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What is the difference between definition and axiom? For instance, Newton's Definition 1 reads: (Cohen p. 403)

Quantity of matter is a measure of matter that arises from its density and volume jointly.

Here, Newton defines density: Density = Quantity of matter/ Unit volume

And Newton's Axiom 1 reads: (Cohen p.416)

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

In Axiom 1 Newton defines his concept of inertia but he classifies his definition of inertia as an axiom (not as a definition).

And also, the title of the chapter is, "Axioms, or the laws of motion." So Newton, defines his axioms as laws of nature.

Any thoughts about the difference in meaning between definition, axiom and law in Newton's time?

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    $\begingroup$ There are some tricky cases (e.g. when it is implicitly asserted that what is defined exists), but the OP examples are straightforward. The quantity of matter passage makes no assertions, it just names something "quantity of matter". So it is plainly a definition. In contrast, Axiom 1 does not define "inertia" or anything else, it makes a substantive assertion about behavior of bodies - "every body perseveres in its state..." One can then add that such a state is to be called "inertial", and that would be a definition, but that is not what the axiom says. $\endgroup$
    – Conifold
    Aug 30, 2022 at 20:46
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    $\begingroup$ Although I cannot speak authoritatively about what I. Newton might have had in mind by "definition", ... in my experience the best way to think about a "definition" is not as a claim about any facts, but just a naming convention. That is, here is a name/label, which may or may not apply to various situations, but does (at least allegedly) apply to phenomena/objects of interest in the moment. "Naming". $\endgroup$ Aug 30, 2022 at 20:52
  • $\begingroup$ You can see this similar post $\endgroup$ Aug 31, 2022 at 12:03
  • $\begingroup$ And see the post What is the difference between an axiom and a definition? $\endgroup$ Aug 31, 2022 at 12:05
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    $\begingroup$ But Ax.1 does not define the concept of inertia: the word "inertia" does not occur in it. Ax.1 is clearly not a definition: it states a universal property: the behavior of bodies in presence/absence of forces. $\endgroup$ Aug 31, 2022 at 13:02

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Borrowing from another post: Are Newton's laws just definitions?

"Newton follows the Euclidean tradition of presenting mathematical proofs by first providing a set of definitions which are then followed by a set of axioms (i.e., laws) that are assumed to be true. Definitions in themselves do not admit the existence of anything, but rather provide terminology that is later recognized as a particular condition as a result of a set of axioms.

Newton presents his laws of motion based on a collection of known observations regarding motion (in particular, he cites Galileo's inclined plane experiment and hypothesis that objects only slow down due to air resistance or other friction forces, which is counter to Aristotelian physics), and carries forward with the logical consequences of these laws in a series of propositions, which so happens to match experimental evidence.

Once the laws are provided, the particular effects of the laws can be matched with the definitions he provided. For example, his definitions of quantity of matter, quantity of motion, vis insita, etc., only make sense within the framework provided by his laws.

...the laws are not "definitions" since definitions are only understood within the assumed truth of the laws."

In other words (and to directly answer your question), "Definitions" provide a specific term used to compactly identify something that arises from some set of axioms or propositions.

As a loose example, in calculus, we define an "inflection point" as being a specific point where the second derivative of a function equals zero. This definition does not declare the existence of the inflection point as an axiom or proposition but instead provides an identifying term for when it does occur based upon a set of assumed truths.

In Axiom 1 Newton defines his concept of inertia but he classifies his definition of inertia as an axiom (not as a definition).

Newton's First Law (i.e., Axiom 1) does NOT define inertia, rather, it declares an axiom of motion that is assumed to be true. The term "inertia" is nowhere mentioned in the law: "Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impress'd thereon." Wikisource: Newton's Laws.

