Math Mutation 65:  Sacrificing A Goat To Calculus

    The differences between mathematics and religion are pretty clear,
right?  In math, we only believe in clearly proven consequences of
elementary assumptions.  In religion, we must take things we don't
really understand on faith.  But is it always that way?  In fact, when
new mathematical ideas are initially being developed, we often
understand them in a vague or imprecise form.  Like in religion,
mathematical speculation often begins with intuitive ideas about how
something should be; rigorous definitions and proofs might be left
until later.  A classic example of this case is the initial definition
of calculus.  In the early 1700's, the philsopher Bishop
Berkeley wrote a famous critique, claiming calculus was really a type
of religion.  And in some ways, he actually had a point!
    To start with, let's review some of the basic concepts of
calculus.  You probably recall that a "derivative", or what Newton
called a "fluxion", is the slope of a curve at a single infinitesmal
point.  But does this make sense?  After all, ever since Euclid, we've
known that a point can have lines through it at any angle, and there's
no reason to prefer one over another.  Just because the point happens
to lie on a curve doesn't change this.  Newton defined it by looking
at the slopes of smaller and smaller lines crossing pairs of points on
the curve, as you get closer to a single point-- if they seem to
converge to a known value, as is the case with most common geometric
curves, than that can be said to be the slope at the point.  
    Bishop Berkeley's scathing critique of this method was called "The
Analyst:  A Discourse Addressed to an Infidel Mathematician".  He
pointed out that if you are assuming you have small intervals to
obtain a slope, then compress those intervals to a single point, you
have violated your initial premise-- so you no longer can believe your
information about the slope.  He closes with a long list of final
questions to ask the infidel mathematician, such as: 
    Query 4:  "Whether men may be properly said to proceed on a
scientific method, without clearly conceiving the object they are
conversant about, the end proposed, and the method by which it is
pursued?" 
    Query 16:  "Whether cerain maxims do not pass current among
analysts which are shocking to good sense?"
    Query 63:  "Whether such mathematicians as cry out against
mysteries have ever examined their own principles?"
    Query 64:  "Whether mathematicians, who are so delicate in
religious points, are strictly scrupulous in their own science?
Whether they do not submit to authority, take things upon trust an
believe points inconceivable?  Whether they have not their mysteries,
and what is more, their repugnances and contradictions?"
    It is questionable whether Berkeley was sincerely critiquing
calculus at the time, or just trying to make a forceful case that if
calculus were acceptable, an equal case could be made for
Christianity.  At the time, he was concerned about defending his
religion against the Deists.  And he was right that the concept of a
limit had not been clearly and rigorously defined-- it was not until
the 19th century that Bolzano, Cauchy and Weirstrass would precisely
define limits using the concept of epsilon-delta proofs.  So, in a
sense, scientists of the day were accepting something they had not
fully defined or proven, in order to use Newton's fluxions in
calculations.  However, Berkeley seems to have been glossing over the
critical point that calculus provided experimentally verifiable
results, and thus while not as rigorously sound as other branches of
mathematics at the time, had a clear and demonstrated connection to
reality.  So his comparison with religion seems a little strained.
Over the next few decades a number of books and pamphlets were
published by Berkeley's contemporaries, refuting him point-by-point. 
    But this episode does serve to remind us that the "ideal" view of
math we often see in school, where the edifice of proofs is slowly
built up to precisely define everything we know today, is really just
a small part of the story.  Without dreamers taking intuitive leaps
ahead of what they can really prove, humanity would never have
achieved nearly as much in the mathematical arena.  
    And this has been your math mutation for today.


References:

<li><a href="http://en.wikipedia.org/wiki/Bishop_Berkeley">Bishop
Berkeley at Wikipedia</a>
<li><a href="http://en.wikipedia.org/wiki/The_Analyst">The Analyst at
Wikipedia</a>
<li><a href="http://en.wikipedia.org/wiki/Limit_of_a_function">Limits
at Wikipedia</a>
<li><a
href="http://www.project2061.org/publications/rsl/online/TRADEBKS/REVS/WORLMATH.HTM">Review
of 'The World of Mathematics'</a>, where I first read an extended
excerpt from "The Analyst".