# Common sense in mathematics

Are there any claims and counterclaims to mathematics being in some certain cases a result of common sense thinking? Or can some mathematical results be figured out using just pure common sense i.e. no mathematical methods?

I'd also appreciate any mentions relating to sciences, social sciences or ordinary life.

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What do you mean by "mathematical method"? Does it include, e.g., finger counting? Where do "mathematical results" start for you? – Gregor Bruns Nov 7 '12 at 21:17
This link on numeracy (or innumeracy!) may be of interest to you. – amWhy Nov 7 '12 at 22:03
Define common sense. – Rudy the Reindeer Nov 8 '12 at 10:47

"Common sense" in mathematics is not very common.
Many things seem very anti-intuitive, at least until you train your intuition properly. The untrained intuition is lost when dealing with, for example, infinite sets, or geometry in more than $3$ dimensions. However, one example of "common sense" that does come to mind is the Pigeonhole Principle in combinatorics.

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The pigeonhole principle is a good example of common sense in math, but applying it is not always obvious. For example, if $A$ is a set with $n + 1$ integers, it is far from obvious to show that it is always possible to choose two numbers, $a$ and $b$, from $A$ such that $a - b$ is divisible by $n$ using the pigeonhole principle. – glebovg Nov 7 '12 at 22:10

There is this saying among mathematicians, that you don't really understand something until it becomes obviously trivial. So, in that sense, all of mathematics is "common sense thinking".

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The common sense is the backbone of whole mathematics.

It is fair to say that nowadays all branches of mathematics are axiomatic theories. To start building an axiomatic theory you must decide what are your axioms, what are your axiom schemes, what are your rules of inference. When you finished setting up those things you can forget, in some sense, about common sense. But to make a right (right=at least interesting, usually you know what is right or what you need) choice of axioms, axioms schemes and rules of inference you will need a common sense because it is your only tool at that moment of the very beginning! You cannot create something from nothing (unless you are God :), you cannot start from nowhere. The common sense is the right starting point for mathematics, even if mathematics is capable of taking you far, far beyond it.

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Many basic theorems can be proven using common sense, not to mention that almost all axioms in mathematics, except for axioms of set theory are based on common sense. According to MathWorld, an axiom is a proposition regarded as self-evidently true without proof, which is just another way of saying it is based on common sense. The reason why I mentioned set theory is because common sense leads to numerous paradoxes in naive set theory, hence the name. In general, common sense does help, for example, understanding what a limit or a continuous function is, or why a certain theorem is true, but it is not enough. Everything in mathematics must be rigorous and every word is important. If you study or read books about epistemology and metaphysics, you should realize that it is very difficult to define common sense, so I think your question is very hard to answer indeed.

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I would like to write about the problem of "expression", from my own experience. In the 1960s, it seemed to me that, from a commonsensical viewpoint, there should be some way of expressing that

in the above diagram the big square should be the "composition" of all the little squares. Then I found that Charles Ehresmann had defined double categories, which did the job.

The next question was: what is a commutative cube? For a square with sides $a,b,c,d$ the answer might be $ab=cd$. But if we want the "faces" of a cube to commute? For all this to be significant one has to move from sets with a total operation to sets with partial operations!

The point I am trying to make is that one function of mathematics is to develop language for rigorous expression, deduction and calculation, and this may take a while to develop. For example Descartes' notion of a graph of a function is now a commonplace, even common sense; but it may take an intellectual leap to make something into "common sense".

Another example is the introduction of the zero, and Arabic numerals.

Are higher dimensions than $3$ common sense? See the book "Flatland"! (downloadable).

Worth discussing is: have there been revolutions in mathematics? See discussions on the work of Thomas Kuhn on "Revolutions in Science".

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