In the aftermath of the dispiriting comments following last week’s post on the Parapsychological Association, it seems worth spelling out in detail the claim that parapsychological phenomena are inconsistent with the known laws of physics. The main point here is that, while there are certainly many things that modern science does not understand, there are also many things that it does understand, and those things simply do not allow for telekinesis, telepathy, etc. Which is not to say that we can prove those things aren’t real. We can’t, but that is a completely worthless statement, as science never proves anything; that’s simply not how science works. Rather, it accumulates empirical evidence for or against various hypotheses. If we can show that psychic phenomena are incompatible with the laws of physics we currently understand, then our task is to balance the relative plausibility of “some folks have fallen prey to sloppy research, unreliable testimony, confirmation bias, and wishful thinking” against “the laws of physics that have been tested by an enormous number of rigorous and high-precision experiments over the course of many years are plain wrong in some tangible macroscopic way, and nobody ever noticed.”
The crucial concept here is that, in the modern framework of fundamental physics, not only do we know certain things, but we have a very precise understanding of the limits of our reliable knowledge. We understand, in other words, that while surprises will undoubtedly arise (as scientists, that’s what we all hope for), there are certain classes of experiments that are guaranteed not to give exciting results — essentially because the same or equivalent experiments have already been performed.
A simple example is provided by Newton’s law of gravity, the famous inverse-square law. It’s a pretty successful law of physics, good enough to get astronauts to the Moon and back. But it’s certainly not absolutely true; in fact, we already know that it breaks down, due to corrections from general relativity. Nevertheless, there is a regime in which Newtonian gravity is an effective approximation, good at least to a well-defined accuracy. We can say with confidence that if you are interested in the force due to gravity between two objects separated by a certain distance, with certain masses, Newton’s theory gives the right answer to a certain precision. At large distances and high precisions, the domain of validity is formalized by the Parameterized Post-Newtonian formalism. There is a denumerable set of ways in which the motion of test particles can deviate from Newtonian gravity (as well as from general relativity), and we can tell you what the limits are on each of them. At small distances, the inverse-square behavior of the gravitational force law can certainly break down; but we can tell you exactly the scale above which it will not break down (about a tenth of a millimeter). We can also quantify how well this knowledge extends to different kinds of materials; we know very well that Newton’s law works for ordinary matter, but the precision for dark matter is understandably not nearly as good.
This knowledge has consequences. If we discover a new asteroid headed toward Earth, we can reliably use Newtonian gravity to predict its future orbit. From a rigorous point of view, someone could say “But how do you know that Newtonian gravity works in this particular case? It hasn’t been tested for that specific asteroid!” And that is true, because science never proves anything. But it’s not worth worrying about, and anyone making that suggestion would not be taken seriously.
As with asteroids, so with human beings. We are creatures of the universe, subject to the same laws of physics as everything else. As everyone knows, there are many things we don’t understand about biology and neuroscience, not to mention the ultimate laws of physics. But there are many things that we do understand, and only the most basic features of quantum field theory suffice to definitively rule out the idea that we can influence objects from a distance through the workings of pure thought.
The simplest example is telekinesis, the ability to remotely move an object using only psychic powers. For definitiveness, let’s consider the power of spoon-bending, claimed not only by Uri Geller but by author and climate skeptic Michael Crichton.
What do the laws of physics have to say about spoon-bending? Below the fold, we go through the logic.
- Spoons are made of ordinary matter.
This sounds uncontroversial, but is worth explaining. Spoons are made of atoms, and we know what atoms are made of — electrons bound by photons to an atomic nucleus, which in turn consists of protons and neutrons, which in turn are made of quarks held together by gluons. Five species of particles total: up and down quarks, gluons, photons, electrons. That’s it.
There is no room for extra kinds of mysterious particles clinging, aura-like, to the matter in a spoon. That’s because we know how particles behave. If there were some other kind of particle in the spoon, it would have to interact with the ordinary matter we know is there — otherwise it wouldn’t stick, it would just zip right through, as neutrinos zip right through the Earth nearly undisturbed. And if there were a kind of particle that interacted with the ordinary particles in the spoon strongly enough to stick to the spoon, we could easily make it in experiments. The rules of quantum field theory directly relate the interaction rates of particles to the ease with which we can create them in the lab, given enough energy. And we know exactly how much energy is available in a spoon; we know the masses of the atoms, and the kinetic energy of thermal motions within the metal. Taken together, we can say without any fear of making a mistake that any new particles that might exist within a spoon would have been detected in experiments long ago.
