The electron is not in any state until it's measured, so there is no contradiction. One could think of it as the potentiality of states, or, there is the probability that it will be in this state or that state when measured. So, it's not that it's both up and down at the same time.

This reply is only meant to address some of your concerns. It doesn't address your comments or questions about Godel.

The electron is not in any state until it's measured, so there is no contradiction. — Sam26

But no one can say what a measurement is.

Whether a measuring instrument is enough, which is heavy enough, or whether only a consciousness completes the measurement nobody could answer until today.

The measuring can happen in a variety of ways. It seems to be a kind of disturbance, conscious or otherwise. It's a disturbance of the wavefunction. The main point, though, is that there aren't contradictions happening, as far as I can see.

in its current form, is consistent...or not. — TheMadFool

Well, what Godel demonstrated is that consistency is unprovable within the system. Further, that there are true statements also unprovable within the system. The "within the system" making all the difference. The unprovable true statements being, when added, just the axioms of an augmented system.

In referring to "mathematics," then, and depending on the exact topic and point being made, it becomes even necessary to make clear exactly what "mathematics" is meant.

And to say that a system is unprovable is not to say that processes within the system are not provable, or in any way inconsistent. 2+2=4 being a simple example of a true and consistent expression from within mathematics.

To be up and down at the same time is a frank, outright contradiction. — TheMadFool

Sure, in English/language/logic. But whoever said that those were in fact adequate to describe nature?

consistency is unprovable within the system — tim wood

And unprovable from other systems of a certain kind.

to say that a system is unprovable is not to say that processes within the system are not provable, or in any way inconsistent. 2+2=4 being a simple example of a true and consistent expression from within mathematics. — tim wood

Systems are not things we look at for being probable or unprovable. Maybe you meant this: That a particular system can't be proven consistent by certain means does not entail that the system is inconsistent.

The electron is not in any state until it's measured, so there is no contradiction. One could think of it as the potentiality of states, or, there is the probability that it will be in this state or that state when measured. So, it's not that it's both up and down at the same time.

This reply is only meant to address some of your concerns. It doesn't address your comments or questions about Godel. — Sam26

Schrödinger's cat?



A = Axiomatic system of math we're currently using (could be unknown)

S = Schrödinger's equation (superposition)

C = the cat is both dead and alive (Schrödinger's cat) = the spin is both up and down [English language equivalent of Schrödinger's equation]

1. A $\rightarrow$ S [premise]
2. S $\rightarrow$ C [premise]
3. A [assume for reductio ad absurdum]
4. S [1, 2 MP]
5. C [2, 4 MP; contradiction]
Ergo,
6. ~A [3 - 5 reductio ad absurdum]

~A = There's an inconsistency in the axioms of math we're currently using.

I'm not sure whether a contradiction in the English language equivalent of an equation in physics (Schrödinger's equation) is a contradiction in math though. Thrice removed from the real McCoy, like art according to Plato.

About Gödel's Incompleteness Theorems

Gödel sentence (G): This (mathematical theorem) is unprovable (within given system)

P = G is provable (G has a proof)

Gödel's argument

1. P $\rightarrow$ ~P
2. P [assume for reductio ad absurdum]
3. ~P [1, 2 MP]
4. P & ~P [2, 3 Conj]
Ergo,
5. ~P [2 - 4 reductio ad absurdum]

~P = G is unprovable: Incompleteness theorem

However, take a look at statement 1:
6. ~P v ~P [1 Imp]
7. ~P [6 Taut]

In other words, 1. P $\rightarrow$ ~P = 7. ~P

Gödel's argument becomes:

1. ~P [substituting P $\rightarrow$ ~P with ~P]
2. P [assume for reductio ad absurdum]
3. P & ~ P [1, 2 Conj; contradiction]
Ergo,
4. ~P [2 - 3 reductio ad absurdum]

But the conclusion (4. ~P) appears in the premises (1. ~P). Circulus in probando.

