What is the number 2?

Question

My friend told me that he took a course in the philosophy of mathematics and said that they defined the number 2 to be "the set of all sets with two elements." I may be remembering wrong, but this is what I think he said. This is an incorrect definition, mostly because the collection of all sets with two elements is not a set (it is a proper class). This all I "know" about what philosophers think of the number 2, and I have a hard time believing philosophers are so misled.

The thing that seemed to bother the philosophers (according to my friend) is that the number 2 could be a natural number, an integer, a rational number, a real number, and a complex number. To me this is not a problem, and indeed there are infinitely many other different things that I call "2," many of which are also called "0" (this happens in a ring of characteristic 2). If I think about it, though, it is only in trivial cases where something I call "2" could also be called "3," and that is when they are both zero.

I am interested in hearing how this apparent problem is resolved from a primary source who does not, as do my friend and I, basically consider this pointless speculation (no offense intended). What is the number 2?

Answer 1

I think it might be best to define 2 to be the result of 1+1, because that leaves underlying mappings to set theory or anything else wide open. The thing that distinguishes the meaning of the symbol "2" is its behavior in arithmetical operations.

In that case, when you define integers via set theory, such as von-Neuman ordinals, You take care to define 0 and 1, then the definition of 2 is 1+1, which is trivial to show as {0, 1}. However, remember that von-Neuman ordinals define just one construction of natural numbers. When you extend this to rational numbers, the definition is still 1+1, though now that definition has become an equivalence set.

If you wish to narrow it down further, Peano arithmetic could change 1+1 to S(1). In Peano arithmetic, that is "simpler" than 1+1. In other arithmetic, it might not be.

Remember, "2" is just a symbol. Its a curved line intersecting with a low horizontal line. It's only the rules of arithmetic that give it meaning beyond that.

Answer 2

The above answers present very interesting points, yet it seems to me that they touch mere top of the iceberg. Philosophical problems behind numbers are far more fundamental and subtle.

First of all, no competent lecturer of philosophy of science would ever make such a bold statement and give a reputedly ultimate definition of a number, or of anything else, and for sure no one would have ever done that having provided no alternative view. Philosophy is not a science, we have different purposes. Obviously, there have been hundreds of accounts of the aim of philosophy throughout the ages, but I would risk to claim that contemporary analytic philosophy of science considers itself a sort of a metadiscipline assessing what has to be taken for granted by other fields of research for them to work. The question of the character of numbers is a perfect example of this approach.

Thus philosophers proposing well-argued terms and definitions contrary to common scientific practice are not misled. It is their duty to think out of the paradigm. And not just philosophers, the theory of mathematics or axiology are also concerned with those problems. Even more practical mathematicians do make use of them, perhaps unconciously, when switching between paradigms, like from point-based to noncommutative geometry.

The point here is not about a functional, mathematical definition of numbers and how they work in mathematical frameworks. What a philosopher is interested in is usually their ontic status. Namely, for instance, is number 2 an abstract object that can be labelled and referred to, like mental states, etc., according to some theories. Or is it rather a certain concept we use as a part of a given language-game. Or maybe it is a scheme, a pattern, non-existent on its own but nevertheless determinable? The stake here is unbelievable - the very subject and role of mathematics.

There have been many approaches to the problem. From Plato's ideas, through Kant's relations in space, Frege's extensions of concepts based on the relation of equipotency, to moderns times of Russell, Whitehead, or Wittgenstein. The topic is huge, and there is no single, nor all the less simple answer. For preliminary reading I would suggest Frege's "The Foundations of Arithmetic", Russell and Whitehead's "Principia Mathematica", and Wittgenstein's "Remarks on the Foundations of Mathematics". There are also nice, original clues in Kripke's "Wittgenstein on Rules and Private Language".

Mathematicians usually do not care about it, I have had several opportunities to find it out myself in the faculty. But hey, this is no surprise for philosophers! (;

Answer 3

As Kronecker might say the Natural Numbers are God given, the rest are the work of man. My interpretation of this is that the Counting number are objects of intuition derived from experience, other "numbers" are constructed my man, and none have an existence outside of minds and culture. The axiomatic definitions/constructions of the Naturals (set theoretic, Peano etc) are attempts to capture our intuition in a formal system. Within this structure of Natural Numbers we identify a particular element as the Number 2. We extend the Natural Numbers to the Integers by extending the Naturals so they are closed under subtraction. Now we identify the Integer 2 with the Natural 2 in the obvious manner, but note we are already using 2 in two different senses, I what follows I will be talking about the Integer 2 rather than the Natural 2.

We define the Rational Numbers as the set of ordered pairs (a,b) of Integers a and b (b not equal to 0) with no common factor other than 1 (or rather of equivalence classes of ordered pairs of Integers under a suitable equivalence relation). With "arithmetic" operations defined to be in some sense analogous to what we require to mimic everyday calculation with fractions. Within this new structure of Rational Numbers we identify a substructure, the set of Rationals of the form (a,1) which is isomorphic to the Integers. Now by slight of hand we identify this substructure of the Rationals with the Naturals and refer to the element (a,1) as a.

So when we are talking about the Number 2 in the context of the Rationals we are really talking about (2,1) which while not an Integer can with little risk be talked about as though it were.

