# I Is there a violation in logic when we define notions of mass and force?

#### fxdung

Which part of Newton's second law is definition and which part is law content?It seem that there is a violation in logic, because we define the notion of force through the notion of mass, then we define the notion of mass through the notion of force when we consider the second law.

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#### PeroK

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Which part of Newton's second law is definition and which part is law content?It seem that there is a violation in logic, because we define the notion of force through the notion of mass, then we define the notion of mass through the notion of force when we consider the second law.
The second law is essentially a definition of force. Mass could be defined informally as a property of matter that resists acceleration.

Physics isn't logic and there is a certain circularity in Newton's laws.

#### fxdung

Is it because the force is a reality,it exist before there is a definition of it,so we do not need a logic argue here(we need not avoid a circularity here)?

#### PeroK

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Is it because the force is a reality,it exist before there is a definition of it,so we do not need a logic argue here(we need not avoid a circularity here)?
I'm sorry to say I cannot make any sense of what you've written.

#### fxdung

I do not understand why a science subject not follow a principle of logic?

#### PeroK

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I do not understand why a science subject not follow a principle of logic?
Because it's an empirical science. It's not a branch of logic.

Physics is logical, in the sense that statements must follow logically from each other. But, there is no way to define (inertial) mass or force empirically without reference to each other. In an empirical science, that's the best you can do.

#### fxdung

So we define notions referencing to each other,then we prove experimentally the definition is not self contradict in all cases, then it is OK?

#### PeroK

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So we define notions referencing to each other,then we prove experimentally the definition is not self contradict in all cases, then it is OK?
Yes, the true test is that it matches experiment.

When Newton devised his laws, he was trying to explain physical phenomena. Previously it was assumed (Aristotle) that objects had a natural tendency to slow down and stop. Newton recognised that this was not true and that, instead, he gave his first law:

1) An object will remain at rest or in uniform motion in a straight line unless acted upon by an external force.

And, that there was a proportional relationship between the force and the acceleration:

2) $F = ma$

These laws don't define "velocity" or "acceleration" or "straight line". And, they introduce two key concepts: "inertial mass" ($m$) and "force".

You could, of course, try to introduce physics from a more abstract point of view. And, in fact, in something like quantum mechanics the Schroedinger equation is much more abstract.

But, that's not what Newton's Laws are trying to do. There's an underlying assumption that we know what we mean by "object", "straight line" and that we can measure acceleration etc.

The next issue is that, because force and mass are introduced together, we need a standard unit for one or the other. The solution for many years was a platinum cylinder in Paris that defined $1kg$. That provided the means to measure masses and forces in standard units.

This is how an empirical science works.

#### fxdung

If we define force as product of mass and acceleration,then when we compare the test mass with standard mass(1kg) we must use the same force and measure accelerations.But how do we define the same force acting on standard and testing masses?If we use a spring to determine the same force then we do not use the definition of force as product of mass and corespondent acceleration?

#### PeroK

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If we define force as product of mass and acceleration,then when we compare the test mass with standard mass(1kg) we must use the same force and measure accelerations.But how do we define the same force acting on standard and testing masses?
Gravity would do as a standard force, for example. You can balance two masses on a set of scales. That takes you into the practical world of weights, measures and equipment calibration.

#### PeroK

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Gravity would do as a standard force, for example. You can balance two masses on a set of scales. That takes you into the practical world of weights, measures and equipment calibration.
PS One point to note is that this works. We're able to build aircraft or whatever based on classical physics and a practical approach to measurement.

Your point would only be valid, really, if this stuff didn't work. If we couldn't work out how much stress an aircraft wing could sustain, because there was some logical impasse at the heart of classical physics. There isn't a logical impasse. We can and do test equipment and build planes using Newton's laws. We know the mass of things and can measure the forces on them.

Physics works, in other words. It's not a branch of logic, as I've said before.

#### A.T.

So we define notions referencing to each other,then we prove experimentally the definition is not self contradict in all cases, then it is OK?
Yes. We postulate several quantities and the relationships between them. If it matches observation we use it.

#### Dale

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Science almost never has to worry about circularity like this. All you have to do is define something experimentally. In this case, we could define mass as the thing measured by balance scales. That has the added advantage of tying our important quantities directly to experiment.

#### FactChecker

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Even in Newton's time, there were ways to measure and define force that did not reference mass. Spring deflection is one. And in modern times, there are other ways to accurately define mass using atomic properties.

#### DrStupid

Which part of Newton's second law is definition and which part is law content?
Newton's second law is law content only. All related terms are defined seperately:

mass: definition 1
momentum: definition 2
force: definition 4

However, these definitions alone are not suficcient to use the terms. They just clarify what Newton is talking about in general. For a full understanding the laws of motion and additional restrictions (e.g. isotropy) are required as well.

It seem that there is a violation in logic, because we define the notion of force through the notion of mass, then we define the notion of mass through the notion of force when we consider the second law.
No, there is no violation of logic. Definitions don't need to be explicit - not even in mathematics (see Hilbert's axioms).

#### Mister T

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Which part of Newton's second law is definition and which part is law content?It seem that there is a violation in logic, because we define the notion of force through the notion of mass, then we define the notion of mass through the notion of force when we consider the second law.
It is possible to take mass as a fundamental entity and use Law II as a definition of force. It is also possible to take force as a fundamental and use Law II to define mass. Different textbooks have tried the different approaches, with those textbook authors and others arguing in the literature as to which is better. I would say that the former approach has tended to win the day. Especially since the SI defines the unit of mass as fundamental and the unit of force as derived.

#### fxdung

So are there (possibly) two types of definition: theoretical definition(e.g definition force as through acceleration and mass) and experiment definition(E.g definition of force as through the deflection of spring using in experiments)?(But we consider them equivalent)

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#### FactChecker

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In the modern age, there are incredibly precise ways to define or measure force experimentally. Your original question referenced Newton's Principia. Things were very different then. The English translation can be found here. Read the definitions that @DrStupid referenced in post#15 on pages 79-80 of the pdf. You will see how difficult things were then to be precise. And the understanding of logic then was very immature. Newton said that he was "standing on the shoulders of Giants", but those shoulders were vague and squashy.

#### Mister T

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So are there (possibly) two types of definition: theoretical definition(e.g definition force as through acceleration and mass) and experiment definition(E.g definition of force as through the deflection of spring using in experiments)?(But we consider them equivalent)
Except I see no reason to refer to one or the other as either theoretical or experimental.

"Is there a violation in logic when we define notions of mass and force?"

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