Here is continuation thread that was spanned out from that is it possible to solve go. Since it went a few googolplexes off the topic, better to continue here.

It started here and continued few posts with increasing word count and increasing off topic content.

http://lifein19x19.com/forum/viewtopic.php?f=18&p=29695#p29695

If replies to quantum logic discussion is someone's interests, it is better to do it here.

Liisa -- Your comments in the orginal thread missing something vital about mathematics/logic.

Not to the point that one of the alternative logics would be "better" (in some way). The mathematical statements being made were within the context of the usual logic used (its axioms) and the axioms of mathematics. It can make sense to argue whether one of these systems is "complete", whether it is "consistent", etc. It does not make sense to argue that the system is "false" (or "true"). Keeping in mind that the terms like "cpmplete", "consistent" etc. have specialized meanings.

Nothing about the physical universe involved.

It's difficult to approach the first part of this post because my perspective is compatible already with the example you gave. Further, your citation of observation of Jackdaws is confusing. Can you be more precise about what non-contradictory things they have done and what the contardictory counterpart to those things that they did not do would be? Then can you explain to me how anyone can tell that the Jackdaw is generalizing its experience in the form of a logical axiom?

When observing quantum phenomena I fail to see how non-contradiction is violated (or any less obvious) since we have no reason to believe that its probabilistic description of entities is anything more than a predictive model, its fuzziness produced by inherent limits in measurement and not by actual, real essential properties of the particles themselves. This is, I think, where we disagree fundamentally. I do not buy your assumption that quantum mechanics does anything other than function as a very good (or even best possible) predictive model of very small phenomena. I'm also not convinced that it couldn't be rephrased with no loss whatsoever in its descriptive capacity in a manner compatible with classical logic, where the supposed 'contradictions' produced are merely the result of the misapplication of the notion of identity. For instance, using an example later in your post:



Surely we could simply either A) Deny that the Jackdaw is actually the same thing, and thus there is no contradictory behavior or B) Reject that the law of non-contradiction has anything do with spatiotemporal location. In other words, the Jackdaw is perfectly capable without any contradiction whatsoever of having its presence spatiotemporally operate as a wave and a particle. After all, it was not classical logic that made us suppose that being a wave and being a particle were mutually exclusive, it was classical physics.



The "perfect game" can actually exist either abstractly or naturally. It definitely exists abstractly, though we do not know what it is. Since Go is defined in terms of these rules it is capable of being instantiated in an actually existing game of Go. The chances of this are immensely unlikely, but not impossible.




I don't share your certainty that it is not allow for large enough computations. However, I don't mind ceding this point particularly much. Even if it's not possible to ever compute the perfect game of Go on a 19x19 board, I think it's a perfectly valid object of discussion and contemplation.



This is a rather large section so I'm going to isolate some key points:



You continue to assert this but have yet to justify it, shortly below is the closest you've come besides the very unclear Jackdaw example:



How? Does this include itself? If it does include itself as being derived from experience, then how do you know that we will not later derive something from experience that demonstrates it to be false? Even if it is capable of self-justification isn't it potentially rendered vulnerable to Godel-type propositions? You would know better than I would what the particular properties of the system are.

Of course, even if that were the case since you reject the principle of non-contradiction, there would be no problem having a complete but inconsistent system.

However, if it can contain and allow for contradictory propositions then clearly it allows for the proposition (in probabilistic form) that it is itself false. Which brings things back around to the problem of radical empiricism. Unless the truth of empricism is transcendental, empiricism itself becomes only a product of finite inductive reasoning and thus will be subject to falsifiability. So, let's proceed to your remark concerning verification:



I don't follow this one bit. How precisely is verification possible in the framework of probabilistic logic?

Let's say that I consider a six-sided die to have a probability of producing a 6 when rolled of 1/6.

How do I verify this? I cannot. No matter how many times I roll it, no matter what the results are, they are all compatible with the hypothesis that the probability of rolling a 6 is 1/4 or 2.3/9. Some will be more or less probably compatible, but this probability is itself subject to the same unverifiability. Further, considering that you believe the universe to be a closed and finite system I wouldn't even think the sample size would be sufficient at all for verifying propositions regarding its nature or the nature of its contents as a whole.

