Finite musings on infinity

Finite musings on infinity

by Eashwar Thiagarajan

 

Abstract: This essay briefly starts with reflection upon the notion of Mathematical infinity in west and in India. Mathematical intuition of infinity in any era, is shaped by the dominant philosophical discourse on infinity. A case-in-point would be the influence of Brahmavāda (ontology of  Advaita Vedanta) on Bhaskara-II’s articulation and mathematical intuition of infinity. The essay briefly compares the Vedantic view of infinity or Brahman with the Mathematical notion of infinity (stated earlier in the essay), before it concludes with scope for further studies

I: Western notion of Mathematical Infinity

The word infinity brings to our mind a notion of vastness in quantity, as in something that stretches beyond our known comprehension of measurement. Figure 1 shows the linguistic roots of the English word from oxford languages.

 

Figure1: Linguistic roots of the word infinity (Google search)

The Greek word apeiria which means endless, is the inspiration for the French word infinite. As Rudy Rucker observes in his essay - Infinity and the Mind [1 - link]

‘The Greek word for infinity was apeiron, which literally means unbounded, but can also mean infinite, indefinite, or undefined.’

Georg Ferdinand Ludwig Philipp Cantor is considered by many to be the father of Set theory [1845-1918]. In Set theory the notion of infinity is well explored.  Let us briefly have a taste for it.

Say if there is a set A consisting of positive integers starting from 1 till m, i.e. A = {1,2,3...m}. Say there is another set B consisting of all positive, negative integers and zero i.e. B = {m…3, 2,1,0,-1,-2, -3, -4...-m}. For finite value of m, set A has <  ½ as many elements as set B. Clearly, they are not equally sized. However, if m → ∞ both sets become infinite sets, i.e. A= {1,2,3…. ∞} and B = {-∞….-3, -2,-1,0,1,2,3….∞}. But are they of the same size?  We can show that for every element in Set A there is an element in set B, which we can map to, as shown in table 1 for illustration purposes.

A	B
1	0
2	-1
3	1
4	-2
..	..

Table 1: Set A to Set B – One to one mapping

According to Cantor’s hypothesis (in 1874 in Crelle journal and [2]), this implies both the sets have the same size (as m → ∞). This does not imply all infinite sets have same size. Mathematicians have ably demonstrated that there can be infinite sets which are unequal size. A good example would be a set R consisting of real numbers (rational/irrational) which would by definition include a set of integers also in them. So, in effect set R and set B can both be infinite but set R by definition is larger than set B. (There is a more rigorous way of arriving at the proof, which is beyond the scope of this essay). This can raise many philosophical questions surrounding the idea of how multiple infinities can co-exist, and what does it mean to have infinites of unequal size?

This whole introduction to the notion of infinity in set theory, is to highlight the nature of infinity, as viewed in western Mathematics. It is essentially a quantitative notion, which accommodates the possibility of multiple infinite sets to co-exist, with equal or unequal size, as the case might be.  There are several such concepts in Mathematics – like cardinal numbers, transfinite numbers etc. But essentially all of these are  thought formulations, riding on the notion of infinity being an endless quantitative vastness.

II: Mathematical infinity in India

Long before Cantor would make his colleagues uncomfortable with his mathematical notions of infinity, India attained peak comfort levels with infinity. Many of us may know, have read or might have heard from various sources about India’s exploration with the number 0, and its placement in the Hindu numeral system, as being one of its original contributions to the world of mathematics. But ancient Hindu mathematicians were also equally comfortable with zero’s twin namely infinity.

There is no better or simpler mathematical example than the following, to illustrate the point.

Let us define f(x) = |1/x|  ----- (1)

If x tends to 0, i.e. x → 0, what will be the value of f(x)?

