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Definitions of Terms Commonly Used in Higher Math

The following is a guide to the weary student of mathematics who is often confronted with terms which are commonly used but rarely defined. In the search for proper definitions for these terms we found no authoritative, nor even recognized, source. Thus, we followed the advice of mathematicians handed down from time immortal: "Wing It."

CLEARLY: I don't want to write down all the "in- between" steps.

TRIVIAL: If I have to show you how to do this, you're in the wrong class.

OBVIOUSLY: I hope you weren't sleeping when we discussed this earlier, because I refuse to repeat it.

RECALL: I shouldn't have to tell you this, but for those of you who erase your memory tapes after every test...

WLOG (Without Loss Of Generality): I'm not about to do all the possible cases, so I'll do one and let you figure out the rest.

IT CAN EASILY BE SHOWN: Even you, in your finite wisdom, should be able to prove this without me holding your hand.

CHECK or CHECK FOR YOURSELF: This is the boring part of the proof, so you can do it on your own time.

SKETCH OF A PROOF: I couldn't verify all the details, so I'll break it down into the parts I couldn't prove.

HINT: The hardest of several possible ways to do a proof.

BRUTE FORCE (AND IGNORANCE): Four special cases, three counting arguments, two long inductions, "and a partridge in a pair tree."

SOFT PROOF: One third less filling (of the page) than your regular proof, but it requires two extra years of course work just to understand the terms.

ELEGANT PROOF: Requires no previous knowledge of the subject matter and is less than ten lines long.

SIMILARLY: At least one line of the proof of this case is the same as before.

CANONICAL FORM: 4 out of 5 mathematicians surveyed recommended this as the final form for their students who choose to finish.

TFAE (The Following Are Equivalent): If I say this it means that, and if I say that it means the other thing, and if I say the other thing...

BY A PREVIOUS THEOREM: I don't remember how it goes (come to think of it I'm not really sure we did this at all), but if I stated it right (or at all), then the rest of this follows.

TWO LINE PROOF: I'll leave out everything but the conclusion, you can't question 'em if you can't see 'em.

BRIEFLY: I'm running out of time, so I'll just write and talk faster.

LET'S TALK THROUGH IT: I don't want to write it on the board lest I make a mistake.

PROCEED FORMALLY: Manipulate symbols by the rules without any hint of their true meaning (popular in pure math courses).

QUANTIFY: I can't find anything wrong with your proof except that it won't work if x is a moon of Jupiter (Popular in applied math courses).

PROOF OMITTED: Trust me, It's true.

From: ncbauers#NoSpam.ndsuvax.UUCP (Michael Bauers)

Note: This entry was inspired by something I once read in NUTWORKS (The Computer Humor Magazine.)

This is a guide to translating the language of math textbooks and professors.

1) It can be proven...

This may take upwards of a year, and no shorter than four hours, and may require something like 5 reams of scratch paper, 100 pencils, or 100 refills (For those who use mechanical pencils). If you are only an undergraduate, you need not bother attempting the proof as it will be impossible for you.

2) It can be shown...

Usually this would take the teacher about one hour of blackboard work, so he/she avoids doing it. Another possibility of course is that the instructor doesn't understand the proof himself/herself.

3) It is obvious...

Only to PhD's who specialize in that field, or to instructors who have taught the course 100 times.

4) It is easily derived...

Meaning that the teacher figures that even the student could derive it. The dedicated student who wishes to do this will waste the next weekend in the attempt. Also possible that the teacher read this somewhere, and wants to sound like he/she really has it together.

5) It is obvious...

Only to the Author of the textbook, or Carl Gauss. More likely only Carl Gauss. Last time I saw this was as a step in a proof of Fermat's last theorem.

6) The proof is beyond the scope of this text.

Obviously this is a plot. The reader will never find any text with the proof in it. The Proof doesn't exist. The theorem just turned out to be usefull to the author.

7) The proof is left up to the reader.

...sure let us do all the work. Does the author think that we have nothing better to do than sit around with THEIR textbook, and do the work that THEY should have done?

8) Sample Proof: . . .

4.7 At this point we assume that x is an element of the set S, and therefore...We know this according to L. Krueger[pg. 71]

Question...has anyone ever bothered to see if these type of references exist. Come on...we all know what happens when we are writing a fresh- man english composition and run out of sources...how better to prove your thesis with a little blurb from some obscure, and nonexistant source

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From: "dcoble" <dcoble#NoSpam.gateway.net>
                        Mathematics Glossary
                  Rainer Koch  (UNI011 @ DBNRHRZ1)
Any  student who ever  sat or slept  trough a  mathematics course knows
that  certain words  and phrases  occur very frequently.  This glossary
might eliminate some confusion.