It's a misinterpretation of Schrodinger's cat to think the cat is both dead and alive. You can never observe the cat in both states. And, once you do observe it, it's then in one state or the other, so no contradiction.

It's a misinterpretation of Schrodinger's cat to think the cat is both dead and alive. You can never observe the cat in both states. And, once you do observe it, it's then in one state or the other, so no contradiction. — Sam26

Somehow I don't believe you and in a certain sense I think you're right.

I am right, this is a common misunderstanding.

I am right, this is a common misunderstanding. — Sam26

Maybe.

Man, if you don't believe I'll just die. Just do a little more reading, and you too, will agree. lol

Man, if you don't believe I'll just die. Just do a little more reading, and you too, will agree. lol — Sam26

:ok:

Godel does not use circular reasoning in the incompleteness proof. The proof can be given by means of assumptions no greater than finitistic combinatorial arithmetic and within intuitionistic logic. Contrary, to a post above that begins in abysmal ignorance with mentioning, in a contradiction in terms, a theorem that is not provable.

1. How on earth does math state, as a mathematical equation probably, the contradiction referred to above (bolded/underlined) without itself being contradictory? This is diabolical sorcery: Here is a mathematical equation :point: Schrodinger's equation perhaps and it's not a contradiction BUT when translated into English, :point: both up AND down, it is a contradiction. — TheMadFool

The issue is only that something was lost in the translation from math to English. Paraphrasing SMBC, "Quantum superposition... It doesn't mean spin-up and spin-down at the same time. At least, not the way you think.
...
It means a complex linear superposition of a spin-up state and a spin-down state. You should think of it as a new ontological category: a way of combining things that doesn't really map onto any classical concept."

2. Is it possible that quantum mechanics reveals that math is inconsistent? — TheMadFool

No.

I'm of the opinion reality can't be understood, even with a language so esoteric as math.

You can't possibly make things more self-evident than they make themselves via any language.

Math is really just an area of wizardry, so to speak. Useful, but not sufficient.

The truly best understanding is just that feeling in your soul, in the pit of your stomach. If you could write mathematical poetry, that might be something, cause poetry is closer to the ephemeral heart. And reality actually just happens to be just as ephemeral, as much as we were hoping for immediate solutions.

There's so much more to this all than any current area of expertise.

The issue is only that something was lost in the translation from math to English. Paraphrasing SMBC, "Quantum superposition... It doesn't mean spin-up and spin-down at the same time. At least, not the way you think.
...
It means a complex linear superposition of a spin-up state and a spin-down state. You should think of it as a new ontological category: a way of combining things that doesn't really map onto any classical concept." — Andrew M

Then why all the hullabaloo about Schrödinger's cat? There's something odd about quantum mechanics, that's for sure.

I like the recommendation to introduce "...a new ontological category..." It seems necessary and thereby hangs a tale I suppose.

No — Andrew M

Why? Using only the axioms of math, whatever they are, we've arrived at a contradiction. What's the next step?

Then why all the hullabaloo about Schrödinger's cat? There's something odd about quantum mechanics, that's for sure. — TheMadFool

The hullabaloo is about how to interpret the math, not the math itself. Many Worlds is closest to treating a superposition as a conjunction with the caveat that the opposite spin states are indexed to different worlds: thus no contradiction.

Why? Using only the axioms of math, whatever they are, we've arrived at a contradiction. What's the next step? — TheMadFool

No, the math doesn't imply a contradiction. Here's an example of a superposition in Dirac (Ket) notation:

$|\psi\rangle = 0.6|up\rangle + 0.8|down\rangle$

The '+' in Dirac notation is not a logical 'and'. To link the formalism to observation, square the coefficient for a state to calculate the probability that that spin state will be observed (e.g., 0.6*0.6=36% probability of observing spin-up).

You would need additional assumptions to derive a contradiction. See, for example, Bell's Theorem.

