The Paradox of Infinity

THE PARADOX OF INFINITY

PREFACE

infinity triangleFrom our observation of the physical reality, we may draw one inescapable conclusion: the everlasting quality in our Universe is impermanence, caused by the contingent and entropic nature of all material things in it. Any belief or act contrary to this fact of life, exposes itself to an illusion and a conflict with physical reality.

Aristotle states: ‘One may ask: What is physical reality; is it merely a conglomeration of physical substances recognised by their familiar attributes? Is it an image that our mind creates in our consciousness, as being immanent, (i.e. in-dwelling)? Alternatively, is it something outside ourselves that transcends beyond our limited understanding? The paradox at issue is whether our consciousness “contains more than it actually contains”.’ (Aristotle, Nicomachean Ethics, 1120b)

Could something, which is immanent and visible in the physical reality, simultaneously possess transcendent and metaphysical attributes? For example, when we look at a block of marble (Aristotle’s favourite thought) that has the potentiality of an immanent quality of a statue in it, we are at the same time able to visualise with our mind a transcendental idea of that very same statue.

The concept of non-material infinity is a transcendental idea of the purest form, and yet, this idea is represented through a wide range of diversities and of various types in our daily life. While some modern-day cosmologists attribute even physical sizes and quantities, as realities, to the concept of non-material infinity – at least by implication – its openly indiscriminate use (i.e. the actual, physically realisable, form of infinity) is gaining enthusiastic acceptance by the modern scientific community. Such ‘infinities’ are nowadays found everywhere, even in the Standard Cosmological Model, but the meaning of infinity has not been defined.

Our present-day, western concept of infinity came from Aristotle (384-322 BC), who was convinced that one could reach an absolute truth only by means of general notions, which he equated first with true theorems of mathematics. Aristotle reasoned that mathematics, with its abstract numbers, values, theorems and figures was leading to truth itself, which transcends the physical world of ideas. He introduced inadvertently, and to much of his astonishment, the metaphysical concept of (non-material) infinity, the ‘apeiron’.

Although Aristotle initially disliked the word apeiron immensely, he had to contend with it as it was the result of his very own logical thought process. However, he still maintained his dislike for the concept of actual infinity, as he stated:

‘There is something inherently absurd in the idea that an infinite task has been accomplished or an infinite distance traversed, for these suggestions force us to think in terms of an actual infinity of finite things, lying complete and whole in finite space and time.’

The controversial idea of both transcendental as well as actual infinity have appealed to many mathematicians, physical scientists and philosophers ever since, while others have rejected them both outright. While we may have a limited capacity to perceive transcendental ideas – through analogy at least – but never actual infinity, we still have no scientific proof for or against the physical existence of an infinity of any kind; hence cosmologists can only believe or disbelieve that infinities do exist.

INTRODUCTION

In both mathematics and philosophy, the concept of transcendent infinity is used mostly in its purely abstract form, where the operational need for actual (realisable) infinity would not ordinarily arise.

On the other hand, in the physical sciences, the concept of actual infinity in any shape or form would not often appear as a noun in isolation by itself; hence, one can find its use lately in its adjectival form only to describing various physical entities and events as being actually (erroneously) infinite.

How can one describe or define infinity? The conflict in describing any kind of infinity lies in trying to set a physical limit to it, which is a contradiction in terms. Also, without first knowing the essence of infinity – i.e. to form a mental image of it – one is unable to set limits to it; hence, we find ourselves in a vicious circle when trying to define infinity.

Galileo asserted that problems arise only ‘when we attempt, with our finite mind, to discuss the infinite, assigning to it those properties which we give to the finite and limited.’

While in this article I cannot resolve the present-day problems of infinity, I will nevertheless attempt to consider both – the transcendent infinity in mathematics and philosophy and the actual infinity in the physical sciences – in the light of the general usage of the terms, and according to the definitions of the Systematic Aristotelian Philosophy.

ACKNOWLEDGEMENTS

I used the following sources for this article:

  • The Computational Beauty of Nature, by G.W. Flake, in The World Treasury of Physics, Astronomy and Mathematics, edited by Timothy Ferris.
  • Out of Chaos, by Louis J. Halle.
  • Modern Philosophy, by Roger Scruton
  • The Philosophy of Physics, by Lawrence Sklar.
  • Infinity and the Mind, by Rudy Rucker.
  • Cosmology-Philosophical Issues, by Stoeger, Ellis and Kirchner.

CONTENTS

1. Transcendental Infinity in Mathematics

2. Actual Infinities in the Physical Sciences

3. Transcendental Infinity in Philosophy

4. Conclusion

 

1) TRANSCENDENTAL INFINITY IN MATHEMATICS

‘A technique succeeds in mathematical physics, not by a clever trick, or a happy accident, but because it expresses some aspect of a physical truth.’ (O.G. Sutton)

Mathematics, the ‘language of our Universe’, is a way of thinking, based on axioms and developed through logic. One wonders why logic works at all and what its origin is. The mathematical language has been continually refined throughout the ages to fit new requirements and results. It started with numbers, then the arbitrariness of measurements, through trigonometry and calculus, etc.

a) Certainty of Mathematics

Mathematicians pursued the truth over thousands of years with an almost endless achievement of certainty. This marvellous creativity was largely due to the method used by deductive reasoning from self-evident principles (called axioms) that, if true, guaranteed the truth in reality. Mathematics is regarded therefore as an effective tool for describing physical reality. This effectiveness points to a mystery, as if we are performing transcendental extrapolations with a human-made imperfect tool.

Mathematics can be used as a criterion of logical correctness; however, as they say, what appears correct today may prove wrong in the next application. On the other hand, if mathematics could be proven illogical, then it would be almost useless as a tool for producing reliable facts about the Universe. Fortunately, experience proves otherwise.

The use of infinitely small and infinitely large quantities of numbers in calculus was soon replaced by the limit process, nevertheless mathematicians still preferred to think in term of actual infinities. Mathematical infinities gained increased acceptance by the physical sciences, where either the infinitely small (such as the concept of the Big Bang singularity) or the infinitely large (the actual infinite size of the Cosmos) filled the gap adequately in the scientific process of deduction; the Standard Cosmological Model is a good example.

b) Uncertainty in Mathematics

Uncertainty in Mathematics brings us to a deeper problem with mathematics, suggested by these three statements.

