2

I am trying to improve my ability to reason about proofs. To accomplish this, I am studying the textbook, "How to read and do proofs", by Daniel Solow.

In the textbook, Solow makes the claim that "...everything that you are assuming to be true is the hypothesis.". One of the examples given is:

Example 2: The quadratic equation $ax^2 + bx + c = 0$ has two real roots provided that $b^2 − 4ac > 0$, where $a\not= 0$, $b$, and $c$ are given real numbers. Hypothesis: $a$, $b$, and $c$ are real numbers with $a\not= 0$ and $b2 − 4ac > 0$. Conclusion: The quadratic equation $ax^2 + bx + c = 0$ has two real roots.

Given that my previous interpretation of 'hypothesis' was purely from other scientific disciplines, I find this interpretation difficult to understand:

"...everything that you are assuming to be true is the hypothesis.".

Is the author's claim correct? In mathematics, is the hypothesis defined as everything that you are assuming to be true?

I would appreciate it if some of the more experienced mathematicians could definitely clarify this for me.

Thank you.

yoyostein
  • 19,608
The Pointer
  • 4,182

1 Answers1

2

In mathematics, many statements are stated in the form: "If $A$ is true, then $B$ is true."

In this case, you can say $A$ is the "hypothesis".

Though the actual word "hypothesis" is rarely being used in practice, except perhaps in mathematical induction, where there is the "induction hypothesis".

yoyostein
  • 19,608
  • 1
    @user251257 That may not be completely exact, but even then: axioms are not "statements" in the sense the OP, and apparently yoyo, mean. – DonAntonio Nov 23 '16 at 09:51
  • @DonAntonio notice that statement has a precise logical meaning. Not every statement is an implication. – user251257 Nov 23 '16 at 09:56
  • @user251257 I'm not sure how precise that meaning is... Neither Suppes nor Halmos define it formally, Mendelssohn uses "Statement form" and statement letter wrt propositional connectives in prop. calculus, etc. Even Wiki mentions statement as the same as "declarative sentence or what follows from a declarative sentence". Anyway, I meant more the fact that axioms may have antecedent in the sense that one could claim that "if so and so then axiom (A) is so, otherwise axiom (A) is so@ . This would be weird, I think, but it could happen. – DonAntonio Nov 23 '16 at 10:06
  • You cited a Wikipedia article, which is filled by examples of statements which aren't implications. – user251257 Nov 23 '16 at 11:01
  • @user251257: yoyostein said ‘many statements’, not ‘every statement’. In any case the original question is clearly about theorems, not axioms; context matters. \ Sentence has a precise meaning in formal logic; statement, in my experience, does not. – Brian M. Scott Nov 23 '16 at 18:18
  • @BrianM.Scott sorry. Somehow I have read every instead of many. My bad. – user251257 Nov 23 '16 at 18:33