This one is addressed to everyone:

Guys, I think we all find this discussion more pleasant and productive if we all will tone down the adversarial overtones. I want to remind as much to myself as to every single one of you that there's no competition or battle taking place here. We are all on the same team and we share our common passion and interest in physics. Let's celebrate it with collaborative communication. All objections, different views and rigour stays the same. I ask only to reconsider the emotional tone. Just a pinch of kindness and collaboration might have very surprising effect on overall productivity of this engagment.

5 hours ago, TheVat said:

What ontological claims do you then think mathematics is making? Just to clarify this tragedy you speak of.

Mine or anybody else thoughts or opinions are irrelevant here. What we have here is a different operational understanding of mathematical language between us. And I would like to understand your position clearly. Lets take this simple example where 2 equations are solving for the same value but, as I see it, having radically different ontological meanings:

\( v_{a} = \sqrt{\frac{GM}{r}} \qquad\text{vs.}\qquad v_{b} = \frac{2\pi r}{P} \)

Can you see the ontological difference?

13 minutes ago, Anton Rize said:

va=GMr−−−√vs.vb=2πrP

Can you see the ontological difference?

Not every algebraic manipulation deserves an brand-new ontological coinage.

4 hours ago, MigL said:

Not sure if anyone else has noted that whenever @Time Traveler is asked a particularly 'difficult' question, his answer always includes highlighted text, and it begins with "Thank you for the sharp and critical counterarguments" or "I appreciate you holding my feet to the fire".
Almost as if he poses your questions to an AI, then copies and pasts the AI's reply here.

@exchemist is a pretty good early-warning system for this kind of thing. He's clued us all in a while ago.

12 minutes ago, joigus said:

Not every algebraic manipulation deserves an brand-new ontological coinage.

You making it very hard for me not to sound adversarial right now hahaha!
If you will look at this two equations as an algorithms one has to preform in order to obtain the values of v_a and v_b - you will find that this are categorically different algorithms. They lead to radically different physical actions that lead to different measurements of different phenomenon. Is it reasonable to call it "algebraic manipulation"?
Maybe if you would actually engage with the argument and spend just few minutes thinking about it before writing your comment - we could better understand each other? Would you agree?

1 hour ago, Anton Rize said:

This one is addressed to everyone:

Guys, I think we all find this discussion more pleasant and productive if we all will tone down the adversarial overtones. I want to remind as much to myself as to every single one of you that there's no competition or battle taking place here. We are all on the same team and we share our common passion and interest in physics. Let's celebrate it with collaborative communication. All objections, different views and rigour stays the same. I ask only to reconsider the emotional tone. Just a pinch of kindness and collaboration might have very surprising effect on overall productivity of this engagment.

Which do you want to discuss - ontology, or physics?

1 hour ago, Anton Rize said:

Mine or anybody else thoughts or opinions are irrelevant here. What we have here is a different operational understanding of mathematical language between us. And I would like to understand your position clearly. Lets take this simple example where 2 equations are solving for the same value but, as I see it, having radically different ontological meanings:

va=GMr−−−√vs.vb=2πrP

Can you see the ontological difference?

But is there ontology of the math, or does ontology only come into play when you add the context of physics to it?

Isn't Ontology a question for Epistemologists ?

Seems to me that Mathematics, unlike Language, needs clearly stated definitions for everything, so as to avoid issues like the above mentioned. But sometimes those definitions can only 'dig down' so far before they start getting circular.

But I'm a simple Physicist ( ? ) and out of my depth here.

I would be interested in @Eise 's take on it.

Edited October 19Oct 19 by MigL

@swansont Thank you for splitting the thread. It was branching hard out there.

2 hours ago, swansont said:

Which do you want to discuss - ontology, or physics?

I genuinely can not understand how can you split them apart. As soon as we claim that our math has any connection to empirical physical reality we engaged with ontology. Due to physics domain is empirical physical reality the split ontology or physics seems logically contradicting. Do you have any solid basis justifying this split?

2 hours ago, swansont said:

But is there ontology of the math, or does ontology only come into play when you add the context of physics to it?

As far as I know pure abstract math doesn't demand any ontology. Ontology starts when we assign physical meaning to mathematical expressions. And if we not keeping track on what math expressions are physically real (Kepler's third law for example) and forced by the physical properties of the Universe; and what math expressions and solutions are just a consequence of the specific formalism we use to describe it (singularities for example) - we might find ourselves so far deep in the artifacts that nobody can agree anymore on what is actually ontologically real. Sounds familiar? Its the current state of the modern physics.

