Re: ​Mathematics with the potential infinite

[Vaughan Pratt]

So far it seems to me that every objection to my proposal (repeated below)
concerning gapless geodesics depends on the assumption/axiom/whatever that
a nonempty linear order with no greatest (or least) element must be an
actual infinity.

To me, this has the flavor of Zeno's paradox, that movement from 1 to 0,
say in units of meters, must be impossible because it must pass through the
infinitely many points 1/2, 1/4, 1/8, ...

Now it is certainly a fair point that obtaining zero as the limit as n goes
to infinity of 2^-n is the limit of an infinite sequence.  What is
bothering me is the idea that because the sequence is infinite, it is
therefore not something we can experience.

Now if we were to spend one second at each of those points, we'd be less
than one Planck length away from zero in about two minutes.

Yet we still wouldn't be right at zero, not even a billion years later.
(Math is not physics.)

The existence of a model in which we can reach 0 in one second by traveling
at a constant speed of 1 meter per second shows that there is nothing
inconsistent about dropping the axiom that a nonempty linear order with no
least element is impossible to experience in practice.  And we reach
uncountably many other points along the way, with the transcendental ones a
set of Lebesgue measure 1, all in unit time.

Cantor is not to blame for the paradoxes arising from viewing the unit
interval as consisting of infinitely many points.  After all, Zeno was from
the fifth century BCE.  As far as paradoxes go, the fact that Cantor
identified more points than Zeno is somewhat of a red herring.

We can find Zeno's paradox elsewhere, namely in the transcendence of pi,
which is how we prove pi is not a constructible number, i.e. not a quantity
that can be "computed" with a straightedge and compass.  (Which might seem
overkill, surely it should be easier just to prove that if pi is algebraic
it must be of degree at least 3, yes?)

Today we can define the Euclidean plane very slickly as the two-dimensional
vector space over the field of reals, made an inner product space realized
with the usual dot product, a bilinear operation.

In Book I of his *Elements*, Euclid accomplishes this bilinear aspect of
his geometry by working with quantities of two types, lengths and areas.
For example the penultimate Proposition 47 concerning right-angled
triangles asserts the Pythagorean Theorem, that the square on the
hypotenuse equals the sum of the squares on the other two sides, while the
last, 48, asserts the converse, that such an equality implies a right angle.

The paradox here is that even though pi is transcendental, an area of size
pi can be drawn with a compass set to unit radius, and moreover the
perimeter of that circle is 2 pi.

In this case we can approach pi as a length with a simple infinite
construction due essentially to Archimedes, though starting with a square
rather than a hexagon, and only computing the outer polygons, which takes
about the same effort as the mean of the outer and inner polygons.

Start with X = (1,0), O = (0,0), and Y = (0,1).  XOY is a right isosceles
triangle, four of which can be distributed evenly about O to make a
four-slice pie with a square boundary.

At step n = 1, move Y to the midpoint of XY, namely (x,y) = (1/2,1/2),  and
shorten OX (along itself without moving O, i.e. move X left) to make XOY
isosceles again.  X is now at (sqrt(x^2 + y^2), 0) = (0.7071..., 0).
However the angle XOY has now been halved, to 45 degrees.  2^{n+2} = 8
copies of this prototype triangle can be distributed evenly about O to make
an eight-slice pie with an octagonal boundary.

At step n = 2, halving the 45-degree isosceles prototype in the same way
produces a 22.5-degree isosceles triangle, 2^{n+2} = 16 of which make a
regular 16-agon.  And so on, with the base of the isosceles prototype
becoming more vertical at every step.  The length of the base therefore
tends to 2^-n, and the perimeter of the 2^{n+2}-agon therefore tends to 4.

In the limit we obtain a circle of perimeter 4.

Now given a flexible line of length 4, if we bend it into a circle and
connect its endpoints so as to have constant curvature throughout, we have
constructed a circle of radius 2/pi.  We were able to do this with a
Zeno-like approach involving infinitely many steps with just a compass and
straightedge, but with this flexible line as an alternative to a compass
for creating a circle, we were able to construct 2/pi (or 4/pi if we prefer
to measure the diameter) in finite time.

In Cartesian coordinates, the midpoint of XY is a point (x,y).  Observing
that y halves at each step, we can take y_n = 2^-n.  We can then define the
x coordinate of the successive midpoints as

        x_1 = 1/2
 x_{n+1} = (x_n + sqrt(x_n^2 + (2^-n)^2))/2

For n from 1 to 28, the first 16 digits of 2/x_n after the decimal point
are as follows.

4.0000000000000000
3.3137084989847603
3.1825978780745281
3.1517249074292560
3.1441183852459042
3.1422236299424568
3.1417503691689664
3.1416320807031818
3.1416025102568089
3.1415951177495890
3.1415932696293073
3.1415928075996445
3.1415926920922543
3.1415926632154084
3.1415926559961970
3.1415926541913941
3.1415926537401934
3.1415926536273932
3.1415926535991932
3.1415926535921432
3.1415926535903807
3.1415926535899401
3.1415926535898299
3.1415926535898024
3.1415926535897955
3.1415926535897938
3.1415926535897933
3.1415926535897932

(Newton computed pi to this precision in 1665, though with a method much
more different from Archimedes' than the above.)

At n = 2,  x_2 = (1 + sqrt(2))/2 = 1.20710678118...  .  Now  2/x_n is
constructible for all finite n, but the limit is the transcendental
quantity pi, with the precision increasing by almost exactly two bits at
each halving of our isosceles triangle.

The form taken by Zeno's paradox here is that with infinitely many
constructible numbers, presumably taking infinite time, we can construct a
radius r = 2/pi such that sweeping out a circular arc of length 1
constructs a right angle, and of length 4 a circle, of radius 2/pi.  Or we
can do it in finite time with a flexible line of length 4, as our
two-dimensional counterpart of smooth motion from 1 to 0.

Vaughan

On Sat, Jan 28, 2023 at 12:13 AM Vaughan Pratt <pratt at cs.stanford.edu>
wrote:

>
> Define a *geodesic* to be a nondegenerate linearly ordered group (G, *, 0,
> <=).  (Although G is not assumed abelian, the linear order makes it
> abelian.)  Examples include the integers, the dyadic rationals, every field
> between the rationals and the reals, and many non-Archimedean extensions
> thereof.
>
> Call a geodesic G *gapless* when (i) it is dense, and (ii) for every
> nonempty suborder (U, <=) of (G, <=) having no least element, and every
> nonempty suborder (L, <=) of (G, <=) with L < U and having no greatest
> element, such that there is at most one element of G between L and U; then
> there exists an element of G between L and U.
>
> I claim that every gapless geodesic is isomorphic to R with the above
> structure.
>
> (Proof outline: Take any element x of G with 0 < x and pair 0 and x with 0
> and 1 in R.  Pair the integers in R with the subgroup of G generated by x,
> cyclic and therefore abelian.  Repeatedly divide the intervals in (n, n+1)
> in G into two equal parts and pair the results with the dyadic rationals in
> (0,1), a dense set.  Pair each dyadic irrational q in R with the unique x
> given by the gaplessness condition for any L and U in G whose counterpart
> in R converges to q from each side.  Lastly, G must be Archimedean or there
> would be an empty gap between the finite and infinite elements of G.)
>
>
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