Incompleteness, Undecidability, and Uncertainty

[sjastro]

Ironically in another field of biology or more precisely biochemistry, the science behind photosynthesis involves quantum mechanics.
In this case the Heisenberg Uncertainty principle ∆x.∆p ≥ h/2 where ∆x and ∆p are the uncertainties in the measurement of position and momentum respectively is not applicable.

Since photons participate in photosynthesis and travel at the speed of light, there is no quantum mechanical position operator x which can measure the position of a photon in spacetime at some given time t.
Without going into the gory details if such a mathematical operator existed then photons can travel backwards in time and violate causality.
In this case photosynthesis could occur before photons had reached the chlorophyll sites.
It’s one of the main motivations in the development of quantum field theories where photons are excitations of the quantum field and not as particles localized in spacetime.

The study of photosynthesis involving quantum mechanics is part of the emerging field of quantum biology.

 

[FrumiousBandersnatch]

Ironically in another field of biology or more precisely biochemistry, the science behind photosynthesis involves quantum mechanics.
In this case the Heisenberg Uncertainty principle ∆x.∆p ≥ h/2 where ∆x and ∆p are the uncertainties in the measurement of position and momentum respectively is not applicable.

Since photons participate in photosynthesis and travel at the speed of light, there is no quantum mechanical position operator x which can measure the position of a photon in spacetime at some given time t.
Without going into the gory details if such a mathematical operator existed then photons can travel backwards in time and violate causality.
In this case photosynthesis could occur before photons had reached the chlorophyll sites.
It’s one of the main motivations in the development of quantum field theories where photons are excitations of the quantum field and not as particles localized in spacetime.

The study of photosynthesis involving quantum mechanics is part of the emerging field of quantum biology.

Yes, there are also indications that quantum effects are relevant in the sensitivity of some birds to magnetic fields, and in the olfactory sense, where relatively few receptor types can accurately distinguish an extremely wide range of scent molecules. Jim Al-Khalili wrote a book about this, 'Life on the Edge: The Coming of Age of Quantum Biology'.

 

[durangodawood]

I made an offhand comment here, but now it's going to bug me. I might as well get it out of my system.

It seems to me that if science were necessarily tied to mathematics in an existential way, a scientific incompleteness theorem would simply be an extension of the mathematical one (i.e. Godel). However, that seems impossible to establish.

Maybe something based on measurement, as in physics' uncertainty principle, would be a more realistic possibility. That is, maybe some measurement in biology (etc.) would imply a biology uncertainty principle.

What do you think? Is there, maybe, an incompleteness, undecidability, or uncertainty principle for sciences other than physics?

I think science does rest on certain unprovable but intuitively precious axioms. Like: this stuff is real and Im not a brain in a vat. Etc.

 

[FrumiousBandersnatch]

I think science does rest on certain unprovable but intuitively precious axioms. Like: this stuff is real and Im not a brain in a vat. Etc.

They may seem like axioms, but they are empirically derived (or effectively meaningless); we find that our perceptions fall into various patterns of 'events', many of which are consistent and repeatable - we infer some source for these which we call 'reality'. We have no more reason to think we're a brain in a vat than any other conceivably contrived situation, so there is no good reason to suppose that we are.

But it is interesting that we do seem to be in a situation that is analogous to being a brain in a vat - we are brains in the 'vat' of the skull, kept alive and fed with information by a biological machine. Of course, this is only a superficial analogy, lacking the crucial idea of being helpless victims of a fiendish experiment, kept alive at the whim or pleasure of some superior intelligence - this ought to strike a chord with the theists out there...

 

[Tanj]

Apologies for the frenetic pace, but in checking publications that referenced the above paper, I found this:

Unsolvable Problems of Biology: It Is Impossible to Create Two Identical Organisms, to Defeat Cancer, or to Map Organisms onto Their Genomes

I think it's fascinating that someone is claiming unsolvable problems in biology. How widely the idea is known or accepted is a different matter.

