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TOPIC: The Experimental Verification of Mathematics via an Axiom of Measurement by Christopher Levi Duston [refresh]

TOPIC: The Experimental Verification of Mathematics via an Axiom of Measurement by Christopher Levi Duston [refresh]

In this essay we argue that the partial abstraction used to describe reality in physics can be extended to a complete formal system by the addition of an axiom of measurement, elevating physics to a branch of formal mathematics. Once we have established parity between the two fields, we discuss how one might conduct the experimental verification of important mathematical results, such as the Riemann hypothesis. Using this axiom, we are provided with a framework for the definite separation between objective reality and physical reality, with the precision of our measurement system working as a continuous ladder between them.

Christopher Duston is a mathematical physicist whose current research focuses on the representation of 3- and 4-manifolds as branched covering spaces to construct models for the gravitational field. He has also worked on exotic smooth structures, semiclassical gravity, loop quantum gravity, and cosmic strings. He holds a B.S. in Astrophysics from the University of Massachusetts at Amherst, an M.S. in Astrophysics from the Pennsylvania State University, and a Ph.D. in Theoretical Physics from the Florida State University. He is currently an Assistant Professor at Merrimack College in North Andover, MA.

Christopher –

I appreciate that your essay is well written and clearly thought-out, but it seems to me you gloss over a basic difference between physics and mathematics. You acknowledge that the axiom of measurement is unusual, in that there's unavoidable complexity in specifying each particular measurement arrangement. But I think the issue goes deeper than that. What gets measured...

view entire post

I appreciate that your essay is well written and clearly thought-out, but it seems to me you gloss over a basic difference between physics and mathematics. You acknowledge that the axiom of measurement is unusual, in that there's unavoidable complexity in specifying each particular measurement arrangement. But I think the issue goes deeper than that. What gets measured...

view entire post

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Hey Conrad,

Thanks for checking out my paper and bringing up these points. I'll check yours out as well, but for the moment I'll try to comment on your observations.

I didn't address the relative importance of certain axioms for exactly the reason you identify - the diversity of meanings makes the discussion quite complex. A single axiom might even have different importance...

view entire post

Thanks for checking out my paper and bringing up these points. I'll check yours out as well, but for the moment I'll try to comment on your observations.

I didn't address the relative importance of certain axioms for exactly the reason you identify - the diversity of meanings makes the discussion quite complex. A single axiom might even have different importance...

view entire post

What do your consider quantum mechanics (which is a probabilistic model not a interaction or objects model) to be?

report post as inappropriate

report post as inappropriate

Hey John,

Quantum Mechanics certainly includes interactions - that's how particles move in a lattice for instance, by interactions with the effective potential. The abstraction for quantum mechanics includes the probabilistic interpretation. For example, in a model for the hydrogen atom you might have the state function for the electron (as "the object") interacting with the Coulomb potential, with kinematics governed by the Schrodinger equation. Measurements rely on verifying the probability amplitude, like .

So, "objects" need not be physical objects (like cars or chairs or baseballs), just "the things that interact to produce the phenomena we care about".

Quantum Mechanics certainly includes interactions - that's how particles move in a lattice for instance, by interactions with the effective potential. The abstraction for quantum mechanics includes the probabilistic interpretation. For example, in a model for the hydrogen atom you might have the state function for the electron (as "the object") interacting with the Coulomb potential, with kinematics governed by the Schrodinger equation. Measurements rely on verifying the probability amplitude, like .

So, "objects" need not be physical objects (like cars or chairs or baseballs), just "the things that interact to produce the phenomena we care about".

Dear Professor Duston,

Thank you for courageously responding to my comment. Other credentialed essayists at this site have reported my truthful comments as being inappropriate and have had them removed. They could not handle the truth.

The goals of the Foundational Questions Institute's Essay Contest (the "Contest") are to:

• Encourage and support rigorous, innovative, and influential thinking connected with foundational questions;

Abstract objects in abstract space/time cannot travel at the same speed. Only real surface can travel at the same constant speed. I have used the terms “light” “stationary” and Universe correctly. Mathematics and Physics use these words abstractly.

I am honored that you at least read my comment.

Joe Fisher

report post as inappropriate

Thank you for courageously responding to my comment. Other credentialed essayists at this site have reported my truthful comments as being inappropriate and have had them removed. They could not handle the truth.

