Consider the statement: If n exists then n+1 exists

Now if ‘n’ in this statement refers to a written numeral, then this statement is probably false because as far as we know only a finite amount of numerals have ever been written. Also, if brains are mere finite biological machines of some sort, then to conceive of a number might be comparable to a computer process accessing a symbol in a memory location. And if numbers are merely physical symbols at various locations inside brains, then just like written numerals, it is likely that only a finite amount of them can ever exist.

But we like to believe that natural numbers are not physical in any way. We might say that for any given natural number (n) its successor (n+1) must exist and therefore infinitely many natural numbers must exist. This is a cyclic argument because we cannot show that numbers have their own out-of-brain existence. We must first assume that infinitely many non physical numbers exist before we can make this argument to show why infinitely many numbers must exist. This belief in the existence of non physical or ‘abstract’ objects is called ‘Platonism’ after the Ancient Greek philosopher Plato.

Many modern mathematicians try to distance themselves from Platonism by calling themselves ‘formalists’. They claim to reject the notion that numbers are out there somewhere, mysteriously existing in a non physical form. Instead of claiming that all numbers ‘exist’, they say that mathematics is just a game we play with symbols and rules. For example, with just ten symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and a finite number of rules (such as Rule_1: Each symbol by itself is a number; Rule_2: If you append one or more symbols to a number, where the left-most symbol is not allowed to be zero, you get another number) then this allows for an infinite number of expressions.

The claim is that these symbols and rules are not subject to physical limitations. Yes we can only use them to construct a finite quantity of numbers, but this doesn’t matter. Merely by appreciating what can be done with these ten symbols and these two rules we can say that we have conceived of something that ‘defines an infinite number of expressions’. Bizarrely the claim is that we can conceive of the infinite by describing how to do something and saying it can be done repeatedly.

But again, if we relate these rules to written numerals then the argument fails. The rules merely provide instructions on how we might construct other numbers, they do not tell us how we can achieve a quantity of them that is not finite. Even if we were to add a third rule that says to keep constructing different numbers endlessly then all it would be saying is to keep increasing a finite quantity to a bigger but still finite quantity.

However, formalists might argue that we can instruct a computer to set x = 0 at time 0, then set it to 1 at time 1/2, 2 at time 3/4, 3 at time 7/8, 4 at time 15/16 and so on. It follows that when time = 1, the computer will have set x to all (infinitely many) natural numbers. The formalists might argue that it doesn’t matter that no real world computer can ever be built that can perform this task because we should not be constrained by physical considerations. They will claim that every aspect of this hypothetical scenario is well defined and the computer could theoretically do an infinite amount of things within a finite amount of time.

Here the formalists have chosen to ignore the obvious problem. The hypothetical computer is performing one step after another, and so once the process has stopped, we know for certain that there must have been a last step that it performed. In other words, there must have been a ‘last number’. This logic is as simple and as obvious as logic can get. It does not matter when we stop the process, if the process has stopped then there must have been a last number processed. This contradicts the concept of ‘infinitely many’, as this concept requires there to be no last number.

These arguments and counter arguments are not new. This ‘last part’ argument is essentially the same argument used by the Atomists of Ancient Greece over 2,000 years ago to show why any theoretical line cannot be divided into infinitely many parts (see the videos below). And so it is merely a rerun of the classical confrontation between Atomism and Platonism.

The formalists say they are manipulating syntactic forms whose shapes and locations have no meaning unless they are given some interpretation (or semantics). They claim that Platonism only occurs when the symbols are interpreted in a certain way. In this respect they claim that formalism is separate from Platonism. For example, when we look at the earlier example of the ten single digits together with two simple rules, we have to interpret the words of the two rules. The anti-Platonist might interpret these ten symbols and two rules as a set of instructions that tell us how we can construct as many ‘number’ symbols as we like until we run out of physical resources. The symbols are physical, the written rules are physical and if we follow the instructions to construct more numbers, then everything involved in this process would be physical. This would be a non Platonic interpretation that would appear to demonstrate that formalism does not need to contain Platonism.

But formalism consists of two parts: 1) the manipulation of symbols and 2) the assertion that we are not constrained by physical considerations. The first part does not appear to be connected to Platonism but the second part is a blatant Platonic assertion. Formalism is merely a re-branding of Platonism to make it sound more acceptable. Words like ‘rigourous’, ‘consistent’ and ‘well-defined’ are often thrown in to make the formal approach sound good and proper. In the formalist’s world we are no longer allowed to interpret the ten symbols and two rules as a set of instructions about physical things; they don’t even have to relate to a process. We are now forced to accept the bewildering Platonic assertion that ‘this allows for an infinite number of expressions’.

Platonism won the argument over 2,000 years ago because nobody dared to deny that the Greek gods existed and were perfect forms. In the interviening years many mathematicians have had their own motivations for accepting Platonism. Kurt Gödel claimed his ‘ontological proof’ was a formal argument for God’s existence. Georg Cantor believed that the theories he’d discovered had been communicated to him by God. He equated actual infinity directly with the concept of God. Alan Turing believed that when the body dies, the spirit finds a new body. Of course, the supernatural beliefs of great mathematicians are in no way arguments against the logic of their mathematics, but it may provide some insight as to why they embraced the Platonic foundations upon which their arguments relied.

Just because the predominantly religious mathematicians of the past 2,000 years mostly accepted Platonism, it doesn’t mean we all should keep on accepting these beliefs. There have been dissenting voices during this period but they were usually drowned out (and often ridiculed) by the followers of the prevailing Platonic viewpoint. Everyone who learns mathematics through academic study is indoctrinated with the Platonic approach. There is no academic route available to the would-be anti-Platonist mathematics student. For anyone seeking academic success it is Plato’s way or the highway. Sadly this will guarantee the continued success of Platonism for many years to come.

So all you need to do is to accept Platonism (disguised as formalism) and you will know what a number is. You will be able to imagine that it is some non physical concept, like the notion of ‘three-ness’, and you will have the ability to conceive of the infinite. Good luck with that!

Related Videos

1. What is a Number? Can Infinitely Many of them Mysteriously Exist?

What is a number? How can numbers exist? Regarding existence, does infinity exist? How can infinitely many numbers exist? And what does it mean anyway? Could all of maths be wrong? This video examines the arguments for and against Platonic existence. It describes how Plato won the battle to ensure that in today’s world, most people believe in the Platonic existence of infinitely many abstract objects.

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2. A Static Zeno’s Paradox Defies Infinity and Infinite Divisibility

What is mathematics? Can a line or anything else be infinitely divisible? Could all of maths be wrong? This video examines the core argument concerning the validity of the concept of ‘infinitely many parts’ involving a finely-tuned version of Zeno’s Paradoxes that appears to indicate that everything must be granular in nature. So does infinity exist? Take this tour through the history of mathematics to see how the metaphysical foundations were established, and how the whole of mathematics is built upon whatever other-worldly set of rules our mathematicians choose to believe.