Discuss:Mathematics, Logic, Philosophy:Continuum Hypothesis
From da Vinci Concept
Chat on Monday, Sept 25 11:31:00 2006
User:Henrik Nordmark: Hey
User:Sunny: Whats up?
User:Henrik Nordmark: So you are moving to Albuquerque to study CS?!
User:Henrik Nordmark: Do you have time?
User:Sunny: Yep. I'm starting graduate school in the spring.
User:Henrik Nordmark: cool
User:Sunny: I've got some time. We are just hanging out. Maybe go surfing.
User:Henrik Nordmark: okay...
User:Henrik Nordmark: so...
User:Henrik Nordmark: I am hoping that an interesting discussion can arise since I know you sympathize with objectivism.
User:Sunny: Er...sympathize....?
User:Sunny: lol
User:Henrik Nordmark: I am hoping that if the quality of the discussion is good we can make it into something a bit more formal
User:Henrik Nordmark: perhaps pedagogical even.
User:Henrik Nordmark: well... I wasn't sure whether you identified yourself as being an 'objectivist'. I thought that may be too strong a statement.
User:Henrik Nordmark: Thus, I preferred to to tread with caution and use a slightly weaker statement.
User:Sunny: Right.
User:Henrik Nordmark: Anyways, are you up for it then?
User:Sunny: Absolutely.
User:Henrik Nordmark: ok great...
User:Henrik Nordmark: I feel as if I would like to believe in objective truth.
User:Sunny: A stronger way to say that would leave out the emotional response:
User:Henrik Nordmark: However, I find it more and more difficult to take that position.
User:Sunny: I have every reason to think that there is objective truth.
User:Sunny: There is no evidence against and every evidence for objective truth.
User:Henrik Nordmark: I consider it possible that objective truth may still be possible for statements about the physical world.
User:Sunny: Er, possible? I'm not sure you could come up with an example that was not 'objective'.
User:Henrik Nordmark: However, when it comes to mathematics, I feel more and more inclined to believe that there may not be something we can cal objective truth.
User:Sunny: Perhaps if you claim something ethereal, which can't be measured, but then that is meaningless because there is no evidence for its existence.
User:Henrik Nordmark: That is not what I mean.
User:Sunny: There is a fundamental premise hiding in your question of the objective truth of reality versus the objective truth of knowledge which appears independent of reality (i.e. mathematics)
User:Henrik Nordmark: Independent of whether or not there is some Platonic realm where mathematical objects live, I do not seem to find a reasonable way in which one can sustain that there are objective truths in mathematics.
User:Sunny: How so?
User:Henrik Nordmark: In the old days, I used to believe that mathematical truths simply depended on them being derivable from axioms that we had determined to be true.
User:Sunny: Well, your derivation must also be correct and not break any laws of mathematics or physics.
User:Henrik Nordmark: However, as we both know, Gödel's Incompleteness Theorem states that not all mathematical truths are derivable from axioms.
User:Henrik Nordmark: Yes of course
User:Henrik Nordmark: I always assume that the inference rules of logic are respected at all times.
User:Sunny: That isn't quite what Gödel states. Isn't it that not all mathematics truths are derivable from the axioms of the system in which you currently reside.
User:Sunny: If you escape the system then you can then prove the previously unprovable theorem.
User:Henrik Nordmark: Yes, that is true.
User:Henrik Nordmark: However, that is not very comforting for then you cannot solely rely on your axiom system to determine the truth value of a mathematical statement.
User:Sunny: Its a problem of self-description. There are always truths that can't be described using the faculties which are doing the describing.
User:Henrik Nordmark: You are forced into adding stronger and stronger axioms to determine the truth values of previously undetermined statements.
User:Henrik Nordmark: The problem then becomes: How does one choose these stronger axioms?
User:Sunny: Alternatively you can apply your mathematics to a problem in physics or geology and let the physical world determine the correctness or form that a previously undecidable statement takes.
User:Henrik Nordmark: You can indeed take as an axiom the undecided statement and this will trivially determine it's truth value.
User:Sunny: I don't think that incompleteness is a problem. I don't think that it undermines mathematics.
User:Henrik Nordmark: The problem is that the physical world generally does not provide us with any sorts of intuitions as to whether these statements should be true or false.
