Democritus proposed the Greek word atomos, which means
uncuttable, to indicate the smallest bit of matter that could exist. [1]
Throughout history, humankind has wrestled with the concepts of “biggest” and
“smallest,” and at every stage, we have concluded that there might be something
either smaller, or bigger, than that which we already know about. And while physics has mostly concerned itself with the concrete
(things) and how they behave, mathematics has traveled ahead from being a mere
method of counting to creating levels of abstraction which can be used to
explain our universe as well as universes that may or may not exist. In what
follows, I will trace the basics of that journey, which I think both
mathematicians and physicists consider to be a useful partnership.

It might be, however, that some people would rather the two
fields go their separate ways. As an example, I would like to draw attention to
an article from February 20, 2015 by Max Tegmark, in

*Discover Magazine*, entitled:*“Infinity Is a Beautiful Concept – And It’s Ruining Physics.” Quoting from it:*“Physics is all about predicting the future from the past, but inflation seems to sabotage this. When we try to predict the probability that something particular will happen, inflation always gives the same useless answer: infinity divided by infinity. The problem is that whatever experiment you make, inflation predicts there will be infinitely many copies of you, far away in our infinite space, obtaining each physically possible outcome; and despite years of teeth-grinding in the cosmology community, no consensus has emerged on how to extract sensible answers from these infinities. So, strictly speaking, we physicists can no longer predict anything at all!” [2]

I have one quibble with the language of
that paragraph. Physics, as is the case with all science, should ideally be
able to predict things, but the wording above might have it confused with
fortune-telling. Scientists do, in fact, make predictions, then perform experiments
or make observations to verify that their understandings are correct. But due
to the speed of light, most of our observations of the universe consist of
peering into the past. Thus, physicists are regularly

*predicting*the outcome of observations of events which transpired billions of years ago. I also don’t believe that all of theories of the universe require that it be infinite, but certainly some do involve that possibility. A blanket statement that multiverse theories are untestable may be premature at this point. [3]
Additionally, the author, Max Tegmark,
does not appear to know the difference between the theory of infinite expansion
and multiverse or string theory, and assumes that people simply think that
there are copies of themselves far away in the same universe. (More on this is
a bit.) Furthermore, “infinity” as a mathematical concept actually makes it
easier to measure things, not more difficult. (Consider calculus, which relies
on infinite processes, or conceptualizations in infinity, and these models are
tremendously valuable for hard physics. [4])

“Infinity” is incredibly important in cosmology and
astrophysics; without it, general relativity, and special relativity, cannot
work out mathematically. It is likewise an integral part of thinking about the
relationships of objects in space (e.g., “infinitely small”). That is, for
instance, how we must imagine the “state of the universe” prior to the Big
Bang. As another example, the average density in a region of space is equal to
the mass contained in that region, divided by the region’s volume. If that
region contracts to zero volume, the density becomes infinite.

(I believe that there are newer theories that attempt to
quantize this, such as loop quantum gravity, which I, admittedly, am not an
expert on, although I do know the fundamentals of it such as the Planck length
and the fascinatingly complicated relationship between density and volume. In a
way, I suppose, it is, at the very least, something of an attempt at a unified
theory.)

Infinity is also used to measure grand-scale events in
cosmology and astrophysics, and to predict the various probabilistic outcomes
of quantum events and quantum mechanics experiments. It literally led to a
scientific revolution, providing us with a number of answers which we didn’t
have before. (Infinity is likewise what makes concepts such as event horizons,
black holes, and gravity possible.)

More importantly, there is nothing that I can detect in the
article which places the blame for discord among physicists at the feet of our
concept of infinity. Bearing in mind that the article is excerpted from a book,
it may simply be that this is intended as a teaser to encourage readers to
purchase the book (nothing wrong with that, as it is common).

Interestingly, in another article from 2013 called
“Everything in the Universe Is Made of Math – Including You” [5], Tegmark
discusses the notion that some physicists (and philosophers) share (with
numerous themes and variations) that there is no physical substance to the
universe at all! I am not attempting to discourage anyone from buying his
books. As we will see, physicists are extremely divided on the nature of our
universe. Considering how long (or briefly) humankind has been considering the
problem, I don’t think that we have anything to be embarrassed about (although,
some individuals might disagree, as we will also see).

But first, a much abbreviated bit of history on the
development of counting and “mathematics” in general:

If we travel back to times of pre-history, we know that
human beings had already learned to count things. At minimum, they counted
domesticated animals in their possession, they traded items of value with one
another which necessitated a counting of them, and there are various pieces of
evidence that they could record these counts in different fashions (which evolved gradually).

