The Paradox of the Proof
<p><pre><code>  "For centuries, mathematicians have strived towards a
  single goal: to understand how the universe works, and
  describe it."
</code></pre>
I cannot resist engaging that loaded premise. It is a thought that I've wanted - for some time - for someone to skilfully dissect and lay bare for easy comprehension.<p>The closest I've come to seeing it, is the following - a concise and accessible yet well-rounded explanation of the relevance (or lack thereof) of mathematics to the fabric of our reality.<p>Alex Knapp, a science writer at Forbes :<p><pre><code>  In the midst of a rather interesting discussion of the
  notion of Aristotle’s Unmoved Mover, Leah Libresco went on
  a mild digression about the philosophy of mathematics that
  I couldn’t let go of, and feel compelled to respond to.
  
  She says:
   
    I take what is apparently a very Platonist position on  
    math.  I don’t treat it as the relationships that humans
    make between concepts we abstract from day to day life.  
    I don’t think I get the concept of ‘two-ness’ from
    seeing two apples, and then two people, and then two
    houses and abstracting away from the objects to see what
    they have in common.

    I think of mathematical truths existing prior to human 
    cognition and abstraction.  The relationship goes the
    other way.  The apples and the people and the houses are
    all similar insofar as they share in the form of two-
    ness, which exists independently of material things to
    exist in pairs or human minds to think about them.

  The notion that there’s something special about math –
  that it has some sort of metaphysical significance – only
  makes sense if you ignore the history of how we uncovered
  math to begin with. It was, despite Leah’s protestations,
  exactly just the abstraction of pairs and triplets and  
  quartets, etc. The earliest known mathematics appear to be
  attempts to quantify time and make calendars, with other
  early efforts directed towards accounting, astronomy, and
  engineering.

  Mathematics is nothing more and nothing less a tool that’s 
  useful for humans in solving particular problems. Math can
  be used to describe reality or construct useful fictions.
  For example, we know now that the spacetime we live in is
  non-Euclidian. But that doesn’t make Euclidian geometry
  useless for everyday life. Quite the contrary – it’s used
  every day. You can use mathematics to build models of
  reality that may not actually have any bearing on what’s
  real. For example, the complicated math used to describe
  how the planets moved in the Ptolemaic model of the solar
  system – where everything orbited in circles around the
  Earth – actually produced very accurate predictions. But
  it was also wrong. There aren’t actually trillions of
  physical dollars circulating in the economy – there are
  just symbols for them floating around.

  The bottom line is that human beings have brains capable
  of counting to high numbers and manipulating them, so we
  use mathematics as a useful tool to describe the world
  around us. But numbers and math themselves are no more
  real than the color blue – ‘blue’ is just what we tag a
  certain wavelength of light because of the way we perceive
  that wavelength. An alien intelligence that is blind has
  no use for the color blue. It might learn about light and
  the wavelengths of light and translate those concepts
  completely differently than we do.

  In the same way, since the only truly good mathematicians
  among the animals are ourselves, we assume that if we
  encounter other systems of intelligence that they’ll have
  the same concepts of math was we do. But there’s no
  evidence to base that assumption on. For all we know,
  there are much easier ways to describe physics than
  through complicated systems of equations, but our minds
  may not be capable of symbolically interpreting the world
  in a way that allows us to use those tools, any more than
  we’re capable of a tool that requires the use of a
  prehensile tail.

  Math is a useful descriptor of both real and fictional
  concepts. It’s very fun to play around with and its
  essential for understanding a lot of subjects. But it’s
  just a tool. Not a set of mystical entities.
</code></pre>
This explanation is very satisfying yet disillusioning at the same time.<p>In short, his explanation allows for some (or a very large number of) mathematical truths (according to the consensus of mathematicians and what appeals to their logic) to be just that -- figments of numerical imagination that neatly sit in the confines of our logic.<p>Nothing necessitates all mathematical truths (much less conjectures) to be corresponding to some aspect (however minute or however large) of our reality.<p>Some truths might (purely out of happenstance), but nothing mandates that all math truths correspond to some facet of our physical reality.<p>So, some (or a lot of) math is just hocus pocus.<p>Non-mathematicians will never know owing to the very nature of peer-review and the consensus-building aspect of modern research and scholarship.<p>A few questions:<p>What other conclusions can be drawn if one were to find this explanation appealing?<p>Are there other explanations of the relationship between math and our reality, that you've found appealing?<p>Is there a consensus among mathematicians as to what higher-order math, essentially is in pursuit of or should be in pursuit of?<p>Is it an exercise of random shooting of darts hoping that some "mathematical truth" sticks and corresponds to some observable phenomenon?<p>Source:<p>Does Math Really Exist?<p><a href="http://www.forbes.com/sites/alexknapp/2012/01/21/does-math-really-exist/" rel="nofollow">http://www.forbes.com/sites/alexknapp/2012/01/21/does-math-r...</a><p>Edit: Clean-up and rewrite.