  • Firstly, mathematical logic does not necessarily equate with truth, or with physical reality. Hence, it was found lately that although some theories were set out logically, not all their axioms were representing the truth in reality. New cosmological theories – with their so-called ‘elegant solutions’ – caused according to R. Scruton ‘with reliance on mathematical infinities, inadequate and even illogical understanding of some of the scientific hypotheses, resulting in need of rigor of scientific proof.’
  • Secondly, Einstein made clear his doubts about the applicability of pure mathematics in speculative scientific theories. Of mathematical successes of the 20th century physics he said they ‘wandered so far from the physically realistic theorising that the new-age theories could only be understood by meditating deeply on their mathematical formalism to reach the conclusion however wildly bizarre they may have been.’
  • Thirdly, in addition to the difficulties surrounding the relationship between the mathematical models filled with hypothetical use of ‘actual infinities’ (and their interpretations backed up with further hypotheses) there are mutually exclusive ways to axiomatise mathematics. Cosmological theories that rely for proof on such mathematics are courting with trouble. The Scientific American referred to some such ‘sensational’ theories that offer final solutions to problems, as to ‘being the latest in cold fusion or closer to cosmological confusion.’

c) Infinity in Mathematics

Infinity in mathematics corresponds to an operation that is never complete, i.e. no matter how many new members you list, there are still more to come. To understand infinity in mathematics, a first step may be to consider the numerical expressions in their adjectival use, in a similar way as they are used in physical sciences (as described above).

The reason being, according to Russell, is that: ‘Our ordinary activities of counting and calculating tacitly assume that there is something being counted and calculated, some subject matter other than numbers themselves, the real object of our attention.’

There are several genuine paradoxes about infinity in mathematics. Among them is one of Zeno’s paradoxes, the Achilles-tortoise foot race. As Scruton describes: ‘Zeno’s argument was that Achilles could never overtake the tortoise over a long track ever. This theory was supposed to prove the existence of actual infinity through the application of infinitely many small quantities of distance that would add up to an infinitely large quantity, by a swift use of the magic of numbers. In reality, however, proof to the contrary was produced through an infinite summation of fractions, showing that Achilles, at the speed of 1 km/sec, could indeed overtake the tortoise at exactly the 2 km mark. The proof against infinity was thus produced similarly by the use of the same magic of the infinite summation of numbers. This paradox proves that we may have an idea of how to begin an infinite series of numbers, but often not of how to end it.’

According to Aristotle, the mathematical infinite may have only two basic alternatives, namely:

  • the transcendental infinite, which is potential or conceptual and refers to a set or a process without any potential limit; or
  • the physical or realised infinite, which could be a physical entity or event, without any equal physical limit.

Aristotle states that, however, in the physical reality there are only potentially infinite series of addition or division that can never be completed.

The cosmologist and philosopher W. Stoeger argues that: ‘Actual infinities are not possible as applied to physical entities or events, because any realised entity or event requires a specifically definite limit to its extent in its number, space, etc… Infinite sets are not constructible, there is no procedure one can in principle implement to complete such a set, they are simply in-completable. But, if that is the case, then infinity cannot be arrived at or realised, in a concrete physical setting.’ However, as he continues: ‘There is no conceptual problem with an infinite set, countable or uncountable in so far as they are only being possible or conceivable.’

Also, D. Hilbert points out that: ‘A really existing infinite set is not possible, because in Set Theory it directly or indirectly leads to well recognised, unsolvable contradictions …The infinite is nowhere to be found in reality, no matter what experiences, observations and knowledge applied.’

There is yet another problem worth mentioning: We know the laws that govern the macrocosm, but physics at infinitesimal scales of microcosm seems to indicate, according to G.W. Flake, that the sub-atomic particles are not divisible infinitely, and that they represent a discreet amount of particles. He further adds that a continuous computational process, based on a perfect representation of real numbers, exists only in theory and will never be realisable.

G.W. Flake made the paradoxical statement, based on his findings, that whenever attempting to make a simple measurement of a self-similar (fractal) length, like a coastline, the apparent length of a coastline seemed to increase proportionately with the reduction in the length of the measuring rod. This curious phenomenon becomes obvious when maps of different scales are compared. Are the lengths of coastlines infinite? Interestingly, he does not make clear whether a coastline represents a transcendental or an actual infinity.

Mandelbrot analysed the same problem and found that, when the logarithm of the length of the measuring rod was plotted against the logarithm of the total length of a coastline, the points tended to lie on a straight line. This appears logical, he remarked, but again, he does not clarify how this affects the total length of that coastline. So the problem remains.

I would like to conclude this discussion of infinity in mathematics with a few unanswered questions that are often asked by some philosophers of science, among them Sklar:

  • Do numbers exist as pure transcendental ideas and is a mind required to bring them into realisation?
  • Can numbers exist in the absence of physical objects and events to count?
  • Do ‘empty sets’ exist (J.R. von Neuman’s idea); for example, a limitless collection of ‘Nothing at all’, from which the mind elicits the numbers?
  • Is the transcendental faculty of mind more fundamental than numbers?
  • How can an infinite series of numbers add up to a finite sum?
  • Are the infinitely many parts in an infinite series of numbers also infinite?

d) The Numbers in Mathematics

Although we know quite a lot about the origin of numbers, it seems that, as J.D. Barrow said: ‘Numbers appear to be so mysterious that I would be more comfortable if someone could prove that numbers were human invention.’