Again: imagine you have invented a new kind of particle relevant to the dynamics of spoons. Tell me its mass, and its interactions with ordinary matter. If it’s too heavy or interacts too weakly, it can’t be created or captured. If it is sufficiently light and strongly interacting, it will have been created and captured many times over in experiments we have already done. There is no middle ground. We completely understand the regime of spoons, notwithstanding what you heard in The Matrix.
- Matter interacts through forces.
We’ve known for a long time that the way to move matter is to exert a force on it — Newton’s Law, F=ma, is at least the second most famous equation in physics. In the context of quantum field theory, we know precisely how forces arise: through the exchange of quantum fields. We know that only two kinds of fields exist: bosons and fermions. We know that macroscopic forces only arise from the exchange of bosons, not of fermions; the exclusion principle prohibits fermions from piling up in the same state to create a coherent long-range force field. And, perhaps most importantly, we know what forces can couple to: the properties of the matter fields that constitute an object. These properties include location, mass, spin, and various “charges” such as electric charge or baryon number.
This is where the previous point comes in. Spoons are just a certain arrangement of five kinds of elementary particles — up and down quarks, gluons, electrons, and photons. So if there is going to be a force that moves around a spoon, it’s going to have to couple to those particles. Once you tell me how many electrons etc. there are in the spoon, and the arrangement of their positions and spins, we can say with confidence how any particular kind of force will influence the spoon; no further information is required.
- There are only two long-range forces strong enough to influence macroscopic objects — electromagnetism and gravity.
Of course, we have worked hard to discover different forces in nature, and so far we have identified four: gravitation, electromagnetism, and the strong and weak nuclear forces. But the nuclear forces are very short-range, smaller than the diameter of an atom. Gravitation and electromagnetism are the only detectable forces that propagate over longer distances.
Could either gravitation or electromagnetism be responsible for bending spoons? No. In the case of electromagnetism, it would be laughably easy to detect the kind of fields necessary to exert enough force to influence a spoon. Not to mention that the human brain is not constructed to generate or focus such fields. But the real point is that, if it were electromagnetic fields doing the spoon-bending, it would be very very noticeable. (And the focus would be on influencing magnets and circuits, not on bending spoons.)
In the case of gravitation, the fields are just too weak. Gravity accumulates in proportion to the mass of the source, so the arrangement of particles inside your brain will have a much smaller gravitational effect than just the location of your head — and that’s far too feeble to move spoons around. A bowling ball would be more efficient, and most people would agree that moving a bowling ball past a spoon has a negligible effect.
Could there be a new force, as yet undetected by modern science? Of course! I’ve proposed them myself. Physicists are by no means closed-minded about such possibilities; they are very excited by them. But they also take seriously the experimental limits. And those limits show unambiguously that any such new force must either be very short-range (less than a millimeter), or much weaker than gravity, which is an awfully weak force.
The point is that such forces are characterized by three things: their range, their strength, and their source (what they couple to). As discussed above, we know what the possible sources are that are relevant to spoons: quarks, gluons, photons, electrons. So all we have to do is a set of experiments that look for forces between different combinations of those particles. And these experiments have been done! The answer is: any new forces that might be lurking out there are either (far) too short-range to effect everyday objects, or (far) too weak to have readily observable effects.
Here is a plot of the current limits on such forces, from the Eot-Wash group at Julianne’s home institution. This particular plot is for forces that couple to the total number of protons plus neutrons; similar plots exist for other possible sources. The horizontal axis is the range of the force; it ranges from about a millimeter to ten billion kilometers. The vertical axis is the strength of the force, and the region above the colored lines has been excluded by one or more experiments. On meter-sized scales, relevant to bending a spoon with your mind, the strongest possible allowed new force would be about one billionth the strength of gravity. And remember, gravity is far too weak to bend a spoon.
That’s it. We are done. The deep lesson is that, although science doesn’t know everything, it’s not “anything goes,” either. There are well-defined regimes of physical phenomena where we do know how things work, full stop. The place to look for new and surprising phenomena is outside those regimes. You don’t need to set up elaborate double-blind protocols to pass judgment on the abilities of purported psychics. Our knowledge of the laws of physics rules them out. Speculations to the contrary are not the provenance of bold visionaries, they are the dreams of crackpots.