The goals of the Foundational Questions Institute's Essay Contest (the "Contest") are to:

• Encourage and support rigorous, innovative, and influential thinking connected with foundational questions;

Abstract objects in abstract space/time cannot travel at the same speed. Only real surface can travel at the same constant speed. I have used the terms “light” “stationary” and Universe correctly. Mathematics and Physics use these words abstractly.

I am honored that you at least read my comment.

Joe Fisher

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Oops,

I meant mathematicians and physicists use abstract words incorrectly.

Joe Fisher

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I meant mathematicians and physicists use abstract words incorrectly.

Joe Fisher

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If I am correct about only surface having the ability to travel at a constant speed, it means that scientists attempting to build a spaceship that would have a physical surface that could travel “faster” than that of a surface of a garbage can are engaged in an act of utter futility.

Warm Regards,

Joe Fisher

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Warm Regards,

Joe Fisher

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Dear Professor Duston,

I was very interested to read your essay as I have thought about this subject in some detail for some time.

Your comment "So, an axiom of measurement must contain both a specification of a machine and a condition upon which we will determine the measurement to be accurate enough. One example might be “The velocity of the car can be measured by the odometer...

view entire post

I was very interested to read your essay as I have thought about this subject in some detail for some time.

Your comment "So, an axiom of measurement must contain both a specification of a machine and a condition upon which we will determine the measurement to be accurate enough. One example might be “The velocity of the car can be measured by the odometer...

view entire post

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Dear Ed,

I appreciate your support - after some though, I don't actually think your thought process is that far off from the topic of the essay.

Implicitly I am claiming that after we define our model and the conditions upon which we will decide that the model can be verified, the rest is computation. So your project to model the orbits is like a microcosm of this entire process, since you can do all the steps - construct the model, define an axiom of measurement, perform the experiment, and verify the results!

I appreciate your views, I would not clearly have seen this connection myself!

I appreciate your support - after some though, I don't actually think your thought process is that far off from the topic of the essay.

Implicitly I am claiming that after we define our model and the conditions upon which we will decide that the model can be verified, the rest is computation. So your project to model the orbits is like a microcosm of this entire process, since you can do all the steps - construct the model, define an axiom of measurement, perform the experiment, and verify the results!

I appreciate your views, I would not clearly have seen this connection myself!

Dear Professor Duston,

Fortunately for everyone's piece of mind, Reality does not need to be modeled.

Warm Regards,

Joe Fisher

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Fortunately for everyone's piece of mind, Reality does not need to be modeled.

Warm Regards,

Joe Fisher

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Dear Dr. Duston,

You wrote a really good essay. I kicked it up the rankings a bit. I also am going to eagerly devour the papers by Cubbit et al. That truly looks fascinating. You might take a look at my essay. I advance the idea that mathematics of a finite, discrete or computational (can be run on a computer) nature is what is most directly relevant to physics. I give a bit of a physical description of Godel's theorem, Turing machines and related matters, which I will honestly confess if I had to do again I would improve some.

Cheers LC

report post as inappropriate

You wrote a really good essay. I kicked it up the rankings a bit. I also am going to eagerly devour the papers by Cubbit et al. That truly looks fascinating. You might take a look at my essay. I advance the idea that mathematics of a finite, discrete or computational (can be run on a computer) nature is what is most directly relevant to physics. I give a bit of a physical description of Godel's theorem, Turing machines and related matters, which I will honestly confess if I had to do again I would improve some.

Cheers LC

report post as inappropriate

Dear Christopher Duston,

I was attracted to your essay mainly because in your title you speak of an "Axiom of Measurement". That would turn Physics into Mathematics. But all I could find is,

*"... an axiom of measurement must contain both a specification of a machine and a condition upon which we will determine the measurement to be accurate enough."*.

Am I missing something? Measurement is so central to all of Science and especially Physics. Surely it requires more than this! Your formulation, in my view, does not encapsulate the core essence of measurement. But gives only an operational definition. Different for different measuring devices. It surely cannot be a fundamental Axiom that can turn Physics into Math.

But Physics can and should be formulated as Math. In my view, there are no Universal Laws of Physics. Rather, all such Basic Laws are mathematical Truisms that describe the interactions of measurements. For example, Planck's Law for blackbody radiation is an exact mathematical identity. And can be derived without making any physical assumptions, like the existence of energy quanta. (see, “The Thermodynamics in Planck's Law”)

I participate in FQXi Contests mainly to engage others in open and honest discussions. I welcome your comments on the above. And on my current essay, ”The 'man-made' Universe”. Where I introduce**The Anthropocentric Principle: our Knowledge and Understanding of the Universe is such as to make Life possible**.