In other words, what is the magnitude of the result,  when you divide any finite number by an infinitesimally small number? Today we know that number is undefinably large. However, this problem was not dealt well in the west, whereas in India, this posed no problem. Hindu metaphysics always has an intuitive appreciation indefinably large quantities (e.g. Śukla Yajur Veda 16.54 mantra starting with asaṁkhyāta sahasrāni, which translates to indefinably large). By the time of Bhaskara-II (1114-1185 CE), this was well developed.  See the Bhījaganita śloka below :-

vadhādau viyat khasya khaṃ khena ghāte

khahāro bhavet khena bhaktaśca rāśiḥ॥

This fraction of which the denominator is a cipher, is termed an infinite quantity” [Bhījaganita 2.18, Also [3] – Chapter 2, Page 71].  

How do we intuitively understand this? The very act of division implies successive subtraction. The quotient of any division process, is the number of times, one successively subtracts, before we arrive at a result. Say, if we attempt to divide 1 by 0, we have to make infinite attempts, before we can achieve something of substance. This was explained by Bhaskara -II, in the aforesaid verse. Similar observations can also be seen in Bhaskara-II’s Leelāvati [e.g. verse 46, not discussed here]

Furthermore, in Bhījaganita 2.20, Bhaskara observes that  : -

asmin vikāraḥ khahare na
rāśāvapi praviṣṭeṣvapi niḥsṛteṣu
bahuṣvapi syāt laya — sṛṣṭikāle
anante acyute bhūtagaṇeṣu yadvat

There is no change in the Khahara (infinity) by adding or subtracting, just like infinite immutable (Brahman) which does not have any effect by the living beings entering or leaving it [4: link]

Clearly Bhaskara-II’s mathematical intuition is influenced by Advaita vedanta darśana’s ontology (i.e. brahmavāda), as we shall see in the next section.  

III: Brahman as Infinity in Advaita Vedanta – Brahmavāda’ s influence on Mathematical intuition.

Bhaskara’s statement above regarding infinity, is very reminiscent of the ancient mantra which appears in several locations in the śukla yajur veda.

pūrṇa̱mada̱ḥ, pūrṇa̱mida̱ṃ, pūrṇā̱t pūrṇa̱mu̱dacyate,

pūrṇa̱sya pūrṇa̱mādā̱ya, pūrṇa̱mevā̱vaśiṣyate –

-          Brihadāranyaka upaniśad 5.1.1.

That (Brahman) is infinite, and this (universe) is infinite. The infinite (this universe) proceeds from the infinite (that Brahman). By taking the infinitude out of the infinite, it (i.e. Brahman still) remains as the infinite alone.

(Note: For those interested, the actual Advaita commentary on this Mantra, can be seen in Sri Śankara Bhaśyam for this mantra – Brihadāranyaka upaniśad 5.1.1. It is not shown here.)

We can clearly see this mantra and associated siddhanta must have had a strong influence on Bhaskara forming his conclusion in Bhījaganita. However, we can also translate the word pūrṇa̱ to mean the “Full” or “Whole”. If done that way, it would translate as follows : -

That (Brahman) is Whole, and this (universe) is Whole. The whole (this universe) proceeds from the Whole (that Brahman). By taking the whole out of the Whole, it (i.e. Brahman still) remains as the Whole alone.

This translation will make you think about how it can possibly apply to Set theory with infinite sets and arithmetic of transfinite numbers!

Advaita Vedanta’s view is that the “apparent” reality of names, forms, ideas (including notion of different mathematical infinities) all rests upon the substratum-reality (adhiśtana-satyam) – namely Brahman, which is of the nature of infinite consciousness. By consciousness we are referring to the Sanskrit term bodham or chaitanyam. All ideas presuppose consciousness; hence consciousness is the ground of all thoughts. Even time, space and causality are notions (thought constructs) appearing in consciousness. Hence consciousness transcends time, space and causality. In this sense and this sense alone is Brahman termed as infinity, in Advaita, not in a quantitative sense but in a non-dual ontological (advaita brahmavāda) sense.

