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2 = +2 = 2/1 = 2.0 = 2.0 + j x 0.0?

Summary:
"Typically 2 the integer is used for counting, whereas 2 the real number is used for measuring. But in higher mathematics there's a technical sense in which integers aren't real numbers -- we say instead that they can be identified with real numbers." -- Timothy Gowers 1.0 Introduction Numbers, in some sense, are only defined in mathematics up to isomorphisms. This post runs quickly through some math to explain what this means. I begin by assuming knowledge of the natural numbers, {0, 1, 2, ...}, as characterized by the Peano axioms. I also assume an understanding of what it means for two natural numbers to be equal, for one to be greater than another, and for two to be added or multiplied together. Other operations could be built on top of this structure, as needed. 2.0

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"Typically 2 the integer is used for counting, whereas 2 the real number is used for measuring.

But in higher mathematics there's a technical sense in which integers aren't real numbers -- we say instead that they can be identified with real numbers." -- Timothy Gowers

1.0 Introduction

Numbers, in some sense, are only defined in mathematics up to isomorphisms. This post runs quickly through some math to explain what this means.

I begin by assuming knowledge of the natural numbers, {0, 1, 2, ...}, as characterized by the Peano axioms. I also assume an understanding of what it means for two natural numbers to be equal, for one to be greater than another, and for two to be added or multiplied together. Other operations could be built on top of this structure, as needed.

2.0 Integers as Equivalence Classes of Ordered Pairs of Natural Numbers

Consider ordered pairs (a, b) of natural numbers. The point of this may be cryptic for a while. Part of the point is to check that no manipulations of these ordered pairs or definitions rely on anything that is not defined in the natural numbers.

I define equality between two ordered pairs:

(a, b) = (c, d)

if and only if

a + d = b + c

At this point, one should check that, by this definition, an ordered pair is equal to itself (reflexive) and that if an ordered pair is equal to another ordered pair, that ordered pair is also equal to the first (symmetry). One also wants to show that this definition is transitive.

if (a, b) = (c, d) and (c, d) = (e, f) then (a, b) = (e, f)

A relation with these three relations is called an equivalence relation. An equivalence relation breaks a set into non-intersecting equivalence classes. All elements of an equivalence class are equal to one another, and no element in the set outside an equivalence class is equal to any element in that class.

Here are three equivalence classes, by this definition: {(1, 0), (2, 1), (3, 2), ...}, { (0, 0), (1, 1), (2, 2), ...}, and {(0, 1), (1, 2), (2, 3), ...}. Is it becoming clear what is going on here?

Next, I want to define a total order:

(a, b) ≥ (c, d)

if and only if:

b + ca + d

I claim that ≥ is well defined. That is, if

  • (a, b) = (a', b')
  • (c, d) = (c', d')
  • (a, b) ≥ (c, d)

then:

(a', b') ≥ (c', d')

I also want to show that ≥ is reflexive, transitive, and antisymmetric. This relation is antisymmetric if and only if for all ordered pairs of natural numbers,

if (a, b) ≥ (c, d) and (c, d) ≥ (a, b) then (a, b) = (c, d)

Furthermore, the relation is complete. For all pairs of ordered pairs, (a, b) ≥ (c, d) or (c, d) ≥ (a, b). With these properties, ≥ orders equivalence classes, just as well as ordered pairs of natural numbers. And one could use the above to define a relation >.

The next step is to define addition:

(a, b) + (c, d) = (a + c, b + d)

This definition also applies to equivalence classes. The sum of elements from two equivalence classes is in the same equivalence class, whichever elements you start with. Notice that:

(a, b) + (0, 0) = (0, 0) + (a, b)

So one might as well define:

+0 = {(0, 0), (1, 1), (2, 2), ...}

The negative numbers are those equivalence classes less than zero. That is, if (a, b) is in an equivalence class where a > b, then that equivalence class is a negative number. The positive numbers are equivalence classes greater than zero. More on this below.

