A never-ending marvel

An always-evolving, never-ending, always-fresh fractal landscape from Vimeo:

Surface detail from subBlue on Vimeo.

The structures that evolve out of this are incredible. This is both jamāl and jalāl, both mysterium fascinans and mysterium tremendum.

Counting, uncounting, and dust, part three

Sequences and convergence
We saw earlier that the rational numbers Q are much more versatile than the natural numbers N for doing a wide variety of mathematical operations. However, there is still one thing the rational numbers cannot do adequately: it is possible to construct sequences of rational numbers that do not converge to a rational number. In formal terms, the rational numbers are not complete.

Why does this seemingly obscure mathematical quirk matter? Because this is the foundation of calculus. The ideas of sequences and limits are fundamental to calculus, and if such limits don't exist meaningfully, then neither does calculus. But there is another, intuitive reason for why completeness matters.

One proof that the square root of 2 is an irrational number

Here is an elementary proof of the fact that \( \sqrt{2} \) is irrational. But what does that mean? An irrational number is, well, a number that is not rational! So let's begin with a definition of the set of rational numbers \( \mathbb{Q} = \{ \frac{p}{q} | p, q \in \mathbb{Z}, q \neq 0 \}\). As we saw earlier, this is to be read: \( \mathbb{Q} \) is the set of all fractions \( \frac{p}{q} \) such that both \( p \) and \( q \) are integers, \( q \) not being \( 0 \), and with \( p \) and \( q \) in lowest terms. That simply means that they have no prime factor in common. (Let us recall that we can always write every natural number uniquely as a product of a number of prime factors.) What this definition means, at least for us right now, is that we're not going to be looking at numbers like \( \frac{12}{10} \); we're going to assume that all numbers have been reduced to their lowest form, like \( \frac{6}{5} \).

Another minor note: by our definition, a number like \( \frac{1.5}{3} \) would not count as a rational number. This is not true, of course, because we know we can simply rewrite \( 1.5 \) as \( \frac{3}{2} \), and then our number becomes \( \frac{\frac{3}{2}}{3} \), which is the same as \( \frac{1}{2} \), which is clearly a rational number. 

So when we say a number is irrational, we're saying it's crazy in some really fundamental way: there really is no way to write it as the ratio of integers. (If you're wondering why we write \( \mathbb{Q} \) for the rational numbers, it's from the word "quotient".)

The proof of \( \sqrt{2} \) being irrational is a proof by contradiction, that is to say that we shall suppose that it is in fact rational, and show that this necessarily leads to a logical impossibility.

Counting, uncounting, and dust, part two

Extending countability
Having looked at the definition of countability and the natural numbers N, it is but natural (no pun intended) to ask how far this concept can be extended. The natural numbers are pretty awesome, after all, but there are a huge number of things you can't do with natural numbers, including something as basic as subtraction.

Integers
The only operations under which N is closed are addition (that is, you can add two natural numbers and the result will be a natural number) and multiplication, but the latter is pretty limited in the sense that it's basically just repeated addition. So all we can do with natural numbers is "go forth and multiply"; there is no way to reduce them. In other words, there is no "inverse"operation to addition under which N is closed.

Counting, uncounting, and dust, part one

I've wanted to write about this for a while but never really got the chance. One of the coolest things about modern mathematics is infinity: we have found all sorts of ways to study and classify different notions of "size" and "number" that can result in the most counterintuitive results imaginable. I want to look at a few examples, starting today.

Counting
Let's begin with the natural numbers, the set N = {1, 2, 3, ... }. This is "clearly" an "infinite" set, as indicated by the ellipsis. What does "infinite" mean in this case? Now we know that we can add natural numbers together to produce natural numbers; more simply, we can add the number 1 to a natural number n to produce a new natural number n + 1. N is infinite in the sense that there is simply no "end" to it, at least in one direction (because there is such a thing as the smallest natural number, which is 1). To rephrase that, there is simply no way to list every single element of N, whereas for any finite set, no matter how large it may be, you can eventually list out every element.

Before we go any further, let's actually get a handle of what counting means. In real life, when we have to count out a bunch of things, we essentially point at each item once (and only once) and call out the natural numbers in order. We are in effect uniquely indexing every item with one (and only one) of the natural numbers. There is nothing twisted or weird about this; this is the most "natural" (pardon the pun) way of counting things.

What this definition of counting allows us to do is to compare the size of different sets: we can know that a sack of three potatoes is the same numerical size as a sack of three apples because we can index both of them with the set {1, 2, 3}. (It should be clear that it does not matter which potato we're pointing to when we call out a particular number; all that matters that we point to each potato only once and we call out the numbers in order only once.) Now so far this sounds like a ridiculously overdone way of stating the utterly obvious (and I've not even pulled out real mathematical terminology or notation!), but the utility of such abstraction will become more and more evident as we journey along.