User:Sunny: I think that it simply provides bifurcation points that enrich the mathematical landscape.
User:Henrik Nordmark: I do not think that incompleteness undermines mathematics at all.
User:Sunny: Sure it does. The physical world is the reason we assume that one theorem is intuitive or not.
User:Henrik Nordmark: However it might undermine the belief that mathematics consists of a body of knowledge that is in some sense 'objective'.
User:Sunny: However, if we haven't used the mathematics to describe the physical world (i.e. it is still just a game and has never really been applied) then we would have no insight into its applications or whether one form is more 'correct' than another.
User:Henrik Nordmark: Let us take a concrete example:
User:Sunny: Objective truth is context dependent.
User:Henrik Nordmark: The Continuum Hypothesis (CH).
User:Sunny: Meaning, if you drop the context it becomes impractical to define or purport that objectivity is possible.
User:Sunny: I can objectively have the most correct answer given the information that I have, my faculty for understanding it, and the tools currently at my disposal.
User:Henrik Nordmark: The continnuum hypothesis states that whenever you take an infinite set of real numbers, then there must be a bijection between this set and the integers or between this set and all of the real numbers.
User:Sunny: If everyone lived rationally, objectivity only really says that we can have a 'best' answer in the context of all of these things.
User:Henrik Nordmark: Another way of stating this is that there are no degrees of infinity between the integers and the reals.
User:Henrik Nordmark: I do not see how the physical world could *ever* provide an answer to such a question.
User:Sunny: Or between one set of reals and another.
User:Sunny: Isn't the CH just stating that the reals and integers are countable?
User:Sunny: Not only countable, but I could count them at the same pace at a one-to-one rate?
User:Henrik Nordmark: X is countable means that there is a bijection between X and the integers.
User:Henrik Nordmark: X is uncountable means that X is infinite and there is no bijection with the integers.
User:Sunny: But the integers are infinite.
User:Henrik Nordmark: correct
User:Sunny: So what else about X makes it different from the integers?
User:Henrik Nordmark: omega, ie the set of all integers is infinite and is the smallest infinite set.
User:Sunny: Or slowest?
User:Henrik Nordmark: By smallest I mean to say that any infinite set that cannot be put into bijection with the integers must either be finite or it must be strictly larger than the integers.
User:Henrik Nordmark: You can of course find subsets of the integers that have the same cardinality as the integers.
User:Henrik Nordmark: You can take the even numbers as an example.
User:Sunny: Its funny to think about actually counting even numbers. At the limit of infinity, I would be twice as far ahead in the even values.
User:Henrik Nordmark: hehehe...
User:Henrik Nordmark: Yes, it seems a bit strange at first.
User:Sunny: Is there a measure of this difference rate at which the counted values change, maybe with integers as the baseline?
User:Henrik Nordmark: However, it is best to think of this as a counting problem in which one does not have access to a notion of numbers to do the counting.
User:Henrik Nordmark: Imagine you want to trade a set of sheep for a set of pigs.
User:Sunny: Why not, we are counting numbers and if a computer were doing the counting each subsequent value would take more space to store.
User:Henrik Nordmark: And for every sheep, you want exactly one pig.
User:Sunny: Sure for comparing cardinality that makes sense.
User:Henrik Nordmark: If you do not have access to numbers, you can simply line up the sheep and the pigs and make sure that the two are in one to one correspondence.
User:Henrik Nordmark: This is exactly what we do with infinite sets.
User:Sunny: But if I know that I can never really count all of my pigs, I'd rather model the rate at which I count so I know whether in the end I'll end up with more pigs or more sheep.
User:Henrik Nordmark: the even numbers have the same cardinality as the integers because you can map
User:Henrik Nordmark: 1 --> 2
User:Henrik Nordmark: 2 --> 4
User:Henrik Nordmark: 3 --> 6
User:Henrik Nordmark: etc...
User:Sunny: My brother Nick says: "What's up!"
User:Henrik Nordmark: CH states that if you take any infinite set X of reals you like then you can either line them up one-to-one with the reals or with the integers.
User:Henrik Nordmark: In other words, it is impossible to find a set X which would be strictly larger than the integers and strictly smaller than the reals.
User:Sunny: But I could probably find sets that grow in numeric size (not cardinality) that lies somewhere in between.
