This question arises, in part, because of the representative nature of language. Words, whether spoken or written, are commonly understood to be symbols which refer to things. Nouns refer to persons, places, objects, or ideas. Verbs refer to actions or states of being. Adjectives and adverbs refer to properties and qualities; prepositions refer to relationships. What about numerals? When I see the numeral ‘3’ written on paper, it is a symbol, and I might well ask if it refers to something, and if so, what that something is.
If numerals refer to something, that something would be numbers. Numerals are physical symbols, consisting of ink on paper. What type of things would numbers be? Numbers would be non-physical things, i.e. metaphysical things, at least according to philosophers like Plato. Plato's view, adopted by a number of other significant philosophers over the centuries, is called dualism and realism. The word ‘dualism’ indicates that Plato is positing two levels of reality, a physical level and a metaphysical level. The word ‘realism’ indicates that Plato posits that mathematical statements are statements about an independently-existing state of affairs, that mathematical words refer to independently-existing things, and that mathematical propositions are true or false to the extent that the correspond, or do not correspond, to such an independently-existing state of affairs. Author Rui Vieira writes:
What exactly are the objects of mathematics, and how do they relate to our knowledge of them? Since Plato (427 BC–347 BC) such questions have been central to the philosophy of mathematics. Plato realized that mathematics seems to involve perfect circles, triangles, and so on. But as Plato also noticed, there are no perfect circles or triangles to be found in the world, only imperfect approximations. Imagine a polygon [shape] with an ever-increasing number of equal sides. As the number of sides approaches infinity, the polygon will become a circle. Thus a perfect circle may be conceived of as a regular polygon with an infinite number of infinitesimally small sides. So no matter how accurate our or our computer’s rendering of a circle may be, it will only be an imperfect approximation. Finite humans and their computers cannot create objects with infinite mathematical features, such as the infinite sides of our ideal circle. Plato concluded that since there are no perfect mathematical objects to be found in the world, the objects of mathematics – perfect circles, triangles, and indeed numbers themselves – must somehow exist as eternal abstract entities beyond space and time in some otherworldly Platonic heaven called the world of Forms (or Ideas). Plato’s particular type of mathematical realism (ie, of attributing objective reality to mathematical objects), has been one of the most prevalent views of mathematics among both philosophers and mathematicians ever since.
Opposed to Plato’s view, however, are several groups of philosophers, called variously ‘formalists’ and ‘conventionalists’ and including brilliant mathematicians like David Hilbert, Heinrich Eduard Heine, and Carl Johannes Thomae. These thinkers regarded numerals as non-referential. Phrased another way, they thought that numerals are numbers and numbers are numerals. A numeral, according to formalism, is a mark - ink on paper - which is related to other marks by means of a set of rules. Famously, the analogy to games arose. Consider a card game. The seven of spades, or the eight of clubs, does not refer to, or represent, anything. Its role in a card game is simply that if it is played, then certain other things must happen in the game.
According to the formalists, 7 + 5 = 12 because the rules of the game are such that when four symbols ‘7 + 5 =’ are written, then the fifth symbol must be ‘12’. By contrast, the mathematical Platonist considers that 7 + 5 = 12 because when the object called ‘7’ is added to the object called ‘5’, another object, called ‘12’, is formed.
The formalists - conventionalists are similar - thought that they had made a contribution to the progress of mathematics, because they had freed mathematicians from mysterious discussions of what type of ethereal thing a number might be. They felt that mathematicians should not have to be hindered by discussions of metaphysical objects.
The Platonists, on the other hand, saw that formalism contained certain weaknesses. If the formalist views were adopted as the foundations of mathematics, and those weaknesses emerged, the entire superstructure might collapse. A leading critic of formalism was Gottlob Frege, who wrote:
The question now forces itself upon us: Is calling these signs numbers enough to ensure that they have the properties of the numbers proper which we have previously been accustomed to regard as quantitative ratios?
To be fair to David Hilbert, he and some of the formalists did not make a blanket ontological assertion that there are no metaphysical objects to which numerals refer. Hilbert was not so much saying that numbers as Platonic objects do not exist as much as he was saying whether or not numbers exist as Platonic objects, we will go ahead and do mathematics based only on numerals as otherwise meaningless symbols manipulated according to the rules of mathematics. The only relevant meaning had by numerals, for Hilbert, is the meaning given to them by their place in the rules. Whether or not they referred to Platonic numbers, and whether or not those Platonic numbers really existed, was for Hilbert not relevant.
There is a detectable difference between formalists and conventionalists. Formalists treat numerals as meaningless symbols to be arranged and manipulated according to a set of rules. Conventionalists treat mathematical propositions, and perhaps also meta-mathematical propositions, as true by agreement: they are the stipulations of the mathematical community. But for the purposes of discussion the ontological status of numbers, formalists and conventionalists are very similar, and we may conveniently lump them into the same category.
In any case, Frege sees formalism and the formalist project as a failure. He alleges that it cannot hold to its own program. While the formalist lays the foundations of the mathematics promising to treat numerals and all other operators as meaningless patterns of ink on paper, and to simply do no more than rearrange those symbols according to a set of rules, the formalist will in fact later attribute meaning to those symbols. Frege asserts that the formalist, having kept meaning and reference out of mathematics at the foundations, will later sneak little bits of meaning and reference into mathematics in its more advanced stages of development. Frege writes:
This attempt at formal arithmetic must be considered a failure, since it cannot be pursued consistently. In the end numerical figures are used as signs after all.
Specifically, Frege noted the use of the operators ‘greater than’ and ‘less than’ and asked, in which way a numeral, as opposed to a number, could be greater or less than another numeral. True, the formalists can reply that numeral are assigned an arbitrary order, and that the operators simply reflect that order; but this line of thought is complicated by the fact that there is an infinity of numerals, in fact an uncountable infinity, and how would an order be assigned to them all? Likewise, the formalists wants to maintain that an expression like ‘3 ÷ 0’ is meaningless, but if the formalists have already declared all symbols meaningless at the foundation, in which way is ‘3 ÷ 0’ especially meaningless? Frege continues:
We saw that terms and expressions were borrowed unconsciously and without explanation from arithmetic that has a content (e.g. ‘larger than’ and ‘smaller than’) and that their role in the calculating game remained obscure, although it seemed to be highly important. Formal arithmetic proved unable to define the irrational, for it had only a finite number of numerical figures at its disposal.
Frege stands with Plato and rejects the formalist interpretation of mathematics. The matter is not resolved. The formalist camp will charge Frege with violating Ockham’s Razor. While Frege has succeeded in demonstrating that there are difficulties for formalism, he has not conclusively demonstrated the existence of Platonic numbers.