Addition has some other properties of interest. But I am going to move on to define multiplication:

(a, b) * (c, d) = (ad + bc, ac + bd)

Multiplication is well-defined for equivalence classes. Multiplication also has an identity element:

(a, b)*(0, 1) = (0, 1)*(a, b) = (a, b)

So define +1 as {(0, 1), (1, 2), (2, 3), ...}.

Now, associate every natural number n with an equivalence class:

f( n ) = +n = {{(0, n), (1, 1 + n), (2, 2 + n), ...}

The function f is one-to-one and onto for the set of non-negative integers. It preserves equality:

If n = m, then f( n ) = f( m )

The first equality is a relation in the set of the natural numbers. The second equality was defined above. The function f also preserves order, addition, and multiplication:

  • If nm, then f( n ) ≥ f( m )
  • n + m = f( n ) + f( m )
  • nm = f( n ) * f( m )

The function f is an isomorphism.

Suppose proofs can be given for propositions stated above. Then I have (unoriginally) constructed the integers out of the natural numbers. Negative numbers are not some mystical entities. In this construction, zero is the additive identity. Every integer has an additive inverse, that is, one can negate every integer. Addition is associative and commutative. The integer +1 is the multiplicative identity, and the additive and multiplicative identities are distinct. Multiplication is commutative and associative. Multiplication distributes over addition. Finally, if (a, b) * (c, d) = +0, then (a, b) = (c, d) = +0.

Any set with two binary operations with these properties is an integral domain. I think the set of polynomials with coefficients that are rational numbers, with the usual definitions of polynomial addition and multiplication, is also an integral domain.

So the natural number 2 has been shown to be isomorphic to the positive integer +2. I think this proof of the first equality in the post title is fairly typical of math. It is not particularly difficult, but requires what is called "mathematical maturity". One has to keep track of what can be inferred from definitions, and not let your intuition leap ahead. Even so, your intution will guide you. When writing the above, I kept thinking of (a, b) as "b - a". I would like to say this is the kind of mathematics that Bertrand Russell amused himself with when the Brits put him in prison during World War I. But this would all come after the proof of 1 + 1 = 2 in Principia Mathematica.

3.0 The Field of Quotients of an Integral Domain

Now consider ordered pairs (a, b) of the elements of an integral domain, where b is not +0, the additive identity for the integral domain. Here, I define equality by:

(a, b) = (c, d) if and only if a*d = b*c

I have put aside the structure of the elements of the integral domain, whether they are equivalence classes of ordered pairs of natural numbers or polynomials with rational coefficients or whatever. Addition is defined by:

(a, b) + (c, d) = (a*d + b*c, b*d)

One can show that this definition is well-defined for equivalence classes. The additive identity is the set of ordered pairs {(0, a), a a non-zero element of the integral domain}. Multiplication is defined as:

(a, b)*(c, d) = (a*c, b*d)

The multiplicative identity is the set of ordered pairs equal to (1, 1). Every non-zero ordered pair (a, b) has the multiplicative inverse (b, a).

Define an isomorphism to a subset of the quotient field defined above:

g( a ) = a/1 = {(c, d), where (a, b) = (c, 1)}

Obviously, I have skipped over a lot of steps. But the above is an outline of how to construct the rational numbers and to prove that +2 = 2/1.

4.0 The Reals as Equivalence Classes of Cauchy-Convergent Sequences of Rationals

The set of real numbers is the set of limit points of all convergent sequences of rational numbers. I guess below is Cantor's approach to constructing the real numbers. Dedekind had another approach, and Rudin (1973) presents this construction based on "cuts".

Consider a sequence of rational numbers (a0, a1, a2, ...) where for all ε greater than zero, there exists a natural number N such that for all n greater than N,

| an + 1 - an | < ε

In some sense, the terms in the sequence get closer together. Such a sequence is known as Cauchy-convergent.