User:Henrik Nordmark: You would first need to define what you mean by a set growing in numeric size.
User:Sunny: Like the set of (n*1/2)
User:Sunny: At ever iteration i each set being compared in its numerical values.
User:Sunny: Perhaps the problem with thinking about infinities in a purely mathematical sense is that we ignore the fact that the numbers are actually getting larger. An infinite set can be talked about in a simple manner using its cardinality.
User:Sunny: We do this often in writing computer programs because it is easier to keep track of than each unique value, or even the largest value.
User:Sunny: I often run into problems where the data is combinatorially explosive and very quickly realize that I could never have enough computer memory to store each value.
User:Henrik Nordmark: hehe
User:Sunny: These problems are usually far worse than counting integers, but the practical solution is to just create bins and start counting.
User:Henrik Nordmark: This seems to be a different issue. This seems to be a problem about the way a given algorithm works in oder to do some counting task.
User:Sunny: But there is no reason why we can't also talk about the growth rate of a set of infinite values. Then I can differentiate between the integers and the even integers or the integers and the reals.
User:Sunny: Finding equivalent sets that not only have the same cardinality, but reach large values at the same rate or a specific rate.
User:Henrik Nordmark: When we talk about the integers, we do not think of this as something that is gradually being constructed over time, but rather it is a set that is already infinite in size. It is not growing, it is static.
User:Henrik Nordmark: At least this is not what we are concerned about when talking about cardinalities of sets.
User:Sunny: Sure. When talking about cardinality that is fine, but when confronted with a question like whether or not we can prove the CH we might need another method.
User:Sunny: It might be an unprovable theorem within our given system.
User:Henrik Nordmark: CH is a statement strictly about cardinalities.
User:Sunny: If we bring in rates of increasing value we might be able to say things like:
User:Henrik Nordmark: There is no reference made as to how such sets get constructed if at all.
User:Sunny: The rates of all reals or subsets of the reals is constant.
User:Sunny: Or no subset of the Reals, if counted sequentially, will increase in its numerical value slower than the Reals.
User:Sunny: Or better, all subsets of any set, if counted sequentially, will grow in numerical value at a faster rate than the set being considered.
User:Henrik Nordmark: I am not saying that the introduction of such notions are not potentially interesting. However, even if we introduce such notions and find adequate formal definitions to do so, this would not shed any light on CH which is strictly about cardinalities.
User:Sunny: It could if it states anything about cardinalities that we didn't already know or know how to prove.
User:Henrik Nordmark: correct
User:Henrik Nordmark: but then you are introducing new axioms
User:Sunny: Thats how you create new systems, right?
User:Sunny: Besides, it might be shown that such axioms are just a reflection of things already known, but just stated in a different light.
User:Sunny: But this new perspective might play a critical role in understanding a more complex aspect of the mathematics under scrutiny.
User:Henrik Nordmark: The problem is that there is no particular reason why one axiom should be introduced rather than another. The choice seems arbitrary.
User:Henrik Nordmark: And hence, truth seems to always be relativistic.
User:Sunny: Just as no single mathematics is the best approach for every engineering problem, no single set of axioms provide the best approach to solving every mathematical mystery.
User:Henrik Nordmark: Truth seems to be strictly dependent on the axiom system you choose.
User:Sunny: Not arbitrary. Just exploratory.
User:Sunny: Because you can't know ahead of time what the solution is or the path to the solution you try new approaches that seem reasonable.
User:Sunny: Science works the same way.
User:Henrik Nordmark: Just as no single mathematics is the best approach for every engineering problem, no single set of axioms provide the best approach to solving every mathematical mystery. <--- I very much like this analogy.
User:Henrik Nordmark: The problem is: What does it mean "to solve a mathematical mystery".
User:Sunny: Oh, and the term is relativism. Relativistic is a term to describe light.
User:Sunny: lol
User:Henrik Nordmark: In engineering you have a very pragmatic justification: does the mathematical approach serve us with what we want to create.
User:Sunny: I liken solving a mathematical mystery to be similar to solving one in physics, simply the exposure and measurement of more intricate details that perhaps were only alluded to or measured/seen indirectly.
User:Henrik Nordmark: Yes!
User:Henrik Nordmark: BUT
User:Henrik Nordmark: what does that correspond to in the mathematical realm?
