If your definition for existence is being able to see it with your eyes, then nothing can exist for a blind person. Sound does not exist. Odors do not exist. Infrared light does not exist. Heat does not exist.
It doesn't matter what your definition for existence is. It doesn't change the fact that the empty set is a useful idea.
Sense it how? Would you consider weak infrared light to exist? No human being could sense it with their senses alone, but they can build instruments that pick up the infrared light and depict its presence to a user via an interface compatible with their senses.
Name:
Anonymous2012-01-13 8:23
>>18 Would you consider weak infrared light to exist? No human being could sense it with their senses alone, but they can build instruments that pick up the infrared light and depict its presence to a user via an interface compatible with their senses.
Yes. It exists as long as you can detect using some device. For example, world in video game exists (even if in computer memory), in as sense, that you can detect it using video display.
Name:
Anonymous2012-01-13 8:26
>>11
my dick. i can't feel it, you can't feel it, but it's out there, somewhere
alright, but now, how do you know if the device is lying to you or not? Or perhaps your senses alone are lying to you. Optical illusions are an example of this. Once you realize the optical illusion, you see it for what it is. But what if you never realize it? How would you know that the illusion doesn't actually exist, even though you perceive it to legitimately exist from what you seen with your senses?
The point is that there are things that you can perceive that don't exist, and there are things that exist that you can't perceive. It might be better to use a different definition for existence, but I'm not going to try to offer one. Getting specific about it just opens doors for inconsistency.
# This codes demonstrates that empty_set exists
try:
empty_set = set()
except:
print 'there is no empty set!'
if empty_set.issubset(set((1,2,3,4))):
print 'empty set is subset of nonempty set'
else:
print 'empty set is not a subset'
>>48
That's like saying "qzmogsdfjzeogwns" corresponds to the number 16 because of its length. The list-to-set conversion you're using is completely arbitrary and probably isn't the only possible one.
>>37
No, you are shit at comparing things. Here's Python's interactive environment doing the same thing:
Python 2.7.2 (default, Jun 12 2011, 15:08:59) [MSC v.1500 32 bit (Intel)] on win32
Type "copyright", "credits" or "license()" for more information.
>>> set()
set([])
>>>
>>53
[cpde]
$ ghci
GHCi, version 6.8.2: http://www.haskell.org/ghc/ :? for help
Loading package base ... linking ... done.
Prelude> id id id id id id id id id id id id id id id id id id id id id id id id id 0
Fuuuuuuuuuuuuu...
Interrupted. [/code]
Name:
Anonymous2012-01-13 21:43
>>54
Lisp | Haskell
----------------------------------|---------------------------------------------
(defun id (x) x) | Prelude> id id id id id id id id id id id id
(id (id (id (id (id (id (id (id | id id id id id id id id id id id id id 0
(id (id (id (id (id (id (id (id | <interactive>: out of memory
(id (id (id (id (id (id (id (id |
(id 0))))))))))))))))))))))))) |
0 |
C
------------------------------------------------------
id(id(id(id(id(id(id(id(id(id(id(id(id(x)))))))))))));
Segmentation fault
Name:
Anonymous2012-01-13 22:08
Do not argue with the ultrafinist, or WWBT! (I should know better, for I'm about to)
He doesn't even believe in the concept of natural numbers, much less in any more complex abstract concepts (such as set theories).
The only thing he seems to believe in are his direct senses, but he refuses to actually infer (by induction or deduction) anything from them.
He claims mathematics is religion, and I'll partially agree with him on this(more on this soon), although mathematics just teaches "If A then B" and it doesn't force you to believe in some particular A. Mathematics proves very useful in the real world and thus the consequences and things it teaches have a good chance of being true. There is another way to look at this: claim that computation exists (in the Turing-equivalent sense), and you also get most mathematical theorems from that. The thing about computation is that it just means that simple finite abstract rules can be followed and will always give the same result in all possible worlds. It can be shown that computation can be encoded in arithmetic and a Turing Machine can also do arithmetic (unbounded). The Chuch-Turing Thesis shows that computation is universal and could even be considered a natural class. If you posit a reality having well-defined physical laws, you'll also have to posit some form of computation (our world is capable of it, at least in the Finite State Machine sense, a restricted Turing Machine). If after some thought experiments you end up subscribing to functionalism or computationalism, you also have to put computation in the ontology (actually it's the only thing that you truly require). Such beliefs, be they in a reality, your own mind or in the fact that computation works, or in mathematical truths are "religious" beliefs, although they do have substatial evidence for them being true - in another way, some parts of math are good things to bet on being true. If you ask me a belief in the Chuch Turing Thesis is a good "religious" belief - a bet on something being true(even if supported by evidence), such a bet taken more seriously can lead you to some forms of Platonism (again, religious belief). Some beliefs are incompatible, for example, the belief in functionalism and a negative belief in computation, or a belief in an ontologically primitive reality and computationalism (the version where one assumes one's mind exists), and one should not hold incompatible beliefs if they want to be correct.
Me, I have no problem in making such bets (about certain things being true) when I have good evidence for them. Why? I risk being more wrong and stand to gain getting closer to more true statements. Of course, a smarter way to go about this is to do uncertain inference: hold confidence values for each belief and adjust them based on evidence.
Too bad our local ultrafinitist doesn't want to have any beliefs, and so he will never ever have a chance of being right or wrong. He may hold various beliefs, but will refuse to acknowledge that he holds them. In other cases, he must bet on some theory, and if he bets or not is itself a choice, and not betting is also a form of "belief" (or lack of).
Good luck to you, ``in LISP''/symta-guy, and I hope you do some more thinking about what beliefs are acceptable to you and which are not, as well as what standards you use to select them. It's easy to pick some ultraskeptical standard which doesn't allow you to hold any beliefs at all, but that limits greatly one's abilities, to the point of never being able to be right or wrong, or making any choices. I think it's great having a chance of being right, and if you're wrong, that's not a problem as long as you retain the ability of recognizing wrong beliefs and removing/replacing them, thus being less wrong as time passes (and now I expect that you'll dispute time's existence - that can be done as well, but again limits your options!).
Name:
Anonymous2012-01-13 22:15
>>57 The only thing he seems to believe in are his direct senses, but he refuses to actually infer (by induction or deduction) anything from them.
No. I'm don't refuse. I just want the inference to be justified by senses too. The inductive proposition that "given N, there exists N+1" cant be justified.
>>58 I just want the inference to be justified by senses too.
The problem with this is that you won't be able to form any theories/metatheories about why you have senses at all, or why physics (or just whatever makes your senses be this or that) behaves the way it does and so on.
That inductive proposition talks about abstract systems, in the real-world, you will have a local limit imposed by physics.
If for example you deduced by induction that reality corresponds to some specific mathematical structure. Why only that structure? Let's say its information limit is some constant k. Why not k-1 or k+1? Occam's Razor suggests that the simplest theory is that all possibilities are realizied, thus also the k+1 world. A belief in only world k is a stronger belief than a belief in all finite worlds 0,1,...,k,k+1,...
It's a stronger belief just like the belief that the sun doesn't exist as long as you don't look at it(such as at night).
Math itself tries to be as general as possible - in that it can also be used by physics, but general enough to not limit itself to only what's locally physically possible. There is no reason why any finite natural number should be the limit, hence why you get a countable infinity of natural numbers.
You could say that all this is abstract non-sense, and that you don't care about why anything behaves like it does.
Consider this then: in the far future, you're very old and have developed some incurable brain tumor, the doctor offers you to get a digital brain replacement (be it gradual or instant, this is a thought experiment, so pick whatever suits you best), do you say yes or no? You have to bet on a theory because your future experiences depend on it, yet without having worked out all the possibilities you cannot make a choice, but you don't have the luxury of not making a choice - a default no also has costs.
>>59
Why are you arguing about math with someone who has no expectation that the sun will rise yet again tomorrow? (I could ask him the corresponding question, but it would be less fruitful.)
Name:
Anonymous2012-01-13 23:20
>>59 The problem with this is that you won't be able to form any theories/metatheories about why you have senses at all, or why physics (or just whatever makes your senses be this or that) behaves the way it does and so on.
Why should we form useless theories, which are completely unrelated to reality?
That inductive proposition talks about abstract systems, in the real-world, you will have a local limit imposed by physics.
Why do we need "abstract systems" or any other religion?
If for example you deduced by induction that reality corresponds to some specific mathematical structure.
I've a better idea! Lets find correspondency to some physicaly sensible structure, and throw the mathematics aways.
Why only that structure? Let's say its information limit is some constant k. Why not k-1 or k+1? Occam's Razor suggests that the simplest theory is that all possibilities are realizied, thus also the k+1 world. A belief in only world k is a stronger belief than a belief in all finite worlds 0,1,...,k,k+1,...
Does Occam's Razor suggests it's own validity. and how do define the "simples theory". For a religious man/woman the term "God" maybe simple (it's much "simpler" than your phyisical theories, Einstein).
It's a stronger belief just like the belief that the sun doesn't exist as long as you don't look at it(such as at night).
It exists in my memory (or machine's memoty, if we talk about AI).