At the dawn of documented history, we learn how peoples in
different parts of the world developed different rules for counting, or more
precisely, for recording what the counts were, and those rules varied widely
from simple marking with a dot or line representing each item being counted to
more sophisticated systems involving a “base.” For instance, most of the world
now uses systems founded on “base ten,” meaning that the rightmost digit of any
integer can consist of one of ten symbols, “0” (zero) through “9” (nine). In
evaluating the number, that rightmost digit is self-explanatory. The other
digits must be multiplied by positive powers of ten: 10, then 100, then 1,000,
and so on. We take this for granted, but numbering system were not always thus.
The Babylonians, using symbols most people wouldn’t recognize today, had a
system based on “base sixty,” such that the rightmost digit for them was one
of sixty symbols representing themselves, while the digit to its left would
have to be multiplied by 60 to ascertain its value. Accordingly, the
Babylonians, employing a two-digit number, could represent a number as high as
3,599. It is no coincidence that there are 3,600 seconds in an hour and 360 degrees
in a circle.

(A story for another time.)

Interestingly, early number systems such as the one the
Babylonians used did not have concepts such as zero, infinity, or even
fractions. Why? Because they didn’t need those ideas for counting sheep. Where
we would utilize a zero in a multi-digit number as a placeholder, earlier
systems might simply use a space. Zero as a concept that we know presently is
estimated to have come into existence for the first time in the seventh
century. [6]

In other words, early “math” was not developed as an idle
pastime or mental exercise, but as a way to solve problems. As the problems
grew more complex, so did the nature of math. The concept of fractions was
discovered rather early, but the notation we use today developed over time. [7] Negative numbers were not conceptualized until the 7th century in India, and as
late as 1758, Francis Maseres was claiming that negative numbers “darken the
very whole doctrines of the equations and make dark of the things which are in
their nature excessively obvious and simple.” [8] Which sounds a bit like the
criticism of infinity that we started with, doesn’t it?

Fractions, as noted, have been around for a long time,
chiefly because, in trade, people often divide things such as “half a loaf” of bread.
In other periods, fractional coinage such as “pieces of eight” were physically
represented by actually cutting coins into smaller sections like a pie. [9]

Finally, while most counting and dividing of things can be
represented by whole numbers and fractions, other quantities that most
mathematicians and physicists would consider essential (such as pi) cannot be
precisely represented in fractional form. Such numbers are aptly named
irrational numbers. While the

*concept*of pi goes back to antiquity, it is not certain how it was dealt with in ancient times, other than that such dealings were not completely precise [10], and the notion that we could never actually represent an exact value for pi was initially expressed by William Jones in 1706, when he first proposed a representation of the value by the use of the symbol that we recognize currently (π). [11] As to other irrational numbers, the history of modern “number theory” is more complex than I wish to cover in this piece, but here is a reference for anyone interested: “The Real Numbers: Stevin to Hilbert” [12]. In addition, I won’t cover “imaginary” and “complex” numbers, other than to state they are foundational to physics, as well as number theory. [13]
Suffice it to say that eliminating the concept of infinity
seems no more rational (no pun intended, really) than it would be to, say, just
use 22/7 for pi and call it a day.

Number theory, as a whole, similarly to physics, is not exactly
“settled science.” Mathematicians are constantly learning new things about how
numbers work, such as that there are an infinite number of prime numbers, even
though the density of prime numbers gradually decreases as they grow larger. If
we take Tegmark’s 2013 book,

*Everything in the Universe Is Made of Math – Including You*, at face value, then discoveries such as the nature of prime numbers should inform our understanding of the universe. Even our ideas of what infinity consists of are still subject to change. [14]
It appears to be a part of human nature, in fact, to want to
understand

*everything*, and modern mathematics, as well as physics, are areas where, in modern times, at least, it seems that our understanding frequently runs ahead of our actual need to know. Indeed, Tegmark points out that we can already use modern computers to calculate any value we need to as many decimal places as we need. For example: “A researcher has calculated the 2,000,000,000,000,000th digit of the mathematical constant pi - and a few digits either side of it.” [15] And I’m not even certain that is the current record.
As I stated, the idea that the universe isn’t even “real,”
or at least real as the word is usually interpreted, isn’t new. Does the
universe and how it is constructed and how it operates completely define what
is possible in mathematics? In other words, if a property can be conceived and
expressed mathematically, does that represent some fundamental aspect of our
universe? Or to the contrary (as would seem more intuitive), is it possible to
define mathematical concepts that have no expression in real world terms? Such
questions have vexed both mathematicians and physicists for as long as the two
fields have existed. A terrific lecture by Robbert Dijkgraaf on how math and
physics have positively influenced one another can be found under reference
[16] (“The Unreasonable Effectiveness of Quantum Physics in Modern
Mathematics”).

