In the other "Godel" thread questions came up concerning the application of the Incompleteness Theorems to Philosophical Systems. I decided to do some research to see if the theorems apply only to arithmetical systems, or if they can be extended. I figure it deserves its own thread. The main questions asked in the thread was "what constitutes a sufficiently complex system," and "does the system have to be arithmetical."
The formal term is "sufficiently strong." A formal theory of arithmetic T is sufficiently strong if it captures all decidable numerical properties. An example of a decidable numerical property is that a number is prime.
In this particular text, properties are considered equivalent if they have the same extension, i.e. if the set of numbers satisfying them is equivalent. So "square root of 4" and "even prime" are both numerical properties which have the extension "2", so under the above definitions the "square root of two" is the same numerical property as "even prime." The only other criterion for something being a property is that it can be written in the language of your formal theory.
This immediately creates some concern for philosophers seeking to exempt themselves from Godel's findings. Suppose you take a reduced English language as your formal language. You could encode a set of objects in your language with the natural numbers, and then objects which satisfied your property would be the extension of that property. For instance, if we encode every single living creature on the face of the Earth at present with a natural number, this would (as far as I know) be finite. If we define "mammalian" as a property, a subset of living creatures would satisfy that property, and thus a subset of the natural numbers would possess this property.
This allows us to extend Godel's theorems beyond Arithmetical Logic to any Logic whose language can be encoded by the natural numbers. The question then is not if "non-numeric" properties can be considered, but if these properties are decidable. I.e. If we can "decide" if an object possesses that property. This is probably, if you don't want the theorems to effect philosophy, what you want to argue against.
Thus, Godel's Incompleteness theorems apply to a philosophical system if: 1. We can, using an algorithm and with infinite time, decide whether or not that property is true for every possible object in your language. This excludes more candidates for a property than one might think. For instance, the property "encodes a theorem in the 'nice' formal Theory T" does not count as a decidable property. 2. Have a list of axioms in the system which, given infinite time and paper, we could list out
I'd be interested to see if anyone with a more formal education in Logic has any comments. I'm also curios if you guys think "possesses a human consciousness" is a decidable property.
"The other which is such for itself is the other within it, hence the other of itself and so the other of the other – therefore, the absolutely unequal in itself, that which negates itself, alters itself. But it equally remains identical with itself, for that into which it alters is the other, and this other has no additional determination; but that which alters itself is not determined in any other way than in this, to be an other; in going over to this other, it only unites with itself. It is thus posited as reflected into itself with sublation of the otherness, a self-identical something from which the otherness, which is at the same time a moment of it, is therefore distinct, itself not appertaining to it as something." —Hegel, Science of Logic §21.107
Formalize this, op.
Carter Torres
>If we can "decide" if an object possesses that property. That seems to be the essence of what you are getting at, right? But I still am not sure what you are getting at.
Do you mean, we 'decide' if a basketball has the property of sphere? or if an orange has the property of sweet?
Aaron Gonzalez
Hegel would have benefited a lot from coherentizing his writing. Using numbers and A, X, Y, letters, and circles, and pie charts, to show what he was actually talking about...but we all know he didnt want to do that, because then he would not have been able to ride on the coatails of mystique and confusion. Because what he says really boils down to simple things, that could be said in half the words he uses.
Andrew Hill
Well you would start by probably reducing his language to something intelligible. Part of the issue is that Hegel uses words which actually imply a relationship already; the word "other" already implies "not something."
But just for this particular passage, I would do this define in my formal language a objects "A" and "B". Then I would define as acceptable operations "negation," "reflection," "identity", "implication", "and" and "sublation."
The passage reads: 1. B identity Negation() 2. B identity B 3. B identity Negation(B) implies B identity Negation(Negation()) 4. B negation(identity) B 5. B negation(identity) B and B identity B 6. B negation(identity) B implies B identity B 7. (6) identity reflection(B) 8. (7) identity sublation(B)
That's not perfect, but it should get the point across.