Math itself tries to be as general as possible
You found the main problem with math. There is no such thing as "general" or "all" or "everything", without some kind of bounded universe, like cons-pair list, one passes to Lisp's `all` and "any" high-order function.
in that it can also be used by physics
Can you imagine a “physical process” whose outcome could depend on whether there’s a set larger than the set of integers but smaller than the set of real numbers? If so, what would it look like? -- Scott Aaronson
There is no reason why any finite natural number should be the limit, hence why you get a countable infinity of natural numbers.
There is no reason why physical world should be the limit, hence why you get a heaven with almighty God.
You could say that all this is abstract non-sense
Mathematicians themself call it nonsense: http://en.wikipedia.org/wiki/Abstract_nonsense
There is good reason for that - it cant be justified by senses.
Consider this then: in the far future, you're very old and have developed some incurable brain tumor, the doctor offers you to get a digital brain replacement (be it gradual or instant, this is a thought experiment, so pick whatever suits you best), do you say yes or no? You have to bet on a theory because your future experiences depend on it, yet without having worked out all the possibilities you cannot make a choice, but you don't have the luxury of not making a choice - a default no also has costs.
How that is related to the question of existence? In both, "yes" and "no" cases, it wont contradict my point of view. Doctor just makes a replacement copy, like you copy audio CDs. No magic or paradoxes.
Name:
Anonymous2012-01-13 23:21
Python, Lisp, and Haskell are shit.
Name:
Anonymous2012-01-13 23:23
>>60 Why are you arguing about math with someone who has no expectation that the sun will rise yet again tomorrow? (I could ask him the corresponding question, but it would be less fruitful.)
Do you have strong evidence, that sun wont rise? Why not? I think you're lying, mathematician!
No. I'm don't refuse. I just want the inference to be justified by senses too. The inductive proposition that "given N, there exists N+1" cant be justified.
Why should we form useless theories, which are completely unrelated to reality?
If you would just fucking study it, you would learn about the discoveries that have been made, and the applications that were found form them years or decades after their discovery. You are using your ignorance as justification for its lack of worth.
>>61 I've a better idea! Lets find correspondency to some physicaly sensible structure, and throw the mathematics aways.
I have no idea how you could talk about it without some form of math. Does Occam's Razor suggests it's own validity. and how do define the "simples theory". For a religious man/woman the term "God" maybe simple (it's much "simpler" than your phyisical theories, Einstein).
You could define it by algorithmic complexity, such as Kolmogorov complexity. God is very complex in popular definitions and almost everyone has different definitions. There is no reason why physical world should be the limit, hence why you get a heaven with almighty God.
There are many more choices than that. There are also concepts which are logically inconsistent thus cannot have any form of existence (except as an inconsistent idea). Consider for example the position that computation is the only thing contained in the ontology. If that is the case, there are ways one could show that other worlds which are non-physical (in the local sense) can exist and can be accessible, if you want an example of how this could be practically realized, you should read Greg Egan's "Permutation City". However, if you privilege only some particular structure with "existence" (typically called physical nature), then that would no longer work, but you'll have a more complex theory that privileges only this single world with existence. Where do you stop? What interpretation of quantum mechanics do you choose? Not all choices are compatible with each other. Each assumption has very specific consequences.
How that is related to the question of existence? In both, "yes" and "no" cases, it wont contradict my point of view. Doctor just makes a replacement copy, like you copy audio CDs. No magic or paradoxes.
Replacement copy? Would 'you' still be conscious? Would you have some form of contintuity of consciousness? Would it change the nature of your senses? If you assume some multiverses (which are actually forced upon on you, if you assume both computationalism and that you have a mind, but since you refused to reason about it, you're not aware of such consequences), do you predict that your future experiences to be in the same universe and that they would be stable (what about changes in measure)?
These things are not trivial. If you assume you have conscious experience/senses, that has consequences. If you assume it's an illusion, then there are different consequences.
If your belief is non-computationalist, you would believe that if you say 'no', you still have a survival chance, but saying 'yes' kills you. If you have a computationalist belief, you can say 'yes', but you can never know for sure that your beliefs are correct (despite appearing internally consistent), thus in either case it's a "religious" bet. Or was your question about what does this have to do with the existence of natural numbers? A belief in computationalism imposes very strict requirements about what one's ontology has to be (due to the Chuch Turing Thesis and what is known as the "Universal Dovetailer Argument" and "Movie Graph Argument(1-3)"). I could elaborate on this, but it would be less effort for both you and me if you just used google and looked up the arguments by yourself as they are way off-topic to this thread.
>>67
I should stop responding, but at times it makes me think that he probably believes what he's saying, which makes me wonder if he realizes that his beliefs might not be consistent(free of contradictions).
Can you have negative one item(s) in a set? (Eg. maybe something that isn't in the set?)
would the empty set have a minimum bound negative infinity / maximum bound 0? (contains everything that ISN'T in the empty set??)
Name:
Anonymous2012-01-14 1:10
>>72
There is no light without dark. You can sense emptiness only relative to something.
>>70 not in my playroom. I have infinite dominoes, and I watch them topple all day long, for eternity.
You have autism. Also, note, in some countries, the term "autism" is considered to be a lighter form of schizophrenia, because it sometimes progresses into it. So you are a potential schizo/psycho.
If you would just fucking study it
How can one study something, he doesn't understand. "Set Theory" is axiom, meaning one should have inborn understanding of it. I dont have. I cant imagine "a set", this highly abstract, invisible, unordered and unbounded entity.
I have no idea how you could talk about it without some form of math.
Lisp's lambda (in it's call-by-name form) allows proving sensible theorems without any math. And lambda is a physical entity. Of course lambda has its "problems", but these problems manifest themselves only in the presence of "infinity", meaning they aren't really problems, but the way systems shows us, that we doing something wrong.
You could define it by algorithmic complexity, such as Kolmogorov complexity. God is very complex in popular definitions and almost everyone has different definitions.
Sorry, I dont understand terms "algorithmic complexity" and "Kolmogorov". The notion of "mathematical complexity" is by itself pretty complex for my little goy brains.
There are many more choices than that.
For me, there are only choices I see. Making hypotheses without confirming data would be a waste of time. Even the most crazy projects, like SETI, have some confirming data.
Replacement copy? Would 'you' still be conscious?
What is "consciousness"? It's a vague term. I'm afraid, you yourself don't see what you're talking about.
I should stop responding, but at times it makes me think that he probably believes what he's saying
I do believe what I'm saying, because I havent seen better theory, than subjective idealism.
which makes me wonder if he realizes that his beliefs might not be consistent(free of contradictions).
"consistency" is a buzzword and in reality, there are no contradictions, except those we invent/define.
Well if you don't like math, then feel free to not study it. Life it too short to base it around things you'd rather not do. But you should acknowledge that there are things about math that you are not yet aware of, both in itself and in its nature. There is nothing concrete about math. It appears very formal, but it was devised by people, and over a very long period of time, was unified and reinforced with a standard vocabulary. Of course, you need to stay very formal or else you will quickly get lost, and you wont be able to get your bearings. Logic forms the ground at your feet. If you get lost, you can look at the ground and see where you can immediately walk. But if you want to get somewhere you will have to see the big picture. But if you can't see the ground you are walking on, you'll probably trip and fall into a cave of incorrectness. Then you need to slowly retrace your steps until you arrive at where you fell, and then ponder what the right step should have been.
>>74 For me, there are only choices I see. Making hypotheses without confirming data would be a waste of time. Even the most crazy projects, like SETI, have some confirming data.
Actually there are physical evidences/confirming data for some such theories, however science can never "prove" anything, it can only disprove some false theories. What is "consciousness"? It's a vague term. I'm afraid, you yourself don't see what you're talking about.
The fact that I have unified experience: sound, images, other senses; that such senses have some continuity over (subjective) time. If you think the term is nonsense, just say so. There is just one problem with this: you infer the material world's existence through your senses (by induction) and then from that inference you see that anyone claiming to be conscious is delusional about it as there's no such thing as mental experience in the material world, except you just used your own senses to infer the material world's existence, so it's self-defeating. There are also more subtle contradictions with this as shown by the "Movie Graph Argument" or "China Brain" thought experiment (the only problem is that in the China Brain example, the author does not see that there is a solution to the problem which is free from absurdities). I do believe what I'm saying, because I havent seen better theory, than subjective idealism.
My favorite candidate for now is a form of "arithmetical"/computationalist neutral monism (in it, both mind and matter are a consequence of "immaterial" natural number relations). Unlike a lot of philosophical/metaphysical theories, it makes some very solid predictions about the shape of physical law (confirming some of the laws of quantum mechanics) as well as future expected experiences (some of which let you set it apart from other theories, thus making it falsifiable). Is it possible that I am wrong? sure, no beliefs or mine are absolute, it's just the one which I have the highest confidence value for now, if it's wrong, then some extra "magic" will have to exist to explain both physical and mental phenomena, and some rather counterintuitive things will be possible (beyond what most scientists expect today). Why this theory and not another? It follows as a consequence of a few rather simple assumptions, but any assumption has a chance of being wrong, no matter how much it makes sense to me right now. Still, I must bet on what makes most sense to me, and I don't have any problem with that.
Anyways, the problem with subjective idealism is that it's not really falsifiable (unlike my form of neutral monism) and it cannot be used to predict anything. What's worse is that it's a solipsistic theory - it only says that you have mind, it doesn't claim that I have mind. In my case, I can predict that both of us have minds and even a wide array of properties of those minds, as well as what the physical world is, why it exists, what is its relation to one's mind, why does that exist and so on.