However, in the "definitions" section that's located prior to his Axioms, Newton defines inertia (Vis inertiae) under Definition 3: "The Vis Insita, or Innate Force of Matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forwards in a right line. This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the Mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this Vis insita may, by a most significant name, be called Vis inertiæ or Force of Inactivity." Wikisource: Newton's Definitions

Thus, "inertia" is a term used to identify the effect (namely the Force of Inactivity) that results from axiom 1. However, "inertia" does not rely on axiom 1 alone because the definition states it "is a power of resisting," thereby invoking an axiom of change (i.e., Law #2) and that it is categorized as an "Innate Force of Matter," which practically speaking, can only be understood through some interaction (i.e., Law #3 where inertia is perceived as a reactionary force in response to a force). Naturally, you might ask, "then why is Law #1 called the 'law of inertia'?" Well, because inertia is still "there" during the absence of any change and external forces; however, if we omit Laws #2 and #3 and place Law #1 in a vacuum, then inertia becomes an absurd concept since there are no rules that allow change and forces that permit change. In other words, the definition of inertia is absurd unless we admit all three laws.

And also, the title of the chapter is, "Axioms, or the laws of motion." So Newton, defines his axioms as laws of nature.

Any thoughts about the difference in meaning between definition, axiom and law in Newton's time?

Just as you said, Newton titles the section "Axioms, or the laws of motion (Axiomata sive leges motus)." Thus, Newton treats "Laws" as being synonymous with "axioms." If I were to guess, he probably chose the term "Laws" in order to categorically distinguish these rules of physical governance from mathematical axioms, but they are functionally the same. It's much like "postulates" versus "axioms" in maths in which they are functionally the same, but "postulates" is a species of "axioms" that act within a specific field of study.

For more examples of how definitions are treated, look at Euclid's Elements Book 1: Definitions are provided first, followed by postulates (i.e., axioms), then common notions, then propositions. Though definitions are provided first, they are only fully understood within the context of the postulates and constructed propositions. Elements, Book I

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    $\begingroup$ Forgive my butting in on your generally interesting answer, it is only to suggest that 'axioms' and 'laws of motion' were not being treated as synonyms: the expressions were less close to synonymy than to apposition. Thus they had the character of laws of motion (at least partly referable to others, as Newton acknowledged in the scholium), and on top of that they also had the characters of axioms, starting-points not deemed to be in need of proof. $\endgroup$
    – terry-s
    Aug 31, 2022 at 21:36
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    $\begingroup$ "Apposition" is much more appropriate! "Synonymous" is perhaps too exactly equivalent and interchangeable. I was attempting to convey that they are functionally the same, yet subtly placed in different categories. $\endgroup$
    – Andrew R.
    Aug 31, 2022 at 21:49
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I agree with the trend of the comments, and believe that in most cases Newton was trying to write straightforwardly even where we might find difficulty, perhaps partly owing to the lapse of time and changes in habits of expression.

Specifically, the definitions appear to be naming conventions (just as Conifold and Paul Garrett said): each consisting of a name and and explanation, hopefully sufficient, of what is to be known by that name.

The axioms are a different matter. The full name of the section is "Axioms or laws of motion". The laws of motion are what Newton thought to be laws, and giving them the added name or label of 'axiom' meant that he also took them to be well accepted or acceptable in view of current knowledge, and thus did not propose to prove them. This is strongly reinforced by the content of the following Scholium, where he wrote

"Hitherto I have laid down such principles as have been receiv'd by mathematicians, and are confirm'd by abundance of experiments. By the two first Laws and the first two Corollaries, Galileo discover'd that the descent of bodies observ'd the duplicate ratio of the time, and that the motion of projectiles was in the curve of a Parabola; experience agreeing with both, unless so far as these motions are a little retarded by the resistance of the air."

Ans then he went on to discuss other points and credit others including Huygens and Mariotte.

It will be seen that Newton did not claim the laws of motion, axioms, to be his own. A fair comparison might be with the axioms of Euclid's Geometry: they were not thought to be in need of proof, they were starting-points on which the ensuing discussions were to build further conclusions. It was only long after his lifetime that others named the laws of motion after Newton.

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