Mathematical Proof and the Principles of Mathematics/Preliminaries/Mathematical Statements
The grist for a mathematician's mill does not consist of numbers or graphs or formulas, but of statements. We write down a statement, write down other statements intended to convince people it's true, then if all goes well we pronounce that the statement is a theorem and start again with a new statement.
So before going on it will be helpful to discuss what we mean by a statement and how we use them. We won't go so far as to say we're defining a statement though, since such a definition would consist of statements and we haven't decided what a definition actually is yet.
What is a statement?
A statement is a meaningful sentence which can be true or false. (In grammar this roughly corresponds to the indicative mood.) For example
- Humans are mortal.
is a statement that happens to be true, and
- The Sun orbits the Earth.
is a statement that happens to be false (at least according to current astronomical theory).
On the other hand, the sentence
- Which two teams are playing in the Cricket World Cup Final?
is a question (interrogative mood), not a statement. There are questions which can be answered either yes or no, but a question itself can't be true or false. Similarly, the sentence
- Go finish your homework.
is an order or request (imperative mode), and
- What a fantastic goal!
is a exclamation; neither are statements.
In mathematics, statements must be about mathematical concepts or objects. For example
- Humans are mortal.
is a statement in biology, while
- The number 4 is even.
is a mathematical statement since it concerns a mathematical object, the number 4, and a mathematical concept, evenness.
In addition, mathematical statements are required to be clear, unambiguous and not subject to opinion. Natural languages are inherently vague and subjective at times, so to a certain extent special mathematical language has evolved to avoid this. Humans have a remarkable ability to correctly interpret ambiguous phrasing and fill in hidden assumptions, but mathematical concepts are often unfamiliar so greater care is necessary.
For example, consider
- My cat is black and white.
which seems like a perfectly simple and harmless statement. But when you try to determine what it actually means then it seems that there is much more to it. First, the sentence can parsed in several ways. For example:
- My cat is black and my cat is white.
- My cat is a mixture of black and white.
- My cat is partly black and partly while.
We know that something can't be both black and white, and a mixture of black and white is normally called grey, so we accept the third interpretation without thinking about it. We also automatically assume that the statement is referring to my cat's fur and excludes, for example, his eyes, which happen to be green. And there is more meaning in that the phrase "my cat" makes, without actually saying so, the statement that I have a cat, and because I say "My cat" instead of "One of my cats" it's implied that I only have one cat. To expand the statement into what it actually says or implies in unambiguous language:
- I own at least one cat.
- I don't own two different cats.
- The cat that I own has fur.
- Some of the fur on the cat which I own is black.
- Some of the fur on the cat which I own is white.
- All of the fur on the cat which I own is either black or white.
Because natural language can be vague, symbolic notation has evolved which is not only more precise, but is also more concise in that it allows complex meaning to be captured in a few symbols. Let's compare a statement from Euclid's Elements before this notation was invented, and how the same statement might be made now.
- Euclid: If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to each other. (Book Vii, Prop. 16, T.L. Heath's translation)
- Modern: If a and b are any two numbers, then ab = ba.
Finally, mathematical statements must not be a matter of opinion. So the sentence
- 4 is pretty.
may be a statement about a number, but it's a statement of opinion rather than fact. Some may agree and some may disagree, so there can be no universal agreement on whether it is true or false. Actually such sentences do occur in mathematics, but only as commentary. For example:
- This formula is very complex.
- This statement seems obvious but is difficult to prove.
Self referential statements
In the episode "I, Mudd" of the TV series Star Trek, the galaxy is about to be taken over by a race of well-meaning but overbearing androids. Their only apparent weakness is that they are easily confused by the illogical behavior of humans. So a plan is hatched, a very silly and irrational skit is performed, and as the coup de grâce, one of the humans tells Norman, the android leader, "Now listen to this carefully, Norman, I am lying." The android, who apparently has no choice by to try to decide if the statement is true or false, falls into an infinite loop of contradictions. The androids are disabled and (spoiler alert!) the galaxy is saved.
The sentence which defeated the androids is based on
- This sentence is false.
which is known as the Liar paradox. If the sentence is true then, since it states that it is false, it must be false. But if it's false then it must be other than false, or true. There are many interpretations and possible resolutions to this paradox, but for mathematical purposes it is enough to disallow the sentence as not being a statement. The cause of the paradox seems to be that the statement is, in some sense, about itself. In mathematics, such 'self-referential' sentences are not considered statements so the whole issue is avoided.
Although statements are the primary type of sentence used in mathematics, it would be silly to insist that no other types appear. Requests and questions are both used to state problems, and requests are used in logical arguments.
A variation of the concept of a statement is that of a predicate. You can think of this as a function whose values are either true or false. For example
- x is an even number.
is a predicate in the variable x. By itself it's not a statement since whether it's true or not depends on the value of x. When x is replaced by a specific value, say 4, you get
- 4 is an even number.
which as a statement.
The idea of a predicate is generalized to allow more than one variable. This is the most general case since you can think of a statement as a predicate in zero variables. Predicates in more than one variable are sometimes relations, and some examples are:
- x = y
- Triangle X is congruent to triangle Y.
- The point R lies between the points P and Q.
Decide whether the following sentences are acceptable as statements. If so then decide if they are clear and unambiguous.
- All equilateral triangles are isosceles.
- Some isosceles triangles are equilateral.
- All isosceles triangles are equilateral.
- Some rational numbers are integers.
- Some rational numbers are not integers.
- Not all integers are rational.
- Between any two rational numbers there is a rational number.