User:Henrik Nordmark: ???
User:Henrik Nordmark: Gödel has an answer for this...
User:Sunny: Just because you can look in two different directions when standing at the cusp between one axiom and its negative and see different landscapes does not pose any real challenge for objectivity.
User:Henrik Nordmark: However, I do not find it very satisfactory...
User:Henrik Nordmark: Just because you can look in two different directions when standing at the cusp between one axiom and its negative and see different landscapes does not pose any real challenge for objectivity. <--- It doesn't?
User:Sunny: I still think that is really about applications, about why the mathematics is relevant. Why should I use one mathematics over the other? Because it is easier to apply, or because it is self-similar to the physical world in some way.
User:Henrik Nordmark: I thought that a requirement for any notion of objective truth is that bivalence holds.
User:Sunny: It is all about context.
User:Sunny: Not necessarily. There may be instances within the physics of subatomic particles where this axiom is not upheld.
User:Sunny: When you talk about physical objects, this truth may be more dependent on the rate of growth of various infinities than it is on cardinality.
User:Sunny: As in programming, it is just as important in most instances on what values you are storing as to how many of them you have.
User:Henrik Nordmark: Before we go any further...
User:Henrik Nordmark: Do you believe in bivalence?
User:Sunny: I don't 'believe' in anything. I either have confidence in it based on experience and inferred knowledge or not.
User:Sunny: There may be an infinite number of electron orbital states, but they do fill up.
User:Henrik Nordmark: From now on, *always* interpret the word 'believe' as having confidence in it based on experience and inferred knowledge.
User:Henrik Nordmark: Do you believe in bivalence?
User:Sunny: No.
User:Henrik Nordmark: okay...
User:Sunny: I think that there are statements that are undecideable because the question is itself flawed.
User:Sunny: Making it neither true nor false, but just null or 'mu'
User:Henrik Nordmark: That seems to be a different issue. I may not know whether it is raining in Moscow or not, however I very strongly believe that either it is raining or it isn't raining in Moscow.
User:Sunny: Moscow either exists or it does not.
User:Sunny: There are statements that follow a bivalent logic, but not all statements are best stated as such.
User:Sunny: The humidity in Moscow and definition of raining is somewhat dependent on experience and the local customs.
User:Sunny: If we assume that we share the same understanding of what is 'rain' versus sleet, mist, haze, drizzle, etc.. then the Moscow proposition is also bivalent.
User:Henrik Nordmark: Yes, of course.
User:Henrik Nordmark: That is not under dispute.
User:Henrik Nordmark: I assume that we use language in the same way and that we have appropriately defined all the concepts we need such as 'raining'.
User:Henrik Nordmark: The question simply is whether once we have established all necessary definitions, does bivalency hold. In other words, are we justified in believing that phi OR not phi is always a true statement?
User:Sunny: Depends on what question you are asking.
User:Sunny: It is not true that objectivity means 'independence on context'
User:Henrik Nordmark: So your answer is: no ??
User:Sunny: It seems reasonable, but we couldn't have possibly covered all of human knowledge, potential knowledge, etc... allowing phi to be anything at all.
User:Henrik Nordmark: You are treading on treacherous waters now... You almost sound like an intuitionist...
User:Henrik Nordmark: Let me ask you this:
User:Sunny: Huh? What do you mean?
User:Henrik Nordmark: Can you conceive of a question in which you could doubt that phi OR not phi is not true?
User:Sunny: Bivalent logic holds if your premised are properly stated.
User:Henrik Nordmark: I apologize for the technical jargon.
User:Henrik Nordmark: Intuitionism was a philosophical movement initiated by Brouwer in the early twentieth century.
User:Henrik Nordmark: Brouwer believed that we are no justified in asserting that phi OR not phi is true, unless we have some method that allows us to show that phi is true or we have some method that allows us to show that not phi is true.
User:Sunny: Is there a tribe of fembots on the surface of venus?
User:Sunny: God exists.
User:Sunny: The soul is independent of the body.
User:Sunny: These are all 'mu' questions.
User:Henrik Nordmark: Because the terms are ill-defined?
User:Sunny: Bivalent logic does not apply because the existence of the entities are not provable, the question is meaningless.
User:Henrik Nordmark: okay
User:Sunny: Might as well be speaking jibberish if there weren't so many cultural artifacts to describe using the broken concepts.
