# Is Newton Circular ?

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This thread was prompted by the surprise expressed in some quarters that Newton’s laws of Motion (N1 to N3) was not only not a circular argument but could be recast in an alternative manner with beneficial results.

The originating thread can be found here:_

In Mathematics we have axioms and we deduce relationships resulting from these axioms in mathematical systems, which obey or follow these axioms.
Note this does not disallow the possibility of mathematical systems that do not follow a givens set of axioms, as well as systems to which any particular set of axioms has no relevance.

In Physics we have Principles, rather than axioms, and we deduce effects resulting from these Principles in physical systems which follow or obey them.
Principles may be explicit for example Huygen’s Principle, The Principle of Relativity (to which I will return)

Or they may be implicit as for example the maxim
”Like charges repel, unlike charges attract.”

So to reinvent Newton we shall start from a stated Principle.
But since we do not (yet) have a definition of mass or force, we cannot use them in our statement of principle, if we wish to avoid circularity.
So instead we start from something more general.

By way of introduction, let us look at what N1 tells us.

N1 effectively introduces Frames, and indeed defines an inertial frame as one in which a body “continues in a straight line.”

So what does that give us?

Well a general frame is a coordinate system in which we can plot the path of a body in motion (or at rest).
Such a frame provides variables of space and time as generating the coordinate axes.

Taking one of each of these we can plot a path taking a space coordinate (s) as a function of time (t).

Then we can derive the velocity (v) as the first derivative of our plot

$\frac{{ds}}{{dt}} = v$

And acceleration (a) as the second derivative.

$\frac{{{d^2}s}}{{d{t^2}}} = a$

Note that these are kinematic quantities, entirely independent of any mass we eventually attribute to the body or any forces we eventually consider as acting.
Also we are discussing a purely mechanical system.

Now to state the first Principle

“If the system contains two otherwise isolated but interacting bodies then, at any point in time, the ratio of the magnitude of their accelerations is a constant.

Further their respective accelerations are oppositely directed.

So for accelerations a1 and a2 and acceleration vectors a1 and a2 we can write the following equations following this principle.

(1)

$\frac{{{a_1}}}{{{a_2}}} = {k_{12}}$

(2)

$\frac{{{{\vec a}_1}}}{{{a_1}}} = - \frac{{{{\vec a}_2}}}{{{a_2}}}$

We have taken the meat from N1 and stated a Principle that nearly leads to N3, however Newton’s laws are a trinity and should be considered as such.

So to complete the job we need to bring in mass and force via N2. Which is the next stage of the development.

There is much to be gained by completing this journey, to whit a conservation law and a version of Newton that is compatible with the Principle of Relativity.

Edited by studiot

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N1 says nothing about accelerations, though. It simply defines an inertial frame as one in which there is uniform motion (which includes being at rest), with the implication being that under these conditions, the rest of Newton's laws will apply.

Absent N2, you don't know what the relationship is with acceleration (other than it must be zero to work), or mass.

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There is a mathematical representation of the three laws that may come in help.

Newtons first law: where r is a position vector. $u=\frac{dr}{dt}=constant$ can be simply expressed free particles move with constant inertia or velocity. This law restricts the reference frame to be non accelerating hence an inertial frame.

Newtons second law : the acceleration $a=\frac{du}{dt}$ is proportional to the force  (F) exerted upon it $F=m_i\frac{du}{dt}$ where m_i is the inertial mass.

Newtons third law if particle 1 acts upon particle 2 the particle 2 acts upon particle 1

$F_{21}=F_{-12}$

a reference to these can be found in Einstein’s General Theory of Relativity by Øyvind Grøn and Sigbjørn Hervik pages 3 and 4. I particularly liked this example as it expresses Newtons laws in a simple mathematical format. It also follows a key postulate under GR in that all reference frames are inertial.

Edited by Mordred

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1 hour ago, Mordred said:

There is a mathematical representation of the three laws that may come in help.

Newtons first law: where r is a position vector. u=drdt=constant can be simply expressed free particles move with constant inertia or velocity. This law restricts the reference frame to be non accelerating hence an inertial frame.

Newtons second law : the acceleration a=dudt is proportional to the force  (F) exerted upon it F=midudt where m_i is the inertial mass.

Newtons third law if particle 1 acts upon particle 2 the particle 2 acts upon particle 1

F21=F12

a reference to these can be found in Einstein’s General Theory of Relativity by Øyvind Grøn and Sigbjørn Hervik pages 3 and 4. I particularly liked this example as it expresses Newtons laws in a simple mathematical format. It also follows a key postulate under GR in that all reference frames are inertial.

That was not the chain of reasoning I was pursuing, and is in grave danger of becoming the circular argument everyone is talking about.

