[reading time 10 minutes]
“...but Numbers are imperfect; and as for the begetting of numbers, it is done by Multiplication and Addition; but Subtraction is as a kind of death to Numbers. The only mystery of Numbers, answered they, concerning the Creation of the World, is, that as Numbers do multiply, so does the World. The Empress asked, how far Numbers did multiply? The Spirits answered, to Infinite.”
-Margaret Cavendish, The Blazing World
We are taught that mathematical concepts, once discovered, are true and fixed forever; that math is in and of itself complete. But perhaps this is not the whole story. Perhaps, even in mathematics, there is room enough for interpretation. Perhaps there is room in Mathematics for Philosophy. Perhaps, even, philosophy will have the final say.
Number as Arithmetic: The symmetry of addition and subtraction, multiplication and division
The four basic operations of arithmetic are addition and subtraction, multiplication and division. Addition and subtraction are paired because subtraction is the inverse of addition; these operations undo one another. The same is true for multiplication and division. Thus a kind of symmetry is introduced to those who learn arithmetic. But if we look just a little deeper, this symmetry falls apart. That “subtraction is to addition as division is to multiplication” does not hold when we consider each operation as it tends toward infinity.
Introducing Modern Infinity
Infinity is a gloriously complex subject. It is strange - deeply ironic even - that we should be allowed to have a single word which attempts to capture it. Infinity is infinite... how could it be comprehended in a single finite word?
Strange too: Georg Cantor (German mathematician whose life bridged the 19th and 20th centuries) showed that there are different sizes of infinity. The set of all counting numbers (natural numbers) is infinite: 1, 2, 3, 4, ... to infinity. But the set of all real numbers (that which includes numbers such as 1.001, or 1.0001, or -1.2345...) is a bigger infinity. His work even implies there are an infinity of infinities. Infinity is so strange, you can replace the word “are” in the last sentence with “is” and neither one is more truly correct.
Very Large Numbers
Since infinity is so strange, I often find it more profound to first concentrate on truly large numbers, and then realize that infinity is always bigger than any of these by an infinite amount. What? Oh, we’re just getting started.
John Locke - philosopher extraordinaire in the field of metaphysics (reality) in the time of Newton - in book II, chapter 17, paragraph 13 of his mighty An Essay Concerning Human Understanding, imagines the above in reverse: “It would, I think, be enough to destroy any such positive [concrete] idea of infinite, to ask him that has it, whether he could add to it or no...”. Without saying it, Locke imagines infinity as a kind of doing rather than an object (which underscores again the strangeness of the word “infinity” as a noun). So let us begin to do infinity here, by considering really, really big numbers.
Consider an example from cryptography. A typical cryptographic SHA-256 hash is a 256-bit number, i.e. it contains 1.1579x10^77 possible values. In seconds, that number is 60 orders of magnitude larger than the age of the universe . That is, if you want to crack (guess) that number (assuming no other algorithmic vulnerability shortcuts), and you were given a computer which could process one trillion values per second, then you were given one trillion of those computers, then you enlisted one trillion nations with the same capability... it would still take all of you many, many lifetimes of our universe to exhaust all possible values.
Now consider Graham’s Number. Using Knuth’s up-arrow notation, Graham’s Number is 3↑n3, where n is the up-arrow exponent, and if you are not familiar with up-arrow notation you should research this right now. For Graham’s Number, n itself is incalculably large, being derived from 63 iterations of n0=3↑↑↑↑3, n1=3(↑n0)3, n2=3(↑n1)3, etc. Graham’s Number is so large, the observable universe can not contain a digital representation of it. The observable universe can not even contain a digital representation of the number of decimal places in Graham’s Number. Furthermore, Graham’s Number isn’t even the largest number imaginable (consider increasing the up-arrow exponent in its up-arrow representation). For all this, no number imaginable even comes close to touching infinity. Infinity is always infinitely far from any concrete number. That is, even Graham's Number is 0% of infinity.
We must ask ourselves, with numbers such as these, what is the need for infinity as a concrete number? Perhaps, there really is no such thing as an actual infinity at all.
Number as Philosophy: The asymmetry of division
Let us return to arithmetic and its four basic operations. Addition can be applied to any number ad infinitum to reach infinity; so too with subtraction. Here the symmetry of these inverse operations is preserved as we explored previously when we introduced these basic operations.
Multiplication can also be applied to any number to reach infinity. That is, multiplication where at least one operand is greater than or equal to unity (the number one), otherwise the operation is akin to division. So what of division? Infinite division does not lead to infinity, but rather it leads to nothing. Continue to divide anything into infinite smallness and you get zero.