Have you ever asked yourself "how many equations that I know and use are actually just artifacts?"? You should. You will be surprised to say the least.

I think the core of this longstanding problem in physics is what I call the mirroring effect:
"As humans, we often project ourselves onto observed phenomena, obscuring the true nature of physical reality behind our anthropic tendencies. In order to prevent this mirroring effect, theoretical development has to establish strong epistemic constraints on scientific reasoning. Without this constrains our resulting theories risking telling us more about ourselves then about the nature of the Universe."

Edited October 19Oct 19 by Anton Rize

A comment from the sidelines.

It seems to me that

Ontology is about isolation. That is what I understand by the being or essence of something.

Physics is about connections comparisons, relationships and interactions.

For instance we can define a circle as a ring shape.

Or we can define it by its connections to something else for instance a collection of tangent vectors as in a string diagram.

The two are not totally separate and there is some crossover between the approaches.

10 hours ago, Anton Rize said:

You making it very hard for me not to sound adversarial right now hahaha!

You can express velocity in terms of average potential energy or in terms of the period of a cyclic motion. There's no 'ontology' to any of those. Newton did a similar trick to show planetary motions require an inverse-square force law from Kepler's laws. There's also a trick involving average potential energy and average kinetic energy by using the virial theorem. There are many, many such tricks. Newton himself implied that he couldn't care less about the underlying 'ontology' (hypotheses non fingo = "I frame no hypotheses"). It's just algebraic manipulations.

You're making it very hard for me to believe you understand elementary physics. Ha, ha, ha!

13 hours ago, MigL said:

I would be interested in @Eise 's take on it.

Doing it again?

15 hours ago, Anton Rize said:

Lets take this simple example where 2 equations are solving for the same value but, as I see it, having radically different ontological meanings:

va=GMr−−−√vs.vb=2πrP

Can you see the ontological difference?

Nope. Seen as mathematical expressions, you could just have written a = sqrt(b/cd) and a = 2 πb/c. It is only in the context of physics, suggested by the symbols you use, that they get meaning and describe different areas of mechanics. So mathematically seen, they are just numerical expressions, equalities.

15 hours ago, Anton Rize said:

If you will look at this two equations as an algorithms one has to preform in order to obtain the values of v_a and v_b - you will find that this are categorically different algorithms.

There are different operations involved, that's all.

15 hours ago, Anton Rize said:

They lead to radically different physical actions that lead to different measurements of different phenomenon.

What 'physical actions'? Those in the brain of somebody calculating? What measurements? There are no 'measurements' in pure mathematical expressions. Again, if you endow them with meaning, physical meaning in your examples, then these expressions are related to physical relationships.

15 hours ago, Anton Rize said:

Is it reasonable to call it "algebraic manipulation"?

If presented as pure mathematics, sure.

11 hours ago, Anton Rize said:

As far as I know pure abstract math doesn't demand any ontology. Ontology starts when we assign physical meaning to mathematical expressions.

Last time I looked, mathematics is used in nearly all sciences. Why are you so obsessed with physics? Take 'compound interest' as an example. Just look at all the mathematics! And maybe you discover a Taylor series in it? But Taylor series are also heavily used in physics. And AFAIK it was a pure mathematical discovery.

13 hours ago, MigL said:

Isn't Ontology a question for Epistemologists ?

Not really. Ontology and epistemology are different, but related disciplines of philosophy. But you can do epistemology very well without doing ontology. The other way round is more difficult, if you do not want to practice ontology completely detached of our daily and scientific worldviews.

13 hours ago, MigL said:

Seems to me that Mathematics, unlike Language, needs clearly stated definitions for everything, so as to avoid issues like the above mentioned.

Let's just exaggerate a little: definitions are the objects in mathematics. And so they must be rigorous, otherwise you get 'no grips' on it with logic. As example take the definition of a vector space.

28 minutes ago, Eise said:

Let's just exaggerate a little: definitions are the objects in mathematics. And so they must be rigorous, otherwise you get 'no grips' on it with logic. As example take the definition of a vector space.

Can you expand on that example please.

Take the definition of a vector space and do what ?

17 hours ago, Anton Rize said:

@swansont Thank you for splitting the thread. It was branching hard out there.

