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Infinity

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Infinity is the concept of the unlimited — of a magnitude, a number, a duration or an extension that has no end. In mathematics it is a working object, handled by rules laid down since Cantor and reproduced in every calculus and set-theory course. In physics it is something else: a quantity that has no meter reading, that no experiment returns, and that appears in the equations of the accepted theories in two very different roles. It appears as an idealisation, harmless and convenient — the infinite straight wire, the infinitely massive source, the limit taken as a variable grows without bound. And it appears as a result, where a theory that was supposed to describe something finite produces a divergence instead: the point-mass singularity of general relativity, the moment of the Big Bang, the infinite self-energy of the electron, the infinite energy density of the zero-point field.

Among the researchers documented on this wiki, infinity is contested ground in both directions at once. Many of them hold that physics has wrongly refused an infinity that is real — the infinite extent and infinite age of the universe — while simultaneously tolerating infinities that are not real, the divergences that arise inside its own equations and are then removed by hand. Others take the opposite view on one or both counts, arguing for a finite closed space, or for a lower limit to divisibility below which no infinity of the small exists. What unites the collection is that infinity is not treated here as a technicality. It is treated as the assumption that decides what kind of universe is being described, usually before any observation has been consulted.

Potential and actual infinity

The distinction that runs under most of these disputes is the classical one between potential and actual infinity. Aristotle allowed the first and denied the second: a series may be extended without ever reaching an end, and a magnitude may be divided further and further, but no completed infinite totality exists as an object. Cantor's set theory in the nineteenth century made the actual infinite a legitimate mathematical object, and mathematics has proceeded on that basis since.

The positions taken on this wiki cut across that line in ways that do not line up neatly. Glenn Borchardt insists on an actual infinity in nature — infinite in both the large and the small directions, and infinite in time — and treats it as the foundational assumption of his whole programme. Stavros T Tassos goes further, holding that "finite does not exist in any physical sense" and that infinity is what he calls the ultimate material actuality. Peter F Erickson accepts absolute space and absolute time but denies the actual infinite in the small altogether: on his account space bottoms out in indivisible elements and "there is no microscopic infinity." Thomas E Phipps leaves the physics aside and attacks the mathematical handling of infinite processes itself, arguing that the Cauchy conception of the value of an infinite series is a convention rather than a discovery.

Antonino Drago made the underlying methodological point explicitly in "The Two Options Generating Incommensurability Among Scientific Theories" (1989). Physics is normally assumed to rest on the rigorous mathematics of Cauchy, Weierstrass and Dedekind as though no alternative existed; Drago argues that the several available foundations of mathematics, and of infinitesimal analysis in particular, make different formulations of one and the same classical physical theory essentially different from each other, and thereby generate incommensurability inside classical physics and not only across the classical-modern divide. On that reading, a choice about infinity has already been made before the physics starts.

Infinity of the universe

Main article: Infinite Universe Theory

The largest single body of work here concerns whether the universe is infinite in space and eternal in time. The dominant position in the collection is that it is, and that the finite, originating universe of standard cosmology is an inherited assumption rather than an observational result.

The most fully developed version is Glenn Borchardt's Infinite Universe Theory, set out in a 2007 paper for the Natural Philosophy Alliance and expanded into a book in 2017; it is treated at length in its own article and only summarised here. Borchardt's argument is that Big Bang cosmology presumes finity and Infinite Universe Theory presumes infinity, that neither assumption can be proven, and that physics adopted the first without noticing it had chosen. Once infinity is granted, he argues, an origin becomes incoherent and cosmogony — the study of the origin of the universe — has no object at all. He put the case against cosmogony directly in "Ten Assumptions of Science and the Demise of Cosmogony" (2004) and "The Scientific Worldview and the Demise of Cosmogony" (2007), the latter opening with the complaint that "the absurd idea that the universe exploded out of nothing is a commonplace among today's mathematicians, cosmologists, astronomers, and physicists." With Stephen J Puetz he extended the picture to a theoretically infinite sequence of nested cycles in "Unified Cycle Theory: Integration Toward a Cause" (2010); Puetz's own "The Unified Cycle Theory: Introduction & Data" (2010) proposes that the longest of these cycles originate outside the observable universe altogether.

The same conclusion is reached here from several independent directions. Lee Coe argued in "Inertial Gravity and Cosmology" (1988) that if inertial effects are gravitational and the laws of nature are the same everywhere, then the universe is probably infinitely great, not expanding, and neither open nor closed. Richard A Waldron's "An Infinite Non-Expanding Universe in Dynamic Equilibrium" (1991), published posthumously from an abstract he did not live to deliver, proposed a universe in dynamic equilibrium rather than expansion. James B Wright treated the infinite as evidential rather than assumed in "The Infinite and the Eternal in Cosmology, from the Evidence" (2000), and in "Cosmology-Galactic Renewal" (2009) described the observable universe as "a tiny sample of an infinite ocean of galaxies" engaged in continuous recycling of galactic debris into new galaxies, with gravity as the motor.

