Everything and More: A Compact History of Infinity
by
by
Thinking Impossibly
It was the Greeks who discovered that numbers, and therefore mathematics, had only the most tenuous connection with the world in which we live. Numbers constitute a separate order of existence. The number 5 for example has no connection with the five apples that might be sitting on my kitchen table, or with the age of my youngest relative. The number 5 is something all on its own. It is constructed out of other numbers, which are made up of other numbers that may in turn be constructed using the number 5. Mathematics, in other words, is a completely self-contained and isolated world we make up.
No one was aware of this quite separate world before the Greeks stumbled across it. They were, rightly, in awe of its implications. The otherworldliness of mathematics suggested an unnaturalness, indeed a supernaturalness, that demanded religious veneration. Mathematics seemed to literally reveal things that were unknowable in any other way. Numbers must be divine, they thought. Numbers were perfect. What we experienced outside of mathematics were imperfect approximations or distorted reflection of numbers. Within this religion of numbers, only two heresies were recognised: zero and infinity. These were demons which had no place in either the divine or the divine ‘word’ of mathematics.
The theological prejudice of the Greeks was tempered a bit in late antiquity. As mathematics inched its way from geometry to algebra, zero was recognised as a useful addition to mathematical doctrine - much like free will later became essential in strict Calvinism to motivate virtue. Zero seemed real enough since it was possible to point to an empty basket of fruit as a purported proof of its existence. But even today, there is debate about whether zero is a number or merely a digit which is useful in mathematical expression - something like a decimal point for example.
Infinity, however, is a different matter altogether. Although infinity is an essential concept in modern mathematics, there is no way to throw shade about what it is. Infinity can’t be pointed to nor represented except by symbols for something that is entirely beyond anyone’s experience. As Wallace’s title so concisely says, infinity is more than everything there is - more than the number of gluons, muons, bosons, and all other elementary particles in the entire universe, for example.
And the distance of infinity from any reality we know only increases when we recognise that there are many ‘orders’ of infinity - infinities that are more or less than other infinities. These higher orders of infinity weren’t discovered until the 19th century. And we appear still to have resisted the implications of these discoveries in the same way that the Pythagoreans did by keeping the indeterminacy of the infinitely long decimal expression of π, the relation between the circumference and the diameter of a circle, as a cultic secret which might undermine faith in mathematics. Infinity for them meant ‘mess.’
And infinity today, although less of a mess, is still very messy indeed about what it implies. Wallace quotes the great German mathematician, David Hilbert, approvingly: “The infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to.” Infinity is an abstraction, the ultimate mathematical abstraction. But an abstraction of what? No one has ever seen an infinitely full basket of anything in order to make such an abstraction.
No, infinity is an abstraction from a system of numbers, which themselves are supposedly abstractions. It is at this point that the ultimate revelation of mathematics takes place: numbers are indeed abstractions but abstractions of each other not of some experience of baskets of various items. Numbers produce each other; they have no existence except in their relationship with each other. 2 + 2 = 4 is not an inductive generalisation of market experience of baskets and their contents; it is an entirely intellectual proposition/discovery/definition. Which of these you choose to describe infinity is a reflection of one’s already established metaphysical position. It fits with and confirms them all.
Here’s the thing: infinity shows that the world created by mathematics has only an obscure and unreliable connection to our experience. This applies not just to the infinite in all its manifestations but also to the number 5 and its colleagues and associates. Like infinity, no one has ever experienced the number 5, or the way it interacts with other numbers to produce itself or yet further numbers. If you doubt this, just try to provide a precise statement of, say, the square root of 5. Numbers don’t cut the world at its joints. Sometimes they don’t even know their own joints. As Wallace summarises the situation: “... mathematical truths are certain and universal precisely because they have nothing to do with the world.”
And the revelations generated by infinity are not limited to mathematics; they extend to that more general realm of which mathematics is a part: language. Wallace nails this too: “... the abstract math that’s banished superstition and ignorance and unreason and birthed the modern world is also the abstract math that is shot through with unreason and paradox and conundrum and has, as it were, been trying to tie its shoes on the run ever since the beginning of its status as a real language.” Mathematics is the most precise language we have. Yet, ultimately it doesn’t know what it’s talking about, except itself.
None of this means that mathematics, or language in general, isn’t immensely useful. Of course it is; but for rather complex and often mysterious reasons. The revelation of infinity is simply that mathematics is not reality. Nor is any other language. Like all language, mathematics can be beautiful, and compelling, and inspirational. But it is never the way the world is. Confusion about this simple fact is something that human beings seem to have a great deal of trouble with. Language especially political language, easily reverts to religion (and vice-versa). Wallace’s little book is appropriate therapy for reducing this confusion.
And by the way, Neal Stephenson’s introduction alone is worth the price of admission.