Cantor himself had once said: -

“... in truth the potentially infinite has only a borrowed reality, insofar as a potentially infinite concept always points towards a logically prior actually infinite concept whose existence it depends on” [1 - link].

One may think Cantor was influenced by Vedanta but it’s more likely that he was influenced by Plotinus (who himself might have been influenced by Vedic thought). Plotinus is known to have said that : -

“Absolutely One, it has never known measure and stands outside of number, and so is under no limit either in regard to anything external or internal; for any such determination would bring something of the dual into it” [1: link].

(Note: We must be careful not to mistake Plotinus’ statement, as being Advaita in tenet or in temperament– as Plotinus never established the primacy and the non-duality of consciousness like Advaita Vedanta clearly does).

Some of the key differences between Mathematical notion of Infinity and Brahman as infinity is summarized herewith for ease of understanding : -

1.        When Vedic seers speak of Brahman as ananta or pūrṇa̱, care must be taken to understand that this “infinity” is nondual reality

a.      As revealed in māndukya upaniśad verse – 7 - ...śāntaṃ śivam advaitaṃ.

b.      As limitless-existence-consciousness [Tait Up 2.1.1, satyam-jnanam-anantam brahma; See also Appendix -I for discussion on this mantra].

Whereas when mathematicians talk of infinity, as in Set theory (Cantor et al) it’s within the relative order of reality, or realm of thought (dvaita).

2.        Another key difference embedded in the discussion above, is that Brahman as infinity is sentience or consciousness, whereas Mathematical notion of infinity or multiple infinities, deals with categories of insentient objects.

3.        Brahman as infinity is ever revealed as the Self (“aham brahmā̱smi - Brihadāranyaka upaniśad -1.4.10), whereas infinity in set theory is a thought construct appearing to the mind of the individual.

IV - Understanding Maya (from Advaita perspective) and dealing with modern problems on infinity, infinite series and paradoxes.

According to Advaita, Maya is the inexplicable power latent in the fabric of reality (Brahman), which gives rise to infinite categories of objects, and knowledge.  In other words, despite the substratum being non-dual (i.e. Brahman alone Is Real, everything else is only an appearance – brahma satyam jagan mithya – [Sri Śankara in Vivekachudamani – verse 20]), the empirical experience of names, forms, time, space and causation cannot be denied simply because they are mere superimpositions on brahman nor can they be explained with a causal linkage to Brahman. Just as there is no causal link between the events in the movie and the TV screen on which it is played. Both pertain to different orders of reality.

It can be inferred for the purposes of explanation or transaction that this duality or plurality presenting itself to human cognition, thought and experience, is an outcome of an inexplicable singularity, whose nature defies description – this is called Maya in Advaita epistemology, derived from Vedas.

The Greeks usage of the word Apeiron comes more closer to the notion of Maya in Advaita rather than the Brahman of Advaita.  Rudy Rucker in his essay - Infinity and the Mind [1] while observing the Greek notion of apeiron says as follows : -

“The original chaos out of which the world was formed was apeiron. An arbitrary crooked line was apeiron. A dirty crumpled handkerchief was apeiron. Thus, apeiron need not only mean infinitely large, but can also mean totally disordered, infinitely complex, subject to no finite determination.”

It is easier to explain the how different categories or set of infinite elements, can co-exist within the realm of Maya, if we understand Maya as the infinite creative potency of Brahman or freely translated into English as Chaos. Greeks themselves may not have been as comfortable with notion of Chaos or Apeiron, as Hindus were. To quote from Randy Rucker’s essay [1] : -

In Aristotle's words, "... being infinite is a privation, not a perfection but the absence of a limit... There was no place for the apeiron in the universe of Pythagoras and Plato. Pythagoras believed that any given aspect of the world could be represented by a finite arrangement of natural numbers (where "natural number" means "whole number"). Plato believed that even his ultimate form, the Good, must be finite and definite. 

