Mistakes Concerning Infinity

G. Stolyarov II
Issue XXIV - July 21, 2004
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This essay is a version of Chapter IX of Mr. Stolyarov's treatise, A Rational Cosmology.

“Infinity” is one of the most frequently encountered terms in the modern culture, and one of the least understood. Too often has its invocation been an attempt to justify mysticism, irrationalism, and contradiction, especially in the natural sciences. It is the province of filosofy, as a foundational science, to set the very framework without which the natural sciences cannot operate. Unfortunately, numerous modern scientists have stepped far outside their field in making generalizations about the nature of existence, and of infinity, deliberations which properly belong in the realm of filosofy and which filosofers must employ to weed absurd and contradictory statements from the natural sciences.

Infinity and Existence

Reality is absolute and every existent has an identity. According to the filosofy of Objectivism, existence and identity are inextricable corollaries. To be is to be something and to be something in particular. To be something in particular means to have a set, deliberate, fathomable nature. It is no coincidence that the word “to fathom” means both “to measure” and “to understand.” In order to be understood by man, a given entity must have attributes that can be measured on some scale, be it a qualitative or a quantitative one. In order to be measurable, an entity must demonstrate a finite quantity of each measurable attribute. A particular given entity, say, a dog, must have finite mass and length, and its fur must reflect light of a finite frequency. A concept, such as dog, is formed by omitting the particular measurements of every dog and claiming that a dog must have dog-like qualities in some quantity, but could have them in any of a range of quantities. To claim that any dog has infinite measurements of given qualities is absurd: if something is infinite, and does not have a set, delimited quantity to be measured, how can it be measurable? If it is not measurable in some manner, absolute or relative, how can it serve as a necessary quality in the definition of a concept? Thus, infinite, that is, limitless measurements of qualities cannot exist if concept formation is to continue to maintain its legitimacy.

Using this as a foundation, we now proceed to investigate prevalent misconceptions and faulty logic in the examination of the notion of infinity and where it is applicable.

Mistake 1: If nothing can be infinite, then everything will have to be destroyed someday.

This in no way follows from the assertion that no entity may ever have an infinite quantity of anything. Let us say that an architect has designed a tower of such durability that no known substance can erode or puncture it. There is absolutely no guarantee that this tower will ever be destroyed. It can be said to be invincible, but it will always have a finite age! After one thousand years, it will be one thousand years old. After one million years, it will be one million years old. No matter how old it becomes, its age can still be measurable, and thus is not infinite. Thus, it is possible for things to last indefinitely, and there is no inherent guarantee that everything will someday be destroyed. While man’s mind cannot envision infinite size or infinite smallness, it can conceive of the possibility of “infinite” longevity of anything: buildings, planets, animals, men, so long as these entities had a certain origin in time.

This fenomenon can be referred to as a chronological infinity, though I use this term with reservation, because it does not truly describe an infinity, for all the measurements concerning it must be in all cases finite. The true infinity, or a simultaneous infinity, concerns either coexistence of infinite and finite measurements or the presence of all infinite measurements within an entity. God has been defined by the religious as an object of allegedly infinite quantities of everything, i.e. omnipotence and omniscience. However, the rational man would need to reject God by this definition, because it implies a simultaneous infinity: the technique of measurement-omission cannot be applied to the formation of the concept, “God,” and, thus, “God” cannot be a legitimate concept unless it is a hypothetical God that does have a finite age, and exhibits delimited qualities and abilities. (And, simply because something is conceivable, does not guarantee that it exists; the existence of such a conceptually legitimate God would still need to be proven to be within the realm of reason.)

Mistake 2: Singularities and black holes exist which have an infinite density.

Filosofy must urgently employ its veto power over the natural sciences to refute this illogical theory. Density is the ratio of mass per unit volume. An infinite density implies the existence of unlimited mass within a limited volume. Mass is not a chronological attribute, and exists all at the same time. To claim that infinite densities can exist is to acknowledge the existence of simultaneous infinities, which immediately renders one’s concept or theory illegitimate.  

One must ask the modern fysicists the following questions: “What properties does a finite volume have to enable it to hold infinite mass without expanding? If a finite volume can hold any quantity of mass, no matter how large it is, does it not follow, then, that each individual unit of mass must occupy zero volume? If one unit of mass has zero volume, and zero multiplied by anything remains zero, then must a singularity, too, not have zero volume? But how can it also be claimed to have a certain finite non-zero volume?” This is a contradiction that cannot be reconciled. The fysicist, if he thinks rationally, will be forced to admit that the singularity does indeed have zero volume, that is, it does not exist.

