Essay Abstract

Mathematics allows us to develop physical theories that make accurate predictions. These theories relate a primitive ontology to nomological variables. All of our physical theories make metaphysical commitments that are directly or indirectly related to mathematical models. After a brief introduction to some of the fundamental issues associated with the use of mathematics in developing physical theories, I discuss how the link between physics and metaphysics was achieved in classical mechanics, special relativity and quantum theory. I argue that mathematics may not be sufficient for developing a theory of everything due to incompleteness of formal systems or the existence of unknowable truths.

Author Bio

Efthimios Harokopos received Bachelor's and Master's degrees from State University of New York at Buffalo, a Master's degree from Columbia University and completed the necessary coursework for a PhD while working for AT&T and Bell Labs. He is an independent researcher in the field of philosophy of science.

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Thanks Sujatha!

Good to be back with another essay after 4 years from last one. This is a great community a an abundant source of interesting ideas. I see that some other authors also claim that math may not be enough. Let me be clear because this may get tricky: math is more than enough if what we are after is some accurate predictions about everyday life phenomena. Math may not be enough if what we are after is a theory of everything because of unknowable truths or incompleteness or formal systems. Thanks to the pioneers that "lifted all the intellectual weight", we already know enough but not nearly enough for a theory of everything. Math is a great tool and it is neither trick nor a panacea.

My best regards to all participants.

Dear Efthimios,

Your comment elsewhere talking about the need to identify the most appropriate out of the different varieties of space or spacetime drew my attention to the fact that here was someone knowledgable about some of the fundamental issues in contention.

I enjoyed your essay very much and the reasoning was impeccable in many places. You however seemed to place too much faith in the correctness of Special relativity (and spacetime) and Quantum theory, which if you had not your logic would have moved us closer towards the truth of physical reality. Your reason for staking your claim is that ALL experiments have verified both, but I can dispute this on another forum since this is all about your essay.

Now, a few of the other areas I would want you to ponder over...

You said, "There is nothing in Newton's second law to demand that when a force is applied to a particle, a distance will be covered in positive time, meaning in the future, or in negative time, meaning in the past".

This smells of Zeno's paradox about motion, particularly the Dichotomy and Arrow paradoxes, viz. "What is in motion moves neither in the place it is nor in one in which it is not", i.e. An object O actually remains in its own place despite motion occurring. I discussed this in my 2013 essay and my current essay.

Since you seem to have a good grasp of the issues involved and seem to be in agreement with Allori that "Any fundamental physical theory must always contain a metaphysical hypothesis about what are the fundamental constituents of physical objects. We will call this the primitive ontology of the theory", you will be most welcome to comment and criticize my unconventional essay.

All in all congratulations on a very cerebral submission.

Regards,

Akinbo

    Efthimios Gia sou, Ciao,

    very interesting essay. Do you think that meta-math will still be expressed in some new math?

    Good work and right on the subject.

    Antonio

      Dear Akinbo,

      Thank you for your good words. I don't pretend to know everything but I'm not aware of any experiment that falsifies special relativity. If you are ware of one I would be very interested in knowing that.

      About Zeno's paradox I talked in detail in my 2011 essay.

      I will read your essay and comment soon.

      All the best.

      Ciao Antonio,

      I have no idea as I wrote in the essay whether it is possible to have a meta-mathematical framework in physics and I cannot speculate of its form. Possibly some sort of new math but I don't know what. Thank you.

      Today after this was posted I show a 9 public rating and now I see a 5.5. Quick calculation shows that someone gave me 2 points. I would appreciate if he can state the reason for that. Then we can all learn something.

      • [deleted]

      • Post #9

      Your essay touches on something that I have pondered. Newton's second law of motion

      F = ma,

      has a vector quantity that is physical equal to a physical scalar times a vector that is purely geometric. The acceleration a = d^2r/dt^2 is a derivative of one geometric object, a vector in space with respect to another geometric parameter called time. In some funny sense we might think of the geometric quantity as converting the physical scalar into a physical vector. The physical vector is a dynamical quantity that results from the multiplication of a physical scalar (kinematic in nature) with the geometric object.

      LC

        Dear Efthimios Harokopos,

        I very much enjoyed your well-written essay. Unless I have forgotten someone, I do not believe any other essays maintain such a clear focus on linking physics and metaphysics through math, and it seems quite an appropriate undertaking. What is most interesting is that you managed to do so using the most basic concepts and beginning with Newton. The math is simple, the ideas are simple, but the significance is very large I think. Your equation (9) is so clean and yet I have never seen it written in this way. Very clearly thought out - I congratulate you. Your equation (10) and your discussion of autonomous versus non-autonomous is also interesting and worth thinking about. And after discussing what you view as possible with regards to the existence of a 'meta-mathematical' framework, you properly leave the question open, and did not cloud the issue with hypotheticals.