Carter Nguyen
Yes, that's what I'm getting at. You represent the basketball as a number. Spherical applies to the basketball, thus we can code it a numerical property. It seems to me we can 'decide' that the basketball possesses that property in a finite (read, effectively enumerable) number of steps, and that determining if an object in general is spherical should take a, if not necessarily finite, effectively enumerable number of steps. Thus the property is property is represented as a numerical decidable property.
Jackson Reed
In the something, being is being-in-itself. Now, as self-reference, self-equality, being is no longer immediately, but is self-reference only as the non-beingExistence of otherness (as existence reflected into itself).
The only thing not intelligible in the paragraph itself is the bit on reflection into self, yet this makes sense in light of the structure that is made apparent in the developments before this section.
>"The other which is such for itself is the other within it, hence the other of itself and so the other of the other – therefore, the absolutely unequal in itself, that which negates itself, alters itself." "Other" as a concept is a concept which immediately doubles itself. What we mean when we say a thing is an 'other' is precisely that there is some other thing to which it is other to. When you take this conceptually as an absolute standpoint, the other is other to the other, thus the concept is other to itself in this opposition of one and the same concept put against itself. Otherness is a concept that always necessitates a split from itself; thus, it is an operating concept of self-negation in the dialectical method.
>But it equally remains identical with itself, for that into which it alters is the other, and this other has no additional determination; but that which alters itself is not determined in any other way than in this, to be an other; in going over to this other, it only unites with itself. It is thus posited as reflected into itself with sublation of the otherness, a self-identical something from which the otherness, which is at the same time a moment of it, is therefore distinct, itself not appertaining to it as something." Take the concept of the 'other', let it do its self-negation, and what does it give you? The other. This is how it remains self-identical in its otherness. It unites with itself because the other posits a negative unity with its other: it always is the other of the other, it is never just the other on its own. The reflection is this structure of "other of the other". It is in itself because taken as absolute, the other of the other has to be within the other. Something is the self-identity of the other as it reflects off of the other, and this is the nature of identity statements when spoken from an inner perspective, i.e. I am I and I=I are made by us through this capacity of self-othering ourselves as objects before ourselves.
Quite wrong since the whole thing is a simple concept analysis in conceptual content terms, it's semantic.
Chase Flores
You wrote literally the exact same thing I wrote in longform.
Isaac Foster
1. All you have is syntax, not semantics. 2. Anti-foundationalism is quite fashionable, as you can find it in Heidegger, Peirce, and the man called Late Wittgenstein the Late, seminal figures in 20th century Western thought, be it in the continental, pragmatic or analytic school of philosophy, respectively.
Noah Brooks
Kvine 2
Robert Ross
Your logic for saying Godels theorem can be extended to non arithmetic systems is rather shaky, since from what I can understand from it you're stating that all reality can be communicated in an arithmetic fashion (ie associating real objects to numbers in such a way so that their numbers interact with each other in the same way the objects do) Is this the case? I don't find it convincing since it seems like trying to fit an infinite amount of objects into a room with infinite space and the logic always gets wonky when you do things like that
Luke Diaz
so what are you wondering, the cases that are not finitely decidable? Do you have any examples? If the color red as we (or I) perceive it will possibly exist in a trillion quintillion years from now?
Jonathan Gonzalez
>"Other" as a concept is a concept which immediately doubles itself. What we mean when we say a thing is an 'other' is precisely that there is some other thing to which it is other to. When you take this conceptually as an absolute standpoint, the other is other to the other, thus the concept is other to itself in this opposition of one and the same concept put against itself. Otherness is a concept that always necessitates a split from itself; thus, it is an operating concept of self-negation in the dialectical method. And could this not have been said thusly: There are multiple things?