Best Wishes,

Constantinos

report post as inappropriate

I was attracted to your essay mainly because in your title you speak of an "Axiom of Measurement". That would turn Physics into Mathematics. But all I could find is,

Am I missing something? Measurement is so central to all of Science and especially Physics. Surely it requires more than this! Your formulation, in my view, does not encapsulate the core essence of measurement. But gives only an operational definition. Different for different measuring devices. It surely cannot be a fundamental Axiom that can turn Physics into Math.

But Physics can and should be formulated as Math. In my view, there are no Universal Laws of Physics. Rather, all such Basic Laws are mathematical Truisms that describe the interactions of measurements. For example, Planck's Law for blackbody radiation is an exact mathematical identity. And can be derived without making any physical assumptions, like the existence of energy quanta. (see, “The Thermodynamics in Planck's Law”)

I participate in FQXi Contests mainly to engage others in open and honest discussions. I welcome your comments on the above. And on my current essay, ”The 'man-made' Universe”. Where I introduce

Best Wishes,

Constantinos

report post as inappropriate

Dear Dr. Duston,

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

report post as inappropriate

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

report post as inappropriate

Dear Christopher,

I find your idea about an axiom of measurement very interesting. Allow me to suggest two ways to formalize your idea a little more:

1. Reformulation you axiom or measurement as a standard set theoretic axiom schema:

Let x be any physical or mathematical object

Let phi_i(x) be a formula which expresses a particular formulation of the axiom of measurement e.g. “x is diffraction limited"

Let S be the set of all objects which satisfy what might be called a "measurement condition" (i.e. they are empirically measurable).

Then, it seems to me, your idea can be expressed in the form of a simple axiom schema:

Notice that purely mathematical objects satisfy neither the left nor the right side.

2. Specifying that urelements are allowed in the set theory and correspond to physical objects

One issue you may want to consider is that standard set theory is built purely out of sets. Your idea would require the pursuit of one of the variants of set theory which are built out of atoms or urelements (like ZFA), so that when x is a physical object, it can be represented as such.

I hope you found my feedback useful.

Best wishes,

Armin

report post as inappropriate

I find your idea about an axiom of measurement very interesting. Allow me to suggest two ways to formalize your idea a little more:

1. Reformulation you axiom or measurement as a standard set theoretic axiom schema:

Let x be any physical or mathematical object

Let phi_i(x) be a formula which expresses a particular formulation of the axiom of measurement e.g. “x is diffraction limited"

Let S be the set of all objects which satisfy what might be called a "measurement condition" (i.e. they are empirically measurable).

Then, it seems to me, your idea can be expressed in the form of a simple axiom schema:

Notice that purely mathematical objects satisfy neither the left nor the right side.

2. Specifying that urelements are allowed in the set theory and correspond to physical objects

One issue you may want to consider is that standard set theory is built purely out of sets. Your idea would require the pursuit of one of the variants of set theory which are built out of atoms or urelements (like ZFA), so that when x is a physical object, it can be represented as such.

I hope you found my feedback useful.

Best wishes,

Armin

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Pardon, as soon as I submitted the post, I realized that the axiom schema is missing something, it should be

The existential quantifier for S is essential, otherwise you cannot prove that such a set exists.

Armin

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The existential quantifier for S is essential, otherwise you cannot prove that such a set exists.

Armin

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Hi Armin,

Yes, I think you are right that it would be nice if this was formulated more precisely in the language of set theory (or a generalization). This was something of a "first forey" into getting my ideas down.

Off the cuff, I would probably have a construction which more explicitly connected predictions with experiment. Maybe start with a set of objects O and interactions I, and a map which generated a prediction p(O,I) (it seems likely that these predictions would have to be decidable). Then I would have your x be an "experimental apparatus", and be the statement of truth ("x has measured the mass to 5%"). The maybe the axiom of measurement would be

This probably requires a bit more thought, but I think this structure might be more amenable to some of the issues I brought up in the paper - like the experimental apparatus, which would be some kind of recursion of measurement axioms once we understood it's operation well enough.

Anyway, thanks for your input, and inviting me to think harder about this.

Chris

Yes, I think you are right that it would be nice if this was formulated more precisely in the language of set theory (or a generalization). This was something of a "first forey" into getting my ideas down.