It was the Greeks who discovered that numbers, and therefore mathematics, had only the most tenuous connection with the world in which we live. Numbers constitute a separate order of existence. The number 5 for example has no connection with the five apples that might be sitting on my kitchen table, or with the age of my youngest relative. The number 5 is something all on its own. It is constructed out of other numbers, which are made up of other numbers that may in turn be constructed using the number 5. Mathematics, in other words, is a completely self-contained and isolated world we make up.
No one was aware of this quite separate world before the Greeks stumbled across it. They were, rightly, in awe of its implications. The otherworldliness of mathematics suggested an unnaturalness, indeed a supernaturalness, that demanded religious veneration. Mathematics seemed to literally reveal things that were unknowable in any other way. Numbers must be divine, they thought. Numbers were perfect. What we experienced outside of mathematics were imperfect approximations or distorted reflection of numbers. Within this religion of numbers, only two heresies were recognised: zero and infinity. These were demons which had no place in either the divine or the divine ‘word’ of mathematics.
The theological prejudice of the Greeks was tempered a bit in late antiquity. As mathematics inched its way from geometry to algebra, zero was recognised as a useful addition to mathematical doctrine - much like free will later became essential in strict Calvinism to motivate virtue. Zero seemed real enough since it was possible to point to an empty basket of fruit as a purported proof of its existence. But even today, there is debate about whether zero is a number or merely a digit which is useful in mathematical expression - something like a decimal point for example.
Infinity, however, is a different matter altogether. Although infinity is an essential concept in modern mathematics, there is no way to throw shade about what it is. Infinity can’t be pointed to nor represented except by symbols for something that is entirely beyond anyone’s experience. As Wallace’s title so concisely says, infinity is more than everything there is - more than the number of gluons, muons, bosons, and all other elementary particles in the entire universe, for example.
And the distance of infinity from any reality we know only increases when we recognise that there are many ‘orders’ of infinity - infinities that are more or less than other infinities. These higher orders of infinity weren’t discovered until the 19th century. And we appear still to have resisted the implications of these discoveries in the same way that the Pythagoreans did by keeping the indeterminacy of the infinitely long decimal expression of π, the relation between the circumference and the diameter of a circle, as a cultic secret which might undermine faith in mathematics. Infinity for them meant ‘mess.’
And infinity today, although less of a mess, is still very messy indeed about what it implies. Wallace quotes the great German mathematician, David Hilbert, approvingly: “The infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to.” Infinity is an abstraction, the ultimate mathematical abstraction. But an abstraction of what? No one has ever seen an infinitely full basket of anything in order to make such an abstraction.
No, infinity is an abstraction from a system of numbers, which themselves are supposedly abstractions. It is at this point that the ultimate revelation of mathematics takes place: numbers are indeed abstractions but abstractions of each other not of some experience of baskets of various items. Numbers produce each other; they have no existence except in their relationship with each other. 2 + 2 = 4 is not an inductive generalisation of market experience of baskets and their contents; it is an entirely intellectual proposition/discovery/definition. Which of these you choose to describe infinity is a reflection of one’s already established metaphysical position. It fits with and confirms them all.
Here’s the thing: infinity shows that the world created by mathematics has only an obscure and unreliable connection to our experience. This applies not just to the infinite in all its manifestations but also to the number 5 and its colleagues and associates. Like infinity, no one has ever experienced the number 5, or the way it interacts with other numbers to produce itself or yet further numbers. If you doubt this, just try to provide a precise statement of, say, the square root of 5. Numbers don’t cut the world at its joints. Sometimes they don’t even know their own joints. As Wallace summarises the situation: “... mathematical truths are certain and universal precisely because they have nothing to do with the world.”
And the revelations generated by infinity are not limited to mathematics; they extend to that more general realm of which mathematics is a part: language. Wallace nails this too: “... the abstract math that’s banished superstition and ignorance and unreason and birthed the modern world is also the abstract math that is shot through with unreason and paradox and conundrum and has, as it were, been trying to tie its shoes on the run ever since the beginning of its status as a real language.” Mathematics is the most precise language we have. Yet, ultimately it doesn’t know what it’s talking about, except itself.
None of this means that mathematics, or language in general, isn’t immensely useful. Of course it is; but for rather complex and often mysterious reasons. The revelation of infinity is simply that mathematics is not reality. Nor is any other language. Like all language, mathematics can be beautiful, and compelling, and inspirational. But it is never the way the world is. Confusion about this simple fact is something that human beings seem to have a great deal of trouble with. Language especially political language, easily reverts to religion (and vice-versa). Wallace’s little book is appropriate therapy for reducing this confusion.
And by the way, Neal Stephenson’s introduction alone is worth the price of admission.
posted by The Mind of BlackOxford @ August 14, 2013
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