• Danny Raj M

The infinitude of nature vs the nature of infinity

As scientists, we make observations of physical or biological phenomena and posit naturalistic explanations. These are explanations that satisfy the presumed laws of nature, are empirically verifiable (to some level), and often reproducible under a set of attainable conditions. Examples range from understanding why the planets move the way they do, to finding out the conditions under which a disease (like COVID) becomes a pandemic.

Naturalistic explanations do more than just explain a phenomenon at the observed level. Very often it could provide a deeper understanding of the implications associated with the phenomenon, way beyond the space and time scales of what was observed. For instance, in one of the problems we study, we ask: How do individual fish interact to produce the beautiful schooling phenomenon at the level of the group, that is both orderly and cohesive? Exploring the properties of schooling helps one understand the fish’s ability to survive in the wild—to do efficient foraging, predator evasion, etc.

However, recently I have been asking myself:

Can naturalistic explanations exhaustively explain the observations? For instance: We still do not understand how molecules self-assembled to form the first self-replicating system (like a cell). Our current understanding of the processes underlying this assembly reveal that the probability associated with such an assembly (from abiotic materials), is staggeringly small. Now, when the chances of an event to occur are extremely low and when we know that such events have successfully (and in some sense, robustly) come to pass, how sure should we be about the sufficiency of natural explanations in explaining the causes of such events.

Put in other words, we can ask: Is reality, as we perceive it, comprehensively explainable and (or) composed only of natural processes?

If the answer to the above question is in the affirmative, then science is already doing its best in trying to find out the most reasonable natural explanation. However, if it is not, then in our search for purely naturalistic explanations we may be missing out on more fundamental causes that are at play. This may limit our understanding or even mislead us from taking a more holistic approach to the process of learning and ultimately appreciating reality.

How do we resolve this? Do we go about testing the sufficiency of natural explanations for every observed phenomenon? That would be an extremely hard task to do. Natural explanations seldom give us a full understanding of a given process. It gives us only a partial understanding; but one that is substantial and sufficient to make one place their trust on the enterprise of explaining the observed phenomena through natural explanations. Now, the missing knowledge about a phenomenon could be due to the yet to be discovered natural explanations or causes that are beyond the ‘natural’. Disentangling the two would be an incredibly hard task to do. Therefore, we need to find a different route to answering this question that is not specific to a particular phenomenon.

Alternatively, one could ask:

Was there ever a phenomenon in time, space, or matter for which natural explanations were not possible?

By answering this question, we could try to establish if there is at least one scenario where natural explanations are insufficient. If it is so, then we have reason to believe in the existence of causes or explanations beyond natural which could imply that the assumption that science could eventually explain all of reality, is incorrect. If it is not the case, then we can rule out the need for anything more than what we already know and regard as natural.

Thinking about this took me a very old question that philosophers have debated over a thousand years: Is our universe eternal (had always existed) or created? If we assume that the truth of the origins of the universe is knowable, then we can think of two (most obvious) answers that to this question:

A. It has always existed from eternity past or

B. It began to exist at a certain point in the past.

Now, if the universe is eternal, then one could in essence argue that natural explanations that are part of the fabric of our universe are also eternal. This could also mean that any event with a non-zero probability, given enough time will eventually come to pass.

However, if the universe began to exist at some point in the past, then natural explanations could also have a beginning. This would take us to a point in time, at or beyond which the so called natural may not explain the state of being.

Hence, we could repose our original question, ‘are natural explanations sufficient?’, as, ‘is the universe or reality as we know it, eternal or finite?’.

Let us start with each of the answers—eternal or finite—to find out how well each answer corresponds with reality.

What if past was eternal?

If the universe had existed forever, then in principle an ‘infinity’ of time (or discrete time-events) has passed before the 'present'—to reach the 'now' that we are in. But infinity, is not simply some large number; it is, as large a quantity can be. You may decide to put a million zeros after one and you will still be left with a number that is 'smaller' than an infinity. This translates to the question: how long should one wait to cross an infinity of events to reach the present? It appears that no matter how long one waits, it may never be enough. So, assuming an eternal universe requires one to deal with the nature of infinity.

But is it ever possible to cross an ‘actual’ infinity of things?

Consider an interval [a, b], where a and b are two time points. The point a could be for example 9 AM on a Sunday morning and b could be 10 AM on the same day. If any snapshot of time may be regarded as an 'event' where a snapshot is simply a time point corresponding to a real number between a and b, then there are an infinite such points between a and b. However, life goes. We move, in time, past many such a and b, where we cross an infinity of “events”.