The hullabaloo is about how to interpret the math, not the math itself. Many Worlds is closest to treating a superposition as a conjunction with the caveat that the opposite spin states are indexed to different worlds: thus no contradiction. — Andrew M

That's what I was getting at. What about Schrödinger's cat thought experiment? I suppose it's a veiled criticism of the Copenhagen interpretation which is open to so-called quantum weirdness.

No, the math doesn't imply a contradiction. Here's an example of a superposition in Dirac (Ket) notation:

|ψ>=0.6|up>+ 0.8|down>

The '+' in Dirac notation is not a logical 'and'. To link the formalism to observation, square the coefficient for a state to calculate the probability that that spin state will be observed (e.g., 0.6*0.6=36% probability of observing spin-up).

You would need additional assumptions to derive a contradiction. See, for example, Bell's Theorem. — Andrew M

Yes, despite my math illiteracy, I can tell, it's safe to assume, that there's no mathematical contradiction. The question then is, why do people, scientists, Schrödinger himself for example, resort to analogies that are frank contradictions (the cat is both dead and alive)?

What's up with that?

Just an observation: electrons are not really 'particles' but rather localized excitations in the electron field.

Just an observation: electrons are not really 'particles' but rather localized excitations in the electron field. — Photios

So, the way out of this maze is to change perspective? I like that. Thank you!

That's what I was getting at. What about Schrödinger's cat thought experiment? I suppose it's a veiled criticism of the Copenhagen interpretation which is open to so-called quantum weirdness. — TheMadFool

Yes, Schrödinger posed it to highlight the consequences of accepting a "blurred reality" at the microscopic level. Specifically, that we should then also expect to observe a "blurred reality" at the macroscopic level. But we don't - we observe either alive cats or dead cats, not both at the same time. Thus, for Schrödinger, the apparent "blurred reality" at a microsocopic level is similarly refuted. Our picture is merely shaky or out-of-focus, not a picture of clouds and fog banks. As Schrödinger concludes:

It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks. — The Present Situation in Quantum Mechanics, 5. Are the Variables Really Blurred? - Erwin Schrodinger

Yes, despite my math illiteracy, I can tell, it's safe to assume, that there's no mathematical contradiction. The question then is, why do people, scientists, Schrödinger himself for example, resort to analogies that are frank contradictions (the cat is both dead and alive)? — TheMadFool

In Schrödinger's case, he presented his thought experiment to point out what he considered the absurdity of the prevailing view of quantum mechanics (that is, the Copenhagen Interpretation).

But the thought experiment does not by itself express a contradiction. That requires additional assumptions, such as the "blurred reality" model above and the results of observation (that are not blurred).

:up: I'll have to take your word for it. Still, as per the Wikipedia article on Schrödinger's cat, there is something fishy going on and that wierdness comes into sharp focus once we causally link the microscopic to the macroscopic (Schrödinger's cat).

↪SolarWind
The measuring can happen in a variety of ways. It seems to be a kind of disturbance, conscious or otherwise. It's a disturbance of the wavefunction. The main point, though, is that there aren't contradictions happening, as far as I can see. — Sam26

The funny thing is that there is never a contradiction if you only look at what arrives in the consciousness. Every consciousness carries out its own private collapse of the wave function (->Wigner's friend).

If every observer has his own wave function according to his state of knowledge, then the contradictions are cancelled.

The funny thing is that there is never a contradiction if you only look at what arrives in the consciousness. Every consciousness carries out its own private collapse of the wave function (->Wigner's friend).

If every observer has his own wave function according to his state of knowledge, then the contradictions are cancelled. — SolarWind

Contradictions are about statements or propositions, reality itself is not contradictory. Contradictions only occur in language, i.e., when using concepts.