It's fascinating you find fundamental, basic aspects of biology fascinating. You should keep reading, I am keen to learn where you go if you consider the very simple start point "fascinating".

 

[durangodawood]

They may seem like axioms, but they are empirically derived (or effectively meaningless); we find that our perceptions fall into various patterns of 'events', many of which are consistent and repeatable - we infer some source for these which we call 'reality'. We have no more reason to think we're a brain in a vat than any other conceivably contrived situation, so there is no good reason to suppose that we are.

But it is interesting that we do seem to be in a situation that is analogous to being a brain in a vat - we are brains in the 'vat' of the skull, kept alive and fed with information by a biological machine. Of course, this is only a superficial analogy, lacking the crucial idea of being helpless victims of a fiendish experiment, kept alive at the whim or pleasure of some superior intelligence - this ought to strike a chord with the theists out there...

Id say the axioms of formal arithmetic are also empirically derived. So there's really not a great distinction there.

 

[Tinker Grey]

Id say the axioms of formal arithmetic are also empirically derived. So there's really not a great distinction there.

Agreed. I've a saying: All axioms are arrived at inductively including this one.

 

[J_B_]

It's fascinating you find fundamental, basic aspects of biology fascinating. You should keep reading, I am keen to learn where you go if you consider the very simple start point "fascinating".

It is easy to take for granted what has been given us without our effort. I try not to mistake "fundamental" for simplistic or easily achieved. In mathematics the term is "elegant" - those things that, though they appear simple, are made from layers and layers of complexity.

 

[FrumiousBandersnatch]

Id say the axioms of formal arithmetic are also empirically derived. So there's really not a great distinction there.

Yes; although different mathematics can and have been devised by deliberately changing the standard axioms.

 

[durangodawood]

Yes; although different mathematics can and have been devised by deliberately changing the standard axioms.

Interesting. I wonder how much practical application those maths have?

At any rate it sounds like were back to something like: the axiomatic statements that underpin scientific reasoning are empirically derived (and completely reasonable in the colloquial sense) but unprovable in an absolute sense.

 

[J_B_]

Interesting. I wonder how much practical application those maths have?

I could say something about that, but it's probably much too obvious to everyone here assembled, and therefore not worth the waste of the electrons.

 

[durangodawood]

I could say something about that, but it's probably much too obvious to everyone here assembled, and therefore not worth the waste of the electrons.

Ummm not obvious to me. Go on, "burn" a few electrons.

 

[J_B_]

Ummm not obvious to me. Go on, "burn" a few electrons.

The obvious example that many will reference is Euclid's Parallel Postulate, which has been a contentious mathematical problem for millennia and involved many famous names. From it stemmed the debate regarding whether space is flat, hyperbolic, etc.

But that's so water under the bridge, so 20th century, so freshman physics. There couldn't possibly be anything interesting to discuss about the basics of dimensionality.

 

[Speedwell]

The obvious example that many will reference is Euclid's Parallel Postulate, which has been a contentious mathematical problem for millennia and involved many famous names. From it stemmed the debate regarding whether space is flat, hyperbolic, etc.

But that's so water under the bridge, so 20th century, so freshman physics. There couldn't possibly be anything interesting to discuss about the basics of dimensionality.

 

[durangodawood]

The obvious example that many will reference is Euclid's Parallel Postulate, which has been a contentious mathematical problem for millennia and involved many famous names. From it stemmed the debate regarding whether space is flat, hyperbolic, etc.

But that's so water under the bridge, so 20th century, so freshman physics. There couldn't possibly be anything interesting to discuss about the basics of dimensionality.

Freshman physics was as far as I got.

Not sure that non-Euclidian geometry is quite the departure that, say, revising the associative/commutative axiom of arithmetic would be. Would love to know what neat real world applications that would have.

(In fact, I cant even imagine what an alternative to the associative/commutative axiom would look/work like..... although in fairness to myself I only spent a few mins trying.)