Let us see if there is any more interest in the subject.

1 hour ago, swansont said:

N1 says nothing about accelerations, though. It simply defines an inertial frame as one in which there is uniform motion (which includes being at rest), with the implication being that under these conditions, the rest of Newton's laws will apply.

Absent N2, you don't know what the relationship is with acceleration (other than it must be zero to work), or mass.

Ah, I missed that post.

Thank you.

That is why we replace N1 to N3 with a Principle, (as stated).

We end up with N1 to N3, we don't start with them because that way does lead to circularity.

Do you disagree with that principle?

Edited by studiot

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Isn't there supposed to be a logical fallacy inherent in a circular logic ? If I understand circular logic correctly  its a form of logical fallacy one based upon an assumption in many cases ie if A is true then B must be true. Where it is assumed A is true

I would be curious how one would find a logical fallacy in Newton's laws of inertia.

Edited by Mordred

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14 hours ago, Mordred said:

Isn't there supposed to be a logical fallacy inherent in a circular logic ? If I understand circular logic correctly  its a form of logical fallacy one based upon an assumption in many cases ie if A is true then B must be true. Where it is assumed A is true

I would be curious how one would find a logical fallacy in Newton's laws of inertia.

It's a fallacy if you use them to derive each other. e.g. N1–>N2–>N3 and then N3–> N1. But it's not clear that this was the case. If you merely state them as being true it's not a problem.

15 hours ago, studiot said:

That was not the chain of reasoning I was pursuing, and is in grave danger of becoming the circular argument everyone is talking about.

Let us see if there is any more interest in the subject.

Ah, I missed that post.

Thank you.

That is why we replace N1 to N3 with a Principle, (as stated).

We end up with N1 to N3, we don't start with them because that way does lead to circularity.

Do you disagree with that principle?

I do.

Since N1 says nothing about acceleration, other than it having to be zero, you do not know the form of N2.

What if force actually had a term in it that depended on the second derivative of velocity? Then your principle would not hold.

IOW, your principle assumes the second law. You are not deriving it from the first law.

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20 minutes ago, swansont said:

I do.

Since N1 says nothing about acceleration, other than it having to be zero, you do not know the form of N2.

What if force actually had a term in it that depended on the second derivative of velocity? Then your principle would not hold.

IOW, your principle assumes the second law. You are not deriving it from the first law.

No I was not deriving the Principle from the first or second or third laws.

It is not necessary to even introduce any of them  at this stage.

That was just a preamble to place things in context, since we have these laws and are trying to discuss them.

Do you disagree that the first and seciond derivatives of a position - time graph are purely geometrical?

IOW acceleration is a purely geometrical property, that can be considered quite independently of forces or masses or temperatures or other properties of a body.

That is not to say what the causes of that acceleration are, simply its measurement.

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2 hours ago, studiot said:

No I was not deriving the Principle from the first or second or third laws.

It is not necessary to even introduce any of them  at this stage.

That was just a preamble to place things in context, since we have these laws and are trying to discuss them.

Do you disagree that the first and seciond derivatives of a position - time graph are purely geometrical?

IOW acceleration is a purely geometrical property, that can be considered quite independently of forces or masses or temperatures or other properties of a body.

That is not to say what the causes of that acceleration are, simply its measurement.

I see.

It should be noted that your 1st principle (and all of kinematics) implicitly assumes that you are in an inertial frame.

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4 hours ago, swansont said:

I see.

It should be noted that your 1st principle (and all of kinematics) implicitly assumes that you are in an inertial frame.

I claim no originality for this approach, it was introduced to me by Roy Turner of the department of Theoretical Physics at Sussex University.

I don't think he claims originality either, but allow it is very useful.

My take is that it is a kind of separation of the Constitutive and Compatibility relations.

Maybe the next installment will make things more clear, I was disappointed that folks appeared to only want to mock, so I held back on that today.

The Principle enunciated place more severe limits than general kinematics, although it does imply non discontinuous paths, due to compatibility requirements.

N1-N3 amopunt to the constitutive relations which come next.

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I'm not sure how your getting mocking, as I only see the three of us involved in this thread. I look forward to the net installation, not that I ever concern myself too much in philosophical debates lmao. That is one topic I have little skill in, its still a good exercise to compare the ramifications equations can have in how we interpret them.

For one thing I'm still uncertain how to distinguish a circular logic from a math equation in terms of the LHS and RHS of an equal sign. Hence my questioning the fallacy requirement earlier. Considering physics never assumes something as being absolutely true how does one distinguish the assumption basis ? Granted in this case we certainly have enough evidence supporting the three laws of inertia that one can safely assume a high degree of accuracy. However on a philosophical basis is this ever sufficient ?

suffice it to say I am uncertain as to when an argument becomes a circular argument in terms of numerous math expressions if applicable in this case.