This is in complete contrast to Hindu mindset, which was since the days of Rig Veda or before was comfortable with the notion of Chaos. See Rig Veda 10:129 – nāsadīya sūktam for more on this. An excerpt is provided below : -

..táma āsīt támasā gūháḷam ágre 

At first there was only (chaos) darkness wrapped in (chaos) darkness. 

This discomfort of the Greeks and subsequent western thinkers with Chaos, might have resulted in the delayed development of comfort level with Mathematics of infinity (in comparison to the Hindus).

As per Advaita Vedanta, when the individual attempts to view Brahman, through the lens of Maya (i.e. time-space-causation) we have the vision of Ishwara or Saguna Brahman or God. This epistemological basis for Ishwara, has profound philosophical value to explain or perhaps inspire further Mathematical and scientific discoveries. Any conundrum involving time-space be it the quantum entanglement (EPR paradox) or Zeno’s race course paradox (i.e. summation of infinite tasks leading to finite result) or its modern illustration namely the “Thompson’s lamp puzzle” – can be demystified, once we postulate the presence of an omniscient, omnipresent, omnipotent being identical with the fabric of empirical reality (of time and space).

Take the example of Thompson’s lamp puzzle. (The ensuing discussion is inspired by reference [4 – page 120]). Imagine an electric lamp with push-button switch. Imagine a period of finite time say an hour, within which the push-button switch is turned on and off, successively but in vanishingly small-time intervals like 1/2 an hour, 1/4^th  of an hour, 1/8^th of an hour and so on. At the end of the hour, we will know for a fact that Thompson’s lamp is on or off. But this raises two uneasy questions : -

a)       How can we predict the lamps final state ahead of the hour? (in other words what is the last state in an infinite series)

b)     How can infinite set of supertasks (supertask here is turning lamp on-off, in infinitesimally small chunks of time) can be completed in a finite time? (similar to Zeno’s race course paradox)

Let us take question (b) first. After an hour the lamp will be off or on – it is a bounded possibility with two discrete states. This suggests that the super-task of turning the lamp on/off at infinitesimal chunks of time, is within realm of possibility. This in turn demands the possibility of a being, who can complete that supertask. This being ought to be God or God like ~ omniscient, omnipotent, omnipresent. Thus, God as the instrumental cause (nimitta kārana) for the completion of the sequence of infinite supertasks, is a necessity to explain the bounded outcome of the experiment (i.e. lamp is either off or on after an hour).  With this postulate, we can also say God (the super task performer) will know (ahead of time) the final state of the lamp, at the end of infinite sequence – the answer to the question (a).

Similarly, if we take quantum entanglement, the question of how information traverses faster than the speed of light will be raised. This is a mystery only if we stick with the notion that the two particles are “apparently” separated by time and space. The fact that they are “entangled”, can be explained if we hold the Advaita view that time-space itself as being identical with Ishwara’s Sakti (i.e. Maya). This implies that the two particles are always instantaneously connected in and as identical with Ishwara.

The discussions in this section – IV, are simple thought experiments with regard to infinity, paradoxes etc., riding on the philosophical insights, enabled by Advaita Vedanta.

V: Concluding comments and scope for further study

So, when Vedas says pūrṇa̱mada̱ḥ, pūrṇa̱mida̱ṃ… pūrṇa̱mevā̱vaśiṣyate it clearly has more profound implications than mere set theory-based arithmetic of transfinite numbers !

Having said that we can see clearly that Vedas have had unmistakable influence in Bhaskara’s Mathematical intuition.  It is this strong Vedantic underpinning in India, which had led to profound discoveries in Mathematics and more significantly the great philosophical ease with which the Hindu mathematical intuition tackled difficult topics such as infinity (unlike western thinkers).