Contradictions are about statements or propositions, reality itself is not contradictory. Contradictions only occur in language, i.e., when using concepts. — Sam26

Yes, and that's also Schrödinger's position in the "Are the variables really blurred?" quote above. It's worth noting that Aristotle formulated both a logical and an ontological version of the LNC:

The traditional source of the law of non-contradiction is Aristotle's Metaphysics where he gives three different versions.[14]

1. ontological: "It is impossible that the same thing belong and not belong to the same thing at the same time and in the same respect." (1005b19-20)
2. psychological: "No one can believe that the same thing can (at the same time) be and not be." (1005b23-24)[15]
3. logical (aka the medieval Lex Contradictoriarum):[16] "The most certain of all basic principles is that contradictory propositions are not true simultaneously." (1011b13-14) — Law of non-contradiction - Wikipedia

Challenge To All

Most posters have denied that there are contradictions in quantum mechanics.

Despite my doubts I'll give you that, ok, there are no contradictions in quantum mechanics.

Suppose this :point: E is the equation for the superposition of spin states of a particle.

Your task: Translate E into English.

Suppose this :point: E is the equation for the superposition of spin states of a particle. — TheMadFool

You appear to be operating with two related agendas: 1) to find a solid bridge between language and reality, and I am pretty sure that, as language is ever descriptive, it can not ever be real, or, 2) to be satisfied that at some level of smallness, reality itself ceases to be real.

And this reflective of a dissatisfaction with your own understanding - some variety of which in some or another application we all feel. But what I think you have got, and all that you have got, is that language takes in the fringes of reality clumsily and with difficulty, and at the extremes not at all, until someone invents new language to cover it. That is, I am very sure that the reality you question has itself no problem with being real.

Suppose this :point: E is the equation for the superposition of spin states of a particle.

Your task: Translate E into English. — TheMadFool

For an English translation, consider a coin. It has two possible states: heads or tails. There are three operations we can perform.

O1. Place the coin in an initial state (either heads or tails).
O2. Flip the coin.
O3. Measure the coin's orientation (either heads or tails).

We can also choose a machine to do the coin flipping - either a classical flipper or a quantum flipper.

Consider an experiment with the following steps:
1. Place the coin in a heads state.
2. Flip the coin.
3. Measure the coin's orientation.

Over many runs, the observed statistics of heads/tails will be 50%/50% regardless of whether a classical or a quantum coin flipper is used.

Now consider the following experiment:
1. Place the coin in a heads state.
2. Flip the coin.
3. Flip the coin again.
4. Measure the coin's orientation.

In this case, the observed statistics of heads/tails using the classical coin flipper will still be 50%/50%. But using the quantum coin flipper, the observed statistics of heads/tails will be 100%/0%.

Quantum mechanics formalizes that result. It represents the state of the coin after step 2 as a linear combination of heads and tails (i.e., a superposition) and which can be treated mathematically like any definite state. The way this is done is by applying the mathematical operation to each component of the superposition separately and then combining the results. The following rules apply to a quantum flip:

R1. quantum flip(heads) = heads + tails
R2. quantum flip(tails) = heads - tails

Applying this to the earlier experiment:
1. prepare: heads
2. quantum flip(heads) = heads + tails
3. quantum flip(heads + tails) = quantum flip(heads) + quantum flip(tails) = (heads + tails) + (heads - tails) = heads + heads
4. measure: heads

So that particular mathematical formalism correctly predicts what is measured in the experiment. However it doesn't say what it physically means for the coin to be in superposition. That's the job of interpretation.

You appear to be operating with two related agendas: 1) to find a solid bridge between language and reality, and I am pretty sure that, as language is ever descriptive, it can not ever be real, or, 2) to be satisfied that at some level of smallness, reality itself ceases to be real.

And this reflective of a dissatisfaction with your own understanding - some variety of which in some or another application we all feel. But what I think you have got, and all that you have got, is that language takes in the fringes of reality clumsily and with difficulty, and at the extremes not at all, until someone invents new language to cover it. That is, I am very sure that the reality you question has itself no problem with being real. — tim wood

Very perceptive of you to diagnose my condition accurately.

What I see is the problem how, math is a language, a perfectly sensible expression (equation of quantum superposition) in math when translated into another language (natural languages like English), most who do so end up with a contradiction? I can't wrap my head around that, sir/madam, as the case may be.

:ok: However, the equation is just one line.