"For every science that concerns itself solely with logical relations between given objects according to given rules is mathematics"  Albert Einstein ( 1895  1955 )
"... we are regarding the function of the mathematician as simply to determine the truth or falsity of propositions"  Alan Turing ( 1912  1954 )
Newton's Principia Mathematica  published July 5, 1687
Sir Isaac Newton ( 1643 – 1727 )
DEFINITIONS
by Isaac Newton:
§ Absolute, true, and mathematical time, of itself and from its own nature, flows equably without relation to anything external, and by another name is called "duration"; relative, apparent, and common time is some sensible and external ( whether accurate or unequable ) measure of duration by the means of motion, which is commonly used instead of true time, such as an hour, a day, a month, a year.
§ Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces, which our senses determine by its position to bodies and which is commonly taken for immovable space; such is the dimension of a subterraneous, an aerial, or celestial space, determined by its position in respect of the earth. Absolute and relative space are the same in figure and magnitude, but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes; at another time it will be another part of the same, and so, absolutely understood, it will be continually changed.
§ Absolute motion is the translation of a body from one absolute place into another, and relative motion the translation from one relative place into another.
[ note: translated from the original Latin by Andrew Motte ( 1729 ), as revised by Florian Cajori ( Berkeley, University of California Press, 1934 ) ]
AXIOMS, OR LAWS OF MOTION
by Isaac Newton:
Original 1729 English translation ( from Latin ) by Andrew Motte two years after Newton's death:
LAW I
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 impressed thereon.
Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motion, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.
LAW II
The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
If a force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion ( being always directed the same way with the generating force ), if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.
LAW III
To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse ( if I may so say ) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also ( because of the equality of the mutual pressure ) will undergo an equal change, in its own motion, toward the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of the bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made toward contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.
[ note: Axioms are self  evidently true á priori statements and hence inherently unprovable. ]
MATHEMATICAL INTERPRETATION OF NEWTON'S CONCEPT OF FORCE
by this author:
In circumstances relative to an inertial frame of reference of straightline, unmodified motion of a body or particle in either direction or speed, there is no existing force present. On the other hand, where there exists a modification of the motion of a body or particle relative to an inertial frame of reference, in either direction or speed, there is present in such a system an existing force which makes such modification of a body's or particle's motion possible. Force, therefore, is a "modifying action" made upon a body or particle which changes either ( or both ) the direction and/or speed of the body or particle relative to an inertial frame of reference. And according to Newton's 2nd Law, the magnitude of the external force applied to a body or particle to modify its motion is given as
Furthermore since velocity and acceleration both possess direction and magnitude, they therefore are vectors and hence force also becomes a vector quantity as follows:
Example: Newton's 2nd Law of Motion^{∗}
A body of mass 1,000 g ( 1.0 kg ) is initially at rest in a spacetime vacuum but then is subjected to a straight  line constant net force of 50,000 dynes ( 0.5 Newton ) for a time duration of 1,000 seconds.
(i). The amount of acceleration is also constant since the applied net force is given as a constant and is thereby derived from Newton's 2nd Law of Motion as follows:
(ii). The final velocity starting from rest is given as:
(iii). The total distance travelled, starting from rest, is therefore derived as follows:
^{∗} note: these examples are used in the future upcoming Relativity Science Calculator Mac application
Derivation of acceleration without time
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