An individual might prefer to know what is right, but that’s just not an option – at least not the strong sense I was referring to earlier. Ultimately, he has got to make his best guess. There are no guaruntees against failure. The undying conceit of the philosopher is that one day he’ll discover a way to prove that he has not erred. Goedel’s theorems seem to poor cold water on that dream; other arguments do similarly, but its rigourous presentation and implications for mathematics seem to afford it more respect. This is why Godel’s theorems make so many people uneasy; it seems to undermine the very goal of their investigations. It’s like discovering that the concrete you were walking on has suddenly turned into quicksand and you’re scared to move. Whether that is an accurate understanding or appropriate response to the theorems is another matter altogether.
The distinction become important when discussing a language which can (and most do) contain an infinite number of propositions. If a proposition is finite (I think most/all? systems define propositions to be finite), then no proposition can be constructed which contains all the propositions in a language with an infinite number of propositions.
Hmm, at this point I suspect we’re using different semantics. My “consistency” is probably the same as your “truth”, and my “overall consistency” is probably the same as your “consistency”. Does that make sense?
On another level, we seem to have no problem embodying an inherently infinite proposition in a finite amount of space. “The set of all positive integers” refers to an infinite concept, but isn’t itself infinite. (I hope this doesn’t derail things.)
I did it because “Why or why not” implies the law (or rather principle) of excluded middle, which (surprize!) I reject.
I have no problems with excluded middle in finite systems - you can always check all the cases and decide whether the statment holds or not. I do not think though it is justified to extend our intuitions or formal rules abstracted from finite and extend them to infinite ones. For P to be true (or false) you either have to believe in existence of world of natural numbers (in some platonic sense) or you have to use some formal system and prove P (or not P) from axioms. I reject the former (though it is a philosophical choice and you are free to take it) and (as Goedel showed) consistent formal systems contain quite a few sentences that it can neither prove nor disprove, and P may well be one of them. If you agree to Goedel, I do not see how can you defend excluded middle for infinite systems.
“The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo–Fraenkel set theory”
Do you still believe it must be either false or true, even if it can neither be disproved nor be proved?
How are you defining an infinite system? Potential infinity or actual infinity? Now my knowledge of set theory is practically nothing however its solution to the problem of zero and infinity seem specious. Zero is the empty set allegedly but if that set actually exists then it must be a set (ie. 1 set). Any notion of actual infinity is absurd. See Hilbert’s Hotel
It seems to me that rather than formal systems and axioms are not basic since they really just hide that they’re based upon natural numbers. How many axioms are you using in a formal system? So yes I would take the Platonic view of the existence of a world of natural numbers. The discarding of a proper metaphysics (or in other words a literal identity) has led to some really odd epistemology.
Confuse not the set with the contents of the set. The set of all animals is not an animal. The empty set exists, but its contents don’t. There is nothing in that set. Just because you have no paper money, you might still own a container for that money, a wallet. The wallet may be empty, but it exists.
The hotel paradox just shows that we humans are unaccustomed to thinking correctly about infinity. Like the savages who see a movie of an oncoming train and run off in panic, thinking they will be run over. Their experience did not include trains that dont run people over; our experience does not include infinite hotel rooms. So they both take getting used to.
It’s strange that you take Hilbert’s Hotel as a reductio ad absurdum, it’s not meant to be. It’s meant to illustrate the behavior of ordinal numbers. There’s nothing absurd about transfinite (infinite) ordinals… simply define a number omega which has the property that it is greater than any finite ordinal. Transfinite ordinals are well-behaved.
Formal systems aren’t hiding anything. The encoding of formal systems into arithmetic statements wasn’t meant to obscure anything but, rather, meant to reduce complex operations to simpler ones. It turns out that this technique does, in fact, work and that complex operations in formal systems can be reduced to a small set of simpler ones, provided that definitions of the complex operations in terms of the simpler operations is given. In other words, all you need to implement any formal system is a Turing machine that accepts a prefix describing another Turing machine it is to simulate. This is known as a Universal Turing Machine. In the computer age we find it unremarkable that any one computer can simulate the operation of any other (perhaps more slowly) but back in the '30s it was a truly revolutionary concept.
I’m shy about the ontology of numbers… I know what I mean when I use a number and (most of the time) I know what other human beings mean when they use a number but I’m sketchy about the idea that “numbers exist” in the same sense that a rock exists (whether or not there is a human there to think about it). It seems to me that numbers are more of an aid or device which our brains find useful in reasoning about the physical world and derive their consistency from the consistency of the physical world to which they were originally devised to correspond. It’s not so much a fact about numbers that 1+1 = 2 as it is a fact about the physical world that one rock and another rock makes two rocks (conservation of mass and other physical properties) and we have devised a formal system or symbol-manipulation system to aid us in calculating (predicting) the outcome of physical processes.
