# The Mathematical Image

The proof is elegant and the result profound. Still, it is typical mathematics; so,
it’s a good example to reflect upon. In doing so, we will begin to see the elements
of *the mathematical image*, the standard conception of what mathematics is. Let’s
begin a list of some commonly accepted aspects. By ‘commonly accepted’ I mean
that they would be accepted by most working mathematicians, by most educated
people, and probably by most philosophers of mathematics, as well. In listing
them as part of the common mathematical image we need not endorse them. Later
we may even come to reject some of them – I certainly will.With this caution in
mind, let’s begin to outline the standard conception of mathematics.

*Certainty* The theorem proving the infinitude of primes seems established
beyond a doubt. The natural sciences can’t give us anything like this. In spite of
its wonderful accomplishments, Newtonian physics has been overturned in
favour of quantum mechanics and relativity. And no one today would bet too
heavily on the longevity of current theories. Mathematics, by contrast, seems
the one and only place where we humans can be absolutely sure we got it right.

*Objectivity* Whoever first thought of this theorem and its proof made a great
discovery. There are other things we might be certain of, but they aren’t discoveries:
‘Bishops move diagonally.’ This is a chess rule; it wasn’t discovered; it
was invented. It is certain, but its certainty stems from our resolution to play the
game of chess that way. Another way of describing the situation is by saying
that our theorem is an objective truth, not a convention. Yet a third way of making
the same point is by saying that Martian mathematics is like ours, while
their games might be quite different.

*Proof is essential* With a proof, the result is certain; without it, belief should
be suspended. That might be putting it a bit too strongly. Sometimes mathematicians
believe mathematical propositions even though they lack a proof.
Perhaps we should say that without a proof a mathematical proposition is not
justified and should not be used to derive other mathematical propositions.
Goldbach’s conjecture is an example. It says that every even number is the sum
of two primes. And there is lots of evidence for it, e.g. 4 = 2 2, 6 = 3 3, 8 =
3 5, 10 = 5 5, 12 = 7 5, and so on. It’s been checked into the billions
without a counter-example. Biologists don’t hesitate to conclude that all ravens
are black based on this sort of evidence; but mathematicians (while they might
believe that Goldbach’s conjecture is true) won’t call it a theorem and won’t use
it to establish other theorems – not without a proof.

## Notes:

How mathematics provides certainty, objectivity,

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Mathematics (0.957418): dbpedia | freebase | opencyc

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Four color theorem (0.468168): dbpedia | freebase | yago

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**Philosophy of Mathematics**

**Books, Brochures, and Chapters>**

**Book:**Brown, James Robert (2012-10-12)

*, Philosophy of Mathematics*, Routledge, Retrieved on 2013-06-18

**Folksonomies:**mathematics philosophy