"consistency" is a buzzword and in reality, there are no contradictions, except those we invent/define.
I didn't say that reality is inconsistent. I think that whatever reality is, it cannot be inconsistent. It's not a buzzword, it has a very specific meaning: a contradiction or paradox in beliefs - it happens when you prove that something is both true and false starting from some assumption or axiom. In the real world, it could be something like someone holding contradictory beliefs, such as "this person is my father and is not my father", where each of the terms in that sentence have un-ambigous meaning within that person's mind. except those we invent/define.
Of course contradictions are in our own mind - they just show that some of our reasoning or beliefs are wrong and we must re-examine them.
"Set Theory" is axiom, meaning one should have inborn understanding of it. I dont have. I cant imagine "a set", this highly abstract, invisible, unordered and unbounded entity.
If you can't while others can, then you are defective.
Name:
Anonymous2012-01-14 3:21
>>76 Of course contradictions are in our own mind - they just show that some of our reasoning or beliefs are wrong and we must re-examine them.
But mathematicians love to "define" contradictions. When I hypothesise that I can move hand into some occupied space, and then fail to do this - that is an invented contradiction. When mathematician proposes that moving hand in some (un)occupied space is a contradiction (by sole definition), that would be a defined contradiction, that has no sense, because it's in strict contrast with senses.
Name:
Anonymous2012-01-14 3:25
>>79 you are defective.
You mean I'm not "ashkenazi"? I'm perfectly okay with that.
>>80
It's easy to get contradictions or paradoxes in natural language - just look at the liar paradox, or various self-refernece sentences.
Contradictions and paradoxes are not always obvious, even when talking about formal systems. For example, naive set theory is inconsistent. We don't even know if modern set theories are consistent. By Godel's incompleteness theorms, we cannot even know if arithmetic is consistent, and that any consistent system containing (or being stronger than) arithmetic cannot ever claim its own consistency as a theorem (if you do, the system is inconsistent). Claiming consistency as an axiom is okay(not that you'd really know if the new system is consistent), but any belief about arithmetic's consistency is a matter of "religion" (since it cannot be known, despite that it would be utterly strange for it to not be consistent) - you cannot know that in a countable infinitity of inferences (in well-defined first order logic) there is no contradiction. Still, such belief is common in almost all scientists, along with belief such as any finite sentence written in the language of First Order Logic + Peano Arithmetic has a truth value (true or false). In more complex systems, such a belief may not be warranted - some statements can be undecidable or independent in the system, this is especially common with infinitary set theories, examples of independent axioms: Continuum hypothesis, Axiom of Choice, various "high" non-constructive cardinal axioms.
You define something, but you don't know if it's sound or consistent(free of contradictions), or if it's true or false.
Let me give a more complex example, consider Goldbach's conjecture "Every even integer greater than 2 can be expressed as the sum of two primes". It can be defined in arithmetic. It can be computably verified (given unbounded resources) by a Turing Machine, yet if it's true, the process will never terminate (it will never find an integer which isn't the sum of 2 primes). Which means that to prove or disprove Goldbach's conjecture (and many other mathematical hypotheses) one must show if one particular process halts or not.
Some set theories makes bets about the behavior of the infinite, or in this case, infinite processes, and if those bets are correct, you will be able to say if some process terminates or not, even if you would not know of that given only the "beliefs" of Peano Arithmetic (definitions of successor, addition, multiplication and an induction axiom; if you don't like the induction axiom, you can avoid it, and you'll get Robinson Arithmetic instead, but in RA, even if you can still compute anything, you can prove much less, not even simple sentences like: "x + y = y + x").
Consider Goodstein's theorem ( http://en.wikipedia.org/wiki/Goodstein's_theorem ), it states that some particular (computable) sequence terminates, yet it cannot be proven in Peano Arithmetic, but can be proven in some stronger systems (such as some set theories). Of course, we do know that the Halting problem isn't generally solvable, but the idea here that with stronger systems, more things can be proven and more (particular) cases of the problem can be solved. The downside of such betting is that the stronger you make the theory (by adding axioms) the more you risk adding contradictions and making it inconsistent (can prove anything, thus it no longer talks about truth).
Mathematicians cannot always know if a theory has contradictions or not ("always" knowing will require you to have a solution to the halting problem), despite knowing its axioms, but they can hope to discover such contradictions if they exist (otherwise they will work on a false theory).
Name:
Anonymous2012-01-14 5:40
>>83 One with more imagination is greater that one without the capacity to imagine.
How "greater"?
Name:
Anonymous2012-01-14 5:59
>>84 By Godel's incompleteness theorms, we cannot even know if arithmetic is consistent
As I said "there are no contradictions, except those we invent/define."
Mathematicians are trying to solve nonsense problems, they created themselves by some wrong definition of "consistency" in a poorly defined framework, that rests on some crazy axioms (see ZFC), purpose of which a layman wont get without tracing full 20st century history of mathematics, which is full of hacks and silly conventions.
You define something, but you don't know if it's sound or consistent(free of contradictions), or if it's true or false.
It's consitent, when it's parts are consistent, together with their composition.
they can hope to discover such contradictions if they exist (otherwise they will work on a false theory).
Like the Banach-Tarski Theorem, which postulates that given single orange you can transform it into two oranges, by the sole power of applying Set Theory axioms. Behold The Wonder of Infinity's Creation!
>>86 Mathematicians are trying to solve nonsense problems, they created themselves by some wrong definition of "consistency" in a poorly defined framework, that rests on some crazy axioms (see ZFC), purpose of which a layman wont get without tracing full 20st century history of mathematics, which is full of hacks and silly conventions.
I'm sorry, but to me consistency makes perfect sense, all these terms are well-defined. It makes sense within computer science, first order logic and the various arithmetical theories.
I do partially agree that the syntax of some parts of math is a bit messy and it could be done in a more easily parsable manner, but this doesn't change the fact that the semantics would still stay the same even if you chose a better syntax (look at some theorem provers if you want examples of more "parsable" syntaxes).
I'm not an expert on Set Theory, but as far as I understand it, originally there was the simpler naive set theory, with rather intuitive axioms, but it proved inconsistent (see Russell's paradox), so they had to choose some axioms that limit what sets are and how they can be defined. The axioms themselves may seem a bit strange at first as some of them do appear non-constructive. If that bothers you, you could look at a constructive or iterative approach of building sets, one example that I like is described in the first chapters of Boolos' "Logic, Logic, Logic" - it takes a few simple axioms which are intuitive enough (as opposed to the ones in ZFC which make you wonder where they appeared from) and then it shows how sets are built in it and also shows that most ZF axioms are theorems in that particular theory! You might not like it because it has infinity in it, but it can't be avoided, even with only natural numbers: the process of listing the numbers obtain from 0 and a successor function never terminates - the list/set is infinite (countably). Instead, just understand what infinity is and where it comes from and then you can even look at things like ordinals which are well-defined (constructively even, and a computer/theorem prover can talk about them too, despite being 'infinite'!). It's consitent, when it's parts are consistent, together with their composition.
How do you know the parts are consistent?
Consistency in math means that starting with some axioms and some logic (inference rules), you will never prove any syntactically valid statements in that language to be BOTH true and false. It's fairly simple and well-defined. You can even write a program that could find an error in some inconsistent axiomatic system (of course, for most modern ones, this is highly unlikely and your theorem prover which is searching for an inconsistency will never halt if the theory is consistent, but you'll never know this!). Like the Banach-Tarski Theorem, which postulates that given single orange you can transform it into two oranges, by the sole power of applying Set Theory axioms. Behold The Wonder of Infinity's Creation!
I don't find that paradox that much stranger than saying that the cardinality of some continous real interval like [0,1] has the same as the cardinality of all real numbers, or that there are as many even natural numbers as natural numbers or as rational numbers, yet there are more reals than naturals. Infinity is counterintuitive like that, but if you can show a bijective function between 2 sets, then their cardinality is the same. It may be strange, but as I said before, there is no reason why you should set a higher bound for natural numbers, and if you do, you're only inviting trouble for your theories (I don't even know of a single consistent ultrafinitist theory, but maybe you know more about them than me). Not that anyone forces you to work with infinity directly, computable functions only work with finite numbers, just that one has to acknowledge that there's a countable infinity of them (both computable functions and natural numbers).
Name:
Anonymous2012-01-14 7:25
>>78
Set A is a subset of A, it just isn't a proper subset.
>>86 Like the Banach-Tarski Theorem, which postulates that given single orange you can transform it into two oranges, by the sole power of applying Set Theory axioms. Behold The Wonder of Infinity's Creation!
This is what Jesus used to feed all those people.
>>68 Like the Banach-Tarski Theorem, which postulates that given single orange you can transform it into two oranges, by the sole power of applying Set Theory axioms. Behold The Wonder of Infinity's Creation!
Also worth noting that those apples are nothing like our apples. Our apples are made of finite atoms occupying finite space and that you can't really divide them too much (up to subatomic particles, although thinking of them like that might be wrong). On the other hand, a "set theory" sphere can be cut with infinitely precise detail, that is, it would take infinite bits to specify where you want to cut (that's a real for you) and how. This obviously wouldn't work with a finitely detailed apple (or one which only works with rationals, which still allow for unbounded "detail").
tl;dr: Set theoretical universe sphere != physical sphere made of atoms.