Yes, thought precedes communication, but communication situations are where we use these “laws of thought.” I guess you could also use them in your own mind, but since you would be putting everything into words (else no chance to use the laws of thought, which are about wordings), it is a situation of communicating with yourself. It’s more direct to ask for clarification (even from yourself), because ever since Heisenberg there have been a lot of people who seem to think embracing contradictions might be OK sometimes. The law-of-thought approach prompts such a response, but asking for clarification doesn’t. You either get clarification or they refuse to clarify and then at least there is temporary closure to the matter.
We just call some of our experience “reality” to distinguish it from some other parts of our experience (such as dreamed or imagined things or states of affairs) that don’t matter so much, are not verified by “other people,” are less permanent, less connected to other parts of our continuing experience that we care about, etc. I am hence noncognitive to “ultimate reality,” which I guess falls under the purview of your “there’s no way to know whether what we call ‘reality’ is the ‘ultimate reality’.”
Like all statements, strictly speaking I can only interpret “infinitely many” in the praxeological sense of guidance for my actions. So it would render to something like, “I’m sure I will never run out of twin primes no matter how high I look.” My answer to the first question is then “it seems likely.”
I can only interpret “true” praxeologically, so yes, either the belief “I’m sure I will never run out of twin primes no matter how high I look” pays rent in guiding my actions, or it doesn’t.
“Either P or not P is provable” in this context can only mean "the axioms of arithmetic can be semantically transformed in a standard way (based on English grammar if loosely worded, or based on the rules of logic if written formally) to yield either:
“I’m sure I will never run out of twin primes no matter how high I look.”
“I’m not sure I will never run out of twin primes no matter how high I look.”
Whether or not the axioms can tell me which of these is the case, it is patently obvious - and I need no law to be sure - that one of them has to be the case. Either I am sure or I am not sure. It doesn’t take any principles or laws to make this clear as long as we understand that all beliefs and all statements are only interpretable as praxeological ones, that is, as guideposts for one’s action. It is only the fact that many statements appear to be “objective” that people ever need to be reminded of the Law of the Excluded Middle (and the others). I’d be surprised if anyone seriously held that they can be neither “sure” nor “not sure” of something at the same time (though word confusion makes anything possible).
Good point (the one I think you’re making). The proposition “A and not A” - if we call it a proposition - is incoherent, so it can’t guide one’s actions. If someone says, “A and not A,” I will first respond: “Please unequivocate your A’s.” If that doesn’t work, I will respond, “Why should I care?” It means nothing to me, in the deepest sense of the word.
If we call incoherent/contradictory propositions “propositions,” then we can either arbitrarily label such propositons “true” or “false,” or we are stuck breaking the Law of the Excluded Middle by assigning such proposations a third label. Either way the reader immediately starts to doubt the usefulness of the labels true and false, because they don’t tell the reader anything about what actions he should take to get what he wants. He either loses interest and says, “So what?”/“Please clarify,” or he starts getting a feeling of mysteriousness that he starts to enjoy, and he heads down that path. I recommend the former.
Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms.
[edit]Finitely many new guests
Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests.
This is silly, because the underlined “and so on” can never be completed.
To start from the beginning, praxeologically an “infinite” hotel is simply a hotel with so many rooms that I’m sure I’ll never see the end of them, and I’m sure that everything I ever do with the expectation that I will “run out of rooms” or “do something to every last room” will result in disappointment. (If “infinite hotel” meant anything else, I would have to ask how I can possibly find any meaning in it.)
But for the very same reasons, I can be equally sure that the act of moving each guest will never be completed. So the final sentence in the Wikipedia article pays no rent (no pun intended), since I can most definitely not expect to make room for any new guests. I could do it by some sort of broadcast to all guests at once, so that they all move in unison, but by hypothesis (in bold above) I cannot ever be sure my message reached all the rooms, as I would have to experience the impossible to do so: I would have to experience a flow of information that I could never expect to end.
When “infinity” and “infinite set” are rationally interpreted in a way that actually means something in terms of anticipated future experience, there are no eery paradoxes, just normality.
So you just define Omega into existence? This is begging the question by smuggling in the word “number” before “Omega.” The hidden claim residing in that definition is: “Omega is both a number and not a number (not a finite ordinal).” It’s just going to be a word game, because you’ll have to say that the terms number and finite ordinal refer to “different things” in order to clear up the contradiction. But notice that is just what you’ve got to prove.