Constantin Antonopoulos attacked the temporal side of the question logically rather than observationally in "A Bang into Nowhere" (2003): a first ever moment of time, lacking any moment before it, lacks a lower barrier and so recedes inconsistently into an infinitely remote past. On his account there can be no beginning of time, only a beginning in time.

Other infinite or eternal cosmologies collected here include Peter F Browne's "Cosmology Based on a Hierarchy of Finite Isolated Systems in an Infinite Cosmos" (2000), which places finite closed subsystems inside an unbounded whole; Eit Gaastra's "An Astronomy Model within an Infinite Universe" (2004), edited by Cynthia Kolb Whitney, in which stars in a universe infinite in space and time cycle repeatedly between Population I and Population II states; Henry P Dart's photon-decay alternative to the Big Bang (1993); Arnold G Gulko's Universe Cycle theory (2008); Mitch Emery's closed-loop cyclic model (2010); Mogens True Wegener's survey of cosmological models (2002); and Bob Ticer's observation in "The Cosmic Coincidence" (2012) that once Einstein had inserted the cosmological constant, Friedmann was able to conclude that an infinite number of cosmological models were possible.

Stavros T Tassos arrived at the question from geophysics. His "An Evolutionary Earth Expansion Hypothesis" (1994) opens by saying plainly that a seismologist attempting to answer whether the universe is finite or infinite, static or dynamic, open or closed, may gain intuitive insight into problems closer to home. The answer he developed, with David Ford, is Z-infinity Space (2005): space is the necessary infinite source of all mass, "finite does not exist in any physical sense," and zero, vacuum and empty space are not physical entities at all. He restated the framework in "The Solid, Quantified, Growing and Radiating Earth" (2007).

Dissent within the dissent

Not everyone here argues for an infinite universe, and the disagreements are direct. Tuomo Suntola's Dynamic Universe is built on the opposite premise: space as the spherically closed three-dimensional surface of a four-dimensional sphere, finite but without edges, in a zero-energy balance between motion and gravitation. His "New Cosmology Model Shows Relativity in Universal Time and Distant Observations in Euclidean Geometry" (2001) states the case for finitude as the natural one, and "From Local to Global Relativity" (2008) contrasts it explicitly with the Newtonian picture of a space that is "Euclidean until infinity" with no overall limits to physical quantities. Gerardus D Bouw proposed in "A New Look at the Aether" (1987) a plenum-type aether — a firmament — finite in extent and in time but physically behaving as though it were infinite. Robert Marion LaFollette and Lou Ellen LaFollette began their 2011 paper from the declaration that theirs "envisions a universe which is finite and competitive rather than infinite and harmonious."

Infinite divisibility and the very small

The second front is the infinitely small: whether matter, space and time can be divided without limit, or whether division terminates.

Peter F Erickson has given the question more sustained attention than anyone else in the collection, in a series of papers running from 2005 to 2012 and in the book Absolute Space, Absolute Time, & Absolute Motion (2006). His doctrine is that space consists of spatial infinitesimals — points of location, without area, shapeless, indivisible, continuous in all directions — and that there is accordingly no microscopic infinity. Time likewise consists of infinitesimals, the instants, each discrete yet leaving no gap. From this he argues that irrational magnitudes within the unit, asymptotes and "infinite series" become comprehensible rather than mysterious, and in "The Hidden Opportunities in the Derivative" (2007) that the limit concept applied to the derivative conceals the problem that the derivative usually differs from dy/dx instead of solving it. In "George De Bothezat's Teaching on the Infinitesimal" (2012) he examined an earlier statement of the same doctrine in de Bothezat's Back To Newton.

The atomism-versus-continuum question is put in its long historical frame by David L Bergman in "Atoms and Void" (1999), which traces twenty-five centuries of argument over the nature of matter and space and the weakness of the deductive methods brought to bear on it. Bert Schreiber reduced it to a single fork in "Time" (2005): the problem, as he traces it back to Pythagoras, is whether time and length come in granular bits or are continuous, and his charge is that later scientists had a fifty-fifty chance and chose wrong.

