Kurt Gödel & Quantum Physics

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.

Quoting Sam26

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

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.

Reply to 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.

This user has been deleted and all their posts removed.

Quoting tim wood

consistency is unprovable within the system

And unprovable from other systems of a certain kind.

Quoting tim wood

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.

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.

Quoting Sam26

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.

Schrödinger's cat?

Reply to tim wood

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 [math]\rightarrow[/math] S [premise]
2. S [math]\rightarrow[/math] 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 [math]\rightarrow[/math] ~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 [math]\rightarrow[/math] ~P = 7. ~P

Gödel's argument becomes:

1. ~P [substituting P [math]\rightarrow[/math] ~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.

Reply to TheMadFool 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.

Quoting Sam26

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.

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

Reply to TheMadFool I am right, this is a common misunderstanding.

Quoting Sam26

I am right, this is a common misunderstanding.

Maybe.

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

Quoting Sam26

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

: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.

Quoting TheMadFool

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.

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."

Quoting TheMadFool

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

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.

Quoting Andrew M

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."

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.

Quoting Andrew M

No

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

Quoting TheMadFool

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

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.

Quoting TheMadFool

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

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

[math]|\psi\rangle = 0.6|up\rangle + 0.8|down\rangle[/math]

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.

Quoting Andrew M

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.

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.

Quoting Andrew M

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

|?>=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.

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?

Reply to TheMadFool

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

Quoting Photios

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

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

Quoting TheMadFool

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.

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:

Quoting The Present Situation in Quantum Mechanics, 5. Are the Variables Really Blurred? - Erwin Schrodinger

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.

Quoting TheMadFool

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)?

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).

Reply to Andrew M :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).

Quoting Sam26

?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.

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.

Quoting SolarWind

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.

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

Quoting Sam26

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

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:

Quoting Law of non-contradiction - Wikipedia

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)

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.

This user has been deleted and all their posts removed.

Quoting TheMadFool

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

Your task: Translate E into English.

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.

Quoting tim wood

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.

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.

Reply to Andrew M :ok: However, the equation is just one line.

Quoting TheMadFool

However, the equation is just one line.

It's expressed in rules R1 and R2.

Quoting TheMadFool

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.

Popular science writing is both a blessing and a curse...

Reply to Andrew M I'm not satisfied with your answer. Thank you for taking the trouble to explain it though. G'day.

Quoting Andrew M

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

I don't understand. Can you please write that in the usual bra–ket notation?
Especially the minus sign in R2 is strange.

Quoting TheMadFool

I'm not satisfied with your answer. Thank you for taking the trouble to explain it though. G'day.

You're welcome! What in particular were you not satisfied with?

Quoting SolarWind

Can you please write that in the usual bra–ket notation?
Especially the minus sign in R2 is strange.

Sure. The minus sign represents a 180 degree phase shift which differentiates the two superposition states. Adding them results in destructive interference for the tails component.

The coin (or particle) can be represented by a qubit where heads (or spin-up) is defined as:

[math]\hspace{10 mm}|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}[/math]

and tails (or spin-down) is defined as:

[math]\hspace{10 mm}|1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}[/math]

The quantum coin flipper is implemented by a Hadamard (H) operation:

[math]\hspace{10 mm}H = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}[/math]

1. The coin is prepared in a heads state:

[math]\hspace{10 mm}|0\rangle[/math]

2. Perform first quantum coin flip:

[math]\hspace{10 mm}H|0\rangle = \frac{1}{\sqrt2}(|0\rangle + |1\rangle)[/math]

3. Perform second quantum coin flip:

[math]\hspace{10 mm}\begin{aligned}H(\frac{1}{\sqrt2}(|0\rangle + |1\rangle)) &= \frac{1}{\sqrt2}(\frac{1}{\sqrt2}(|0\rangle + |1\rangle) + \frac{1}{\sqrt2}(|0\rangle - |1\rangle)) \\ &= \frac{1}{2}|0\rangle + \frac{1}{2}|1\rangle + \frac{1}{2}|0\rangle - \frac{1}{2}|1\rangle \\ &= |0\rangle\end{aligned}[/math]

4. Perform measurement:

[math]\hspace{10 mm}|0\rangle[/math]

Quoting Andrew M

You're welcome! What in particular were you not satisfied with?

It just doesn't feel right to me.

Here's an equation/formula, a rather simple one, F = ma (Newton).

In English F = ma can be rendered as the magnitude of a force F acting on a mass m that imparts an acceleration a is equal to the product of m and a.

Do something similar with the equation for superposition (Schrödinger's?).

