Incompleteness, Undecidability, and Uncertainty

[durangodawood]

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.

For future reference: you wont go broke underestimating my math knowledge.

 

[J_B_]

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 ...

Not that I have any chance of changing your mind, but I do disagree, and I take some comfort in not being alone. Rather than mounting an argument, I'll simply quote from the Foreward of Longo's book, which was written by the Oxford biologist Denis Noble.

My apologies for the length of it.

During most of the twentieth century experimental and theoretical biologists lived separate lives. As the authors of this book express it, "there was a belief that experimental and theoretical thinking could be decoupled." This was a strange divorce. No other science has experienced such a separation. It is inconceivable that physical experiments could be done without extensive mathematical theory being used to give quantitative and conceptual expression to the ideas that motivate the questions that experimentalists try to answer. It would be impossible for the physicists at the large hadron collider, for example, to search for what we call the Higgs boson without the theoretical background that can make sense of what the Higgs boson could be.

From there he goes on to make several interesting comments about biology. For example, that evolution "does not make specific predictions in the way in which the Higgs boson [does] ... evolution is more that of a general framework within which biology is interpreted." Further, while biology does have theories - many theories - they are not subject to any "theoretical construct", they are "not formulated [as theories]", rather they are "presented as fact, a fait accompli."

He concludes then, that "There is a need for a general theory of biological objects and their dynamics. This book is a major step in achieving that aim."

With respect to this thread and the articles referenced, it was not specifically the work of Strippoli from 2005 that intrigued me, but what Sverdlov did with it (and other information) in the second paper in formulating "unsolvable problems" in biology. In the same way, the uncertainty principle of physics is only interesting because of the unsolvable problems it identifies.

I'm not trying to establish some exact equivalency between the two, but as soon as one establishes a principle, it not only tells you what you can do, but what you can't do. That was one of Ana Soto's original complaints about her cancer research. Researcher 1 uses data set 1 to make conclusion 1, which conflicts with what researcher 2 concluded from data set 2. And there was no theory of biology that could mediate the conflict. It was just a battle of numbers with no end in sight.

I've had similar frustrations in my own career. A test is run and numbers generated. The numbers don't make sense. The suggested action is, "Let's run another test." No thinking, no analyzing, just test and test and test until you get numbers you're comfortable with.

 

[FrumiousBandersnatch]

Not that I have any chance of changing your mind, but I do disagree, and I take some comfort in not being alone. Rather than mounting an argument, I'll simply quote from the Foreward of Longo's book, which was written by the Oxford biologist Denis Noble.

My apologies for the length of it.

During most of the twentieth century experimental and theoretical biologists lived separate lives. As the authors of this book express it, "there was a belief that experimental and theoretical thinking could be decoupled." This was a strange divorce. No other science has experienced such a separation. It is inconceivable that physical experiments could be done without extensive mathematical theory being used to give quantitative and conceptual expression to the ideas that motivate the questions that experimentalists try to answer. It would be impossible for the physicists at the large hadron collider, for example, to search for what we call the Higgs boson without the theoretical background that can make sense of what the Higgs boson could be.

From there he goes on to make several interesting comments about biology. For example, that "[evolution] does not make specific predictions in the way in which the Higgs boson [does] ... evolution is more that of a general framework within which biology is interpreted." Further, while biology does have theories - many theories - they are not subject to any "theoretical construct", they are "not formulated [as theories]", rather they are "presented as fact, a fait accompli."

He concludes then, that "There is a need for a general theory of biological objects and their dynamics. This book is a major step in achieving that aim."

With respect to this thread and the articles referenced, it was not specifically the work of Strippoli from 2005 that intrigued, but what Sverdlov did with it (and other information) in the second paper in formulating "unsolvable problems" in biology. In the same way, the uncertainty principle of physics is only interesting because of the unsolvable problems it identifies.

I'm not trying to establish some exact equivalency between the two, but as soon as one establishes a principle, it not only tells you what you can do, but what you can't do. That was one of Ana Soto's original complaints about her cancer research. Researcher 1 uses data set 1 to make conclusion 1, which conflicts with what researcher 2 concluded from data set 2. And there was no theory of biology that could mediate the conflict. It was just a battle of numbers with no end in sight.