Name:
Anonymous2012-01-14 8:22
>>90
What makes you think, you cant create the atoms or subatomic particles themself out of nothing? If you believe in N+1, then nothing would stop you from believing in creation.
>>91
That would depend on the particulars of the laws of physics. Someone existing within some particular mathematical structure cannot actually change it, as they are themselves part of the structure. Of course, if whatever laws of physics allow for such an act, sure, it just doesn't seem like our own have any such possibilities (although technically, there's the quantum foam).
A belief in computability (thus natural numbers) in the ontology just means that there's a lot of possibilities as far as laws of physics go, as well as an inevitable first-person indeterminacy regarding in which particular state (or universe, if you want to use that term) you happen to be. It even tells you how you can change your "viewpoint" to one which you might like more (read that novel I mentioned for an example). However, getting one particular mathematical structure to be something else is of course impossible, for example: given the the usual definitions of Peano Arithmetic within the standard interpretation of arithmetic, it's impossible for 1+1=3 to be true (if PA is consistent). For some other system with very different definitions and semantics of 1,3,+,= or what one means by truth, that sentence may very well be true.
Belief in some countable infinity doesn't mean that suddenly the world breaks or that you can do impossible things.
Name:
Anonymous2012-01-14 8:51
>>87 I'm sorry, but to me..
Everything is subjective.
originally there was the simpler naive set theory, with rather intuitive axioms
In which way they were "intuitive"? How can the property of being "unordered" be intuitive?
it proved inconsistent (see Russell's paradox)
Usingly a nonsensical construction, based on unrestricted quantifier "all".
The axioms themselves may seem a bit strange at first as some of them do appear non-constructive.
They are just hacks, whose sole purpose is to patch their naive theory and hide problems under the carpet.
How do you know the parts are consistent?
Using senses? Decomposition should always terminate into sensible physical terms.
I don't find that paradox that much stranger than saying that the cardinality of some continous real interval like [0,1] has the same as the cardinality of all real numbers
both are nonsense.
>>92 Belief in some countable infinity doesn't mean that suddenly the world breaks or that you can do impossible things.
Just like the belief in any deity wont make it suddenly appear. Still, religious fanatics are a danger to society.
Cantor's obsession with mathematical infinity and God's transcendence eventually landed him in an insane asylum.
For the Hindu math genius Ramanujan an equation "had no meaning unless it expresses a thought of God."
The prolific Hungarian mathematician Paul Erdos imagined a heavenly book in which God has inscribed the most elegant and yet unknown mathematical proofs.
If a `religion' is defined to be a system of ideas that contains unprovable statements, then Godel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one. -- John D. Barrow, Between Inner Space and Outer Space, Oxford University Press, 1999, p 88.
Suppose we loosely define a religion as any discipline whose foundations rest on an element of faith, irrespective of any element of reason which may be present. [Atheism], for example, would be a religion under this definition. But mathematics would hold the unique position of being the only branch of theology possessing a rigorous demonstration of the fact that it should be so classified. -- H. Eves, Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1969.
The mind of man being finite, when it treats of things which partake of infinity, it is not to be wondered at if it run into absurdities and contradictions, out of which it is impossible it should ever extricate itself, it being of the nature of infinite not to be comprehended by that which is finite. -- George Berkeley
It's what I call "mental masturbation", when you engage is some pointless intellectual exercise that has no possible meaning. -- Linus Torvalds
>>93 Everything is subjective.
Not entirely sure what that's supposed to mean, but if you insist... In which way they were "intuitive"? How can the property of being "unordered" be intuitive?
Unordered just means you don't care about ordering. Imagine you have a list of items and you only test if an item is in the list and don't care about the order of the items on the list. They are just hacks, whose sole purpose is to patch their naive theory and hide problems under the carpet.
I'm not so sure. The axioms of ZFs were a bit unnatural, but the iterative version does make sense to me. Using senses? Decomposition should always terminate into sensible physical terms.
For a subjective idealist, you sure use the term 'physical' a lot. Physical is just an indexical property, and saying that "only this" exists is a stronger and more complex belief than "this" exists. Some would even argue that using that name is not too different from calling it magic. >>94
I'm not talking about whatever current laws of physics we have found by induction. I'm talking about whatever the universal law is (which we might have a chance of discovering at the end of our inductive process), regardless of our knowledge of it - we can observe that the world behaves by precise laws, even if we don't know all their exact specifics (for now).
Newtonian physics is correct given the right context, but not correct in more extended contextss.
Same goes for quantum mechanics and general relativity. The 2 of them aren't even compatible in their current form and finding a way to reconcile them is a challenge for theoretical physics. >>95 Just like the belief in any deity wont make it suddenly appear.
Of course. However, nobody is having you work with infinity directly, only that you at least not place a particular bound on naturals, otherwise you are severly limited about what you can talk about or think about.
Still, religious fanatics are a danger to society.
Popular religions are silly and some of them can lead to bad behavior. However, any belief which cannot be directly proven is of "religious" nature. There are many such beliefs which can be justified, even if not directly proven, in the end, you have to bet on some things being true or false, otherwise you can't even talk about anything at all, or make any choices. You do know that a system without any axioms, or one where all the valid sentences are theorems is an inconsistent one? It cannot talk about anything at all and can prove any falsity. You have to draw a line somewhere and have some initial assumptions - that's what "religion" actually is. You're free to use your best reasoning and meta techniques to pick whatever theory has the best chance of being true - that's what science is about. Saying that "religion" is bad is only because most people have very nonsensical and irrational religious beliefs. To many regular people the term just means anti-epistemology, where they will gobble up anything they hear as truth, but that's not what I'm talking about. The moment you realize that you must hold some (at least tentative) beliefs to be able to do anything or reason about anything, that's mostly "religion", even if those beliefs are perfectly rational. I could even argue that one should be very careful about what beliefs they have and understand how they test, verify or falsify them.
Name:
Anonymous2012-01-14 9:28
>>98 Unordered just means you don't care about ordering. Imagine you have a list of items and you only test if an item is in the list and don't care about the order of the items on the list.
Algorithm have to check doubles them in some order.
>>97 If a `religion' is defined to be a system of ideas that contains unprovable statements, then Godel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one. -- John D. Barrow, Between Inner Space and Outer Space, Oxford University Press, 1999, p 88.
That much I agree with, however it is a religion that almost all scientists and mathematicians subscribe to, wether they do it consciously or not. Without it, we can hardly get anywhere. I have no problem with "religious" beliefs as long as they are probably true. This doesn't mean that all of them are - if you dislike set theory and think it's false, you're welcome to find inconsistencies in it, surely you won't be the first ultrafinitist to try!
Name:
Anonymous2012-01-14 9:33
>>98 For a subjective idealist, you sure use the term 'physical' a lot. Physical is just an indexical property, and saying that "only this" exists is a stronger and more complex belief than "this" exists. Some would even argue that using that name is not too different from calling it magic.
I use it as a synonym to sensible, the same way I use term "reality" to refer to the things I can sense. To change one's world, one have to change language.
>>102
Oh, I see, so you're using a different definition that I am.
Not that I'm not partially guilty of that too, the meaning of the term "religious" belief that I mostly used was in the sense of provably unprovable statements which nonetheless can posses a truth value (false or true).
>>104
Sure it can, but you could only choose to define APIs on it which never take into account the ordering. In which case, the list is de-facto unordered (as long as its internals are not exposed).
>>103
In English, the word "agreement" can have many meanings, such as the act of agreeing, an understanding, or a contract; in Scientology the word means the agreement of two or more people about reality, which is said to exist only when there is agreement that it exists.
>>104
This is your daily reminder that the statement every subset of a countable set is also countable is absolutely true and never fails under any condition.
Keep denying the truth you mental midget, I'm sure it will change some day. Oh wait...
Unordered just means you don't care about ordering. Imagine you have a list of items and you only test if an item is in the list and don't care about the order of the items on the list.
is totally clueless about the concept of a 'list'.
>>115
Actually, now that I think about it, the person who wrote
>Unordered just means you don't care about ordering. Imagine you have a list of items and you only test if an item is in the list and don't care about the order of the items on the list.
ts clueless about unordering. I can cite several trivial programming examples where an unordered list gets rearranged when performing a delete() operation.
>>115-116
I was talking about what the concept 'unordered' means in math. Of course, all memory is ordered on a computer, and memory itself corresponds to some particular circuit with a particular structure and so on. Unordered in programming means that there is a "contract" where you don't use the order, or merely don't expose the order through some interface - the internals of how you hold the items is irrelevant. Unordered in math is just that, but without looking at the implementation (if you do, you're talking about a particular interpretation of some concept).
I wanted this discussion to be civil, so I refrained from resort to restating your past history as >>113 did, but it's hard to do so when you keep confusing interfaces with implementations, and the "what should be" with "what is".
>>118 Of course, all memory is ordered on a computer, and memory itself corresponds to some particular circuit with a particular structure and so on.
I never said nor implied anything about the memory you halfwit. I was talking about the value(stored) in that memory.
>Unordered in programming means that there is a "contract" where you don't use the order, or merely don't expose the order through some interface - the internals of how you hold the items is irrelevant.
Bullshit. There is no contract involved if I fill an existing array with a series of random values.