Scope for further studies: Some of the works like Yoga Vasiśta, can inspire mathematicians and scientists alike to consider the impact of how field of unitary consciousness shapes the empirical notion of reality. For example, those interested in strong ontological basis for multi-verse can find them in Yoga Vasiśta. Another Vedic term which maps to the word infinity is the word bhūma, seen in chāndogya upaniśad. This teaching – i.e. bhūma vidya can offer interesting insight into further exploration of “infinity”.  The understanding of the anirvachanīya khyāti (inexplicable phenomena) of Maya (as śakti of Brahman) in Advaita siddhanta can offer further philosophical impetus to sharpen or shape our mathematical/scientific intuition

References

1.      Rudy Rucker “Infinity and the Mind – The Science and Philosophy of the infinite”

https://math.dartmouth.edu/~matc/Readers/HowManyAngels/InfinityMind/IM.html

2.      Edward R. Scheinerman, “The Mathematics Lover’s Companion: Masterpieces for Everyone”, Yale University Press, 2017 – Chapter 8.

3.      Charles Seife “Zero – The biography of a dangerous idea”,  Viking publications, 2000.

4.      Bhijaganita translation from

https://medium.com/@resanskrit/the-concept-of-infinity-explained-by-bhaskaracharya-81772037e4b7

5.      Gary Hayden and Michael Picard, “Paradoxes – adventures in the impossible”, Metro books New York, 2014

 

Appendix – I – Brief discussion on Brahma svarūpa lakśana (as per Advaita Vedanta)

Brahman or the ultimate reality revealed in Vedas, has the svarūpa lakśana (essential indicators) as outlined in Taitireeya upaniśad – namely satyam, jnānam, anantam brahma (Tait Upanisad 2.1.1).  Familiarity with this Brahmavāda, shall help understand better Advaita view of infinity. So, this Appendix section -I, briefly outlines the meaning the terms – satyam, jnānam, Anantam, as used in Taitireeya Upanisad 2.1.1. 

1.       Here the words satyam implies – existence (astitvam), or substratum (adhiśtana) of all names and forms. When we say the “Pot is there on the table”. The “is”-ness of the Pot, is not an essential attribute of the pot, but of Brahman, which as the substratum of name and form, lends a notion of reality to the name and form (e.g. like the TV  screen being the substratum to the movie playing on it) [for more discussion see BG 2.16 Śankara Bhaśyam, Tait Up 2.1.1 Śankara Bhaśyam]. The reality of the TV cannot be denied by anything that appears on it, similarly satyam i.e. pure existence is never negated by human reason, time, space and causality. This is a key difference between Advaita and other schools of thought, which lend themselves to the notion that phenomenal existence, has intrinsic reality of its own

2.       The word jnānam implies – the principle of consciousness, underlying every act of cognition (Kena Upanisad – 2.2.4). In other words, Brahman is pure consciousness, on which notions of names, forms, time, space, causation appear, subsist and disappear. The word jnānam conveys to us a principle of intelligence that is has the potentiality for manifestation. Brahman is pure consciousness, i.e. purity here implies unqualified existence of the nature of pure awareness, enabling but transcending all human conceptualization, as it is the very consciousness that illumines the mind of the thinker.

3.       Next comes the word anantam implies –endless. This word Anantam is often translated as infinity in English.  But as discussed in the article thus far, it’s not quantitative infinity (as described in Mathematics). Here it implies it is not limited by time-space and causation. As the very notion of time-space appears in consciousness, it (i.e. consciousness) transcends time, space and hence its infinity in an eternal-changeless sense.

To summarize the Brahmavāda of advaita : -

·       satyam is jnānam

(Existence is Awareness/Consciousness);

·       jnānam is anantam

(Consciousness is beyond limitations);

·       satyam is anantam

(Existence is beyond limitations, conditions)

Any one of the indicators by itself can reveal Brahman to us, as the Infinite Reality – eternal and changeless Self (“aham brahmā̱smi - Brihadāranyaka upaniśad -1.4.10). This Self-revelation leads to freedom from all suffering and limitations.