OK... but I don't see how that's relevant to the post it was responding to - what has Noble's forward to do with whether the 'axioms' of science are foundational or necessary?

 

[J_B_]

OK... but I don't see how that's relevant to the post it was responding to - what has Noble's forward to do with whether the 'axioms' of science are foundational or necessary?

Because, repeating and paraphrasing part of what I quoted from Noble above, they give quantitative and conceptual expression to the ideas that motivate the questions that experimentalists try to answer. It's impossible to search for a scientific object without the theoretical background that can make sense of what it could be.

Flat space is different than hyperbolic space. Changing an axiom results in a change from one to the other.

 

[FrumiousBandersnatch]

Because, repeating and paraphrasing part of what I quoted from Noble above, they give quantitative and conceptual expression to the ideas that motivate the questions that experimentalists try to answer. It's impossible to search for a scientific object without the theoretical background that can make sense of what it could be.

Those are theory models & hypotheses, not axioms.

Flat space is different than hyperbolic space. Changing an axiom results in a change from one to the other.

Flatness may be an axiom of Euclidean geometry, but the curvature or flatness of space is not an axiom of physics - we make observations to discover the geometry of space.

 

[Ophiolite]

It's impossible to search for a scientific object without the theoretical background that can make sense of what it could be.

If we consider that geology proper began in the early part of the 19th century then for most of the time geology has been practiced it has been either in the absence of any theoretical background, or using theoretical backgrounds that were largely bogus. Despite these supposed limitations a vast body of observations were gathered that allowed for the emergence of plate tectonic theory in the 1950's and 1960's. That theoretical resource now illuminated a wealth of extant observations beyond those on which the theory was founded. This experience contradicts your assertion.

 

[J_B_]

Flatness may be an axiom of Euclidean geometry, but the curvature or flatness of space is not an axiom of physics - we make observations to discover the geometry of space.

I disagree. You're inventing a history of science that never happened to fit your philosophy of what should have happened.

As I said, I knew I wouldn't be able to change your mind. I've mentioned some references to get you started if you decide to look into it further. Based on this last post, I suppose I should add some historical references as well. To start with, maybe: Einstein's Jury: The Race to Test Relativity by Jeffrey Crelinsten or Einstein's Masterwork: 1915 and the General Theory of Relativity by John Gribbin.

Good luck.

 

[Speedwell]

I disagree. You're inventing a history of science that never happened to fit your philosophy of what should have happened.

As I said, I knew I wouldn't be able to change your mind. I've mentioned some references to get you started if you decide to look into it further. Based on this last post, I suppose I should add some historical references as well. To start with, maybe: Einstein's Jury: The Race to Test Relativity by Jeffrey Crelinsten or Einstein's Masterwork: 1915 and the General Theory of Relativity by John Gribbin.

Good luck.

I'm inclined to agree with you--"common sense" Newtonian space was certainly taken for granted in the early days of physics. But I have lost track of the point you were trying to make with it.

 

[J_B_]

If we consider that geology proper began in the early part of the 19th century then for most of the time geology has been practiced it has been either in the absence of any theoretical background, or using theoretical backgrounds that were largely bogus. Despite these supposed limitations a vast body of observations were gathered that allowed for the emergence of plate tectonic theory in the 1950's and 1960's. That theoretical resource now illuminated a wealth of extant observations beyond those on which the theory was founded. This experience contradicts your assertion.

I don't think it does. Good science requires a lot of rigor, and we need to be careful in separating observations, definitions, data, and theories.

If I call a certain insect a "bee" and describe what it looks like, that is a definition. It's not a theory of bees. If I observe that bees fly, that is an observation, not a theory of insect flight. If I take data on the shapes of bee wings, flight speed, altitude, the effect of wind, that is data, not a theory ... even if I package all that data together to statistically describe when bees can fly and when they can't. If I took that data with the belief that fairies help bees fly, then later use that data to investigate aerodynamics, it doesn't mean the aerodynamic theory of bee flight started when I took the data.

