A taste for infinity

Curry is such a varied cuisine that it may feel infinite…. but is it?

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And while we’re on the subject… does infinity exist?

In the distinguished company of Archimedes – and Professor Ian Rumfitt, Senior Research Fellow, All Souls College Oxford – Good Korma travels the path to infinite knowledge.

 

 

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When Archimedes experienced his famous ‘Eureka’ moment, he was at home, in Sicily ­- lying comfortably in his bath.

When I experienced my micro-Eureka moment, I was in Chennai Dosa restaurant in Tooting Bec – staring at a glass of Kingfisher.

It was a Saturday evening, and while billions of people were going about their daily lives, I was fretting about my relationship with infinity.

For months, I’ve wanted to write a curry blog, titled ‘A taste for infinity’. I want the blog to explain that part of my love of Indian food is driven by the sheer limitlessness of South Asian cuisine… by the existence of hundreds of original Indian ingredients, captured in hundreds of thousands of cookbooks, cooked via tens of millions of recipes – and interpreted by over a billion Indians.

For me, exploring Indian cooking is like looking at the Milky Way… and I want to share this sense of wonder with other curryphiles.

infinity

Even with one hundred lifetimes dedicated to cooking curry, I’d still only be scratching the surface.

 

But here’s the snag – whenever I sit down to tackle the ‘infinity’ blog, I realise I am totally unqualified to write it. Leaving aside the question of how much I do or don’t know about curry, the fact remains that – as a human being – I simply don’t ‘get’ infinity.

Let me unpack this…

My instinct is that asking a homo sapiens to explain infinity is like asking a snowflake to explore the nature of heat, or asking a kitchen sieve to understand the properties of water. The one is just not equipped to understand the other.

Let’s face it… as a species, we human beings are so gloriously finite!

Am I alone in thinking that the whole of human civilisation might turn out to be a match-flame in a minor galaxy… with a single human life counting for less than a ten-billionth of that flicker? Mankind may hunger for the infinite, but all the evidence is that we are as transient as mayflies.

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And even if we were presented with the gift of infinity, would we know what to do with it?

In his magnificent History of the World in 10 ½ Chapters, Julian Barnes uses the final chapter Dreams to ask if there is any pleasure, any stimulus, any goal that could sustain a human being through eternity.

If life on Earth were followed by an infinite existence somewhere else – wonders Barnes – could we hack it?

The novelist thinks the answer is ‘no’. Whatever your passion, whatever your thirst for the absolute – Barnes argues that even the intellectual giants of human history would find infinity too much to bear in the end.

The following are the last lines in the book – a conversation between Barnes and his heavenly guide:

‘So … even people, religious people, who come here to worship God throughout eternity … they end up throwing in the towel after a few years, hundred years, thousand years?’

‘Certainly. As I said, there are still a few Old Heaveners around, but their numbers are diminishing all the time.’

Human beings, it seems, just don’t have an appetite for infinity.

 

The next question is whether human beings have the grey matter to understand what infinity is, or might be?

It’s not obvious.

Here’s a very simple example that I read recently in the press – two smart people debating the nature of infinity. The point they agree on is that numbers are infinite. They also agree that within any group of numbers, there will be a smaller total of prime numbers (e.g. 168 primes up to number 1,000). The point where they fall out is that SmartPerson A is convinced that composite (non-prime) numbers must somehow be ‘more infinite’ than primes. “They happen more often; there’s got to be more of them.”

Smart Person B disagrees, stressing that both composites AND primes are infinite: “There are no ‘big infinities’ and ‘small infinities’. Infinity means infinity.”

 

For me, their debate suggests than even our simplest instincts about infinity may be wrong.

But does their conversation also point towards a wider question: is infinity simply too big a thought to fit inside human heads?

Oddly enough, Archimedes himself could probably have helped.

The famous physicist, engineer, inventor, and astronomer (287 to 212 BC) not only gave us the famous Eureka moment – but also built the foundations for a lot of modern mathematics. He anticipated modern calculus, as well as proving the area of a circle via an accurate approximation of pi. And when he wasn’t designing innovative gadgets like the screw pump – and war machines to protect his home city of Syracuse – Archimedes was working on a system for expressing very large numbers.

But I’m more than two thousand years too late to talk to him. And whenever I sit down to write the ‘infinity’ blog, I stop.

 

Until my micro-Eureka moment in Chennai Dosa.

I am reading a thoughtful, heart-felt piece in the Guardian by Karl Ove Knausgaard ‘What makes life worth living?’ in which the writer praises the nature of everyday objects – in the form of a letter to his unborn baby.

Ranging from the quiet beauty of plastic bags to the solace of beds, Knausgaard delivers some pretty good stuff. And as I read the final paragraph of the last of the seven sections (on Faces), he socks it to me:

“Whatever is human is changeable, it is mobile, and it is unfathomable.”

