Pointfull Conjectures:
Notes on Truth
(This started out as notes on A. Einstein's “The Special and General Theory”)
Notes on Truth
(This started out as notes on A. Einstein's “The Special and General Theory”)
“In your school-days most of you who read this book made acquaintance with the noble building of Euclid's geometry, and you remember – perhaps with more respect than love -- the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of our past experience, you would certainly regard everyone with disdain who should pronounce even the most out-of-the way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you:
"What, then, do you mean by the assertion that these propositions are true?"
Let us proceed to give this question a little consideration. Geometry sets out from certain conceptions such as "plane," "point," and "straight line," with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as "true."
Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct ("true") when it has been derived in the recognized manner from the axioms.
The question of "truth" of the individual geometrical propositions is thus reduced to one of the "truth" of the axioms.
Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called "straight lines," to each of which is ascribed the property of being uniquely determined by two points situated on it.
The concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves. “
A. Einstein
The Special and General Theory
A. E. considers it preferable to describe the statements that are logically derived from axioms as “logically consistent”, instead of “true”. Axioms themselves, however, are not “derived”. An axiom is perceived as “true” without referring to anything. It is self-evident.
Would A. E. then assign the word “true” to axioms?
No.
Of Euclidean geometry, A.E. writes:
“(...) it has long been known that [the truth of the axioms] is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning.”
He further writes about geometry in general:
“The concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object”.
By “real” object A.E. undoubtedly refers to something ultimately perceivable (using instruments or not) through the senses.
A.E. must thus also deny 'truth value' to mathematical statements. A statement like “1+1=2” for example, is to be referred to as something along the lines of “logically consistent with itself and the logical system of which it is part”. Only statements referring to physical stuff can be “true”.
OK, but I am somewhat puzzled by:
“Geometry sets out from certain conceptions (...) with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as "true."
One could theoretically make “simple propositions (axioms)” at will. Where does the difference lie between those which we are inclined to accept as 'true' and those which we are not inclined to accept as 'true' (or those which we are inclined to accept as not "true”)?
Also, to derive logical consistencies from axioms (themselves described as “without true meaning”) is quite useful in revealing how “real” stuff behaves. But by what magic does that happen? Why do “real” objects relate to each other with the same logical links as conceptions? Should a process (“ …not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of ideas among themselves ...”) describe laws for objects of experience?
A. E. tells us that “Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas.”
He then writes that “Geometry ought to refrain from [referring to objects in nature], in order to give to its structure the largest possible logical unity”.
But he then re-assigns, to geometry, correspondence to objects in nature: “Geometry which has been supplemented in this way is then to be treated as a branch of physics”.
Here, A. E.'s position is that, even though objects in nature originally caused the genesis of geometrical ideas, pure geometry, a logical field dealing with internal consistency, no longer corresponds to these objects of nature. A.E. thus views “pure” (non-physical) geometry as irrelevant to “truth”. To make it truth-relevant, A. E. proposes to systematically check the ideas of geometry against physical “reality”. And when in conflict, the results of observation (of objects in nature) must be preferred. This basically positivist perspective tells us that, if and when the abstracted mental objects of logic do not behave as the real ones, it is logic that must be changed.
Interestingly, it follows from this view that if we ever came to observe objects in the world not to comply with mathematical equations, “1+1=2”, for example, we would also have to determine these equations to be 'false'. But what would it even mean for objects in the world not to comply with “1+1=2”? Let us, for example, imagine some law of the universe that forbade ever observing 2: No two things of any kind could ever be seen. If one observed one object and then brought in another into view, one would disappear, or, alternatively, a third would appear. There would be no 'logic' to this, and there wouldn't need to be. This could simply be “how things are”.
Could such a state of affair be possible? Probably not. A number is never observed in the universe as it is. It makes no “physical” sense to say that 2 can be forbidden by some law of the universe. We observe stars in the sky. And these are indeed “objects in nature”. But, at any point, we can mentally select some of them and create mental groups of 2 or 3 or 4 etc. But these numbers are not observed. They are not out there in the universe to be prohibited by a physical law. They are in our mind. Similarly, neither “1+1=2” nor any other mathematical equation (logically consistent or not) could possibly be forbidden by the observable universe.
This doesn't mean, however, that the observed physical universe is not indeed behind the genesis of numbers. Imagine that we gained consciousness in an empty universe. There is nothing to observe. We can never see any “objects in nature” for us to apply number-concepts to. Would we, under such circumstances, invent ideas like 1 and 2 and statements like 1+1=2? Again, probably not. Our mind seems to need something to work with.