A similar line of reasoning would apply to telepathy or other parapsychological phenomena. It’s a little bit less cut and dried, because in the case of telepathy the influence is supposedly traveling between two human brains, rather than between a brain and a spoon. The argument is exactly the same, but there are those who like to pretend that we don’t understand how the laws of physics work inside a human brain. It’s certainly true that there is much we don’t know about thought and consciousness and neuroscience, but the fact remains that we understand the laws of physics in the brain regime perfectly well. To believe otherwise, you would have to imagine that individual electrons obey different laws of physics because they are located in a human brain, rather than in a block of granite. But if you don’t care about violating the laws of physics in regimes where they have been extensively tested, then anything does in fact go.
Some will argue that parapsychology can be just as legitimately “scientific” as paleontology or cosmology, so long as it follows the methodology of scientific inquiry. But that’s a slightly too know-nothing attitude to quite hold up. If parapsychologists followed the methodology of scientific inquiry, they would look what we know about the laws of physics, realize that their purported subject of study had already been ruled out, and within thirty seconds would declare themselves finished. Anything else is pseudoscience, just as surely as contemporary investigation into astrology, phrenology, or Ptolemaic cosmology. Science is defined by its methods, but it also gets results; and to ignore those results is to violate those methods.
Admittedly, however, it is true that anything is possible, since science never proves anything. It’s certainly possible that the next asteroid that comes along will obey an inverse-cube law of gravity rather than an inverse-square one; we never know for sure, we can only speak in probabilities and likelihoods. Given the above, I would put the probability that some sort of parapsychological phenomenon will turn out to be real at something (substantially) less than a billion to one. We can compare this to the well-established success of particle physics and quantum field theory. The total budget for high-energy physics worldwide is probably a few billion dollars per year. So I would be very happy to support research into parapsychology at the level of a few dollars per year. Heck, I’d even be willing to go as high as twenty dollars per year, just to be safe.
Never let it be said that I am anything other than open-minded.
Anyone who assumes that this “sustained record of negative results” is a given, might read the contravening evidence in the “Out of the Gates..” book by Damien Broderick who approaches this whole question empirically and comes to a quite different conclusion.
For follow-on reading, I also suggest revisiting “Flatland” by Abbott.
Then I suggest we all beef up our tolerance for ambiguity which almost certainly will be stretched in many ways over the next few decades while we watch a number of “spooky” things start to seem ordinary once explained.
over the next few decades while we watch a number of “spooky” things start to seem ordinary once explained.
Huh. Just curious, what “things” did you have in mind?
Spooky things = paranormalism, what else could it be?
The problem is that the the argument is being made that to dismiss these things is “closed mindedness.” This is a tactic used by creationists, who will charge that acceptance of evolution as the scientific operative theory for interrelatedness of species is “closed minded,” for this precludes their quasi-theory summed up in the first three chapters of Genesis.
It is not a matter of closed mindedness, it is just that certain things just don’t appear to function in the world, from UFOs, to angels, ghosts and … . If something pops up which demostrates something along these lines then I will pay attention, but I am not holding my breath.
Other forces — Sean outlines above measurements of departures from gravity at small distances, which might show up as a departure from the Einstein equivalence principle. So far no cigar, gravity appears well behaved or there are no extra forces. Why there are the forces in the universe which exist is a deep subject. It has something to do with maximal supersymmetry, orbifold compactifications, Calabi-Yau spaces and other matters a bit beyond the scope here. I am working on these things with so called sporadic groups, which are stranger than the exceptional groups employed for some string sectors. A particularly interesting sporadic group is called the monster group (Griess-Fischer group), which has 196884 dimensions with order ~ 8.8×10^{53}! It is the root extension of the Leech lattice in 24 dimensions which nicely embeds 11 dimensional supergravity. Maybe new forces or states of mass-energy will materialize sporadically or monsterously. But I offer no hope for any paranormal “physics” based on this.
Lawrence B. Crowell
Lawrence,
So would you say that time is the basis of motion, or an emergent description of it, such as is temperature?
Well,
Consciousness manifests as an emergent property of interactions of neurons. At what level does this compare to the interaction of individuals that is civilization. We are nodes in a network. Is the network an illusion, or do we lack the perspective to understand its function? Is our individuality one extreme of a larger spectrum of consciousness. As a younger child in a large family, these are not political issues for me, but personal ones.