I genuinely can not understand how can you split them apart. As soon as we claim that our math has any connection to empirical physical reality we engaged with ontology. Due to physics domain is empirical physical reality the split ontology or physics seems logically contradicting. Do you have any solid basis justifying this split?

On some earlier occasions where someone insisted that physicists must incorporate more philosophy I asked which school would help me align a laser into an optical fiber. They were unhelpful in explaining.

Perhaps you can explain the ontology involved.

In my many years of learning physics, there was never any discussion of ontology. I know that some people do investigate it, but you seem to be insisting that you must. I’d like you to explain that.

17 hours ago, Anton Rize said:

As far as I know pure abstract math doesn't demand any ontology.

So your objection to The Vat’s statement is moot. It’s only when you added your own spin on it that there’s something to discuss. There’s nothing inherently wrong in the discussion, but for someone decrying confrontation, it seems like an interesting choice to springboard the discussion off of a manufactured disagreement

Wow there's a lot to unpack here... Ok let me roll up my sleaves.

This is addressed to everyone:
I want to remind to all participants that our goal here is NOT to prove that one side is wrong and other side is right. The goal is to gain deeper insight on physics and our methods of doing it. The fact the our views are so different is an opportunity for both sides to learn and it is crucial for both sides remain ready and open for it. Any dogmatic unchallenged beliefs will create an obstacle in our mutual understanding. So I asking all participants to carefully examine the commitments they subscribed to before approaching any analyses. Thank you.

---

Instead of trying to explain philosophy to old school physicists I will demonstrate the ontology in action:


@Eise asked which physical actions I meant, and @swansont asked me to show the ontology involved. Here it is in one example.
I will use one rule: every parameter that enters the math must be traced to the action that produces its reading. I call this operational reasoning.

Take a circular orbit of radius r and period P.

\( v_b = \dfrac{2\pi r}{P} \)

• P - is a local clock reading (discreet benchmark cycles count) of the duration of an orbital cycle.

• r - we do not measure distance directly. Instead it defined through light travel. I write \( t_d = r/c \) for the time light needs to cross the radius. Since 1983 the number c is fixed by definition, so c is a chosen unit conversion.
  
  Dividing by c gives the speed as a pure number:

\( \beta = \dfrac{v_b}{c} = 2\pi \dfrac{t_d}{P} \)

β is a ratio of two durations read on one clock. The unit of time cancels.

Now the second expression:

\( v_a = \sqrt{\dfrac{GM}{r}} \)

No action on the orbit reads G or M. Setting \( v_a = v_b \) gives the only route to GM:

\( GM = \beta^2 \, t_d \, c^3 \)

Dividing by the unit conversion \( c^3 \):

\( \dfrac{GM}{c^3} = \beta^2 \, t_d \)

The right side is a duration built from two clock readings. This duration is the entire physical content of GM. Multiplying it by \( c^3 \) adds no physical meaning.

Is there another route to GM?

GM has units of length cubed over time squared, so every reading of it combines lengths and times only. A kilogram could enter only by weighing the body on a balance, and nobody weighs a star. Every method in use confirms this. Planetary ephemerides use radar times, Doppler shifts and periods. Gravitational-wave signals give the binary's scale directly as a duration, from the wave frequency and its rate of change.

The speed can also be read from light alone. If z is the transverse Doppler shift of a spectral line from the orbiting body, then:

\( \beta^2 = 1-(1+z)^{-2} \)

So the orbit offers three ontologically real readings: the light radius \( t_d \), the period P, and the shift z. Any two of them fix the system's scale, and the third one tests it:

\( 2\pi \dfrac{t_d}{P} = \sqrt{1-(1+z)^{-2}} \)

This line equates a ratio of durations with a ratio of frequencies. These are two different types of readings, and the equality can fail. @joigus , this is why I do not call \( v_a = v_b \) an algebraic manipulation. With GM taken from the same orbit, it is an identity that tests nothing. With operational readings, it is a physical law. Can you see this difference? That what ontology is, and there's no need to escalate adversarial tones. We are better then this.

For an elliptical orbit the same chain holds, with r as the semimajor axis and v as the speed at the point where the radius equals the semimajor axis.

This is the ontology in action.
The readings are durations and frequency ratios. G and M enter only when we convert the result into kilograms, a unit we invented. Relativists already drop G in practice by setting
\( G = c = 1 \) and writing mass as a length, and the chain above shows why that convention works. Yet the way we teach orbital mechanics starts from the Sun's mass in kilograms and the value of G, and derives the orbital speed from them. The real physics operational order is the reverse: the speed is read, and the kilogram mass is computed from it.

