In this post we shall embark on our maiden voyage, asking the mother of all questions:
Are space and time fundamental aspects of the universe? In particular we shall focus on space, by investigating the standard model of particle physics from the perspective of philosophy of science. Our investigation will be carried out by use of intuition (inductive reasoning), logic (deductive reasoning), and a pocket calculator (arithmetic). Some mathematical analysis is necessary, but will be kept to a minimum in order to facilitate reading and because the author is a mathematical idiot. The meaning of individual ideas and treatments covered in this investigation should be easily understood, though a few do rely on (or are the product of) sophisticated derivation.
Let us then weigh anchor and set the sails of this ship of fools. We begin our voyage with the last recognized universal thinker, Henri Poincaré.
Logic necessitates intuition
Jules Henri Poincaré
“We believe that in our reasonings we no longer appeal to intuition; the philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies; it could create nothing new […]. In one sense these philosophers are right; to make arithmetic or […] any science, something else than pure logic is necessary. To designate this something else we have no word other than intuition.”^{(1)}
Mathematics and experimental observation are the standard investigative tools of theoretical physics. Poincaré believed that the use of intuition along with logical formalism is justified as an investigative method. I tend to agree, due to the fact that mathematics and formal logic have emerged from geometry, which is a product of philosophy, which ultimately is a product of intuition. It should therefore not come as a surprise that the sciences necessitate the use of intuition, yet in modern times inductive thought is viewed as the black sheep of science, outcast in favor of the logic machine.
Table 1. Wonky logic – an argument that adheres to the rules of formal logic but is never the less nonsense.
Natural form | Abstract form | |
Dogs are Pets | D = P | |
Cats are Pets | C = P | |
therefore Dogs are Cats | ∴ D = C |
These two forms are equivalent; the natural language form is obviously nonsense, yet its abstraction is mathematically correct, logically valid, sound^{(note 1)}, and true. Crucially, there is no logical way of finding and defining the fault in this obviously faulty argument.
Logic is a valuable tool, about as useful as a screwdriver or a roll of sellotape. However, when looking for truth, tools are no more important than observation and critical thinking. In the most successful of endeavors people use all three.
Measurement vs reality
“The map is not the territory”
Alfred Korzybsky
An orange tree grows n^{(note 2)} fruits but n does not equal the fruit. A quantification of n, including a description of the energy and mass as well as the physiological and environmental ‘currency’ involved in the production of the fruit is still not real fruit. We are thus forced to take n as a representation or model of the oranges which a tree has produced. Similarly, we may assume that space is continuous (infinite), though it may equally be said that space is deep (finite). We may assume there are 3 dimensions of space, or only 2 or even 1, just as well as there may be many (a branch of string theory called M-theory, for multiple dimensions theory, uses 9, 10, 11… dimensions of space).
An arbitrary number and kind of space dimension, is possible because the manner in which we measure space is arbitrary (by use of standardized but arbitrary units such as the meter, inch, parsec, cubit^{(note 3)}…). The time dimension is also measured in arbitrary units (month, second, light year, Káshthá^{(note 4)}…). Together, space and time are assumed to form spacetime and this arbitrary complex is taken as the fundamental universal fabric within which matter and energy interact, in accordance with an equivalence principle (E = mc^{2}).
Interestingly but not surprisingly, after publishing the general and special theories of relativity, Einstein returned to the idea of ether.
“[…] according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable matter, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.”^{(2)}
It is clear that although he could not devise a method of observation and measurement, Einstein had resolved his inductive thoughts about the fundamental nature of the universe; arbitrary spacetime is a good model but necessitates a physical (fundamental) ether.
The current standard model of particle physics describes four different interactions, each mediated by a fundamental particle (or characteristic set of fundamental particles). Though it could be argued that fundamentally all is energy, for the purpose of this discussion it seems worth noting that each of the four fundamental interactions in the standard model manifest in one of two ways, that is either as energy or as matter.
Table 2. Standard model of particle physics – four fundamental interactions, their mediating particles, types of measurable manifestation, and units of measurement.
Interaction and Force | Particle | Measurable manifestation | Unit |
Electromagnetism (visible and invisible light, electricity and magnetism) | Photon | energy | eV |
Gravitation (attraction of massive bodies, space-time curvature) | Graviton (not measured) | energy | eV |
Weak (nuclear radiation) | Boson | matter | eV |
Strong (holds atomic nuclei together) | Gluon | matter | eV |
Physicists use the electron-Volt (eV) as the standard unit of measurement for mass-energy (remember that mass and energy are interchangeable under special conditions, and thus viewed as fundamentally the same). Interestingly, the standard unit of measurement for space-time is the inverse electron-Volt^{(note 5)} (eV^{-1}). Because matter may be interpreted as a complex of energetic interactions of fundamental particles, we may assume that a change of state does not necessitate matter, though obviously matter can undergo change.