It is conceivable that an object may have a very high density, exhibiting a very large mass to volume ratio. It is also possible that there exist what are now called black holes and singularities, and that they can explode outward or attract matter into themselves. But, a “singularity” can only release some very large amount of matter in an explosion; it cannot be an inexhaustible fount of matter. A “black hole” cannot be said to have an infinite holding capacity for objects, either.  Throughout its existence, it must have attracted some finite quantity of objects into it, which quantity affects its mass. But, if it also has some finite density, any intake of matter must also have amplified its volume in some manner. Even if this matter were to become compacted to an immense extent in the “black hole,” it would not be possible to compact this matter infinitely.

This is what filosofy tells us in regard to modern cosmology. It informs us what propositions must be false, but it does not guarantee that even a conceptually feasible notion of black holes and singularities is true. Such proof would be a task for empirical observation to undertake. Filosofy can, however, catch scientists spouting senseless generalizations and propositions, and inform them whenever they venture into a realm for which the filosofically unsystematic scientist is quite ill-equipped.

Mistake 3: If any entity must have a finite age, then the universe must also have originated at some point in time.

Yes, any entity must have a finite age at any point in time. The mistake here, however, is quite simple: the universe is not an entity! It is a mere collection of everything that exists. The purpose of the term “universe” is to serve as intellectual shorthand that substitutes listing every single existent when one wishes to speak of universal principles that are applicable to everything (such as the axioms of existence and identity). The term “universe” is not in itself a legitimate concept. If the sum of Chicago, Quasimodo, a telescope, and a hippopotamus cannot be a legitimate concept, how can the sum of Chicago, Quasimodo, a telescope, a hippopotamus, and everything else be a legitimate concept?

If one were to say that Chicago, Quasimodo, a telescope, and a hippopotamus had a certain single origin in time, the statement would evidently be ludicrous, from any perspective. The more expansive such a statement becomes, however, the more reverence is given to it in contemporary academia! Rationally, though, it must be all the more ludicrous for it. There is no such one thing as “everything,” nor even “the potentiality of everything.” If there is no such one thing, it cannot have a single origin in time. (It cannot, per se, have any quality, not being a single thing!) Thus, all the modern cosmological speculations about a Big Bang that occurred to “initiate everything” and a Big Crunch to occur that will “destroy everything” are sheer blunders, caused by the inability to understand the limitations of a term such as “everything” (or its equivalent, “universe.”) Filosofy instructs the rational man to reject these superstitions right away.

The universe cannot have a beginning or an end, for the term “universe” is synonymous with “existence.” Existence exists. Existence can never not exist or not have existed. A=A.

Mistake 4: Matter is infinitely divisible.

I could take a sheet of metal and slice it in two pieces. It could be said then, that the metal is divisible by two. I could, using advanced futuristic technology, dismember it into its constituent atoms. It can then be said to be divisible by about 6.022 * 1023 (assuming we have a mole of metal to begin with). I may also be able to extract the cores of these atoms and separate them into their constituent nucleons, and, subsequently, split those nucleons into the quarks that comprise them. Matter can be divisible by a very large factor, and this factor may be far greater than we presently even suspect. Only science can inform us of the precise extent of matter’s divisibility. But can matter ever be infinitely divisible?

Can we ever have an infinity of particles originating from some finite object? Just like having infinite mass in a finite volume, this is a simultaneous infinity, and is thus impossible. After all, this would imply that each of these particles would have zero volume, and would thus simply not exist. How one can form an existent piece of metal out of non-existent particles, no matter how many of them there are, is beyond rational comprehension. I will wager my life savings that nobody will ever be able to do this!

Thus, matter cannot be infinitely divisible. We do not know the extent of matter’s divisibility, and we may be able to continue dividing it for vast periods of time, and still find new division to be possible. But we will only know matter to be divisible as far as we will have divided it. Since simultaneous infinities cannot exist, we will never reach a state where infinite divisibility can be empirically verified. Thus, it is not a legitimate proposition, scientifically or filosofically.

Mistake 5: Division by zero gives infinity. Thus, infinite quantities must exist.