Since mathematical infinities appear mainly in numbers, their brief review may be of interest:

  • Natural Number: A standard counting number, i.e. positive integers, having finite numbers between them. There are an infinite number of natural numbers.
  • Prime Number: A natural number that can be evenly divided only by itself and one.
  • Real Number: A number that can be represented with a potentially infinite decimal expansion. Natural, rational, irrational and in-computable numbers are all real numbers.
  • Rational Number: A rational number (or fraction), is a number that can be represented as a finite decimal expansion, i.e. the ratio of two natural numbers.
  • Irrational Number: A real number that cannot be represented as a fraction, (e.g. square-root of 2 or Pi), and has an infinite decimal expansion without pattern. Irrational numbers can be computable, incomplete and in-computable (i.e. a solution does not exist).
  • Infinity in a mathematical equation leads either to zero or to an undefined real number, i.e. a meaningless result, because infinity is neither a real nor rational number. Infinity is an abstract concept only. Conversely, if a number can be defined, it cannot be infinite.
  • The Set Theory was invented by Georg Cantor and he was the first to prove that not all infinities are equal. A Set is a collection of a form of possible thoughts that can be mentally handled and are entities or numbers. Georg Cantor proved that an infinite set may indeed exist. He named the size of the natural numbers as countable infinite, while he referred to the size of all the real numbers as uncountable infinite.

Cantor was a profoundly religious man, hoping to find the true nature of the infinite. Once he realised the apparent contradiction with his religious belief, he fell into a bewildering mental crisis, from which he never recovered. (Note the somewhat similar case to Aristotle). Surprisingly though, Cantor’s concept of an infinite set did not have any contradiction with belief, as he thought. We shall see this in the Philosophy section of this article.

Platonism maintained that: ‘Mathematics describes a realm or system of real and independently existing objects and numbers, which are entities over and above the proofs…’

There are many reasons for being alarmed by the above statement, as pointed out by R. Scruton: ‘Not only are numbers very strange objects, if they are objects at all; there is a terrifyingly large quantity of them… There is simply no formula or procedure which identifies the totality of the numbers, not even a formula that is applied infinitely many times.’ And ‘So what are numbers? The answer is that there is no answer. But maybe that is the answer… Whatever numbers are, they are not objects. There is no object which is the number three.’

On the metaphysical level, Gödel tried to put some life back into Platonism with his famous Incompleteness Theorem, which was an attempt to replicate the ‘liar paradox’ in purely mathematical terms. Scruton’s comment to that was: ‘If there can be unprovable truths of mathematics, then mathematics cannot be reduced to the proofs whereby we construct it. There is a realm of mathematical truth, whether we can gain access to it through our own intellectual procedures. And the extraordinary thing is that this too is something that we can prove.’

All the above phenomena appear to suggest that the idea of an infinite mathematical task can be accomplished in our mind only and through the use of terms of actual infinity of finite (physical) things, which represents a contradiction in terms. R. Scruton remarked that ‘…it, however, is not absurd to think that one can begin a seemingly infinite mathematical task that has no end or will remain incomplete.’

Scruton concludes ‘Although we have a strange attraction to the concept of infinity in all its types and forms, the path to understanding it, especially physically real infinity, appears to be closed to the human mind. Hence the mental effort of avoiding those paths is often the prime mover of our logical argument.’

2) ACTUAL INFINITIES IN THE PHYSICAL SCIENCES

‘If the atoms were infinitely old, then all radioactive uranium and thorium would have already turned to lead. Their very existence tells us that they were formed at a finite time past.’(Owen Gingerich)

a) Infinity of Energy and Matter

There are two things that are commonly known to all physical scientists, namely:

  • The fundamental basis of cosmology in reality is physics.
  • Physics is incompatible with any actual infinity in theory as well as in reality.

Should the inevitable conclusion from these two statements be unacceptable to any cosmologist, then there must be a gross error in either, or both, the above statements.

To put it in another way, according to our every-day scientific experience of reality: could any physically measurable entity possess infinite mass, and could any physically measurable event possess infinite energy, without ever having any change to their quantitative and qualitative properties? Furthermore, if such physically measurable entities or events would exist, considering our finite capacity of abstracting a mental image from reality, how could we identify infinity; and how could we measure infinity to satisfy our scientific requirements?

After all, our finite measuring devices produce only finite results, i.e. ‘only a finite number will exceed the sensitivity threshold of any detector.’

In his article The Decay of the Proton (Sciam, 1981), Steven Weinberg points out that protons may be unstable, with an average lifetime of 1030 years each. If this is true, then (the quality of) atoms do not last forever. This, however, does not mean necessarily that although a Universe may not be infinite, it could still be eternal (e.g., a hypothetic, cyclic Universe that had a beginning, but will have no end).

If an infinite material entity or event would exist (or even if it would be immaterial, which according to quantum physics of atoms and particles is quite feasible), the question of ‘who or what did generate it, and who, what and how keeps it in an unchanging energy-state of 100% efficiency to do work for all eternity?’ remains.

Purely for argument’s sake, let’s ‘assume’ the following logical conclusions that may follow from the above cosmological dilemma:

  • physical science may be endorsing the (infinite version of the) existence of ‘perpetual motion’
  • physical science may be denying the validity of the second Law of Thermodynamics.

Should these conclusions be incorrect, the premises described in Section a) ought to be reconsidered in order to avoid a true contradiction in physical sciences, including the Standard Cosmological Model.

b) Modelling of Physical Systems

When scientists want to model finite physical systems, they begin with conceptual systems, and sometimes apply mathematical infinities. ‘Conceptual systems are made up of proven, observed or scientific principles and hypothetical ideas that are generated by computerised calculations of variable values and algorithms, all of which result in an evidential conclusion.’

In this type of modelling technique, the first thing physicists do, is eliminate the mathematical infinities from their calculated physical quantities, as e.g. ‘ large-scale physical predictions should not depend on infinitesimal, point-like length scales, whereby they achieve finite answers to problems of finite nature.’ It is also a known fact that a microscopic change in cosmology can often create vast changes on a macroscopic scale. A typical example is the hypothetical concept of ‘singularity’ of the Big Bang, upon which the concept of the age and development of the entire Universe is built.

c) Assumed Proof Against Classical Physics

Despite the fact that cosmology is based on the use of physical sciences – which try to eliminate mathematical infinities from their physical theories – it has been found recently that the use of actual (realisable) infinity in cosmological theories is gradually creeping in and becoming an important part of the solution. It is an understood practice that a scientific theory or model often contains hypotheses, which is not the subject of the present problem. The problem here is that the concept of actual infinity, apart from it being in conflict with reality, is used often and indiscriminately not as a mere hypothesis but as a scientifically accepted, and in fact, tacitly proven fact.