In order to teach number-concepts to those who do not yet possess them (children), we must refer to actual or imagined physical experiences. For the concept 1, for example, we usually imagine observing some object, let's say an apple, located in a selected region of space (on a table-top for example). Similarly, for the concept 2 we imagine another apple next to the first. For 0 we imagine looking at the same region of space (the table-top) but seeing no apples there. We might then replace the apples with pencils, and repeat the exercise, to show that the aspect of the experience we are interested in is the number, not types of objects.
It might be fun to try such a process at a quantum scale. Let us take some region of 'empty' space, observe it, and determine, first of all, if it is indeed empty. We use 0 to refer to the absence of any object and some positive number to refer to the presence of one or more such objects. According to 'quantum fluctuation', however, in a region of completely “empty” space, particles are observed to constantly appear and disappear without cause. In terms of apples, it would be like seeing 1 apple and then 0, then 2, then none again and then 3. We wouldn't be able to state how many, if any, apples are on the tabletop at any point. The number would “fluctuate” without any obvious explanation.
However, at the quantum scale, objects are not observed as big, solid, clear-cut objects, like apples. Quantum objects, seem to be more akin to tiny “mists”, “clouds” or “fogs” than apples. Telling how many, if any, objects are to be found in a selected region of space is like telling how many mist-clouds there are in a humid region of the atmosphere. Mist-clouds are observable as hazy objects that tend to appear and disappear, literally “out of thin air”. Just as a mist is a typically unseen part of the atmosphere that only becomes observable when the humidity in the atmosphere condenses, with quantum fluctuations, stuff “condenses and de-condenses in and out of existence” so to speak. Instead of humidity, however, we learn from physics that it's something called “energy” that condenses into matter.
Fine, so let's see what this “energy” is.
“Energy” is defined as the ability to do “work”. So, the random movement of molecules, light, a horse, an internal combustion engine, a star and a human all … “contain energy”. Simple location with respect to something else can either give or take energy away from an object: A rock on a ledge, for example, contains gravitational potential energy, simply because it might fall down some day. Exactly the same rock on the ground doesn’t. How peculiar!
How did physicists ever manage to make “the ability to perform work” an essential feature of “objects in nature”? As natural philosophers, they seek to understand the fundamental nature of things, and there doesn’t seem to be anything “fundamental” about such a notion. It's as if engineers, who are interested in practical machines (things that perform tasks), made up a utilitarian concept, and then managed to convince physicist that it actually referred to something. In any case, “laws” about “energy” are considered very important in physics. And so, physicists spend a lot of time determining the “energy states” of systems.
When pressed, physicists, will admit that: “It important to realize that in physics today, we have no knowledge of what energy is.”
(R. P. Feynman, 1963)
And yet, the “law of conservation of energy”, for example, is viewed to be “one of the most basic laws of physics”. And it has never been broken. Physics also tells us that, for example, “kinetic energy” (it appears when an object is observed to be moving with respect to another object, or vice-versa) creates actual observable particles.
And so, we cannot accept being content with something like “we have no knowledge of what energy is” to then go on happily calculating energies all day. We must insist: What is “energy”? Is it “real” or a ”concept” or something else?
In an earlier example we saw that our mind imposes numbers upon a numberless universe. Might “energy” refer to such a conceptual abstraction?
We often project concepts on the observable. Let's illustrate this process: Tough far from being considered a fundamental aspect of objects in nature, in common parlance, one might describe a person or a skyscraper as being “tall”. We can thus alternatively say that something possesses “tallness”. To make it appropriate for physics, “tallness” would need to be translated into something measurable by specific units (“meters” in this case) like “height”. One would then need to clarify the concept even further because “height” does not simply refer to “length” or “extension in space”. It implies a starting-point and a direction, as defined by a chosen coordinate system. Having been invented by humans living on the earth, “height” is typically measured “up” in the direction away from the surface (and centre of mass) of the planet . But we can use “height” in any other situations by adapting the concepts of “ground”, “down” and “up' accordingly.
A debate might ensue as to whether the vague “tallness” and the scientific “height” are equivalent concepts (given that “tallness” has an apposite, “shortness”), but the point is that by formally specifying our terms, we can arrive at some scientific meaning. We can unequivocally determine the “height” of anything. Even of a sheet of paper flat on the ground (not something that would originally have been thought of as “tall”), we can agree that it objectively possesses a height of 0.1 mm.