Quoting TheMadFool

Do something similar with the equation for superposition (Schrödinger's?).

It seems to me that you're thinking of a superposition as a kind of law (like F=ma). It's not. It's just a particular kind of state that a quantum system can be in.

In classical mechanics, the definite position of a system can be calculated. The classical expectation is that it will be in that definite position whether or not it is measured.

In quantum mechanics (using the Schrödinger equation), the wave function for a system is calculated. The wave function is then used to calculate the probability of finding the system at any definite position. For example, if the wave function has amplitude (i.e., height or depth) at positions' 2 and 10 and those amplitudes are equal, then there is a 50% probability of finding the system at position 2 and a 50% probability of finding the system at position 10. This combination of potentially measurable positions prior to measurement is termed a superposition.

Quoting Andrew M

It seems to me that you're thinking of a superposition as a kind of law (like F=ma). It's not. It's just a particular kind of state that a quantum system can be in.

Even if we take Schrödinger's equation as just that and not a law which I think it is, we should be able to come up with a one line description of it.

Take this equation: [math]y = 2x + 7[/math]. In English it would be y is equal to twice x increased by 7.

Quoting TheMadFool

we should be able to come up with a one line description of it.

Take this equation: y=2x+7. In English it would be y is equal to twice x increased by 7.

Perhaps this is what you're looking for?

Quoting Andrew M

Perhaps this is what you're looking for?

:up: Yes and No.

The link you provided to a science forum does lead me to the exact same question I asked but it still seems deeply immersed in mathematical concepts that don't have corresponding, matching ideas that are part of normal, non-technical discourse. The question then is, is math invented or discovered?

Quoting Andrew M

For example, if the wave function has amplitude (i.e., height or depth) at positions' 2 and 10 and those amplitudes are equal, then there is a 50% probability of finding the system at position 2 and a 50% probability of finding the system at position 10. This combination of potentially measurable positions prior to measurement is termed a superposition.

Such a wavefunction does not exist. Dirac deltas are not eigenfunctions. They are distributions.

Quoting TheMadFool

The question then is, is math invented or discovered?

Obviously invented and projected upon physical reality. It's reasonable that math is effective. Math is derived from structures in the physical world.

Quoting GraveItty

Obviously invented and projected upon physical reality. It's reasonable that math is effective. Math is derived from structures in the physical world.

Invented and derived. :chin:

Perhaps we forgot what we had invented. This happens for real I believe. Andrew Wiles (mathematician who proved Fermat's Last Theorem) must've all but forgotten his proof.

Don't take derivation litterally, as if physical structures posses mathematical structures. A circular structure doesn't possess a math formula for a circle. The circles on top of my soup can induce (maybe that's a better word than derivation) a math formula for a circle. It's not an inherent formula of the circle. At least, not for me. A Platonist would disagree, although he places the formula in an extra worldly heavenly realm. Some place it in the objects themselves. People like Deutsch and Texmark, who, especially the former, try to base QM on a formal math basis. Without any knowledge of what the math describes I think that's a fruitless effort. David Bohm's hidden variables, though the nature of these variables stays obscure (but hey, isn't the Nature of reality obscure?, come closer to a physical reality, explaining the probalistic nature of QM. It's a pitty the people at the Copenhagen conference gave the fiat to the probability interpretation! How different the standard texts on QM would have looked!

Quoting GraveItty

Don't take derivation litterally, as if physical structures posses mathematical structures

Why would physical structures not possess mathematical structures. When I look at an ordinary die, I see a cube and when I look at the sun/moon, I see a sphere.

Quoting TheMadFool

Why would physical structures not possess mathematical structures. When I look at an ordinary die, I see a cube and when I look at the sun/moon, I see a sphere.

Of course. No doubt about that. It is their mathematical formulation and their abstraction in the mathematical realm (a mathematical sphere is a different sphere as a physical one) that they don't possess. There are in fact
no structures in the physical world corresponding exactly to mathematical structures unless prepared in a very precise way as to accommodate a precise math form. A temporally finite piece of music (its sound pattern, that is), an arbitrary pulse of sound, or an infinite periodic pattern, can be written, but only approximated as an infinite sum of sine waves of sound with appropriate coefficients (a procedure like the one used in the epicycle approach to the motion of celestial bodies). There corresponds no exact mathematical form in the realm of math. Only an approximation. If the piece or pulse become too extended then the pattern can't be approximated by math. Though the pattern is actually there. Only single sine soundwaves and finite combinations of them have exact correspondences in the mathematical kingdom. You can question if an approximation corresponds to the exact pattern of the piece of music. There is no exact or even approximate mathematical structure of the musical sound pattern if the piece is too long. So eventhough there is a physical pattern, there is corresponding pattern in math. Of course you can transpose the non-functional, non-approachable sound pattern of the piece of music to the realm of math as a form. But this form can't be expressed as a function (other sets of base-functions could be chosen, but this doesn't change the argument). There is in fact not much you do with the transposed form. You can construct tangent lines (or planes and volumes if the sound pattern is 2- or 3-dimensional. But that's about it. The pattern is non-reducible.

