The fundamental theorem of space is Pythagoras’ Theorem. This states that for a right angled triangle the square on the hypotenuse is the sum of the squares on the other two sides.
On a plane Pythagoras’ Theorem gives:
This was the first real law of physics to be discovered.
This theorem takes on huge significance when it is combined with Cartesian Geometry. Decartes, also known as Cartesius, developed an algebraic method of handling geometry called Cartesian Geometry. This uses three independent axes (up-down, left-right, forward-back – abbreviated to (x,y,z) to plot the positions of any object in space.
In Cartesian Geometry Pythagoras’ Theorem is used to relate the three independent directions for arranging things in space (but see note 1 below).
In a three dimensional space:
In English this means that the length of anything projecting straight out from a point can be calculated from its projections along the 3 independent axes.
Suppose we have a plastic ruler and move it around in space, the projections of this ruler on the axes of our three dimensional coordinate system will change but as one projection gets smaller another will get larger. The value of "r", the length of the ruler in the formula above, will stay the same. In mathematics it is said that the length of the ruler is "invariant". The length, "r", is the real thing and and x, y and z are just ways of together calculating "r".
Einstein's great discovery was that the spatial lengths of objects were not actually invariant, if objects are moved at high speed they appear to get shorter. Einstein asked whether this shortening of objects was related to how we measure objects. A similar thing would happen if we only used projections on a two dimensional plane to calculate the length of a ruler because if we tilted the ruler out of the plane its projection on the plane would shrink. Einstein realised that the missing dimension was time. Include time as a fourth dimension and we can always get the correct value for the length of a ruler no matter how fast it is moving.
The fourth dimension is sometimes called "dimensional" time and each second of dimensional time is 300,000 km long (300,000 km in a second is “c”, the speed of light). The new form of Pythagoras’ Theorem is:
"s" is called the "spacetime interval". It is invariant and is the real length of objects such as the ruler discussed above. The real length of an object consists of a combination of how long it extends in time as well as how far it extends in space.
A consequence of this is that things can be at a single point in four dimensional “spacetime” but separated in space and time.
If an object obeyed this equation it would be zero distance in spacetime from a point but also spread out in time and space. The time dimension, (ct), is sometimes called a "negative dimension" (Weyl). This equation shows how something could conceivably be at a point but also out there, in Experience.
If we were going to explain the time extension of Experience with world geometry we would also need to add a fifth coordinate axis (a fifth dimension) because the brain is too small to host time extensions of more than about a tenth of a nanosecond. Perhaps there is a fifth dimension and the problem of consciousness is partly due to our failure to detect it in the physical world or perhaps Experience is due to some other physical phenomenon. Experience does indeed involve geometry but there are many possible geometries known to mathematicians so maybe Experience is not the geometry discussed here or maybe it is.
This discussion does NOT mean that the observation point is necessarily described by the equation given above, it just shows that a point such as the observation point is not impossible according to modern physics. An observation point is conceivable.
If you are amused by maths you can use simple algebra to get from the equations for the spacetime interval given above to the basic equations of Relativity.
Suppose John is on a planet and travels past Bill who is floating in space at v metres per second for t seconds on Bill’s clocks and x is the distance travelled by John which is vt metres long according to Bill. John thinks he has been travelling for T seconds when Bill’s clocks read t seconds.
John and Bill are both describing the same thing, namely John’s motion. John, being on a planet, thinks he has only travelled in time and Bill is doing all the moving but Bill thinks John has travelled in space and time. The overall projection in space and time “s” from Bill to John and vice versa is the same for both of them. In Bill’s system of measurements:
Bill describes John's motion by:
In John’s system this interval is described by:
The interval “s” is the same for both so:
Which is the Relativistic expression for time dilation between two observers and one of the major discoveries of the twentieth century.
Cartesian Geometry is simple for mathematicians but horribly complicated for people who do not like maths. What Descartes, also known as "Cartesius", discovered is that the positions and lengths of objects can be represented by their displacements within a three dimensional coordinate system. The bottom of the red line below would be given the coordinates X1,Y1,Z1 and the top would have coordinates X2,Y2,Z2.