Is time fundamental? – or does it emerge from momentum?
Any change of state necessitates energy. Any action by (or interaction of) energy necessitates a period of time in which the action (interaction) occurs. We may thus assume that energy and time are linked in some intimate way, possibly even that time is simply change. For a detailed discussion of the arbitrary nature of time please see Rovelli (2008)^{(3)}.
Is space fundamental?
Experimental observations have shown that a change of energy state can occur in a point particle such as an electron or photon. As a point consists of no space dimensions, we may assume that space is not necessary in order for something to happen.
3-D space as an emergent phenomenon of point-particles
Before going any further and at risk of sounding pedantic, allow me to provide a glossary of symbols:
MeV/c^{2} is a unit of mass-energy which happens to be a convenient size to measure fundamental particles.
M stands for Mega (M = 1,000,000).
eV stands for electron Volts (1 eV = 1.60217653 ×10^{-19} joules) and is a measure of energy.
c stands for velocity of light in a vacuum (c = 299,792,458 meters per second).
Σ stands for sum (as in the sum of 1+1+1 is 3).
m stands for mass.
Let us now continue to assume that dimensions (and thus units) of space are fundamentally arbitrary but that they emerge from some fundamental physical mechanism. My own intuition is that space is an emergent phenomenon of matter, that is to say, matter creates the space in which it sits. Fascinatingly, science does not understand mass, which is assumed to be an emergent property; so matter may somehow create both, space and mass. More troubling is the fact that the mass of a proton (call this mP)^{(note 6)} is far greater than the sum of masses of its constituent quarks (call this ΣmQ)^{(note 7)}.
A quick calculation shows that ΣmQ accounts for just over 1% of mP.
((2*3.3)+6.0)/938.272013 MeV/c^{2} *100 = 1.3%
Arbitrary dimensioning of ΣmQ (assigning dimensions of space to mass-energy values) has the following fascinating results:
Cubing (ΣmQ^{3} = 2000.4 MeV/c^{2}) gives about 200% of mP.
Squaring (ΣmQ^{2} = 158.76 MeV/c^{2}) gives about 17% of mP.
By experimenting with a pocket calculator, we find that mP lies close to 3 dimensions of ΣmQ, at very nearly 2.7 dimensions (more about this later).
ΣmQ^{2.7} = ((2*3.3)+6.0)^{2.7} = 935.406694 MeV/c^{2}.
This is good because it fits quite nicely with our intuitive view of the physical universe, as occupying three dimensions of space. There is at least one problem though. Unitary analysis shows that the units (dimensions) are inconsistent. Allow me to explain.
Imagine that some physical entity is measured by esteemed scientists, using a shiny, new, sophisticated and expensive piece of apparatus. The value 1 z is found. Then some unrecognized whack-job with below average IQ comes along, toting a pocket calculator. Fiddling about with some ideas and numbers, fully ignorant of and consciously disregarding the customary doctrine(s), the idiot finds that the measured value 1 z is quite likely to be 1 z^{3}, which seems to make intuitive sense though nobody understands why it should be so.
By accepting the arithmetical treatment above, one must assume that the experimentally observed mass-energy value for the proton is a cubic value (or nearly so) and that the discrepancy between the sum of dimensionless constituent particles (quarks and gluons) and the proton is due to measurement and logical formalism devoid of critical intuitive thinking; i.e. observations were made and formal tools were used to measure a value, but inductive thought about the meaning inherent in the measured value was omitted.
Surprisingly, this kind situation is not at all strange. Modern people tend to not think quite often, preferring instead to rely on machines and on tried and tested doctrines. The trouble of course is that neither machines, nor doctrines, nor thoughts are infallible. Indeed the foundation of science is not proof, but disproof. So as with all aspects of science, experimental apparatii, doctrines, and the people who use them must be kept under constant scrutiny for error.
Assuming that ΣmQ^{2.7} is significantly close to the truth, must we then assume to live in a universe of 2.7 dimensions? Taking the dimensions of space to be arbitrary, this figure should not bother us, though intuitively it would be nicer to see the other 1/3 of a dimension accounted for. Perhaps the remaining 1/3 of a dimension is created through quark interaction?
– a kind of shared or common space?