I wanted this discussion to be civil, so I refrained from resort to restating your past history as >>113 did, but it's hard to do so when you keep confusing interfaces with implementations, and the "what should be" with "what is".
This has nothing to do with interfaces or implementations. This is abstracting shit you fucking idiot.
Name:
Anonymous2012-01-14 11:54
>>118
With that, shut up and go scrub another fucking toilet.
>>119-120
I'll shut up, because you don't seem to understand some very simple concepts and there is no reason for me to waste my time on an argument which isn't ``even wrong''. As for scrubbing toilets, I thought that was your new job?
Name:
Anonymous2012-01-14 12:14
>>121
No, you clearly have no clue what you're talking about. The concept of "contract" comes from OOP. What I'm talking about extends to your non OOP languages.
So again, what happens if I do the following in the Java main() method...
for (int i = 0; i < mylist.length; i++) {
int index = (int) (Math.random() * mylist.length)
}
Where is the "contract" in this case? Now what happens if I want to delete the 3rd item in this array. This means all the unordered numbers are in the same order up until the point in quesetion. What happens to the numbers after that? Exactly.
You don't know because you have zero fucking clue as to what you're talking about. Again, you're stupid. And again, you have no possible future as a computer programmer.
>>122 So again, what happens if I do the following in the Java main() method... for (int i = 0; i < mylist.length; i++) {
int index = (int) (Math.random() * mylist.length)
}
Not the guy you're flaming but what happens is that you assign some int to index each iteration of the loop and do nothing with it and then the loop terminates.
>>122 The concept of "contract" comes from OOP. What I'm talking about extends to your non OOP languages.
It is used with that meaning in some OO languages, but what I was talking about is more general, for example, in more low-level languages like C or assembly, you could imagine a header file that states that some linked list or array is to be treated as unordered by all code and anyone depending on the ordering is doing so at his own risk (because other functions that operate with it may change it in ways which are not specified in this "contract"). In the context of programming, it's merely what a programmer promises about the behavior of the code he's written and what the intended usage is - the language is irrelevant.
I'm not entirely sure what the argument in the rest of the post is about, the java code you posted can be optimized away (to nothing) as the code does not change any state, nor does it return any values.
Again, you seem to be forgetting that this whole conversation was only tangentially related to programming and that some particular implementation of some concept isn't the same thing as the concept.
To better explain this, consider 2 examples:
1) Some programmer defined a file format or a network protocol and implemented it, the implementation is proprietary and closed-source, but some parts of the format(s) are freely available for all to see. His implementation is mostly correct, but in some cases, he made some mistakes which lead to his programs behaving a bit differently than he promised they would in the documentation - this isn't yet known to either him or the audience (such as the users of the programs). I notice some peculiarities in the way the program constructs the data (which is supposed to fit the official document), I then proceed to disassemble his closed-source software and I'm not aware of all kinds of implementation details that not even the author was aware of. I can now write a document describing exactly what the program does, not what the programmer intended for the program to do. I can also point out how the "is" differs from the "should" in this particular case. A lot of programming bugs happen when one confuses the should with the is. There's many deeper questions about should vs is, outside the programming-related meaning I just explained, but I'll stop because it would be widely off-topic.
2)Numbers define progressions. In some interpretation I could use sticks to talk about natural numbers, in another I could use a Turing Machine to encode them, in another I could use C, Lisp, Java, whatever works. In a less usual interpretation I could even consider non-standard numbers which represent infinite quantities, yet still obey Peano's Axioms, but are most certainly not what one means by the "standard interpretation" of arithmetic. They are all valid interpretations that match a valid "contract" (in this case some axioms), but they are also very different things, some so different that even different properties and truths are possible, while still matching the "contract".
>>124
Right. But the point is that each element in this array now has some random number. These random numbers may or may not be in any kind of order. Now the problem arises when I want to delete say, the third item in this array.
What happens to all the random numbers after the deletion? Do I preserve the preexisting order of all the random numbers or not? Exactly.
>>125
>you could imagine a header file that states that some linked list or array is to be treated as unordered by all code and anyone depending on the ordering is doing so at his own risk (because other functions that operate with it may change it in ways which are not specified in this "contract").
A linked list preserves an order of a list. Even if the items in that list are unordered. Cripes, get a freaking data structures book and look how insert() and deletion() are done on the list.
>>128
I know very well how a linked list works, or how an array or various other more complex data structures work. That is however irrelevant as far as what one promises to do with some particular data structure. I could easily imagine someone who instead of inserting an element on the head of a linked list (O(1)) decides to insert the element in the middle or at the end (up to O(n), where n is the length of the list) - this isn't about efficiency, but about how someone - anyone - uses your data. For a programmer such as yourself that seems to pride himself in working in an ENTERPRISE, you sure know very little about keeping your own code abstracted and separate from other people's code as well as respecting their own contracts and expecting them to respect your own contracts.
Name:
kodak_gallery_programmer2012-01-14 12:59
>>130
The order of the list depends on the abstract data type you fag. What happens if the ADT is a Linked List? Then the order of the list would have to be preserved when I do operations line insert() and delete(). Now whast happens if the ADT is a bag? Then I don't have to work about the order being preservd when doing stuff like insert() and delete().
>>131,133
Only if you're directly working with it and not through some other interface. It of course makes sense when you are the one writing the implementation. >>133
In set theory, when someone wants order in an unordered set they can just make a set of ordered pairs, in the sense:
ordered {a,b,c} - would be {(1,a),(2,b),(3,c)}, of course this isn't that much different from just defining a function f:{1,2,3}->{a,b,c}.
Name:
kodak_gallery_programming2012-01-14 13:20
>>134
It's pretty much a no brainer to convert an exisiting list to a set. Seriously. All I would have to do is something like...
for(int i = 0; i < size; i++) {
if(!newList.contains(data[i])) {
newList.add(data[i]);
}
}
Name:
Anonymous2012-01-14 13:23
>>135
Yup, gotta keep one step ahead of the filters by changing my name ).
>>135
Now what gets kind of interesting is when I don't know the exact size. How would I remove all the duplicate items in this set? In other words, OMFG, could it be. Yes. There is a finite set of duplicate items in an uncountable set!
>>138
Yeah. The point is you don't need any kind of "contract". Now I'm suspecting that the minimum wage idiot who wrote this was buzzing from the pinesol or whatever else the janitors use these days.
Name:
Anonymous2012-01-14 17:39
>>144
I didn't read any of the other crap you or any of the other people wrote, I'm a programmer so I looked at the code, it's the only thing that interests me.
>>146
Have you ever written a single line of code in your entire life you idiot? Probably not. Here's an example from work you dumb mental midget. I want to delete an item from the array. The ADT is such that I don't care about the order.
The list looks something like the following
[doug, alice, nancy, ted, vandee, sue]
Now I want to remove alice from the list. Since I don't care about order, I can just move sue to alice's spot and then put null in sue's place. After this happens, the list looks like
[doug, sue, nancy, ted, vandee, null]
Now what happens if I would have imposed a defined order? Well you stupid fuck, the list would have been
[doug, nancy, ted, vandee, sue, null]
Now shut up and go scrub another toilet.
Name:
Anonymous2012-01-14 20:31
>>146
And of course this would have been easy to see if you would have written some actual code instead of "googling" shit. Oh wait, you're too fucking stupid to code. That's why you work as a janitor!
>>125 implementation of some concept isn't the same thing as the concept.
Please, "implement" the concept of "God" or the concept of "Infinity".
I see it as nonsense to talk about concept without prior implementation.
Name:
Anonymous2012-01-14 21:35
>>125 programmer defined a file format or a network protocol and implemented it, the implementation is proprietary and closed-source,
You can reverse engineer it.
>>156
One way to look at infinity is just to look at a non-terminating process. ``God'' I could give one definition which fits in a very small program (such as Schmidhuber's conception of the Universal Dovetailer, which is a trivial scheduler + interpreter (such as an universal turing machine)), but unfortunately for you the program is also non-terminating and increases in its (always unbounded but finite) size each internal time step, so you obviously won't like it as you don't like anything which grows in size unboundedly, even if finite at each step (actually our universe seems to do just that as well). I could also give another definition which is non-computational, but mathematically well-understood, but yet still partially accessible to senses, but I'm not going to go into that definition for that discussion could stretch for a long time. I cannot give you any popular religions ``God'''s implementation for they either have impossible properties (showing they are false concepts that can never exist in any possible world and still retain all the claimed properties) or they are far too complex (surely we have yet to digitize a human mind or create an artificial general intelligence, you can't expect me to be able to do that for now). So stop asking someone to implement something on which nobody can even agree on a definition. >>157
Which is what the rest of the post does. It just shows that there was a specification showing a programmer's expectations, there was an implementation that didn't do exactly what the programmer wanted, but still mostly fit the specification, and there was a reverse engineer who could create a specification which describes exactly what the program does, as opposed to what the programmer intended the program to do.
So stop asking someone to implement something on which nobody can even agree on a definition.
That's no excuse! Hell, computer programmer's can agree on '=', yet they still implement it!
Name:
Anonymous2012-01-14 22:54
>>170 One way to look at infinity is just to look at a non-terminating process.
Don't know of such process. Any known process either terminates or repeats itself.
there was a specification showing a programmer's expectations, there was an implementation that didn't do exactly what the programmer wanted
What if specification had error, and implementation was right? It is common to have outdate comments or mismatched interfaces.