The length of the line is then:
This was abbreviated in the text above as:
Why Relativity Contradicts Presentism
The Lorentz Transformation Equations show that moving observers have their time axis tilted over in the direction of motion and also have their X-axis tilted upwards. The X-axis includes all of those objects that are at the same time as the observer (simultaneous with the observer).
This means that any extended object will have parts that are at different times for one observer than for another. Both observers believe that their object is fully present. At normal velocities the difference in time of the parts of an object between observers is to be measured in picoseconds or less but even that small amount would prove in principle that objects extend in time. (The amount of the displacement in time is vx/c2 where x is the distance away from the joint observation point in the direction of motion).
The moment that there is an object that extends in space it will inevitably extend in time.
More about time
There are two popular, modern ideas of time. In the first idea of time objects exist now, and at no other time, and rattle around the time and space of the universe, this is part of the family of ideas known as presentism. In the second idea of time objects persist as evolving objects that span time and space. (There are yet other ideas, such as Barbour's "Platonia", that are discussed below.) In the first idea of time the concept of travelling back in time to meet our great great grandfathers is impossible because the past is gone. In the second idea of time our great great grandfathers are still there in the past and the past is just like another place.
There are three problems with presentism. The first is that it
contains the paradox that objects extend for no time at all and existing
for no time at all is the same as not existing. The second problem
is that, given there is no motion during "no time at all", nothing
moves. The third problem is that according to physics objects that
are extended in space do not have fixed, simultaneous parts (see
above). This forces us to alter our idea of presentism to the idea
that it applies to very small objects and these objects are extended in
time by a tiny amount so that they can exist and move. This then leads
to the argument that time exists as a direction for arranging events but
is only permitted to have a very tiny extent. We then have a model
of an object such as an electron where it extends in space and time by a
tiny amount and it is its extension in time that gives it the
possibility of existence and movement. We can then introduce
"energy" as a measure of how the object's extension in time can cause
movement in other objects. It also allows us to understand the
energy-time version of the Uncertainty Principle.
So we can see that presentism is forced to capitulate to a limited form of persistence through time. There is no obvious reason why, having allowed some limited persistence in time, there should be any limit to persistence.
Although we have events ordered in what we call time it is possible that this ordering is an artifact of selection from a much wider set of events. Physicists such as Julian Barbour have proposed versions of this idea (Barbour calls a universe where this might happen "Platonia). It is fundamental to all physical theories that they reproduce well verified results, such as those that Relativity Theory predicts, so for all practical purposes they do not abolish time. However, Barbour's theory does permit a point observer (what Leibnitz called a "monad") and so do provide some support for the descriptions given in this book.Quantum Theory, Time and Free Will
Historically there were two interpretations of Quantum Theory. The first held that the act of observation could turn waves into particles with definite locations or momenta and the second that the act of observation splits the observer into numerous copies so that each copy observes a separate patch of energy (the particle) at a particular location. Recently it has been pointed out that the first explanation is just what you would expect from the second explanation. (See Zeh 2013).
The latest theory of the origin of time is that of Zurek (2018) in which the observer is unable to reverse any quantum processes it observes. This implies that observation fixes the events in our own world.
The nature of things
If a rod is being rotated round and round it might look like it is changing from a dash to a dot and back again. In two spatial dimensions the rotating rod might not look like a rod at all. Two spatial dimensions do not capture the whole of the rod. However, if all three spatial dimensions and time are brought into account it becomes obvious that the object that is extending then contracting to a dot is a rod of constant length rotating away and toward the observer.
It turns out that the length of a rod can be calculated from the projections that it makes in the three dimensions in space. The constancy of the length of a rod (h) in three dimensions is given by a simple formula that uses the extension of Pythagoras' Theorem to three dimensions:
h2 = x2 + y2 + z2Where h is the length of the rod and x,y,z are projections of the object in the three independent directions in space.