But I think that most people would agree that the fact that
we can mathematically conceive of infinity does not automatically mean that the
universe is infinite, or vice versa (i.e., that if we

*prove*that the world is finite, that does not mean that we can dispense with the mathematical concept of infinity with no loss to other aspects of mathematics). So, infinity cannot have ruined physics because the two concepts are fundamentally distinct; or to phrase it another way, the two concepts are orthogonal to one another.
I have mostly talked about how mathematics has evolved when
it comes to counting things (and dividing them into smaller pieces) in our
everyday lives. At the other extreme, Georg Cantor first conceived of levels of
infinity and the idea that it can be demonstrated that one sort of infinity is
“larger” in some sense than another. Likewise, we start with the definition of
“universe” as something which is all-encompassing. Yet in modern physics, we
have had to invent terms such a “multiverse” to describe the possibility of
universes that are outside of our ability to perceive them. When Hubble first
realized that the fuzzy objects in his telescope were not clouds of dust, but
other galaxies, humankind first realized that the thing we refer to as the
Milky Way was not in fact the whole universe.

The apparent movement of these
billions of other galaxies away from us caused a simple thought experiment to
commence. By running time backwards, it was conceived that, at one time in the
distant past, all of the matter in our “universe” must have been together in
what would have been an unstable “singularity.” Hypothetically, this would be a
single point with no dimensionality. (Just one more thing that is arduous for
us to conceive intuitively.)

From this, it is hypothesized that a “Big Bang”
occurred, hurling all matter in every direction. This is not to be confused
with an ordinary explosion in that it was not just matter exploding into space,
but space exploding into “nothingness,” another concept which we can hardly
imagine. Today, we can observe what we think are the edges of our universe in
the form of the cosmic background radiation, which we might see in the form of
static on an old television set.

And yet physicists have conceived of many variations on this
seemingly tall tale. Suppose the singularity from which our “universe” emerged
was not alone? Our universe may be one of many, or even an infinite number of
universes in a sea of such things which will be forever out of our reach.
Alternatively, the Big Bang might be only one of a sequence of such occurrences.
Such multiverse visions have been compared to bubbles in a pot of boiling
water, or the bubbles that form when bread is baked.

One factor that all of these theories have in common is that
they, concurrently, include elements of finiteness with that of the infinite.
As difficult as it might be to imagine infinity, it seems that our minds also
rebel against the notion of finiteness when it comes to “everything.” We must
ask: “What, if anything, is outside our universe?” For more on the Big Bang and
related topics, check reference [17] (“The Physics of the Universe”).

If you think that physicists are rapidly zeroing in on the
answers to such conundrums, you would be wrong. In fact, ironically enough, in
1997, at a conference entitled “Fundamental Problems in Quantum Theory,” held
at the University of Maryland, one of the attendees decided to conduct a survey
of the group to find out which theory of “everything” was the most popular.
That person was this same Max Tegmark. The poll was, in his own words, “highly
informal and unscientific.” Nevertheless, that informal poll caused some other
physicists to wonder what a more carefully conducted poll would reveal. In a
recent YouTube video, physicist Sean Carroll somewhat sarcastically accused the
current state of quantum mechanics of being “an embarrassment.” The video is
titled: “Quantum Mechanics (an embarrassment).” [18]

In it, Carroll cites a paper: “A Snapshot of Foundational
Attitudes Toward Quantum Mechanics” by Maximilian Schlosshauer, Johannes
Kofler, and Anton Zeilinger which makes in its conclusion a similar statement:

“Quantum theory is based on a clear mathematical apparatus, has enormous significance for the natural sciences, enjoys phenomenal predictive success, and plays a critical role in modern technological developments. Yet, nearly 90 years after the theory’s development, there is still no consensus in the scientific community regarding the interpretation of the theory’s foundational building blocks. Our poll is an urgent reminder of this peculiar situation.” [19]

A key question asked was: “Question 12: What is your
favorite interpretation of quantum mechanics?” For which the answer spread was:

● Consistent
histories: 0%

● Copenhagen:
42%

● De
Broglie–Bohm: 0%

● Everett
(many worlds and/or many minds): 18%

● Information-based/information-theoretical:
24%

● Modal
interpretation: 0%

● Objective
collapse (e.g., GRW, Penrose): 9%

● Quantum
Bayesianism: 6%

● Relational
quantum mechanics: 6%

● Statistical
(ensemble) interpretation: 0%

● Transactional
interpretation: 0%

● Other:
12%

● I
have no preferred interpretation: 12%

I won’t expand on the definitions of each of the above
theories. What is noteworthy is that there is no consensus that exists, and
none that seems to be forming. The oldest theory, forged from 1925 to 1927 by
Niels Bohr and Werner Heisenberg in Copenhagen (hence its name, the Copenhagen
interpretation) is shedding adherents in favor of the alternatives. If ever
there was an unsettled science, the origin and nature of our universe is it.