Daniel Johnson
Ok so like: well the thing is there is top down and bottom up:
Start with atoms/the elements/molecules? number them, and then show how all the numbers interact? The problem is, maybe numbers arent the best to use, and then we immediately see something I realized a while ago but just recalled: first let me say why numbers arent the best to use: because they already have a specific purpose, purely the relations of interacting quantities in spaces:
If I say basketball = 1 basketball player = 2 basketball court = 3 hoop = 4 Sphere = 5 orange = 6 bumps on basketball = 7 black ridges on basketball = 8 the concept of dribbling = 9 the concept of dribbling with the force of x= 10 the concept of dribbling with the force of x+.001 =11 the concept of dribbling with the force of x+.002 = 12 the concept of movement direction degree 1 = 13 the concept of moving direction degree 1.00001= 14 moving at speed .0001 mph = 15 the concept of speed = 16 etc. the concept of fans = bleachers = cheering = coach =
What are you saying, to use this information in what way: if you try to push these numbers together to attempt to depict the reality as truly and vividly as reality is: a basketball player dribbling up the court hearing directions from the coach approaching a defender and trying to get by them:
You are going to be 5 + 4 divided by 3 time 7 - 2 in time frame 4 in space 3 + 15 .......
And now what I recalled about my past insight: language is math.
Instead of using numbers to detail all that: we use letters, words.
What is the point in introducing numbers as labels for things? When the meaning of numbers as labels is interaction of purely quantity (hint, what is the nature of quality, can it be purely expressed in terms of quantity: love, fear, sweetness, (maybe, x type (type, quality?) amount of chemicals etc.). Numbers are labels of quantity, reality is more than pure abstract quantity: because there are types of quantity:
1 is = to 1 but 1 proton is not equal to 1 electron nor 1 apple = to 1 orange
but you are saying: let apple = 1 and orange = 2
But it is being said: let "that round red thing = A P P L E; and let that round orange thing = O R A N G E"
And let these lists of qualities equal those lettered terms ____________
Justin Lee
You would think it's shaky, but it's actually the same thing we do for set theory in general.
In set theory, we call the empty set zero, the set containing the empty set 1, the set containing 1 and the empty set is 2, etc. Essentially, power sets of empty sets are what we code to the natural numbers, and this is sufficient to extend Godel's theorems.
If we start with objects instead of the empty set, we get a similar extension for "reality."
The relationships between the objects don't necessarily need to have an "arithmetic." Just because we code "empty set" as zero and "the set of the empty set" as 1, doesn't mean that "empty set plus the set of the empty set" is the "set of the empty set". There is a relationship that does that, you can take the union of those two sets, but that's not quite "addition" as we think of it; thus the relationship does not need to be a true representation of addition or any such "arithmetical" property.
Jackson Harris
Yes that's exactly right, the only point of my coding is to get around the claim that language is somehow substantially different from an arithmetic theory, and thus exempt from Godel's theorems. All the coding does is allow us to change "properties" in the sense we think of them to "numeric properties".
The desire is not to conflate quantity and quality. It is different to say "2 has a certain cardinal size" than it is to say "2 is prime," but for the purposes of the theorems both of those are properties of a number, even though they're different types of properties.
I think a substantial argument could be made that certain qualitative properties are not decidable, although that's sort of admitting that your system is already incomplete.
Jacob Long
>Start with atoms/the elements/molecules? number them, and then show how all the numbers interact? You might want to check this out plato.stanford.edu/entries/frame-problem/
Mason Nguyen
No, because the other is the explicit functional concept which generates distinction. 'Things' are already self-othered unities, that's why something comes after other as the standing thing which contains a being within itself that is and is not itself.
Adam Ortiz
I think it's possible to argue that there is no finite set of operations which would allow us to determine whether something possesses human consciousness, for example. Thus "possessing human consciousness" is possibly not a decidable property, and would fall outside the capacities of any philosophical system.
Eli Perez
1. The fact that the syntax is present means that any error in semantics expresses the failure of language, and any philosophical system by extension, to represent a property. For instance, if the property "even prime" somehow related to something broader than the number 2, but in our system it only represented the number 2, this would indicate an incompleteness on the part of the language to capture the property "even prime." So it is possibly that our language cannot represent certain ideas, but this would simply be admitting that our theory is incomplete already.
2.Anti-Foundationalism is not quite what I'm going for. I think we ought to establish foundations, but just realize we can never prove the consistency of such foundations, and we must realize that they'll always be incomplete; they will either be unable to represent properties, be unable to assign properties, or be subject to Godel's Incompleteness Theorems.