@swansont , aligning a laser into a fibre is a different shape of question and needs none of this. But the laser itself came from exactly this kind of reasoning. In 1917 Einstein published "On the Quantum Theory of Radiation". In its section on hypotheses about energy exchange through radiation, he asked which processes between molecules and radiation must physically exist for both to stay in thermal equilibrium. Absorption and spontaneous emission alone could not reproduce Planck's law. So a third process had to exist: emission induced by the incoming radiation. That process is stimulated emission, and the laser is named after it. Every laser rests on an ontological question asked forty-three years before the first one was built.

Here I would usually supply direct links to colab notebooks that you could run yourself and verify every claim, but because @swansont is convinced that all of this is elaborate marketing act that I preform to sell something - in order not to get another ban from the local thought police I will leave this comment link free. You can always DM me for links anyway.

9 hours ago, swansont said:

So your objection to The Vat’s statement is moot

This is correct only if Vat was commenting on pure mathematics. If he did so then he's comment was off topic. So the only logically valid contradiction-free reading of he's comment "Math makes no ontological claims." is "Math (the one we use in physics) makes no ontological claims.". So the "manufactured disagreement" exists only in your context depraved narrow reading.

15 hours ago, Eise said:

Last time I looked, mathematics is used in nearly all sciences. Why are you so obsessed with physics? Take 'compound interest' as an example. Just look at all the mathematics! And maybe you discover a Taylor series in it? But Taylor series are also heavily used in physics. And AFAIK it was a pure mathematical discovery.

This is a good point. I shouldn't limit my view with only physics. And it brings to questioning my statement "Pure math demands no ontology". The irrational constants like e, pi, sqrt2 actually might have some deep ontology behind them after all... But this demands its own ontological investigation that will join my never ending list of things to do.

Im not sure about your comment on Taylor series though. Yes it is used in physics as approximations but its not physically real. So Taylor series do not have a direct link to real physical readings and therefore ontologically not real as far as I understand.

And on mathematical discoveries that predate their physical application: we need to be carful here. Yes all physics mathematical, but not all mathematics gets to be physics and not all mathematics that we use in physics is ontologically real. That's kind of the whole point of this threat.

5 hours ago, Anton Rize said:

Yes all physics mathematical,

Can I take it that you really meant "Yes all physics is mathematical," ?

If so do you not find this to be an overstatement ?

If you meant something else, please clarify.

I do however fully agree with your version of the converse (all mathematics is not physics)

5 hours ago, Anton Rize said:

but not all mathematics gets to be physics

13 hours ago, Anton Rize said:

@swansont , aligning a laser into a fibre is a different shape of question and needs none of this.

My point exactly. Not all of physics requires ontology.

Here I would usually supply direct links to colab notebooks that you could run yourself and verify every claim, but because @swansont is convinced that all of this is elaborate marketing act that I preform to sell something - in order not to get another ban from the local thought police I will leave this comment link free. You can always DM me for links anyway.

You should probably steer clear of trying to guess at what I’m convinced of.

11 hours ago, Anton Rize said:

This is correct only if Vat was commenting on pure mathematics. If he did so then he's comment was off topic. So the only logically valid contradiction-free reading of he's comment "Math makes no ontological claims." is "Math (the one we use in physics) makes no ontological claims.". So the "manufactured disagreement" exists only in your context depraved narrow reading.

My take on the statement was that any ontology is found in how you use the equations, i.e. trying to model what we observe, not in the math itself.

I have no idea how that would be off-topic for the discussion that was taking place.

21 hours ago, Anton Rize said:

But the laser itself came from exactly this kind of reasoning. In 1917 Einstein published "On the Quantum Theory of Radiation". In its section on hypotheses about energy exchange through radiation, he asked which processes between molecules and radiation must physically exist for both to stay in thermal equilibrium. Absorption and spontaneous emission alone could not reproduce Planck's law. So a third process had to exist: emission induced by the incoming radiation. That process is stimulated emission, and the laser is named after it. Every laser rests on an ontological question asked forty-three years before the first one was built.

Something almost circular about saying there's an ontological question merely by saying something physically exists. Physics is by definition a study of the physical - hence the name. A presumption of realism is baked into the discipline. What else would physicists be measuring, if not the interaction of physical entities? Ghosts, ectoplasm, fairy dust, thought forms? Math is the descriptive symbol system used for this study, and can hardly carry any ontological burdens. It is measuring and predicting observed patterns, not making any claims about some intrinsic substance. It only touches on properties, or bundles of properties, not on things in themselves.