“Tallness”, formalised into “height”, might even be treated as “real”. But of course, nothing of the sort can actually be found in any “object in nature”. Objects in nature do not really have either “tallness” nor “height”. If this is the case, this must also be the case for any extension in space. An object's “length” is about as real as its “tallness”. The same must apply to “distance”, and not just the units of measurement, but the very concepts.
Here we begin to answer why real objects relate to each other with the same connections that ideas have among themselves. When we observe “objects in nature”, we cannot do so without translating them into concepts. It just isn't possible. We then manipulate these concepts. And lo and behold: “real” stuff seems to follow the same rules as concepts! It also becomes clear that which concepts we choose to describe “objects in nature” can either open or close paths of understanding.
In the atmosphere, sound always travels at the same speed, when measured with respect to the air, regardless of the motion of the emitting source. So, whether something is moving towards or away from us, as it emits a sound, that sound is measured to arrive towards us at the same speed. The reason this occurs is that, unlike a bullet for example, sound is of the same 'substance' as air. Nothing material travels, only an impulse, a distortion of air, a “wave”.
In the vacuum of space, the speed of light is similarly always measured to be the same. So naturally, a light-carrying aether, similar to the sound carrying air, has been conjectured.
Now, in order for the speed of sound to always be the same with respect to the air, a sound must travel at different speeds with respect to the moving body emitting that sound. As a moving (with respect to the air) body emits a sound, and the sound-waves travel away from it (at the speed of sound) this body moves towards the sound-waves ahead and leaves even faster the waves behind. Sound must therefore be measured, with respect to the emitting body, to go slower in the direction of motion and faster away from it.
The same ought to happen to light in an aether. In the often-mentioned Michelson-Morley-Miller experiments, however, light was not measured to move slower in the direction of motion. Some aether theories are compatible with its observation (the Lorentz Aether Theory for example), but A.E.'s elegant special theory of relativity offered a description that had not need for any aether. The S.T.R. is derived from only two stated principles: 1- the principle of relativity (physical laws are the same in all inertial systems of coordinates)” and 2- the principle of the constancy of light (light is measured to propagate at c in regardless of one's inertial system).
In the special theory, A. E. thinks of “space” as a relational concept, not 'something' on its own. About motion 'in space', A.E. writes: “ … we entirely shun the vague word "space," of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by "motion relative to a practically rigid body of reference."
Nonetheless, A.E. had to give certain characteristics to space (such as isotropy and homogeneity), and to describe it (in the GT) as being 'bendable'. Giving “nothing” some characteristics is slightly odd. This, however, is explained to mean that it is the observable effects of space being bent that are real, whereas space itself is not. In other words, space is neither isotropic nor homogeneous nor being distorted. It's just that real stuff affects other real stuff as if it were.
To be considered 'something', an 'object in nature' has historically been thought to have extension (in space) and mass. Descartes viewed “extension” as the foremost characteristic of matter whereas Newton viewed mass as its essential characteristic. Not surprisingly, the Cartesian coordinate system is concerned with extension, whereas Newton's theory of gravity deals with mass.
Now, in modern physics there are some objects ('singularities'), that are understood to be point-like objects with (often tremendous) mass but no extension in space. These are thought to be usually hidden by an event-horizon. But under certain circumstances 'naked' singularities are though to be observable. A 'singularity', thought it is called “point-like” is not even a point. In terms of spacial measurements, it is nothing. As far as extension goes, it doesn't exist. Though, in terms of gravity there is something there.
Let us postulate that, though unobservable, space is something with extension but no mass (just as a singularity has mass but no extension).
Let us furthermore interpret this to mean that space is made of unobservable regularly spaced massless singularities. But for its mass, a singularity is already described as almost nothing. Retain the idea of a singularity but remove its mass. Massless and without extension such singularities cannot be observable.
Let us postulate that the positions of such points can be made to change, but that they nonetheless desire an equilibrium, both pushing and pulling each other toward some Euclidean rest state. This elasticity would be necessary to explain why an area of space seems to return to a euclidean state once we remove the influence of mass. Otherwise, when a mass has been in an area, that area would remain 'bent' even if the mass were no longer there.
We could then start imagining various situations.
We could, for example, imagine that for an object to make room for itself in space, it pushed, displaced or repelled the normally equidistant, massless-singularities (just as an object in water displaces the water around it in order to make room for itself). By pressing 'space stuff' outwards thus, an object would create an area of tension around itself. This constant push outward, created by the mere presence of something, could be felt by the space around it as a constant acceleration of the object outwards: 'gravity'. Gravity and acceleration being equivalent (as in the G. T.) would then seem perfectly natural. Of course, for this to make sense, the displacement should somehow depend on mass, not extension, as is the case. Furthermore, is doesn't explain why gravity attracts objects to each other as opposed to repelling them. This, therefore, is probably a dead end.