A credit meltdown in the derivatives market?
28
Someone didn’t read the original post. Scratch that, a lot of people didn’t.
33
Yes… and the data all say there are no effects of that sort. If, you know, you use the data that was actually taken in controlled, double-blind situations. So honest parapsychologists gave up. As they say, “There’s no PEAR there.”
41
Yep. All the cases he could test. In fact, all the cases anyone could test for hundreds of years. And non-Newtonian effects are not on the human scale.
60
No, you are not. You cannot be a scientist if you don’t have a basic understanding of empiricism.
87
First, they don’t happen. Second, are you kidding? Physicists (and scientists in general) have the highest ambiguity tolerance I know of. Most people get to believe that they know things for certain. Scientists never have that luxury (and most I know wouldn’t even want it).
“Keeping an open mind is a virtue, but not so open that your brains fall out.”
— James Oberg
Your claim that “Spoons are made of ordinary matter” is very offensive, because the same arguments you make about spoons would equally apply to humans. And if humans are made of ordinary matter, without any extra magical particles, than how can we have an immortal soul?
Everybody take note….
PEAR did not die, but is now reincarnated as the International Consciousness Research Labs, Inc:
http://www.princeton.edu/~pear/
And, a spin-off will sell you a REG w/ops & analysis SW for those who desire a home test of the PSI abilities !
The only catch is the $350 samoleans they want for it:
http://www.psyleron.com/info/
Have fun & try not to levitate without a helmet !
@ anon #96:
Ehh.. are you deaf? Scores of animals make use of the medium air (or water) to communicate with sound waves…. perfectly consistent with known physical laws. But damn, the range is so limited… so why didn’t they switch to telepathic communication, just why?
Sean:” We can’t, but that is a completely worthless statement, as science never proves anything; that’s simply not how science works…
I think it’s useful to reserve the word “science” for the particular type of contingent, empirical knowledge about this actual world that we obtain through hypothesis testing, observation, and experiment. The type of logical truths revealed by mathematics (and amenable to proof) seem very different. There are obviously similarities, but the distinction is worth emphasizing — especially because too many people suffer under the misimpression that physics and biology actually do “prove” that certain things are true or false.”
I suffer under impression that you ask questions in form of mutually contradicting statements. The statement “science never proves anything” is deadly wrong. The science (physics) exists if and only if it is proved by the empirical and the mathematical evidence. It has nothing to do with the unconstrained imagination (land escape).
First of all, the physical theory must comply with all known experimental facts during last 2500 years (geometry included) without single exception. If you don’t consider that already prove, I have no further arguments. To achieve a progress, one should find the microscopic crack in the concrete wall and to demonstrate that it is a wide door into the new universe of knowledge. Only perfect mathematical structures are capable to do that and each step should be accompanied by the perfect mathematical prove (as in your example). Who make that is not important, it happens almost randomly. That one usually doesn’t understand much more than the others and his/her original motivation even often happens to be wrong. That leads to another criterion that we call the predictive power of the theory. It is not appearance of the artifacts that require addition effort to remove. It is something completely unexpected by anyone, something that we didn’t know before and will study from new math. Finally, that prediction must be unambiguously confirmed through the empirical verification.
What I say is supported de facto by the history of science, but I can’t “prove” that. Sorry if what I say make “cosmology” more close to parapsychology (using old fashion notion of the metric spaces) than to the physics (you may use JB addition/multiplication table for the calculations).
Regards, Dany.
P.S. I do not consider anything except physics to be a science… yet.
Science is not about proving anything, except that a theory is false. A theory generally predicts certain things about the world and we look for those predictions in observations and measurments. This may support a theory, and a good theory ends up with lots of data behind it. Yet, no theory is ever proven, which is one reason that classical physics used the world “law,” but modern physics does not.
What is interesting is that the Intellegent Design “theory” is a statement on the falsifiability of evolution. If something could be shown to be truly irreducibly complex then evolution is falsified and either needs modification or to be replaced or supplanted by a better theory. Of course this is not a theory, for a falsification thesis of a theory is not itself a theory.
The only domain of human endevour which involves proofs is mathematics. In science a theory is regarded as tentatively true, or true beyond a reasonable doubt — to use a legal concept. But in the future reasonable doubts might occur as new data comes in.
Lawrence B. Crowell
lawrence,
Let me rephrase an earlier question; Could temperature be considered a dimension?