You are probably thinking that this is a crazily complicated way of measuring the length of a rod but the calculation is actually about the independent dimensions attached to each of us, the observer. That the length of a rod can be calculated from how it lies in three dimensional space shows that there are no hidden directions in which the rod might be tilted: the rod seems to be wholly present in the three dimensions of space.
Until the twentieth century everyone thought that the three dimensions in space were all that was needed to describe the length of an object. Whichever way a rod is placed in space, the three dimensions of space that it occupies appear to always give the same value for the length of the rod. It is said that the length of the rod is "invariant" in three dimensions. This process of calculating a constant length from the projections of a rod on the dimensions in space is important because it shows that we seem to have captured all the possible ways in which a rod can be arranged.However, in the twentieth century physicists made a monumental discovery, they found that the three dimensions in space are not sufficient to describe the length of a moving object. It was necessary to include time as a "negative" dimension and use a conversion factor "c" to express seconds as metres.
s2 = x2 + y2 + z2 - (ct)2
The new, four dimensional length of the rod is "s" and remains the same, provided all four dimensions are taken into account, it is invariant however the rod is moved and also invariant between observers. "s" is known as the "space-time interval".
(The version of Pythagoras' Theorem given above is closely related to what physicists call the "metric of space-time" (which is ds2 = dx2 + dy2 + dz2 - (cdt)2). )
It is the nature of things to be four dimensional objects in a four dimensional universe.
The most interesting space-time interval is an interval of zero which permits simultaneous observation at a point:
0 = x2 + y2 + z2 - (ct)2
This interval is sometimes called the "light cone" because it describes how photons of electromagnetic radiation converge to, or diverge from, a point. If space and time actually exist it really describes which parts of the universe are at any point. Anything on the lines converging to a focus as described by the equation above will measure itself to be no distance from the observation point because it is going at the speed of light. We know from physics that a small part of any object, such as the electric fields that define the everyday properties of objects, will lie on the lines pointing at the apex of the light cone. Our observation is much like a volume of brain tissue that has parts that fall on the cone described by the "light cone", these parts are no distance from a common point in space-time even though they are separated in space ( x2 + y2 + z2) and separated in time (-(ct)2) from the point according to an outside observer. The "being at a point" of items lying along the light cone should be considered as a connection of no length that takes no time to traverse rather than as the conjoint presence of material objects because the objects are always separated in space and in time from the point, though not separated in spacetime.
This may all seem very esoteric but given that the universe is indeed, at least, four dimensional the only reality is four dimensional objects such as the observation point which includes all of those events on the light cone around it. Observable reality is four dimensional. This sounds very mysterious but it is what physics tells us. Whether our own observation point is due to four dimensional spacetime is an open question.
The light cone is actually two cones:
The observation point (labelled "observer" above) is separated in time from the content of the observation which lies on the surface of the light cone.
If we are to treat the geometry of the light cone as a real geometry rather than a model of dynamics then we must accept that when a photon is directed at the observation point an observer on the photon itself is truly no distance and no time from the point but also at its place and time of origin according to any other observer, the light cone then becomes a direct connection from one place in spacetime to another. The apparent spatial separation of the observation point from the contents of Experience would then be a separation in time rather than a separation in space. The "seeing of" the view would be the future state of the brain being in contact with its past state in this model.
The reason for introducing the light cone was to examine the nature of observation in a four dimensional world. Our observation cannot simply be the light cone because objects moving on such a (hyper-)surface would not be sufficiently extended in time. Any geometry that could describe Experience would need to include highly time extended objects.
A note on Idealism versus Physicalism
Summary: Science is the collection of possible relations between things. Everything we currently know is in our Experience. Although we can know the relations between things we cannot know any things in themselves except our current Experience. The relations between things imply that most of the universe is outside our Experience. The relations between things extend into our Experience and it is conventional to describe all things with connected relations the "physical universe".