Alternatively, taking into account that, at the quantum scale, the difference between 'objects' and empty space is unclear, we can imagine that matter is composed of the same points as space, but denser. In other words, as there doesn't seem to be, at the quantum scale, a clear difference between 'something' (an observable 'object in nature') and 'nothing' ('empty' space), mass' could thus simply be a matter of the density of space-points. Using the mist analogy, it would then be possible for the apparent 'nothing' of space to condense into becoming observable and thus 'something'. The essential point is that matter would be of the same 'substance' as space. Space-point-singularities, though never observed, would be THE fundamental particle, composing both matter and space.
None of this explains most of the phenomena, such as the duality of electromagnetism, that we observe in the universe. Nonetheless, in a reversal of the scientific method, we are going from principles to consequences (as opposed to from observations to principles), so we simply don't care at this point.
Since we conjectured that our space-points desire an equilibrium, as if connected by rubber-bands, the closeness of points in matter would then create a pull upon the surrounding space points. The more numerous and denser the points, the stronger the pull. This would pull everything (space and other 'objects') towards it, again manifesting itself as 'gravity'.
Acceleration, could be understood as a compression of point-distances in the direction of acceleration, both of the space ahead and of the points composing the object itself, thus creating a pull behind it. This elastic resistance to acceleration would manifest itself as inertia. But once acceleration ceased, an equilibrium would again be reached with this “compressed space” object continuing to smoothly glide along, wavelike, through regular space.
An interesting aspect of this analogy is that, one single process, an increase in 'density', would be behind the formation of everything with mass, from the quantum scale to the galactic one.
Imagine a cluster galaxy. From a very great distance, it is observable as a luminous point-like object. If one zooms in, it becomes fuzzy, the closer, the fuzzier it becomes. Able to observe galaxies quite closely, we know that a galaxy is not really one single 'thing'. It is only perceived as such on a huge scale. At a closer range, it is a process. It is hydrogen, under the influence of gravity, condensing into gigantic hydrogen clouds, then stars then clusters of stars etc. At the galactic scale, a hydrogen atom is so completely unobservable as to be nothing. However, with enough of them condensing into structures, these 'nothings' give rise to stars and eventually to phenomena that become observable at the galactic scale.
Let us alternatively imagine that we cannot see the light of galaxies, only their mass. And let's say that we can only perceive great concentrations of mass, anything below a certain threshold being stuck in uncertainty. The only things that we would consider to be 'objects' would thus be great concentrations of mass. A major part of the mass of a galaxy would be found at the core (at and around the giant central black-hole). So we would definitively perceive this core as an 'object'. Where it get interesting is what happens when we try to determine things about our galaxy beyond that core. There would be enough mass there for us to perceive an object. But the location of that perceived 'object' would be the weirdly uncertain centre of mass arising from hundreds of millions of unobservable interacting clouds, stars and planets. It would be a strangely behaving 'object'.
Through an admittedly clumsy analogy, we can imagine some strangeness to arise from the large scale difference between an observable 'actual object', and its unseen constituent parts. It would all be just a question of scale. And it would make things incredibly simple, as similar processes are at work at all scales (its like zooming into a Mandelbrot set).
Whereas, at the quantum scale we believe that we had reached the limits of the small (fundamental particles, like electrons, are considered to have 0 extension in space), we might be referring to things that are galactically large compared to the actual fundamental components of the universe.
Trying to describe quantum-scale phenomena might thus be like trying to deduce the formation of a galaxy without the hydrogen that explains the birth of stars. Trying to explain the formation of galaxies at the galactic scale cannot be done. At a scale where the largest of stars is so minute as to barely be anything at all, 'objects' seem to condense out of nothing for no causal reason.
We can imagine that a space-point-singularity, compared to even the 0 sized “elementary particles”, is of a scale of a hydrogen atom compared to a galaxy. The number of space-point-singularities required to form even the tiniest 'thing' would, beyond astronomical, be almost inconceivable. At this scale, space-point-singularities would not even need to be massless. The mass of a hydrogen atom compared to that of a galaxy is tiny, but not zero. According to this analogy, large chunks of “empty” space would thus have a significant mass. There would be a need for neither dark matter and nor dark energy.