Off the cuff, I would probably have a construction which more explicitly connected predictions with experiment. Maybe start with a set of objects O and interactions I, and a map which generated a prediction p(O,I) (it seems likely that these predictions would have to be decidable). Then I would have your x be an "experimental apparatus", and be the statement of truth ("x has measured the mass to 5%"). The maybe the axiom of measurement would be

This probably requires a bit more thought, but I think this structure might be more amenable to some of the issues I brought up in the paper - like the experimental apparatus, which would be some kind of recursion of measurement axioms once we understood it's operation well enough.

Anyway, thanks for your input, and inviting me to think harder about this.

Chris

Dear Chris,

It is always gratifying when one's suggestions are taken seriously. Yes, I naturally assumed you were referring to set theory because in your paper you made a statement that related your axiom to set set theoretical axiomx. BTW I have more than my own share of unorthodox ideas about set theory, so it might not be so surprising that my reaction is positive.

Some feedback on your proposed axiom:

1. I did not say this, but the index letter 'i' on phi is an element of an index set which turns the axiom into an axiom schema (an infinite number of axioms having the same structure). Without it, you have only one axiom which contains a formula that specifies only the condition 'measured mass to 5%'.

2. It seems to me the kind of structure you are attempting to specify in your axiom is a function. If so, then as written it won't work because what you have in place of a domain is a formula, but what you need is a set (formulas are not sets). Supposing this is what you want, then the domain of your function is the set of ordered pairs of which the first coordinate is an element of the set of observations and the second an element of the set of interactions, and the range is the set of predictions.

3. If you want to directly relate membership in the domain to some statement like "x has measured the mass to 5%" then the only way I know how to do it is to posit an equivalence between describability by the formula and membership in the domain, similar to what I wrote in previous response.

I hope you found this useful.

Best,

Armin

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It is always gratifying when one's suggestions are taken seriously. Yes, I naturally assumed you were referring to set theory because in your paper you made a statement that related your axiom to set set theoretical axiomx. BTW I have more than my own share of unorthodox ideas about set theory, so it might not be so surprising that my reaction is positive.

Some feedback on your proposed axiom:

1. I did not say this, but the index letter 'i' on phi is an element of an index set which turns the axiom into an axiom schema (an infinite number of axioms having the same structure). Without it, you have only one axiom which contains a formula that specifies only the condition 'measured mass to 5%'.

2. It seems to me the kind of structure you are attempting to specify in your axiom is a function. If so, then as written it won't work because what you have in place of a domain is a formula, but what you need is a set (formulas are not sets). Supposing this is what you want, then the domain of your function is the set of ordered pairs of which the first coordinate is an element of the set of observations and the second an element of the set of interactions, and the range is the set of predictions.

3. If you want to directly relate membership in the domain to some statement like "x has measured the mass to 5%" then the only way I know how to do it is to posit an equivalence between describability by the formula and membership in the domain, similar to what I wrote in previous response.

I hope you found this useful.

Best,

Armin

report post as inappropriate

Hey Armin,

I'm thinking of those functions as "maps to statements", so given a model, I can make a prediction about a given observable, so the map is from the observable to a statement about it's value. Maybe my notation is not correct, but I don't think there's a mathematical inconsistency there.

More to the point, I've tried several times to improve on my initial attempt, and I keep getting statements which are close to what you wrote originally. So I think you got pretty close to the mark on what my intention is - if something is observable, it should satisfy an axiom of measurement, and vise versa. What is missing is the structure behind the observable (derived from the model) and the measurement statement (which is derived from an experimental apparatus that satisfies it's own set of axioms of measurement). Perhaps these extra details are not required at the level of the axiom.

I'm thinking of those functions as "maps to statements", so given a model, I can make a prediction about a given observable, so the map is from the observable to a statement about it's value. Maybe my notation is not correct, but I don't think there's a mathematical inconsistency there.

More to the point, I've tried several times to improve on my initial attempt, and I keep getting statements which are close to what you wrote originally. So I think you got pretty close to the mark on what my intention is - if something is observable, it should satisfy an axiom of measurement, and vise versa. What is missing is the structure behind the observable (derived from the model) and the measurement statement (which is derived from an experimental apparatus that satisfies it's own set of axioms of measurement). Perhaps these extra details are not required at the level of the axiom.

Dear Chris,

Having myself first learned to think more like a physicist and only in the last 1.5 years or so more like a mathematician (though I find thinking like a physicist still more intuitive, or, I should say, easy to come by), I think I can well appreciate where you are coming from.