However, the above is not an apt comparison to our eternal universe problem. While we have surely established that when we move through time, we traverse an infinity of events, the time itself is bounded (between a and b in this case). Just imagine the interval boundary 'b' moving towards the right (+ time direction) in a speed faster than the 'speed of time'. Then no matter how long one spends, 'b' can never be reached. Similarly, 'a' can be moved to the left and one will soon have a 'larger' infinity of past events that could have not been crossed to reach the 'now' of the present. Alternatively, without any loss of generality, one could have discretized the time between any interval ‘a’ and ‘b’ (which was otherwise continuous). That will allow one to see very clearly the difference between an eternal past which is an infinite interval and the time in-between a finite interval like [a, b].

From the above arguments, it appears that an eternal past may not be feasible in a real world since it would require one to deal with an ‘actual’ infinity of events. This is not entirely surprising since infinity is a mathematical concept to denote a boundless quantity and bringing it into the physical world we live in, often produces absurdities. Another famous illustration to show the counter-intuitive-ness of the infinity in a real physical reality is, ‘the paradox of the grand hotel’, introduced by the famous mathematician David Hilbert.

Taken from: https://www.scienceabc.com/pure-sciences/what-is-the-infinite-hotel-paradox-definition-examples.html

Hilbert’s paradox of the grand hotel

Hilbert’s grand hotel has infinite rooms, which are 'all' completely filled with customers to begin with. The paradox arises, when new customers come to the hotel and the skilled manager of the hotel finds room for his new customers in the already filled hotel. Let us look at a number of different scenarios.

Scenario 1:

A new customer drops by, and the hotel manager says, 'our rooms are filled but there is still room' and makes room number 1 available for the customer by asking every one of his existing customers to move to the next room. A person in room is asked to move to room .

Scenario 2:

A bus filled with an infinite number of customers arrives, each person with an identity of increasing natural numbers (Ex: 1, 2, 3, etc.). The hotel manager still finds room for all infinity of his new customers by asking the existing customers to move to a new room whose number is twice that of the current room: , which produces an infinite number of empty rooms for the new ones to occupy.

Scenarios 1 and 2 show how bringing a real infinity into the physical universe can result in absurdities like a hotel that can be both fully filled and have space for new customers at the same time. Does this mean, this hotel can always be filled? Let us look at scenario 3 for an interesting case that demonstrates the limit to filling the rooms in the ever-accommodating grand hotel.

Scenario 3:

A bus comes with an infinity of new customers. But this time each person has an identity corresponding to a real number between 0 and 1 (ex: 0.520012342…, 0.2323232… etc.). Any possible real number between 0 and 1 that one could imagine would correspond to a unique person in this bus. This time the manager says that he will not be able to find room for all the new customers. This is because, no matter what strategy the manager uses to reallocate his existing customers, he understands that he will never be able to produce enough rooms for his new customers.

This has to do with the difference in the nature of the two infinities here: the infinity produced during reallocation and the infinite new customers. While the former is what is called a countable infinity, the latter is an uncountable infinity. If you are wondering why an infinity is called “countable”, here is one way to think about it. Say, if you start counting natural numbers in increasing order, 1, 2, 3 and so on. You will be able to cross any arbitrary bound in a finite time. Say, if your bound is 10000. You will be able to count 10000 in a certain time; it could be 10000 seconds if you count one number per second. However, try counting real numbers. You will never be able to reach a bound in a finite time (irrespective of how small you set your bound to be). Since, you keep finding more and more numbers by adding numbers after the decimal point (like, 0.9, 0.99, 0.999, 0.9999, …).

While scenario 3 is finally an example where it is impossible to find rooms in the hotel, it reveals the ever so mysterious nature of infinity. Also, as a side remark: we find the unphysical, counter-intuitive effects of infinity to be present even when we do not consider other effects in the physical world such as the time required to reallocate existing or register new customers.

Hence, it appears that infinities could be troublesome when we bring them into our reality. Does this imply that infinities can never really be used when we deal with the physical universe?

We use infinities all the time.

The difficulty in understanding real infinities in our physical world does not really prevent us from using them in our analyses. We use infinities in chemical engineering very often, and almost all the time, it is used to denote the 'largeness' of one quantity when compared to another. Calling it an infinity instead of simply a large number allows one to analytically simplify the governing equations, which makes the analysis meaningful at the interested asymptotes.