Kevin Morris
>The relationships between the objects don't necessarily need to have an "arithmetic." water + soil + sun + seed = seed growth
Anthony Thompson
How do we decide what the center of the earth looks like? Or the center of the sun? How do we decide what a photon looks like in and of itself as it actually is before we detect it and interact with it?
Oliver Bell
Numbers are different in a sense, because of infinity: If someone said: there are only 1000 numbers! and everyone agreed and believed them, would Godels incompleteness theorem hold?
Or is that preceisely the point, they would be incorrect in asserting? Because you can always +1...+1....+1...+1...+1....+1....+9999^9999... +999999999^999999999.....+1.....+1....
So the scope and frame of the possible interactions and tricks between numbers are infinite.
Now we can also say that we are unaware of the full scope, quantity, and quality and nature of Substance Of The Universe, and all the possible forms and interactions it can take, on earth, and other planets, in the past, and in the future.
And even 'of the subtle substance'... like cartoons for instance, how different qualties can continously be combined, and not seen in physical reality, but represented with forms, color, shape, implications: griffin, dragon, spongebob, catdog, dracula, etc...
Before these things were came up with it was possible they were never in an earthly human mind before: it is possible in the history of absolute reality those were the first times those concepts were conceieved.
And so to cavemen, the potential of substance and minds relation to it, was certainly incomplete: and we can only figure, there will be more things created on by humans in 100 years then we can imagine: though sci fi likely has covered some bases.
Jose Bennett
I assume you are trolling and/or mentally disturbed: but...No.
Multiple things exist. Some are more similar to each other, some are more different. Some things, are people, they have minds that can see and think, they can recognize that they are not a tree or rock (other).
Why could he not have said that?
Carson Powell
>the failure of language, and any philosophical system by extension, to represent See, nobody it the philosophy of language takes representational theory seriously anyway.
Anglos have this habit of making fun of people who "argue semantics" as if meaning was a bad thing, and the formal logic their philosophy loves so much would rather "argue syntax", and I don't see that big of an improvement.
If language isn't seen as the solution to everything anymore, it's because of the Indeterminacy of Translation by Quine, not Gödel's Theorem.
>Anti-Foundationalism is not quite what I'm going for. Then why did you start this thread? To wallow in misery and hopelessness?
David Walker
So with this statement, regardless of what you think and believe, you are suggesting its possible for every grain of sand to be an individual consciousness? And we cannot develop machines to detect what we think we are familiar with as consciousness reactions taking place in animals and human heads?
Brody Martin
For the center of the Earth or Sun, there is a process by which, given infinite resources and time, we could determine what these look like, so there's no problem there.
As for the things like the photon's exact position and velocity, that's a good example of a non-decidable property. We could determine the position, or the velocity, but not both, so the entire "state" of the photon cannot be captured. Thus it can't be described by the system, and the system is already incomplete.
I don't really care if a particular property is not decidable. If it's not decidable, that basically means your system is already incomplete.
Easton Wood
>2.Anti-Foundationalism is not quite what I'm going for. I think we ought to establish foundations, but just realize we can never prove the consistency of such foundations, and we must realize that they'll always be incomplete; they will either be unable to represent properties, be unable to assign properties, or be subject to Godel's Incompleteness Theorems. Imagine if the main reason God made the universe was to try to figure this stuff out
Julian Walker
>and the system is already incomplete. which is pretty much what was meant by Kants Noumena
Reality doesn't owe us to impress into our minds the totality of actual truth: we are fortunate we exist at all (especially as anything other than maggots) and can detect a single quality: we then are extremely fortunate that we can piece together what appears to be quite a lot of durable ""approaching truths"" of reality
Austin Jackson
burmp up the jam burmp it up while your feet are burmping
Michael Sanders
well?
Isaiah Morris
...
Caleb Butler
whats it have to do with?
Gavin Rivera
huuuhhh huuuuhh
Samuel Gutierrez
>Veeky Forums is a slow board! Please take the time to read what others have written, and try to make thoughtful, well-written posts of your own. Bump replies are not necessary.
>Bump replies are not necessary
>not necessary
Was this necessary?