Name:
Anonymous2012-01-14 22:56
>>172 That's no excuse! Hell, computer programmer's can agree on '=', yet they still implement it!
They first implement it, then trying to agree, creating some junk theories about it in the process. Haskell would be a good example of language based on junk theory.
>>173 Don't know of such process. Any known process either terminates or repeats itself.
Only because you're looking at finite state machines which could technically repeat themselves given enough time (if sometimes much longer than the expected lifetime of the universe).
In my example, the UD is simple, imagine you have a scheduler which each time step creates a new process, the initial program that it runs is defined by index 0, the next process will be the previous number + 1, and so on. Thus at t=0, you have process 0 running, t=1, you have process 0 running for 1 time step, and process 1 being started, at t=2, you have process 0 running for 2 time steps, process 1 running for 1 time step, and process 2 just started, and so on. If you can't think of abstract machines which have unbounded memory (which can grow as it sees fit), just consider that your machine is constantly upgraded with more and more memory. You will obviously say that the machine "terminates" along with the universe, but then you have to define the universe and other things which will require such unbounded concepts which you're trying so hard to avoid. What if specification had error, and implementation was right? It is common to have outdate comments or mismatched interfaces.
It's possible. It just shows that there was a standard that promises something and a program written by the standard writter which wasn't quite compliant to that standard, even though 'it worked'.
Oh, and when I said "process x", where x is some index value, that x is to be interpreted as the program's code in some Turing-equivalent language/encoding (even if most programs will be invalid, all valid programs will run in this "infinite"/non-halting run).
Name:
Anonymous2012-01-14 23:12
>>173
A continous measurement of time doesn't repeat itself.
Name:
Anonymous2012-01-14 23:19
>>176 sometimes much longer than the expected lifetime of the universe
How do you know the "lifetime of the universe" wont itself repeat, when the universe exhausts all uniq state?
>>178
how do you know the time wont make a loop, when the universe exhausts all uniq state?
>>179 How do you know the "lifetime of the universe" wont itself repeat, when the universe exhausts all uniq state?
Sure it could. That could be true if the universe's structure happens to be a bounded/finite one. Evidence for this is lacking, but it's not an impossibility - there is more evidence for the unbounded (but still locally finite) view.
Here's something for you, the subjective idealist to wonder about: It may be that the notion of time and even the notion physical law is directly connected (I'm not going to use the term 'caused', because it's not that simple) to the nature of consciousness. For consciousness (or forming memories or having experiences), you require an arrow of time. For computation, you require an arrow of time. Without an arrow of time, the "universe" is just a static mathematical or computational structure. What about physical sturctures? You can only observe those that fit your particular inner mental/'physical' structure: you will never observe an universe where (the arrow of) time doesn't exist, as tht is required for your subjective time to exist, you will never observe an universe where various laws of entropy don't apply (although this can be relaxed greatly), and if computationalism is to be taken seriously, you will never observe an universe where some form of quantum indeterminism doesn't apply.
Name:
Anonymous2012-01-14 23:50
>>180 you will never observe...
Why would you care about the things you'll never observe? They are just your fantasy anyway. Now I'm sure you're a potential schizo, who sees things.
Name:
Anonymous2012-01-14 23:52
I'm a 16 y/o male and for years I've been paranoid that someones always watching me and that they'll try to kill me. I have tried keeping the windows in my room shut. I check the closet every night. I even check under my bed at night. I sleep with the bedroom door shut and locked and I find it hard if not impossible to sleep at other peoples houses. I also can't sleep in the same room as anyone or i can't sleep at all. Then recently I've started to hear a voice which reminds me of one of those horror movies like saw telling me to hurt myself and other people. Its cause me to get into physical altercations with my mother, father, and older brother. It has also caused me to call Children and Youth Services and report my parents for neglect and abuse. It's caused me to lie to the people I care about and hurt them emotionally. I'm starting to fear that if it keeps telling me to do it I'll hurt or even possible kill myself or someone I care about. I have even had some strange hallucinations that I wasn't really worried about because I thought they were just day dreams but they've started to turn violent and I'm getting worried. I don't drink, smoke, do any drugs, or Self Injure and there isn't any other possible cause that i can think of. So if there is any sort of advice you have I really would like to know before I go marching into a doctors and claim I have Schizophrenia.
>>181 Why would you care about the things you'll never observe? They are just your fantasy anyway.
Why does a scientist care about what his theory predicts and what it doesn't predict? Now I'm sure you're a potential schizo, who sees things.
My observations so far have been of a most stable lawful physical world.
Why does a scientist care about what his theory predicts and what it doesn't predict?
Why does a scientist care why his theory doesn't predict the existence of God?
My observations so far have been of a most stable lawful physical world.
Yep! You observe "evidence for the unbounded view", but fail to present such evidence to the humble goyim.
Why does a scientist care why his theory doesn't predict the existence of God?
Limitative results are just as important, if not more important, science usually can show theories wrong, but it cannot show them right - at best they are just confirmed by experiment. If you can show that certain things are impossible, that is progress forward. Yep! You observe "evidence for the unbounded view", but fail to present such evidence to the humble goyim.
Depends on which unbounded view you mean - there are quite a few. In cosmology you have the concept of "eternal inflation". In some multiverse theories, you almost always end up with some unbounded parameters. In multiverse theories which even vary the laws themselves, you again end up with the infinity of structures or possible computations - this infinity is almost always at the meta-level, not at the local level which you experience, however this is important, if the "everything" was limited in entities, it would be a (mathematical) miracle (highly improbable event) if anything like you or me would ever have any chance of existing at all. In the computational theory of mind, one yet again ends up with this globally unbounded, but locally finite ontology which predicts that the shape of physical law will always have certain properties (mostly confirmed by quantum mechanics so far). Terms to google for each case:
1) "eternal inflation" ( http://arxiv.org/abs/astro-ph/0101507 )
2) "Ultimate Ensemble" ( http://arxiv.org/abs/0704.0646http://arxiv.org/abs/1008.1066http://arxiv.org/abs/gr-qc/9704009 ), "Algorithmic theories of everything" ( http://arxiv.org/abs/quant-ph/0011122 ), "Universal Dovetailer Argument"/"Movie Graph Argument" ( http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.htmlhttp://iridia.ulb.ac.be/~marchal/publications/CC&Q.pdf ) or if you want an easier to grasp understanding of this concept, read the novel "Permutation City", you might even like it as it takes a partially ultrafinitist view near the end.
Note that I intentionally avoided the cases where scientists put real numbers in their physical theories - you would reject such theories out of hand, however the problem is that in almost all cases where you take reality seriously, you end up with at least the countable infinity - this is not easily avoidable, even if it can be locally avoided, just never globally.
However, don't dispair - if you do show arithmetic inconsistent, everyone will be forced to completly rethink their theories, but for now, almost no-one doubts that arithmetic or computation leads to inconsistencies. If I ever saw an inconsistency proof of arithmetic, I would be shocked and it would shake my world-view, but then I would read it and understand it, and if it was correct, I would accept it and continue to move on from there as now our knowledge is more complete and we can form better theories. However, for now, I'll continue my religious bet in the consistency of arithmetic (the same as almost any scientist or mathematician) and reason from there.
Name:
Anonymous2012-01-15 1:47
>>185 Limitative results are just as important, if not more important, science usually can show theories wrong,
So, for an average scientist God is important? Nice. That explains why I hate modern science, which is full of hypocrisy.
In cosmology you have the concept...
In some multiverse theories...
You mean some religious cosmologists made up a "concept", then built crazy theories on top of it, and now we have to accept it as an evidence?
in almost all cases where you take reality seriously
"Reality" is just what I see.
it would be a (mathematical) miracle (highly improbable event) if anything like you or me would ever have any chance of existing at all.
What is "probability"? What is a "chance"? Please, care to avoid buzzwords. I maybe a goy, but such blatant bullshit is offensive even for me.
you end up with at least the countable infinity
"Cantor's obsession with mathematical infinity and God's transcendence eventually landed him in an insane asylum."
http://arxiv.org
A site full of crazy theories and unconfirmed data! What could be better?...
read the novel "Permutation City"
A shiny sci-fi novel! Even better than arxiv.org! Thank you, honorable rabbi, I'll study it like Torah.
So, for an average scientist God is important? Nice. That explains why I hate modern science, which is full of hypocrisy.
What? What does this have to do with God, unless you're afraid that some kid will be able to show your concept of deity impossible because it features some trivial contradictions.
You don't seem to understand what science is - it's a methodology which allows one to define, test, falsify theories. "Reality" is just what I see.
That's a limited point of view. For me, reality is what I obtain by deduction and induction through senses and thought. If my inductive beliefs tell me that a star exists some million light years from home, I don't have a problem assuming that, but if you only went by your senses, it would just be a tiny blip of light and nothing more. "Cantor's obsession with mathematical infinity and God's transcendence eventually landed him in an insane asylum."
Just understand someone's work by itself and judge it with your own mind. You cannot take others beliefs as your own - that's dumb - you have to understand what they are and you can only take them once you understood them. Cantor did pioneer an important mathematical technique - diagonalization and whatever his personal problems and beliefs is irrelevant to the validity of his mathematical work, even if motivated by non-mathematical goals. A site full of crazy theories and unconfirmed data! What could be better?...
