Most people who take math classes that don't go past Calculus III hate proofs. They don't understand them, they dont want to understand them, and yet we expect them to understand logical statements on a deep enough level to work with them and manipulate them in freshman/sophomore mathematics classes at college. If anyone in these classes has done a proof prior, it was in a high school geometry class; students have very little to work with to understand math proofs even on a basic level, let alone with all the complicated math terminology.
Part of the problem is that English, and perhaps other languages, are not, by and large, logical languages. It has logic within it; a great deal of arguments people make on all sorts of topics have an inductive (evidence based extrapolation) or deductive (purely formally logical) logic underlying them, but what I'm saying in particular is that English is a very bad language for communicating these ideas. Let's get into an example using one of the most common formal logic constructs in English, an "if... then..." statement:
If I pass Mathematical Structures, then I will become a mathematics major.
Simple enough to understand, right? Actually, not really. There's ambiguity here about what type of logical structure is being used. If we were in a formal logic course, then this sentence structure would be interpreted as a normal conditional statement, i.e. that if the antecedent "I pass Mathematical Structures" is true, then the consequent "I will become a mathematics major" is true. We can also argue that if the consequent is false, then the antecedent is false; this is called the contrapositive of the statement. So, if I will not become a mathematics major, then I did not pass Mathematical Structures. Nothing else is guaranteed or being stated by this sentence. If that was how the statement was meant, then if I do not pass Mathematical Structures, there is nothing that can be concluded about whether or not I will become a math major. It would be perfectly reasonable for me to not pass Mathematical Structures in that case and still become a mathematics major. Similarly, if I become a mathematics major, nothing can be concluded abot whether or not I passed Mathematical Structures; that is not covered by the scope of the statement.
However, that may not have been what you interpreted the sentence as. You may, instead, have interpreted the sentence as what's called a biconditional, where, if I pass Mathematical Structures, I will become a mathematics major, but if I don't pass Mathematical Structures, I will not become a mathematics major. With a biconditional statement, you're effectively stating that the two statements on each end of the biconditional are equivalent to one another: there is no case where one of the two statements is true and the other is not true, and there is no case where one of the two statements is false and the other is not false. I would argue that this is actually a more natural interpretation of this English sentence, even though if we were speaking completely formally this sentence structure is reserved for the conditional. Hence, the ambiguity is my major argument for English not being a logical language.
However, the language of mathematics — proofs — does not have this ambiguity. Sentence structure and logical constructs are meant in their strictly logical interpretation, and students are expected to know this before actually taking any logic courses (unless they also happen to take some introductory philosophy courses during their degree, which, I would argue, most do not).
Let's look at an example from calculus. Don't worry if you don't understand the math - let's just focus on the logical form.
Theorem: If \( f \) is a function such that \( f \) is continuous on the closed interval \( [a,b] \), \( f \) is differentiable on the open interval \( (a,b) \), and \( f(a) = f(b) \), then there exists a point \( c \) in the open interval \( a, b \) such that \( f'(c) = 0 \).
All of knowledge in mathematics — every technique or property you learn of equations, and every statement of behavior in mathematics — is based off of formal logic. In that way, mathematics is really a form of applied philosophy, or applied logic, specifically within the world of numbers, shapes, and other mathematical constructs ("math" has a very broad definition, so we're simplifying a little bit). That is, this previous ambiguity we brought up in English does not apply to statements in mathematics. So, when we look at the above theorem, at its most core level this is an "if... then..." statement interpreted in the formal logic way of a normal conditional statement. That means that, if the antecedent is true, the consequent is also true, or if the consequent is false, the antecedent is false. Nothing else. For those that have taken Calculus I, this is Rolle's Theorem, and there are (infinitely) many examples of functions such that the antecedent is false but the consequent is true (for example, \( f(x) = x^{3} \) has a derivative of \( 0 \) at \( x = 0 \), but that does not mean that any two points \( x = a \) and \( x = b \) of the function surrounding \( x = 0 \) have to have equal functional values to one another; in fact, the function has no places where the function gives the same output for two different inputs). We would call these counterexamples to the interpretation of this as a biconditional statement. However, interpreted as a conditional statement, as it was intended, this is a logical, proven truth of math: if all of the conditions in the antecedent are true, then the conclusion is also true.
So, how does mathematics express a biconditional? With the construct "... if and only if...":
Theorem: The system \( A \vec{x} = \vec{b} \) has a solution if and only if \( \vec{b} \) is a linear combination of the columns of \( A \).
This statement from linear algebra, in contrast to the previous statement, is saying that the antecedent and the consequent, in essence, mean the same thing. Any time you determine one of them is true or false, you can immediately cite this theorem (or any proven theorem using a biconditional) to conclude that the other part of the biconditional is also true or also false. Put another way: which statement is the antecedent and the consequent doesn't really matter, you can swap them and arrive at a statement which is saying the exact same thing. This is not true of a conditional.
It may be easy to mix these two up as a student of math, and you can easily become confused about what the statement is saying without a rather deep understanding of formal logic. This, to me, is where students fall apart when interpreting math, and understanding why things are true and how things work in precalculus onwards. (How can a student understand a proof written in a purely logical language when they have had no previous introduction to logic?) So, we will begin with logic before moving on to "actual math" as most would interpret it on this site, as it is necessary to understand properly exactly what is meant by mathematical statements.
My advice to students is to interpret the sentence structures that I will bring up as meaning certain logical structures to only tentatively apply to English sentences, but absolutely apply to mathematical statements. If you would like to apply the logic learned here to English, a best practice is what's called the principle of charity, or charitable interpretation: you should always interpret what someone says in the way which makes the most logical sense if there is any ambiguity in the statement. So, if someone for example uses the first example above, in conditional form, to mean a biconditional, then you should interpret it as a biconditional and not get too caught up in the style of writing. With mathematics, however, you are encouraged to rigorously apply logical constructs to the sentences in the exact way they are written, as that is the style of mathematical proofs and mathematical statements.
September 30, 2023