There is no such operation in the real world called “division by zero.” I can split a pie into three pieces, or five thousand pieces (if I have a microscopic cutting tool). I cannot split it into zero pieces. Matter does not originate ex nihilo, nor can it be annihilated. The fundamental fabric of that which exists (i.e. matter) cannot all of a sudden stop existing for no apparent reason. The scientific principle of matter conservation is in fact a filosofical proposition which must be true in order to exclude magic from the realm of science.

Division by zero is in fact not even a valid mathematical operation, but rather the description of a trend: the magnitude of the quotient is inversely proportional to the magnitude of the number by which the dividend is divided. Similarly, all other uses of infinity in mathematics are mere convenient shorthand notation to the identification of trends. For example, a quantity “approaching infinity” is the same as a quantity increasing without bound. At any particular time, it will still be a finite quantity.

Mistake 6: If infinite quantities cannot exist, then space itself is finite.

All quantities are attributes of existents. Space is not an existent. It is a mere positional relation of existents with respect to each other. There cannot simultaneously exist an infinite amount of existents, but space itself cannot be said to be finite or infinite. It cannot be said to be. Something, i.e. an existent, is. Nothing, i.e. space, is not.

This is why all coordinate systems are inherently relative: they must presume an arbitrary origin at some point. But, just as an entity can be conceived to exist at (0,0,0), so can it be conceived to exist at (1087, 9*1065, 2.79*10988757), which is just a set of numbers describing its relation to an entity that could exist at (0,0,0). A spaceship with recyclable fuel could be equipped to distance itself from other existents indefinitely. At any time, it will still be a measurable distance from those existents, and its distance would be finite. No matter how large this distance is, however, it could always become larger. 2.79*10988757 +1 is a conceivable number, but infinity is not.

Space is neither finite nor infinite, but it can be said to be indefinite.

Mistake 7: If everything is finite, time must have had an origin.

Time, too, is not an entity. While space is a relationship in three dimensions, time is a quality that enables the establishment of relationships in the fourth dimension. It can be measured by any uniform standard we deem fit, and something can be chronologically remote to something else in either direction to any finite quantity. Though this quantity must be finite, there is no limit to how large this quantity can be. Like space, time is neither finite nor infinite, but rather indefinite in two directions (earlier and later).

Here, it is fitting to note that each dimension (and there are only four) describes a particular relationship, and is indefinite in two directions: time (earlier and later), height (up and down), length (front and back), and width (left and right). This is a filosofical insight that science cannot nullify by any amount of theorizing or observation.

Mathematics, being a sister foundational science to filosofy, calls this truth in the three spatial dimensions Euclidean space. Perhaps it would be fitting to refer to it in all four dimensions as Euclidean space/time, which is based on arbitrarily designated uniform units. Euclidean space/time is to science a metafysical given that mathematics must accept if it is to function in this world. Neither science nor mathematics can legitimately claim the existence of more than three spatial dimensions and one chronological dimension. Thus, dimensions with numbers like 6, 2.34, or p+ 3/4 must be immediately rejected as unreal and logically absurd.


Consistent and rational application of filosofy can indeed tell us many things about the nature of existence: indefinite Euclidean space-time, the impossibility of simultaneous infinities, the possibility of indefinite, but not infinite, measurements of all qualities, including the four dimensions of Euclidean space-time, and the nonexistence of infinite divisibility. Filosofy can also help alleviate senseless scares about the “inevitable end of everything,” which threaten, by no legitimate logical basis, to render the long-term purpose of existence itself meaningless.

Whenever one uses the term “infinity,” one treads a thin line (though not an infinitely thin one!). I make no apologies for the term’s existence, however; like “universe,” it can be a convenient intellectual shortcut to lengthier expressions of mathematical and natural trends. It can also be used to point out logical impossibilities. It is convenient, for example to inform an opponent in debate, “You claim the existence of a simultaneous infinity. This means you have committed a logical fallacy.” But, in the vast majority of cases, the term “indefiniteness” is far more suitable to describing an entity or fenomenon than “infinity.” The latter term suffers from improper cultural use, and has far exceeded its boundaries, ironically enough. It is time to constrain the term, “infinity," to its proper limits.

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Learn about Mr. Stolyarov's novel, Eden against the Colossus, here.

Read Mr. Stolyarov's comprehensive treatise, A Rational Cosmology, explicating such terms as the universe, matter, space, time, sound, light, life, consciousness, and volition, here.

Read Mr. Stolyarov's four-act play, Implied Consent, a futuristic intellectual drama on the sanctity of human life, here.