Eureka!

Just stick the word ‘infinitely’ into that sentence– and I have my answer:

“Whatever is human is infinitely changeable, infinitely mobile, and infinitely unfathomable.”

As human beings, we may not understand infinity as a concept – but we embody it in our infinite changeability and infinite unknowability.

BOOM.

Infinities R Us.

And as an infinitely unfathomable human being, I feel entitled to write my blog on the infinity of curry.

 

Postscript…

With the blog written, and my finger hovering over ‘publish’ – the Guardian chooses to blow me away with new research suggesting that the number zero (and therefore the concept of infinity itself) is an Indian invention!

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Deciphering a parchment document (dated via radio carbon technology to the third or fourth century AD) British scholars have recently proved that a dot on the text is the birth of the concept of zero:

“The development of zero as a mathematical concept may have been inspired by the region’s (India’s) long philosophical tradition of contemplating the void and may explain why the concept took so long to catch on in Europe, which lacked the same cultural reference points.

“Despite developing sophisticated maths and geometry, the ancient Greeks had no symbol for zero, showing that while the concept zero may now feel familiar, it is not an obvious one.

“The development of zero in mathematics underpins an incredible range of further work, including the notion of infinity, the modern notion of the vacuum in quantum physics, and some of the deepest questions in cosmology of how the Universe arose – and how it might disappear from existence in some unimaginable future scenario.”

Not only is infinity legitimate fodder for a curry blog… it turns out that India invented infinity itself!

 

A brief guide to infinity

Ian Rumfitt, Senior Research Fellow, All Souls College, Oxford, shines objective light on a subjective blog.

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“The topic remains difficult. I confine myself to some remarks about the debate between the two ‘smart people’ that Adam recalls reading about.

“Debates of this kind go back to antiquity but a famous example is found in Galileo’s dialogue, Two New Sciences, of 1638. To establish that two finite collections have the same number of members (i.e. the same size or ‘cardinality’), it suffices to show that there is a one-one correspondence between the members of the two collections. Thus, to show that the collection of canonical gospels has the same size as the collection of Brahms’s symphonies, it suffices to pair St Matthew’s Gospel with the C minor Symphony, St Mark’s with the D major, St Luke’s with the F major, and St John’s with the E minor. In Two New Sciences, Salviati (who is the spokesman for Galileo’s mature views) observes that apparently paradoxical results arise when the same method is applied to infinite collections. Consider two such collections: the positive integers, 1,2,3,…; and the perfect squares, 1,4,9,… There is a one-one correspondence between these collections: 1 pairs with 1, 2 with 4, 3 with 9; quite generally, n pairs with n2. Applying the method, then, we infer that the collection of positive integers has the same size as the collection of perfect squares. Salviati, however, finds this result paradoxical, for the collection of perfect squares is a proper part of the collection of positive integers. Every perfect square is a positive integer, but not every positive integer is a perfect square. The result that the two collections have the same size, then, contradicts Euclid’s Fifth Axiom: ‘the whole is greater than the (proper) part’.

“The moral that Salviati, alias Galileo, draws from the paradox is that it makes no sense to assign sizes to infinite collections. In a way, this conclusion makes sense: to call a collection ‘infinite’ is precisely to deny that it has any definite size. In the 19th century, however, the German mathematician Georg Cantor argued that some collections—among them, the positive integers—do not conform to Euclid’s Fifth Axiom. Yes, the positive integers contain the perfect squares as a proper part, but precisely because there is a one-one correspondence between them, the two collections have the same size or cardinality.

“On this basis, Cantor developed a theory of precise sizes or cardinalities, each of which was larger than any of the natural numbers. Respecting the point that ‘infinite’ implies a denial of definite size, he called these cardinalities ‘transfinite’. The cardinality of the natural numbers, which Cantor called ‘Aleph Null’, À0, is the smallest of them. Cantor postulated an unending series of larger and larger alephs.

“Fellow mathematicians were initially sceptical but were gradually won round as it became clear that Cantor’s theory was not only consistent but could be used to solve problems that had been formulated before its invention. However, the series of alephs remains in some crucial respects mysterious. By way of his celebrated ‘diagonal argument’, Cantor argued that the collection of real numbers was strictly larger than that of the natural numbers. He showed, in fact, that the reals have cardinality 2À0. The question then arises where this cardinality stands in the sequence of alephs. Cantor conjectured that it was the next largest aleph after Aleph Null. That is, he hypothesized that 2À0 = À1. However, this ‘Continuum Hypothesis’ has been shown to be undecidable on the basis of currently accepted mathematics.

“Basic questions about the transfinite, let alone the genuinely infinite, lie beyond the range of the methods mathematicians now have at their disposal.”

 

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