The "philosophy of mind" has been tormented for centuries by the conflict between Idealism and Physicalism. Idealism is the belief that all things are ultimately mental whereas Physicalism is the belief that the world is largely composed of non-mental, physical things. The debate is absurd because it ignores the evident properties of Experience. Imagine that our Experience is something physical, like an electric field, and because we are extended in time the field at one time can interact with itself at another time. The field might then "know itself". On the other hand, suppose we are a field of "non-physical thoughts" and because this is extended in time it might then know itself. So whether the field of events that is Experience is physical or mental it would make no difference except in nomenclature.
Given that there are relations between the parts of Experience as described in this book, Idealism will be faced with the same problem of explaining these relations as Physicalism. Idealists cannot simply say "its all supernatural" when their Experience contains consistent relations between events that could be used to create a science. Were Idealists to investigate the relations between events in their "minds" they would end up with a science that is identical to that of the Physicalist.
In this book Experience is what it is like to be a set of ordered events, ordered in time as well as space. Science, the study of the relationships between events, has amassed a body of consistent relations between events that suggests strongly that there are events outside our current Experience: that there is a physical world that also has ordered events but is not in Experience. The most likely description is that Experience is a set of physical events in our brains and that Physicalism is correct but that science is not yet complete because we have not identified the relations that correlate with Experience. What science must eventually explain is how the relations between the spatial and temporal parts of Experience can occur as events in the brain.
Science is a description of the relations between events. The descriptions are not the events themselves. The only events that we can know in themselves are the direct contents and form of Experience. No events outside Experience can be known directly. Although some scientists might imagine that they observe the physical world directly this does not happen except in the single case of the events that are Experience and these events are in our brains. We can imagine physical objects in our Experience but these imaginings will always differ from Experience because they are objects within it. Imaginings are less than our Experience in extent and content so what we imagine to be physical things can never be the same as Experience. So it should be no surprise that the imagined physical world does not supervene on Experience. The best we can say of a successful physical theory of Experience is that it contains relations that are exactly like those in Experience.
(The argument given above touches on the philosophical argument from "Supervenience" where physicalism might be dismissed because we cannot imagine how it could supervene on the mental or vice versa. As pointed out above, we can never imagine a physical thing that is the same as Experience, we can only check that we know all of the relations of Experience so that measurements can be used to identify the physical things that are Experience in the brain).
Another argument in favour of physicalism relies on imagining the progressive replacement of the brain by devices that perform the same functions as the parts removed. This argument usually has the hidden proposition that the parts perform functions that are no more than a succession of instantaneous, three dimensional states. It is not obvious that the parts of the brain all perform functions that are no more than a succession of three dimensional states or that solely three dimensional parts will do the job (a substituted part might be a four dimensional part that needs to be primed over a period of time etc). The fact that the argument is a non-sequitur does not disprove physicalism.
Studies have shown recruitment during inner speech of areas associated with overt speech production and comprehension, such as left inferior frontal gyrus (IFG), supplementary motor area (SMA) and the superior and middle temporal gyri (McGuire et al., 1996; Shergill et al., 2002; Aleman et al., 2005).
Dennett and Kinsbourne( 1992) published a highly influential paper which summarises many of the claims that Dennett made in his book "Consciousness Explained".
The authors used the phi and cutaneous rabbit illusions to argue that we only report what we think we sense and do not actually have any model of this in our brains that constitutes our experience:
"..our model claims that the brain doesn't bother "constructing" any representations that go to the trouble of "filling in" the blanks"
In the past 15 years MRI scanners have advanced to the state where even small areas of cortical activation can be visualised. The best test of whether the brain goes to the trouble of representing illusory motion is now to actually look at the brain. Both the phi illusion (Muckli et al 2005, Larsen et al 2006) and the cutaneous rabbit illusion (Blankenburg et al 2006) are accompanied by the filling in of brain activity to represent the "illusory" motion. Dennett and Kinsbourne's hypothesis has been falsified by experiment.