You know something is wrong when you find yourself having to prove a definition.
However, there is a way out. You can just take the first-person perspective and not talk about unexperiencable notions. If you phrase everything - including infinity - in terms of experience, I think all the math that matters will still work out, and we won’t have to deal with the risk of people misusing or misinterpreting Godel and friends.
@AJ: That’s ridiculous. There is no reason to limit one’s mathematics to experience or purely practical concerns. There was absolutely nothing practical about imaginary numbers (complex numbers) for the first, oh, 400 years of their existence. Then came James Maxwell and his predecessors and showed how electromagnetism can be mathematically modeled on complex numbers. The computer you’re typing on is a device which would never have been possible without the “useless” doodlings of mathematicians regarding “impossible” numbers over the course of several centuries.
Zero, negative numbers, real numbers, complex numbers… these are all very ordinary tools of mathematics, yet each of them presents serious ontological problems. What, precisely, is zero? What is it? What in the physical world can possibly correspond to zero? The development of zero was held up by this very problem and was resolved by Indian mathematicians and was later transplanted to the Ottomans from whence Europeans would eventually learn its use in place-value numbering systems. Or what is a negative number? To what practical/physical object can a negative number possibly correspond? What is a real number? And so on.
Transfinite ordinals are no different than these other classes of “impossible” or “useless” numbers. Yes, we are overloading the word “number” to refer to things very different than ordinary whole counting numbers which you can correlate to your fingers and toes. So, no, a transfinite “number” is not precisely the same thing as a counting number nor is the use of the word “number” to describe it meant to be a surreptitious attempt to smuggle transfinite numbers into the realm of counting numbers. The two are clearly distinct sub-definitions. Mathematics heavily overloads many words (look up “fixed-point”, for example). Transfinite ordinals simply happen to be a generalization of finite ordinals in the same way that the infinite geometric plane is a generalization of the bounded geometric plane. I don’t see why this should be problematic.
Just like 0 is the “number” which has the property that it is less than every natural number, so omega is the “number” which has the property that it happens to be greater than every natural number. What’s the hold-up?
I was hoping that I could skirt around your take of things, that only that which can help us unscrew a pickle jar is genuine knowledge.
I agree that if we grant that assumption, there is no infinite hotel, and everything’s cool.
But as Clayton wrote, why limit ourselves to this, if our intellect can understand more?
And why should we assume that our intellect is guiding us wrong, if it all seems consistent?
I can but repeat Clayton’s point, that all of the modern [=starting with Newton] physics and engineering triumphs we have at our disposal all assume infinities of various sorts. You can’t have a derivative without it, for instance.
For that matter, Euclid asumes the parallel postulate, that there exist parallel lines that will never meet. Mankind did pretty well for itself with that.
Have you studied advanced Math say to the level of proving something using epsilon-delta reasoning, and arrived your philosophy after seeing how the other half lives? Or are you working without that knowledge of the “enemy” position?
So the final sentence in the Wikipedia article pays no rent (no pun intended)
I suspect your statement holds in a sense that the procedure for accomodation of new guests is not constructive unless the logic we use contains axiom of choice (broadcasting, as you said) or its equivalent (again, I just suspect this, I am not a logician, I am justa programmer). But cannot we solve this by changing our contract a bit? Let’s introduce an element of self-service (otherwise we run out of staff) - all guests agree to move to the next room upon a notification from management or upon a guest from the previous room moving in. You do not care about all the infinite number of current guests moving in a finite time, you care about a finite number of the first rooms being freed for the new guests (the problem of overcrowded halls is not part of the model).
There was absolutely nothing practical about imaginary numbers (complex numbers) for the first, oh, 400 years of their existence.
I agree with your analysis.
Now, is it unimaginable that in future someone comes up with a grand unified theory which fully explains all the experiments to date, but requires natural numbers to behave slightly differently from Peano model for very big numbers?
At the risk of sounding naive, I don’t see how classical logic itself is a finite system. Perhaps I don’t know what you mean by the phrase “finite system”.
On another note, I think you’re getting mixed up between empirical truth and logical truth. To me, the former concerns what is observed in reality, while the latter “only” concerns what is consistent with a given set of premises/axioms. That said, since the Law of the Excluded Middle is a reflection of the Law of Non-Contradiction - and both are reflections of the Law of Identity - I don’t see how you can reject the Law of the Excluded Middle when it comes to logic.
