In a message dated 9/13/2015 4:06:47 P.M. Eastern Daylight Time,
donalmcevoyuk@xxxxxxxxxxx quotes from Pears
"it is established empirically, like the conjecture that every even number
is the sum of two prime numbers [or it is taken] to be provable, as it is
hoped that the arithmetical conjecture will be proved one day."
And McEvoy was wondering if this was a slip of D. F. Pears's tongue.
It ain't.
Pears had the original idea of providing a COMPLEX example (recall he went
to philosophy after a 'lucky accident' involving gas). He chose none other
than Goldbach's conjecture, about which much as been written.
It all started in 1742, when Goldbach wrote a letter to Euler.
Dear Euler,
I hope this finds you well. I've been thinking of late.
Would you agree that all even numbers greater than 2 are expressible as the
sum of two primes?
I hope you're thriving,
Love,
Goldbach.
Over the following two and a half centuries, mathematicians have been
unable to prove Goldbach's Conjecture -- and the fact that Euler never replied
is not, perhaps a good sign (but then Goldbach had moved, so perhaps Euler's
response was undeliverable).
To irritate Popper, Goldbach's conjecture has been verified for many
billions of examples, and there appears to be a consensus among mathematicians
that the conjecture is most likely true.
Below is a partial list (as of October 2007) showing the order of magnitude
up to which all even numbers have been checked and shown to conform to GC.
Bound Date Author
1 × 103 1742 Euler
1 × 104 1885 Desboves
1 × 105 1938 Pipping
1 × 108 1965 Stein & Stein
2 × 1010 1989 Granville
1 × 1014 1998 Deshouillers
1 × 1018 2007 Oliveira & Silva
Despite this vast accumulation of individual positive instances of GC,
aided since the early 1960s by the introduction—and subsequent rapid increases
in speed—of the digital computer, no proof of GC has yet been found --
which should not implicate that it has been lost, either.
Not only this, but few number theorists are optimistic that there is any
proof in the offing.
Pears knows this, and uses this example to motivate his Oxonian reader
(although the thing was reprinted by London-based Routledge Classics).
Fields medalist Alan Baker stated in a 2000 interview:
"It is unlikely that we will get any further in proving GC without a big
breakthrough."
"Unfortunately there is no such big idea on the horizon."
Also in 2000, publishers Faber & Faber offered a $1,000,000 prize to
anyone who proved GC between March 20 2000 and March 20 2002, confident that
their money was relatively safe.
What makes this situation especially interesting is that mathematicians
have long been confident in the truth of GC.
Hardy & Littlewood asserted, back in 1922, that "there is no reasonable
doubt that the theorem is correct," and Echeverria, in a recent survey
article, writes that "the certainty of mathematicians about the truth of GC is
complete" (Echeverria 1996, 42).
Moreover this confidence in the truth of GC is typically linked explicitly
to the INDUCTIVE (i.e. almost, anti-Popperian, as it were) evidence.
For instance, G.H. Hardy described the numerical evidence supporting the
truth of GC as "overwhelming."
Thus it seems reasonable to conclude that the grounds for mathematicians'
belief in GC is the enumerative inductive evidence.
One distinctive feature of the mathematical case which may make a
difference to the justificatory power of enumerative induction is the
importance of
order.
The instances falling under a given mathematical hypothesis (at least in
number theory) are intrinsically ordered, and furthermore position in this
order can make a crucial difference to the mathematical properties involved.
As Frege writes, with regard to mathematics:
"The ground is unfavourable for induction; for here there is none of that
uniformity which in other fields can give the method a high degree of
reliability. (Frege, Foundations of Arithmetic, translated by Pears's friend,
J.
L. Austin)
Frege then goes on to quote Leibniz, who argues that difference in
magnitude leads to all sorts of other relevant differences between the numbers:
An even number can be divided into two equal parts, an odd number cannot;
three and six are triangular numbers, four and nine are squares, eight is a
cube, and so on. (Frege, Foundations of Arithmetic -- tr. by Austin -- a
copy was possessed by D. F. Pears)
Frege also explicitly compares the mathematical and non-mathematical
contexts for induction:
In ordinary inductions we often make good use of the proposition that every
position in space and every moment in time is as good in itself as every
other. … Position in the number series is not a matter of indifference like
position in space. (Frege, Foundations of Arithmetic, tr. by J. L. Austin,
Pears's copy).
As Frege's remarks suggest, one way to underpin an argument against the
use of enumerative induction in mathematics is via some sort of non-uniformity
principle: in the absence of proof, we should not expect numbers (in
general) to share any interesting properties. Hence establishing that a
property
holds for some particular number gives no reason to think that a second,
arbitrarily chosen number will also have that property.
Rather than the Uniformity Principle which Hume suggests is the only way to
ground induction, we have almost precisely the opposite principle! It
would seem to follow from this principle that enumerative induction is
unjustified, since we should not expect (finite) samples from the totality of
natural numbers to be indicative of universal properties.
A potentially even more serious problem, in the case of GC and in all other
cases of induction in mathematics, is that the sample we are looking at is
biased. Note first that all known instances of GC (and indeed all
instances it is possible to know) are—in an important sense—small.
