Infinity is an important concept in mathematics. Strangely, unlike most other branches of mathematics, it doesn’t represent anything in the natural world – unless it tells us about God himself.

Mathematics isn’t really a science. It is an entirely independent intellectual pursuit that just happens to reflect reality. When new branches of mathematics arise, they initially look like mere games with numbers. However, eventually they turn out to be useful and often invaluable for advancing some new area of discovery or invention. It seems that all the mathematics that we “invent” in our imagination are actually a discovery about something that already exists. For example, fractals, which require computers to explore properly, don’t merely produce pretty patterns; they describe the way that plants grow – and three-dimensional fractals are found in the distribution of blood vessels in our lungs.

^{1}

This kind of tie between mathematics and the real world is no longer surprising. Galileo first spotted the fact that “the universe … is written in the language of mathematics.”

^{2}Some terms in this language are so common that they have their own symbol – such as infinity (∞), the ratio of a circle’s circumference to diameter (π), the exponential growth constant (e), and the “imaginary” square root of −1 (i). We can demonstrate practical uses for most of these: π for calculating circles, e for calculating compound interest, and i to calculate the electric current needed to boil all the kettles during a World Cup halftime.

^{3}We might reasonably expect that every new discovery in mathematics will eventually lead us to a new understanding of some aspect of the universe – but infinity remains an exception.

## 5-minute summary(More videos here) |

### Is infinity absurd?

Infinity is a mathematical necessity, but it has no practical correspondence to reality. Infinity is important in the Bible for describing the eternity of God and his infinite greatness (e.g., Exod 15:18; 1 Kgs 8:27). And infinity is implied by any system of numbers that can continue without limit. But in the world described by the sciences, there is nothing that corresponds to infinity.We used to think that space was infinite, but the consensus now is that the matter in the universe spreads across about ninety billion light years.

^{4}The empty space beyond may technically be infinite, but this is a nonpractical example of infinity, because space without matter is not really anything except direction.

Time is also finite. For time to exist, things need to happen, to mark time. An expanding universe, such as the one we live in, will eventually cool down until no energy or movement remains. If the density of matter is high enough to make the universe collapse again into a singularity, it might then burst out in a new big bang and go through this cycle indefinitely. However, time itself still wouldn’t be infinite, because there is no continuity from the universe before a big bang into the universe after a big bang. That is, there is no information or memory of anything or even a single atom that existed in the previous universe that will be present in the next. Everything literally starts again, including time.

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Even the concept of the infinitely small has been abandoned. All matter is made of atoms about 10−10 meters in diameter, which are made of quarks and gluons; even light is made of discrete units (photons), so nothing can be shaved smaller infinitely. The exception may be time, though ancient Greek philosopher Zeno (in the fifth century BC) demonstrated the absurdity of infinitely small units of time in the paradox behind the fable of the tortoise and the hare.

^{6}The hare gives the tortoise a head start, but he never catches up because every time the hare has reached where the tortoise was, the tortoise will have walked a little farther. As the distances get infinitely smaller, the intervals of time will become infinitely smaller too, so that the hare will never be able to catch up with the tortoise. This conclusion is clearly absurd and therefore wrong, and the simplest explanation is that there is something wrong with the concept of infinity.

A modern mathematician, David Hilbert, added a new set of paradoxes to show that the concept of infinity results in absurdity; these were based on a hotel with an infinite number of rooms (it is worth watching the animation).

^{7}Some mathematicians (such as Doron Zeilberger) have similar suspicions about infinite numbers. They suggest there must be a maximum number after which the next number is zero. For a brief period, when my daughter was about four years old, she though there weren’t any numbers beyond sixty-three – perhaps this was a significant mathematical insight that has yet to be proven.

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### Zero solves infinity

Interestingly, the number zero has an entirely different history. Whereas infinity was discussed by ancient Greek philosophers, zero wasn’t discussed by scholars until about a thousand years later, by Indian mathematician Aryabhata in about AD 500. The mathematics of zero was gradually developed by others, including Brahmagupta a century later, who initially rejected it as a number, because dividing any number by zero always gave the same answer – infinity. However, zero was useful in practice, unlike infinity, so zero survived. Actually, zero had already been used as a place marker in practical calculations by merchants since Sumerian times (about 6000 BC), so in this case it seems practical applications preceded the mathematics, rather than the other way around.^{9}

Newton benefited from this work on zero and combined it with his new mathematics of calculus to solve Zeno’s paradox. Instead of trying to calculate an infinite series of time divisions, he combined them into a single set that includes constant change. Newton also developed the mathematics of gravity, which showed that Galileo’s insight was right: the universe really does run according to mathematical rules. Newton didn’t conclude from this that God isn’t needed to run the universe. Instead, he concluded that God created the universe in a mathematically perfect way. However, he rejected the suggestion of Leibniz that God simply wound up the universe and left it to run alone. Newton concluded: “This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent Being. … This Being governs all things, not as the soul of the world, but as Lord over all.”