If the concept of infinity plays such an essential part in those theories, it would prove that the assumptions as detailed in Section a) could unfortunately be correct, and the proof against classical physics may be vindicated. Alternatively, the paradox of actual infinity ought to be defined clearly, to comply with the evidence-based requirements of physical sciences.

d) Assumed Uncertainty in the Physical Reality

The logic of physical sciences demands that rational standards for the assessment of scientific evidence must always be based on objective results of observations, verifiable experiments and/or models containing scientific deductions from first principles for the un-observables.

P.C.W Davies states that ‘There is an unwritten rule in science that when anything potentially observable is predicted to become infinite, it is a sure sign that the theory itself is in need of fixing’.

Notwithstanding the above, recent technical publications are filled with sensational scientific theories, ranging from the microscopic singularities to actual macroscopic infinities. According to the perceived scientific bases of these publications, our view of the physical reality is slowly changing from that of a Newtonian certainty to the metaphorical quantum uncertainty. While Einstein challenged Newton‘s gravitational theory by the introduction of the relativity theories, quantum mechanics, in turn, challenged Einstein’s theories with the conceptual role of the observer, the uncertainty principle and the existence of ‘uncaused’ events in the physical world. These theories of course involve the acceptance of the concept of actual infinity, in both the microcosm and the macrocosm.

Irrespective of the controversial nature of the, as yet unproven, ideas, some of them have already been accepted as quasi-theories, because of the unqualified claim that they resolved certain scientific issues in an ‘elegant’ way. And, they can prove it with glossy diagrams too. However, this fact of events may question or invalidate the above statement about the requirements for evidence-based physical sciences.

Sklar comments about the value of speculative science: ‘What is it about the Universe that gives us certainty of knowledge we claim to have about its nature generated by reason alone?’

The nature of ‘uncertainty’ in the physical reality can be reduced to a scale of physical entities and events, whose scale is either measurable and necessarily finite, or immeasurable (i.e. without any physical dimensions or change, and therefore assumed as being actual infinite). It is therefore expected that the evidence hitherto acquired about the observable Universe, including its origin, age, size, mass and changes in it, which are all physically measurable phenomena, would comply with the well-established scientific methods, and without any contradiction with the philosophical concept of being physically finite in every respect.

Although only a finite part of the Universe is observable by us at any finite time, according to Frank Shu: ‘…if the Universe is open, space/time and the number of stars in it would be actually infinite. In this scenario of, say a ‘multiverse’, only the capacity of the human mind could limit the infinite possibilities.’ Furthermore, in his opinion, ”The alternative possibility of a closed Universe is almost as bizarre according to our normal conceptions, because in case of a finite volume of real space it still wouldn’t possess boundary.’

Whichever way we look at the observable Universe, there are two principal laws that govern the physical world, namely the universal Law of Gravity and the second Law of Thermodynamics – upon which the present-day Standard Cosmological Model is built. After all, the present problems have been resolved, a future successful model would determine the shape and size of the Universe as well as, hopefully, delete the infinities, and will fix the location of the observable Universe within the total Universe.

3) TRANSCENDENTAL INFINITY IN PHILOSOPHY

‘He who has not contemplated the mind of nature, which is said to exist in the stars, is not able to give a reason of such things as have a reason.’ (Plato)

In this section, I will consider several items purely from the viewpoint of philosophy; among these are the concepts of infinity and space-time that are a fundamental part of systematic philosophy.

There has been a contest of abstract thinking. Scruton describes that from the mid-seventeenth century onward, there has been a battle going on between the keepers of the ancient ideas and the so-called enlightened philosophers, as well as scientists, over the intellectual supremacy of a new way of thinking about the role of science and philosophy. I have summarised this ‘battle’ in the following sections.

a) The Revisions by the ‘Enlightenment’

The revisions by the ‘Enlightenment’ produced two important concepts:

  • the physical reality
  • the concept of infinity.

Ever since, philosophers have applied these concepts to the physical Universe and everything in it.

As Scruton suggests: ‘It is needless to say that: …in the end the enlightenment won the contest.’

b) The Knowing of Reality

It is known, in general, that the foundation of our ‘common sense of reality’ is based on the following three desirable axioms:

  • the physical nature of the Universe, whose language is mathematics
  • mathematics, whose language is logic
  • logic, whose language may lead to the objective truth in reality.

‘Modern cosmologists pretend to be outside of the Universe. Their quest is to achieve “objectivity”, without any attempt ever made to know the all important and inescapable concept of “reality”, which is needed in order for the object to be known.’ (Prc’ls of Ibn al-Arabi.s Cosmology)

The path of personal experience of what is reality, in terms of what that term is meant to represent through our ‘common sense’ understanding, is the most difficult path to follow, because it is blocked by modern scientific sensibilities. The reason being that science relies on empirical verifications first, but as soon as one asks what is meant by the concept ‘real’, scientists often quote other authorities, without having any idea themselves.

As mentioned earlier, if in a statement the first principles of philosophy are true, then the provable ideas or facts will also be true. This of course does not mean that every truth can be proven, or that whatever is provable will also be true necessarily; this is mainly because primarily we often may have no clear idea of what truth is all about in the reality. Philosophers can only generalise about it; however, the following definition by Scruton appears to epitomise (or rather generalise) its essence: ‘Truth is a kind of absolute, abstract entity, having its basis in logic; it is the one, mysterious human quality that is directing our mind in its different choices of many axioms, trying to approximate the perception of reality.’

c) The Knowing of the Infinite

Our limited mental capacity to ‘comprehend the incomprehensible’ is like a tool we employ and it determines what we will discover. This makes our mind, according to Eddington’s analogy, ‘Just as the mesh of the net that determines what kind of fish we pull from the sea.’

Non-material Infinity

The concept of infinity means, in general, to be without limit, i.e. an unending entity or event, which cannot change ever. Notwithstanding our human inadequacy to comprehend the philosophical concept of non-material infinity, the following philosophical attributes attempt to describe it, through analogy only, of its logically ‘presumed’ attributes.

