Logic - 2018-10-28

user21820

09:58

@user525966 Just because we only have perceptions does not mean there is no reality. My claim is solely that reality obeys classical logic, and this has nothing to do with our perceptions. Your comment about quantum mechanics is based on a common logical fallacy. Classical logic does not at all fail when describing quantum mechanics. In fact, classical logic is needed for the very foundations of quantum mechanics. So please don't propagate the common misconception.

@user525966 Truth in the absolute sense of course cannot be defined in an absolute way. However, there is no issue with my claim that truth (though not perfectly accessible to us) obeys classical logic, where "truth" here is the philosophical notion and not the mathematical one.

In case it was not clear, when I said "truths about the real world clearly obey classical logic", I did not mean this can be proven absolutely, but just that it is clearly so based on the empirical evidence in its favour.

user21820

10:20

@user525966 Also, your assumption about accepted mathematical axioms is historically correct but no longer so. Indeed, foundational systems used to be based on "self-evident truths", but as time went by it became more and more a product of choice of elegance over truth. For example, after naive set theory (based on self-evident 'truths') was discovered to be severely inconsistent, logicians simply hacked the axioms, restricting the specification axiom, then adding axioms to recover what was lost.

The result was still not the current accepted foundation! It was only ZC set theory, lacking replacement and regularity. Logicians then realized that they couldn't do something that they wanted to do without having something like replacement, so they accepted it too. Later still, they found that regularity makes the 'set-theoretic universe' look pretty, so they added it (but never even accepted it on a self-evidence basis).

As I said to user1414 earlier, the replacement schema cannot be non-circularly justified, and that includes empirical justification. More precisely, ZC plus replacement has almost zero additional predictive power more than ZC. Logically it is true that ZC plus replacement proves consistency of ZC, which is an arithmetical statement, but it is not of the 'natural' kind whose empirical verification carries weight, since one can add consistency statements to any reasonable foundational system.

Furthermore, all of modern physics as known today that can be formalized can be handled within much less than ZC.

user21820

11:02

Worse still for regularity. The one who proposed it never even considered it to be true for all 'sets', and himself explored sets that failed regularity. And not surprisingly, because one can show in ZFC−Reg that if ZFC−Reg has a model then the ∈-well-founded sets in that model forms a model of ZFC.

user131753

14:33

Wow. Nice philosophical discussion going on here.

user525966

16:00

@user21820 I don't know what you mean about the quantum mechanical bit. Almost everyone I read / listen to on the subject (such as the video I linked) says the other way around. The use of classical logic (such as with respect to probabilities) breaks down and you have to treat things like "and" and "or" differently because they no longer act the way that you might expect when extrapolating from a macro-scale environment.

user21820

@user525966 Then I'm quite sure you haven't studied quantum mechanics. I have.

user525966

Do you disagree with the instructor in the video?

user21820

Popular science is very often full of junk, and I don't have the time to wade through tons of videos just to pinpoint where the errors are.

user525966

It's a professor of physics from IIT Madras, not some junk-science thing

We also have en.wikipedia.org/wiki/Quantum_logic

user21820

There are many professors in top universities who cannot even produce a correct proof of a simple theorem using mathematical induction. There are also well-known cranks (common knowledge of the mathematical community) who are professors at relatively reputable universities.

@user525966 If you don't understand what I am saying, please refrain from just randomly citing articles. I have studied quantum mechanics and I know what I'm talking about. If you want I can give you a loose analogy of the logical fallacy, but if you don't believe me then you will have to ask proper experts in both logic and quantum mechanics what the foundational system required to establish quantum mechanics needs, not rely on popular science.

user525966

16:11

I mean I can cite several more -- I just didn't want to spam the room -- but I spent a fair amount of time researching it and they all seem to be in relative agreement on the point

so I disagree it's just "citing random articles" but rather trying to see what the consensus is among experts in the field

user21820

@user525966 They are not experts at all. Let me say very clearly that most of them do not know logic and whatever they say on logic is faulty.

user525966

You would not consider the top answer of this question physics.stackexchange.com/questions/141567/… to be from an expert?

Physicist (MSc and PhD in Theoretical Physics) and a mathematician, full professor of Mathematical Physics, head of the Doctoral School in Mathematics

user21820

@user525966 That answer does not say what you think it says.

I have read it, and it is consistent with what I have said.

In particular, quantum logic can be easily described within the classical foundational system we already have for mathematics.

user525966

Yes but you are changing how they are used / how they apply, fixing the system / holding it constant

same as mentioned in the video with respect to probabilities / probability amplitudes

user21820

As I said earlier we only need a small fragment of ZC set theory to capture practically all mathematics that is relevant to the real world, including quantum mechanics.