And yet, Sean Carroll, at the end of his video, backs away
from the attention-garnering title. As I have noted on several occasions in
other works, journalists (and their editors) love to grab us with a headline
that regularly bears no resemblance to the article itself. Perhaps scientists
seeking funds, and book authors seeking readers, can be forgiven for doing
likewise.

If there is any reason to think that the existence of the
concept of infinity has somehow stifled our theories of the cosmos, the article that
I linked at the top does not explain it. If anything, maybe all this
demonstrates is that we suffer from too many physicists, even if not an endless
supply of them.

It is generally accepted that the cosmic background
radiation is the earliest thing that we can “see” of our universe, and that
there is little hope of ever being able to see beyond it (if there is anything beyond
it to see). Thus, we may never know if the universe is infinite or not, but
regardless of that, there are uses for the concept of infinity, just as there
are uses for imaginary and complex numbers, pi to many decimal places, an
understanding of how prime numbers come about, and other mathematical concepts
which don’t represent anything that we can see or touch.

It can be anticipated that we will
continue to learn more about our universe, even if we may never understand the
totality of it. Furthermore, our knowledge of mathematics will certainly
increase with time. Perhaps someone will present a theory of everything that
doesn’t involve infinity, but given that there are numerous concepts in math
which depend on infinity, and a multitude of concepts in physics that depend on those
derived mathematical concepts, it is difficult to see how.

Author: Krista Milburn [@Femitheist]

____________________________________

__NOTE:__If you liked this post, feel free to share it, and if you have any thoughts on it or the general subjects discussed within, or even merely semi-related topics, please leave them in the comments below. I always enjoy reading the feedback of others whether they liked and agreed with what I had to say or not.

____________________________________

References (Last Accessed on March 22, 2016):

[1] NPR. “Science Diction: The Origin of the Word ‘Atom’”

[2] Max Tegmark. “Infinity Is a Beautiful Concept – And It’s
Ruining Physics.”

*Discover Magazine*. February 20, 2015. http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/#.Vr0a4_k9671
[3] “IS THE UNIVERSE A BUBBLE? LET'S CHECK”

https://www.perimeterinstitute.ca/news/universe-bubble-lets-check
And:

[4] School eBook Library, “Calculus”

[5] Max Tegmark. “Everything in the Universe Is Made of Math
– Including You.”

*Discover Magazine*. November 4, 2013. http://discovermagazine.com/2013/dec/13-math-made-flesh
[6] Ask History. “Who invented the zero?”

[7] Liz Pumfrey. “History of Fractions.”

*Enriching Mathematics.*
[8] Leo Rogers. “The History of Negative Numbers.”

*Enriching Mathematics*
[9] “Pieces of Eight.”

*History.org.*
[10] David Wilson. “The History of Pi.”

[11] Patricia Rothman. “William Jones and his Circle: The
Man who invented Pi.”

*History Today*Volume 59 Issue 7. July 2009. http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pi
[12] J J O'Connor and E F Robertson. “The Real Numbers:
Stevin to Hilbert.”

*St Andrews University.*2005. http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Real_numbers_2.html
[13] “Complex Numbers in Quantum Mechanics.” The University
of Illinois at Urbana–Champaign. https://courses.physics.illinois.edu/phys580/fa2013/susy_v2.pdf

[14] Natalie Wolchover. “To Settle Infinity Dispute, a New
Law of Logic.”

*Quanta Magazine.*November 26, 2013. https://www.quantamagazine.org/20131126-to-settle-infinity-question-a-new-law-of-logic/
[15] Jason Palmer. “Pi record smashed as team finds two-quadrillionth
digit.”

*BBC News*. September 16, 2010. http://www.bbc.com/news/technology-11313194
[16] Robbert Dijkgraaf, “The Unreasonable Effectiveness of
Quantum Physics in Modern Mathematics.” Perimeter Institute for Theoretical
Physics. March 5th, 2014. https://www.youtube.com/watch?v=6oWLIVNI6VA

[17] Luke Matin. “The Physics of the Universe.”

[18] Interview with Sean Carroll. “Quantum Mechanics (an
embarrassment).”

[19] Maximilian Schlosshauer, Johannes Kofler, and Anton
Zeilinger. “A Snapshot of Foundational Attitudes Toward Quantum Mechanics.”
Cornell University. January 6, 2013. http://arxiv.org/abs/1301.1069

____________________________________

Supplementary
References (Last Accessed on March 22, 2016):

Why Can’t We See Beyond the ‘Horizon’ of the Universe?

https://ask.fm/Femitheist/answers/132874421995
Promotional Video (“Infinity and Physics”):