What does it mean to literally divide something an infinite amount of times? Consider a length of string in the palm of your hand. You may divide this string as many times as you wish - to infinity. What are you then holding? You are not holding nothing, since the string is still the sum of its parts, which is a finite thing in your hand. But neither are you holding an infinity of string pieces in your hand - an infinity of any mass however small would weigh an infinite amount of grams, and no one will argue that such a mass is holdable in the palm of your hand. You can not hold an actual infinity.
But let us consider that what we are doing when we conceptually divide a string into infinite pieces is just that - making conceptual pieces of string that each weigh nothing. This is exactly the point. Infinity as a conceptual object is perfectly acceptable. Infinity as an actual object is not. In other words, there is no such thing as an actual infinity.
Taking this at face value gives an instant solution to Zeno’s Paradox: if we divide any distance into fractional lengths, each of which take some small amount of time to traverse, then as we continue to subdivide each division into more divisions, would we not take an infinite amount of time to traverse any distance? The answer is obviously no. If we take infinity to mean something which is only conceptually real, not actually real, we can see why.
So who says there is such a thing as actual infinity anyway? For one: the arithmetic of addition, subtraction, multiplication, and division which you learned as a child. Indeed, actual infinity is supposed by all of mathematics, at least since Georg Cantor. But this was not always the case.
Actual Infinity vs. Potential Infinity
Certainly, from the ivory tower that is the 21st century, we may see farther than the ancient Greeks. This is true for many things, including physics, medicine, biology, chemistry, etc. But, perhaps, in some ways, we have forsaken Philosophy for the glory of the others. Aristotle in his wisdom, thousands of years ago, made the distinction between actual and potential infinity which is largely lost today, today when infinity is largely taken to be real.
From book IX, chapter 6 of his Metaphysics, "For the fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately.” That is to say, infinity as a mathematical or physical object does not exist, rather, infinity is a conceptual doing which can never be fully achieved. If this version of infinity unsettles you, and you’d rather return to the familiar world of 21st century mathematics tied up with a bow... consider Zermelo-Fraenkel set-theory.
ZFC and Gödel
The hallmark of mathematics is that it is rule-based; mathematics is logical. Underlying all of mathematics is a set of rules in the general language of set-theory. These rules are called the Zermelo-Fraenkel axioms, of which there are 8, unless the “axiom of choice” is included, in which case there are 9, and these nine are called Zermelo-Fraenkel Choice, or ZFC for short. As an example of an axiom, the “axiom of extensionality” states (informally) that two sets are equal (they are the same set) if they have the same elements.
But why are there not always 8 axioms, or always 9? Because in any logical system of rules, Kurt Gödel showed that there will always be some arithmetic truths which cannot be proved, unless another axiom is added, in which case proving that the new system does not lead to any contradictions (i.e. is consistent) will require an additional axiom, etc.
Thus there will never be a complete set of universal laws for mathematics. Choices need to be made. Some things will be left out in favor of others. This leads to disputes, even in the logical realm of mathematics. The ultimate dispute involves the very nature of infinity itself (see this Scientific American article for further discussion). Reintroducing potential infinity (as opposed to actual infinity) into set-theory could provide a robust answer. Mathematics rests upon Philosophy after all.
Infinity is the Santa Claus of mathematics: it exists only in the minds and hearts of men and women and children, but it is no actual thing. As a concept, it is more akin to a verb than a noun; it would be better to say “to infinity a number is to make it ever larger” rather than “this number shall be made large to infinity”. The former is exactly correct as a concept; the latter leaves us with an erroneous reality. To apply Philosophy in this way, to change the fundamental way in which we imagine and interact with infinity, from the colloquialisms of our speech to the rigors of our Mathematics, would be to bring the way we think of our universe more in-line with actual reality. And when we live, comfortably, with more accurate representations of the way things truly are, that is when - and only when - we can improve upon our wisdom, and see further than those who came before.
As a final demonstration of the far-reaching cultural shifts this new way of viewing infinity implies, consider the word “forever”. Forever is a type of actual infinity. In concluding that there are no actual infinities, we must also conclude that there is no “forever” - in physics and in spiritual contexts. That is to say, for example, the popular ideas of heaven and hell also disappear. Oh, we do not know everything - far from it. But if an after-life remains at all, it would be by definition radically different from these notions.
So where to go from here? Live. Live fully as best as you can, here, on this Earth. Live fully, without the need for infinity.