@user525966 I am not changing anything. I am being precise. Your original claim was that I cannot say that classical logic underlies the real world. That is not supported by the Physics SE post you just linked to.

Specifically, you said "in quantum mechanics at least, our usual classical frameworks start failing and you have to create another sort of language to describe what's going on (particles vs. waves, how we interpret probabilities, etc)", which is simply false.

If you don't understand why this is false, you need to wait until you learn how the foundational system works. We will come to that after you finish learning PA.

user525966

16:26

"Here a twice possibility arises. One can ignore this fact and exploit it as a merely technical opportunity. Alternatively one can assume that these pseudo connectives are the connectives of the quantum world which, in that precise sense (connectives with different properties), satisfy a logic different from the classical one. This was the idea of von Neumann and Birkhoff, who started the first investigation of that mathematical world. Nowadays, quantum logic is a (wide) research area more proper of logicians rather than physicists. I mean, for several reasons even of practical nature, von …

^this feels consistent with the point I was trying to make

user21820

@user525966 No.

user525966

My point here being that you can either hold the system fixed and focus on different things (such as the IIT video) or like in this comment refer to "pseudo connectives" that satisfy a different logic from the classical one, and even here my overall point being that the logic and mathematics follows from the reality and perception, which I think is a perfectly reasonable claim to make

user21820

I don't know why you're insisting on your point, when it is an undeniable fact that ZC set theory (part of the current standard foundations) is capable of reasoning about all of quantum mechanics. ZC is a system based on (classical) first-order logic. So there cannot possibly be an argument based on our current understanding of quantum mechanics that shows that reality does not obey classical logic.

user525966

I don't disagree that "Just because we only have perceptions does not mean there is no reality" -- my contention is that "reality obeys classical logic, and this has nothing to do with our perceptions" seems awfully strong and I don't see how one can make this argument rigorously

user21820

6 hours ago, by user21820

In case it was not clear, when I said "truths about the real world clearly obey classical logic", I did not mean this can be proven absolutely, but just that it is clearly so based on the empirical evidence in its favour.

user525966

16:31

That seems incompatible with your claim of "nothing to do with our perceptions"

empirical evidence is defined specifically by virtue of experience / perception

user21820

@user525966 Empirically verifiability is orthogonal to human perception.

user525966

https://en.wikipedia.org/wiki/Empirical_evidence

"Empirical evidence is the information received by means of the senses, particularly by observation and documentation of patterns and behavior through experimentation.[1] The term comes from the Greek word for experience, ἐμπειρία (empeiría)."

"Empirical evidence is information that verifies the truth (which accurately corresponds to reality) or falsity (inaccuracy) of a claim. In the empiricist view, one can claim to have knowledge only when based on empirical evidence (although some empiricists believe that there are other ways of gaining…

user21820

@user525966: I'm no longer sure whether you're actually interested in understanding my point, or whether you're just trying to argue.

user525966

I am interested in it, yes, but I don't agree that your counterpoints refute any of this

Nor do I see how this point is upheld

user21820

The basic point is that you still fail to distinguish between reality and our perceptions of it.

user525966

16:35

in particular "reality obeys classical logic, and this has nothing to do with our perceptions" again I think is a super strong claim and I don't see how you can justify this

user21820

Reality is independent of our perception, and so whether it obeys classical logic or not is a factual matter and independent of our perception.

That is what I meant, but you keep quoting me out of context.

Malice Vidrine

Guys, guys, such conflict.... [throws in some swords and bucklers and preps the popcorn]

user525966

I don't think it's out of context, I can repaste the whole thing if needed, it's all saying the same thing. I agree that reality is independent of perception, and whether it obeys some logical system is a factual matter, but this is a very different claim that saying it does obey that system regardless of perception (a much stronger claim)

user21820

@user525966: Let me put it again. Either reality obeys classical logic, or it does not. I claim that it does, not because it can be proven absolutely, but because all empirical evidence are completely in line with my claim. There is zero basis to claim contrary.

And the empirical evidence I refer to is not very subjective, contrary to your quotes of wikipedia.