So, we are back to the two-pronged dilemma: Either Quantum Logic takes the form of a transcendental in which case you are not an empiricist (at least in the sense that Kant is not an empiricist) or Quantum Logic takes the form of a probability and is no longer anything resembling a certainty. In which case, we do not have any particular reason to believe that it is more than a predictive mathematical model of reality (EDIT: That I can see, at least. Perhaps you have a very good reason, but I haven't seen any clearly in your posts thus far).

Despite Mike's assurance, this is a type of formalist interpretation of mathematics that is not universally shared, not by any stretch of the imagination. It's more common among "ordinary" mathematicians, computer scientists, etc, but far less universal among set theorists, logicians and people interested in the foundations of mathematics.

(This is not meant to bear on the specific points made by liisa or whoever else--I stopped reading the other thread a while ago).

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Occupy Babel!

This is curious! Pape what do think what is the relationship between mathematics and reality? Monadology made their point rather clear, but can you say something about relation?

If it is not clear, my boldest claim is that mathematics and contemporary logic are just product's of human imagination where real world has been used as a loose reference. Such as it is the case with Euclidean geometry. Mathematics and logic have been at least in historical perspective derived from experiences. Modern mathematics is perhaps more free product of pure imagination and the correspondence to the reality is vague.

Idea is that since we can choose freely rules and axioms for mathematics (and for logic) and we cannot make any difference what is better set of rules and axioms. (My assumption is that mathematicians are trying to find simplest set of rules and axioms). This is rather queer situation. I am definitely not familiar with higher mathematics than differential equations, but I have a feeling, that the justification for mathematics is rather consensus of Maths society than finding a correspondence to the real world.

I guess that this is the general consensus, and people do indeed have altogether discarded naturalist interpretations of science, logic and maths. I find this attitude intellectually very lazy, because one and only thing that we can know for sure is that nature do exist out there. Although philosophers have been pretended to be like there is no nature or at very least they have been dead quiet of the topic. Luckily natural sciences do not require philosophy in order to triumph, so it is of course perfectly ok to discard philosophy as nonsense. But I find it as a lazy attitude.

I will return more accurately to Gottfried's post hopefully in few days.

I look forward to your response.

Just one comment on your most recent post. Be careful about generalizations regarding philosophy as a discipline. The common joke certainly applies here: put ten philosophers in a room and you will get eleven different points of view.

In particular, I know of a recent and interesting (to me) movement referring to itself as "speculative realism," which has strong realist and materialist views regarding 'nature,' allying themselves around a rejection of the so-called Copernican revolution of Kant and its consequences in philosophy since, taking a lot from Gilles Deleuze and Alain Badiou who are both, I think, not the kind of lazy philosophers you describe (though you'd probably hate Badiou )

This is just not true! I think we agree about this much:

1) Mathematics is, at its core, proofs based on axioms (and logical rules or rules of inference)

2) People can (and do!) use any set of axioms they wish and still do mathematics with them.

But then you say:

3) Mathematicians choose axioms without any concern about a correspondence to the real world.

How could physics possibly exist without mathematics? I think you believe that quantum mechanics accurately describes experimental results (as I do also). But how could quantum mechanics have been described without advanced and abstract mathematics?

I am not a physicist, I'm a computer science phd student. But as such I make much use of, and enjoy, mathematics with very real practical application. I could certainly do math with no significance whatsoever, and sometimes I do, but I don't think science or engineering could exist without math.

Monadology, have you heard of Bayesian inference? It's a mathematical approach to solving the problem you describe. Basically we have some prior guess as to the probability of an outcome, and update our estimate as we gain new data.

Yes, this is the situation as I understand it. You can arbitrarily select axioms and come to a variety of conclusions.

But you can approach life in the same way. At some point, you select some axioms that you believe in and can make logical inferences based on those axioms. But somewhere down the line, you are just talking based on some leap of faith. But it's inevitable since we are not gods that know of some sort of universal truth.