The following philosophical attributes may apply to metaphysical infinity, through analogy only:

  • It is an undivided unity, having no constituent parts – as if we have reached, by definition, an ‘irreducible simplicity’ of existence itself.
  • It is a self-subsistent entity, with completeness in it.
  • It is absolute in every real sense of the word; it is without any conditions, non-subjective and non-relative to anything. It is an undivided entity by itself, and has the reason for existence in itself.
  • It has the highest degree of perfection and completeness.
  • Its immensity is unknowable by the rational human mind.
  • It transcends the concept of physical reality.
  • It transcends the concept of any finite space.
  • It transcends the concept of any finite time.
  • It is immutable, without any change, and without a beginning and an end in time.

From the above attributes, as described by Aristotle, we reach a quintessential concept of non-material infinity, which is considered, rightly, a metaphysical stumbling block for our limited human way of reasoning, because it necessarily requires simplicity and completeness.

This metaphysical infinity is synonymous with a supernatural supreme being. Aristotle, who had no monotheistic religion, held that this ‘Prime Mover’, which he called ‘The Absolute’, must have created the material world from eternity. From this, we could conclude that this transcendental being is the self-subsistent immensity of all-perfection, and, simultaneously, conclude that this is existing, for all eternity, as an immanent personal deity (whom all monotheists believe to be God and the creator of the physical Universe).

In consequence, when extrapolating from the above with philosophical logic, this concept of non-material infinity is an immanent, supreme agent of all material subsistence. This agent is in operation and maintenance of all material states; of the finite energy, matter, particles and fields that exist universally throughout in the physical reality, and yet, without contradiction, is believed to be (at the same time) a sublimely transcendental and absolute being.

At this point, the concept of non-material infinity would reach the human boundary of comprehending through logic, giving us an intuition only of how physical cosmology appears to reach confluence with philosophical cosmology, where it all began 2400 years ago.

Let us not pretend, the philosophical problems with the concepts of non-material and material infinities are a formidable challenge to the human mind, from the earliest times up until today; at least to anyone adequately constituted intellectually. Through this reasoning process we have reached the ultimate and irreducible simplicity of the concept itself, beyond which point, without faith, not even philosophers can progress any further.

Material Infinity

A material infinite entity or event may exist either as an idea of a purely abstract transcendental infinity (that transcends reality) or could exist physically as an actual (realisable) infinite being, such as infinite energy, matter, infinite space and time or space-time, etc.

The following types of infinities are in practical use nowadays by the various disciplines:

  • abstract (transcendental) infinity – as used in mathematics and philosophy
  • actual (realisable ) infinity – as used by some scientific theorists and cosmologists
  • absolute (transcendental) infinity – as used by some philosophers to identify the one and only non-material infinite – as described above.

d) The Identification of Infinities

Aristotle identifies the following infinities:

  • Potentially infinite (as in mathematics and philosophy) – This was introduced by Aristotle in order to make a clear distinction from the actually infinite, whose material existence, such as in the case of energy, matter, space and time, he outright rejected. Aristotle did agree that one could divide a physical entity or a physical event, as unending, and that it would be only potentially infinite. The very same applies to a potentially infinite in addition. Therefore, in an abstract sense, no matter how long a certain process has run, we always can observe or continue the next step in that process. This is why this infinity is called potentially infinite, as it involves finite, physically measurable steps in space and time.
  • Actually infinite (as used by some modern cosmologists) – Philosophy says that if infinity could exist in the reality, our mind would fail to comprehend it, because in the reality we can only assume to move progressively from one physical point to another without an end in sight in space and in time; this infinite progress, however, is potential only and not actually infinite.

In other words, if we can assume the above – i.e. an infinite entity or event could be visualised or counted in a finite number of ‘measurable’ steps in the physical reality – the entity or event is already a physically finite or only a potentially infinite entity or event, and can never actually be (physically) infinite.

e) The Difficulties with Actual Infinity

The difficulties with actual infinity are, among other things, the following:

  • We are unable to describe actual infinity because we know nothing about it.
  • We still tend to accept new cosmological theories and ideas based on this unusual concept of actual infinity and, as described above, those theories appear to be in direct contradiction with the known laws of physics and the scientific requirements of evidence-based logic.
  • It seems that by the repeated use of actual infinity we are introduced to an arbitrary choice of a hypothetical concept that, without any questions or objections by the peer-groups or the public, has ironically slowly become a part of our perceived way of thinking about the physical reality.
  • R. Stoeger points out that: ‘It is important to recognise that infinity is not an actual number we can ever specify, determine or reach; it is simply a code-word for “it continues without end”.’ and ‘…something that is not specifiable or determinate in quantity, extent or event, is not materially or physically realisable.’ ‘Whenever infinities emerge in physics, such as in the case of singularities, we can be reasonably sure, as is usually recognised, that there has been a breakdown in our models. Any accepted infinity within a physical parameter is tantamount to specifying what is essentially unspecifiable.’
  • R. Stoeger offers another argument against actual infinity, which, in outline, is that if an actual infinite entity or event physically exists, it must be contingent, as it came into being by some generating process other than by itself. In this case, he argues, there are two hypothetical possibilities:
    • ‘It acquired actual infinity by some process of successive finite addition, in all systems involved. This is impossible, because we can never arrive at infinity by successive addition, and with specific coordination of all systems, i.e. there is no physical process whereby we could achieve such an in-completable state of an entity in space or an event in time.’
    • ‘It was produced as an actual infinity all at once. This assumed process is also impossible because it demands ‘all at once’ simultaneity, which is again totally coordinate-dependent as the above alternative, with respect to all other infinite systems involved.’

f) The Question of Space

The following brief introduction characterises the difficulties involved. There are scientifically identifiable, abstract concepts that are possible and realisable, but inconceivable. Descartes said: ‘The conceivable is a test of the possible.’

Scruton states: ‘Quantum physics defies our imagination. If we say that they are nevertheless conceivable, that is because we are using possibility as a test of conceivability, rather than vice versa. This same ‘inverted’ reasoning may apply to the acceptance of the real space and real time as both being actual infinite. Otherwise, in physical or philosophical terms they would make no sense at all.’