An opposite tendency runs through the aether papers, where infinite divisibility becomes an infinite regress of scale. William R Jones's "Gravitino Ether Hypothesis" (1970) fills space with tiny high-velocity particles, and then fills the space between those particles with a sub-aether of sub-gravitinos, and so on downward; he developed the infinite aether further in "How the Ether Replaces Relativity" (1987) and "The Infinite Aether" (1991). Charles Kenneth Thornhill derived the Planck black-body distribution from a gas-like aether containing "an infinite variety of particles" whose masses are integral multiples of a unit particle mass (1983). Robert L Stilmar characterised the aether of the quantitative analogies as incompressible, frictionless and "infinitely polarizable" (1998), and Robert A Kerr argued that the digital nature of light requires "a virtually infinite population of particulate digits" to account for visual acuity (2000).

Ivor Catt made the practical objection to infinitesimals in engineering. "Signal Transmission in Digital" (1969) notes that all the literature treats a transmission line as a series of infinitesimally small inductors and capacitors, and offers instead an approach in which one never has to consider infinitesimally small segments of the line at all.

Singularities and infinities in physical theory

The most concentrated criticism in this collection is directed at infinities that arise inside accepted theory. The common charge is that when a calculation returns an infinite result, the theory has failed at that point; treating the infinity as a feature of nature, or removing it by a technique invented for the purpose, substitutes bookkeeping for physics.

Stephen John Crothers has pressed this case against general relativity more insistently than anyone here. His argument is that the so-called Schwarzschild solution is not Schwarzschild's solution at all but a corruption of the Schwarzschild-Droste solution due to David Hilbert in December 1916, in which the quantity r is wrongly taken to be a radius that can be run down to zero, producing an infinitely dense point-mass singularity that the original solution never contained. On his reading the Kruskal-Szekeres coordinates do not extend the solution or remove a coordinate singularity but generate counter-examples, and the black hole and the Big Bang alike are artefacts of an incorrect analysis rather than predictions. He set out the general solution for the point-mass in a series of papers from 2005 onward and drew the cosmological consequences in "The Black Hole, the Big Bang: A Cosmology in Crisis" (2010), which disputes the claim that cosmology became a genuine science with general relativity.

The quantum side of the objection appears in several forms. David W Talmage and Richard J Sanderson proposed in "Energy is Everything" (2002) to avoid "the infinities that have plagued previous attempts to quantize gravity" by changing the causal sequence through which the quantum field produces attraction. Alexander L Kholmetskii set out a classical electrodynamics of point-like charges without divergences (2006), attacking the problem at its source rather than renormalising it. Peter F Browne took the reverse approach in "Newtonian Cosmology with Renormalized Zero-Point Radiation" (1994), showing that the infinite energy density of zero-point radiation can be renormalised to a finite value by including gravitational self-potential energy, and extended the same gravitational renormalisation to the self-energy of the elementary particle in "Universes, Black Holes and Elementary Particles" (1994). Harold E Puthoff treated the divergent vacuum energy as a resource rather than an embarrassment in "The Energetic Vacuum" (1990). Friedwardt Winterberg made the broader complaint about the standard model in "Elementary Particle Physics: Science or Dogma?" (1998).

Charles William Lucas listed the symptom alongside its companions in "The Trouble with Modern Physics and the Solution Based on Logic and Metatheory" (2016): modern physics, he argues, "has many problems with infinities, dark matter, dark energy, black holes, too many adjustable parameters, and logical inconsistencies," and the trouble is traceable to postulates known experimentally to be false rather than to any shortage of mathematical ingenuity. Compare the parallel debates on Dark Matter and the Big Bang.

A distinct and more technical objection concerns infinities introduced as idealisations. Jan Olof Jonson has returned to this repeatedly. In "The Electromagnetic Force between Two Parallel Electric Currents of 'Infinite' Length" (2010) he observes that Ampère's famous case is not physically possible at all — every current must be guided back to its origin — and analyses a real circuit in its place using Coulomb's law. In "Implications of Infinite Current Densities at Idealized DC Generator Poles" (2008) he takes the appearance of infinite current densities at idealised generator poles as a result to be explained and applied rather than dismissed, and in a 2016 paper he addressed the standard claim that exceeding the speed of light would require infinite energy. Thomas E Phipps made a related point in "Inertial Modulation of Electrodynamic Force" (1997): the full formula force of Newton's second law exerts observable ponderomotive action on a test element only if the force-exerting element is infinitely massive, which is to say never exactly.

Infinity in mathematics and its use in physics

Several researchers here argue that the difficulty is upstream of physics, in the mathematics of the infinite itself, and that infinities have been imported into physical reasoning on mathematical authority alone.