That said, mathematics is just a tool which we can use to help us with events in the real world.

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However, you CAN make a difference between what's a "better" set of rules and axioms. That's called modeling.

To use this tool, though, you must make a model of what you want to analyze by constructing a mapping between your closed mathematical system and the outside world.

It could very well be that you don't model reality well with mathematics. That's also something to work on in itself.

So deductions based on mathematics are only as good as the fundamental axioms upon which they rely, along with how well they are modeled to the real world.

So making conclusions still involves some degree of uncertainty. But everything has some degree of uncertainty (even this statement), so it's unavoidable.

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be immersed

I have heard of it, and I don't believe that is a satisfactory resolution of the problem.

Don't get me wrong, I'm not any kind of skeptic regarding probabilistic logic or probabilities. I'm not a skeptic about science, the existence of the real world or many other things.

However, unless I have misunderstood, Liisa is placing probabilistic logic in an epistemological position that I do not think is tenable and I don't think Bayesian inference can help even if it is excellent at resolving difficulties on a different level (like the actual practice of scientific observation and statistical determination).

Bayesian inference can be thought of in terms of the scientific method. You have some hypothesis about the world, you test it, you look at the results, and see how confident you were about the hypothesis. This has its own problems, because investigating an issue typically brings about new questions - but that's a different story.

The reason we have a prior with Bayesian inference is because we have some sort of "prior belief" about how the thing we are studying works. In the case of the example with dice, we may have a prior belief that this particular die will land on 3 about 1/6th of the time. However, it's possible that you have a weighted die. Or you might be rolling the die in a weird way. Or the wind may affect how you throw the die. These can all affect reality. By repeating a roll many, many times, you can come to new predictions (hypotheses) about how often the die will land on a 3, for example.

It's because the real world is complex that we have to do such an analysis.

Now suppose that you could somehow capture all of the physics behind throwing a single die. You knew exactly what gravitational forces acted upon the die. You knew what momentum and angle you threw the die with. If you could capture everything about how you threw the die, it may be possible to determine with 100% certainty what number the die lands on. If you know EVERYTHING about the situation, then it's no longer a probability question - it's just a question you know the answer to. In the real world, especially with dice, this is not feasible.

With the game of go, there are many complexities. We currently cannot innumerate the number of possible board states, and cannot determine a proven winning strategy to the game of go. There are many unexplored factors, which prevent us from doing this.

But if there were some way to prove a winning strategy in go (assuming more advanced techniques from future technology, for example), it would mean that we have considered all of the relevant factors of the game.

The question is, "What is a relevant factor of the game?". If we are confident that we've covered all of our bases, we can say that we have proven a solution to the game of go. Of course, it's possible that we've made a mistake, but that's possible with any "truth" that you "know".

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A slightly separate topic is the issue of mathematical modeling. It's important to identify if the math that you've proven accurately captures what's happening in the real world. If someone proved a solution to go, for example, we'd have to ensure that it accurately captures the reality of the game. We can be "certain" that it captures the reality, but it's always possible that everybody is wrong.

For example, though it may be proven that a^2 + b^2 = c^2 in a right triangle, it's possible that, after all of these years, we were applying the equation incorrectly to reality.

But if we have enough confidence to say that something is true, as much as we say anything else is true, I think it's enough.

I can say, "I'm sitting on my couch". I'm pretty sure it's the case. I might be wrong, but I still say it like it is truth.

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be immersed

You know how when you first learn the rules of Go, it seems completely bizarre that the score is decided by mutual agreement? Like, how would you know who's right? And where to end the game? How does this game even make any sense?

And then you learn more about it and it turns out that everything is perfectly simple and unambiguous, except for some really rare corner cases that are so irrelevant to practice that only a couple nuts in the whole world care about them.

It's the same with math and axioms. Issues like who decides on the axioms and whether they reflect reality sound important but turn out not to be. This is why the overwhelming majority of mathematicians doesn't care about them. People who do care about them, foundations freaks, are the equivalent of ruleset freaks in Go. What they do might be entertaining to themselves but it's not important or insightful to anyone else.