Godā Stuti, conclusion and English translation

kavi-tārkika-siṃhāya kalyāṇa-guṇa-śāline |
śrīmate veṅkaṭeśāya vedānta-gurave namaḥ ||

Godā Stuti, 29

iti vikasita-bhakter utthitāṃ veṅkaṭeśāt
bahuguṇa-ramaṇīyāṃ vakti Godā-stutiṃ yaḥ |
sa bhavati bahumānyaḥ śrīmato raṅga-bhartuḥ
caraṇa-kamala-sevāṃ śāśvatīm abhyupaiṣan || 29 ||

Godā Stuti, 28

śatamakha-maṇi-nīlā cāru-kalhāra-hastā
stana-bhara-namitāṅgī sāndra-vātsalya-sindhuḥ |
aḷaka-vinihitābhiḥ sragbhir ākṛṣṭa-nāthā
vilasatu hṛdi Godā Viṣṇucittātmajā naḥ || 28 ||

Resilience and complex systems

Another article in SEED Magazine, by Carl Folk, on the ecological term resilience applied to other complex systems like the global economy. The author defines resilience as "the capacity of a system—be it an individual, a forest, a city, or an economy—to deal with change and continue to develop," in other words, to respond to perturbations and even catastrophes in innovative ways.

So what does this mean? For one,

"systems are understood to be in constant flux, highly unpredictable, and self-organizing with feedbacks across multiple scales in time and space. In the jargon of theorists, they are complex adaptive systems, exhibiting the hallmark features of complexity."

This ability of a complex system to adapt to variations does not necessarily mean that such adaptation is smooth or predictable. Indeed, the non-linearity of feedback loops means that complex systems can behave in what seems erratic (from the perspective of a linear model). This includes wild oscillations between dramatically different positions, such as a lake fluctuating between hyperoxygenation and algal bloom, or for that matter the Earth going from polar regions covered in dense vegetation (as during the Mesozoic era) to iced-over tropical regions (Snowball Earth).

The risks of interconnectedness

This is a tremendously interesting article in SEED Magazine, titled "On Early Warning Signs", by George Sugihara that draws on the author's background in biology to depict some of the limitations of the modern tendency to use linear systems (plus perturbations) to model fundamentally non-linear systems. Among other things, non-linear systems show high levels of interconnectedness that tends to fluctuate (technically heteroskedasticity), which makes predicting their behavior using linear models very difficult.

The author further notes:

"Leading up to the crash, there was a marked increase in homogeneity among institutions, both in their revenue-generating strategies as well as in their risk-management strategies, thus increasing correlation among funds and across countries—an early warning. Indeed, with regard to risk management through diversification, it is ironic that diversification became so extreme that diversification was lost: Everyone owning part of everything creates complete homogeneity. Reducing risk by increasing portfolio diversity makes sense for each individual institution, but if everyone does it, it creates huge group or system-wide risk. Mathematically, such homogeneity leads to increased connectivity in the financial system, and the number and strength of these linkages grow as homogeneity increases. Thus, the consequence of increasing connectivity is to destabilize a generic complex system: Each institution becomes more affected by the balance sheets of neighboring institutions than by its own."

An interesting analogy drawn from nature, this time from plants and pollinators:

"[T]he same hierarchical structure that promotes biodiversity in plant-animal cooperative networks may increase the risk of large-scale systemic failures: Mutualism facilitates greater biodiversity, but it also creates the potential for many contingent species to go extinct, particularly if large, well-connected generalists—certain large banks, for instance—disappear. It becomes an argument for the “too big to fail” policy, in which the size of the company’s Facebook network matters more than the size of its balance sheet."

Very interesting stuff.

Godā Stuti, 27

jātāparādham api mām anukampya Gode
goptrī yadi tvam asi yuktam idaṃ bhavatyāḥ |
vātsalya-nirbharatayā jananī kumāraṃ
stanyena vardhayati daṣṭa-payodharā ’pi || 27 ||

Godā Stuti, 26

raṅge taṭid-guṇavato ramayaiva Gode
kṛṣṇāmbudasya ghaṭitāṃ kṛpayā suvṛṣṭyā |
daurgatya-durviṣa-vināśa-sudhā-nadīṃ tvāṃ
santaḥ prapadya śamayanty acireṇa tāpān || 26 ||

Godā Stuti, 25

Gode guṇair apanayan praṇatāparādhān
bhrū-kṣepa eva tava bhoga-rasānukūlaḥ |
karmānubandhi-phala-dāna-ratasya bhartuḥ
svātantrya-durvyasana-marma-bhidā nidānam || 25 ||

Godā Stuti, 24

ārdrāparādhini jane ’py abhirakṣaṇārthaṃ
raṅgeśvarasya ramayā vinivedyamāne |
pārśve paratra bhavatī yadi tatra nāsīt
prāyeṇa Devi vadanaṃ parivartitaṃ syāt || 24 ||

Godā Stuti, 23

arcyaṃ samarcya niyamair nigama-prasūnaiḥ
nāthaṃ tvayā kamalayā ca sameyivāṃsam |
mātaś ciraṃ niraviśan nijam ādhirājyaṃ
mānyā manu-prabhṛtayo ’pi mahīkṣitas te || 23 ||

Godā Stuti, 22

dūrvā-daḷa-pratimayā tava deha-kāntyā
go-rocanā rucirayā ca rucendirāyāḥ |
āsīd anujjhita-śikhā-vaḷa-kaṇṭha-śobhaṃ
māṅgalyadaṃ praṇamatāṃ Madhu-vairi-gātram || 22 ||

Godā Stuti, 21

Raṅgeśvarasya tava ca praṇayānubandhāt
anyonya-mālya-parivṛttim abhiṣṭuvantaḥ |
vācālayanti vasudhe rasikās trilokīṃ
nyūnādhikatva-samatā-viṣayair vivādaiḥ || 21 ||