Thomas E Phipps wrote the most technical treatment in the collection, the three-part series "On Infinite Process Convergence" (1993). His starting point is that renormalisation — the physicist's habit of overcoming a divergence by subtracting it out — can be generalised by reconceiving what the value of a discrete infinite process means. The standard conception, due to Cauchy, approaches the value by successive partial sums; Phipps's method of terminal summation instead uses an asymptotic approximation to the remainder term at each stage, which speeds convergence for convergent series and forces a value for series divergent in the Cauchy sense. The third part applies the same difference-equation viewpoint to continued fractions. He pursued the argument in "Divergence: What to Do Till the Mathematician Comes" (2001), where he proposes that definitions be chosen a posteriori to reflect mathematical existence rather than a priori to prove it — the inversion, on his account, that the question of divergence requires.

Velimir Abramovic went directly at Cantor. In "How 'Many Infinities' Are There in Mathematics?" (2008) he reports that reading the diagonalisation argument convinced him that nothing in it can be taken for granted and that it must be walked through statement by statement and symbol by symbol in the classical manner. His "Introduction to the Ontology of Time" (2008) makes infinity ontological rather than quantitative: time, on his account, is "a fundamental, real, and unique infinity," and unique precisely because it is non-spatial and inextensive.

Jamie Rose located the obstacle in the same place in "Surmounting Unwanted Infinities" (2010), arguing that a holistic account of how the universe organises its many scales is blocked by foundational misunderstandings within mathematics rather than by any shortage of physical data. Peter Ripota's "The Garden Fence Paradox" (2008) is a puzzle built on an infinitely long fence, of the kind used to test whether reasoning about completed infinities is trustworthy. Michael H Brill worked a legitimate infinite limit carefully rather than polemically in "Infinite-Rydberg Limit of the Hydrogen Atom" (2011), solving the radial Schrödinger equation at E = 0 for the limiting wavefunctions as the principal quantum number grows without bound. Marvin Eli Kirsh approached the theme philosophically in "Evolution at the Surface of Euclid: Elements of a Long Infinity in Motion Along Space" (2011).

Infinities also turn up as an argument against relativity. Shao-Zhi Xu and Xiang-Qun Xu showed in "A Reexamination of the Lorentz Transformation" (1992) that there exists an infinity of linear transformations comparable to and including the Lorentz transformation, so that the Lorentz form is not singled out by the conditions usually imposed on it. Robert S Neiswander defined simultaneous events in "Simultaneity, Absolutely" (1996) as spatially separated events connected by an infinite-velocity signal, and traced the consequences of special relativity's refusal of such signals. Michael Jefferson Lawrence argued in 2013 that the familiar relativistic velocity-addition formula is only the two-variable case of a formula admitting an infinite number of variables.

Not all of the mathematical scepticism here is aimed at professionals. Pal Asija, who does not have a biography page on this wiki, presented a long sequence of talks to the Natural Philosophy Alliance on the counter-intuitive behaviour of the infinite, including "From Zero to Infinity and Back in Zero Time" (2010) and "Math Myths & Mysteries" (2011), the former arguing for a unity of zero, one and infinity.

Infinity, time and eternity

A smaller group treats infinity as a question about time and existence rather than about extent. Velimir Abramovic's ontology of time belongs here as much as to the mathematics. Bernard Guy argued in "Penser ensemble l'espace et le temps" (2010) that space and time should be founded not as substances with characters of their own but relationally, each defined in opposition to the other. John Linus OSullivan divides energy into two kinds in "Space and Time" (2013) and "Electromagnetic Wave Energy and Time" (2015): finite energy, which is mass energy and carries time, and infinite energy, which is energy without time, without beginning or end — on which account the question of how the universe came into existence cannot properly be asked. Helmut Hansen looked for archetypal structure behind the space and time concepts (2007), and David G Yurth traced self-organizing criticality "from the zero point to infinity" (2006).

Researchers on this wiki

Researchers whose work in this collection engages the question of infinity directly. Names are grouped by the aspect they chiefly address; several appear in more than one debate.

Infinity and eternity of the universe

The finite and closed alternative

Infinite divisibility, the infinitesimal, the very small

Singularities, divergences and renormalisation

Infinity in mathematics

Infinity, time and philosophy

Pal Asija, a frequent NPA presenter on zero, unity and infinity, has no biography page on this wiki.

Papers on this wiki

A selection from the papers in this collection that bear on infinity, grouped by theme. Many more mention it in passing; searching the wiki for infinite, infinity or infinitesimal will turn up well over two hundred.

Infinity and eternity of the universe

The finite and closed alternative

Infinite divisibility, the infinitesimal and the aether

Singularities, divergences and renormalisation

Infinity in mathematics and its use in physics

Infinity, time and philosophy

See also

External links

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