Numbers, by contrast, are not physical, and not directly detectable by the senses. Numbers are ideas. Numerals refer to numbers. Numerals are the names for numbers. The English numeral one, the German numeral eins, and the French numeral un all refer to the same number.
So what kind of a thing is a numeral, if it’s not physical? This question has occupied philosophers for more than 2,000 years.
In Greece, Plato began wrestling with this question around 300 B.C., and later philosophers who embraced and refined his ideas asserted that numbers are metaphysical objects. What are metaphysical objects? They’re non-physical objects.
Normally, one thinks of objects, or things, as physical — as detectable by the human senses (sight, sound, taste, touch, and smell).
But those philosophers who argue for the existence of metaphysical objects would point to things like ideas, memories, emotions, and minds as being objects. A person can ask for five pounds of anger, or two gallons of childhood memories — metaphysical objects aren’t measured in the way in which physical objects are.
So, according to the Platonists, numbers can be referred to as ‘things’ or ‘objects.’ But they are a different kind of thing or object than, e.g., a sandwich or a tree.
The Platonic understanding of numbers is called ‘realist’ because the Platonists believe numbers to be as real as any physical object. The Platonic view is also called ‘dualist’ because Platonic philosophers assert that there are two types of reality — the physical and the metaphysical — both of which are different and both of which are real.
The dualist view can be extended from numbers to geometry. The dualists would argue that the circles, triangles, points, and lines in a geometry textbook are merely physical symbols for the idea of circles, triangles, points, and lines.
Opposed to the Platonists are other philosophers, called ‘formalists’ and ‘conventionalists’ — they argue that numerals do not refer or represent numbers. Rather, they argue that numerals are part of a system of symbol manipulation, a system called ‘arithmetic’ or ‘mathematics.’
These anti-Platonists argue that there is no deeper, higher, or mystical reality beyond the symbols called ‘numerals’ — that the numerals are the numbers.
In this debate, each side has its reasoning: The Platonists say that, in order for mathematics to have any meaning, it must be “about” something, that it can’t simply be patterns of ink on paper; the Platonists argue that the anti-Platonist view would reduce mathematics to meaninglessness, and daily experience in practical application shows that mathematics has real meaning.
The anti-Platonists, meanwhile, argue that for practical purposes, mathematics works: it allows people to build bridges or calculate interest or keep score at a game. If it works, the anti-Platonists say, then that’s all that is needed from mathematics: there is no need for any deeper meaning. The anti-Platonists accuse the Platonists of violating a rule of logic called ‘Ockham’s Razor.’
William of Ockham, sometimes spelled Occam, was a logician who worked around 1300 A.D., and was mildly anti-Platonist. He wrote:
Numquam ponenda est pluralitas sine necessitate.
Which is roughly:
Plurality must never be posited without necessity.
With regard to mathematics, this would mean that there is a choice between two systems: one system has both numbers and numerals, while the other system has only numerals. Ockham might advise that, if the simpler system works, it should be chosen.
In another formulation, Ockham wrote:
Frustra fit per plura quod potest fieri per pauciora.
Which can be rendered as:
It is futile to do with more things that which can be done with fewer.
Ockham’s Razor is sometimes phrased this way: “Entities should not be multiplied beyond necessity.”
In any case, Ockham’s logic is part of the debate about the reality of numbers.
The question of the reality of numbers is a discussion which taps into multiple levels of philosophical questioning. The first question is: Are numbers independently-existing objects? The second question is a broader form of the first question: Are there really any non-physical or metaphysical objects? The third question asks about how the truth of a statement is evaluated, in this case, the truth of a statement about numbers: Is a proposition true if it works in a pragmatic sense, to calculate in real-life situations, or is it true if it represents the way things actually are?