So what's my point? My point is that forms in the mathematical realm owe their existence to the physical reality.

Quoting GraveItty

It is their mathematical formulation and their abstraction in the mathematical realm (a mathematical sphere is a different sphere as a physical one) that they don't possess.

Why? Are you trying to say that, for instance, Brad Pitt, Albert Einstein, Abraham Lincoln, (basically men) are not men?

Quoting GraveItty

Only an approximation.

I see, a Platonic point of view as far as I can tell. Did Plato ever consider the imagination? Did he not realize, I'm sure he must've, that, in a certain sense, perfection exists only imagination? How does he tell apart imagination and the world of forms? Imagination = World of forms? :chin:

Quoting GraveItty

My point is that forms in the mathematical realm owe their existence to the physical reality.

Why can't it be the other way round? The physical seems to be obeying mathematical forms.

Quoting TheMadFool

Are you trying to say that, for instance, Brad Pitt, Albert Einstein, Abraham Lincoln, (basically men) are not men?

I'm not sure about Brad, and coming to think about neither about Einstein. Of course they belong to mankind. Though coming to think about it... No, seriously, that is not what I am trying to say. They are men without a mathematical equivalent somewhere because their mathematical equivalent doesn't exist, as it doesn't for the soundwave pattern. That is, not expressable in mathematical terms. For example, a sinus function can be expressed graphically as a wavy line in the plane. But what about a line that can't be functionally expressed? It simply doesn't exist in the mathematical realm. Nevertheless, I can draw the line. It exists physically.

Quoting TheMadFool

Why can't it be the other way round? The physical seems to be obeying mathematical forms.

This question is exactly the reason I argued like I did and I answered it already. If a physical form (the drawn or imagined line) can't be expressed as a math formula, there is no counterpart of the form in the math space.

Quoting TheMadFool

I see, a Platonic point of view as far as I can tell. Did

Not at all. Plato imagined a rmathematical heaven, like the math formalists. The place of math is simply the mind. Like intuitionists think. Approximations don't have nothing to do with being a Platonist or not, although Plato indeed thinks that real physical forms are approximations of the mathematical ones (which are not the same as physical forms).

Quoting GraveItty

They are men without a mathematical equivalent somewhere because their mathematical equivalent doesn't exist

How do you know that? What's nonmathematical about man?

Quoting GraveItty

This question is exactly the reason I argued like I did and I answered it already. If a physical form (the drawn or imagined line) can't be expressed as a math formula, there is no counterpart of the form in the math space.

But there were times, I believe, when many mathematical topics today were once never thought to have been mathematical. You seem math-literate, I'm sure you can think of such an instance. How about the mathematical turn to natural philosophy flagged off by Copernicus/Galileo/Newton?

Quoting GraveItty

The place of math is simply the mind

You don't know that.

Here's some food for thought: Is the theory of a mathematical universe an illusion, a bewitchment by language (re: Wittgenstein)? Not everything in math, to my knowledge, is about numbers/shapes (arithmetic/geometry) but when someone claims "it's all math" he means to say that arithmetic/geometry is involved.

Thus what we have here is a discipline/field (math) whose expansionist behavior is gobbling up other fields/disciplines but the catch is, its (math's) definition is also being altered to factor this in until what we have today, as I'm led to believe, math is a study of patterns. I'm sure Thales, Pythagoras, Archimedes would raise pertinent objections to this point of view. I dunno!

Quoting TheMadFool

What's nonmathematical about man?

They can't be represented by a formal math system, whatever the system. Just like the line that can't be represented by a math expression has no correlate in the mathematical world, a man thinking about math expressions and pondering upon non-math-expressable or non computable forms like a random but non-random line, will have no correlate too.

Math is just a language and certainly not the language of Nature, as is so commonly claimed by physicists. And like all languages it has limitations. It's also called a universal language by many scientists (and the parrots, quasi intelligently, reciting their words). BS! It's a constraining and confining net thrown on physical reality, thereby darkening many facts, though in some situations it fits perfectly and with reasonable effectiveness.