        I invite you to read my essay and invite your comments.

        Edwin Eugene Klingman

        Dear Edwin Eugene Klingman,

        I truly appreciate your good words about my essay. However, I have a problem with your invitation to read your essay as follows:

        Before your comments my community rating was at 6.0 based on two votes. Afterwards, it dropped to 5.0 based on a single vote. This means that someone rate it at 3. There are two possibilities then

        (1) Either you rated my essay at 3 despite your good words, or

        (2) You did not rate my essay despite your good words.

        In either case, I will not read you essay and as a matter of fact I will refrain from judging other essays because such judgment cannot be objective anyway. I will read several essays but not rate.

        Thank you. I am not here to get a prize but just to participate and convey my thoughts. I know the process since last time I participated in 2011. My essay was first for two weeks and then in just a few days I got a series of low marks. It ended up in the 35 essays sent for review but to see how some people acted was quite disappointing. I just wish FQXi would use an independent panel of judges and pay no attention to ratings. I think they are smarter than that.

          Dear Efthimios,

          I just replied you on my blog. Also I just read your 2011 Essay and your discussion of Zeno's Dichotomy paradox. I see that you desire to make spacetime as the foundation of your proposed solutions. Please be careful about this.

          When you say, "I'm not aware of any experiment that falsifies special relativity", are you aware of observations and experiments that falsify the postulates that form the basis for special relativity theory? I mean special relativity for example implies that motion of an observer has no effect on the arrival time of incoming light, i.e. the resultant velocity of light is constant. This postulate depends on what Michelson and Morley found out in their 1887 experiment.

          However, are you not aware that in some other experiments, like Pulsar timing observations, Lunar Laser Ranging, Global Positioning System, motion of the observer affects the arrival time of incoming light signals?

          How can these two discordant findings be reconciled? Can an observer's motion hasten incoming light arrival or not? You may check my humble suggestion HERE in your free time. Please note that I am not quarreling with your statement, "...testable predictions that have been verified in laboratories numerous times. So far, none of the predictions made by special relativity have been falsified in a laboratory experiment", but with the postulate on which SR rests on, which if false there is no basis for SR to be held as true. Or can a postulate of a theory be false and the theory be true?

          Finally, I would like to ask in view of our cosmological model, the Big Bang, whether your 4-dimensional block universe can perish or is it eternally existing?

          Regards,

          Akinbo

          *Sorry about the rating system in the essay contest. It is absurd to receive a 10 and a 1 on the same essay as I received myself. Let us just continue and benefit from the intellectual exchange.

          Efthimios,

          I think something has happened to your .pdf file. I have attempted to download your essay several times and it is blank. Obviously from comments above, your essay was present in the system. I have downloaded other essays, so I am not sure what has changed. Perhaps the file has become corrupted or perhaps my Adobe Reader has become corrupted. Also, I have Windows 8. It is junk.

          Regarding rating ... get over it. I have been given 3 ones and 1 two. I have also been given a ten, an eight, and enough other positive votes to average a five point zero. Some people give ones just to knock scores down ... no other reason. And they don't bother giving comments ... after the one votes I received there were no comments to state what they did not like or agree with ... I doubt if Dr. Klingman voted you down. He has BEEN voted down himself but he has gotten enough positive votes to over-come the haters.

          I'll keep trying to get your essay.

          Best Regards,

          Gary Simpson

          Efthimios,

          False alarm on my part. It appears that somehow my Adobe Acrobat .pdf reader became corrupt.

          In your statement of Newton's 2'nd Law, you implicitly assume that the force and the resulting change in motion are collinear. This is the standard understanding. What would you do if they were not collinear? The reason for my question is that I can use Hamilton to produce an equation of motion that is circular but requires no force. I had not considered a constant kinetic energy reference frame.

          As a minor point, near the middle of page 3 you reference (Newton, 1952). Is this a typo or is there another guy named Newton?

          Your Figure 1 captures the essence of this essay contest I think.

          Best Regards and Good Luck,

          Gary Simpson

            Hi Gary,

            By "change in motion" it is meant change in momentum vector mv. This is the meaning of "motion" in Newton's law. In circular motion the change in momentum dp/dt is along the direction of the center of the path. Thus, a centripetal acceleration is generated and a centripetal force. By definition, these are collinear.