Nobody is going to chew the food for you. Believing in science because by authoritarian argument is wrong, instead you should read and understand something by yourself and then judge if it's true or not. However, if you do insist on the authoritarian argument, it's not like the people that wrote that work are unknown and you cannot check their credentials and reputations: almost all are professors and have Ph. d's., and more than half are even well-known within their fields. A shiny sci-fi novel! Even better than arxiv.org! Thank you, honorable rabbi, I'll study it like Torah.
I figured that if you're not educated enough to understand some of those papers (even though some are not hard at all to understand), a novel from a good hard sci-fi author on the subject would at least make it easier for you to understand certain concepts. The author is so serious about his work being as accurate as possible that he even wrote a manual on general realitivy and quantum mechanics to supplement some of his novels. He's also known to have written some Loop Quantum Gravity papers - this isn't your average fiction writer, but an actual scientist with good physics and mathematics background.
And I will repeat it again - even if some author was some bum living under a bridge, that would not diminish the validity or invalidity of some work - the work must always be judged on its own merit.
>>186 What is "probability"? What is a "chance"? Please, care to avoid buzzwords. I maybe a goy, but such blatant bullshit is offensive even for me.
Consider that you have a particular memory limit for everything that can possibly exist. No number greater than that and so on.
Now consider all the possible Turing-equivalent programs that define some laws of physics, up to your favorite upper bound. Given that you claimed some such limit, you can now decide very exactly which of them will result in our own laws of physics. Some such program may represent the laws of physics of our world, but you see, there's a memory limit, so it will only run a finite number of steps (or will loop). There is also another problem: the first-person indeterimism and the quantum laws that would follow would become severly limited (given the UDA view) and would be likely to contradict our current observations (of course, you try to rise the limit just high enough to prevent that, but you see, you will have to keep rising the limit as time progresses to keep the laws agreeing - you end up with the unbounded view that way). There is of course a different view which requires one to eliminate the notion of senses or consciousness as well as claim that quantum randomness is the result of PRNGs, not random oracles (which result naturally within the UDA view, or in the more restricted MWI view), that could salvage your theory partially, but wait, you're a subjective idealist, thus you will reject any theory that rejects the mind's existence, meaning that you are stuck in a tight corner - either ultrafinitism has to give or the subjective idealism has to give, at least if you don't want to keep increasing the bound endlessly just to satify all observations - there is even an experiment in the future which we could perform to show that such a bound is non-sense, but it's a bit too early to talk about that and I'm not going to bother talking about it either as you refused to even read anything I referenced.
The notion of probability is well-defined in the ultrafinist case, even much more well-defined than in the classical finitist case. Technically you could even compute it if you had... just more resources than your current upper bound (but still finite), ahahaha.
Name:
Anonymous2012-01-15 2:32
>>187 What? What does this have to do with God
Why does a scientist care why his theory doesn't predict the existence of God?
reality is what I obtain by deduction and induction through senses and thought.
How did you obtain the "deduction" and "induction" themself?
Cantor did pioneer an important mathematical technique - diagonalization
"important" for whom? I dont see its importance or relevance to the observable reality.
it's not like the people that wrote that work are unknown and you cannot check their credentials and reputations: almost all are professors and have Ph. d's., and more than half are even well-known within their fields.
Sorry, but Phd wont confirm your theory. Phd is just a title, you can buy if you have enough money and/or time.
I figured that if you're not educated enough to understand some of those papers (even though some are not hard at all to understand), a novel from a good hard sci-fi author on the subject would at least make it easier for you to understand certain concepts.
There is a reason I'm not educated - I cant enter education due to disagreement with concepts of Set Theory and modern mathematics. A sci-fi novel wont fix that.
The author is so serious about his work being as accurate as possible that he even wrote a manual on general realitivy and quantum mechanics
What did you say?! The novelist is so serious, he even wrote comments on other sci-fi novels to supplement his Electric Boogaloo? Sounds like a typical case of copyright infringement. Einstein's family will sue him for stealing fantasy worlds and characters. It's like when you take Lord of The Rings and make sequel, without having license from the owner.
Name:
Anonymous2012-01-15 2:43
>>188 Now consider all the possible Turing-equivalent programs that define some laws of physics, up to your favorite upper bound.
Sorry, you're using unrestricted quantifier "all" again.
Some such program may represent the laws of physics of our world, but you see, there's a memory limit, so it will only run a finite number of steps (or will loop).
What wrong with looping? It's a very robust way to make something work well without imposing artificial borders.
first-person indeterimism and the quantum laws that would follow would become severly limited (given the UDA view) and would be likely to contradict our current observations
I dont understant what is "first-person indeterimism" or "the quantum laws", but some time ago I heard that science is just a bunch hypotheses based on experiment data. You're fast to make a dogma from a hypothesis.
The notion of probability is well-defined in the ultrafinist case, even much more well-defined than in the classical finitist case. Technically you could even compute it if you had... just more resources than your current upper bound (but still finite), ahahaha.
In my definition, the probability is just a ratio of "hits" to the total "shots". But mathematician define probability as some crazy calculus construct.
>>189 Why does a scientist care why his theory doesn't predict the existence of God?
This has nothing to do with God. A limitative result could show for example "no theory of physics may have these and these properties", this result would be useful as it would lead to narrowing down on what the theories can be. The goal is to improve one's accuracy and negative results do that quite well. As for God, I already said, it's too undefined or too personally defined, it's not even worth talking about it scientifically, unless you give a particular definition. How did you obtain the "deduction" and "induction" themself?
There are precise definitions of those terms formally and mathematically, but there is also the more common sense meaning of those terms. For example about induction: someone observes that a certain pattern is always followed by another - such as dropping a metal crate will cause a sound when the crate hits the floor. Or that you will observe that the sun rises each day and conclude that in all your past experiences it was all the case, thus you will hold that belief for now. Inductive reasoning actually comes rather naturally for humans and there are theories in cognitive sciences which show many parallels between our neocortex's hierarchical structure and some forms of bayesian/probabilistic networks. A more layman's introduction to some such high-level concepts could be found in the book "On Intelligence". I could also find some links to actual serious papers on the subject, but since you seem to be hostile to reading papers, I'm not going to waste more of my time locating copies of old papers I've read.
To put it differently, it's a both innate skill contained throughout our most basic reasoning facilities and also a learned skill when talking about formal versions. "important" for whom? I dont see its importance or relevance to the observable reality.
Because you don't frame your theories about reality in math. Too bad most of your theories don't actually give any predictions whatsoever, so they're about as useless as the usual ``God'' hypothesis. Besides results which don't touch reality in any direct way, it can also be used to show the universality of the notion of computable function. Sorry, but Phd wont confirm your theory. Phd is just a title, you can buy if you have enough money and/or time.
I never claimed so. I decide on what to believe in by my own judgement, and so should you. There is a reason I'm not educated - I cant enter education due to disagreement with concepts of Set Theory and modern mathematics. A sci-fi novel wont fix that.
Maybe, but in this particular novel, he doesn't touch any set theoretic notions. He even entertains an ultrafinitistic view as a possible theory to explain one important event in the book, so you might like it (he also wrote a shorter story where he tries to consider the implications about what could the inconsistency of arithmetic mean in some such "impossible world"). What did you say?! The novelist is so serious, he even wrote comments on other sci-fi novels to supplement his Electric Boogaloo? Sounds like a typical case of copyright infringement. Einstein's family will sue him for stealing fantasy worlds and characters. It's like when you take Lord of The Rings and make sequel, without having license from the owner.
Haha. Except most scientific results are open and free (leaving aside silly paywalls). Also his book supplement is freely available on his site, not that you'll need it for this particular book.
>>191 Sorry, you're using unrestricted quantifier "all" again.
All within your define bound, it's a finite "all". Because you insist on using ultrafinitism, and I'm considering what consequences it would have to some theories. I dont understant what is "first-person indeterimism" or "the quantum laws", but some time ago I heard that science is just a bunch hypotheses based on experiment data. You're fast to make a dogma from a hypothesis.
Not dogma, but I look at it like this, you have an established and well-verified physical theory. Now you consider a theory compeltly unrelated at first glance to it and you notice it gives the same consequences in a constructive manner without ever having asked for them or having built the theory artifically to give you those results. There is no dogma - every theory is a hypothesis, except when you have to rely on it for actual results, then you bet on it and you risk being wrong in some cases. In my definition, the probability is just a ratio of "hits" to the total "shots". But mathematician define probability as some crazy calculus construct.
Extended forms of probability are like that, but here it's just a finite ratio, although possibly very big and intractable to calculate, yet still finite. You do get an uncomputable (thus your calculus construct) one when you do use an unbounded ontology though, but that's unavoidable in most non-trivial theories.
just poppin in to say I like ultra finitist in lisp sympta guy. What is your opinion on the natural numbers? How do you resolve always being able to add one to a number?
Name:
Anonymous2012-01-15 4:58
>>193 ultra finitist in lisp sympta guy
I'm not ultrafinitist.
What is your opinion on the natural numbers?
Haven't seen them.
How do you resolve always being able to add one to a number?
What is "always"?
>>191 This has nothing to do with God. A limitative result could show for example "no theory of physics may have these and these properties", this result would be useful as it would lead to narrowing down on what the theories can be. The goal is to improve one's accuracy and negative results do that quite well. As for God, I already said, it's too undefined or too personally defined, it's not even worth talking about it scientifically, unless you give a particular definition.