If that last sentence is true, then plate tectonic theory began when Adam picked up the first rock and offered it to Eve as food.

I'm not familiar with the history of geology, so forgive my source, but according to Wikipedia, observations of the matching geography of continents are recorded as far back as 1596. The idea of continental drift was suggested in 1912. The first evidence supporting the hypothesis came in the 1950s. That makes it appear that the idea preceded the supporting data - that someone proposed the idea (based on observations, sure), but then they had to go looking for data. The data they had to date was insufficient, because they didn't know what corroborating data to look for until someone suggested the phenomena: observed, asked a question, formulated a hypothesis, and designed a test. Sound familiar?

Just because some old data may have been used in that process doesn't mean plate tectonic theory "existed" in that old data. Plate tectonic activity - the ding an sich - may have affected that old data, but the theory didn't exist in that data. To suggest such a thing would be to suggest numbers exist in the things of nature.

I'm not saying you're suggesting any of those things. I just started rambling and decided to cover all the bases (or at least the ones that came to mind).

 

[J_B_]

But I have lost track of the point you were trying to make with it.

That is a problem in Internet forums.

Some are arguing science has no axioms. They seem to equate empirical observation with truth. If we see it and name a principle after it, that's a truth, not an axiom.

Ain't never gonna buy that.

... Or did you mean way back to the OP?

 

[J_B_]

There are many historical examples and psychology experiments that show phenomena are not inherently obvious in what we observe. It's not the data, but how we think about the data that matters (with regard to our ability to master said phenomena).

Here's an example from the U.S. military during WWII I happened to stumble across:

 

[FrumiousBandersnatch]

I disagree. You're inventing a history of science that never happened to fit your philosophy of what should have happened.

I'm not talking about the history of science, but about whether or not what are often taken to be its axioms really are axioms. I'm open to reasonable argument about that.

As I said, I knew I wouldn't be able to change your mind. I've mentioned some references to get you started if you decide to look into it further. Based on this last post, I suppose I should add some historical references as well. To start with, maybe: Einstein's Jury: The Race to Test Relativity by Jeffrey Crelinsten or Einstein's Masterwork: 1915 and the General Theory of Relativity by John Gribbin.

My mind isn't made up, it's open to change if I get a reasonable argument. At present, I suspect that what are commonly taken to be the axioms of science are not foundational.

If you disagree, present what you believe to be a foundational axiom of science and let's discuss it.

What you have posted appears to me to be unrelated to the question of axioms and when pressed, rather than clarify, you accuse me of re-inventing the history of science - as if responding to a different post entirely... ¯\_(ツ)_/¯

 

[Ophiolite]

I just started rambling

That was my impression. Here is why.
I offer agian this assertion of yours:

It's impossible to search for a scientific object without the theoretical background that can make sense of what it could be.

Here are some of the "scientific objects" related to plate tectonics that were observed before the the theory of plate tectonics was proposed. (There are a bunch more.)

• Restricted distribution of volcanic activity
• Benioff zones
• Mid-ocean ridges
• Discontinuities of tectonic zones across oceans

Either your assertion is refuted by this, or your assertion has been poorly phrased and thus misunderstood by me. Feel free to restate it, or acknowledge the refutation. I'm comfortable with either.

Incidental aside:
Continental drift and plate tectonics are not synonymous.
Evidence for plate tectonics was in place, recognised, but not understood, well before the 1950s.
You note:

The data they had to date was insufficient, because they didn't know what corroborating data to look for until someone suggested the phenomena: observed, asked a question, formulated a hypothesis, and designed a test. Sound familiar?

Unfortunately for your argument this does not reflect what occurred. For example, no one was looking for evidence of ocean floor spreading. Magnetic striping ( a scientific object) was discovered with no explanation for it till the work of Vine & Mathews and another pair whose names temporarily escape me.
Same for ocean trenches. Same for Benioff zones. Same for heat flow anomalies. The list goes on.

 

[FrumiousBandersnatch]

I'm inclined to agree with you--"common sense" Newtonian space was certainly taken for granted in the early days of physics. But I have lost track of the point you were trying to make with it.