Some instances include the 'semi-infinite approximation', where certain dimensions in space are much longer than the others, 'long-time limits' - where the dynamics after a long wait are modelled (where the time of wait is much greater than the time scale of the system), 'creeping flow approximation'—an implicit use of infinity where the value of Reynolds numbers is assumed to be zero (ratio of inertia to viscous; viscous effects are infinitely more stronger than inertial)—which renders the equations describing fluid flow, linear, leading to interesting consequences.

In all these examples (and in many more), an infinity is used in place of a really large number, and it helps in simplifying the equations describing the phenomenon. And the infinity does not imply that there are actual infinity of either (matter) space, time, or the energy available.

Taken from Wikipedia

Well, it all started with a big bang!

Our approach so far, to the thinking about the possibility of an eternal universe, has been mathematical and philosophical. To get a scientific perspective, we can look at what modern day cosmology must tell us in this regard.

Technological developments in the Twentieth century has made it possible for researchers to make precise calculations of distances of galaxies from the earth and detect small amplitude fluctuations of the background radiation. This has helped cosmologists to discover two important phenomena: red-shift and cosmic microwave background radiation. These measurements support the so-called ‘expanding model’ for the universe, which was earlier regarded as an artefact in other theories, such as Einstein introducing a cosmological constant to forcibly achieve an invariant universe.

Now, taking the expanding universe back in time, only to shrink it, we eventually reach a singularity, a point at which space and time, as we know, began to exist. This singularity is the first point in time, at and beyond which, all our nature’s laws are non-existent. Most modern cosmologists now believe that the best guess for the origin of the universe is via a singularity that expanded at very high rates in the beginning of time and space, called the Big-Bang. They take the theory forward to note that the rapidly expanding universe will eventually suffer a heat death and fizzle out.

Hence, according to modern day cosmology, our universe is finite in age, having come into existence at a singular event. Note: there are other competing theories for the origin of the universe exist, like the cyclic model of the universe put forth by Paul Steinhardt, or the multiverse theory famously advocated by Sean Carroll.

Logic and Reason as a means to the Truth—a digression

Before we go any further to understand the implications of a finite universe, let us pause to review what we have done so far. Just with logic and the idea of an actual infinity, it was possible (for Philosophers) to predict the finiteness of the universe, which after a thousand years, many scientists accept as reasonable based on empirical evidence.

Isn’t this rather remarkable?

Herein, I do not imply that philosophy is “greater” or “all that is needed” etc. as philosophers can get their predictions or explanations wrong—since not everything that is logically consistent is expected to exist in reality. However, what is striking to me is the fact that the truth in some part is conceivable to mankind through the use of logic and reason, even in the absence of purely empirical evidence.

Now, coming back…

Looks like the universe began to exist.

If the universe began to exist, then it leads one to ask: what caused the universe to exist? Since, our common understanding (intuition) of reality is that it is causal, i.e., anything that begins to exist has a cause. Here, it is important to remark that this ‘common understanding’ we are referring to, is part of our physical reality and need not be valid beyond its inception. However, if we were to assume a cause-less beginning for our universe our discussion would end right here, leaving a curious mind rather unsatisfied! Hence, I think it is reasonable to move forward and ask: if a cause exists what are the necessary properties of such a cause?

For starters, it should be outside the universe, beyond space and time and the so-called natural. Secondly, it should be the source of all energy that is present in this reality: since our reality seems to be bound by energy conservation. And thirdly, it must have logic and reason, as the workings of the universe appear to follow laws, and these laws, or the order at the most fundamental level and some of its effects can be found using the faculties of our mind.

Several interesting questions arise at this point, such as,

  • What is this cause?

  • Would we even be able to comprehend anything apart from the natural, about this cause?

  • If something apart from ‘natural’ exists in the cause, then is there not a possibility of our world being more than just natural?

  • And does this cause continue to affect our natural world?

If it does affect our reality, shouldn’t we be looking at the very definition of science and its philosophical foundations and redefine what we mean by the word ‘natural’?


Thinking about nature surely took me down a rabbit hole! I am off now, back to my research, to study the spontaneous formation of order in schooling fish.


Kalam cosmological argument: [Much of what I have discussed was inspired from the writings/lectures of Dr William Lane Craig. See link below for more information]


Hilbert’s hotel: [Veritasium YouTube channel explains this paradox well]


Big Bang theory:



End of the universe:


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