Quoting GraveItty

The question then is, is math invented or discovered? — TheMadFool

Math is derived from structures in the physical world.

Originally perhaps. Not so much these days. Abstractions and generalizations abound. After many years I have concluded that math is both created and discovered. When I define a new (but tiny) math object, that is more creative than discovered. Afterwards one can discover what follows from such a definition. But generally speaking most mathematicians pay little attention to the question.

Quoting GraveItty

There corresponds no exact mathematical form in the realm of math.

I recently defined an "LFT form" and "attractor form" for linear fractional transformations. These are "exact" mathematical forms. However, "form" can take on other meanings in math. I'm not talking about Platonic forms.

Quoting GraveItty

So what's my point? My point is that forms in the mathematical realm owe their existence to the physical reality.

Some do.

The scope of mathematics today is astounding, with an unknown number of areas of investigation. Each day about 250 new papers arrive in ArXiv.org, to be roughly categorized in about 35 general areas. Awhile ago I challenged Alexandre to find out how many math articles exist on Wikipedia.

Quoting GraveItty

It's a constraining and confining net thrown on physical reality, thereby darkening many facts

Possibly. Mathematicians working in pure math ask, Why throw a net?

Quoting GraveItty

Such a wavefunction does not exist. Dirac deltas are not eigenfunctions. They are distributions.

Yes, I was envisaging a wave function that just approximates those statistics. Also, welcome to the forum!

Quoting GraveItty

Obviously invented and projected upon physical reality. It's reasonable that math is effective. Math is derived from structures in the physical world.

And what is structure? I notice you give a nod to Aristotle in another post, who defined things in terms of matter and form.

Quoting jgill

Originally perhaps. Not so much these days. Abstractions and generalizations abound. After many years I have concluded that math is both created and discovered. When I define a new (but tiny) math object, that is more creative than discovered. Afterwards one can discover what follows from such a definition.

I can remember that Mandelbrot said in an interview that his realization of the set named after him (stemming from the iterations in the complex field of the simple zexp2+k function, giving rise to these colored pictures on which you can zoom in ad infinity, or ad nauseam) felt more like a discovery than an invention. Well, if that's the case for him then why not. I just don't think this is so. Math can't be discovered (say I indeed, but not holding it for an absolute truth). The field of math is of course very rich. I don't think it is. And of course many mathematicians don't throw it on the physical universe. You can find mathematical models (say the different quantum field models) that have no counterpart in physics. These models obviously don't have a physical counterpart, but they describe the same physical structures with non-existing parameters. All math corresponds to physical structures, though they are abstracting them to a high level. The spin 1/2 of the farming can be represented by a spinor or a vector traveling twice around a M?bius band, but this isn't what an electron truly is. Neither is it a mathematical point or a collection of charges on a 4d curved Planck-sized structure seemingly 3d in our 3d universe (though the last comes close). Even "the great Feynman" said that the cause of the electrons spin is a mystery, though we'll described by physics. How the hell can an electron 's spin rotate once if you rotate the electrons twice (which is closely connected to anti-symmetric wavefunctions), a fact I have seen beautifully illustrated by rotating one of two interfering electron fields once, which resulted in an inversion of the interference pattern. Somehow the anti symmetry and the spin half are connected. Of course they are connected by math (like spin 1 vectors and symmetric wavefunctions are connected) but what happens in reality remains a mystery. I have a gut feeling about that though. But this is not the place to get into that. A M?bius band is a form that can be realized in the physical world. Certain properties of it can be described by math. But if we make the band sufficiently erratic (while keeping its overall form) math can't describe anymore (like the erratic line I mentioned earlier, without a functional form can't be described by a mathematical formula and as such doesn't exist in the mathematical world. You can of course measure the line's distance to a n origin but it still can't be described by a formula, though you could of course claim that it can be described by an infinity of real numbers, though not being calcuable). However it may be be, for me math is a human invention and it can reasonably be applied to the physical world. It would be far more unreasonable if it couldn't. It certainly is not an inherent property of Nature but can sometimes describe it very well. Again, the simple example of a physical line form, say a long sewing thread, can't be calculated. Measured yes.

Quoting GraveItty

a fact I have seen beautifully illustrated by rotating one of two interfering electron fields once, which resulted in an inversion of the interference pattern.

So we've settled on waves now? Interference happens when the behavior is waves. Is this right?

Quoting Caldwell

So we've settled on waves now? Interference happens when the behavior is waves. Is this right?

If you ask so, then we can settle on them. Fasten your seatbelts though for some heavy destructive interference can be expected. Which basically answers your second question.