Edited by Mordred

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6 hours ago, Mordred said:

For one thing I'm still uncertain how to distinguish a ﻿circular logic from a math equation in terms of the LHS and RHS of an equal sign.

I have no idea of what you guys are talking about, but this I understand.

A circular argument validates itself; for example, we know "God" exists because it would take a "God" to create existence. In this argument "God" is validated by the reality of existence. THAT is a circular argument.

On the surface, math can look like a circular argument; for example, 3+4=7 seems to validate itself and can be proven because 7-4=3 and 7-3=4. But it is not a circular argument because math is just another language, and that language represents other things. So 3 red apples plus 4 green apples equals 7 apples, which will make a nice pie, but 3 pumpkins and 4 cups of flour will make a mess of 7 things -- even if the math is correct.

If we don't follow all of the rules of math, or if we are wrong regarding the things that it represents, then we could conceivably create a circular argument and even prove it valid. I think I may have done that a few times back in high school. My thought is that if Swansont can see something and Studiot can see something, there might be something there to see, but it is way past my comprehension, so I am out of here.

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Hi Studiot,

I admit I have some problems following your line of thought. I hoped to get the gist of your argument when it is complete, i.e. I also see the results. But obviously you want to build up some tension... So I just give some random thoughts.

On 2/6/2019 at 3:36 PM, studiot said:

N1 effectively introduces Frames, and indeed defines an inertial frame as one in which a body “continues in a straight line.”

There is nothing about inertial frames in Newton's original formulation:

Quote

Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.

I think this must be interpreted against Newton's idea of absolute space and time. This is the inertial frame for Newton. Newton claims universal validity of his laws of movement, and so for him this single one, universal inertial frame must exist.

Now when I look at modern version of the first law:

Quote

In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.

Does it define an inertial frame of reference, or does it define how objects in an inertial frame of reference move when they do not interact with anything at all? My claim would be the latter. And I really wonder if there is not already a circularity in this definition.

• What is an inertial frame of reference? It is a frame of reference in which objects move uniformly or are at rest, when no force acts upon them.
• What is uniform movement? It is the way objects move in an inertial frame when no force acts upon them.

Can this circularity of definition be avoided? (Without introducing another one...) And if I remember some of the definitions of inertial frames, like 'an inertial frame of reference is a frame of reference in which Newton's laws of motion are valid', the circularity is even more obvious.

On 2/6/2019 at 3:36 PM, studiot said:

If the system contains two otherwise isolated but interacting bodies then, at any point in time, the ratio of the magnitude of their accelerations is a constant.

Hmmm... Sound like a nice principle. But isn't the concept of 'force' hidden in the concept of 'interacting'?

On 2/6/2019 at 9:19 PM, Mordred said:

Isn't there supposed to be a logical fallacy inherent in a circular logic ?

It is not about logical argumentation here: it is about the circularity of definitions.

21 hours ago, studiot said:

IOW acceleration is a purely geometrical property, that can be considered quite independently of forces or masses or temperatures or other properties of a body.

That seems correct to me. In f = ma the 'a' is the less problematic concept, exactly for the reason you give here.

I hope you will complete your train of thought soon.

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On 2/8/2019 at 8:55 AM, Eise said:

Hi Studiot,

I admit I have some problems following your line of thought. I hoped to get the gist of your argument when it is complete, i.e. I also see the results. But obviously you want to build up some tension... So I just give some random thoughts.

There is nothing about inertial frames in Newton's original formulation:

I think this must be interpreted against Newton's idea of absolute space and time. This is the inertial frame for Newton. Newton claims universal validity of his laws of movement, and so for him this single one, universal inertial frame must exist.

Now when I look at modern version of the first law:

Does it define an inertial frame of reference, or does it define how objects in an inertial frame of reference move when they do not interact with anything at all? My claim would be the latter. And I really wonder if there is not already a circularity in this definition.

• What is an inertial frame of reference? It is a frame of reference in which objects move uniformly or are at rest, when no force acts upon them.
• What is uniform movement? It is the way objects move in an inertial frame when no force acts upon them.

Can this circularity of definition be avoided? (Without introducing another one...) And if I remember some of the definitions of inertial frames, like 'an inertial frame of reference is a frame of reference in which Newton's laws of motion are valid', the circularity is even more obvious.

Hmmm... Sound like a nice principle. But isn't the concept of 'force' hidden in the concept of 'interacting'?

It is not about logical argumentation here: it is about the circularity of definitions.

That seems correct to me. In f = ma the 'a' is the less problematic concept, exactly for the reason you give here.

