Newton on the aim of natural philosophy. Newton does not merely adopt the mechanical version of natural philosophy completely, translating Descartes' (and others') claims into mathematical language. He also claims that the fundamental objects of study for a natural philosopher are forces.
"all the difficulty of philosophy seems to consist in this-- from the phenomena of motions to investigate the forces, and then from these forces to demonstrate the other phenomena."What sort of forces are these?
"the phenomena of nature... may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, ... or are repelled and recede from each other."So particles have these forces or powers of attracting or repelling each other -- but we do not know the cause of the attractive force or repulsive force.
1. Quantity of matter ("mass"). Density times volume. Differs from Descartes (and other mechanical and atomist philosophers), who thought extension alone fully characterized the quantity of matter.
2. Quantity of motion ("momentum"). Mass times volume. Because of the difference in definition 1, Newton's 'quantity of motion' also differs from Descartes'.
3. vis insita or innate force of matter ("inertial force"). This force is responsible for every object's tendency to remain in its current state: at rest if it is currently resting, or having a certain velocity v if it currently is moving at v. It always operates in right lines, never in curvilinear lines (cf. Descartes' 3rd Law). This force is only exerted when the body meets some other force. [Think about riding in the car at a constant speed of 60 mph. Your body has an inertial force, but you don't feel anything-- until you slam on the brakes and your body wants to keep moving at 60 mph, but the seat belt in the car won't let you.]
4. Impressed force. Any force that "tries" to change a body's motion.
5. Centripetal force. Any force that draws bodies towards a point.
Examples: Gravity (heavy bodies are drawn towards the center of
the earth).
Magnetism (negatively charged bodies are drawn towards a positive pole)
The force that prevents the stone in a sling swung circularly from
flying away, and keeps the stone moving in a circle.
Because gravity is a centripetal force, if we could make a bullet travel fast enough, that bullet could travel all the way around the world and establish an orbit around the Earth. Here we find the first hint of the central insight of the Principia: the force that causes heavy bodies to fall in straight lines towards the center of the Earth is exactly the same force that causes the moon to trace out its orbit around the Earth, and the planets around the Sun. That is, the moon is continually "falling" towards the Earth, but its velocity is great enough that it does not spiral downwards into the Earth (and the velocity is not so great that it flies away from the Earth altogether). BUT at this point, as Newton is fully aware, Newton has not established that the moon can be treated like the various heavy things on Earth-- does the moon have weight? Finally, Newton says that it is the job of the mathematician to determine the law governing the force for any given orbit; and vice versa: given a force law. the mathematician should calculate what the trajectories will be for bodies of various masses and velocities.
After Def.8. In Books I and II, Newton proceeds purely mathematically: "For I here design only to give a mathematical notion of those forces, without considering their physical causes and seats." ... "considering those forces not physically, but mathematically: wherefore, the reader is not to imagine, that by those words [for various forces], ...that I attribute forces, in a true in physical sense, to certain centres (which are only mathematical points), when at any time I happen to speak of centres as attracting, or as endued with attractive powers."
Absolute space is also completely independent of any bodies or interactions in the universe. Even if every body in the universe maintained exactly the same distances from each other, and thus appeared 'from the inside' to be completely unchanging, all these bodies could still be moving with respect to absolute space -- even though, in such a hypothetical case, each body would not be moving with respect to relative space. Relative space always refers to some object, to which motions of other bodies are referred. So, for example, all the books sitting on a bookshelf are NOT moving through the relative space determined by the bookshelves, but they ARE moving through absolute space, since the Earth is rotating and revolving with respect to absolute space. Place, which is just space that is occupied by a body, will be absolute or relative depending on whether the space in question is absolute or relative.
Absolute motion occurs when a body changes its absolute place, i.e., when it moves from one portion of absolute space to a different portion. Relative motion is change of relative place. So the books on the bookshelf are in absolute motion, but they are not moving relative to the bookcase. There may be in our world no body that is truly at rest.
The order of the parts of absolute time cannot be changed: the moment of time when you woke up this morning cannot be 'switched' with the moment you turned off the alarm. Similarly for the parts of space: we cannot swap bits of space. Why? If portions of absolute space and time could change or move, then we would lack any standard to measure absolute motion.
Distinguishing absolute from relative motion.
~ What determines the difference are the forces acting on
bodies. A body's true motion can only be changed by a force acting on
it; a body's relative motion can be changed without any force acting
on that body. (For example: if you and I start out at absolute rest,
but then I get on the bus and start moving, you are in relative motion
with respect to me, even though you are still at absolute rest, and no
force has acted upon you.) Similarly, a body's relative motion can be
unchanged, even though its true motion changes (imagine applying a
force to the whole bookcase, with all the books in it).
