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My question is related to what is the physical content of infinitesimals. In mathematics the infinitesimals as hyperreals (Robinson 1960) are introduced as numbers greater than 0 but smaller than any real number. Can a physical quantity in real nature increase with a such infinitesimal number?

This must be outside physics because it is impossible to measure (each measurement gives a real number - even a rational decimal number). In the end infinitesimal increase can not be proven.

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  • $\begingroup$ Related: physics.stackexchange.com/q/92925/2451 , physics.stackexchange.com/q/70376/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Jul 12, 2021 at 20:39
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    $\begingroup$ Physics is not math. Mathematical models are used to provide a slight (or sometimes huge) simplification of a physical process, but the mathematical models are not reality. $\endgroup$
    – David White
    Commented Jul 12, 2021 at 21:15
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    $\begingroup$ This falls within philosophy. Are logical concepts like math part of nature? Is nature just physics plus time and probabilities? You might get better answers in philosophy.stackexchange.com $\endgroup$
    – lurscher
    Commented Jul 12, 2021 at 22:03
  • $\begingroup$ Basically, infinitesimals as well as negligible values are something that is insignificantly small compared to what is relevent. $\endgroup$
    – Steeven
    Commented Jul 13, 2021 at 9:28

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Numbers, by themselves, do not have any physical content whatsoever. Physics is about creating models that accurately predict the outcome of experiments. So the physical content is in the model, not the numbers used by the model.

That said, the most successful models that we have developed, the ones that produce the best experimental predictions, all of those models can be based on infinitesimals.

However, the same models can also be based on limits. So people who don’t like infinitesimals can use the same models but based on limits instead of infinitesimals. Infinitesimals and limits are mathematically equivalent for these models. It is the models that have the physical content and not the underlying numbers.

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  • $\begingroup$ In terms of answering the question of "physical content" you seem to be making a distinction without an actual difference. You did not define what a "model" is, nor did you explain why the theoretical components of a model do not have "physical content" even though the model itself does. For example, is there no physical content in the concept of an electron, even though its just part of the model of an atom?... Is there no physical content in the values of energy attainable for an electron in a certain atom? Why not? We certainly interpret it to have physical content. $\endgroup$
    – Daddy Kropotkin
    Commented Jul 13, 2021 at 13:12
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    $\begingroup$ @DaddyKropotkin there is no need to define every term in an answer. The term “model” is well known. If you do not know what a model is then please ask a question on the topic. The physical content of a model is in its experimental predictions. I cannot make a physical prediction with only infinitesimals and nothing else from the model. I stand by my assertion. Regarding the electron, what experimental prediction can you make using only the electron field in the standard model? If there is one then it justifiably has physical content by itself. Otherwise the physical content is in the model $\endgroup$
    – Dale
    Commented Jul 13, 2021 at 13:31
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    $\begingroup$ Please provide the rigorous definition of a "model." $\endgroup$
    – Daddy Kropotkin
    Commented Jul 13, 2021 at 14:29
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    $\begingroup$ Please post that as a question. Comments are not intended for asking questions $\endgroup$
    – Dale
    Commented Jul 13, 2021 at 14:31
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    $\begingroup$ And that's exactly my point. There is no rigorous definition of a "model." Instead, you just point to a quote that supports whatever you claim. That's pretty suspicious, especially since "model" is something that is debated heavily in philosophy of science still. $\endgroup$
    – Daddy Kropotkin
    Commented Jul 13, 2021 at 15:10
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The existence of planck length and plank time seems to show ourselves that the world is discrete.... So i think we can answer "no" to your question... Everything in nature appears to be quantized at fundamental level

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    $\begingroup$ The Planck length and Planck time are just special distances in space and time just like how an electron volt is a special amount of energy; they aren't discrete chunks of anything. $\endgroup$
    – BioPhysicist
    Commented Jul 13, 2021 at 7:49
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As @Dale has correctly pointed out, infinitesimals are a mathematical concept. They are relevant to physics as much as the fields and other mathematical objects in physical theories can be considered continuous, and therefore described in terms of differential equations. Specifically:

  • On a microscopic level not everything is continuous - at least not all the time. Still, infinitesimals have their mathematical role: e.g., discrete spectra can be calculated and analyzed using continuous Schrodinger equation.
  • Many fields of physics, notably the theory of elasticity and the macroscopic electrodynamics, operate with a physically small volume - that is the quantities in these theories are understood as averages over volumes containing $N_A$ number of atoms/molecules - not truly infinitesimal, but which ate treated as infinitesimally small in mathematical equations.
  • Coarse graining is the term specifically used to describe the type of procedures that introduce continuous description of microscopically discrete problems, notably in the theory of the critical phenomena.
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  • $\begingroup$ It is good to mention coarse graining in this context. +1 $\endgroup$
    – Dale
    Commented Jul 13, 2021 at 13:57

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