I hope you will complete your train of thought soon.

I have been putting my time here into a couple of other threads, but I need to push on here as well so I had crack on.

I fear you are still looking at things in the order N1, N2, N3.

You also worried about circularity of the definition of mass, which is why I included the opening paragraph in the Stanford discussion on Principia post in the now defunct thread.

Quote

Stanford

The definitions inform the reader of how key technical terms, all of them designating quantities, are going to be used throughout the Principia. In the process Newton introduces terms that have remained a part of physics ever since, such as mass, inertia, and centripetal force. The emphasis in every one of the definitions is on how the designated quantity is to be measured, as illustrated by the opening definition: “Quantity of matter [or mass] is a measure of matter that arises from its density and volume jointly.” (Because a primary measure of density was then specific gravity, no circularity arises here.)

I have underlined why they think no circularity arises from mass  = volume times density.

Density in those days (like other mathematics) was comparative.

They really meant the density compared to a standard (water) i.e. specific gravity.

This was because they wrote in terms of ratios and proportionality.

When you make the comparison you end up with a ratio of ratios and the standard cancels out.

This  exactly the same process to move on from ratios of accelerations to atios of masses.

This is achieved by introducing a third body to the other two already noted, and comparing the acceleration ratios agains each of the original two.

That is all I have time for tonight, the (still simple) maths follows next.

Edited by studiot

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On 2/9/2019 at 11:42 PM, studiot said:

I have underlined why they think no circularity arises from mass  = volume times density.

Density in those days (like other mathematics) was comparative.

They really meant the density compared to a standard (water) i.e. specific gravity.

I think you are slipping from conceptual definitions into operative definitions. My claim is that scientific theories always have circularity in their conceptual definitions, and that this is no problem. So critique on a scientific theory cannot be supported by 'its definitions are circular'. But argo did seem to use such an argument, and it is also an argument I have heard against Darwinian evolution.  But I am happy to wait with my 'final judgment' until you completed your argument.

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49 minutes ago, Eise said:

I think you are slipping from conceptual definitions into operative definitions. My claim is that scientific theories always have circularity in their conceptual definitions, and that this is no problem. So critique on a scientific theory cannot be supported by 'its definitions are circular'. But argo did seem to use such an argument, and it is also an argument I have heard against Darwinian evolution.  But I am happy to wait with my 'final judgment' until you completed your argument.

Surely a definition is only of worth for the purpose for which it is made?

Clearly it must be fit for that purpose.

It surely cannot be guaranteed for any other purpose.

So here, a definition of mass must be made fit for purpose by posing it in a way that can be included in mathematical derivations employing it.

Any other theoretical considerations are irrelevant.

Edited by studiot

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On 2/9/2019 at 5:42 PM, studiot said:

I have underlined why they think no circularity arises from mass  = volume times density.

Density in those days (like other mathematics) was comparative.

They really meant the density compared to a standard (water) i.e. specific gravity.

This was because they wrote in terms of ratios and proportionality.

When you make the comparison you end up with a ratio of ratios and the standard cancels out.

IOW, they are comparing it to a standard. That's not really a definition in the way were discussing. As I had stated in the other thread "All definitions are ultimately circular, unless you have some sort of axiom, postulate or artifact that you can refer to."

Here you are using the density of water.

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2 hours ago, swansont said:

IOW, they are comparing it to a standard. That's not really a definition in the way were discussing. As I had stated in the other thread "All definitions are ultimately circular, unless you have some sort of axiom, postulate or artifact that you can refer to."

Here you are using the density of water.

The measurement of all quantities in Physics, except one, are by comparison with a standard.

And a form of comparison creeps into the use of the one exception (that I can think of).

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28 minutes ago, studiot said:

The measurement of all quantities in Physics, except one, are by comparison with a standard.

And a form of comparison creeps into the use of the one exception (that I can think of).

Measurements, of course. But we're discussing definitions.

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Is there any reason why a definition cannot also provide a scale of measurement?

Surely that is a more useful definition.

Let us say a body A exhibits a certain physical phenomenon.

Let us observe that a different body B exhibits the same phenomenon but to a different unspecified amount when the phenomena are placed side by side.

An in some (dare I say most?) cases it is all we have.

So we choose a standard, say body  C, and compare the phenomenon due to A and B with that of C.

Now we can remove C from the discussion and try to establish the relative values of that phenomenon between A an B.

The postulate  comes in to Newton's analysis that this phenomenon is one of direct proportion.

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47 minutes ago, studiot said:

Is there any reason why a definition cannot also provide a scale of measurement?

Perhaps not, but "what is a kilogram" and "what is mass" are two different questions, with two different answers.