Two sequences (a0, a1, a2, ...) and (b0, b1, b2, ...) are equal if and only if for all ε greater than zero, there exists a natural number N such that for all n greater than N,

| an - bn | < ε

This definition will yield equivalence classes. Addition of sequences is defined termwise. Should multiplication also be defined termwise? What about order?

Anyway, for any rational number r, the equivalence class of the sequence (r, r, r, ...) is isomorphic in the reals to that rational number.

5.0 Complex Numbers as a Field Extension

I have already stepped through a construction of the field of complex numbers.

6.0 Observations and Higher Mathematics

So that is what numbers are, in some sense. In each of these constructions, I have attempted to preserve some properties of the more primitive domain. I do not know how much freedom one would want to permit me for the properties of, say, equality addition, or multiplication.

Does "+", the symbol for addition have the same meaning when adding natural numbers, integers, rationals, real numbers, and complex numbers?

"This discussion prompts us to ask: do the arithmetical operations have the same meaning in each of these calculi? For instance, in the domain of integers, is subtraction the same operation as in the domain of natural numbers? Furthermore, what do we mean here by 'the same'? If this implies that the operations must satisfy the same conditions, then the questions must be answered negatively. For in the domain of natural numbers the expression a - b is admissible only if a > b, while in the domain of integers this restriction is removed; obviously this is an important distinction. Consequently, there is not, strictly speaking, one subtraction but as many different operations with this name as there are domains of numbers. We should not be deceived regarding this situation by the fact that we use the same signs +, -, :, etc., at the various levels. If we put the statements of these concepts side by side, it becomes clear how far the analogy between them goes and where it stops." -- Friedrich Waismann (1951: 61).

I am not at all sure, however, that Waismann would answer the question in the post title in the negative:

"In the construction above, the integers were first constructed and then rational numbers. Is there an innate necessity for this sequential order? Couldn't we first introduce the rational numbers without signs and then the distinction between positive and negative numbers? Certainly! We would not thereby obtain another system of rational numbers; rather the system so constructed would prove to be isomorphic to the one considered above, since every relation of one system could be mapped on a similarly constructed relation in the other, and conversely." -- Friedrich Waismann (1951: 65).

Waismann famously drew on Wittgenstein in developing his views, though Wittgenstein might have come to disavow them.

Some advanced mathematics raises further questions about how symbols can have meaning. Gödel's first incompleteness theorem seperates the notion of provable from a set of axioms from the question of truth. According to his second incompleteness theorem, consistency, if true, cannot be proven within a sufficiently interesting system of axioms and derivation rules. The Löwenheim-Skolem theorem shows that axioms are not enough to fix the meaning of mathematical objects. I was explaining to a colleaque that a power set is always "bigger" than the original set. And I raised the question of whether the set of all subsets of the natural numbers can be put in a one-to-one correspondence with the reals. (I can understand what the quesion is, at least, given a naive acceptance of the construction of the reals.) I told my colleague that it was not clear what the answer was or even what it would mean for the question to have an answer. Apparently, Gödel thought that mathematicians might one day come to agree on an answer.

I think how mathematical terms are used in these systems affects the meaning of mathematical terminology in ordinary life. Some strict separation of these uses cannot be drawn, as far as I am aware. I conclude by noting that some classics (for example, Putnam 1981 and Kripke 1982) of Anglo-American analytical philosophy argue difficulties in understanding how mathematics means extends to (much of?) the remainder of language.

References
  • John B. Fraleigh. 2003. A First Course in Abstract Algebra, 7th edition.
  • Kurt Gödel. 1947. What is Cantor's continuum problem? American Mathematical Monthly 54: 515-525. Reprinted in Kurt Gödel: Collected Works, Volume II.
  • Saul A. Kripke. 1982. Wittgenstein on Rules and Private Language Harvard University Press
  • Hilary Putnam. 1981. Reason, Truth and History Cambridge University Press.
  • Walter Rudin. 1976. Principles of Mathematical Analysis, 3rd edition
  • Friedrich Waisman. 1951. Inroduction to Mathematical Thinking: The Formulation of Concepts in Modern Mathematics.

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