While I have no idea of where the "empirically" comes from, goldbach conjecture
was not and is not to be established by counting or by the bullshit of these
clowns called philosophers. Famously a Hungarian (otherwise not silly)
philosopher, Imre Lakatos, attempted a popperian (the popper experts here have
to judge) style in mathematics, in a text called proofs and refutations (in
1976 the text was published in Cambridge, if I recall rightly) and it is
frankly a nit embarrassing. At that time Popper was alive but I have no idea of
what he thought. I thought even the title of the text was meant to mock
conjectures and refutations (the main text by popper, that introduced the
From: lit-ideas-bounce@xxxxxxxxxxxxx [mailto:lit-ideas-bounce@xxxxxxxxxxxxx] On
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Sent: 14 September 2015 03:40
Subject: [lit-ideas] Re: Copy of The Philosophy of Logical Atomism, 1918
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.
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.
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,
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
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
"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,
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
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
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,
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
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
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
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
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
Minuteness can make a difference.
What about the seeming confidence that number theorists have in the truth of
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
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,
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
However it does allow us to give an interesting twist to the argument of the
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,
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*?
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