A similar process takes place when we construct the Reals from the Rationals, we treat the natural substructure isomorphic to the Integers as though it is the set of Integers (and the substructure isomorphic to the Rationals as though they are the Rationals), and we repeat this slight of hand when we go to the Complex Numbers.

A slightly different slant on this is required if instead of constructing the Ration, Real and Complex Numbers in this way we define them but the general idea holds, there are substructures in these isomorphic to the Integers which are referred to and treated as though they are the Integers.

In summary the Counting Numbers and the number 2 in particular, is an object of intuition, then we formalise our intuition and call the formalised system the Natural Numbers, and the element playing the part of 2 we call 2. From there we extend the system assigning 2 to the element that plays the part of 2 in the extension.

The extreme degree of "hand waving" that I have had to employ is to obviate the extended discussion needed to explain precisely what I think is going on in this almost universal, useful, abuse of terminology.

An explanation of the some of the main ideas of the Philosophy of Mathematics may be found in the relevant Stanford Encyclopedia of Philosophy Article on the same.

Answer 4

It's probably useful to note that the very appealing "definition" of Frege for 2 as "the set of all sets with 2 elements" is still a useful philosophical guideline, even though the Fregian logic itself turned out to be unsound.

In particular, the theory of cardinalities starts by defining a cardinal as an equivalence class of sets, which itself may be larger than a set, and then identifying a special representative of that class, which is a particular set, usually an ordinal. The advantage of ordinals is that they are somewhat canonical, though one must fix a particular construction of ordinals to avoid similar issues.

So morally 2 is the set of all sets with 2 elements, though one may choose the ordinal {0,{0}} as a canonical representative.

There are consistent theories which do allow the impredicative definition, e.g. System F, where 2 is the function which takes a function and an argument and applies the function twice to that argument, or Quine's New Foundations.

Answer 5

• Numbers are mathematical objects, therefore, as such, they exist only under the domain of human understanding.
• Sets exist as factual entities (three atoms that interact permanently forming a molecule) and as ideal concepts (the idea of the H2O molecule): philosophically, an objective fact and a subjective interpretation. It is easy to understand the subjective interpretation, but the objective fact, not: why my favorite gloves pertain to this set? they don't have any physical relation! Even if the relation exist only in my mind, the gloves exist as physical objects. And on an atomic level, the relation between one glove and the other is exactly the same as between this glove and that tree. The relation can be only mental. Remember the two favorite cities you knew on vacations? they are on the group, but their relation is the same as with that tree.
• A person who never had contact with humans can have the idea of sets of two things, but be incapable of reading the symbol "2". What is the difference between him and us? That we share the idea of this symbol. The point is that we associate the set of two things not only with pairs of things, but also with the symbol "2". Therefore, 2 is also a symbol, the number.
• In consequence, 2 is just another symbol, like any musical object on the score is just a symbol. How does it differs with other numbers? Because we associate it with a property, like the symbol Dm7, D minor seventh chord is associated with a chord sound. The property applies to the set 2 and conditions the size of the set to a couple of objects. It is the reason of 2 being different to 3.
• You are right about 2 being a proper class. But I also will use set, due to this definition is really nice to be used in daily life (I prefer telling my kids that "the set..." instead of "the proper class..."). That's why your friend used that term.

Therefore: 2 is... 1. Customarily a set, or formally, a proper class; 2. a subjective interpretation (an idea of the set); 3. a symbol related to the subjective interpretation; 4. a distinctive property of the set (having a couple of things inside) 5. an objective fact (the physical existence of two objects sharing a relation, even if the relation is only mental -the subjective interpretation-).

"The set(1,2) of all sets(1,2) having two(3,4) elements(4,5)": personally, I take it.

Answer 6

This is a "what is" question about the origin of a concept. There are many ways to define 2, but the Formalist who defines 2 by means of its use in arithmetical operations has missed the point: humans are not the only animals who can sense 2, neither do all the animals who can sense 2 know arithmetic. And there is good reason to believe that early humans conceived the notion of 2 before they knew arithmetic.

The meaning of 2 is abstracted from sensations.

Two is what all two-things have in common: a pair shoes, a pair of socks, two knocks on the door, two smells, two ears, two eyes, two hands, two swans, two otters, etc; they all have the common quality which we call two.

It is easy to imagine two-things (e.g., a pair of shoes), but no one can sense or imagine 2 detached from things; 2 is actually a universal; it is what all the sensations of sensing two things have in common, i.e. 2 is a common property of all couples.

The logical extension of the above sensible experiences is this: 2 is the set of all couples because a set represents a property that defines its extension.

Red is a what all red sensations have in common, thus an instance of seeing a red dog is an instance of red. Red is a colour, but "an instance of seeing a red dog is a colour" is not true, thus the hierarchy of universals is like this: colour is the class of particular colours; a particular colour is a class of sensations each of which has this colour. It follows that colour is a universal just like 2 is a universal: both are class of classes. *

The rest of the story is told in Whitehead & Russell's Principia Mathematica, where using class of classes to represent a number is shown to be sufficient to deduce ordinary mathematics.

*Note: A class in PM is simply a collection of things of the same type. There is no such notion as "proper class" in PM.