Genius post.
It is a great strength of Homo sapiens that we can, better than any other species in the world, learn to model the unseen. It is also one of our great weak points. Humans often believe in things that are not only unseen but unreal.
I am dealing with this in another thread.
I might as well jump in at this point.
How do you know the proposition in question is true if it can’t be proven using those things? This is really a question about the definition of “true” being used in the Incompleteness Theorems.
How do you know the proposition in question is true if it can’t be proven using those things?
We don’t, the reasoning is more like (quoting myself):
we say, either this sentence is true, and then the formal system contains some true but unprovable sentences, or this sentence is false, and then the formal system contains some false but provable sentences.
So the system is either incomplete or inconsistent. In other words, every consistent system is incomplete (or if you prefer, every complete system is inconsistent).
it took a while for mathematicians to get this, but all that we humans do, even when studying math, is use our noodle and our common sense. What constitutes common sense and correct logical thinking [for a mathematician] is what is accepted by the mathematical community. Of course sometimes a genius may come along and bring to light some way of thinking that wiull take a while to be accepted, but that is a detour to our main subject.
what Godel did was show that the sentence he had in mind was true to the eyes of mathematician looking at the sentence from outside the system [=axioms and rules of deduction] he was studying, while at the same time showing that the system under study could not prove the sentence.
This might sound naive, but if the proposition isn’t provable either way, then why are the conclusions different between saying the proposition is true vs. saying it’s false? In other words, why does saying it’s false mean the formal system contains some false but provable propositions, instead of meaning that it contains some false but unprovable propositions?
In other words, why does saying it’s false mean the formal system contains some false but provable propositions, instead of meaning that it contains some false but unprovable propositions?
Let’s read the essence of that sentence again:
[this sentence] says that in this system this sentence cannot be proven
So if this sentence is false, the this sentence CAN be proven in this system.
Sorry, I didn’t realize you were talking specifically about that proposition. At this point, I share the position that you outlined here.
Actually, the Godel sentence doesn’t assert its own truth, it asserts its own unprovability which creates a dilemma: either the Godel sentence is true (and, therefore, unprovable) or it is provable (and the formal system in which it was constructed is inconsistent). So, you have to chooose… either all true statements in your formal system are provable or your formal system is consistent (this is Godel’s second incompleteness theorem). Since it’s difficult to imagine inconsistent truth, we reject the former possibility as even intelligible.
Clayton -
I look at it differently… we can perform arithmetic on small numbers and from these arithmetics we can extract the underlying mechanics by which we are manipulating the numbers. Given these underlying mechanics, the question is whether there is any reason why the mechanics cannot be extended indefinitely. Consider addition, for example. I need only an addition table and to understand how to carry in order to be able to add any two digits from two numbers. So, given that I can easily and quickly add numbers having as many digits as my patience will endure and that the operation of addition does not change from digit to digit, what reason is there to object to the idea of addition of numbers with more digits than my patience could endure?
The same is true of the non-visible portions of a pile of grains. If you stack a few grains on a table, you can spread them with your hand and see that there are as many grains as you started with. You might be able to do this for a larger stack. After a certain number of grains, your patience could not endure the counting of grains to ensure you had the same number of grains after putting them in a stack as you had before putting them in a stack. But it would be an implicit denial of a great deal of what we consider to be “real” to suggest that if you put a really large stack of grain down that the number of grains might increase or decrease a little bit … after all, you’d never count them to be sure. My response to such misplaced skepticism would be to ask why we should suspect that the number of grains might be more or less and, absent any reason for suspecting this, there is no good reason not to assume that the properties we observed in the small (that the number of grains remains the same before and after being placed in a stack) hold in the large.
Clayton -
Forgive me if this sounds rough around the edges. My thinking on this issue is that any self-referential system necessarily contains multiple levels of consistency. For example, “This sentence cannot be formulated” is certainly grammatically consistent, but we wouldn’t say that it’s semantically consistent.
I don’t think it’s possible to combine these multiple levels of consistency into a single level. What I guess that means is, any self-referential system lacks a necessary standard for “ultimate truth”.
there is no good reason not to assume that the properties we observed in the small (that the number of grains remains the same before and after being placed in a stack) hold in the large.
There was no good reason for quantum or relativity theories until very specific experiments were conducted. Newtonian laws worked just fine for the usual set of experiences people call “real”.
It depends on your definition of the word “know”. But it sounds like your definition is necessarily experiential in nature. So if we’ve never counted more than a few hundred real objects, we don’t know, by that definition (i.e. from experience), whether the axioms of arithmetic correspond to reality. Technically, however, we could never know it in that way, since there’s no formal bound to the set of positive integers.
Fortunately, we have a way around this issue. It’s called mathematical induction.
Fortunately, we have a way around this issue. It’s called mathematical induction.
…which was given to Moses along with the ten commandments? Or rather codifies some experience? This principle is exactly the essence of natural numbers, and if one does not believe in properties of natural numbers of any magnitude, why should he use mathematical induction?
At the risk of taking the bait, I’ll say that mathematical induction is valid a priori, so there’s no real need for experience. Then again, even the laws of thought themselves (pace AJ :P) were derived from experience.
Then again, even the laws of thought themselves (pace AJ :P) were derived from experience.
…and as experiences differ, some people do not believe in law of excluded middle. And similarly, different logicians have different ideas of induction.
Keep in mind that I’m talking about mathematical induction, not inductive reasoning. The two are different.
Some people don’t believe in the law of the excluded middle? Really? Why not?
Some people don’t believe in the law of the excluded middle? Really? Why not?
Schrödinger’s cat.
Wasn’t the thought-experiment of Schrödinger’s cat intended as a reductio ad absurdum against those quantum physicists who were forsaking the law of the excluded middle?
Wasn’t the thought-experiment of Schrödinger’s cat intended as a reductio ad absurdum against those quantum physicists who were forsaking the law of the excluded middle?
How does answer to this question invalidate the statement that some people reject the law of the excluded middle?
Also, may I suggest looking up “Is logic empirical?”?
I hope you have no problems with me using only questions in my post, do you? 
I regard “laws of thought” like the Law of the Excluded Middle the same way as I do the Law of Non-contradiction (and the “Action Axiom” for that matter) - as redundant and unhelpful responses to failed communication attempts.
As for what constitutes a valid definition of knowledge, that is a fairly deep rabbit hole, so we should expect plenty of disagreement until quite careful definitions are agreed upon.
Schrödinger’s cat is just nonsense, since a belief that the cat is “neither dead nor alive” or that “the universe hasn’t decided” does not pay any rent in terms of anticipated experience. In other words, it is not a belief at all, just a bunch of uninterpreted words that do not do anything to guide my actions - they are meaningless in the deepest sense of the word.
As for Gödel’s theorems, the Gödel sentence doesn’t communicate anything to me at least until it is put in its physical form, which I think has something to do with Turing machines and/or the halting problem - someone might refresh my memory on that.