User:Henrik Nordmark: this is the logical-positivism stance.
User:Henrik Nordmark: I very much sympathize with this stance.
User:Sunny: So the statements become useful in understanding archeology and culture, but otherwise are just blowing smoke.
User:Henrik Nordmark: However, let me now ask you this: Do you believe CH is a mu statement?
User:Sunny: I disagree with the logical positivism stance because it assumes the analytic-synthetic dichotomy.
User:Sunny: There is no such cusp.
User:Henrik Nordmark: Forget about logical positivism. I was simply stating that they are also in favor of dismissing statements that are meaningless.
User:Henrik Nordmark: Do you believe that CH is a mu statement?
User:Sunny: I don't think so. It is speaking to real objects.
User:Sunny: The existence of the set of reals is verifiable.
User:Henrik Nordmark: If you consider infinite sets of real numbers to be 'real objects', then yes you are speaking about 'real objects'. However, this sounds like you are in favor of Mathematical Realism.
User:Sunny: God is not.
User:Sunny: We have every reason to think of the real numbers as real object, but no reason to think of God as anything but a cultural artifact or a computer programmer.
User:Henrik Nordmark: Are you in favor of Mathematical Realism?
User:Sunny: As I've stated before, mathematical objects are real objects, as are all ideas to some extent; (i.e. either as the object itself or as the fact-of object as in a fantasy)
User:Henrik Nordmark: Let me restate the question:
User:Henrik Nordmark: Do you believe in a Platonic realm in which mathematical objects reside?
User:Sunny: No. I reject the analytic-synthetic dichotomy and any Platonic or ethereal realm.
User:Henrik Nordmark: okay...
User:Sunny: Mathematical realism seams reasonable, but there is no reason to have a special place in which these mathematical objects exist.
User:Henrik Nordmark: You then need to explain what it is you mean when you say that "mathematical objects are real".
User:Sunny: The physical world/reality is large and diverse enough to hold all of it.
User:Sunny: Mathematical objects, or their physical manifestations are latent within the physical world, the mind, the structure of thought, and the structure of the universe.
User:Sunny: I'd even argue that you can't attempt rejection of this without using the faculty of the mind that is such a confluence of mathematical objects.
User:Sunny: This is not the same as allowing every mental fantasy as a real object.
User:Henrik Nordmark: Mathematical objects, or their physical manifestations are latent within the physical world, the mind, the structure of thought, and the structure of the universe. If that were really the case, it would seem that the physical universe should be able to provide an answer to CH. However, I do not see how the physical universe could *ever* provide a truth value for CH.
User:Sunny: As I was saying before, look at the applications of CH in computer science.
User:Sunny: Most of science is indirectly measured.
User:Sunny: Surely there are applications that might test such a hypothesis, if not directly, then indirectly.
User:Henrik Nordmark: If you have a concrete way of determining the truth value of CH by looking at applications of CH in computer science, please let me know immediately.
User:Sunny: I don't, but I think that it is worth considering.
User:Henrik Nordmark: I find it highly unlikely for that to be possible. To me, this would be analogous to making a claim about whether or not the physical universe is discrete or continuous.
User:Henrik Nordmark: Okay... I think I have reached some sense of closure in this discussion. Is there anything else you would like to say before we end the discussion?
User:Sunny: What are the potential consequences for proving the Continuum Hypothesis?
User:Henrik Nordmark: I would like to mention that Hartry Field holds a somewhat similar position to yours in the sense that he believes that all mathematical statements correspond to some statement about the physical universe.
User:Henrik Nordmark: consequences...
User:Sunny: Other than mathematical rigor, why is it important?
User:Henrik Nordmark: It has implications regarding the structure of the set-theoretic universe.
User:Henrik Nordmark: Historically speaking it was important in the sense that this was the first more or less natural mathematical question which was discovered to be independent of wikipedia:ZFC.
User:Henrik Nordmark: Cantor tried very hard to prove that CH was true.
User:Henrik Nordmark: Gödel then managed to show that we cannot show that CH is false using the axioms of set theory.
User:Sunny: The Hartry Field premise is likely false, unless you allow mathematics as an aspect of the physical universe, and then it elegantly just describes itself.