I know just enough not to have any real views.

More seriously, I'm impressed by the ongoing research into the truth/falsity of the continuum hypothesis, despite its independence from previous axiomatizations of set theory. In my opinion, that work means that you have to be very careful to state any kind of formalism that's not a dead end.

Also, I think Palapiku exaggerates a little bit. It's true that most math gets by just fine without foundational research, but even our friendly set of non-lesbegue measurable reals (a result you'll see in a first or second course in analysis) depends on your set-theoretic axioms (unless I've gotten myself muddled).

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Occupy Babel!

There are some terminology problems (we aren't all using terms the same way). And there are also some wrong assumptions abiout what some of us are saying, taking these as saying more than they are.

a) I was not meaning to imply "there is one correct logic" of "one correct mathematics" or "one correct geometery", etc.

b) When it was noted that for a right triangle the sum of the squares of the sides equals the square of the hypetenus this is (for example) not true for all geometries. I would not consider the question "which best matches the physical universe" having anything to with the geomtetries themselves (doesn't affect the theorems within them). What you might use a geometry for (or a system of logic for or a system of methematics for).

c) I am perfectly aware of the disagreements within mathematics, which axioms to include or not, etc. Similary with logic. It's not being in a "nut faction" to be interested in one of these alternatives and to study the resulting theorems within that system. The "nut" factions" are those who want to go farther and try to argue "but my preferrred set is the right, true, etc. set of axioms" (its "real mathematics", "real logic", and the others are wrong)

As an example of thinking mathematically -- somebody said the positions of stones on a go board wasn't "numerable". Meaning of term? For me that would have nothing to do with time but whether a specified relation could be set up with the "integers" (I could specify a rule that listed the possible posiitons in order) and that there were less than the integers (finite).

That's easy --- consider the 361 intersections as a linear array (start in one corner, go across, then the next line). The number of legal go positions cannot be more than the numebr of ways it is possible to have a black stone, a white stone, or no stone on an intersection. The you can consider the board to be a representation of the integers in base three form zero up to 3**361 - 1 That's a big number but it's finite and the order of possible board positions specified (that of the number represented). The legal boards (possible in go) are a subset of this (and for any "number" easily determined by the rules of go wheither this is a legal arrangement at the end of a move).

It is a separate matter whether practical to carry out the computation (given speed of hardware, etc.) But in theory of computations the first issue is "finite" or not and not whether practical. Though that is of interest too. You don't begin by trying to implement but first demonstrate that you have a finite algorithm (one that will terminate giving a definite answer).

I was wondering when you'd show up in this thread.

And just about any Go player, yes

Oddly enough, although at the time they did it the mathematicians were just "entertaining themsleves" turns out fairly common that about 100 years later some area of math that appeared to be abstract turns out very useful for something of other.

To find cases where areas of math were developed in order to be useful (because needed) perhaps have to go back to the time of Newton.

Mike, my understanding (and I'm venturing a bit beyond areas where I'm that knowledgable) is that you can't just say "well, choose whatever axioms you like and study the consequences" because the notion of consequence is explained in terms of models of the axioms, and models just end up being sets (in most cases), so you need to have an intended model of set theory, but there's more than one hanging around.

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Occupy Babel!

Monodology: In response to your comment about the "fuzziness" being due to the measurement inaccuracy rather than a probabilistic nature of the underlying particles.

The Copenhagen Interpretation would strongly dispute this,

And Bell's Inequality would out-and-out refute it.

Not saying that it doesn't feel right (After all, you're in good company, Einstein felt the same way), But this is at odds with what we know about quantum.

EDIT: Woops, meant Monod, not Liisa

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Tactics yes, Tact no...

As I understood it, the experiments confirming Bell's theorem have not yet entirely ruled out hidden variables on a non-local scale. I was also under the impression that hidden variables were not the only potential problem (something to do with a deterministic model of the universe, but it's been a while).