If you ask an average person on the street what is the highest level of mathematics, the most common answer would probably be Calculus. There might even be a few throwing College Algebra out there as fairly advance. However, if you ask a math major or engineering student the question of what is the lowest level of mathematics, the foundation of the mathematics they use, the most common answer would likely be Calculus. Why the disparity?

The mathematics that is taught from kindergarten through secondary is often limited to procedural techniques to solve specific problem types without recognizing that advanced mathematics is all about recognizing patterns and using axioms, definitions, and theorems to formalize those patterns thereby leading to new patterns.

So, of course, in upper level mathematics we spend a great deal of time moving from procedural mathematics, to proving theorems, to developing new theorems. Unfortunately, the rigor is sometimes lost in the classroom for many reasons. I’m sure I’ve used just about every one of the following invalid proof techniques.

**Proof by obviousness:**"The proof is so clear that it need not be mentioned."

**Proof by general agreement:**"All in favor?…"

**Proof by imagination:**"Well, we’ll pretend it’s true…"

**Proof by convenience:**"It would be very nice if it were true, so…"

**Proof by necessity:**"It had better be true, or the entire structure of mathematics would crumble to the ground."

**Proof by plausibility:**"It sounds good, so it must be true."

**Proof by intimidation:**"Don’t be stupid; of course it’s true!"

**Proof by lack of sufficient time:**"Because of the time constraint, I’ll leave the proof to you."

**Proof by postponement:**"The proof for this is long and arduous, so it is given to you in the appendix."

**Proof by accident:**"Hey, what have we here?!"

**Proof by insignificance:**"Who really cares anyway?"

**Proof by mumbo-jumbo:**

**Proof by profanity:**(example omitted)

**Proof by definition:**"We define it to be true."

**Proof by tautology:**"It’s true because it’s true."

**Proof by plagiarism:**"As we see on page 289,…"

**Proof by lost reference:**"I know I saw it somewhere…."

**Proof by calculus:**"This proof requires calculus, so we’ll skip it."

**Proof by terror:**When intimidation fails…

**Proof by lack of interest:**"Does anyone really want to see this?"

**Proof by illegibility:**

**Proof by logic:**"If it is on the problem sheet, it must be true!"

**Proof by majority rule:**Only to be used if general agreement is impossible.

**Proof by clever variable choice:**"Let A be the number such that this proof works…"

**Proof by tessellation:**"This proof is the same as the last."

**Proof by divine word:**"…And the Lord said, ‘Let it be true,’ and it was true."

**Proof by stubbornness:**"I don’t care what you say- it is true."

**Proof by simplification:**"This proof reduced to the statement 1 + 1 = 2."

**Proof by hasty generalization:**"Well, it works for 17, so it works for all reals."

**Proof by deception:**"Now everyone turn their backs…"

**Proof by supplication:**"Oh please, let it be true."

**Proof by poor analogy:**"Well, it’s just like…"

**Proof by avoidance:**Limit of proof by postponement as it approaches infinity

**Proof by design:**If it’s not true in today’s math, invent a new system in which it is.

**Proof by authority:**"Well, Gauss says it’s true, so it must be!"

**Proof by intuition:**"I have this gut feeling."