User:Sunny: i.e. false in that likely not every mathematical proof has a physical manifestation.
User:Henrik Nordmark: And finally, Paul Cohen showed in the 1960's that CH cannot be shown to be true using the axioms of set theory.
User:Sunny: Is that sort of the birth of Gödel numbering and incompleteness?
User:Henrik Nordmark: Not exactly.
User:Henrik Nordmark: Gödel numbering and incompleteness was proven prior to CH being not provably false.
User:Sunny: Sorry, it just seems like the methods would be similar since we are talking about counting.
User:Henrik Nordmark: However, prior to the discovery of the incompleteness theorem, mathematicians had not considered the possibility of statements being independent like CH.
User:Henrik Nordmark: The incompleteness theorem showed the existence of very weird mathematical statements being not determined by axioms.
User:Henrik Nordmark: However, CH was the first non-weird example of an independent statement.
User:Sunny: Hmm... Like a mathematical statement that is akin to the 'mu' statements.
User:Sunny: Not CH, but maybe some of the others.
User:Henrik Nordmark: 'weird' is of course a very subjective description of a mathematical statement...
User:Sunny: The reason that I asked about consequences is that unless CH can be proven (either via mathematics or experimentation) it may be a mu statement.
User:Sunny: If it is, then we cannot prove it, ever.
User:Sunny: Therefore we need to judge where to put our effort based upon its possible consequences and uses if it were to be shown to be true or false.
User:Henrik Nordmark: Let me be bold and tell you how I see things:
User:Sunny: I bet a mathematical form of the statement 'God exists' would look very strange.
User:Sunny: Of course, such a statement cannot be known because the existence of God is (as far as we can ascertain) a mu statement, and I don't think it is decomposable into a mathematical statement.
User:Henrik Nordmark: I believe that CH is completely irrelevant to mainstream mathematics and to questions in science and engineering. Moreover, I do not believe that there are any good mathematical arguments to accept or reject CH. However, I do not feel it would be fair to dismiss CH as being a meaningless mu statement. CH seems to be a very concrete statement. It may not be a very relevant statement, but it still seems like something that is meaningful, ie it is not non-sensical. And I believe that in the end, mathematicians should accept CH if it is useful for the research they are doing and they should accept not CH if that is useful for the research they are doing. In other words, I see myself as a hardcore pragmatist.
User:Sunny: Hmm...Well, take that pragmatism a bit farther and you've got objectivism. Just have to state that it is not arbitrary whether the statement is true or false; that it is not the assumption of truth or falsity that makes the research work, but the research that determines the truth or falsity of the statement.
User:Henrik Nordmark: I will have to think about that...
User:Henrik Nordmark: Anyways, it is getting late here and I want to get up early tomorrow.
User:Sunny: If I use a wrench to unscrew a nut, I've found the proper application of the tool.
User:Sunny: If try to use it as a hammer it might sort of work, but its non-optimal and definitely not what the tool is best at doing.
User:Sunny: Some mathematical tools may very well only be like wrenches, human constructed and only really useful to deal with other human constructions.
User:Henrik Nordmark: What you say seems plausible, however I think I need more time to think about it carefully to see whether I fully agree with it or not.
User:Henrik Nordmark: Thank you for conversing with me.
User:Henrik Nordmark: It is much appreciated.
User:Henrik Nordmark: I enjoy having conversations with other people who I consider to be rational but who may have slightly different insights and perspectives.
User:Sunny: I always enjoy your insights and questions as well Henrik. Thanks for the great conversation.
User:Sunny: Another one for davinci?
User:Henrik Nordmark: Yes, that was my intent since the very beginning.
User:Sunny: Awesome.
User:Henrik Nordmark: Once again, I would like to spend some time cleaning it up.
User:Sunny: Did you see where the last chat was posted?
User:Henrik Nordmark: I would like to make the discussion readable and educational for other eople as much as possible.
User:Sunny: I concur. I spent some time linking the last one into WikiPedia and MathWorld.
User:Sunny: The last thread is posted at: Discuss:Mathematics, Logic, Philosophy:Infinite Games
User:Henrik Nordmark: Actually, no. I haven't had a chance to look at that yet. I have been traveling a lot. I just came back from Germany. I went an Epistemology of Mathematics workshop in Berlin. It was *VERY* interesting.