I am willing to agree that abstract concepts like numbers and their interrelationships would exist in a different way from physical objects. But that hardly brings us to needing some kind of deep existential questioning to study radiation absorption and emission, and lasers.

Not that someone couldn't take, say, a panpsychist position and posit that matter is fundamentally conscious and that abstract concepts like number are somehow part of the essence of the physical. But such positions, giving math some kind of independent existence (like Max Tegmark's universe "made of math") are in the realm of metaphysics. I am not seeing a case here being made that math itself, as we use it in sciences, is making any such claims or has leverage to do so.

It seems to me that math is descriptive and not prescriptive. Much like a math equation defining an observed relationship between two particles, we also use sounds that conjure a wealth of sensory data: words like "wet" to describe a phenomenal experience. I went to the lake, it was wet. My saying the word "wet" doesn't cause water to gush from me and start pooling around me. It's just a way to describe something. The word "wet" doesn't have to have an ontology or be somehow fundamental. It's just something to do with ducks and soggy trouser legs.

@swansont , then we agree on the core: ontology starts when we assign physical meaning to mathematical expressions.

We still differ on "not all of physics requires ontology". By your own account, ontology enters whenever we use equations to model what we observe. Every physicist does that.

Aligning a laser into a fibre does not ask an ontological question. It inherits the answer to one. The device you align exists because in 1917 Einstein asked which processes between matter and radiation must physically exist, and found stimulated emission. Every laser lab works downstream of that question.

Daniel Dennett framed this precisely, and @Eise keeps it in his signature: "There is no such thing as philosophy-free science; there is only science whose philosophical baggage is taken on board without examination."

So the question is not whether a physicist makes ontological decisions. Every physicist makes them the moment an equation is given physical meaning. The question is whether those decisions are made blindly or examined with the rigour physics demands.

Rereading my orbital example, I found a logical gap in my own post, and it shows exactly the problem this thread is about. I wrote that with operational readings,
\( v_a = v_b \) becomes a physical law. That sentence assumes the speed read from the spectrum is caused by G and M. No reading supports this.

The physical law is:

\( 2\pi \dfrac{t_d}{P} = \sqrt{1-(1+z)^{-2}} \)

Neither side contains G or M. The readings fix one scale of the central body. Splitting that scale into a kilogram mass and G, and calling the mass the source of z, is an assumption added on top of the readings.

Nobody in this thread flagged that step, and neither did I when I wrote it. That is the point. An unread term slips into an equation, and everyone, including me, accepts it as physics. Tracking which terms correspond to readings is the only check that catches it.

@TheVat I will skip your comments on the claims I never subscribed to. You wrote that mathematics measures and predicts observed patterns. Which observed pattern does M describe?

To everyone: on what basis do you hold that M is the source of z? Which reading shows it?

6 hours ago, TheVat said:

It seems to me that math is descriptive and not prescriptive. Much like a math equation defining an observed relationship between two particles, we also use sounds that conjure a wealth of sensory data: words like "wet" to describe a phenomenal experience. I went to the lake, it was wet. My saying the word "wet" doesn't cause water to gush from me and start pooling around me. It's just a way to describe something. The word "wet" doesn't have to have an ontology or be somehow fundamental. It's just something to do with ducks and soggy trouser legs.

I recall an episode of QI in which it was said that humans do not sense "wetness". What we perceive as "wetness" is really changes in temperature due to the heat conductivity of water compared to air. However, it was also said that there are creatures that do genuinely sense "wetness", and therefore it is not simply a matter of semantics to say that humans do or do not sense "wetness".

4 hours ago, Anton Rize said:

ontology enters whenever we use equations to model what we observe.

Not necessarily. For example, different interpretations of quantum mechanics use the same equations to describe the physics being observed. However, the different interpretations represent different ontologies, so the mathematics in this case does not indicate the ontology of quantum mechanics. Specifically, both the Copenhagen Interpretation and the Many Worlds Interpretation are described by the Born Rule, but they are ontologically quite different unless one accepts the ontology of the Many Worlds Interpretation and regards the Copenhagen Interpretation as the solipsistically subjective perception of the Many Worlds Interpretation, which I do. One thing to note is that the mathematics of the Born Rule does not actually contain wavefunction collapse, and it is my belief that mathematics cannot explicitly model intrinsic randomness.

