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 in physics is always a specific parameter such as mass or spin or distance or the gravitational constant. Each of these diverse parameters has a very different significance in physics. And even though the meaning of each parameter can be expressed through its mathematical relations with other parameters, this is no way reduces the special role each one plays in the physical world. Likewise each one requires its own particular kinds of measurement arrangements, so that measurement processes have the same kind of irreducible diversity.

Do you think there's anything in mathematics that plays a similar role? In geometry we have points and lines, lengths and areas, circles and triangles, etc. I can see the argument that since mass and electric charge, spacetime intervals and momentum, etc. all show up in the equations as mathematical quantities, they're essentially the same as geometric entities and properties, only more complex. But how lines and triangles work together in geometry doesn't really seem parallel to how atoms and molecules interact in physics.

I proposed in my essay that while the language of physics is surely mathematical, it takes a very special combination of many complex mathematical structures to support something like a physical world, where each parameter is definable and measurable in the context of other such parameters. So I don't disagree with your thesis, that physics can be considered a form of mathematics. But I think it's more important to understand how this unique kind of self-defining mathematics differs from the various systems we construct on the basis of undefined elements and formal axioms.

Thanks for the chance to respond to your quite interesting and intelligent paper.

Conrad

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

      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 [math]|< \phi | \phi >|^2[/math].

      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".

      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 in different subfields of physics ("atoms are molecules" is a good example). But I don't think the existence of these different meanings has any real bearing on my basic thesis.

      * I'm implicitly using the idea that we are not modeling the "real" universe, but rather "the observable" universe. So, the spin of an electron may be a very important quantity in the model, but the importance of that axiom to the underlying pattern of nature is unclear - or even nonexistent. So we wouldn't have QED without the internal symmetry properties of the electron, but we wouldn't have plane geometry without the parallel postulate. It seems to me they have similar roles here.

      * I'll just quickly comment that there are certainly mathematical axioms which are of differing importance - at the moment, the Axiom of Choice comes to mind. Without it, some basic topological results are not true (product of compact spaces are compact), but with it, you can get some strange results (Banach-Tarski, for instance).

      * As for the interactions that you bring up, I think it's fair to say interactions in mathematics are more diverse than in physics. Interactions in physics are essentially governed by partial differential equations, so they lie in realm of Analysis. There are topological interactions as well, which also belong to mathematics. But how the axioms of Euclid (lines and points) interact to give us theorems is certainly different than how we use the interaction of particles to give us predictions - essentially just applied calculus. So *why* these axioms exist come from outside mathematics - from nature - but with the axiom of measurement you can bring them into the formal system.

      Again, thanks very much for reading and commenting. Your thoughts are worth further consideration to be sure, but my first instinct is that since I'm not actually assigning importance to the abstraction beyond the observable universe, the internal symmetry of an electron has no more significance than something like the parallel postulate.

      Chris

      Joe,

      Thanks for the comments, and actually I completely agree with your correction. I am suggesting that not only is physics based on abstraction, but measurement is as well. Therefore, a complete abstraction of physics is possible if one carefully defines an axiom for the measurement process.

      However, I'm not sure I follow your statement directly following it - or at least don't understand how it is related. If you want to take the position that every single observation of every single thing is unique, you are essentially using abstraction without any of it's power. Since you and I can differentiate between things which physics cannot ( this letter -> a and this letter -> a), you are essentially using an abstraction which is exactly equal to reality. This is how I understand the first part of your statement.

      You lose me in the second part. Why must all surfaces travel at the same speed? We already know that all objects in spacetime travel at the same speed - is this what you are referring to? And several of your statements can be easily disproven by experiment, such as "Real light must be the only stationary substance in the real Universe" (unless you are using the words "light", "stationary", and "Universe" in an unconventional manner). I'm not sure how this part is related to mathematics and physics, the topic of the essay contest. It looks like you are playing linguistic tricks, rather than physical ones.

      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

        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

          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

          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!

          Dear Professor Duston,

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

          Warm Regards,

          Joe Fisher

          5 days later

          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

          8 days later

          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

          16 days later

          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

          16 days later

          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

            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

            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

            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

            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.

            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