Perhaps it would be worthwhile to step back for a moment and think about exactly what role you want this to play in the connection between physics and mathematics. The spectrum ranges on one end from something like a philosophical or physical principle to the other as a rigorously formulated mathematical axiom. Where you consider your idea to fall on this spectrum determines how precisely you need to state it. Your choice of calling it an "axiom" led me to believe that you are in fact considering it as something much mores imilar to the latter. In that case, expressions like "maps to statements", which, I agree, are perfectly acceptable in physics, require some work to be intelligible to mathematicians. Your expression reminded me that set theory is not the only way you can try to incorporate your idea into the foundations of mathematics. Although I know at this time still very little about category theory, from what I do know I have the impression that "maps to statements" might be more easily accommodated there than in set theory.

As a final note, part of my interest in this probably reflects the fact that I am myself struggling with incorporating a general philosophical principle as a rigorous axiom into the foundations of mathematics.

Best,

Armin

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Having myself first learned to think more like a physicist and only in the last 1.5 years or so more like a mathematician (though I find thinking like a physicist still more intuitive, or, I should say, easy to come by), I think I can well appreciate where you are coming from.

Perhaps it would be worthwhile to step back for a moment and think about exactly what role you want this to play in the connection between physics and mathematics. The spectrum ranges on one end from something like a philosophical or physical principle to the other as a rigorously formulated mathematical axiom. Where you consider your idea to fall on this spectrum determines how precisely you need to state it. Your choice of calling it an "axiom" led me to believe that you are in fact considering it as something much mores imilar to the latter. In that case, expressions like "maps to statements", which, I agree, are perfectly acceptable in physics, require some work to be intelligible to mathematicians. Your expression reminded me that set theory is not the only way you can try to incorporate your idea into the foundations of mathematics. Although I know at this time still very little about category theory, from what I do know I have the impression that "maps to statements" might be more easily accommodated there than in set theory.

As a final note, part of my interest in this probably reflects the fact that I am myself struggling with incorporating a general philosophical principle as a rigorous axiom into the foundations of mathematics.

Best,

Armin

report post as inappropriate

Dear Chris,

I noticed via Researchgate that recently it appears you took an interest in some of my articles.

Thank you for your interest, and I would like to note that my earlier papers, (pre-2015) particularly pertaining to the foundations of quantum theory, do not yet reflect an appreciation that (in my current view) the challenges in this field have their origin in the foundations of mathematics, particularly in the fact that mathematics, in its current standard form, does not have sufficient expressive power to describe in a formal language certain kinds of distinctions that are important in quantum theory.

I only began to acquire such an appreciation toward the end of 2013, and have accordingly shifted much of my focus to this area to learn more, having in the process found that the same distinctions have much broader applicability than I originally thought.

My essay in this contest did already outline many of the ideas that I am referring to, but since I wrote it, my aim has changed somewhat: I am trying to use as little additional logical machinery as possible in order to achieve essentially the same outcome. The reason for that is simple: The less you need to use, the more the new ideas that are introduced will be palatable to a wider audience, thereby increasing the chances of their general acceptance. Also, the less likely it is that the new ideas introduce new problems in some unanticipated way in other areas of mathematics.

Should you have any questions about any of the works, feel free to contact me.

Happy holidays,

Armin

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I noticed via Researchgate that recently it appears you took an interest in some of my articles.

Thank you for your interest, and I would like to note that my earlier papers, (pre-2015) particularly pertaining to the foundations of quantum theory, do not yet reflect an appreciation that (in my current view) the challenges in this field have their origin in the foundations of mathematics, particularly in the fact that mathematics, in its current standard form, does not have sufficient expressive power to describe in a formal language certain kinds of distinctions that are important in quantum theory.

I only began to acquire such an appreciation toward the end of 2013, and have accordingly shifted much of my focus to this area to learn more, having in the process found that the same distinctions have much broader applicability than I originally thought.

My essay in this contest did already outline many of the ideas that I am referring to, but since I wrote it, my aim has changed somewhat: I am trying to use as little additional logical machinery as possible in order to achieve essentially the same outcome. The reason for that is simple: The less you need to use, the more the new ideas that are introduced will be palatable to a wider audience, thereby increasing the chances of their general acceptance. Also, the less likely it is that the new ideas introduce new problems in some unanticipated way in other areas of mathematics.

Should you have any questions about any of the works, feel free to contact me.

Happy holidays,

Armin

report post as inappropriate

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