In a very real sense, there are no large numbers: Any explicit integer can
be said to be “small”. Indeed, no matter how many digits or towers of
exponents you write down, there are only finitely many natural numbers smaller
than your candidate, and infinitely many that are larger (Crandall and
Pomerance 2001, 2).
Of course, it would be wrong to simply complain that all instances of GC
are finite. After all, every number is finite, so if GC holds for all finite
numbers than GC holds simpliciter.
But we can isolate a more extreme sense of smallness, which might be termed
minuteness.
Definition:
A positive integer, n, is minute just in case n is within the range of
numbers we can write down using ordinary decimal notation, including
(non-iterated) exponentiation.
Verified instances of GC to date are not just small, they are minute. And
minuteness, though admittedly rather vaguely defined, is known to make a
difference.
Consider, for example, the logarithmic estimate of prime density (i.e. the
proportion of numbers less than a given n that are prime) which is known to
become an underestimate for large enough n.
Let n* be the first number for which the logarithmic estimate is too small.
If the Riemann Hypothesis is true, then it can be proven that an upper
bound for n* (the first Skewes number) is 8 × 10370.
Though an impressively large number, it is nonetheless minute according to
the above definition.
However if the Riemann Hypothesis is false than our best known upper bound
for n* (the second Skewes number) is 10↑10↑10↑10↑3.
The necessity of inventing an ‘arrow’ notation here to represent this
number tells us that it is not minute.
The second part of this result, therefore, although admittedly conditional
on a result that is considered unlikely (viz. the falsity of RH), implies
that there is a property which holds of all minute numbers but does not hold
for all numbers.
Minuteness can make a difference.
What about the seeming confidence that number theorists have in the truth
of GC?
Echeverria discusses the important role played by Cantor's publication, in
1894, of a table of values of the Goldbach partition function, G(n), for n
= 2 to 1,000 (Echeverria 1996,29–30).
The partition function measures the number of distinct ways in which a
given (even) number can be expressed as the sum of two primes.
Thus G(4) = 1, G(6) = 1, G(8) = 1, G(10) = 2, etc.
This shift of focus onto the partition function coincided with a dramatic
increase in mathematicians' confidence in GC.
What became apparent from Cantor's work is that G(n) tends to increase as n
increases.
Note that what GC amounts to in this context is that G(n) never takes the
value 0 (for any even n greater than 2).
The overwhelming impression made by data on the partition function is that
it is highly unlikely for GC to fail for some large n. For example, for
numbers on the order of 100,000, there is always at least 500 distinct ways to
express each even number as the sum of two primes!
However, as it stands these results are purely heuristic.
The thirty years following Cantor's publication of his table of values
(described by Echeverria as the "2nd period" of research into GC) saw numerous
attempts to find an analytic expression for G(n).
If this could be done then it would presumably be comparatively
straightforward to prove that this analytic function never takes the value 0
(Echeverria 1996, 31).
By around 1921, pessimism about the chances of finding such an expression
led to a change of emphasis, and mathematicians started directing their
attention to trying to find lower bounds for G(n).
This too has proved unsuccessful, at least to date.
Thus consideration of the partition function has not brought a proof of GC
any closer.
However it does allow us to give an interesting twist to the argument of
the previous section.
The graph suggests that the hardest test cases for GC are likely to occur
among the smallest numbers; hence the inductive sample for GC is biased, but
it is biased against the chances of GC.
Mathematicians' confidence in the truth of GC is not based purely on
enumerative INDUCTION (Yes, Popper, there is such a thing in analytic
mathematics! and successful too!).
The values taken by the partition function indicate that the sample of
positive instances of GC is indeed biased, and biased samples do not—as a
general rule—lend much support to an hypothesis.
But in this particular case the nature of the bias makes the evidence
stronger, not weaker.
So it is possible to argue that enumerative induction is unjustified while
simultaneously agreeing that mathematicians are rational to believe GC on
the basis of the available evidence.
Note that there is a delicate balance to maintain here because evidence for
the behavior of the partition function is itself non-deductive.
However the impression that G(n) is likely to be bounded below by some
increasing analytic function is not based on enumerative induction per se, so
the justification—while non-deductive—is not circular.
The upshot of the above discussion, albeit based on a single case study, is
that mathematicians ought not to—and in general do not—give weight to
enumerative induction per se in the justification of mathematical claims.
To what extent enumerative induction plays a role in the discovery of new
hypotheses, or in the choice of what open problems mathematicians decide to
work on, is a separate issue which has not been addressed here.
More precisely, the thesis is in two parts:
(i) Enumerative induction ought not to increase confidence in universal
mathematical generalizations (over an infinite domain).
(ii) Enumerative induction does not (in general) lead mathematicians to be
more confident in the truth of the conclusion of such generalizations.
Pears, being a genius, knew his simile should motivate the Oxonian. Note
that it is a simile in that he writes: "like"; and as he notes, ain't it
ironic that, of all places, Russell had to publish his "Logical Atomism"
originally in "The Monist"? Couldn't he have chosen something like
"Divisibility
Bi-Weekly" *or something*?
Cheers,
Speranza
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