~ We can distinguish true circular motion from merely relative
circular motion because true circular motion forces bodies away from
the center of the circle (e.g., a spinning bucket with water in it:
once the water starts spinning, it starts moving up the walls of the
bucket).
~ Note that Newton has a note on interpreting Scripture in this
section: "they do strain the sacred writings, who there interpret
those words [space, time and motion] for the measured quantities
themselves." Newton's point is that the usual or common meaning of
the words 'space,' 'time,' and 'motion' are meant in the sense of
sensible, i.e., relative motion. And the sun does have a relative
motion with respect to the Earth -- and that relative,
commonly-understood motion is how we should understand the Scriptural
passages that imply that the Sun is moving and the Earth is resting.
The Earth, although it is in true, mathematical motion, it is at rest
with respect to us humans. Newton was intensely interested in theology and the interpretation of Scripture: for example, many of his contemporaries remarked that he had extremely detailed knowledge of the Bible, and Newton was alos interested in calculating the beginning and end times of the world (according to his calculations, 2060 will be the end).
Law II. We today learn this as "Force=mass times acceleration" (F=ma). But that is not exactly what Newton said. Rather, he says that the total change of the quantity of motion is proportional to the total force applied. So if I give a body at rest a push, and that body starts moving at 20 mph, if I give it another push of exactly the same strength, it will move at 40 mph. (In more modern terms, we would say that force is proportional to change in momentum, aka impulse.) Also, every impressed force acts along straight lines.
Law III. "To every action there is always opposed an equal reaction." This is another 'counter-intuitive' idea: does it really seem that, when I impart a force to a soccer ball by kicking it, that the ball imparts an 'equal and opposite' reaction to my foot? To make this more intuitive, think about pool: a cue ball hits the 8-ball, and the 8-ball moves off at the same speed the cue ball previously had, and the cue ball stops completely. It is obvious that the cue ball imparted a certain amount of force to the 8-ball; but we see that the 8-ball impressed an equal amount of force, in the other direction, on the cue ball -- because the cue ball stopped, and a certain amount of force is necessary to stop the cue ball's movement.
Corollary I: decomposition of forces. Here we have another
case in which the theory is both (i) far from our immediate experience
and (ii) fully mathematized: we don't see the motion broken down into
two components, in any sense.
~ The decomposition of motions is essential to Newton's analysis of planetary behavior. For Newton breaks down the planets' motions into two components: a part due to inertial forces (which is always directed along the tangent of the planet's trajectory) and a part due to an attractive (centripetal) force, acting at a right angle to inertial force of the planet. So where the Ancients saw perfect simplicity and unity in the curvilinear motions of the planets, Newton (because of his First Law) sees complexity, and breaks down this complex motion into two component parts.
F=1/(distance from object to focus^2)So, since the planets travel along elliptical paths with the Sun at one focus, they must be attracted to the Sun according to the above equation. But we are still a long way from universal gravitation.
The Moon. Later in his life, Newton recounts that, as a young man, he wondered how high up gravity -- 'heaviness' -- extended above the surface of the Earth. He knew that it was slightly less on mountaintops than valleys, but he wondered whether it might reach up as far as the Moon. Newton, in the Principia, shows that the same force that pulls bodies towards the Earth keeps the moon in its orbit, i.e., the moon is constantly 'falling towards' the Earth.
Universal Gravitation. Newton goes on to show that several apparently different phenomena can be explained by this single inverse-square law:
1. Jupiter's moons (and Saturn's too) obey Kepler's Third Law [D^3/T^2= constant]. So the central force emanating from the sun is also present in Jupiter and Saturn.
2. Falling bodies on the Earth obey the inverse-square law.
3. Comets' paths (which had only very recently been suspected of being ellipses) follow the universal gravitation law.
4. The tides can be explained by it too, as the Moon's gravitational attraction of the oceans' water.
Newton vs. the Mechanical Philosophers As you might expect, the mechanical philosophers did not like Newton's idea that matter has some sort of inherent attractive power: it smelled too much like the Aristotelians' notion that matter tended towards its natural place, and it seemed like an occult quality, one of the 'sympathethies' of the magician-alchemists. The general sentiment on their behalf was that Newton had shown that matter obeyed this inverse square law, but that we needed some further explanation in terms of bodies coming into contact with one another, instead of mysteriously acting at a distance.
~ When he was young, Newton had read the Mechanical philosophers very closely, but as time went on, he had serious misgivings about it. Thus he was not completely wedded to its principles, and could think of physical matter in a somewhat different way. One explanation for Newton's conception of matter as not completely passive, but having the active power to attract, stems from Newton's interest in alchemy. He saw various chemical reactions that he believed showed matter was not completely passive or inert. (Two examples: mixing cold chemicals that became hot, without applying any heat to the reaction; the "tree of life," in which simple reagents mixed to create a complex, branching structure that resembles a tree or bush without leaves.)