            You said:

            "The reason for my question is that I can use Hamilton to produce an equation of motion that is circular but requires no force"

            There is no way that I know to get circular motion without a force that is directed towards the center of motion, the centripetal force. But I do not claim to know everything.

            It is often confusing because when a particle is on circular motion there is no applied external force but there is a force generated due to the motion that is directed towards the center of the path. This force has a "stretched" causality with acceleration and this is what has prompted some scientists to question the causality in Newton's Law. Specifically, in his graduate text Principle of Dynamics, Greenwood specifically mentions this problem.

            (Newton, 1952) is the book written by Newton (Principia) translated in 1952. It also means that Newton will be always alive (in physics):)

            Thank you for your comments.

            I do not understand the part about power and cause-free motion but the rest is an honest attempt to answer the essay questions. First time I read a concise explanation of relativity theory. Good essay.

              Dear Efthimios Harokopos,

              I saw your complaint about scoring on your thread before I commented on your essay, so I knew that you were already upset about scoring. What I had not realized was that you think you can, in general, correlate comments with voting behavior. It is upsetting when low scores are received for no apparent reason. During the first weeks of this contest I was the top paper with a 10 (every time I looked at it I reminded myself that there was nowhere to go but down) and then I received at least two 1s and a couple of 2s. I was of course not happy about this.

              There are no rules for how one "should vote". After several essay contests I have a voting strategy that I think is most effective. I typically wish to see all essays before I decide how they should be ranked. I think it is presumptuous for you to assume that my voting behavior should match your ideas of voting but if you are hostile over this point it's probably best you not read my essay.

              You have written an essay in an earlier contest, and you should therefore be aware that no one is happy with FQXi voting. There is always some vote trading going on and the best policy is not to discuss votes in comments and not to combine the timing of comments and votes to 'send messages'.

              Your implication that I would give high praise and low scores is unwarranted, and there is no basis for you even to suggest this. The fact is that I have not scored your essay, nor most essays, as I have my own policy or voting strategy. I think your complaint about voting is valid, but your assumptions about how I should vote are invalid and not appreciated.

              Edwin Eugene Klingman

              PS, After you posted your unhappy comment on my thread, someone gave me a 3 vote. By your logic I should suspect you, but I don't think your logic is good. I also note that the other top-rated essays seem to have received low votes at the same time, so it's probably someone either trying to pull the top down as a way of raising their own status, or it could be simply someone who resents anyone having a higher score. My point is, there is no way of knowing, and it is inappropriate to make the suggestion you did that I would give high praise and low scores. That is personalizing an issue in a very uncalled for manner.

              Dear Akinbo,

              I agree with you that if any of the two main postulates of SR are falsified the theory will be falsified. However, I am not aware of conclusive evidence that falsifies either. As I said I do not know every little detail. Please note that regarding SR and QM I have not expressed personal views but those that are accepted. However, my objective was not to promote any theory but to discuss the role of mathematics that is the same regardless postulates. Note that I admit the SR predictions are equivalent with a large class of theories that do not adhere to the speed of light constancy. Therefore, as I note in my essay and maybe not too successfully, the metaphysical commitments vary. I also made a note that many scientists are not ready to adhere to the metaphysical commitments of SR. Also I want to be honest and tell you that I believe that Einstein was correct about the light postulate but I hope he will be proven wrong for some reasons that are hard to explain now but deal with free will. I am open to all possibilities but I have a high regard for Einstein. Even if he was wrong, he challenged us with alternatives. According to pessimistic meta-induction, at the end, we will be all wrong with high probability.

              Since the relativity subject is controversial, it may be better to stick to the essay subject, which is about the role of mathematics in physics. I hope there will be another contest about relativity and we can discuss these matters in more detail. Thanks.

              Note that F = ma is referred to as Newton's second law but it is not, according to the statements in Principia. I think Euler stated that law and it happens to correspond to Newton's law when the mass is constant, F = dp/dt

              I am not sure time is a geometric parameter. Actually, I have no idea what it means other that it is a reading of a clock. Thus, there is some physics in both sides of the equation but also metaphysics. I see nothing wrong about that as long as it works and it does in weak-field limit. After that one must use general relativity or equivalent and maybe there are no forces, i.e., the metaphysical commitments may change.

              Thanks LC

              Thanks. The explanation of relativity is not mine. See related reference in the essay that gives credit where it is due.