I'm not talking about what was commonly assumed at one time or other, but whether there are any foundational axioms without which science can not proceed.

 

[Speedwell]

That is a problem in Internet forums.

Some are arguing science has no axioms. They seem to equate empirical observation with truth. If we see it and name a principle after it, that's a truth, not an axiom.

Ain't never gonna buy that.

... Or did you mean way back to the OP?

In that case, you undermine your own point that flat space was ever an axiom, and given your response, it appears that I mean all the way back to the OP after all.

 

[Speedwell]

I'm not talking about what was commonly assumed at one time or other, but whether there are any foundational axioms without which science can not proceed.

Indeed, but what's the point being made?

 

[J_B_]

That was my impression.

My apologies that you found it excessive.

Here are some of the "scientific objects" related to plate tectonics that were observed before the the theory of plate tectonics was proposed. (There are a bunch more.)

• Restricted distribution of volcanic activity
• Benioff zones
• Mid-ocean ridges
• Discontinuities of tectonic zones across oceans

Either your assertion is refuted by this, or your assertion has been poorly phrased and thus misunderstood by me. Feel free to restate it, or acknowledge the refutation. I'm comfortable with either.

I do believe the term "scientific object" was coined by Noble, probably without the intention of making that specific term a subject of debate.* Regardless, what I am saying is that you are referring to observations and definitions, not a theory, not a "scientific object". You even use that phrase yourself. Just because the guy who observed it was wearing a white lab coat doesn't make it a "scientific object". It was just an observation. People observe all kinds of things with no idea what it means. Those observations can't be the object of a theory if the theory did not yet exist. It is only when meaning is assigned that it becomes a scientific object.

In the end, it seems rather tautological to me.

Still, there is no way I'm going to be able to argue geology with you. I'm not equipped for it. You've declared your point made, so all I can do is move on.

[*Note]: If I'm wrong about "scientific object", please correct me.

My mind isn't made up, it's open to change if I get a reasonable argument. At present, I suspect that what are commonly taken to be the axioms of science are not foundational.

If you disagree, present what you believe to be a foundational axiom of science and let's discuss it.

Likewise, I'm not equipped to argue biology. Neither of us are biologists (I believe), and so this has already devolved into a semantic debate. What is an axiom vs. a foundational empirical observation? The intent of this thread was, first, to discuss uncertainty principles (incompleteness, undecidability, etc.) and, second, to share some recent information I had found. I don't believe any actual biologists have weighed in on that information (maybe I'm wrong). If they do, much like @Ophiolite 's comments on geology, all I can really do is shrug and say, "OK."

I'm not sure we have any scientific common ground upon which we can discuss your proposed question of axioms. If we discuss something from your specialty, it most certainly won't be mine, and if we discuss something from my specialty, it most certainly won't be yours.

But if you want me to throw something out for discussion, how about "force"? It would be interesting for me to see it discussed by you (@FrumiousBandersnatch ), @Ophiolite , and @Speedwell , but I imagine you're not going to let me sit on the sidelines and observe.

 

[essentialsaltes]

I think there's a useful distinction between an axiom of Science aka the scientific method. And the axioms or assumptions that are built into any particular model of some aspect of natural phenomena.

 

[J_B_]

I think there's a useful distinction between an axiom of Science aka the scientific method. And the axioms or assumptions that are built into any particular model of some aspect of natural phenomena.

An example being?

 

[Ophiolite]

[*Note]: If I'm wrong about "scientific object", please correct me.

It's impossible for me to tell at this stage. When you introduced the term in passing, without explicit definition, I had to infer its meaning from your comments. It appeared as if the term was equivalent to observations, experimental results and the like. It now emerges from your comments, still with no definition, that it is closer to being a theory. I base this inference upon this:

Regardless, what I am saying is that you are referring to observations and definitions, not a theory, not a "scientific object".

It seems you are establishing a direct equivalence of theory and scientific object. The trouble with that is you then seem to be saying "we can't have a theory unless we have a theory", which doesn't seem to take us anywhere.

What am I missing?