Supposedly quantum activity can be described without time as a factor, yet wouldn’t it be impossible to describe any such activity that lacked some form of temperature?
Obviously temperature is a description of motion, but it seems that both space and time are also methods of describing motion. We describe time as a dimension because we reductionistically describe it as linear, yet is it? It would seem the effect of the passage of time is an effect of the relative interaction of particles, ie. a consequence of the activity described by temperature.
This raises the question of just what a dimension is. Are they fundamental, or a descriptive modeling tool based our own personal linear motion. A reductionistic description of some more pervasive sea of activity.
Temperature is not a “dimension” in the sense of space that objects exist in. Temperature is a parameter in the Mawell potential equations, so in an abstract sense it is a “dimension.” Temperature in the equipartition theorem is related to energy
E = 3/2kT
and energy is conjugate to time in quantum mechanics. Though this conjugation of energy/time is not on the same footing as momentum/position in a classical-quantum correspondence — but that is for another time.
Anything which can be called “space” is something which has a geometric interpretation. This may involve 11-dimensions for the space directions for gauge potentials we measure as affine connections on the base spacetime. Other parameters do not define direction in space(s) per se, though it has been said that a dog smells a thousand dimensional space. Yet I think few of us would consider odor as a property which defines a vector space direction.
Lawrence B. Crowell
Lawrence, your definition of space is a bit archaic. Now, whatever can be logically related in the way space is (even “pure information” that is arrayed in some form of matrix) can be considered a “space.” What makes “real space” of objects more “real” than such so-called abstractions? Nothing we know for sure, other than the relative means of access and representation. One could cogently argue that physical space is just a structure of relation that appears as “real space” to us because of how we access it (and supported by its plasticity in terms of Lorentz transformation, curved space, etc.) We access other data spaces, such as the visual representation inside our own brains, and this appears to us as a filed of view with content. Anyone not a naive realist appreciates, from e.g. focusing a telescope and finding their image of the moon going sharp and blurry (the moon isn’t doing that!) that the world is represented to us, not given or shown as such. It has even been suggested that time isn’t fundamentally real, but our experience of it is a relative product of brain configurations (in the “4-d structure” of world lines) relationship to the world lines around us.
(PS – I know you said, “anything … which has geometric interpretation” but then you apparently wouldn’t accept space-like representational schemes, and momentum, temperature etc. “spaces” are really spaces IMHO per relational definition, they just don’t relate to us in the same way as general space.)
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Lawrence,
The point is whether time is a dimension, “in the sense of space that objects exist in.” Or whether it is a method of measuring the activity of those objects. I was using temperature to put the question in context. Energy is what is conserved across time, therefore it doesn’t remain in the past, or it wouldn’t be conserved. So it would seem energy is the reality and time is the measure of states this energy proceeds through.
Several threads ago, I was in a discussion with Jason as to whether space is properly three dimensional, or whether that is the coordinate system of a specified point and that space as such, is infinitely dimensional.
One of the problems I have with the geometric description of space is that it doesn’t seem to properly account for zero. Points, lines and planes are said to have zero dimension, yet anything by zero is zero, so it would seem what they really have is a virtual dimension. Instead of applying zero to any specific geometric reference, zero would be the potential, not the actual reference. Thus for geometry, zero would be empty space.
Tell that to a hound.
Would you say that dimensions are the essential nature of space, or a model of it? We model time as linear, but is that due to modal bias? Would a plant model time as linear, or as cyclical expansion/contraction of energy? Which brings us back to temperature.
Space or spacetime are ultimately models. We can’t take them that seriously as something ontological. A quantity is geometric if it fits within certain mathematical models which describe a “space.” This can involve Kummer surfaces with
$latex
(x^1~+~y^2~+~x^2~+~w^2)^2~=~lambda pqrs
$
for pqrs functions of xyzw. BTW, this is an elementary form of a K3 space which are associated with Calabi-Yau spaces.
Space and time are kinematical quanitites really, even with general relativity as the “dynamics of space.” Ultimately these are of value if they tell us something about the dynamics of particles and things that cause a detector to go “bing!”
Lawrence B. Crowell
Lawrence,
So the Max Tegemark type hypothosis, that the equations and their concepts and consequences are the reality, is mistaking the map for the territory?
