dude I was listening to in a Podcast not too long ago Godel merely wanted to bring in a phenomenological approach to mathematics. Why, because like Plato he viewed numbers as real objects that one could come in contact with.

[in?]Sincerely , Aion Phanes, Your loony friend writing from the empty set that includes itself as well as all other possible and impossible sets.

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...I'm thinking that Gödel's Incompleteness Theorem implies that we aren't going to be able to do, or to build, everything we would like to.

I'm no expert...but I'm guessing it means we won't be 'folding space' any time soon.

AionPhanes said:

What, if any, light can Godel's Incompleteness theorems shed on philosophy? Specifically we could discuss epistemology or even ontology? Other areas welcome too. I might not be able to pontificate like I normally enjoy doing on this specific subject and I'm putting my trust in the philosophy geeks and masterminds here to help me out. Math isn't my thing.

For those who don't know Godel's two famous mathematical theorems they can be explained as follows:

1. Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistant and complete.In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,but not provable in the theory (Kleene 1967, p. 250)​

and

2. For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, if T includes a statement of its own consistency then T is inconsistent.​

Now, one might think this could lead to nihilism but interestingly Godel was a strong Platonist of sorts. According to the philosophy professor dude I was listening to in a Podcast not too long ago Godel merely wanted to bring in a phenomenological approach to mathematics. Why, because like Plato he viewed numbers as real objects that one could come in contact with.

[in?]Sincerely , Aion Phanes, Your loony friend writing from the empty set that includes itself as well as all other possible and impossible sets.

Click to expand...

...I'm thinking that Gödel's Incompleteness Theorem implies that we aren't going to be able to do, or to build, everything we would like to.

So, (as a non-expert)...I'm guessing this means we won't be 'folding space' any time soon.

AionPhanes said:

What, if any, light can Godel's Incompleteness theorems shed on philosophy?

Click to expand...

I've asked this question about both philosophy and science. The answer a minimalist will give is that it doesn't apply because math is not philosophy; math is not science. I would disagree.

And when disagreements occur, I think the best way to prevent talking past one another is to formalize the discussion. It forces the participants to commit. If they're not willing to commit to a formal discussion, might that be an indication they want to play games to insure they "win"? Or at least that they don't lose?

Once someone commits to a formal discussion, one begins to approach mathematizing the discussion.

Once something is mathematical, it is guaranteed (per Godel) to be either incomplete or inconsistent. I find that a very profound endpoint.

From the human perspective it means we can never "know" - never "prove" - anything, whether through science or any other human construct. It verifies the age-old precept that philosophy is not about determining what is true, but what is possible.

Resha Caner said:

Once something is mathematical, it is guaranteed (per Godel) to be either incomplete or inconsistent. I find that a very profound endpoint.

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This is probably true regardless of whether something is mathematical.

All definitions are going to be incomplete. It means all description will be non-absolute.

variant said:

This is probably true regardless of whether something is mathematical.

All definitions are going to be incomplete. It means all description will be non-absolute.

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Yes. I consider the qualitative to be less definitive than the quantitative, so a corollary to what I said would be that if no way is found to make a definition quantitative (or mathematical) that would imply it is also incomplete.

It's just that unless Godel is hanging out there, someone can always speculate that maybe it's possible for us to produce something that is complete. The only answer would be the skepticism of experience, which people sometimes write off as overly cynical and negative - defeatist.

Resha Caner said:

Yes. I consider the qualitative to be less definitive than the quantitative, so a corollary to what I said would be that if no way is found to make a definition quantitative (or mathematical) that would imply it is also incomplete.

It's just that unless Godel is hanging out there, someone can always speculate that maybe it's possible for us to produce something that is complete. The only answer would be the skepticism of experience, which people sometimes write off as overly cynical and negative - defeatist.

Click to expand...

Defeatism seems like it would be jumping to a conclusion. Observation and description for a user of concepts can be fundamentally incomplete, but dwelling on that just means that you are upset with reality because it can't live up to an imaginary absolute standard.

I think that the incompleteness of any description is probably necessary to have any concepts at all so it's an odd thing to be upset about.