When the instructor says   He really means
------------------------   ---------------
trivial                    The student might be able to
                           do it in three hours or so.

simple                     An "A" student can do it in a week or so.

easy                       This topic would make a good master's thesis.

clear                      The instructor can do it (he thinks).

obvious                    The  instructor is sure it is
                           in his notes somewhere.

certainly                  The instructor saw one of his instructors do
                           it, but has completely forgotten how it
                           was done.

left as an exercise        The instructor lost his notes.
for the student

is well known              The instructor heard that someone once did

can be shown               The instructor thinks it might be true,
                           but has no idea how to prove it.

the diligent student       It is an unsolved problem -
can show                   probably harder than Fermat's Last Theorem.

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Conclusion /nm./: the place where you got tired of thinking.

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From: ptwahl#NoSpam.aol.com (Patrick T. Wahl)

More recently, I learned another useful term: "modulo errors." This is used, for example, as "Q.E.D., modulo errors." One fellow often applied this to his blackboard proofs, meaning: "This is a representative of an equivalence class of proofs, one of which is correct and all of which look sort of like this one. At least one such proof is correct, but it might not be the one I wrote down."

I don't mean to be critical here; the lectures were quite good. The point is, a proof "modulo errors" presents the important ideas, and we have better things to do today than criticize the details.

For example, many of us who have lectured know the sinking feeling, "ten or fifteen minutes ago and two blackboards back, I should have called that variable something besides 'm', because now I'm stuck calling two things by the same name." One can rewrite the whole thing, or insert hokey primes or subscripts. Or, one can take pity on the students, who after all are paying $20 an hour to see the show. In the latter case, just say "modulo errors," and move on.

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From: mstueben#NoSpam.pen.k12.va.us (Michael A. Stueben)


by Michael Stueben (November 7, 1994)

Today it is considered an egregious faux pas to speak or write in the crude antedated terms of our grandfathers. To assist the isolated student and the less sophisticated teacher, I have prepared the following list of currently fashionable mathematical terms in academia. I pass this list on to the general public as a matter of charity and in the hope that it will lead to more refined elucidation from young scholars.

thinking:                                       hypothesizing.
proof by contradiction or indirect proof:       reductio ad absurdum.
mistake:                                        non sequitur.
starting place:                                 handle.
with corresponding changes:                     mutatis mutandis.
counterexample:                                 pathological exception.
consequently:                                   ipso facto.
swallowing results:                             digesting proofs.
therefore:                                      ergo.
has an easy-to-understand, but hard-to-find solution: obvious.
has two easy-to-understand, but hard-to-find solutions: trivial.
truth:                                          tautology.
empty:                                          vacuous.
drill problems:                                 plug-and-chug work.
criteria:                                       rubric.
example:                                        substantive instantiation.
similar structure:                              homomorphic.
very similar structure:                         isomorphic.
same area:                                      isometric.
arithmetic:                                     number theory.
count:                                          enumerate.
one:                                            unity.
generally/specifically:                         globally/locally.
constant:                                       invariant.
bonus result:                                   corollary.
distance:                                       metric measure.
several:                                        a plurality.
function/argument:                              operator/operand.
separation/joining:                             bifurcation/confluence.
fourth power or quartic:                        biquadratic.
random:                                         stochastic.
unique condition:                               a singularity.
uniqueness:                                     unicity.
tends to zero:                                  vanishes.
tip-top point:                                  apex.
half-closed:                                    half-open.
concave:                                        non-convex.
rectangular prisms:                             parallelepipeds.
perpendicular (adj.):                           orthogonal.
perpendicular (n.):                             normal.
Euclid:                                         Descartes.
Fermat:                                         Wiles.
path:                                           trajectory.
shift:                                          rectilinear translation.
similar:                                        homologous.
very similar:                                   congruent.
whopper-jawed:                                  skew or oblique.
change direction:                               perturb.
join:                                           concatenate.
approximate to two or more places:              accurate.
high school geometry or plane geometry:         geometry of the Euclidean plane
                                        under the Pythagorean metric.
clever scheme:                                  algorithm.
initialize to zero:                             zeroize.
* :                                             splat.
{ :                                             squiggle.
decimal:                                        denary.
alphabetical order:                             lexical order.
a divide-and-conquer method:                    an algorithm of logarithmic
student ID numbers:                             witty passwords.
that bitch secretary in the math dept:          the witch of Agnesi
numerology and number sophistry:                descriptive statistics

Special thanks to Peter Braxton who got me started writing this stuff and who contributed five of the items above.

