**Mistakes Concerning Infinity**

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 * 10^{23} (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 (10^{87}, 9*10^{65},
2.79*10^{988757}), 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*10^{988757 }+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.

**Conclusion**

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.

**___________**

**G. Stolyarov II is an actuary, science fiction novelist,
independent philosophical essayist, poet, amateur mathematician,
composer, contributor to Enter Stage Right, Le Quebecois Libre, Rebirth of Reason, and the Ludwig von Mises Institute, Senior Writer for The Liberal Institute, former weekly columnist for GrasstopsUSA.com, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress. Mr. Stolyarov’s blog, ****The Progress of Liberty**,
offers a combination of commentary, multimedia presentations,
educational materials, and suggestions for effective activism in favor
of individual freedom. Mr. Stolyarov also publishes his articles on Helium.com and Associated Content
to assist the spread of rational ideas. He holds the highest Clout
Level (10) possible on Associated Content. Mr. Stolyarov has also
written a science fiction novel, **Eden against the Colossus****, a non-fiction treatise, A Rational Cosmology, and a play,**

**Implied Consent.****You can watch his YouTube Videos.**

**Mr. Stolyarov can be contacted at gennadystolyarovii@yahoo.com.**

** **

**This TRA feature has been edited in accordance with TRA Statement of Policy.**

**Click here to
return to TRA's Issue XXIV Index.**

**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.