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### Infinity in the Bible

This surprising lack of infinity in any scientific field is contrasted by the Bible, where infinity is a very important concept. However, even there it was disputed by the Sadducees, who said that there was no such thing as eternal life because the Old Testament didn’t mention it. Their theological opponents, the Pharisees, did find eternity in Exodus 15:18, which said the Lord would give the Israelites a home where he “reigns for ever and ever.” This text said that God was eternal, but the concept of eternal life was applied also to individual believers (Ps 52:8; 145:1; Dan 12:3). The Sadducees argued that these phrases only referred to the whole of this life. However, the Pharisees replied that if this were so, it would only say “for ever,” and the addition of “and ever” showed that it means the everlasting next life too.This was such an important discovery for the Pharisees that they inserted the phrase from Exodus 15:18 into the start of their daily prayers. Modern Judaism is descended from the Pharisees (the Sadducees all died in the destruction of Jerusalem in AD 70), and modern practice holds a clue about the ancient usage of this phrase. When Jews say this phrase in prayer, the custom is to speak it in a semiwhisper even to this day – though no one knows why. My theory is that it wasn’t in the original prayer said by both Sadducees and Pharisees. So when the Pharisees added it, they did so under their breath as a kind of rebellion. Christians also adopted the phrase “for ever and ever,” and it became an important statement about the eternity of Jesus (Gal 1:5; Heb 1:8; 13:21; 1 Pet 4:11; 5:11; Rev 1:18; 4:9-10; 5:13, 14; 11:15) and the reward of individual believers (Rev 22:5).

Perhaps mathematics and the Bible can help each other solve the conundrum that infinity is a necessary element of mathematics and yet it has no role in practical science. If mathematics does reflect reality (as it appears to do, in a remarkable way), we might expect there to be something that does correspond to infinity in reality. We have seen that infinity is an important concept in the Bible, so could it be that God is the reality that is witnessed to by the mathematics of infinity?

The universe has intimations of infinity in both the unimaginably tiny components that make up matter and its vastness of space and time, and yet it lacks actual infinity. But if we add the factor that it was created by an infinite God, who will perpetuate elements (or persons) that belong to this universe forever, then the concept of infinity gains a role in the reality of this universe.

I don’t mean that infinity is such an awesome concept that the existence of this idea proves we have been inspired by an infinite God – this was the basis of Descartes’ ontological proof for the existence of God. We aren’t really awed by infinity anymore, so the argument no longer works. What I mean is that infinity is an inescapable aspect of our mathematics and our thinking, and yet there is nothing corresponding to it in reality unless we acknowledge the reality of God.

Mathematics appears at times to be simply playing with numbers. But there is a remarkable correspondence between reality and the strange and complex mathematical concepts that have been discovered. It seems, then, that mathematics may be not only a silent witness to the complexity of God’s creation, but a witness also to God himself.

### Summary

• Mathematics is a theoretical invention that happens to reflect reality.

• Even strange aspects such as imaginary numbers eventually find practical applications.

• Infinity is an exception: it has no correspondence in practical science.

• Proposal: Infinity describes some aspects of God but no aspects of the physical universe. This implies that God is part of the reality of the universe as described by mathematics.

^{1^}See Wikipedia, “Fractal” (tinyurl.com/WikiFractal).

^{2^}See Wikipedia, “The Assayer” (tinyurl.com/WikiAssayer).

^{3^}See Chris Budd, “Complex Electricity,” Plus, November 24, 2017 (tinyurl.com/ComplexElectricity).

^{4^}See Chris Baranuik, “It Took Centuries, but We Now Know the Size of the Universe,” BBC Earth, June 13, 2016 (tinyurl.com/SizeUniverse).

^{5^}See Stephen Hawking, “The Beginning of Time” (tinyurl.com/HawkingBeginning).

^{6^}See Wikipedia, “Zeno’s Paradoxes” (tinyurl.com/ZenoPara).

^{7^}See the fun video by Jeff Dekofsky, “The Infinite Hotel Paradox,” Ted Ed (tinyurl.com/p84db45).

^{8^}For a detailed discussion of whether actual infinity occurs in reality, see Bradley Dowden, “The Infinite,” Internet Encyclopedia of Philosophy (tinyurl.com/Real-Infinity).

^{9^}See “The Development of Zero,” 3010tangents (blog), February 23, 2015 (tinyurl.com/FindingZero).

^{10^}Isaac Newton, Principia 3, discussed in Wikipedia, “Religious Views of Isaac Newton” (tinyurl.com/NewtonReligious).

^{This was previously published in a similar form in Christianity magazine }

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