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From: Kiwini Motutu <shegosay#NoSpam.yahoo.com>

In class I was asked to give the definition of these words.  Here are the
words and my answer.
Polyhedron: She is the mother of all geometry.
Polyhedron Properties: I put Land she bought with the mother and fame she
achieved from becoming the mother of all geometry.
Polyhedron Compounds:  The structures she built on her properties.
Polyhedron Operations: The Name of the business she opened in her compounds
on her properties.
Uniform Polyhedron: She manufactured her own clothing line from her
businesses in her compounds on her properties.
Miscellaneous Polyhedron: She diversified
Dual Polyhedron: She later married and had twins.
Tetrahedron: A life threatening sickness you will catch if you don't get a
tetanus shot.
Releaux Tetrahedron: The first person diagnosed with this sickness around
1610 in Scandinavia.
Tetrahedron Compound 2: coughing
Tetrahedron Compound 3: scratching
Tetrahedron Compound 4: physiologic calcium imbalance 
Tetrahedron Compound 5: tonic spasm of muscles 
Tetrahedron Compound 6: deficient parathyroid secretion 
Tetrahedron Compound 10: loss of hearing, sight, hair and income followed by death all within a 15 second period.
Comedy material written by Edmund Johnson Feb. 23rd 2003  

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From: mini-air 1999-01-10       Questionable Math

Mathematicians, as well as their opposite numbers, have responded
eagerly -- and repetitively -- to our essay question "Why is it
only mathematicians who say 'Why is this obvious?'"

More than 80% of the respondents said, "Because." When respondents
answered anything with other than "Because," it usually "Why not?"
Several of the other answers stood out, for various reasons:

"Because mathematics is the only profession in which the
practitioners are intelligent enough to realize that every person
on the planet is, basically, an idiot, and therefore might need
some time in order to comprehend the perfectly obvious."
        -- Investigator J.C. Jamison

"The assertion is patently false. Why is this obvious?"
        -- Investigator B. Kallick

"Given that the abstract algebra professor has red hair, and
teaches ring theory, then this is clearly a red hair ring."
        -- Investigator L. Sherman

"Missing comma. The quote should have been: "Why, is this
obvious?" Much more in line with a mathematical professor's
        -- Investigator Felix Finch

"Because math is the only subject where anything is *allowed* to
be obvious. In any other science, you have to get a grant, run an
experiment, write an excruciatingly equivocated research article,
and have it peer-reviewed and published and cited in at least 3
literature overviews. THEN it's obvious."
        -- Investigator David Lantz

"Q:  Why is this obvious?
 A:  That depends on what your definition of 'is' is."
        -- Investigator T. Rose

"I could tell you why
and it would thrill you.
I could tell you why
and it would chill you.
"Please tell me why,
O will you, will you?"
I could tell you why
but I'd have to kill you."
        -- Investigator Ray Orrange

Special Category: Godfrey H. Hardy
Februari 7
December 1

1999-01-11      Classic Obviousness

Obviously, there is a rich history to this matter of
mathematicians and the obvious. It is necessary and sufficient to
present three examples:

This is a certifiably non-original story I tell to all math majors
I encounter: One mathematician was showing his new theorem to
another. The colleague pointed at the chalkboard and asked how the
theorem went from one step to the next. The first mathematician
said, "That's obvious." The second went to a second blackboard,
spent an hour filling it up with complex calculations, then
stepped back and said, "You're right, it IS obvious."
        -- Patrick Lenon

It's worth recalling the story of the very famous mathematician
G.H. Hardy, who in a lecture said about some detail in a proof:
"This is obvious." After a pause, he went on: "Hmm, is it really
obvious?" After another pause he left the room to consider the
point, returning 20 minutes later with the verdict: "Yes, I was
right, it is obvious."
        -- J.R. Partington

"The world's most famous mathematician, Humpty Dumpty, speaking
for fellow mathematicians everywhere, said: "When I use a word, it
means precisely what I choose it to mean, neither more nor less".
Mathematicians always say what they mean, even though they do not
mean what they say.  Obviously.
        -- Dirk Laurie

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