 

[J_B_]

Freshman physics was as far as I got.

Sometimes my snarkiness gets away from me, but it wasn't aimed at you.

Not sure that non-Euclidian geometry is quite the departure that, say, revising the associative/commutative axiom of arithmetic would be.

I think it is, but the way these things shook the world is often lost on a modern audience. There are quite a few of them: Galileo's agony over the lack of an absolute spatial reference and it's connection to affine geometry; the controversy over the material constitutive equation and it's relationship to tensor math; the debates among Descartes, Hooke, Newton, Leibniz, D'Alembert, et. al. about the nature of energy and force and it's expressions in calculus; and so on and so forth.

These things look ridiculously simplistic to us, but defending why they are the way they are is tied to what we think reality is.

Would love to know what neat real world applications that would have.

https://www.youtube.com/watch/-EtHF5ND3_s

Eventually in this video he gets around to explaining that the associative and commutative properties are not as universal as we are led to believe in our high school math classes. The implications are critical to infinite series, and thereby calculus, which is used daily in engineering, etc.

 

[FrumiousBandersnatch]

Interesting. I wonder how much practical application those maths have?

Not much, I suspect, outside of academic mathematics and logic.

At any rate it sounds like were back to something like: the axiomatic statements that underpin scientific reasoning are empirically derived (and completely reasonable in the colloquial sense) but unprovable in an absolute sense.

I'm not sure that they're really axioms (i.e. that they're necessary) - we make observations, find patterns in the data, and make and test explanatory models. Fundamentally, we can't help but make observations and detect patterns in the data - science formalises that process and attempts to make it more reliable.

Speculating (I haven't really thought about this), it seems to me that it's only in the hypothesising or model-making stage that we find may find concepts like an external reality to be useful, though not essential; the whole process would work just as well if we were Berkelyites, believing that there was only mind, and that the mind of God was responsible for the consistency of our perceptions...

 

[bhillyard]

Interesting that the associative law does not necessarily hold for computer arithmetic.
Add a very large number to a very small number - depending on the precision of the system the result will be like adding zero to the large number. Repeat the process many times - still leaves the result unchanged.
However if you first add the many small numbers you can end up with a number commensurate with the large number. So the final addition gives an answer different to the first way.

 

[durangodawood]

Sometimes my snarkiness gets away from me, but it wasn't aimed at you.

Oh no offense taken at all. I realize my science knowledge is probably just below the average of an interested lay person.

I think it is, but the way these things shook the world is often lost on a modern audience. There are quite a few of them: Galileo's agony over the lack of an absolute spatial reference and it's connection to affine geometry; the controversy over the material constitutive equation and it's relationship to tensor math; the debates among Descartes, Hooke, Newton, Leibniz, D'Alembert, et. al. about the nature of energy and force and it's expressions in calculus; and so on and so forth.

These things look ridiculously simplistic to us, but defending why they are the way they are is tied to what we think reality is.

To my mind scientific axioms are more basic that even the ideas overthrown by various scientific "revolutions". Things like: are the laws of nature stable across accessible time and space, which, if not, then every finding would be presumed unreliable.

https://www.youtube.com/watch/-EtHF5ND3_s

Eventually in this video he gets around to explaining that the associative and commutative properties are not as universal as we are led to believe in our high school math classes. The implications are critical to infinite series, and thereby calculus, which is used daily in engineering, etc.

Very interesting

 

[essentialsaltes]

Not sure that non-Euclidian geometry is quite the departure that, say, revising the associative/commutative axiom of arithmetic would be. Would love to know what neat real world applications that would have.

(In fact, I cant even imagine what an alternative to the associative/commutative axiom would look/work like..... although in fairness to myself I only spent a few mins trying.)

Quantum field theories sometimes make use of Grassmann variables. Numbers that are anticommutating, i.e. xy = -yx

With the interesting result that xx = -xx = 0, for all Grassmann numbers.