Max Tegmark’s physics = math idea is something we can’t ever know about anyway. Mathematics is probably infinite, and we will never know it all. So the idea is a bit outside physics and more metaphysics. There is a strange relationship between physics and mathematics, but I suspect we will never understand it. I think understanding something of this sorts is outside of empiricism, and I suspect it is something we can only ponder over pitchers of beer, leading to then scotch and cigars.
Lawrence B. Crowell
Lawrence,
You are far more tuned into all of this, but I don’t see the mystery in math. It’s a model. The more complex our understanding of reality, the more complex the models we use to describe it. Sometimes the mathematical relationships seem almost magical in the inter-relationships, but how is that any different from all the strange happenings in reality. If we don’t kill ourselves off, the potential for human development is only just beginning. How long before we reach the stage of conscious understanding that biological evolution reached eons ago?
I do think that the physics establishment has developed a hard crust it is going to have to eventually shed, in order to expand to the next phase of insight. All the loose ends are not going to be tied up and never will be. The problem is that some of the knots need to be cut on occasion.
Math as biology, rather then mechanics. It’s a really big step.
At the core of this is the question, “What is mathematics?” Most mathematicians consider themselves realists, in that what they study has some aspect of reality. This might be the case, at least as some epistemological reality. Of course quantum mechanics has its epistemological interpretation of the wave function. Yet mathematicians can’t really prove somehow that mathematics is real in any sense. Proofs can be made about the relationships between mathematical “objects,” but proofs on the existential nature of these objects are not forthcoming. Godel’s theorems on the truth of unprovable propositions in mathematics touches on this in some ways, but to break out this would require a long post here.
When it comes to the relationship between physics and mathematics, a sports analogy might come in handy. We play billiards or baseball because the number of possible games is infinite, or nearly so. We play any particular game based on the configuration of events and the situation at the moment. Using mathematics as a model system in physics might be compared to playing a particular game, say a baseball game with a score 5-3 in the 6th inning, 2 runners on base, 2 outs and … . The mathematician might be concerned with the set of possible games, say the set of possible configurations on a billiards table. Obviously the two disciplines overlap in their practice, but I think the approaches are in many ways dissimilar.
Remember that with physics the game of using mathematics to find involutory systems as models of conservation laws and … , is a human endevour. In effect we are imposing these structures we borrow from mathematics, or maybe even research out new mathematics. But nature does not necessarily operate by physical laws. What are laws? In many ways these are model systems we employ to make some logical connections with empirical science. Yet nowhere do we ever prove that nature “obeys” any physical principle or what used to be called laws. Mathematicians are faced with a sort of converse issue or unknown. Mathematicians may never be able to prove that anything about mathematics is anything more than a formal game. I don’t tend to think that is the case, but I also doubt any proof to the converse may ever be found.
There appear to be some metaphysical questions which we just might never know the answer to. But that doesn’t keep us from pushing on with scientific discovery.
Lawrence B. Crowell
Lawrence B. Crowell:” But nature does not necessarily operate by physical laws. What are laws? In many ways these are model systems we employ to make some logical connections with empirical science. Yet nowhere do we ever prove that nature “obeys” any physical principle or what used to be called laws.”
I guess you are not familiar with:
1) A.Einstein, “Zur Elektrodynamik der bewegter Körper”, Ann. Phys., 17, 891 (1905);
2) E.Schrödinger, “The present situation in Quantum Mechanics”, Die Naturwissenschaften, 48, 807 (1935); 48,823 (1935); 48,844 (1935);
3) E.P.Wigner, “Unitary Representations of the Inhomogeneous Lorentz Group Including Reflections” (1962).
By the way, they are considered the greatest math-ph of the 20-th century.
Regards, Dany.
Lawrence,
Nature is law. The problem is that it presents a closed set that is constantly breaking down and regenerating. The definition of set gives it shape, but also confines it. Definition is limitation and limitation is definition. So there is this constant tension between being and non being. Matter and void. Contracting mass and expanding energy, like grass pushing through the concrete. Science tries to understand, but by doing so, falls on the closed set side of the equation and constantly runs up against the conclusion of finiteness…
The more certain we become, the harder we fall.
LOL. dude, you’re wasting your life… and for what?
anyways, i’ve never red such a “seriously” funny post. ever 🙂
plu,
If that’s addressed to me, I’m not wasting my life, I ride horses for a living. Most people have to pay lots of money to do that for fun. This is what I do for entertainment.