Edited Wednesday at 08:43 AM4 days by KJW

Here is an interesting quote from Hermann Weyl.

As far as I can see he was talking about the ontology of Physics and/or Mathematics, though he doesn't specifically mention this.

It gives much food for thought.

I have to ask myself what philosophers have contributed to the revolution he speaks of.

Edited Wednesday at 10:54 AM4 days by studiot

On 10/20/2026 at 5:58 AM, Anton Rize said:

@Eise asked which physical actions I meant

No, I did not. I remarked that as pure mathematics, there are no different ontologies behind your mathematical expressions. Only interpreted as description of physical processes the question if there are different ontologies involved here makes sense. But I think the answer is 'no'.

I had a look at your 'ROSS' document, and it does not make any sense to me. It seems a clear example of crackpottism to me.

And I regret that you are misusing the citation of Dennett (which you probably took from my disclaimer). Dennett was a very science oriented philosopher, but his (scientific) interests were neurology, cognitive science, evolution; his main topic was understanding of consciousness, not physics. He has never published something about ontology of mathematics or physics.

9 hours ago, Anton Rize said:

@swansont , then we agree on the core: ontology starts when we assign physical meaning to mathematical expressions.

We still differ on "not all of physics requires ontology". By your own account, ontology enters whenever we use equations to model what we observe. Every physicist does that.

Aligning a laser into a fibre does not ask an ontological question.

Indeed

9 hours ago, Anton Rize said:

It inherits the answer to one. The device you align exists because in 1917 Einstein asked which processes between matter and radiation must physically exist, and found stimulated emission. Every laser lab works downstream of that question.

But one need not know anything about lasers in order to use one, nor must one contemplate the ontology in order to use an equation.

Ontology starts when we assign physical meaning to mathematical expressions. And if we not keeping track on what math expressions are physically real (Kepler's third law for example) and forced by the physical properties of the Universe; and what math expressions and solutions are just a consequence of the specific formalism we use to describe it (singularities for example) - we might find ourselves so far deep in the artifacts that nobody can agree anymore on what is actually ontologically real. Sounds familiar? Its the current state of the modern physics

A person can use an equation without knowing, or contemplating, its physical meaning. Some equations are developed while not understanding the mechanism behind the behavior they describe. And there are the many examples where the equations are just mathematical conveniences that do not refer to things that physically exist. Equations describe behavior of what we observe. There’s no requirement that the model describe reality.

19 minutes ago, swansont said:

And there are the many examples where the equations are just mathematical conveniences that do not refer to things that physically exist.

This is a particularly good category.

For instance Maxwell's Mesh Current equations.

All the 'currents' may be fictitious but the correct manipulation produces a set of the actual physical currents flowing.

Another example would be the method of fictitious forces eg D'Alembert's method.

11 hours ago, Anton Rize said:

Rereading my orbital example, I found a logical gap in my own post, and it shows exactly the problem this thread is about. I wrote that with operational readings,
va=vb becomes a physical law. That sentence assumes the speed read from the spectrum is caused by G and M. No reading supports this.

Thank you for noting this.

Now perhaps you would respond to my query about your other mistype.

6 hours ago, KJW said:

I recall an episode of QI in which it was said that humans do not sense "wetness". What we perceive as "wetness" is really changes in temperature due to the heat conductivity of water compared to air. However, it was also said that there are creatures that do genuinely sense "wetness", and therefore it is not simply a matter of semantics to say that humans do or do not sense "wetness".

Thanks, KJ. Not sure what QI is standing for. But I will look for it.

11 hours ago, Anton Rize said:

will skip your comments on the claims I never subscribed to. You wrote that mathematics measures and predicts observed patterns. Which observed pattern does M describe?

I don't know why you are dragging M into this. The equation is, what, for a transverse doppler effect in relativistic framework? So they do Mossbauer rotor experiments to verify time dilation in SR. So how does this not fit with what I said? Did you actually read and understand my previous post? Where does ontology have to come into this?

With very few exceptions, most Mathematical tools are developed in the 'abstract', without reference to any Physical systems.
They are, then, applied to Physical systems to develop a model for those systems.

When does the Ontology develop between those two steps, @Anton Rize .
( just expanding on Swansont's and studiot's posts; all three of us would like an answer )

Four of us. Still here. Just posted, in fact. 🙂