The notion of ‘occupying physical space’ is far harder to understand than we might imagine. Space cannot be resolved without geometry, i.e. the spatial organisation of the Universe. Philosophically speaking, the objects (real entities), with their physical extensions are envisaged as ‘occupying space’, of which we form a mental image, but which we do not perceive as a real being.

Even if the concept of space and the physical objects ‘in it’ are theoretically convertible, we measure finite objects with finite instruments that do not suggest necessarily that space should be infinite.

‘Action at a distance’ is a good example to highlight the difficulties of an actual infinite space. We may assume by now from cosmological theories that space is not empty; the entire Universe is filled with energy/matter, particles, radiation and fields. It is also known from science that both the gravitational force and the electromagnetic radiation traverse space in the form of waves. A wave is a form that matter assumes, which means that there has to be matter to assume it, because one could not have a wave of ‘nothing’; i.e. there has to be ‘something’ material that waves.

The present-day assumption is that the Universe filled with energy-fields of all types in a four-dimensional continuum. These energy-fields are concentrated (clumped), like aerosol particles in the atmosphere, around every single object – from the tiniest molecules on Earth, to meteorites, to the stars and galaxy clusters in the farthest corners of the Universe. The concentration of these energy-fields can be acted upon and be caused by gravity and electromagnetism.

Therefore, we can deduce from the ubiquity of gravitation an electromagnetism in the Universe that space cannot be empty anywhere. If, however, space is not empty and is infinite, should energy/matter in it also be infinite? Furthermore, if space and energy/matter are infinite, would it not imply that perhaps the theory of ‘action at a distance’, which is physically limited by the known laws and forces of nature (which are equally limited in their capacity) should also be revised?

The above philosophical considerations may have to be reconciled with physics. Nevertheless, the concept of infinite space remains one of the unresolved problems in the Standard Cosmological Model.

g) The Question of Time

First, when considering the physical theory of time from a purely philosophical view point, time can be treated as a contingent (durational) dimension and a direction (arrow of time), simultaneously. It is of course, not time that has these properties, says Scruton, but events that change in time.

Second, ‘We have no “position-related” choice in time’ – we are all swept along inexorably by the physical changes that are happening to us and around us.

Third, ‘We have no “personal-related” privileges in time’ – everyone is subject universally to exactly the same contingent dimension and direction towards exactly the same end indiscriminately.

Fourth, ‘We have in time only its component of present available for our direct observation. Time, unlike space, is lacking of any physical dimensions; hence it is a mere concept of the mind. Its reality depends on the special faculty of memory that is so highly developed in us that we can expand on it through our verbal and written records.’ (Scruton)

Time presents itself to our consciousness as having a unique direction, without any possibility of reversing its course. Our instinctive knowledge of time therefore, to an extent, depends on the subjective faculty of memory. Using this memory, we refer to something as ‘past’ when we recall or reminisce about it and, by extrapolation, we refer to something as ‘future’ when we plan or dream about it. The presence, which is the instant in time when something is realised, will immediately become ‘past’. This independence of time from physical reality is incomparable with our observation of and dependence on spatial dimensions.

Plato believed that the ultimate reality is timeless. Aristotle, however, as if anticipating every future argument, named reality as one of the sources of the mystery – the little word ‘now’. Aristotle’s further thoughts on the subject were: ‘One part of time has been and is not, while the other is going to be and is not yet.’ So, with the exception of the fleeting moment of the ‘now’ part of time, the remaining part can hardly have any share in physical reality.

Aristotle continues by arguing that we form our idea about time as an abstract concept of measure, i.e. an ontological concept and not a physical being. One may reason that real time is finite, for it is measuring finite change and motion (an event) based on a purely arbitrary human convention – initially, the duration of the planet Earth’s orbit around the Sun, then lately the rotation of the Earth in a day – both in a finite world, with finite instruments. The bases of these measurements gradually developed, through progressive atomic refinements, into a fine art in themselves.

Unlike space, the opposite of time is eternity, which is immeasurable and ‘forever-now’, without having ‘before and after’, which is the essence of time.

Scruton contemplates about ‘eternity’ and sees it as ‘where nothing ever changes with time’ and of which its unworldly majesty seems to tempt us to believe in the unreality of time. Or at least, as he puts it, we may see the physical world under the aspect of eternity as that what changes in time of physical reality, and that of which all truth is eternal truth, which is the adequate idea of reality.

The Simultaneity of Time

Einstein did interpret violations of our common-sense knowledge of the physical Universe as ‘proof’ that common sense was false. He assumed as a fact of nature that the speed of light in vacuum was 299,792,458 km/second, which was confirmed by Romer. Light that could circle the Earth eight and a quarter times in a second, is constant and universal, irrespective of the direction and magnitude of motion in space of any observer. Einstein replaced the absolute frame of reference in space with that of being merely ‘relative’ to whatever reference was chosen. The result of this special relativity theory was that the Newtonian absolute simultaneity of time, based on absolute distance in space, had to be abandoned.

Experiments proved, using atomic particles, that uniform motion relative to an observer will contract with an increase in velocity and slow down time; so that, when an object approaches the speed of light, its physical weight will increase to infinity and its length will diminish towards zero. This appears to be the reason why reaching the speed of light is impossible.

This contraction of distance and dilation of time is either an accepted fact or the actual infinite space and actual infinite time would have to be revised. The reason for that is that the simultaneous contraction of distance and dilation of time are pointing towards physically measurable properties of space and time that contradict the nature of anything that is physically (actual) infinite. The above paradoxical phenomena ought to be reconciled with physics. Nevertheless, the question of infinite time remains still an unresolved problem in the Standard Cosmological Model.

h) The Question of Space-time

The question of space-time is complicated enough, even without the question of actual infinity, because the philosophical issues involved here are subtle and problematic. There are five steps to this problem:

  • There are scientific theories, based on unproven assumptions about point-like singularities of space-time, in which space curvature becomes infinite – such theories slip easily from physical reality into the realm of metaphysics, like an Ontological Argument; but more about this later.
  • One of the peculiarities of space-time is highlighted by the theory that the original micro-cosmically infinite ‘singularity’ at the moment of the Big Bang was assumed to be in an equilibrium condition, as opposed to the present state, where stars and galaxies keep loosing their heat energies to cold space. This early entropic increase of the Universe from energy/matter is said to be paid for by the energy decrease of the gravitational field and/or space/time itself. However, if space-time with all its properties is infinite, how can it lose energy as a payment for an increase in entropy of the Universe?
  • While we cannot imagine the physical existence of space-time, as there is nothing to observe, we keep referring to it as something that is actual infinite (or absolute) space-time. Does this mean that we accept, once again, the Newtonian concept of ‘absolute’ space and ‘absolute’ time, which contradicts Einstein’s theory of relativity? The question remains: Is space-time absolute or relative?
  • We call once again on the problem of entropy to clarify the infinity aspect of time. If there is a directionality of time, then there can be no immutability (i.e. a property of infinity) in time. Why is this so? The answer is: because the increasing entropy in a closed system, such as the Universe, marks the ‘direction’ of time. Although this is only a probabilistic law, based on large statistics, still the probability for reversal of such an entropic process would be virtually equivalent to an impossibility. Therefore – since this is one of the strongest scientifically objective arguments we have that makes the reversal of time with correspondingly decreasing entropy less than a total impossibility in principle – one may conclude that time cannot be absolute, immutable or let alone realisably infinite.
  • It follows from the above, that space – just as well as time, when the two are coupled as in space-time – is not an independent phenomenon; that is, it is not a free-standing, absolute, immutable and realisably infinite entity in itself. Its interpretation – that is, its manifestation – is subject to time (whose real infinite nature has already been questioned above), and vice versa.

Hence, logic dictates that both, space and time are the finite, and measurable manifestations of finite energy and matter, which points towards a function of a universal system in which energy/matter undergoing continuous finite, qualitative and directional ‘change’ are governed by the law of entropy. So far, as one can deduct from the above, energy/matter and space/time are only different aspects of one and the same contingent universal order.

It also makes sense that all the above philosophical considerations, despite their alleged logic, would still have to be reconciled with evidence-based physics. Similarly, all the present-day problems that arise from the apparent contradictions in the use of actual infinite space/time may have yet to be addressed by the Standard Cosmological Model. These arguments underscore also the fact that the question of actual infinity in general is not physical in the usual sense, but primarily a conceptual or philosophical problem. Therefore, science may never be able to resolve these problems on its own successfully.

i) The Paradox of Infinity

There are many cosmological theories nowadays in which the concept of ‘actual infinity’ is used skilfully and extensively – as if it is a scientifically accepted term – but which do not provide any evidence that it is scientifically feasible or logically believable; as if it were an intellectual ‘sleight of hand’.

The reason for introducing the concept of actual infinity into a cosmological theory is usually when, in the thought process of a scientific logic (or illogic), an unresolvable problem (a ‘gap’) appears that has to be filled in. It is also used when the word infinity sounds mysterious enough to simply become a part of an elegant solution. Nevertheless, cosmologists seem blissfully unaware that by using this concept they inadvertently create a big problem for themselves.

The nature of the problem is that, whenever the concept of actual infinity is introduced into a cosmological theory, they simultaneously have to make an unavoidable reference to its identifiable and finite physical properties. This is because it is a fact of reality that dictates: without such a reference nobody would know what you are talking about. The above described illogical thought process creates the biggest stumbling block for modern cosmologists – as further described in the following contradictions. These contradictions are, however, almost impossible to recognise by a non-critical reader and include:

  • The First Contradiction: When, in a theory, a physical ‘magnitude’ of an entity or event that has finite and measurable properties is described or implied (even if done covertly) as either ‘infinitely’ small (as in microcosm) or ‘infinitely’ large (as in macrocosm), it is against the logic of the ‘Principle of Contradiction’, which says that ‘What is now measurable and finite, cannot be at once, and simultaneously, immeasurable and infinite.’
  • The Second Contradiction: When, in a theory, a physical ‘magnitude’ of an entity or event that has finite and measurable properties is described or implied (even if done covertly) and those properties are also the attributes of ‘actual infinity’, it is against the logic of the ‘Principle of Identity’, which says that ‘If ‘infinity’ would possess ‘finite properties’, it could no longer be identifiable as being infinite. If these contradiction are fallacious, then, as it is said, the very word ‘fallacious’ might also mean ‘true’.

To define ‘fallacious’: One of the most common ‘fallacious’ thought processes is called ‘the ontological argument’. It occurs when, in a philosophical or scientific argument, an attempt is made to prove from an abstract concept the existence of a physical entity while at the same time omitting to connect the two logically with one another.

This process is called ‘jumping’ from the logical order of ideas to the ontological order of physical reality without comparing the two. This absence of verification of a true connection between the two realms of thought processes (i.e. whether they have anything to do with one another, are truly similar or not) leads to the ontological argument itself, and to misleading conclusions and/or interpretations.

Note: This illogical ‘jumping’ from one order to another may apply in reverse as well; that is, from the ontological order to the logical order.

The Terms of the Ontological Argument

To explain the ontological argument:

  • The logical order deals with purely transcendental ideas, whose sole objectiveness is based in, and from, the reasoning mind. These ideas transcend reality and are irreducible to the complexities of the physical reality. For example, these transcendental ideas usually express the intellectual faculty of an adequate mind, to reason out the logical way of knowing through perception, understanding and recollection of abstract things. This occurs in mathematics, philosophy, sciences and arts. To these abstract ideas also belong emotions and the world of imagination. Ideas such as numbers, the concepts of infinity, life, time, space, goodness, etc. also belong in the same logical order.
  • The ontological order deals with physical entities and events, after the reasoning mind has divested them of all their properties of materiality, in order to be known as mental images of non-material real beings. For example, mental images are formed in our every-day life of all physically measurable material objects, and their physical changes. These images represent also the material objects of all physical sciences. The thought process of proving something through an ontological argument, as described above, is illogical and proves nothing, because:
    • it transgresses from the realm of transcendental ideas to the realm of mental images of physical reality
    • it generally presupposes, erroneously, in one of its premises that which it sets out to prove in its conclusion.