Might it be that this sharing is what holds the quarks together (the gluon)?
– analogous to electron sharing in molecules?
The theoretical mass of the gluon^{(note 8)} (call this mG) = 0, but the experimental limit (whatever the hell that is supposed to mean?!) of mG < 20 MeV/c^{2}
Let us assume to extract mG from the remaining 1/3 dimension.
mG = (((2*3.3)+6.0)^{0.3}) / 3 gluons = 0.712836467 MeV/c^{2}
Assuming mG = 0.712836467 MeV/c^{2} and that 3 gluons bind the 3 quarks in the proton we should be able to say:
mP = (ΣmQ^{2.7} + ΣmG^{0.3}) = (935.406694 + 2.1385094) = 937.5452034 MeV/c^{2}
Though purely speculative, this mass value is tantalizingly close to the mass of the proton and we have obtained it by very nearly filling 3 dimensions. What we are missing is the reiterative 333’s because 1/3 = 0.333…
Interestingly, the missing mass value is similar to that of the gluon.
Missing m = (mP -(ΣmQ^{2.7} + ΣmG^{0.3})) = 0.7268096 MeV/c^{2}
A fresh cup of really hot tea
Intuition tells us that point-particles themselves do not exist as dimensions of space, they require only indeterminate locations (coordinates). Indeed, one can not rely on point-particles to create a line (1 dimension) because quantum theory tells that the precise location of a point is indeterminable! Even if one could somehow get the point-locations to line-up in an orderly fashion the collective state would be a mangled mess the next instant due to random fluctuations.
A much friendlier thought is that energetic point-particles fall in love, bond and live together, raising their masses in a familiar 3 dimensional house.
Bibliography and Notes:
Note 1) I have encountered arguments against the use of the example given here, on the grounds that it is logically unsound. It is noteworthy that soundness is essentially contextual, and that the contexts of the two examples given here, differ. It is this difference of context that creates the nonsense and unsoundness. Deeply interesting about this fact, is that science is rooted in abstractions, truncations, simplifications, models; changes of context.
Importantly, Gödel showed that within a logical system propositions can be formulated that are undecidable with the axioms of the system. Thus, there always exist statements that can be neither proved, nor disproved. Systems of logic are fundamentally uncertain and contradictory.
“It appears to foredoom hope of mathematical certitude through use of the obvious methods. Perhaps doomed also, as a result, is the ideal of science – to devise a set of axioms from which all phenomena of the external world can be deduced.”
and
“Provability is a weaker notion than truth”
– Douglas R. Hofstadter (in reference to Gödel’s theorem of incompleteness)
D.R. Hofstadter, “Gödel, Escher, Bach: an Eternal Golden Braid”, (1979), Basic Books Inc.
Note 2) The symbol ‘n’ is used in algebra to take the place of a variable number. In this case n = the number of fruits that a tree has produced.
Note 3) Earliest standard measure of space, 1 Royal Cubit = 7 palms. http://en.wikipedia.org/wiki/Cubit#The_Egyptian_Royal_Cubit_and_Sumerian_Nippur_cubit
Note 4) Hindu unit of time, 1 Káshthá = 10 blinks of the eye.
http://en.wikipedia.org/wiki/Hindu_units_of_measurement#Time
Note 5) Using scientific notation eV^{-1} is the same as 1/eV.
Note 6) mP = 938.272013 MeV/c^{2}
http://en.wikipedia.org/wiki/Proton
Note 7) ΣmQ = 12.6 MeV/c^{2}
mQ = U = 1.5 to 3.3 MeV/c^{2}
mQ = D = 3.5 to 6.0 MeV/c^{2}
A proton is made up of 3 quarks (2*U)+(1*D). The sum of highest theoretical mass values for the quarks in a proton = [(2*3.3) + 6.0] = 12.6 MeV/c^{2}
http://en.wikipedia.org/wiki/Quark#Mass
Note 8) mG < 20 MeV/c^{2}
http://en.wikipedia.org/wiki/Gluon
1) H.Poincaré, “Intuition and Logic in Mathematics”, (1905), translated by G.B.Halsted, (1907), published online, (2007), http://www-history.mcs.st-and.ac.uk/Extras/Poincare_Intuition.html
2) A. Einstein, “Ether and the Theory of Relativity”, (1920), address at University of Leiden (transcript), http://www-groups.dcs.st-and.ac.uk/history/Extras/Einstein_ether.html
3) C. Rovelli, “Forget Time”, (2008), http://www.fqxi.org/data/essay-contest-files/Rovelli_Time.pdf