Although infinity is a mathematical term, it is very useful in entering the philosophy of what “real” is. To decide whether infinity is real, we must first decide whether any numbers are real and what criteria they either do or do not fit. For that, we need to know what makes something ‘real’? Does it have to be tangible or physical? Does it have to be a part of E=mc²? Does it have to either be or have an impact on energy or matter? Can something be both ‘real’ and paradoxical? Is a conceptual tool actually a part of reality, or a great way to simulate it?
First, some background on infinity. Infinity is the noun for a set that goes on forever, without limit. If you count, there is no “biggest number”, because you can always add 1. The upper limit is infinity (which is the same as ‘there is no upper limit’. There is no referent for infinity, it can only be conceptualised or expressed in terms of concepts; infinity does not count anything. Even the number of subatomic particles in the entire universe is a finite and discrete number. What ever that number is, we can always imagine adding 1 to it, giving us a bigger number: (All the subatomic particles in the universe) + 1. But that number has no referent. There is no real thing it is counting. Is that number real?
I had this discussion with a friend who believes that numbers exist contingent on real things that they are counting. Therefore, there is a biggest number. This number is the number of discrete units (probably subatomic particles) in the universe. All larger numbers are imaginary, have no referent and are not ‘real’.
I disagree. I think all numbers are imaginary. Numbers are only real insofar as they are ideas we have. These ideas are logically sound and in maths we use them as a tool to represent real things. Numbers are a simulation we run in our head and they are no more real than Niko Bellic from GTA:IV. Therefore, I think the number 1 is as real as infinity.
My friend’s idea has the issue of rejecting infinity where she does not reject other numbers. My friend would also reject the number Googolplex (10^10^100) and Graham’s number (could not be written on a piece of paper as big as the universe). A googolplex is thought to be a value greater than the number of all the discrete physical units in the universe. Graham’s number is even bigger again. However, mathematics uses both. In fact, Graham’s number is famous only because it is the largest number ever used in mathematics. But that mathematic equation is about something that is not real: a multidimensional bi-chromatic hypercube. (Google it?). In my friend’s world, if it doesn’t refer back to something real, it is not real.
I have an entirely different issue: I don’t think any numbers are real. I think numbers are tools. These tools are precise, unlike words, and describe objective things. Numbers are also discrete (which is to say they are not continuous, which is to say they are exactly what they are and no gradation up or down). This is obviously not true about words. Words are subjective tools and are continuous (by which I mean they do not have an exact value. “Good” can mean some gradation above and below my precise meaning when I said “I had a good day”). Because numbers have no leeway up or down, their meaning cannot drift. Like my friend, I have no doubt what a number refers to exists: there are 8 bananas. But there is no “8”. Without a unit, “8” doesn’t mean anything.
That entrusts a problem to me: do abstract things not exist? Some of them do. Ideas exist. What ideas refer to do not exist. Pink unicorns and celestial teapots do not exist merely because I think of them. Therefore, that I can think of numbers does not make numbers exist. (Even if you think numbers do exist, you must accept the reasoning ‘that I can think of numbers does not make them exist’). I now have the problem that I believe abstract ideas do exist, but abstract numbers do not. I don’t see this as a problem. Ideas are contingent on something physical: the brain state of the brain having the idea. Certain neurological structures immersed in certain chemicals are what an idea is. What is why you need your brain. Your brain has to change its state for an idea to form, and that takes calories (a reason learning and thinking make you tired). Because ideas emerge out of what does exist (i.e. they are the result of what exists) ideas exist. Numbers are not physical, they are not energy and they are not contingent on anything which is. Numbers cannot be affected or effected, nor can they effect change. Thus, they are not real.
My friend and I are not as at odds with mathematicians as one may think. A simple Google of the question “Does infinity exist” reveals a lot of articles on how infinity is only a tool for understanding mathematical phenomena without real referents. These phenomena are things like geometric progressions (1+1/2+1/4+1/8→2). If you continued that equation on forever, you would get to 2; the number of terms in your equation would be infinite; the last value you added would be infinitely small (equal to 0). Adding 0 would make a difference. This is paradoxical.
A paradox is self-contradictory. The logical law of noncontradiction means that paradoxes cannot exist. If infinity can be shown to be inherently and unavoidably rife with paradoxes then it cannot exist. Before we start, here is an example of a paradox to illuminate the issue: “This sentence is false”. If you accept that the sentence is false then the sentence is telling you the truth. But if the sentence is telling you the truth it is not false. If the sentence which claims to be false is not false, then it is true. But a true sentence telling you it is false necessarily must be false and not true… and so it goes on. It contradicts itself.
Image a hotel with infinite rooms. Each room is occupied. I go to the front desk and I ask if there is a room available. There is not. But the hotel wants my custom, and so it figures out a solution: move the people from room 1 to room 2; from 2 to 3; from 3 to 4… there can never be a room with a guest that can’t move up one. Thus, I move into room 1 and everyone gets a room: ∞+1=∞. Then, everyone in an even-numbered room (which is an infinite number of rooms) leaves. There are still guests in every odd-numbered room, and that is an infinite number of guests -=. Confused and baffled, everyone except me leaves. An infinite number of people leave, leaving only me ∞-∞=1. When the hotel was full, we could all have left at the same time, leaving no one ∞-∞=0. Consider those equations again:
It is self-contradictory that all those equations could be correct. Adding and subtracting natural numbers from infinity does not change it at all. Whereas, adding and subtracting by infinity can do anything. This is self-contradictory. Thus not possible.
There may be a saving grace; there could be a place where the infinite does exist. This is not about numbers, but the space between them: the decimal numbers. How many decimal numbers are there between 1 and 2? It’s obvious the answer in “an infinite number”, because there is no limit to how many numbers you can write after the decimal point. π=3.141…and goes on forever. There is another way of mathematically proving there are an infinite number of decimal places between any two numbers: Cantor’s diagonal argument shows that any given list of decimal numbers cannot be complete by giving rules for how you should change certain numbers to give completely new numbers that cannot be contained in the set. An infinite number of values between any two values is a realistic place for “infinity” to hide. This called continuous data. Length and time and volume are all continuous data; you can measure these things to infinite resolution.
Except you can’t really do that. If, like me, you think numbers are imaginary tools then you cannot force infinity to exist by finding space between two numbers. If, like my friend, you think numbers are only real when they relate to something then each length or time given is a whole unit. A banana is not 9.47929…cm long; the banana is 1 unit in length. Although numbers are real, units are arbitrary.π is a ratio that technically has an infinite number of decimal places. And it represents a real thing: a ratio between the circumference and diameter of a circle. This constant is irrational (i.e. has an infinite number of decimal places) in any integer-base system: base-10, base-12, base-6. Is this consistency a real infinity hidden in a decimal?
You can expect my no based on it still being a number. Interestingly, my friend also says no. Ratios are not real. Ratios are a function of mathematics and are no more real than +, -, ×, ÷. If you want to express how 10 sweets will be split between 3 children, then the real answer is either: 3 sweets each, and 1 leftover (or 3, remainder 1) or 3 sweets and 1 1/3, where a third is a unit and not a fraction. Decimals are not real either. My friend describes decimals as “a malfunction caused by trying to keep 1 arbitrary unit”, explaining that a third of a sweet is a whole unit in of itself, and not really a part of a bigger whole.
And so my friend places “infinity” not in the set of real numbers, or even of the fake numbers (which include minuses, overly big numbers, decimals and imaginary numbers like i). Infinity is a function. Functions are not real. (And neither are numbers.) Infinity is just an idea, it is not real. As an idea, infinity is wrought with paradoxes, and therefore it is not real.