The Experimental Verification of Mathematics via an Axiom of Measurement by Christopher Levi Duston

Joe Fisher

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

Oops,

I meant mathematicians and physicists use abstract words incorrectly.

Joe Fisher

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

Ed Unverricht

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 to an accuracy of 3σ."" may seem obvious to some, but contains some hidden subtleties.

I first ran into this when writing computer programs based on step by step animations of planetary motion using Newtons inverse square gravitational law. Step by step animations of say the earth orbiting the sun, requires incredible accuracy in starting positions and calculations to get the orbit correct (to within a few minutes) after a year given the nature of each calculation being based on the result of the prior calculation. I then went after the orbit of Mercury, hoping to add in the effects of relativity on the forces between objects.

Your comment "If we produce a prediction which does not satisfy the axiom of measurement, we can relax the axiom of measurement until it does, or add another axiom to produce a more appropriate prediction. This is part of the business of doing physics - we expect for new phenomena to emerge which will require adjustments to our models." reflects in my mind what I was up to. I was animating the n-body problem for not only gravity but also relativity!

I realize I am a little off topic to your essay here, but working in animation where you have to choose the accuracy of your time steps (every hour, or every second, or every nano-second) as well as the accuracy of your grid (in kilometers or in micro-meters) before you start and then try to end up with the correct results that match the physics (to an acceptable degree of accuracy) after a reasonable passage of time is very challenging. Your essay caused me to go back in time and rethink a number of issues I had not visited in some time.

Great read, thanks for putting it out.

Ed Unverricht

Christopher Duston

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!

Joe Fisher

Dear Professor Duston,

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

Warm Regards,

Joe Fisher

Lawrence Crowell

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

Constantinos Ragazas

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

Joe Fisher

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

Armin Nikkhah Shirazi

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:

[math]\forall x (\phi_i(x) \Leftrightarrow x \in S)[/math]

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

Armin Nikkhah Shirazi

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

[math]\exists S \forall x (\phi_i(x) \Leftrightarrow x \in S)[/math]

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

Armin

Christopher Duston

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 [math]\phi(x)[/math] be the statement of truth ("x has measured the mass to 5%"). The maybe the axiom of measurement would be

[math]\forall x (\phi(x) \rightarrow p(\mathcal{O},\mathcal{I}))[/math]

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

Armin Nikkhah Shirazi

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

Christopher Duston

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.

Armin Nikkhah Shirazi

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

Armin Nikkhah Shirazi

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

Christopher Duston

Dear Odessa,

Thanks for showing an interest in this - yes, I agree, the next step in this project would be to find a formal system that naturally includes urelements of the sort needed. It's interesting, my main thought when writing the essay was how to carefully define the nature of truth in an experimental context, whereas most people's response has been like yours, worried more about the formal system.

In any case, it is a bit outside of my area of expertise but when I return to this project I feel that there are two major pieces:

1) Determine which variant of set theory is appropriate and carefully formulate it.

2) Use language within the formal system to carefully define the axiom of measurement. This would likely be a recursive sort of definition, since the truth in experiments only comes from the truth in earlier experiments. The choice of set theory would have to include such axioms.

After that, maybe try to fit a few physical theories (GR, QFT, etc) into this framework to illustrate how it can be used in practice to formalize these systems. Beginning with axiomatic field theory might be natural, since that philosophy is rather built-in already.

Again, thanks for you interest and your comments.