Platonism’s Logical Blunder
Many mathematicians and logicians love the notion of abstraction - taking an initial idea and tweaking it so that nothing only the absolute fundamental essence of it remains.
That’s an admirable ideal - but you have to be careful that when you do that, you don’t throw out the baby with the bathwater. Unfortunately, in some cases, that is precisely what mathematicians have done. For example, they appear to believe that the definition of a formal system can be divorced entirely from any reference to the physical world. But can it?
In standard descriptions of formal systems such as the Peano axioms it is assumed at the outset that it is self-evident what a variable is. For example, the very first Peano axiom is commonly stated as: (Footnote: e.g. Wikipedia: Peano axioms, Wolfram: Peano axioms, Encyclopedia Britannica: Peano axioms.)
0 is a natural number
Here “natural number” is a variable, but here we are, supposedly, at the very the beginning of the definition of a formal system. Yet there is no description whatsoever of what a variable is, what a natural number is, or even of what a domain of a variable is. And this is only the first axiom.
Suppose that one were trying to explain this to someone who had been a castaway on a desert island since early childhood, before he had any acquaintance with the written word. Presumably our castaway would have some sort of mental concept similar to the notion of variables in relation to physical objects, so that when he would say, “that’s a fish”, he means that there is a physical thing that he is referring to that satisfies his notion of what a fish in general is like. Or he could have some sort of mental concept similar to the notion of variables in relation to abstract things, such as when he ascribes an impending storm to one god, and the subsequent arrival of sunshine to another god - so that when he refers to a god, he is referring to some abstract notion that he has conceived, but for which he does not have any good evidence for its actual existence.
But it is most unlikely that he would have an innate concept of the mathematical notion of a variable as it occurs in the Peano axioms. So when he is told “0 is a natural number”, he would presume that “0” is a name for either a physical thing or for a non-physical concept like a deity; and that either “natural numbers” are some sort of physical things or that they are some sort of non-physical concept.
However, with the Peano axioms, the term “natural number” does not refer to a physical object like a fish. But it does refer to physical things that are sequences of symbols of a certain type. So, really, before even beginning the Peano axioms, in order to define a formal system, we should be defining what symbols are, what symbol sequences are, and what we mean by a variable whose domain is symbol sequences.
But what happens if one tries to explain what a symbol is without making any reference at all to physical objects? Can it be done within a purely formal system? Can it be done at all? I don’t think so. And after explaining what a symbol is, you then have to define a sequence of symbols.
So can we simply ignore the physical concepts that are inherent in the Peano axioms, and come up with a definition of numbers that are totally abstract, in the same way that our castaway has an abstract notion of gods?
Platonists (See also The Myths of Platonism, Platonism’s Logical Blunder, and the posts Moderate Platonism and Descartes’ Platonism) like to try to divest all physical references from mathematics. For a committed Platonist, the term “natural number” refers not a physical symbol sequence, but to an abstraction of the above concepts. By abstraction, the Platonist can come up with a set X of rules that will be satisfied by a certain type of entity that is a natural number, and where the set X of rules makes no mention of the physical world. The Platonist then asserts that that set X of rules is the ultimate definition of what a natural number is. He will say that every natural number that we know of satisfies this set of rules.
But just because we know that a natural number n satisfies a certain set of rules X, that does not necessarily mean that the set of rules X includes every condition that is required for n to be a natural number.
To assume that it does is a classic logical blunder. Simply because A (in this case that a particular n which we know is a natural number) implies B (in this case that n satisfies a set of rules X), that does not necessarily imply the converse, that is, that B implies A. It doesn’t. It may happen that B applies, but it ain’t necessarily so.
And in this case, it isn’t so - the converse would be that if m satisfies a set of rules X, then it is necessarily the case that m is a natural number. The converse most certainly doesn’t apply - because the Platonist set of rules X omits any reference to the physical world that definitions of a formal system such as the Peano axioms necessarily require, even if they are not explicitly stated - but they should be.
The result is the common perception that numbers can be a purely abstract notion totally divorced from physical reality, which is based on an assumption that has no logical basis. This notion ties with the belief that there “exist” real numbers which somehow belong to the domain of a variable, but at the same time cannot ever be substituted for that variable - a belief arising from the confusion of different levels of language - see, for example The Diagonal Proof.
And these Platonist notions give rise to something I have often encountered; some people claim there is an essential difference between a symbol sequence of a formal system and a symbol sequence that is claimed to not itself actually belong to a system, but it is only a reference to an abstract concept belonging to an abstract system. For example, they might claim that “x + 6 > 3” is not itself a number-theoretic relation, but is only an expression that refers to an abstract number-theoretic relation. But at the same time, if they are talking about formal systems, they insist that a symbol sequence such as “x + SSSSSS0 > SSS0” belongs to a formal system, and that in this case “x + SSSSSS0 > SSS0” is not an expression that refers to an abstract number-theoretic relation - although they will admit that there is some sort of isomorphism between it and an abstract number-theoretic relation (which they never actually define).
But of course there is no essential difference between “x + 6 > 3” and “x + SSSSSS0 > SSS0” - they simply belong to slightly different systems.
But it is quite often the case that someone will claim that there is a difference and that one (“x + 6 > 3”) refers to abstract concepts, while the other (“x + SSSSSS0 > SSS0”) is simply a symbol sequence of a formal system.