Ayden Cook
No, I started it because the question came up in the other thread and I thought I would share what I found, which is that Godel's theorems can be extended to philosophical systems which aspire to actually represent something about our world
Bentley Hernandez
Well for grains of sand we can determine that they are not human, but if we take a human we may not ever be able to decide that it possesses consciousness. My inclination is that we could, but I'm unwilling to make a definitive statement.
Eli Jackson
Oh there are definitely systems which admit their own incompleteness, like Kant's. It's one of the reasons I like him, he has a very measured approach to philosophy.
Lucas Sanders
Conciousness is a word used to point at the 'reasons' (some of which more mysterious than others) that humans can do more than grains of sand. (and I dont mean to implore you to respond 'well., now hold on just a second, I think grains of sand can do quite alot, like erode cliffs and such!') I mean of the conceptual nature of self action, the concept of acting of own accord
All evidence points to there being 'an inner controller' that makes the body get up out of bed in the morning
Isaac Wood
.
Blake Bailey
but maybe what you are suggesting, is the difficulty of pin pointing exactly what consciousness is and how it works, the hard problem? How 'that which sees what is seen in the head' 'exists, and is seeing what it sees', 'how that which sees, and thinks and feels, accesses memories'
Robert Roberts
Axiom 3 is the weak point. The whole thing basically boils down to constructing an ultrafilter P on the set of propositions, but the chain of reasoning before axiom 3 fails to rule out the possibility that G is necessarily false for all x and therefore is not in the ultrafilter.
Alexander Reyes
This is actually easy Hegel. I just gave up on the Phenomenology at the Force and Understanding chapter because he stopped defining terms.
Joseph Mitchell
Is that about substance, numbers, objects, concepts/ideas?
Samuel White
It's Godel's ontological "proof" of the existence of God. Of course it follows 100% logically from the axioms, but the whole question is whether you believe the axioms or not.
William Bell
>philosophy consists purely in building systems
Nathan Rodriguez
in simple terms what is it saying? What is it referring to? Time, substance, first mover, sequence, intelligence/awareness, design/order?
I'm not sure I know what you're asking, but the argument is basically this: Let X be the set of all possible properties, and P be the set of all positive properties. We assume that: 1. If a property p is positive, any property q implied by p must also be positive. 2. A property is positive if and only if its negation is not positive. 3. The property equivalent to "x possesses all positive properties" is positive. From this, it's easy to derive that there is a unique object satisfying all positive properties, which we label "God". The problem is that the third assumption is trickier than it looks. By axiom 2, every conceivable property is either positive or negative. It is entirely possible that the collection P of positive properties is so large and internally inconsistent that they cannot all be satisfied simultaneously. In that case, there is no "God" object and the proof fails. Mathematically, Godel's argument is equivalent to asserting that the set of positive properties forms a principal ultrafilter on the set of all properties. It is trivial to show that a principal ultrafilter has a unique minimal element, corresponding to the "God" of Godel's proof, but there is no convincing reason that I've seen to suppose that the set of positive properties really does form a principal ultrafilter in the first place.
Joshua Jones
hmmm,
So after step 3?
Ok. So. Set of all positive properties [a,b,c,d,e,f,.......]
Then he assumes: If God exists, he would embody those properties, contain them?
And where is the turning point of the proof that says therefore God must exist?
I dont think that 3rd step is that outrageous, or impactful: The argument is whether God exists or not, not necessarily .....
First actually I must do what I should have done before I went ahead and assumed: what is meant by the terms positive and negative here?
Existence and non existence? Or good and bad?
Alexander Reyes
..
Jace Reed
I wonder if there is a modal logic which deals with statements that are unprovable
Dylan Edwards
...
Benjamin Smith
There is literally a version of modal logic where the interpretations of the modalities "necessarily P" and "possibly P" are "P is provable" and "P is not disprovable", respectively.
Joseph Perez
neat
Adam Taylor
.
Jordan Long
Thist thread is too fucking high IQ for me. How do I into math and logic lads I failed at life
Adam Gray
Incompleteness by Rebecca Goldstein is a pretty approachable pop-math/bio on Gödel.
Pick up a cheap Intro to Proofs textbook, there's a ton out there. Should cover most of the basics for logic and post-Calc math.