The subtlety of fallacy of the ontological argument is by far the hardest to recognise in any modernistic scientific or cosmological theory. Scruton also illustrates this: ‘Similarly, this way of argument was so popular in the earlier centuries, despite its misleading conclusions, that it has intrigued many philosophers, so much so that some falling into the trap, and in their enthusiasm they even tried to refine it; notable among those were: Hegel, Descartes, Leibniz and Spinoza.’

Among the many examples for such ‘proofs’ that were based on this sleight-of-hand logic, the use of the ontological argument, were some outstanding ones by Archbishop Anselm of Canterbury (for the proof of the existence of God), Carter (for the ‘design’ argument), Leibniz (for the proof of The Principle of Sufficient Reason) and, finally, not even the Standard Cosmological Model is totally averse to it.

Another example is the case of an often-quoted erroneous concept of ‘infinity’: i.e. one cannot prove logically anything by transgressing from a purely ‘hypothetical’ idea of a material infinity (which in itself is already a contradiction in terms) to the logically unproven existence of a physically measurable ‘material’ infinity, because there is a gap in the logical reasoning. This material characterisation of purely hypothetical concepts of actual infinity (space and time, etc.) makes the modernistic scientific and cosmological theories open to linguistic confusion and conflict, and hence, they could become wholly unscientific.

The following strange questions may arise out of the above linguistic possibilities:

  • First and foremost: What is ‘real’ (i.e. material) infinity? How could it be described and where could we see it?
  • How many actual infinities are there? Would perhaps some infinities be more infinite than others? What would coordinate these infinities? Which infinity would have priority over other infinities? By what criterion should a decision be made, and who or what should decide on which infinity is right and which infinity is wrong at any particular moment, without becoming redundant or contradicting one another?
  • Are the five stable elementary particles of electrons, protons, neutrons, neutrinos and photons (the speed of light) – and also matter and energy – all infinite?
  • Are the three fundamental forces of nature still in control of this multiplicity of infinities and, if so, should these forces themselves be immeasurable and infinite?

Physics and cosmology have proved successful so far, not so much for their hypothesis-based theories as for their science-based accurate measurements of fine-structure constants, which backed up their hypotheses. Should, however, physicists and cosmologists scientifically prove that actual infinity is a universal reality, it would be expected also that the Standard Cosmological Model would by now be able to resolve the paradox of infinity in all its details, including possibly the finite forces and laws of nature, and including the highly measurable universal constants.

And finally, since organic life, according to the respected F. Shu: ‘… can be reduced to the laws of quantum electrodynamics…’, one wonders if modern cosmology would consider this statement as reference to yet an other infinity? If the answer is ‘Yes’, then, firstly, the universal laws and forces that govern biological life itself should also be revised. Secondly, one may ask if this statement by F. Shu means (hopefully for us) a scientific proof, or at least a promise of an infinite or rather an ‘eternal’ life?

4) CONCLUSION

Because of the deep mystery and complexity that surrounds the concept of infinity, our imagination appears to have a strange fascination with it. We appear to have an innate desire to comprehend it as well, in order to reconcile that which constitutes our ‘contingent’ physical nature with its (philosophically claimed) dimensionless immensity and the immutable eternity.

We also have an experience of wonder and awe whenever we contemplate the simplest finite object in the reality; as Halle illustrates, ‘… such as this paper the article is printed on, and our mind tries to enter into its other kind of reality, the enigmatic latent space, which is filled with billions of atomic particles swirling about in it. Couple this mystical nature of any finite matter with the transcendent reasoning faculty of mind, and it appears suddenly as if we could almost witness the incarnation of the same ‘Logos’ (The Word) that was in the beginning, but not the Logos itself; and we’d see similarly unravelling before our very eyes and at every moment in our lives the grandeur of a performance, not the script.’

On a purely realistic level, while we are attracted to a deeper understanding of the transcendental infinite, with its latent mystery of the physical reality in it, we share in both equally. We also have a perceived role to play, in various degrees, between these two extreme and vital phenomena, because by successfully balancing out these two concepts, we will guarantee the very quality of life we are striving for. We have to act out our human life in a ‘transcendental’ hope, without ever loosing the reasoned acceptance of our physically ‘contingent’ nature, which, according to Halle, ‘… depends on the fundamental law of predictable uncertainties of nature’s pre-determinism, as being the ultimate limit of precision in the finite physical order.’

Strictly on a scientific level, our search for a deeper understanding of both, the nature of infinity in our finite Universe is primarily based on our common sense, whose foundation is our common experience obtained through observations of reality, and backed up by reasoning that we apply to the interpretations of those observations. Our knowledge of the objective truth is therefore limited by those experiences and depends on their scientific interpretations.

However, we should need a genius with a dispassionate objectivity of logic (and an intellectual honesty), to identify the truth for us in all those modern cosmological theories that rely so heavily on the use of newly created and undefined (and often undefinable) abstract ideas and concepts, which a reader (and often the authors themselves) knows so little about.

It is a fact that, apart from the above philosophical considerations, we still do not have, and will never have, any scientific evidence for or against the existence of a physically real or a transcendental infinite.

Modern-age cosmologists are cautioned by the words of Socrates: ‘Let us labour to rid our minds of faulty notions, especially the notion that we are wise and well informed.’

In conclusion, I repeat, for clarity, the following two major, diametrically opposed, reasoning methods a reader may often encounter in the recently published interpretations of actual (material) infinity:

  • Physical scientists, in their search for the objective truth, base their theories and experiments on the reason of logic; thereby they scrupulously avoid any reliance on purely hypothetical ideas or theories, such as actual (material) infinity, that has no evidence-based proof in the reality.
  • Physical cosmologists of the modernistic type, on the other hand, often build their theories on or around a purely hypothetical concept, such as actual (material) infinity, from which they extrapolate to the physical reality, all on the basis of the ontological argument. This method of theorising relies heavily on the reason of a hypothesized physical reality.

Consequently, the obvious difference between these two reasoning methods may hopefully provide you with a clearer idea.

‘To see a world in a grain of sand,
And a heaven in a wild flower.
Hold infinity in the palm of your hand,
And eternity in an hour’
(William Blake).

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