Rule II. 'Same effect, same cause': If we see the same type of events happing in different cases, we should infer that the same type of cause is acting in both cases. (Newton's examples: horses and humans both breathe, so our explanation of why and how horses breathe should not be radically different from our explanation of how and why humans breathe.) Rules I and II together obviously help Newton establish that all behaviors that result from attractive centripetal forces should be given the same explanation: for if we can give a single cause for the attraction of stones to the center of the Earth and the attraction of the moon to the Earth, we should accept that there is just one cause operating for free fall and the lunar orbit, instead of two distinct causes.
Rule III. (Roughly) If every body that has been observed has a certain property, then infer that all bodies everywhere have that property. This rule is absolutely necessary for Newton's law of universal gravitation to be truly universal: for we only see a very limited amount of the universe, and only for a very brief time, but Newton's gravitational law (like his other three laws of motion) is intended to apply to matter everywhere and at any time whatsoever. (Someone could say to Newton: we don't know anything about parts of matter that we've never experienced, so we can't infer that they will have all the properties of matter we are familiar with, such as the property of attracting other bits of matter.)
Rule IV. Even if we can imagine some other explanation ("hypothesis") of our observations besides the one we have shown to be supported by empirical data, that does not mean we should doubt our explanation. We should only give up our hypothesis if we find some experimental/ observational evidence against our explanation, favoring another hypothesis. The basic idea here can be illustrated by an example: Newton explains that the ocean's tides are caused by the moon's gravitational attraction of the water on the Earth. Newton shows that the data match up with his hypothesis. Someone could come along and say: 'But couldn't the tides also be caused by secretive mermaids at the bottom of the sea, who move the water around every 12 hours?' We don't have (and Newton certainly didn't have) any evidence that this is not the case, but we nonetheless want to say that this made-up explanation casts no doubt upon the explanation from gravitation that Newton gives.
Theology. Though the planets' current motions can be explained
completely through Newton's laws, these laws cannot explain why or how
the planets were first established in those orbits, all in the same
plane. Newton says "it is not to be conceived that mere mechanical
causes could give birth to so many regular motions, ... This most
beautiful system of the sun, planets, and comets, could only proceed
from the counsel and dominion of an intelligent and powerful Being."
Another sign that the universe was created by God is that the stars
are placed in such a way that they have not all collapsed in together
due to gravitational attraction. And towards the end of the section:
"Blind metaphysical necessity, which is certainly the same always and
everywhere, could produce no variety of things. All that diversity of
natural things which we find suited to different times and places
could arise from nothing but the ideas and will of a Being necessarily
existing."
~ God's characteristics: He rules over everything, as a "living,
intelligent, and powerful Being." God is also "eternal, infinite,
omnipotent, omniscient" He exists "always and everywhere." How is
that possible? -- is my winter coat part of God, then? No, He is
distinct: "In God all things are contained and moved; yet neither
affects the other: God suffers nothing from the motion of bodies;
bodies find no resistance from the omnipresence of God." "He is
utterly void of body and bodily figure, and can therefore neither be
seen, nor heard, nor touched; nor ought he to be worshipped under the
representation of any corporeal thing." This last phrase hints at one
of Newton's primary theological ideas -- which he kept quite secret.
Newton convinced himself that the Christian doctrine that Jesus is God
is actually blasphemy -- treating Jesus, a 'corporeal thing,' as God
is idolatry, and therefore a corruption of the original and
true theology that pre-dates the Christian church.
~ Human knowledge compared to God's knowledge:
(i) God's understanding is completely different from ours. God is
"all eye, all ear, all brain, all arm, all power to perceive, to
understand, and to act; but in a manner not at all human, a manner not
at all corporeal, in a manner utterly unknown to us. As a blind man
has no idea of colors, so we have no idea how the all-wise God
perceives and understands all things."
(ii) We know nothing of what bodies really are -- and we do not
understand what God really is either: "In bodies, we see only their
figures and colours, we hear only the sounds, we touch only their
outward surfaces... but their inward substances are not to be known
either by our senses, or by any reflective act of our minds: much less
then, have we any idea of God's substance."
(iii) We know God only through His incredible works ("His most wise
and excellent contrivances of things")
~ Last, note that Newton says that discussions of God based on "the
appearances of things does certainly belong to Natural Philosophy."
So Newton thinks the law of universal gravitation and claims about
God's nature inferred from His creation are both part of what we today
call 'science.'
The cause of gravity. Newton "frames no hypothesis" about the cause of gravity. Why? Because he has not found anything in "the phenomena" that would explain gravitational force. Newton refuses to give an explanation of gravity that has no experimental consequences.