You can define "God" as anything you cant comprehend/sense. That way God is equivalent to Infinity/incompletness, which scientists care about.
There are precise definitions of those terms formally and mathematically, but there is also the more common sense meaning of those terms.
Common sense meaning has nothing to do with math. It's limited to our senses.
For example about induction: someone observes that a certain pattern is always followed by another - such as dropping a metal crate will cause a sound when the crate hits the floor.
When it's done in local, controlled and well understood environment. For example, crate wont cause sound, if it's dropped from outer space into the Sun.
you seem to be hostile to reading papers, I'm not going to waste more of my time locating copies of old papers I've read.
I cant understand them, because they are full of these cryptic "sets", together with mysterious "for all" and "there exist" wordings, buried under some ugly curly brace infix syntax, which isn't even context-free.
a learned skill when talking about formal versions.
Just like reciting Torah is a learned skill for a rabbi.
Because you don't frame your theories about reality in math. Too bad most of your theories don't actually give any predictions whatsoever
Wait! What are "my theories"? I do happen to agree with prof. Norman Wildberger on his rational-trigonometry, but he happen to demonstrate that geometry can be done without referencing "Infinity".
in this particular novel, he doesn't touch any set theoretic notions. He even entertains an ultrafinitistic view as a possible theory to explain one important event in the book, so you might like it
Author just recites what had been said already by George Berkeley a few centuries ago?
you insist on using ultrafinitism
You should note, that I'm not an "ultrafinitist", I'm "subjective idealist". These terms are different, like "Atheist" vs "Agnostic"
All within your define bound, it's a finite "all".
Then there is nothing that will allow us to look "outside" the bound.
Not dogma, but I look at it like this, you have an established and well-verified physical theory.
Some time ago Newtonian Physics was a "well-verified physical theory" and it definitely confirmed ability to devide space into "infinitesimals".
that's unavoidable in most non-trivial theories.
The good thing is that "non-trivial theories" are avoidable. http://en.wikipedia.org/wiki/Occam's_razor
>>194 You can define "God" as anything you cant comprehend/sense. That way God is equivalent to Infinity/incompletness, which scientists care about.
Fine then, then we can talk about some "ineffable" stuff, especially if that "ineffable" stuff has consequences. Such kind of stuff usually appears once you keep reducing and reducing things until you can't reduce anymore, and yet it's too simple to be able to explain in terms of anything else, but still can be reasoned about. Common sense meaning has nothing to do with math. It's limited to our senses.
How do you think you learn about math or recursion? When it's done in local, controlled and well understood environment. For example, crate wont cause sound, if it's dropped from outer space into the Sun.
Of course. It's not like we don't build fairly exhaustive models given our observations in a variety of environments. I cant understand them, because they are full of these cryptic "sets", together with mysterious "for all" and "there exist" wordings, buried under some ugly curly brace infix syntax, which isn't even context-free.
Heh, if you're going to avoid talking about real things because you have problems with the the language used (or problems with notions in foundations of mathematics), you'll lose more, because those real things actually have consequences. You don't refuse to talk in English, but English is provably inconsistent as far as what sentences it allows you construct. Just like reciting Torah is a learned skill for a rabbi.
Except one lets me make accurate predictions and the other is just mythology. Wait! What are "my theories"? I do happen to agree with prof. Norman Wildberger on his rational-trigonometry, but he happen to demonstrate that geometry can be done without referencing "Infinity".
I considered your subjective idealism as a hypothesis. Ultrafinitism as a hypothesis. Do you really think that a strong form of ultrafinitism (which posits some finite upper bound for naturals) doesn't have severe limiting consequences as far as what physical law can be? As I said before, there's no problem with assuming some hard limits, but this might contradict some current theories which are (mostly) experimentally verified (such as quantum mechanics or general realitivity). Author just recites what had been said already by George Berkeley a few centuries ago?
Not really. In the story, a highly unexpected/unusual event happens and one of the possible hypotheses that were thrown around was that they might have been wrong about that countable infinity after all and a finite upper bound might exist. (Maybe it would disappoint you, but I can easily see a solution to their problem which does not lead to ultrafinitism, although this wasn't really discussed in the book, it was mostly left as an open problem/cliffhanger; of course, some form of ultrafinitism in the ontology could explain that event as well). If Einstein's theory contains errors, then it could be a work of art.
But pretty much all scientific theories are wrong. Some are just less wrong than the other. The goal of science is to get as close as possible to the least wrong theory, possibly reaching a true one. (They are of course applicable within their context, so they are mostly correct within the right context, but for example, general realitivity is likely wrong (and incomplete) regarding what happens singularities (big bang, black holes, etc), while quantum mechanics is wrong at the large scale as it includes no theory of gravity.) You should note, that I'm not an "ultrafinitist", I'm "subjective idealist". These terms are different, like "Atheist" vs "Agnostic"
I wonder why does everyone consider you an ultrafinitist then? How did you manage to acquire this reputation? Does that mean you're merely agnostic about the existence of an infinity of finite natural numbers? Or that you posit an upper bound? Then there is nothing that will allow us to look "outside" the bound.
Not entirely sure I understand what you mean by this. Given some max constant k and some particular definition of computation, such an ultrafinitist computationalist ontology will have radically different predictions than a classically finitist computationalist ontology. It may seem to you that they are unobservable, but if an upper limit does exist, it will have severe consequeces about what physical law is/can be. Some time ago Newtonian Physics was a "well-verified physical theory" and it definitely confirmed ability to devide space into "infinitesimals".
Newtonian physics is not wrong given the right context, it's just more wrong than general relativity. Newtonian physics is not an absolute physical law - it's a local emergent law, same as most other physical laws we know right now, but that doesn't mean that the goal of physics isn't to find the full, consistent law which explains all local physical phenomena within one coherent/consistent formula/function/structure. Given that law, you have all kinds of other laws which emerge from it which are locally true within some context. Taking Newtonian Physics as ontological primitive would be an error in our world as evidence says otherwise. The good thing is that "non-trivial theories" are avoidable. http://en.wikipedia.org/wiki/Occam's_razor
I use it too. Do you know why Occam's razor is more likely to lead to true theories? It has to do with probability theory - the probability of something having multiple disjoint properties is lower than just having some of those properties, especially if those properties explain exactly the same phenomena - in which case, the other parts become superflous.
When I talked about non-trivial theories I didn't mean that they are higher in complexity. Actually they are much lower in complexity than the ultrafinitistic version. If you want to judge complexity more objectively, you could use some formalized version such as Kolmogorov complexity, but then you probably won't like it because it's not computable (although there are ways to define approximations).
Name:
Anonymous2012-01-15 7:50
Cool new Jeans!
Name:
Anonymous2012-01-15 7:57
>>197 How do you think you learn about math or recursion?
I got recursion through exposure to Lisp system (XLisp at the time). I have never learned any math, because I haven't attened school. Everything I know about math is from wikipedia.org.
if you're going to avoid talking about real things because you have problems with the the language used...
Cant see how language about "infinity sets" could be used to talk about "real things".
Except one lets me make accurate predictions and the other is just mythology.
Cant see how "infinity" is accurate.
You don't refuse to talk in English, but English is provably inconsistent as far as what sentences it
"consistency" is a buzzword and in reality, there are no contradictions, except those we invent/define.
I considered your subjective idealism as a hypothesis
It's not. It's just a program (doctrine if you like), that helps me to avoid junk theories, like your "Set Theory".
Ultrafinitism as a hypothesis
I'm not an "ultrafinitist"
Do you really think that...
I don't "think". I prefer more animalistic way of seeing and interpreting things. That is: I either sense or not. No silly "consiousness"/"thinking".
strong form of ultrafinitism (which posits some finite upper bound for naturals) doesn't have severe limiting consequences as far as what physical law can be?
Can't see this. And I don't know the "phyisical law" to have an opinion.
I wonder why does everyone consider you an ultrafinitist then? How did you manage to acquire this reputation?
No idea.
Does that mean you're merely agnostic about the existence of an infinity of finite natural numbers? Or that you posit an upper bound?
That means I can't see them, thus can't have reasonable opinion about them, except that they are just buzzwords.
Newtonian physics is not wrong given the right context
The problem, there is no "right context" for newtonian physics.
"Newton saw a monotheistic God as the masterful creator whose existence could not be denied in the face of the grandeur of all creation."
When I talked about non-trivial theories I didn't mean that they are higher in complexity. Actually they are much lower in complexity
Trivial| Non-Trivial
-------|------------------------------------------------------------------------
lambda | inference, lambda cube, strongly normalizing, equality-qualified types,
| algebraic types, existential types, phantom types, dependent types,
| higher-kinded types, linear types, inductive types, unique types,
| nominal types, recursive types, type classes, bounded quantification,
| type annotations, principal types, higher-order abstract syntax,
| generalized algebraic types, robinson unification, hindley-milner,
| constrained types, polymorphic recursion, parametric polymorphism,
| equivalence classes, type order, judgments, curry-howard isomorphism,
| system t, system f, products, coproducts, categorial sum, call-by-name,
| inhabited types, higher-rank impredicative polymorphism, covariance,
| subtype polymorphism, ad-hoc polymorphism, predicative types,
| signatures types, contravariance, affine types, structural subtyping...