“Next contestant Sybil Fawlty from Torquay, special subject the bleeding obvious”
– John Cleese, Fawlty Towers
In these posts I want to present a line of reasoning that is an explanation of why this universe exists, and in particular why it started with a Big Bang.
There are a few points to cover so it will require a bit of patience on your part. Along the way I will be attempting to answer three questions in particular:
What do we mean when we say infinity exists?
What do we mean when we say mathematics exists?
What do we mean when we say the universe exists?
A key part of my line of reasoning is the presumption that the universe is finite along the lines of digital physics. To try and persuade you of this I need to first talk about infinity as it is presented in mathematics and this requires me to talk about the axioms of mathematics, where infinity first makes its ugly appearance.
Although I am a mathematician myself, I find the study of axioms is particularly unnerving. It’s as if all my mathematical training has left me completely unprepared to grapple with axioms, as if I’ve ventured into a kind of Alice in Wonderland where things are not as they seem. What are the rules in the branch of mathematics that make the rules?
Axioms can be viewed as mathematical statements that are so simple that they can’t be proved. You’ve got to jump into the pool somewhere and axioms are it. They are the starting point that defines mathematics and you choose them because they are obvious.
The commonly accepted, fairly uncontroversial set of axioms that describe the natural numbers (0,1,2,3,…) are the Peano Axioms. There are only minor disagreements over these, for example people may argue about whether to include zero.
I will take the Wikipedia formulation of the Peano axioms, and what I want to focus on is where concepts of existence appear. Axiom 1 is very very simple indeed:
0 is a natural number.
This is saying that 0 is something that exists, and that it belongs to a collection of things named natural numbers. This innocuous statement is actually quite significant, we are beginning with a presumption of existence and that zero is the name we give to the thing we believe to exist, without any line of reasoning to support it.
The next few axioms aren’t quite so profound, for example axiom 4 says that if m,n and p are natural numbers that exist then:
If m=n and n=p then m=p
There is no presumption of m,n and p existing, it’s just saying that this holds if they happen to exist. After the first five axioms we still have only established the existence of zero. The following three axioms, in a subtle way, axiomatically presume the existence of an unlimited number of natural numbers. We now say that for every natural number (of which we initially know of only one – namely zero) that there exists an associated natural number which we’ll call S(n).
Let me cheat a bit here. S(n) is simply the next number. If n is 7 then S(7) is 8. Now forget what I just said, because you can’t do that when defining axioms. It’s considered bad form. Instead we’re going to define it in what appears to be the most perverse way a committee was able to come up with. Here it is
6. If n is a natural number then S(n) is too.
7. If n is a natural number, then S(n) isn’t zero.
8. If m and n are natural numbers then if S(m)=S(n) then m=n.
I’m going to have a stab at explaining why these collectively axiomatize the existence of an indefinitely large number of natural numbers. Axiom 6 almost says that for every n another brand new natural number S(n) also exists, but not quite because S(n) might already exist by virtue it being the same as S(m) from a previous number m, hence axiom 8 plugs this loophole. Also, if S(n) ever equals zero then there would only be a finite cycle of natural numbers, so axiom 7 plugs this. It only remains now to arbitrarily symbolise every S(n): we denote S(0) by the symbol “1”, S(1) by the symbol “2” and so on. We now have a way of constructing an infinity of natural numbers.
So does the set of all natural numbers exist since we now know how to construct it? What does it even mean to say the set exists? Although we can embark on constructing it, to complete the construction would require an infinite number of steps, so we’d need an infinity in order to construct an infinite set. Not a lot of value added.
The Peano axioms do not, in fact, establish the existence of an infinitely large set. To axiomatize mathematics in a way that includes infinite sets you need to extend beyond the Peano axioms. The most common axiomatic foundations of more general mathematics are the Zermelo-Fraenkel axioms (with the axiom of choice) (ZFC). These are a good order of magnitude more difficult to grapple with. I only wish to refer to the seventh one, the axiom of infinity, an axiom that disturbs me. This axiom basically comes right out and states that the set of all natural numbers exists. Problem solved.
It turns out that much of mathematics beyond natural numbers doesn’t actually require the existence of infinite sets. This has been pioneered by Errett Bishop by the approach of constructivism. This approach asserts that it is necessary to construct a mathematical object to prove that it exists. In other words, infinite sets don’t exist. In practice, constructivist techniques are too gritty to be useful. Also they have their limitations. For example, if you want to take a solid sphere, cut it into 5 pieces, and reassemble these pieces (without changing their shapes) into two solid spheres of the same size then you do need to assume existence of infinite sets. Personally I’d sleep just a little bit easier at night knowing that no one really thinks that’s possible. But that’s just me.
So while the axioms that define finite numbers are obvious in that they formalise ‘ordinary’ mathematics, the axioms involving infinite sets are not obvious. It’s not even obvious what existence of infinite sets means let alone whether it’s true. There is arbitrariness about the axiom of infinity. In fact, you can equally axiomatize mathematics by using ZFC with the axiom of infinity reversed to say no infinite sets exist. Even if the full set of natural numbers doesn’t exist we can still say this: for any really big number N that you give me, I can find more than N natural numbers.
So what else doesn’t exist in maths? What about the number pi? No it doesn’t exist, or more specifically its full decimal expansion doesn’t exist. However, for any really big N, it is possible to calculate more than N decimal places. Also, the concept of pi certainly does exist and is used in mathematics without a problem. The algorithms for constructing its decimal expansion exist. But you will never encounter pi numerically, in mathematics or in the real world. Circles are approximations that take pi to many decimal places but that’s it.
This exists and it equals one. It does not need any concept of infinity either. Just interpret the above as “what is the number that we can get arbitrarily close to by adding ½ + ¼ + 1/8 + … a really long way?” It can be shown rigorously that such a number is 1. There is no infinity in the formula, only a summation sign on top of which there’s an 8 on its side that means something else. Taking the essence of this approach we can still do things like calculus.
To say that the axiom of infinity is wrong would be a mistake (unless it turns out to be inconsistent with the other axioms, something that in polite circles you assume not to be the case). Axioms aren’t right or wrong, they just lead to a different “mathematics”. The mathematics resulting from the ZFC is rich and interesting and I personally feel it is extremely important that it be studied. I just find it can be unhelpful sometimes.
In this post I’ve attempted to suggest that in mathematics presuming the existence of infinitely large sets is somewhat arbitrary, leads to results that don’t really make sense and is not really necessary. So I will answer the first of the three questions I set out at the start. My personal answer:
Infinity (i.e. a set of infinite size) does not exist. Unless you choose it to, and then you should feel responsible for the consequences.
I’ve laid out this philosophical groundwork in the hope that it might now make more sense to think that the universe is finite. In my next post I’ll address this further, and eventually I’ll get to why I think this all naturally leads to a Big Bang to kick off the universe.
I can also now explain the apparently self contradictory name of this blog Finitism Forever. I think that the universe is spatially finite but that time may well continue for more than N years, where N is as large a number as you’d like to throw at me.