As a logician, I do not use the kind of 'empirical evidence' that philosophers commonly do.

user525966

How then are you defining "obeys"? Even when going by empirical evidence (experiment and observation which hold true for everyone doing the experiment), we could either go one fork or the other as by that earlier stackexchange answer where we can use the usual classical connectives but only in certain contexts or only with respect to certain properties, or we make new pseudoconnectives that better describe the phenomena that takes place on this other scale

user21820

16:43

@user525966 This is why I think you have to wait until you learn how the foundational system works. A proper foundational system will not be such that you keep having to 'extend' it every time you come across new things. Rather, you should be able to express every new thing you come across in the same foundational system. And that is exactly what ZC can do for every known application of mathematics to the real world.

user525966

Are you saying this ZC system will handle "anything you throw at it" in the real world regardless of how reality happens to unfold at varying scales?

user21820

That is the whole point I am making. Classical first-order logic has never needed modification since 2000 years ago. It is still the underlying logic for every foundational system, used to formalize scientific applications of mathematics. I was careful to say "you should be able to express every new thing you come across in the same foundational system", because a foundational system is not an intelligent being that can think for you; it's just a language.

No scientific application of mathematics cannot be formalized in ZC. And as far as we know, ZC can prove, not just express, all the theorems needed to justify those applications.

@user525966: Are you familiar with the foundational construction of the integers, rationals and reals? That is a key stepping stone in building real analysis that is required for much of physics and chemistry, and is something you probably can understand after a bit of work. It would give an idea of how very complicated structures can be constructed and their properties proven, within the foundational system without adding any assumptions.

user525966

Vaguely yes

Naturals defined in terms of FOL / set theory, integers extending that, then rationals extending that, reals extending them in terms of "dedekind cuts" or "cauchy sequences"

user21820

Yes, that's right.

Malice Vidrine

(Cauchy sequences are better) [returns to homework]

user21820

16:57

@user525966: Maybe you will be amazed to see that not only reals, but essentially all of measure theory and hence also probability theory can be constructed within ZC.

And hence various special kinds of probability spaces that are needed to construct (representations of) quantum mechanical objects within ZC.

@MaliceVidrine Agreed. =)

user525966

17:14

I think at least for now I need to dip back into FOL to better understand / use that system first

Since I can't really even use it fully yet

user21820

@user525966: Yes please let me know if you need more hints for the exercises. Once you're done with those exercises we can move on straight to PA, and prove a few basic results of number theory. And then move on to the foundational system where we basically have enough to handle all of mathematics.

I have to go. See you next time!

user525966

See you

Malice Vidrine

17:44

Incidentally, this is one of my pet peeves from people I know who sort of know about physics: the idea that superposition is "and".

user525966

@MaliceVidrine What do you mean?

Malice Vidrine

(That and, more generally, the notion that excluded middle gets rid of anything counterintuitive in logic. Though I can usually shut people up on this one by pointing out that, intuitionistically, the operation that swaps two elements on of a set and fixes everything else is not necessarily a permutation)

@user525966 - I've known a few laypeople who say that a superposition of states is an instance of "something and its negation both holding", so quantum physics forces us to rethink non-contradiction.

Which, of course, is not what is going on in superposition.

user525966

I think people end up thinking that because of stuff like Schrodinger's Cat which often gets repeated / twisted into "the cat is both alive and dead at the same time"

Malice Vidrine

The "fuzzy logic" response is of the same sort; some hear the other version, "the cat is neither alive nor dead, but in some intermediate state" and conclude that something is wrong with bivalence.

Granted, science writers are weirdly afraid to ever talk about complex numbers, which is weird.

So weird I said "weird" twice in the same sentence.

user525966

I think it's because complex / imaginary numbers aren't the most intuitive things to people who don't have any math or science background

Malice Vidrine

18:01

No, but we encountered them in a very basic way in high school, so if you said something like "blah blah blah is allowed to take values in the complex numbers instead of just the real numbers" with a one-sentence reminder of what that meant, a reasonably educated reader could at least note that they're dealing with uncommon numerical properties instead of uncommon logical properties.

Ugh, this is reminding me of my pending complex analysis homework.... Would that I could put it off forever....

user525966

@MaliceVidrine True, but even if most high schoolers encounter complex/imaginary numbers with things like say solutions to the quadratic formula, they still don't know what that really means or how those kinds of problems / solutions have any basis in the real world

hence the usual question "When am I ever going to use this?" which often doesn't receive a very satisfactory answer from teachers who are mostly just repeating a curriculum guide without really knowing more than that

Malice Vidrine

Well that's how you understand that it has physical applications.... by seeing it used in a physical context. Which never happens if you sweep them under the rug any time you might mention them in the most cursory manner.

Transcript for

♦ $Mathematics$ Logic

Transcript for

♦ Logic

♦ $Mathematics$ Logic